concrete code check theory enu

Theory Design of Concrete Structures

Concrete Code Check - Theoretical Background

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Concrete Code Check

Theoretical Background

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Table of contents

Part I – 1D members

Concrete Code Check .................................................................................................................................................. 2 Theoretical Background .............................................................................................................................................. 2 Table of contents .......................................................................................................................................................... 3

Part I – 1D members ................................................................................................................................................ 3

Restrictions ................................................................................................................................... 8

Internal Forces .............................................................................................................................. 9

Beams ............................................................................................................................................................................ 9 Shifting of the moment line ..................................................................................................................................... 10 Moment reduction .................................................................................................................................................. 10 Shear force reduction ............................................................................................................................................. 11

Columns ...................................................................................................................................................................... 12 Walls ............................................................................................................................................................................ 13 Plates ........................................................................................................................................................................... 13 Shells ........................................................................................................................................................................... 13

Design of longitudinal reinforcement ...................................................................................... 14

Beams and uni-axially loaded columns ................................................................................................................... 14 Ultimate Border ...................................................................................................................................................... 14 Single parameter reinforcement design ................................................................................................................. 18 Bi-parametrical reinforcement design .................................................................................................................... 19 Basic Reinforcement or REDES reinforcement ..................................................................................................... 20

Bi-axially loaded columns ......................................................................................................................................... 21 Interaction diagram ................................................................................................................................................ 22 Optimisation ........................................................................................................................................................... 22 Ratio ....................................................................................................................................................................... 24 Delta Area .............................................................................................................................................................. 26 Multiple combinations ............................................................................................................................................. 26

Circular Columns ........................................................................................................................................................ 27 Walls, Plates and Shells ............................................................................................................................................. 28

Transformation of inner forces to design forces ..................................................................................................... 29 Reinforcement Design ............................................................................................................................................ 31 Reinforcement design of Walls .............................................................................................................................. 32 Reinforcement design of Plates ............................................................................................................................. 35 Reinforcement design of Shells ............................................................................................................................. 37

Design of shear reinforcement ................................................................................................. 39

Beams .......................................................................................................................................................................... 39 General .................................................................................................................................................................. 39 Composite Section and Arbitrary Sections ............................................................................................................. 40

Columns ...................................................................................................................................................................... 40 Plates and Shells ........................................................................................................................................................ 40

Shear Proof Concepts ............................................................................................................................................ 41 Advanced notes on the Shear Effect concept ........................................................................................................ 41

Design of torsional reinforcement ............................................................................................ 44

Crack Proof ................................................................................................................................. 45

General ........................................................................................................................................................................ 45 Beams .......................................................................................................................................................................... 46 Column ........................................................................................................................................................................ 46


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Plates, Walls and Shells ............................................................................................................................................. 46 Crack Proof after NEN 6720 .................................................................................................................................. 46 Crack Proof after ÖNORM B 4700 ......................................................................................................................... 47

Checks ......................................................................................................................................... 48

Response .................................................................................................................................................................... 48 Capacity ....................................................................................................................................................................... 51

Physical Non-linear Deformations ............................................................................................ 55

General ........................................................................................................................................................................ 55 NEN 6720 ............................................................................................................................................................... 56 Other codes than NEN 6720 .................................................................................................................................. 57

Columns ...................................................................................................................................................................... 57 Composite Sections ................................................................................................................................................... 57 Beams, Plates and Shells .......................................................................................................................................... 57

NEN 6720 ............................................................................................................................................................... 57

Part II – 2D members

INTRODUCTION TO REINFORCED CONCRETE DESIGN OF 2D STRUCTURES ................. 61

Introduction ................................................................................................................................................................. 61 Program objective ...................................................................................................................................................... 61

PROGRAM THEORY AND ALGORITHM ................................................................................... 64

Introduction to the theory and algorithmization of 2D design ............................................................................... 64 Transformation of inner forces of the FEM solution to design forces .................................................................. 64 Reinforcement Design ............................................................................................................................................... 67 Design of Walls ........................................................................................................................................................... 68 Design of Plates .......................................................................................................................................................... 72 Design of Shells .......................................................................................................................................................... 78

SHEAR PROOF ........................................................................................................................... 80

Introduction to Shear Proof ....................................................................................................................................... 80 Shear Proof according to DIN 1045 07/1988 ............................................................................................................ 81 Shear Proof according to ÖNORM B 4200 ............................................................................................................... 81 Shear Proof according to EUROCODE 2 .................................................................................................................. 81 Shear Proof according to CSN 73 1201 and STN 73 1201 ...................................................................................... 83 Shear Proof according to SIA 162 ............................................................................................................................. 84 Shear Proof according to NEN 6720 ......................................................................................................................... 84 Shear Proof according to DIN 1045-1 07.2001 ......................................................................................................... 87 Shear Proof according to GBJ 10-89 ........................................................................................................................ 87 Shear Proof according to BS 8110 ............................................................................................................................ 87 Shear Proof according to ÖNORM B 4700 ............................................................................................................... 88 Shear Proof according to BAEL 91/99 ...................................................................................................................... 88 Shear Proof according to SIA 262 ............................................................................................................................. 90 Shear Proof according to EN 1992-1-1:2004 ............................................................................................................ 90 Shear Proof according to IS 456 ............................................................................................................................... 91 Shear Proof according to ACI 318M-05 .................................................................................................................... 92 Shear Effect ................................................................................................................................................................. 92 Dealing with singularities in Shear Proof ................................................................................................................. 95

REINFORCEMENT AMOUNT CONTROL .................................................................................. 96

Introduction to reinforcement amount control ........................................................................................................ 96

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Maximum reinforcement ............................................................................................................................................ 96 Minimum transversal reinforcement ......................................................................................................................... 96 Compression reinforcement in general .................................................................................................................... 97 Minimum compression reinforcement ..................................................................................................................... 97 Minimum tension reinforcement ............................................................................................................................. 101 Overall minimum reinforcement ............................................................................................................................. 104 Minimum shear reinforcement ................................................................................................................................ 104 Minimum reinforcement of Deep Beams ................................................................................................................ 104

SERVICEABILITY PROOFS ..................................................................................................... 106

Introduction to serviceability proofs ...................................................................................................................... 106 Crack Proof according to DIN 1045 07/1988 .......................................................................................................... 110 Crack Proof according to EUROCODE 2 ................................................................................................................ 110 Crack Proof according to NEN 6720 ....................................................................................................................... 112 Crack Proof according to DIN 1045-1 07.2001 ....................................................................................................... 114 Crack Proof according to GBJ 10-89 ...................................................................................................................... 114 Crack Proof according to ÖNORM B 4700 ............................................................................................................. 114 Crack Proof according to BAEL 91/99 .................................................................................................................... 116 Crack Proof according to SIA 262 ........................................................................................................................... 116 Crack Proof according to EN 1992-1-1:2004 .......................................................................................................... 117 Crack Proof according to BS 8110 and IS 456 ....................................................................................................... 117

PROCESSING OF NON-DESIGNABILITY CONDITIONS ....................................................... 119

Processing of non-designability conditions .......................................................................................................... 119 Indication of the Non-designability Status (NSt) ................................................................................................... 119

REFERENCES ........................................................................................................................... 120


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This manual has been written for Scia ESA PT version 2008. This analysis and design program has been later replaced by Scia Engineer. Most of the Theoretical Background information contained in this document is relevant for Scia Engineer as well.

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Part I - 1D members


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Restrictions The concrete modules are restricted to the following calculations:

Necessary main reinforcement for vertical symmetrical beams and ribs loaded by a combination of normal force, Nx, and bending moment My.

Necessary main reinforcement for rectangular and circular columns beams loaded by a combination of normal force, Nx, and bending moments My and Mz.

Necessary shear reinforcement for vertical symmetrical beams and ribs loaded by a shear force Vz.

Necessary main reinforcement for walls, plates and shells loaded by bending moments mx, my and mxy and membrane forces nx, ny and nxy. For some codes the shifting of the moment line is not taken into account.

Necessary shear reinforcement for plates and shells loaded by shear forces qx and qy.

Crack proof of vertical symmetrical beams and ribs loaded by a combination of normal force, Nx, and bending moment My. For some codes the cross-section cannot contain more than one concrete quality.

Crack proof of walls, plates and shells loaded by bending moments mx, my and mxy and membrane forces nx, ny and nxy.

Quasi non-linear deformations for beams and ribs loaded by a combination of normal force, Nx, and bending moment My. For some codes the cross-section cannot contain more than one concrete quality.

Quasi non-linear deformations for plates and shells loaded by bending moments mx, my and mxy and membrane forces nx, ny and nxy.

Checks of moments and normal force response of any reinforced cross-section.

Checks of ultimate moments and normal force Myu, Mzu and Nu of any reinforced cross-section.

Checks of ultimate shear force Vzu for any reinforced vertical symmetrical cross-section.

Calculation of additional eccentricities for bending moments My and Mz for uni- or bi-axially loaded columns.

The following calculations are NOT performed:

Torsional reinforcement based on moment Mx.

Shear reinforcement for cross-sections loaded by a combination of shear forces Vz and Vy.

Connection reinforcement between different items of the cross-section that are cast during separate construction stages.

Design of reinforcement and checks of moments, normal and shear forces for individual construction stages.

Shear reinforcement for beams loaded at the bottom side of the cross-section.

Shear reinforcement for columns.

Crack proof for columns.

Quasi physical non-linear deformations for columns and walls.

Prestressed cross-sections.

Design of deep beams.

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Internal Forces

Beams

In practise a beam is subjected to a combination of a Normal force, bending moment(s), shear and torsion. For the design of necessary areas of reinforcement of a beam SCIA.ESA PT yet only supports a combination of a Normal force (Nx), bending moment (My) and shear force (Vz). This means that the cross-section must always be vertically symmetrical. The beam calculation is not limited to one concrete quality only, the program allows for the design of necessary areas cross-sections with infinite number of concrete qualities.

z

y

z

y

Remark: The user however can check the response or capacity of any reinforced cross-section for the combination of internal forces Nx, My and Mz using the single check or member check functions. These checks do not support torsion or bi-axial shear forces.


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Shifting of the moment line

The shifting of the moment line is done respecting the national code requirements. In general the truss-model analogy is used to calculate the shifted moment line for My only. The shifted moment line respects the depth of the beam, the angle of the concrete strut and the angle of the stirrups. The angles can be set in the concrete code setup. The depth of the cross-section is depended on the height of the cross-section, the diameters of the stirrups and the main reinforcement.

Remark: Please note that the shifted or reduced internal forces are not used when a ‘single’ check of a cross-section is performed.

Moment reduction

The reduction of the moment line, My, is performed when a beam is supported by either a nodal support or column. Both types of supports have different methods to reduce the moments.

For the column the bending moment is taken at each face of the column (Frame XZ and Grid).

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The theory behind the moment reduction for a nodal support is for most codes similar to that of the Eurocode 2. In this code the bearing load in the support creates a reducing effect on the bending moment over the support.

Rd

Qb

According clause 2.5.3.3. de reduced moment is:

MSd = Rd × b / 8

where:

Rd = Design value of the support reaction

b = Width of the support in the direction of the beam.

Shear force reduction

The shear force reduction is done in a similar way as for the moment reduction for beams supported by columns. 3 types: Type 1 uses the shear force operating in the face of the support or column as design force. Type 2 uses the shear force operating in the face of the support plus the effective height of the beam as design force. Type 3 uses the shear force operating in the face of the support plus a factor times the internal cantilever arm.

Remark

Note that for grids and frames that beams connected to beams do not have reduced shear forces.


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Columns

In practice a combination of normal force and primary and secondary bending moments will load the column. The geometrical and physical non-linear effects will cause the secondary moments. Shifted moment lines in columns are not taken into account in SCIA.ESA PT.

The shear force has sometimes influence in case of accidental collision by a truck or car. SCIA.ESA PT only supports the calculation of necessary areas for the main reinforcement. It does not support the shear reinforcement calculation. For such cases the user must define the column as a uni-axially loaded beam and the shear reinforcement calculation can be done.

As already said the secondary moments can have great influence on the main reinforcement calculation. Some codes have ‘tricks’ to rewrite the primary moments to secondary moments using for instance additional eccentricities. For such codes SCIA.ESA PT allows the user to indicate whether he wants to use tricks or use a more precise approach. Please understand that such tricks do not change the deformations of the columns yet, only the design moments for the reinforcement calculation.

The non-linear solver allows for geometrical non-linear calculations and offers a more exact solution. In the current version (5.0) it is not possible to take into account any physical non-linear effects. For those calculations please refer to ESA-Prima Win 3.60. The non-linear combinations can be used for the design and checks of the main reinforcement.

It is possible to use concrete combinations (used for PNL – deformation calculations) for the design of main reinforcement, but the results should be useless, since the PNL-calculation is a quasi-non-linear calculation using tricks according the national code.

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Walls

Walls are structures loaded by in plane Normal forces, nx, ny and nxy, also called wall inner forces. These normal or membrane forces will be transformed to n1 and n2 principal forces. The concept of the wall finite elements indicates that there will be no difference in reinforcement for the top and bottom side of the wall (z+ and z-). Also there will be no shear reinforcement calculation possible, since the reinforcement mesh carries the shear stress nxy.

The results service of SCIA.ESA PT allows the user to review the dimensional magnitudes. These magnitudes are for user reference only and are not actually used in the design of the necessary areas. The service for the design of the necessary areas uses a more sophisticated approach in which various parameters are taken into account such as the reinforcement mesh angle, number of reinforcement layers, etc.

Moments and shear forces in walls are not automatically reduced above walls, columns or supports.

Plates

Plates are structures loaded by out of plane shear forces, qx and qy, and bending moments, mx, my and mxy, also called plate inner forces. The bending moments will cause principal membrane forces per side of the plate, n1+, n2+, n1- and n2-. Thus the reinforcement will differ per side and per direction of the reinforcement. The shear reinforcement is calculated based on qx and qy. For some codes the shear force is used to calculate the shifted moments. Normally shifting is not taken into account in a plate model.

Moments and shear forces in plates are not automatically reduced above walls, columns or supports. Note that not for all codes the moment line is shifted automatically using the design value of the shear force.

Shells Shells are structures that are really combinations of walls and plates. Therefore the same requirements count for shells as for walls and plates. Principally the only difference between shells and plates is the calculation of the main reinforcement, see chapter reinforcement design of shells.

The results service of SCIA.ESA PT allows the user to review the dimensional magnitudes. These magnitudes are for user reference only and are not actually used in the design of the necessary areas. The service for the design of the necessary areas uses a more sophisticated approach in which various parameters are taken into account such as the reinforcement mesh angle, number of reinforcement layers, etc.

Moments and shear forces in shells are not automatically reduced above walls, columns or supports. Note that not for all codes the moment line is shifted automatically using the design value of the shear force.


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Design of longitudinal reinforcement A task called concrete reinforcement design is used very often in civil engineering. This task has many different forms and there are usually many different ways how to solve it. We would like to show our approach. We see the reinforcement design as an engineering problem, which has many variables to be optimised. Some of the variables are the orientation, shape, number and position of reinforcing bars and also the area of each bar. However, in many cases an experienced engineer can reduce the amount of unknowns to one or two. Some issues are given by constructional principles, some are determined by the applied technology, and some are provided by experience. Very often, if we know the dimensions of concrete cross-section, we usually know the position of reinforcing bars. Thereafter, the only remaining unknown is the reinforcement area.

Beams and uni-axially loaded columns F

M

L

3.600 10.8000 3.600

q1 q3

q2

b×h = 450×600 b×h = 450×800

F10.600

A B C D

Ultimate Border

The method of ultimate deformations is used to calculate the main reinforcement. The principle of this method is to look for the plane deformation in the ultimate limit state for which we evaluate the minimal necessary reinforcement area to reach the equilibrium of internal forces.

Imagine a diagram representing the strain in a reinforced concrete cross-section. The cross-section is symmetric to the z-axis and loaded with a combination of N and My. Then the vector of strain will consist of two nonzero numbers = [0;z;0]. The corresponding plane of strain with corresponding internal forces is depicted in the next figure.

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x

x

c

s

Fc

Fs

NMy

y

z

The previous figure shows a non-specific case, but let us imagine an Ultimate Limit State. Under the Ultimate Limit State, we understand a case, where either concrete or steel is strained to limit value. We can draw some cases in a similar diagram.

x

A.

c,min

s,max

y

z c,max

B. C.

c,min

D.

We can, for example, define our ultimate deformations as shown on the previous figure.

Case A. represents maximal bending moment, where concrete is strained on maximal compression and steel is under maximal tension.

Case B. represents maximal tension in both concrete and steel.

Case C. is the other maximal bending.

Case D. is maximal compression.

All other possible ultimate deformations lie in one of intervals (A.-B.), (B.-C.), (C.-D.), (D.-A.). Of course, we can define different ultimate deformations, for example, in a case of a symmetrical reinforcement we can extend maximal tension on the upper half of the cross-section. Nevertheless, we can use presented diagram as a descriptive example.

Let us now imagine a different diagram. This is a 2D-diagram where the value of z is on one axis and the value of 0 is on a perpendicular axis. In this diagram, each of our cases, A., B., C., D., is projected into one point. These points create vertexes of a polygon ABCD. All the other possible ultimate deformations lie on the edges of this polygon. Each point inside the polygon expresses plane of strain that is within bearing capacity and each point outside this polygon represents state out of bearing capacity. We can see this diagram in the next figure.


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z

A.B.

C.

D.

ok collapse

limit state

We designate the border of such a polygon as the ultimate border of the cross-section. The ultimate border represents all allowable planes of deformation when the cross-section is in the Ultimate Limit State. The ultimate border is an analogy to the interaction diagram drawn in strain co-ordinate system.

The definition of ultimate border has an essential meaning during a reinforcement design. The requirements on a plane of strain in the Ultimate Limit State vary in different standards. Most of these requirements can be effectively implemented through the definition of the ultimate border. For example, some standards allow lower maximal compression in the concrete during a full compression (dominant axial force) than during a bending. This issue can be implemented by inserting additional point E. between vertexes A. and D. (see figure 35). This situation is also drawn in the cross-section diagram, see figure 36.

z

A.B.

C.D.

E.

x

A.

s,max

y

z

c,min,c

D.E

.

point of rotation

4/7h

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From mathematical point of view, the ultimate border represents the definitional set for reinforcement design task. To be able to describe each point of ultimate border, we look at the border as a closed oriented curve described parametrically, where each point corresponds to one value of parameter t. We also define the Ultimate border function fu.

max,0 tt (1)

u = fu(t) (2)

This function returns plane of strain u on the ultimate border corresponding to given parameter t.


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Single parameter reinforcement design

Let us have a cross section symmetrical to z-axis loaded by a combination of axial force N and bending moment My. Let us suppose that the position of each reinforcing bar is known and that each bar has the same diameter. The only unknown is the total reinforcement area As. The task is to find minimal reinforcement area As so the cross section would be able to carry just the load forces RL={N,My,0}. Let us suppose, for a while, that we know the right amount of As. Then, if we load the reinforced cross section with RL, we get the corresponding plane of strain in the Ultimate Limit State u.

In this state the internal forces (representing the bearing capacity) will be equal to (representing the load forces). However, we do not know the right amount of reinforcement yet.

On the other hand, imagine that we have state u' which we consider to be Ultimate Limit State. We are able to determine the required reinforcement area from the balance of internal forces and the load forces.

Ru = RL (3)

Cross section internal forces Ru consist of internal forces in concrete RC;u and internal forces in steel RS;u. So we can write

RC;u + RS;u = RL (4)

When we know the plane of strain u', we can integrate internal forces in the concrete as follows:

RC;u = R(u') (5)

Now we are able to evaluate the forces in the reinforcement

RS;u = RL - R(u') (6)

We can write (6) in a scalar form as

NS;u = NL - NC (7)

MS;u = ML - MC (8)

If u' = u, the strain is exactly the Ultimate Limit State for defined load forces RL, then both (7) and (8) must be satisfied simultaneously. Nevertheless, if u' u, then we can choose an equation for the axial force (7). From this equation we can determine the required area of reinforcement, because

NS;u = As

n

iis

1; (9)

wherein s;i is stress in i-th reinforcing bar, which is given by the u' in the point of the bar and corresponding stress-strain diagram of steel.

s;i = (s;i) (10)

The reinforcement area can be expressed from (7) and (9) as follows

As =

n

iis

CL NN

1;

(11)

When we know reinforcement area, we can evaluate the real internal forces in concrete and steel corresponding to u'. The axial force must be equal to loading axial force, because of (11), but the bending moment MS;u will probably differ from ML-MC;u. Therefore we define a M as follows

M = MS;u - (ML - MC;u) (12)

M = MS;u (u') - (ML (u')- MC;u) (13)

This M is zero only for u' = u. That means, if we find such u' for which M=0, then u' is the Ultimate Limit State u and corresponding As is the required area of reinforcement.

Now we use the ultimate border substitution. We define parameter t as shown in expression (1). This parameter is passed to ultimate border function fu. By inserting (2) into (13) we get M as a function of t.

M (t) = MS;u (fu (t)) - ML + MC;u (fu (t)) (14)

Now we can apply numerical solution of a scalar function M. The unknown parameter is t 1,0 , which must satisfy

following condition

M (t) = 0 (15)

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Here we apply Newton's iteration. This method is supplemented with a homogenous selection of starting points for the iteration.

This means that we start with parameter t = 0.5.

If we do not succeed we try t ;...16

7;

16

5;

16

3;

16

1;

8

7;

8

5;

8

3;

8

1;

4

3;

4

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Bi-parametrical reinforcement design

After the single-parameter design we describe the bi-parametrical design. This case is typical for a cross section symmetrical to z-axis with reinforcement situated near upper edge As1 and reinforcement near lower edge As2. The cross section must be loaded in the direction of its symmetry plane. First we have to realise, that if we have in one cross section two different areas of reinforcement in two different places, we can design As1 and As2 for any ultimate plane of strain so they will satisfy balance of internal forces Ru and load forces RL. Nevertheless, there is only one Ultimate Limit State for which the designed reinforcement areas As1 + As2 are minimal.

For a given parameter t we evaluate u'.

u' = fu (t) (16)

From evaluated u' we get internal forces in concrete RC;u and stresses in reinforcing steel bars s1 and s2.

s;i = (s;i) (17)

From the equation of balance of forces we obtain As1 and As2.

As1 × s;1 + As2 × s;2 = NL - NC (18)

As1 × s;1 × z1 + As2 × s;2 × z2 = ML;y - MC;y (19)

Let us consider steps (16)-(19) as a function Afs (t), which returns As1 + As2 in a dependence on parameter t. We can try to find the global minimum of Afs (t) by means of numerical methods. In this case we sample the function on a sparse regular grid and in the minimal value we follow with Newton's iteration.

Note: Parameter t is on horizontal axis. Blue Afs (t) red As1 and yellow As2 are on vertical axis

The previous figure shows an example of function Afs (t), which is depicted in blue. The numerical solution must find the global minimum of Afs (t), which is a sum of As1 and As2.


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Basic Reinforcement or REDES reinforcement

Prior to the calculation of the main reinforcement the user is able to define a layout of reinforcement bars in the cross-section using the advanced member data or REDES reinforcement. These bars can be respected during the design calculation and the program calculates the additional reinforcement area.

For instance:

The user defines two bars in the upper part of the cross-section section of a single span beam loaded by self-weight only. After the calculation of the main reinforcement the user will notice a slightly different amount of necessary reinforcement and a decreased depth of the compression zone.

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Bi-axially loaded columns F

My

L

Mz

The method for uni-axially loaded columns cannot be directly applied to bi-axially loaded columns, since the location of the reinforcement bars is not known in advance, like in the uni-axial method. Therefore the bars could only be used for carrying the load for one direction which would be extremely conservative.

Asy

Asy

Asx

Asx

It is more realistic to use a method that allows bars to act in two directions.

Asy

Asy

Asx

Asx

SCIA.ESA PT uses this more realistic method wherein the positions of the bars are exactly known during the design calculation. By intelligently increasing the number of bars the required number of bars is designed. This area of reinforcement is always the number of bars times the area of a single reinforcement bar, e.g. 1256 for 4Ø20. Also note that the minimum number of bars is 4; 1 for each corner.


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Interaction diagram

SCIA.ESA uses a method that is also described in some code like the ÖNORM. This method is based on an interaction diagram for the design and ultimate moments per direction of bending.

1

x

uz

dz

x

uy

dy

M

M

M

M

wherein:

Mdy Design moment in y-direction

Muy Ultimate moment for reinforcement in y-direction

Mdz Design moment in z-direction

Muz Ultimate moment for reinforcement in z-direction

x Interaction factor, default value is 1.4

The interaction factor is used to define a linear or exponential interaction between My and Mz. An interaction factor of 1 is a linear interaction between My and Mz. This means that when My is fully “used”, the capacity for Mz is zero. Realistically this is not the case and codes will suggest a value around 1.4 for normally loaded columns.

1.0

1.0Mdz/Muz

Md

y/M

uy

x = 1x = 1.4

x = 2 x =

Also through research one has found out that the interaction factor is also dependant on the Nd/Nu ratio. In ÖNORM B4700 (June 2001) clause 3.4.3.5 the safety factor should be taken relative to the ratio of Nd / (Ab × f'b), see table:

Nd / (Ab × f'b) ≤ 0,1 0,7 1,0

x 1,0 1,5 2,0

Between values a linear interpolation may be done.

The work method for the design with the interaction formula is as follows:

SCIA.ESA assumes a reinforcement layout, e.g. 4Ø20 per side. For this layout of practical reinforcement SCIA.ESA determines the Muy and Muz. Then it fills in the interaction formula and gets a result, e.g. 5.5. Since 5.5 is larger than 1.0, SCIA.ESA needs to increase the reinforcement. The reinforcement is increased using a special routine, which will be explained in a later paragraph. Finally if SCIA.ESA gets a result from the interaction formula, that is less than 1.0, e.g. 0.6, SCIA.ESA stops the calculation and the reinforcement amount from that last layout is the result of the reinforcement calculation.

Optimisation

One of the disadvantages of the column reinforcement calculation is that SCIA.ESA stops the calculation, if the result of the interaction formula is less than 1, e.g. 0.5. This does not necessarily mean that the number of bars is the optimal solution. The optimal solution can be a layout of reinforcement with a less number of bars with a higher result value for the interaction formula, e.g. 0.95.

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If the user uses the optimisation function, after the “Normal” design of the reinforcement bars SCIA.ESA will decrease the number of bars 2 by 2 (1 per edge, 2 per direction) and calculate the result of the interaction formula for each layout of bars.

Example

The result of the main reinforcement design is 16Ø20 bars and the interaction formula has a result of 0.8. After gradually decreasing the number of bars the interaction formulae for each layout is calculated, see table.

Layout Interaction Formula

14Ø20 0.98

12Ø20 1.2

10Ø20 3

8Ø20 5

In this specific case a layout of 14 bars has an interaction formula result closer to 1 than 0.8 and thus it is more optimised.


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Ratio

SCIA.ESA uses a special routine to increase the reinforcement in the column. This works as follows:

Prior to the column calculation:

SCIA.ESA automatically determines the design moment per direction, Mdy and Mdz and for those internal forces it determines the Normal stress at the outermost fibre by dividing the moment by the section modulus for that direction, e.g. y = Mdy / Wy. With those 's per direction it can determine the ratio of moments.

ry = y / (z + y); rz = z / (z + y)

Step 1:

SCIA.ESA checks the reinforcement for one bar per corner.

Step 2:

SCIA.ESA determines the values for Muy and Muz and recalculates the interaction formula.

If the results are less than 1, the calculation is stopped.

Step 3:

According the values for r(y/z) the reinforcement is increased per direction.

Step 4:

SCIA.ESA determines the values for Muy and Muz and recalculates the interaction formula.

If the results are less than 1, the calculation is stopped.

Example:

Modeled in SCIA.ESA PT as a frame XYZ.

Concrete class NEN B45, L = 4.5 [m], b × h = 350 × 350 [mm2]

LC1 Permanent Load

F = 1000 [kN]; My = 50 [kNm]; Mz = 125 [kNm]

LC2 Variable Load, momentaneous factor = 0.5

F = 1000 [kN]; My = 50 [kNm]; Mz = 25 [kNm]

NEN ULS Combination = 1.2 × LC1 + 1.5 × LC2

Fd = 1.2 × 1000 + 1.5 × 1000 = 2700 [kN]

Mdy = 1.2 × 50 + 1.5 × 50 = 135 [kN]

Mdz = 1.2 × 125 + 1.5 × 25 = 187.5 [kN]

Wy = 1/6 × 3503 = 7.15 × 106 [mm3] = Wz

r = 135.0 / 187.5 = 0.72

Step 1

As a first layout SCIA.ESA assumes one bar in each corner.

Step 2

Muy = -133.1 [kNm] = Muz

4.14.1

1.133

5.187

1.133

0.135

= 1.011.4 + 1.411.4 = 1.01 + 1.62 = 2.63 >> 1

Step 3:

SCIA.ESA starts adding bars in the cross-section and rechecks the interaction formula. The results were as follows.

25

Nd -1755 [kN] Nd -1890 [kN]

Mdy -87,7 [kNm] Mdy -94,5 [kNm]

Mdz -121,9 [kNm] Mdz -131,2 [kNm]

Nd -2160 [kN] Nd -2565 [kN]

Mdy -108 [kNm] Mdy -128,2 [kNm]

Mdz -150 [kNm] Mdz -178,1 [kNm]

Nd -2700 [kN]

Mdy -135 [kNm]

Mdz -187,5 [kNm]

The increment routine for the number of bars is as follows:

Step 1 Add one bar for the 'weakest' side.

Step 2 Add one bar for the 'strongest' side plus 1/r bars for the 'weakest' side. In which the value of 1/r is rounded off to integer values.


26

For our example: (1/r = 1/0.72 = 1.39)

Step Real Bars y-direction z-direction

0 4 4 4

1 6 4 4*1.39 = 5.6 = 6

2 10 6 6*1.39 = 8.3 = 8

3 16 8 8*1.39 = 11.1 = 12

4 20 10 10*1.39 = 13.9 = 14

Ratio of step 4: 10/14 = 0.71

Ratio y/z: 4/6 (Real bars)

Delta Area

SCIA.ESA PT bases his column reinforcement calculation for an interaction between normal force and bi-axial moments on real bars. By adding sufficient real bars it will find a solution. For some special cases this may seem incorrect. For those cases SCIA.ESA allows the user to define real areas of reinforcement, e.g. 100 [mm2] or 50 [mm2]. SCIA.ESA uses those areas instead of the defined bar diameter in the dialogue ‘concrete member data’. SCIA.ESA however still uses the location of the defined bar in the calculation. Tip Using this option in combination with ‘optimisation of number of bars’ will give the best results.

Multiple combinations

If multiple combinations (e.g. result class “ALL ULS”) load a column and the combinations require reinforcement in different directions, SCIA.ESA PT combines the reinforcement required for two combinations into a new reinforcement layout and amount.

Example

A column calculated using the NEN code is loaded by two combinations. Combination C1 contains a line load in local y-direction and combination C2 contains a line load in local z-direction.

27

Combination C1 requires a reinforcement amount of 6283 [mm3] of which 40% is required in y-direction (8 × 314 = 2513) and 60% is required in z-direction (12 × 314 = 3770). Combination C2 requires a reinforcement amount of 8168 [mm2] of which 23% is required in y-direction (6 × 314 = 1879) and 77% is required in z-direction (20 × 314 = 6289). Note that although the required reinforcement amount for combination C2 is larger than C1, the required reinforcement amount in y-direction for combination C1 is larger than the reinforcement amount in y-direction for combination C2 (2513>1879). Thus SCIA.ESA PT combines both combinations and gives the reinforcement amount based upon the maximum reinforcement amounts in y- and z-direction (2513 + 6289 = 8802 ≈ 8796).

Circular Columns Circular columns are uni-axially loaded columns. Two possible moments My and Mz will be vectored into one design moment Md. Thus principally the same method for uni-axially loaded columns is used.

My

Mz

Md

The only problem is the location of the reinforcement bars. Whilst increasing the number of bars the locations of the bars will change.

SCIA.ESA has implemented a straightforward method of calculating the reinforcement in a circular column. In the first step the program puts six bars in the cross-section and calculates the ultimate moment. If the ultimate moment is larger than the design moment, Md, the calculation stops and the programs returns a reinforcement area equivalent to the area of the chosen reinforcement bar diameter times 5, e.g. 1571 for Ø20. If the ultimate moment is smaller than the program increases the number of bars by one and recalculates the ultimate moment, etc.


28

Remark The ultimate moment capacity of the cross-section is based on two layouts of reinforcement bars.

Md Md

Walls, Plates and Shells One of SCIA.ESA PT most outstanding features is its ability to deal with two- and three-course reinforcement meshes of deliberate geometry, i.e. the angles closed by pairs of reinforcement directions may be freely specified, however, within reasonable limits. The next figure shows the basic definition scheme of reinforcement geometry: The directions of the 2/3 reinforcement courses specified for design are expressed by angles <0°, 180°) closed with the 1st planar axis xp.

The reinforcement geometry may be specified individually at each of the 2D structure faces, concerning the direction angles and the number of reinforcement courses (2 or 3). So it is, for example, possible, to specify at one face a skew two-course reinforcement net with directions, say, 10°/70° and, at the same time, a three-course reinforcement net with directions, e.g., 0°/60°/120° at the other face. The standard orthogonal reinforcement 0°/90°, allowed by most design programs as the only reinforcement geometry specification, is in SCIA.ESA PT one of all possible constellations, nothing more.

xp

yp

1 2

3

1

2

3

29

Transformation of inner forces to design forces

Once the reinforcement design input data have been read and analysed and the FEM Data Base approached, the SCIA.ESA PT design model can be created respecting all Code rules and restrictions applicable to the active structural model: SCIA.ESA PT distinguishes between the Wall, Plate and Shell structural type. They are different not only as to the principal assumptions about the mechanical properties of the reinforcement concrete medium but, in all Codes, also as to the requirements and restrictions these structural types are subjected to.

The first substantial step of the design procedure to be reported here is the calculation of inner design forces for each item to be designed. SCIA.ESA PT distinguishes two design items: elements and nodes. The design forces transformation procedure outlined here takes thus place at each step of the (multiple) design loop.

The SCIA.ESA PT transformation procedure is based on a general ‘transformation formula’ published by Baumann:

ci = [sinjsinkcosjcosk] / [sin(j - i) sin(k - i)]

(i, j, k = 1,2,3)

In this formula the subscripts i, j, k denote the three reinforcement directions according to previous figure. When applied to a pure bending case with principal moments mI and mII, the variables have the following meaning:

i,,j, k : angles between individual reinforcement directions and the direction of the 1st principal moment mI

: quotient mII/mI; according to the values of mI and mII it can attain negative, zero and positive values

ci : transformation coefficient of the direction i:

mi = ci mI

The formula is equally valid for Walls. In that case, however, the principal moments mI and mII in are to be substituted by the principal membrane forces nI and nII to be valid for Walls, too.

In case of Shells, the combined bending-membrane inner forces {mx, my, mxy, vx, vy, nx, ny, nxy} must first be transformed to virtual membrane forces acting as two formally independent force systems at both structural faces:

px = mx/z + nx/2

py = my/z + ny /2

pxy = mxy/z + nxy/2

In these formulae z represents the inner forces lever calculated, as reference value, for the outermost reinforcement layer. In subsequent design calculations, it is considered that the inner layers have effectively lesser inner forces levers than z symbolised by the previous formulae. The calculation of z is an interesting chapter of the design procedure. In this case, the calculation for the first time must refer to the material properties of the concrete continuum; it is no more „material independent“ like the transformation formula, which is based on the assumption of linearly elastic material.

The reference value of z is obtained as the minimum value of the inner forces lever calculated for three characteristic cases:

mI and associated n nI and associated m nII and associated m

The reference virtual membrane forces {px, py, pxy} are in the SCIA.ESA PT design algorithm formally subjected to the same procedure as normal membrane forces of a Wall structural model. However, there are differences in processing them to the final result; they will not be reported here in full detail.

The transformation formula does not yet represent the final solution of the transformation problem. The transformation coefficients ci, cj, ck thus calculated represent transformation forces in a linearly elastic medium that does not make difference between tension and pressure design forces. Such a solution is generally not applicable to a reinforced concrete medium, where the basic medium concrete can oppose pressure stresses only.

Let us, for general considerations, independent of the structural model, denote the design forces obtained by the transformation as {p1, p2, p3}. In Walls the symbols pi represent the design Normal forces {n1, n2, n3}; in Plates - the design moments {m1, m2, m3} and in Shells - the virtual design forces {p1, p2, p3} corresponding in the Baumann transformation to (px, py, pxy) after the formulae (3). The transformation formulae have a fundamental invariant meaning, whatever the values of {p1, p2, p3} are:

p1+ p2 + p3 = pI + pII = const


30

where pI and pII symbolise, analogously, the principal Normal forces nI and nII (Walls), the principal moments mI and mII (Plates) or, directly, pI and pII (Shells). The Formula (1) yields several solutions satisfying. For SCIA.ESA PT, the solution representing the minimum energetic level is used for the design.

I>II>0

I

II

In an elliptic state of stress, the solution for a three-course reinforcement net is relatively quickly found. In two-course reinforcement nets, which represent, without doubt, the standard use in the building practice, only two design forces can be assigned to reinforcement. The third force of the invariant formula is assigned to the concrete medium. It is clear that its value must be negative, for concrete is not able to resist tension. Only in special reinforcement arrangements or in a strictly circular state of stress, the concrete design force can vanish: the concrete medium performs (theoretically) no mechanical work in that case and may be considered as stress-free.

Of practical meaning, however, is the case with concrete participating in resisting the inner forces of external loads. The function of concrete may thus be explained as stiffening medium of the deformable reinforcement steel net, which would, by itself, deform under the action of tension or pressure forces in its plane. We will call that function of concrete as Stiffening Virtual Concrete Strut“, or, more simply, just Concrete Strut.

The position of the Concrete Strut is, however, generally not identical with any reinforcement direction specified for design. It means that if formula does not yield for a three-course reinforcement net all three positive design forces, at least one of the reinforcement courses is inactive (or two of them); the Concrete Strut does not automatically coincide with one of the reinforcement courses! The assessment of the Concrete Strut position is thus an important optimisation task.

It is an outstanding feature of the SCIA.ESA PT design algorithm, developed by months and years of improvements of theoretical and algorithmic procedures, that the formula can usefully be applied to all possible situations of elliptic (1>2>0), parabolic (1<>0;2=0) and hyperbolic states of stress/strain, i.e. also to elliptic pressure state, thus yielding design forces which enable optimised reinforcement design. With respect to competing design programs, the publication of these algorithms is undesirable in any form.

I>0;II<0

II

I

31

Reinforcement Design

Introductory to this paragraph, dealing with the central topic of SCIA.ESA PT, concepts already discussed above to illustrate the SCIA.ESA PT´s algorithm from a more common point of view will be summarised here and given, if necessary, their special explanation.

Reinforcement concrete 2D structures handled by SCIA.ESA PT - Walls, Plates and Shells - are usually reinforced by two systems of steel reinforcement nets consisting of 2 or 3 reinforcement courses situated more or less close to both faces of the 2D structure. SCIA.ESA PT puts no principal restrictions upon the absolute position of reinforcement courses within the cross-section; its axial concrete cover describes the position of each reinforcement course. However, there are relative restrictions: all concrete covers must fulfil some rules to prevent ambiguousness of the geometric definition of the design task. These rules are described in the part of the SCIA.ESA PT manual.

Yet it must not be forgotten that there might be other, more complex situations in the cross-section than symbolised by the next figure:

1. The crossing reinforcement bars of individual layers do not need to touch each other; they might be placed at larger distances from each other within the cross sections;

2. The surfaces of bars are usually corrugated so that there is, as a rule, a greater distance between two crossing bars than expressed by their characteristic bar diameters;

3. Last but not least, in very thick plates, e.g. foundation slabs, two layers or bars bundles in one layer are used, so that the representative axial distance (of the point of gravity) and the representative bar diameter itself are two independent quantities and qualities, which must be defined independently on input in order to carry out reliable analysis.

In Walls, being (theoretically) subjected to forces acting in their planes, the (by definition symmetric) positions of reinforcement nets are of no static interest; however, the cross-section geometry (concrete covers and bar diameters) is of interest for the Crack Proof algorithm (if implemented). Thus, the Wall design branch comprises the same cross-section input dialog as the Plate and Shell models.

In Plates and Shells, on the contrary, the reinforcement covers estimate the effective static height of the reinforcement courses in the cross-section subjected (also) to bending, thus having fundamental meaning for the design process. The covers are related to the faces. Thus, it is necessary to distinguish them clearly from each other. Because Plates are (still) the structural type most frequently used in the practice, SCIA.ESA PT used originally common terms distinguishing the two faces: upper and lower face. These concepts have to be given mathematically exact meaning, which makes them acceptable for Shells, too: the lower face is the structural plane edge in direction of the positive planar axis Zp; the upper face is opposite to it. Finally, the symbol -Zp appears generally in the output protocol instead of the term upper face; the symbol +Zp symbolises lower face. In Walls, there is no need of distinguishing both structural edges; nevertheless, out of formal reasons (simplification), if the concept of upper face appears in connection with Walls it means both faces.


32

The reinforcement courses are, correspondingly to their relative position in the cross-section, called the outer(most), middle (if any) and inner(most) ones. This verbal distinguishing is in the mathematical formulation replaced by assigning them the ordinal numbers 1, 2 and 3 (if three reinforcement courses are specified at all). The same double identification may be given to other associated terms like reinforcement angles, design forces, effective static heights, inner forces levers, etc. So we can speak, e.g., about reinforcement angles , , meaning the same when alternately indicating 1, 2, 3. There is no indication that this ambiguity of terms should cause confusion; as a fact, there is no ambiguousness for the correspondence of both systems of denotation is clearly defined.

Remark: Note that each reinforcement course can hold up to 10 reinforcement layers.

The terms of the reinforcement concrete theory are used in accordance with the general structural use or they strictly follow the rules postulated by the Norms implemented in SCIA.ESA PT. However, for SCIA.ESA PT deals with several national codes, it is probable that this or that term or formulation would appear somewhat unfamiliar to some readers focused onto the use of one code branch only. It is hardly possible to create a manual text on such special topic for international use being in all respects verbally fully conform to every country’s verbal usage. In doubts, the terminology of Eurocode will be given pre-ference.

The design task and the output of results are performed in basic and derived units of the SI system.

Reinforcement design of Walls

The inner forces {nx, ny, nxy} of the FEM solution are retrieved from the FEM Data Base for each design item (element/node) and transformed by the method outlined above into the design (membrane) forces {p�, p2, p3}.

Once a positive design force has been assigned to its associated reinforcement course, the corresponding statically required reinforcement amount ai is calculated after a Formula like this:

ai = pi / dim (i=1,2 (,3)) [cm2/m]

The previous formula has a symbolic meaning only, for we cannot write down an exact calculation rule for codes implemented in SCIA.ESA PT. The symbol dim stands for design effective steel strength. Both pi and dim may be, according to the code of question, charged with bearing and/or security coefficients. We will not discuss the problem of elementary reinforcement design; the SCIA.ESA PT algorithm follows strictly the rules postulated by the national codes and associated regulations.

Above it was emphasised that the application of the transformation formula to the inner forces of the FEM model yields not yet the final result and that there are formally several solutions fitting the invariance condition. In a class of hyperbolic states of stress (nI > 0, nII < 0) the SCIA.ESA PT algorithm finds, by means of the Baumann formula, an energetic minimum solution of the following quality:

p1 > 0; p2 = 0; p3 < 0

In p1 > 0 is the (positive) reinforcement design force; the second reinforcement course is set inactive (or both remaining ones, if a three-course reinforcement is specified); p3 is the (negative) design force of the stiffening strut.

For a two-course, skew (i.e. non-orthogonal) reinforcement net (representing equivalently also three-course reinforcement nets under hyperbolic state of stress) we will follow the explanations by means of the next figure.

33

In certain hyperbolic stress situations, the SCIA.ESA PT transformation algorithm yields for the skew reinforcement specification according to previous figure a solution of the quality. It is sketched by figure (b): the reinforcement course 1 is assigned a tension design force 1p1 > 0; the reinforcement course 2 is inactive; the pressure stress acting at the cross-section of question is resisted by a relatively large pressure force of the stiffening concrete strut 1p3 < 0.

However, the heterogeneous reinforced concrete medium would hardly be armoured by one reinforcement course only. Even if the state of stress would prevail in extensive parts of the structure, an at least two-course reinforcement mesh would still be necessary to maintain the functionality of such 2D-structure. Due to the Minimum Transversal Percentage requirement, the reinforcement course No 2 deactivated (theoretically) in this case would generally be assigned a portion of the statically necessary reinforcement amount of the reinforcement course 1. Thus, in a practical reinforcement design the reinforcement course 2 would also be assigned a real as value.

In addition, many of national codes implemented in SCIA.ESA PT require a Minimum Pressure Reinforcement proof for reinforcement resisting pressure forces. In the case of figure (b), such proof could formally not be performed for there is no data of the calculation pressure force assigned to the reinforcement: the strut pressure force is not associated with any reinforcement course!

For reasons outlined, the seek of another solution fitting the transformation formula yet assigning a non-zero design force to the reinforcement course 2 seems to be a logical if not indispensable algorithmic step. As a fact, SCIA.ESA PT carries this step out automatically in such stress-situations and yields a second order solution symbolised by figure (c). Using vector arrows of different lengths, the stress vectors of figure XX (b) and (c) express the substantial difference of the two transformation solutions mentioned. In mathematical notation the relations are as follows:

2p1 > 1p1; 2p2 > 1p2 = 0; 0 > 2p3 > 1p3

SCIA.ESA PT makes of these two consistent solutions (they are consistent for they fit the invariance condition and inconsistent final solution by combining them according to figure (d). Analytically expressed:

p1 1p1; p2 2p2 < 0; p3 1p3 < 0


34

The solution (a,b) is extraordinary productive: SCIA.ESA PT extends the design forces set of the first basic solution by the pressure design force for the reinforcement course 2. Experience shows that the real amount of pressure reinforcement calculated by this procedure is generally relatively small; in most cases, the Minimum Transversal Percentage requirement yields a higher value, thus replacing the pressure reinforcement value in the output.

The solution described by figure (d) is inconsistent in the sense of the invariance condition: it is no more fulfilled by the set of forces combined to form the effective solution. To denote this important circumstance, the design pressure force p2 = 2p2 is marked by trailing ‘!’ in the output table of design forces of the printable document, however, only if the design forces table output is activated. For the structural engineer it is of importance that the solution just presented is consistent with national code requirements about Minimum Pressure Reinforcement and represents a good mechanical solution of the design problem.

The preceding observations made it obvious that the virtual stiffening strut of the heterogeneous concrete-steel continuum represents a quite substantial item of the design process. Whereas it is possible (unless the Upper Reinforcement Percentage has not been exceeded) to improve the bearing capacity of the cross-section on the side of the reinforcement by augmenting its amount, the bearing limit of the concrete strut is given by the height of the cross-section and the quality of concrete only; thus its limits are predestined by the input data. The following relation describes the concrete strut bearing capacity limit condition:

Reinforcement

Cracks

- p3 < Ac,dim

c,dim representing the concrete effective design stress, which, according to the dode of question, may comprise a security coefficient. In SCIA.ESA PT it is assessed on the base of 80% of the standard concrete pressure strength. This effective reduction follows the recommendation of Schleich and Schäfer: the bearing capacity of concrete under pressure is unfavourably affected by transversal tension stresses which produce cracks parallel to the direction of pressure; this is typically the stress situation of the stiffening strut.

The cross-section area A in is taken in Walls as the full amount of the unit rectangular cross-section h x 1.0.

The transformation formula may yield, however, in other hyperbolic states of stress direct design pressure forces assigned with reinforcement courses specified. At any case, once a design pressure force, direct or virtual one, is known the pressure reinforcement is calculated after the following general formula:

ai = (- pi - Ac,dim) / dim (i=1,2 (,3)) [cm2/m] (ZZZ)

35

Reinforcement design of Plates

In the Wall model the inner as well as the design forces produce constant presses over the cross-section; thus, there is no necessity to examine the distribution of stresses within the cross-section. For bending in Plates, it is a fundamental characteristic that the stresses are non-linearly and discontinuously distributed over the cross-section. For all of the national Codes implemented in SCIA.ESA PT exclude the tension bearing capacity of concrete out of the reinforcement design concept, in the tension zone („below“ the neutral axis) the only bearing material is the reinforcement steel. The resistance ability of concrete exerts in the pressure-bending zone only.

DIN 1045 introduces the concept of combined parabolic (2°) and constant pressure stress distribution. It is the most complex assumption of all Codes implemented (the so called „Parabel-Rechteck-Diagramm“).

ÖNORM B 4200 does not allow for fully plasticized concrete in a portion of the pressure zone; thus, the pressure stress distribution function is parabola 2°. It is of interest to point out that by this assumption ÖNORM gives for comparable material strengths reinforcement design solutions with higher virtual security than DIN.

EUROCODE 2 allows for all national Norm assumptions. For SCIA.ESA PT serves as design algorithm on international scale it would, strictly considering the situation, be necessary to develop several national mutations of the EC 2 algorithm. Actually SCIA.ESA PT keeps to the basic variant developed under the concrete pressure stress distribution assumption according to DIN 1045 which comprises the assumptions of almost all of the implemented Codes. EUROCODE 2 introduces a new concept of the Shear Proof, which explicitly operates with the concept of the shear virtual strut. It also formulates a new approach to the consideration of the interaction between the bending moment + normal force and the shear force. After this concept, the shear force causes, typically, an increase of the necessary net reinforcement. This phenomenon was analysed by the Author of SCIA.ESA PT and 1999 implemented algorithmically into the EUROCODE 2 design branch as well as into all other design branches following the same (or similar) concept (SIA 162, DIN 1045-1, ÖNORM B 4700 – see below).

CSN 73 1201 introduces the concept of the so called „Pressure Bloc“: the resisting concrete stress is assumed to develop constantly over a portion of the pressure zone only (i.e. it covers not completely the zone between the neutral axis and the compressed face) thus forming a calculation substitute for the reality approximated. Comparative tests showed that there is no substantial difference between this simplified approach and more complex pressure distribution assumptions of other Codes on the side of the design results.

SIA 162 introduces, similarly to CSN 731201, the concept of the „Pressure Bloc“ and, parallelly, the “Parabolic-Constant” stress distribution analogously to DIN 1045. SCIA.ESA PT uses the former assumption. It might thus be expected that the design results of CSN 731201 and SIA 162 would differ slightly from each other for comparable material qualities. However, this proves to be true in situations with vanishing shear forces only! As a fact, SIA 162 was the first of national codes implemented in SCIA.ESA PT (and the first Norm used in practice on international scale) which formulated the impact of shear forces upon the mesh reinforcement on both faces(!), which was given the name Shear Effect (see above, EUROCODE 2). Because SIA 162, in difference to EUROCODE 2, DIN 1045-1 and ÖNORM B 4700, does not regularly allow for a design variant without considering the Shear Effect, the development of the Shear Effect algorithm of SCIA.ESA PT was, as a fact, induced by SIA 162, rather than by EUROCODE 2. This special treatment of the Shear Effect is by the SIA Norm explicitly formulated for beams, i.e. 1D structural members, only. In order to make it applicable for 2D reinforcement models, some special assumptions and algorithmic enhancements had to be made. This SCIA.ESA PT´s genuine development was implemented 1999 and published in [17]. By this SCIA holds priority not only in reporting on this phenomenon but also in having developed and implemented their own 2D algorithm in a design program distri-buted on the international market. Aspects of this phenomenon will be discussed in more detail in the Chapter Shear Proof.

NEN 6720 operates with the assumption of linearly changing and constant branch of the pressure distribution function. In comparison with other codes it can be summarised than NEN 6720 is a sophisticated standard of high engineering value. Especially its concept of Shear and Crack Proof is highly valuable. However, NEN 6720 does not introduce the concept of the Shear Effect (see above).

DIN 1045-1 1998/12 is a mutation of EUROCODE 2 developed as a substitute for the actually valid DIN 1045, 1988/7. It maintains the concept of combined parabolic (2°) and constant pressure stress distribution of its predecessor issue. However, the material strengths are defined in the Eurocode manner. The most distinguishing features to the “old” DIN 1045 are: (a) the Eurocode concept of partial safety factors; (b) Eurocode-like classification of concrete; (c) substantially higher allowed ultimate steel strain; (d) the Shear Proof concept is substantially that of EUROCODE 2; however, some new elements were introduced, not all being a real improvement.

ÖNORM B 4700, declared as “Eurocode-like” Norm, is in its concept very similar to EUROCODE 2 or to DIN 1045-1. It introduces both the concept of the “Pressure Bloc” and the “Parabolic-Constant” stress distribution after DIN 1045-1. SCIA.ESA PT uses the latter assumption. The Crack Proof concept of ÖNORM B 4700 is relatively detailed elaborated.

The statically required tension reinforcement of a steel course is calculated by the following elementary formula:

ai = mi / (zi dim) (i=1,2 (,3)) [cm2/m]

The special moment symbol mi for the design moment associated with the reinforcement course ‘i’ is used instead of the common symbol pi for design force in order to avoid confusion with hasty readers. The stress symbol dim has a comparable quality as that explained with the formula for Walls; it again represents the effective design steel strength for all codes. The inner forces lever zi in makes out the obvious difference between the formulae. As a fact, there is no difference between them, for the quotient mi/zi equals the steel design force Zi, which constitutes with the opposing concrete pressure zone resultant force Di the forces couple representing the design bending moment mi; thus, we formally obtain the formula when substituting pi = Zi = mi / zi.

The previous formula reveals the fundamental and equally elementary meaning of the inner forces lever z for the design algorithm. As a fact, by introducing the transformation formulae (3) for Shells above it was made clear enough that the knowledge of the proper value of inner forces lever is indispensable for the reinforcement design.


36

In SCIA.ESA PT the inner forces lever z is calculated following the following procedures:

For DIN 1045, ÖNORM B 4200 and EUROCODE 2 interpolation formulae yielding the value of z very quickly were developed. The maximum approximation error amounts up to 2%, however, in the region of vanishing bending zone heights; the interpolation Formula is much better fitted to higher stress states where there the approximation error is less than 1%.

For CSN 731201, SIA 162, NEN 6720, DIN 1045-1 and ÖNORM B 4700 analytic integration procedures were develo-ped; they yield exact pressure integrals.

The stiffening function of the concrete medium is not as transparently obvious in Plates as in Walls. In Plates we have to do with force couples described as bending moments. The concrete pressure stresses are not constantly distributed over the cross-section. Thus, a direct application of the concrete strut bearing capacity limit condition (8) was not possible here. SCIA.ESA PT had used some approximate approaches until the best and perhaps most simply formulation of the strut bearing capacity limit was found. It is, however, not simply enough to be described by a few mathematical terms; in SCIA.ESA PT it is formulated algorithmically. Here we give the following verbal explanation of the matter relating to figures (c) and (d):

In Plates the strut design force p3 means the force couple m3. From figure XXX (c) it is obvious that m3 causes basically the same kind of stresses in its direction as the other two reinforcement design moments m1 and m2, however, with ex-changed faces (i.e. m3 is of opposite sign). In this case we are not interested in analysing the situation on the tensioned face; the state of stress in the stiffening strut bending zone is of interest. What is the limit condition of the strut bearing capacity; what calculation value of stress integral force D3 can be taken into account?

The answer to the fundamental question posed under (1) is given by figure (d): SCIA.ESA PT allows for the maximum height of the bending zone xmax in the sense of the design algorithm applied. If at this state of stress the equilibrium in the cross-section is not yet attained, i.e. would strengthening of the pressure zone by (pressure) reinforcement be formally necessary, then this is seen by SCIA.ESA PT as an unambiguous indication of the bearing capacity of the stiffening strut being exceeded. The cross-section is non-designable due to concrete failure (Error number 5).

It is not known to the Author of SCIA.ESA PT that any other design algorithms would deal with this problem at all. Even theoretical publications on reinforced concrete design and the Codes implemented here do not care about the state of concrete in the heterogeneous concrete-steel medium under bending. DIN 1045, DIN 1045-1 and ÖNORM B 4700 (curiously, not the old issue ÖNORM B 4200), give some “standardised” advises as to the geometrical arrangement of reinforcement in reference to the directions of the principal moments; they are concerned with stressed situations which are typically of the hyperbolic type (situation in corners of floor slabs etc.).

The virtual strut bearability is a problem of acute practical interest. Users changing to EPW from other program systems come earlier or later across the design error 5. In discussion with the SCIA´s hotline support they then usually claim: “With our ´old´ program there we never had such a problem. All the time we had been using it, no exhaustion of the concrete bearability was reported”. It requires often quite a lot of patience to explain to them that programs that do not care of a phenomenon cannot give any report of it. If the state of stress of the concrete is not monitored sufficiently, not all of possible critical situations can be realised by the design algorithm. Protests like: “We do not know anything of damages to a structure due to insufficiency of the stiffening function of the concrete”, which we hear from time to time, are of no practical impact. Our structures are built with rather a high security reserves. Underestimating of the bearing capacity of concrete does not cause immediately a crash yet generally a lesser than the required level of security, which we are bound to achieve and maintain by codes and other Standards.

37

Reinforcement design of Shells

In the design of Shells, the ideas and procedures of both the design of Walls and the design of Plates are combined. The code requirements and restrictions, which seldom are formulated individually for Shells, must both be considered both for Walls and Plates. Thus, the Shell design is the most complex design model dealt with by SCIA.ESA PT.

From the mechanical point of view, the stress-strain situation in cross-sections of Shells may develop from a typical „Wall pattern“ with constant stress distribution to a „Plate pattern“ with its characteristic non-linear concrete pressure stress distribution over the bending pressure zone along with a cracked region „below“ the neutral axis where there the reinforcement resists the stresses from inner forces. The special situation depends, however, on the character of external load as well as on the boundary conditions of the structure modelled.

SCIA.ESA PT has to manage all possible stress situations arising between the Wall type and the Plate type state of stress using one unique design model to be able to produce results consistent with quantitatively slowly yet qualitatively abruptly changing states of stress. It would be non-acceptable to have such a Shell design model which yields on one side results fully identical with a Plate solution when there is pure bending acting, i.e. the membrane forces being zero, yet would produce unintelligible results just because the membrane forces differ slightly from zero. Little change in loading must imply also little change in the reinforcement design results.

We must be aware of the fact that all of the code texts implemented into SCIA.ESA PT were drafted with strongly focusing to the problems of 1D structural members, i.e. beams. In SCIA.ESA PT, several requirements and restrictions had to be given a reasonable engineering interpretation or extrapolation to fit to the special character of the 2D-structures of interest. So it was also in the design algorithm itself. Above it was shown that the seek of a representative (in this case the minimum) value of the inner forces lever z may be quite a complex algorithmic task for the directions of the principal moments mI and mII generally differ from those of nI and nII. Additionally, the reinforcement on both faces consists of two mutually independent meshes with 2 or 3 reinforcement courses in different directions. In Shells it is thus not possible to proceed by using the design solutions of the type (m/n) „moment + normal force“ like in the design theory of beams.

SCIA.ESA PT follows the logical approach of creating two sets of transformed design forces assigned to individual reinforcement courses and/or the stiffening concrete strut on both faces of the structural model. In the assessment of the inner forces lever z the Shell design procedure resembles the Plate design. In the creation of equivalent inner forces {px, py, pxy} and their transformations (p1, p2, p3) SCIA.ESA PT follows a typical Wall design approach. Formally, we get two systems of design situations on both Shell faces that must be managed in two algorithmic steps in each cross-section by considering the situation on the other face. In this sense, the Shell design is organised like the Plate design.

The next figure shows symbolically a typical Shell design situation: there is the representative design force pdim assigned to a reinforcement course at the upper face (the same procedure applies, however, to the lower face). In next figure symbol popp is used for the virtual design force on the opposite face going in the same direction as on the actual (upper) face; it is without impact if there is specified a congruent reinforcement course parallel to that on the actual face (associated with pdim). The total normal force in this cross-section is denoted as pvirt (virtual normal force). Analogously, the associated virtual bending moment mvirt is defined to constitute the inner forces couple (mvirt, pvirt) acting in the cross-section of interest. Thus, the virtual eccentricity can be estimated. Its value decides of the cross-section exploitation status.

The figure reveals that the design on a Shell face is typically a Wall design; however, the design force pdim is not applied to the total cross-section area as in Walls, yet to some portion of it. SCIA.ESA PT assigns this portion of A in accordance with the suggestions of Baumann. In the area assignment formula

As = A (ZZZ)

the value of the coefficient varies from code to code in the range <0.35; 0.42>. In some sense, this approach may be compared with the approximation made by CSN 731201 and SIA 162 in formulating the stress distribution in the bending pressure zone using the „Pressure Bloc“ approximation (see above). The reinforcement design goes then analogously after the formulae (ZZZ) and (ZZZ). Also the strut proof is the same as for Walls; it is governed by the formula (ZZZ). However, instead of the total cross-section area, the effective one-face area As is to be substituted into these formulae.


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Design of shear reinforcement

Beams

General

As already stated in the chapter about internal forces, the shear reinforcement design is commonly based on the theory of the concrete truss-model. In this theory a virtual truss-model is imagined in a concrete beam. This truss-model has a set of vertical (or slightly diagonal), horizontal and diagonal members. The vertical bars are considered to be the stirrups; the horizontal bars are the main reinforcement and the diagonal bars are the concrete struts.

All implemented codes postulate a stress level which, when exceeded, indicates the necessity of shear reinforcement to ensure the cross-section resistance to shear, as well as another (higher) level which, when exceeded, signalises the structure becoming non-designable. Following the concepts of DIN 1045 the first (lower) limit restricts the so-called Shear Region 1. In SR1 (symbolical abbreviation), no shear reinforcement is necessary. In the Shear Region 2 (SR2) which comprises all stress states between the two limits, the shear reinforcement is calculated on the base of the design value vdim in (15). It proved as good means of communication to use the concept of the Shear Regions when discussing the Shear Proof algorithm aspects and commenting design results of all Codes. When the upper limit of the SR2 is reached in a 2D structural medium, the shear bearing ability of the cross-section is considered to be exhausted. This limit is the threshold to the Shear Region 3. When SR3 is indicated in the output protocol of the 2D design it means non-designabilty of the cross-section.


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Composite Section and Arbitrary Sections

Composite sections (concrete/concrete) are more difficult to calculate the shear reinforcement for. Principally a set of two truss-models must be defined in order to calculate the reinforcement. One truss-model for the lesser concrete quality and one for the difference between the larger and lesser quality. In SCIA.ESA PT a straight forward of calculation of these sections is allowed. The user is able to define the concrete quality by hand, thus allowing a more practical solution. By default however the lesser quality is always taken in the calculation.

Since SCIA.ESA PT is capable of calculating main reinforcement for any vertical symmetrical section, the shear reinforcement calculation can sometimes be seemingly impossible. In those cases SCIA.ESA PT allows the user to set the value of the shear width himself. Thus the shear area is defined by the static height of the cross-section and the user-defined width. Then a normal shear reinforcement calculation is performed.

Columns Presently SCIA.ESA PT does not support shear reinforcement calculations for columns loaded in two directions. When one wants to calculate the shear reinforcement for a column loaded in one direction, change the beam type of the column to ‘beam’ and the shear calculation can be performed.

Plates and Shells All of the Codes implemented in SCIA.ESA PT have their own mechanical model of how slender structures like Plates and Shells resist the shear stressing and how they can be strengthen to increase their shear resistance. Thus, the Shear Proof is still more diversified than the "pure" net reinforcement design. However, this diversification becomes more relative by near consideration. As a fact, there are many common ideas of the Shear Proof among the Codes implemented.

The shear forces vx and vy in the point of consideration are transformed into the design shear force vdim after the following „geometric sum“ formula:

vdim = (vx2 + vy

2) [kN/m] (15)

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Some of the modern Codes – EUROCODE 2, DIN 1045-1, ÖNORM B 4700 – require in the Shear Region 2 some amount of Minimum Shear Reinforcement. This requirement is regulated by special tables combining the control by the concrete and steel strength class. Not unlike this requirement, CSN 73 1201 formulates an additional shear stress limit, which, if attained, implies the need of the so called Structural Shear Reinforcement in such parts of the structure designed (see below).

Shear Proof Concepts

To enable better understanding of the design results, the most important characteristics of the Norm oriented Shear Proof procedures applied in SCIA.ESA PT are summarised here:

DIN 1045 (1988/7) introduces the concept of the so called “Truss Model” (“Fachwerkmodell”) of the shear stress resistance mechanism. Three Shear Regions are defined corresponding with different levels of cross-section exploitation. In 2D Structures of interest, Shear Region 3 is no more allowed. On the other side, in difference to 1D structural members no shear reinforcement is needed in Shear Region 1. The Shear Region limits are expressed in terms of allowable shear stresses as functons of the concrete strength class. Depending on the continuity of the tension reinforcement in individual spans (i.e. fields from support to support), one of two sets of shear tension limits applies. The categorisation by Shear Regions seamed to the Author of SCIA.ESA PT mechanically and formally so representative that it was generalised to describe comparable design states also in other Codes implemented (see above).

ÖNORM B 4200 defines a Shear Proof concept similar to DIN 1045. In difference to DIN, the continuity of net reinforcement is not a factor of design; on the other side, the amount of net reinforcement at both faces is a bearing capacity increasing factor. For in ÖNORM B 4200 this concept is, as usual, introduced for 1D structural members only, SCIA.ESA PT assumed the following generalisation: as effective values of longitudinal reinforcement the „geometric sums“ (analogously to the Formula (15)) on both faces separately, are taken.

EUROCODE 2 introduces a more advanced Shear Proof concept than DIN 1045 (1988/ 7). Two alternative Shear Proof methods are legal: (a) standard procedure based on the Constant Shear Strut Inclination assumption; (b) shear proof model using the Variable Shear Strut Iinclination concept. Actually, ESA-Prima Win enables the application of both appoaches. Like all Codes applying the modern concept of the Strut Inclination, EUROCODE 2 introduces the Shear Effect procedure (see above). In early SCIA.ESA PT versions (before 1997), the explicit calculation of the impact of shear forces upon the net rein-forcement design was disregarded, on the base of the Article 4.3.2.4.4(6), which allows for the consideration of the Shear Effect by applying constructive measures to the net reinforcement, analogously to the old Norm generation (the concept of the so called “Reinforcement Shift”).

CSN 73 1201 has its special Shear Proof concept, which is based on the concrete tension strength merely than on allowable shear stress limits; both approaches are, however, equivalent. In addition to other Codes, CSN 73 1201 formulates a design situation where the so called Constructive Shear Reinforcement is required. For this reason, the concept of Shear Regions, introduced by DIN 1045 and applied to other Codes as well, has for CSN 73 1201 been extended by the formal introduction of the Shear Region 0. This region is equivalent to SR 1 of other Codes. In SR 1 of the CSN 73 1201 mutatation, Constructive Shear Reinforcement is calculated. SR 2 and SR 3 have then analogous meanings like with other Codes.

SIA 162 works after the concept of the Strut Inclination Method. As a fact, analogously to the mode (b) of EUROCODE 2 (see above), the Variable Shear Strut Inclination method is the standard mode of SIA 162. However, the very special requirement of the Article 3 24 203, which is hardly to controle in 2D structures, made it necessary for SCIA.ESA PT to keep the mode (a) of the Constant Shear Strut Inclination available to the user´s decision. SIA was the first Norm in international scale that introduced the Shear Effect concept into practice; however, SIA 162 did it very consequently: the traditional approach disregarding the Shear Effect mechanism was disabled as a means of Shear Proof. It means that in real situations of the Plate and Shell design, where the shear forces act in combination with moments and membrane forces, an increment of the tension net reinforcement due to shear is a standard design result under SIA 162. Sometimes, also the natural prestress (in pressure zone) may be overridden by the longitudinal shear forces, so that tension reinforcement appears as final result where there would, without the consideration of the Shear Effect, be no tension reinforcement necessary [17]. Thus, the Shear Effect has to be considered in „benchmark“ test examples. Also, in regions nearby (end line) supports, where bending moments ap-proach zero while the shear stresses attain extreme values, the difference between the design results disregarding the >v-m/n< interaction and the results containing the Shear Effect increments of the net reinforcement may been "surprisingly" high! It is mainly to enable such comparative calculations like “benchmark tests” that the alternative mode (a) was introduced into the SIA 162 Shear Proof branch as a non-standard calculation mode.

NEN 6720 uses a fine, sophisticated, modern concept of Shear Proof resembling a combination of the ideas of EUROCODE 2 and SIA 162. NEN 6720 applies the Variable Shear Strut Inclination assumption for the Shear Proof algorithm. Also the concept of Shear Regions (see above) is applicable to the NEN 6720 calculation process as common classification base.

DIN 1045-1 (1998/12) was implemented into SCIA.ESA PT before its final installation into the German Engineering practice. The concept of DIN 1045-1 is based on EUROCODE 2. There are, however, differences, yet as a whole they are not of primary importance. The DIN 1045-1 design branch follows the same ideas in the dealing with the problem of variable/fixed strut inclination method and the consideration of the Shear Effect.

ÖNORM B 4700 is actually the last Norm branch implementation into SCIA.ESA PT. The overall design oncept of ÖNORM B 4700 is based on EUROCODE 2. The Shear Proof concept differs slightly from the original EUROCODE 2 concept. The ÖNORM B 4700 design branch follows the same ideas in the dealing with the problem of Variable/Fixed Strut method and the consideration of the Shear Effect like EUROCODE 2 or DIN 1045-1. It comprises an interesting individual approach to the problem of Crack limitations.

Advanced notes on the Shear Effect concept

The Shear Effect phenomenon, originally introduced by SIA 162 into the Engineering practice, is also a vivid concept of EUROCODE 2, DIN 1045-1 (1998/12) and ÖNORM B 4700. Some users not yet accustomed to the modern Eurocode-based approach to reinforced concrete design may consider this concept considered as controversial. As a fact, the state of stress in


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a Plate or Shell cross-section due to the inner shear force may be regarded as an analogy to the situation in a Wall design model under the effect of shear membrane forces! In Walls, the consequence of such a kind of stressing is that the concrete has to withstand relatively high pressure efforts along with substantial tension stresses in both or two of three, respectively, courses of the reinforcement net. There is really a full analogy between this kind of membrane state of stress and that state of stress that is produced by shear forces in a Plate or Shell cross-section, as reported in [17]. Here, one reinforcement course is represented by the shear reinforcement (stirrups), the other reinforcement course are the bars of the upper/lower reinforcement net of the 2D structure. However, as shown in [17], in high cross-sections (more typically, however, in 1D members) the horizontal, 2nd-course reinforcement must be arranged not only at both faces but distributed along the cross-section height (at bar web faces). When calculating the shear reinforcement (stirrups), it is a natural mental step to think of the estimation of the efforts arising from shear in the net reinforcement.

The concept of what was postulated as Shear Effect in [17] is basically the same in EUROCODE 2, SIA 162 and DIN 1045-1. For SIA 162 introduced this concept as binding (i.e. the only shear design mode) before it was formulated by EUROCODE 2 we will correspond here to the symbolics of SIA 162 in giving a short overview of the method fundamentals.

In Art. 3 24 203 SIA 162 formulates the so called Truss Model of the Shear Proof based upon the concept of the Variable Strut Inclination. By “Strut” a 1D representation of the resisting pressure field of the concrete medium is symbolised; the strut inclination is then the assumed direction of the principal pressure stresses activated in the concrete by the shear (transversal) force v. The concrete cannot, in accordance with the general assumption, resist tension stresses; their equilibration is the task of the reinforcement. One shear reinforcement course constitute the transversal stirrups (the primary issue of the Shear Proof); the other reinforcement course is represented by the upper/lower net reinforcement bars.

The total Shear Effect force ft(v) is calculated according to the Art. 3 24 207 of SIA 162 (analogous relations after EUROCODE 2, DIN 1045-1 and ÖNORM B 4700) by the Formula:

ft(v) = vR cotg [kN/m] (16)

where vR is the required cross-section shear resistance and – the variable strut inclination. The required shear reinforcement amount follow from the Formula:

asw = vR tg / (fy z) [cm2/m] (17)

where z is the inner forces lever from the (m/n) design. We recognise from (16), (17) that the shear reinforcement amount and the Shear Effect force component acting upon the net reinforcement are indirectly proportional (tg = 1/cotg �). The strut inclination may be chosen, according to Art. 3 24 203, free within a quite wide range; in Plates: 25° < < 65°. Because the minimisation of the shear reinforcement is the primary goal of the design, the SCIA.ESA PT design algorithm starts an iteration loop with the lower limit value of min = 25° and, increasing it by 1°, seeks an equilibrium solution ensuring the strut resistance at minimum inclination. From this solution of , the corresponding Shear Effect force component is derived after the Formula (16).

If the Constant Strut Inclination as input control is active or the cross-section is over-tensioned (automatic control), no iterative estimation of the strut inclination is started: the central value of o = 45° is set. In such cases: ft(v) = vR, i.e. the total Shear Effect force equals the shear (resistance) force itself!

The SCIA.ESA PT algorithm proceeds in two algorithmic steps: (1) 1st step is as described above; (2) In the 2nd step, the total Shear Effect force is assigned 50/50 to the upper/ lower reinforcement nets, where it is merged with the bending/membrane forces (mx, my, mxy, nx, ny, nxy). We abstain from describing this essential transformation procedure, characteristic for the high performance of SCIA.ESA PT, in detail; please refer to [17]. As result, a net reinforcement respecting the Shear Effect in a consistent way is designed. By developing and implementing this genuine SCIA.ESA PT algorithm SCIA keep primacy on international scale.

In pressure zones of the cross-section resisting the combination of inner bending moment and normal force, the natural pre-stress is, as a rule, so high that it cannot be over-tensioned by the superposing Shear Effect forces. In such cases, the Shear Effect is no explicitly recognisable in the reinforcement design results of such a face.

In cases of low shear stress, where the cross-section lies in Shear Region 1 (see above), no shear reinforcement (stirrups) is required to ensure the cross-section shear resistance.

In the early stages of development of this part of the SCIA.ESA PT algorithm the Author, being partially mislead by some obscure formulations of the corresponding Article of SIA 162, considered also in Shear Region 1 the longitudinal components of the Shear Effect to be assigned reinforcement. However, the concept of the shear resistance mechanism in SR 1 is merely a linear elastic state of stress where the principal tension stress is supposed to be resisted by the concrete itself – in difference to the common assumption of concrete failing in tension, which is generally applied in the net reinforcement design.

It means that in SR 1, basically no Shear Effect upon the net tension reinforcement is to be considered. However, SCIA.ESA PT was equipped also with a possibility to control this part of the Shear Proof algorithm. There are three control stages provided (for Norm branches which have to do with the Shear Effect at all):

The Shear Effect is not considered at all. For SIA 162 this is, as a fact, an illegal control situation for there are no other alternatives to provide for the Shear Effect. Yet this control offers the possibility to carry out “benchmark” test calculations freed from the Shear Effect, whatever their use might be, also for Norm branches which comprise the Shear Effect phenomenon as standard.

The Shear Effect is considered in Shear Region 2 only. This is the standard case for all Codes involved in the Shear Effect.

The Shear Effect is considered both in Shear Region 1 and 2. This is a non-standard case for all Codes involved in the Shear Effect.

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It was explained that the primary goal of the Shear Proof, the minimisation of the shear (stirrup) reinforcement, is respected by the SCIA.ESA PT Shear Effect design procedure. However, the consequence of the minimisation of the stirrups is a higher increase of the net reinforcement (if any) according to the Formula (16). This circumstance, found quite unusual by traditional designers, caused some eager discussions on the hotline. There is also another factor to be considered . The Art. 3 24 203 (SCIA 162) presents a closing sentence causing some confusion. It reads: "The strut inclination, once chosen, ought to be considered constant over the whole length of the shear region". It is not quite clear what is meant by shear region here (the concept of Shear Region used by SCIA.ESA PT has another meaning clearly defined – see above), yet it may be clear that this sentence is concerned with 1D structural elements (beams) where there geometric relations are better controllable than in 2D structures. This sentence, whatever it may mean, cannot be considered by the SCIA.ESA PT design (EUROCODE 2, DIN 1045-1 and ÖNORM B 4700 do not pose such a requirement).

SCIA.ESA PT offers the possibility of generally prescribing the constant strut inclination of = 45°. If = 45° be outside the 0 interval in a Shell model cross-section (see SIA 162, Art. 3 24 203), the value nearest to 45° is estimated.

SIA 162 formulates the 0 interval control for Shells by considering the magnitude of the tension/pressure normal forces (see Art. 3 24 203). EUROCODE 2, DIN 1045-1 and ÖNORM B 4700 do not formulate such a condition. However, Art. 7.2.6(5) of DIN 1045-1 requires for over-tensioned cross-sections (i.e. with zero axis outside of the cross-section) the application of the inclination angle = 45°. SCIA.ESA PT considers this requirement automatically not only for DIN 1045-1 yet also for EUROCODE 2 and ÖNORM B 4700.

Moreover, in the design branches of DIN 1045 (1988/7), EUROCODE 2, SIA 162, DIN 1045-1 and ÖNORM B 4700 the strut inclination may be set constantly = 45° for the whole structure as input control provision.


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Design of torsional reinforcement Presently the design of torsional reinforcement in beams and columns is not possible. The torsional moments in plates and shells (mxy) are fully integrated in the calculation.

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Crack Proof

General All Norm specific Crack Proof concepts are based on principally the same assumptions about the crack propagation mechanism:

1. High-tension stress in a reinforcement bar causes high steel strain. The adhesion between concrete and the reinforcement bar is disturbed and cracks arise. The higher the ratio of steel stress and the adhesion resistance is, the wider become the cracks along the reinforcement bar. Thus, the larger the representative reinforcement diameter is, the higher is the ratio of the steel stress and the adhesion resistance, for the cross-section area of a bar grows with the square of whereas the surface of a (unit length) peace of bar depends linearly on .

2. Cracks arise, however, not only along the reinforcement bar yet also between the reinforcement bars. Thus the lateral distance of the reinforcement bars is another crucial factor of the cracks propagation, i.e. crack width.

Thus, to limit or diminish, respectively, crack widths (as a fact, not the number of cracks yet the representative crack width is of importance for the Crack Proof) the following measures are to be taken:

Use of small reinforcement diameters

2. Reduction of the representative lateral reinforcement bar distance s.

3. Combination of the measures 1 and 2.

4. Combination of the measures 1 and 2 along with augmenting the statically necessary reinforcement amount. Due to the latter provision the Serviceability state steel stress is reduced to a value that, along with the reinforcement diameter and bar distance provided, causes also the reduction of the crack widths.

In practical calculations the concept according to Pt. 4 is of highest interest: A characteristic bar diameter k and/or a characteristic bar distance sk are specified by the user on input. SCIA.ESA PT carries out the Crack Proof according to the Norm proof concept and increases the statically required reinforcement amount where it is needed to meet completely the Crack Proof requirements.

Thus it is important to involve all of the active Load Cases into the Crack Proof even if they are not declared as of Crack Proof type. The design of the statically required reinforcement has to be carried out before the Crack Proof calculations. The result reinforcement corresponding to the Ultimate state Load Cases (or their extreme Combinations) is saved in the SCIA.ESA PT Data Base and retrieved again at the stage of the Crack Proof calculations, thus constituting a start base for possible augmentations of the reinforcement augmentation as outlined under Pt. 4 above.


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Beams The crack proof for beams is generally a check of the allowable steel stress. SCIA.ESA PT calculates the response of the cross-section for the service limit state using the code-given stress/strain diagrams of concrete and reinforcement. Since the lateral distance of the reinforcement bars is off importance for the crack proof, the crack proof can only be performed for vertical symmetrical cross-section loaded by moment My only. If bi-axial bending is introduced in a beam, the lateral distance between bars cannot exactly be calculated. Additionally not all codes have concept for bi-axial crack proofs.

SCIA.ESA PT allows different kind of environmental classes, adhesion factors, increased covers, etc. All these features can be code-driven and are described in the manual.

Column Presently the crack proof for columns is not supported.

Plates, Walls and Shells

Crack Proof after NEN 6720

The Crack Proof theory distinguishes three kinds of Load Cases for the Crack Proof after NEN 6720:

Load Cases representing External Loads. In the SCIA.ESA PT input system they are assigned the attribute Ultimum. They yield the statically required reinforcement being automatically saved in the Data Base to be retrieved by the Crack Proof pro-cedure. However, they can simultaneously be specified two (or more) times being given the attribute Serviceability and used as Load Cases of the types specified below; then, probably, provided with another LC factor.

Load Cases representing the Force Imposed Deformations. They are Crack Proof Serviceability Load Cases destined for the Crack Proof procedure.

Load Cases representing the Strain Imposed Deformations. For Load Cases of this type NEN 6720 formulates a modified proof procedure. They are special Crack Proof Serviceability Load Cases.

The NEN 6720 Crack Proof branch of SCIA.ESA PT has been equipped with 4 different proof subbranches. All of them are useful means of Crack Proof analysis:

Non-controlled Crack Proof: The Crack Proof calculation is not controlled by any restrictions specified by the user. For each reinforcement course SCIA.ESA PT calculates the Characteristic Average Bar Diameter k and the the Maximum Allowable Reinforcement Bar Distance sk at any point (element and/or node), selected for design, which would be necessary to fit the Crack Proof requirements. By examination of the results (graphic portrayal) the user gain an overview over the development of the characteristic values of k and sk all over the structure. In special cases, e.g. when the Crack Proof requirements are fulfilled at some structural parts for the bar diameters or bar distances anticipated, the Crack Proof for these regions can be finished.

Controlled Crack Proof: The Crack Proof calculation is controlled by the Maximum Bar Diameter specified by the user. SCIA.ESA PT calculates the Characteristic Average Bar Diameter k at any point (element and/or node), selected for design, and compares this value with the input value of . If k calculated is less than the input diameter the reinforcement amount augmentation process described above is started. As a result, a higher reinforcement amount fitting the Crack Proof for the input diameter is saved in the Data Base, thus replacing the original statically required reinforcement amount saved prior to the Crack Proof. This Proof variant enables the Engineer to specify a constant bar diameter, e.g. = 16 [mm] anticipated as structural measure for some regions (macroelements) of the structure. The results of this SCIA.ESA PT Crack Proof variant ensure that the Crack Proof requirements will be met overall, however, using the reinforcement saved in the Data Base (maybe augmented by the Crack Proof procedure).

sControlled Crack Proof: The Crack Proof calculation is controlled by the Maximum Bar Distance s specified by the user. SCIA.ESA PT calculates the Maximum Allowable Reinforcement Bar Distance sk at any point (element and/or node), selected for design, and compares this value with the input value of s. If sk calculated is less than the input diameter s the reinforcement amount augmentation process is started. As a result, a higher reinforcement amount fitting the Crack Proof for the input bar distance s is saved in the Data Base. This Proof variant enables the Engineer to specify a constant bar distance, e.g. s = 200 [mm] for some macroelements. The results of this SCIA.ESA PT Crack Proof variant ensure that the Crack Proof requirements will be met overall by using the reinforcement saved in the Data Base (maybe augmented by the Crack Proof procedure).

Optimised Crack Proof: This is algorithmically the most exacting variant of the NEN 6720 Crack Proof algorithm. The calculation is controlled both by the Maximum Bar Diameter and the Maximum Bar Distance s specified simultaneously by the user. SCIA.ESA PT combines the procedures described under Pt. 2 and 3 above. Following variants may be encountered in course of the calculations: (a) If k calculated is greater than or equal the input diameter the Crack Proof has been met. There is no need of augmenting the reinforcement amount; (b) If sk calculated is greater or equal than the input distance s the Crack Proof has been met; (c) if neither nor s specified meet the Crack Proof requirements a procedure described by Pt. 2 and 3 is started by which the best fit of one of both conditions ( or s) is found by augmenting the statically required reinforcement pre-calculated. “Best fit” means that the fulfilment of one of the or s conditions is sought, that one which implies the lesser reinforcement augmentation of both. This variant yields, generally, the lowest total reinforcement augmentation amount of all three variants described by Pt. 2,3 and 4; this is why it is called the Optimised variant. On the other hand, its disadvantage lies in the fact that generally both of the input conditions, the Maximum Bar Diameter and the Maximum Bar Distance s, must be maintained at every point of the (sub)structure subjected to the Crack Proof.

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Crack Proof after ÖNORM B 4700

After ÖNORM B 4700, the crack limitation is controlled, like with GBJ10-89, by the Calculation Reinforcement Bar Diameter dsr rather than by bar distances. However, it is to realise (concerning both GBJ10-89 and ÖNORM B 4700) that the explicit focusing to bar diameters does not mean that the distance of bars within reinforcement net courses is, under such Crack Proof concepts, of no impact upon the crack control. Besides the bar diameter the reinforcement amount (limiting of steel stresses – see above) is the other control parameter affecting the crack width. The bar distances in a real reinforcement net correspond, naturally, to the reinforcement amount provided: the higher the reinforcement amount provided the lesser are the distances of the bars within their course! The difference between the concepts of NEN 6720 on one side and GBJ10-89, ÖNORM B 4700 on the other side lies thus in the manner how the bar distances are dealt with: the NEN 6720 concept makes them to explicit control parameters; the other Codes use bar distances as implicit quantities of the Crack Proof.

ÖNORM B 4700 introduces the Crack Width as central proof item. The crack widths to be dealt with are assigned to the interval <0.15, 0.30> [mm]. The Maximum Allowable Crack Width value wk desired to be maintained throughout the structure or its part is specified on input. Higher or lesser values of wk are obviously outside the scope covered by the ÖNORM B 4700 stipulations. If there appears wk < 0.15 or wk > 0.30 on input, it is adapted to the nearest limit value (wk = 0.15 or wk = 0.30, respectively).

A characteristic feature of ÖNORM B 4700 is the fact that there are no analytic formulas for wk yet empirical tables describing

the functional relations dgr = f(tr) or dgr = f(tr,sD), respectively for Strain Imposed Deformations or Force Imposed Deformations (see below). SCIA.ESA PT uses a double extra-interpolation process to get Limit Bar Diameter values dgr as

functions of Reinforcement Amount Percentage tr or tr and Characteristic Steel Tension sD out of the Code Tables 7 or 9 (for wk = 0.15) and 8 or 10 (for wk = 0.30).

ÖNORM B 4700 distinguishes several situations of exploitation concerning the Crack Proof procedure and the use of Code Tables 7, 8 or 9, 10, respectively. SCIA.ESA PT distinguishes and deals with such situations by grouping them to Load Cases of particular attributes, like NEN 6720 (see above):

Load Cases representing External Loads. In the SCIA.ESA PT input system they are assigned the attribute Ultimum. They yield the statically required reinforcement being automatically saved in the Data Base to be retrieved by the Crack Proof pro-cedure. However, they can simultaneously be specified several times by being given the attribute Serviceability and used as Load Cases of the types specified below; then, probably, provided with another LC factor.

Load Cases representing the Force Imposed Deformations. They are basic Crack Proof Serviceability Load Cases destined

for the Crack Proof procedure. For dealing with Load Cases of this type the functional relation dgr = f(tr,sD) governed by the Tables 9 and 10 is of fundamental importance. An iteration process equilibrates the necessary reinforcement amount by checking the Reinforcement Amount Percentage tr after the Tables 9, 10 against the Limit Bar Diameter dgr which is closely related to the Calculation Reinforcement Bar Diameter dsr, being specified on input as principal control quantity (along with wk).

Load Cases representing the Strain Imposed Deformations after the Art. 3.2.2(1)-(4). They are special Crack Proof Serviceability Load Cases. Load Cases of this attribute are governed by the functional relation dgr = f(tr) of the Tables 7 and 8. For Strain Imposed Deformations are understand to originate from other causes than direct extern forces impact, the inner force components associated with the Load Cases of this attribute enter the calculations merely by their relative than absolute amplitudes (bending, full tension etc.)

Load Cases representing the Strain Imposed Deformations after the Art. 3.2.2(5). They are special Crack Proof Serviceability Load Cases dealing with non-linearly distributed, self-induced stresses within the cross-section. These situations of exploi-tation allow, according to ÖNORM B 4700, for a reduction of the necessary Reinforcement Amount Percentage tr within the interval <60%, 80%>. On the whole, the iteration resolution process follows the same rules as described under Pt. 3.

Load Cases representing the Strain Imposed Deformations after the Art. 3.2.2(6). They are special Crack Proof Serviceability Load Cases dealing with cracks in an early stage of concrete hardening. These situations of exploitation allow, according to ÖNORM B 4700, for a reduction of the necessary Reinforcement Amount Percentage tr to 50% of the standard value. By this it is assumed that the Average Tension Strength fctm,t attains 50% of fctm (after 28 days). For other percentages linear interpolation is allowed. SCIA.ESA PT respects these special rules. For fctm,t the corresponding percentage is expected on input (active for this type of Load Case only).

Load Cases representing the Strain Imposed Deformations after the Art. 3.2.2(7). They are special Crack Proof Serviceability Load Cases dealing with cracks due to strains caused by the hydration heat flow. These situations of exploitation allow, according to ÖNORM B 4700, for a reduction of the necessary Reinforcement Amount Percentage tr to 70% of the standard value in case of normally hardening cements (e.g. PZ 275, PZ 375). For other cements a special proof is necessary (however, no more specified by the Code).

The results of the Crack Proof are available in both graphic and printed form.


48

Checks

Response Based on the internal forces, concrete cross-section and supplied reinforcement by the user, SCIA.ESA PT is able to calculate the response of a member or a single cross-section. This method uses an iteration routine to calculate equilibrium based on the internal forces, the cross-section, material properties and reinforcement layout. This method however does not calculate extremes like the interaction diagram, but calculates the state of equilibrium for that section (response). The calculation also includes depth of compression zones, curvatures, stresses, strains and forces. This iterative method works for the interaction of the normal force with uni-axial or bi-axial bending moments. In the iteration two things are very important: 1. The number of iteration steps, 2. Precision. Both can be set in the setup of concrete.

Stirrups: 2x FeB 500 (8) Dist= 200 mmConcrete: B 35

Y

Z

450 mm

600

mm

4x FeB 500 (20)

4x FeB 500 (16)

CONCRETE COVER 30 mm

CONCRETE COVER 30 mm

Y

Z

-0.788

2.018

169

mm

Curvature = 0.468 mrad

1.793

-0.573

Strain [*1e-4]

49

Y

Z

-0.95 169

mm

35.87

-11.46

Stress [MPa]

Y

Z

-35.9

11.3

11.3 11.3

11.3

-2.3

-2.3 -2.3

-2.3

Forces [kN]


50

Bi-axially loaded cross-section:

Y

Z

-2.472

-0.160

5.224

2.913

4.465

0.229 3.053

1.641

2.539 -1.744

1.112 -0.316

Strain [*1e-4]

Y

Z

-3.0

-0.2

89.3

4.6

61.1

32.8

50.8

-34.9

22.2

-6.3

Stress [MPa]

51

Capacity The interaction diagram calculates the extreme allowable interaction between the normal force N and bending moments My and Mz. In theory this diagram is a 3D-diagram, but SCIA.ESA PT allows the user to make horizontal and vertical cuts. The axis of the diagram has an axis for the normal force N, the bending moment My and the bending moment Mz

Nor

mal

For

ce

Bending Moment Bending MomentMy Mz

Consequently a horizontal cut displays the interaction between My and Mz for a given normal force. A vertical cut displays the interaction between N and Mu. See figure for an example of a 3D interaction diagram;

Mz My

N VUmin N=0.0 kN My=-23.7 kNm Mz=-172.5 kNm

VUmin (N,M)

V N=0.0 kN My=-23.7 kNm Mz=0.0 kNm

V (N,M)

VUmax N=0.0 kN My=-23.7 kNm Mz=172.5 kNm

VUmax (N,M)


52

The interaction diagram is used to find the extremes for the reinforcement. The extreme can be a value for Nu, Mu, NuMu, Muy or Muz. The following figures show the functionality of these extremes. Note that point ‘V' is the design point and points 'Vu;min' and 'Vu;max' are the extremes.

Extreme values for Nd: (My and Mz are constant)

Mu[kNm]

N[k

N]

VUmin N=-5231.9 kN My=-250.0 kNm Mz=0.0 kNm

VUmin (N,M)

V N=-3000.0 kN My=-250.0 kNm Mz=0.0 kNm

V (N,M)

VUmax N=141.1 kN My=-250.0 kNm Mz=0.0 kNm VUmax (N,M)

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000 -7

00

-6

00

-5

00

-4

00

-3

00

-2

00

-1

00

0 10

0

20

0

30

0

40

0

50

0

60

0

70

0

Extreme values for Md: (N is constant)

Mu[kNm]

N[k

N]


VUmin (N,M)

V N=-3000.0 kN My=-23.7 kNm Mz=0.0 kNm

V (N,M)

VUmax N=-3000.0 kN My=612.1 kNm Mz=0.0 kNm

VUmax (N,M)

-7000

-6000

-5000

-4000

-2000

-1000

0

1000 -7

00

-6

00

-5

00

-4

00

-3

00

-2

00

-1

00

0 10

0

20

0

30

0

40

0

50

0

60

0

70

0

53

Extreme values for NdMd: (eccentricity = constant)

Mu[kNm]

N[k

N]

VUmin N=-4535.1 kN My=-377.9 kNm Mz=-0.0 kNm

VUmin (N,M)

V N=-3000.0 kN My=-250.0 kNm Mz=0.0 kNm

V (N,M)

VUmax N=542.3 kN My=45.2 kNm Mz=0.0 kNm VUmax (N,M)

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000 -7

00

-6

00

-5

00

-4

00

-3

00

-2

00

-1

00

0 10

0

20

0

30

0

40

0

50

0

60

0

70

0

Extreme values for Myd: (Mz and N are constant)

Mz[kNm]

My[

kNm

]


VUmin (N,M)

V N=-3000.0 kN My=-250.0 kNm Mz=200.0 kNm

V (N,M)

VUmax N=-3000.0 kN My=420.9 kNm Mz=200.0 kNm

VUmax (N,M)

-600

-500

-400

-300

-200

-100

0

100

200

300

400

500

600

700 -6

00

-5

00

-4

00

-3

00

-2

00

-1

00

0 10

0

20

0

30

0

40

0

50

0

60

0


54

Extreme values for Mzd: (My and N are constant)

Mz[kNm]

My[

kNm

]


VUmin (N,M)

V N=-3000.0 kN My=-250.0 kNm Mz=200.0 kNm

V (N,M)

VUmax N=-3000.0 kN My=-250.0 kNm Mz=-288.0 kNm

VUmax (N,M)

-600

-500

-400

-300

-200

-100

0

100

200

300

400

500

600

700 -6

00

-5

00

-4

00

-3

00

-2

00

-1

00

0 10

0

20

0

30

0

40

0

50

0

60

0

55

Physical Non-linear Deformations

General The physical non-linear deformations are calculated based on the concept of “quasi”-non-linearity. This means that linear calculations are used to model non-linear behavior of the construction. Four steps are used to perform the calculation.

1. Using the short-term stress and strain diagram for concrete the deformations for ‘creep’-load is determined. The ‘creep’-load is commonly the quasi-permanent load (1.0 × DEAD LOAD + FACTOR × LIFE LOAD). The factor is in most cases around 30%.

2. Using the long-term stress and strain diagram for concrete the deformations for ‘creep’-load is determined.

3. Subtracting the short-term deformation from the long-term deformation the ‘creep’-deformation is obtained.

4. Adding the creep-deformation to the linear deformation caused by the representative load (1.0 × DEAD LOAD + 1.0 × LIFE LOAD), the total quasi-non-linear deformation is obtained.

To calculate the immediate deformation, the deformation of the permanent load is calculated using the short-term stress and strain diagram. Additionally by subtracting the immediate deformation from the total deformation, the programs calculates the additional deformation.

So the calculated deformations calculated in SCIA.ESA PT’s PNL – deformations calculation are:

Elastic deformation: Using the short-term stress and strain diagram and representative load combinations. (1.0 × DL + 1.0 × LL)

Creep deformation: Using the long- and short-term stress and strain diagrams and momentaneous load combinations. (1.0 × DL + × LL)

Total deformation: Elastic deformation + Creep deformation.

Immediate deformation: Using the short-term stress and strain diagram and permanent combination. (1.0 × DL)

Additional deformation: Elastic deformation + Creep deformation – Immediate deformation.

The short- and long-term stiffnesses are calculated using a so-called creep factor. This creep-factor is dependant on the relative humidity, outline of the cross-section, reinforcement percentage, concrete class, etc. is used to divide the short-term stiffness and obtain the long-term stiffness, e.g. (acc. NEN 6720)

E’ = E / (1 + 0.75 × creep)

Thus by taken the concrete stiffness for short- and long-term and the representative compression strength the program calculates the stress and strain diagram.


56

NEN 6720

The deformation according NEN 6720 is calculated using so-called M-N- diagrams. These diagrams representing the relation between a combination of normal force (N) and bending moment (M) and the curvature are used to define the deformation for the individual combinations (Creep, Immediate, Representative/Linear).

NEN 6720 prescribes that the diagram is corrected between Mr (Moment of rupture) and approximately Me (End of elastic part of stress/strain diagram). This correction models the so-called tension stiffening effect, e.g. the concrete between the cracks still has stiffness. SCIA.ESA PT uses by default the corrected M-N- diagram, but is capable of other types as well, e.g.

1. Concrete without Tension Part.

2. Concrete without Tension Part AND Code Correction

57

3. Concrete with Tension Part

4. Concrete with Tension Part and Tension Softening according CUR 94-13.

Other codes than NEN 6720

The deformation for other codes than NEN 6720 is calculated by reducing the stiffnesses using the following so-called Stiffness/Moment diagram:

Euncracked

moment of rupture Mrep ultimate moment

E

Columns There is no non-linear stiffness calculated for the axially loaded members. Thus columns will have the same stiffness in PNL – deformations calculation as in a linear calculation. However when the concept of PNL – internal forces is introduced in SCIA.ESA PT the user is able to calculate reduced stiffnesses for columns.

Composite Sections In case of composite cross-sections (concrete/concrete) a weighted creepfactor will be taken into account:

composite

Beams, Plates and Shells The reduced stiffness for Walls is not calculated when a PNL – deformations calculation is performed. Deformations of beams, plates and shells are calculated by integrating the non-linear curvatures over the length of beam or slab. However if some element has a value of Md larger than Mu, than the stiffness according Mu is taken. Since the finite element method can give large internal forces due to singularities, etc. the calculation is allowed to continue without an error message, but supplies messages after the calculation has finished.

NEN 6720

For plates and shells the standard corrected M-N- diagram is NOT used to calculate the deformations, but the diagram with tension part and tension-softening.


58

When a calculation using M-N- diagrams was implemented in SCIA.ESA PT the calculation would take an awful lot of time, since for each individual element a M-N- diagram would have to be constructed with a lot of points.

59

Part II - 2D members

Dipl.-Ing. Eduard Hobst, Ph.D.


60

61

INTRODUCTION TO REINFORCED CONCRETE DESIGN OF 2D STRUCTURES

Introduction The 2D reinforcement concrete design modules PRC.72.xx (ESA-Prima Win) as well as ESACD.02.xx (SCIA.ESA PT) will be referred to by their generic name NEDIM – the original development name of the 2D design module system, used internally by SCIA developers, testers and supporters for quick communication. Both EPW and ESA PT have been using the same NEDIM package. There are, however, differences in I/O handling; ESA PT allows, generally, for some additional, more advanced I/O control options.

The FEM analysis modules doing their work prior to NEDIM, and delivering their results via the EPW/ESA Data Base as input data to NEDIM, as well as NEDIM itself distinguish between three analysis types : Walls, Plates and Shells. Each model has its special mechanical and structural characteristics, which have to be dealt with by individual algorithmic sub-branches.

From the point of view of the static analysis, the primary distinguishing feature of mechanical models of these analysis types is the character of inner forces developing in the cross-section to resist the effect of external forces. The most general analysis model is the Shell type. Shells are subjected to a combination of bending forces, i.e. moments and shear forces (called also Plate inner forces), and membrane forces (called also Wall inner forces). This categorization of inner forces is an idealization used in the theory of 2D structures. However, the experience confirms that our theories and mechanical models, if reasonably linked with characteristic properties of structures, are able to approximate the reality satisfactorily.

In this sense, Plates and Walls are reduced mechanical models of the Shell model. They are confined geometrically to a plane, being genuine 2D models. Shells, as a fact, are 3D structures; however, they consist of 2D structural parts (down to individual plane finite (sub)elements) in compliance with the FEM theory, which assigns the attribute „two-dimensional“ to structures having very small structural thickness in comparison with the other two (main) geometric dimensions. Thus, it is legal to use a simplifying categorization by introducing the three analysis types – Walls, Plates and Shells – and considering them all as 2D models.

The creation of the FEM model is a genuine task of the structural Engineer. ESA-Prima Win and SCIA.ESA PT are powerful means enabling the user to create and manage efficient mechanical models by using the sub-structuring technique. In the SCIA terminology such structural parts are called macro-elements or members.

For the FEM static analysis as well as for the NEDIM reinforcement design it is without concern what is the shape of the individual structural parts the structure consists of, which are represented by macro-elements. NEDIM, however, raises additional requirements to the modelling process : defining macro-elements of the FEM model, the user should pay attention to the cross-section geometry as well as the reinforcement geometry envisaged in order to make clear conditions for the reinforcement design.

NEDIM keeps strictly to the terminology and the notions of ESA-Prima Win and SCIA.ESA PT. The macro-element is for NEDIM the standard reference unit in establishing the amount and the organization of the reinforcement concrete design. However, the FEM stress-strain analysis runs internally over finite elements and element nodes. Let us remember that the actual FEM solver yields results directly in element nodes instead of centroidal points of finite elements, as it had been with the predecessor FEM solvers. Thus, consequently, the NEDIM design refers to element nodes as design nuclei.

When specifying macro-elements for the reinforced concrete design, EPW selects implicitly all elements and nodes of each macro-element selected. On the other hand, NEDIM, in the stand-alone mode, enables the individual selection of elements or nodes; yet this mode is of no concern for NEDIM processing as an integral part of SCIA software. Advanced aspects of this procedure are discussed further on in this manual.

The input data organization of NEDIM keeps strictly to the requirements and restrictions of the Norms implemented; nevertheless, it is versatile enough to enable reasonable deviations from the Norm parameters. When there is a concern about the legality of a calculation because of non-standard values being defined by the program user, NEDIM places a corresponding warning into the output document file.

The 2D reinforcement concrete design module NEDIM has been developed in close cooperation between SCIA Group n. v. Herk-de-Stad, Belgium and Ingenieurbüro Dr. Hobst, Nuremburg, Bavaria; Dr. Eduard Hobst is Author of the theory, design algorithm and calculation kernel as well as this NEDIM manual. SCIA is responsible for the input dialogue and the graphic portrayal of results as well as the linking of NEDIM to the ESA-Prima Win and SCIA.ESA PT program system.

Program objective The purpose of the module NEDIM is the reinforcement concrete design and serviceability proofs (crack proof) according to national Standards (Norms) of Wall, Plate and Shell structures.

At the actual development stage NEDIM performs the reinforced concrete design task according to the following national Norms :


62

o DIN 1045 07/1988

o ÖNORM B 4200, Teil 8 (4/1971) + Teil 9 (4/1970)

o EUROCODE 2 (ENV 1992-1-1:1991)

o CSN/STN 73 1201 (8/1986 & Amendments 9/1989 + 10/2004)

o SIA 162:1989

o NEN 6720:1995/A3:2004

o DIN 1045-1 (07.2001 & 05.2005)

o GBJ 10-89

o ÖNORM B 4700 2001-06-01

o BAEL 91/99

o SIA 262:2003

o EN 1992-1-1:2004 (Eurocode 2 Novella)

o BS 8110:Part 1:1997 & BS 8110:Part 2:1985

o IS 456:2000 & IS 2210:1988/1998

o ACI 318M-05

The Norm branches active in NEDIM are presented in the natural time sequence of their program implementation. NEDIM had been under development since 1986/7; till 1991 its predecessor versions were in practical use in German engineering practice. Its most spectacular application was the application in the design of ceiling slabs of the Computer Centre of Federal Bureau for Labour (Bundesanstalt für Arbeit), Nuremberg. NEDIM was lanced on the software market 1992 with DIN 1045 07/1988 [5] as the first running Norm branch, closely followed by ÖNORM B 4200 [7], [8] and EUROCODE 2 (ENV 1992-1-1:1991) [9]; in 1993 followed the branch CSN 73 1203 [10] and 1994 – SIA 162 [11]. Since 1994, NEDIM has successfully been operating on international scale, being consecutively extended by implementing other Norm branches as shown above; the branch of NEN 6720 [12] was developed and implemented in 1997, DIN 1045-1 [13] – 1999, ÖNORM B 4700 – 2000 [16], BAEL 91/99 – 2002 [24], [25], SIA 262 – 2004. Most recent NEDIM implementations (2005 – 2007) have been the design branches of EN 1992-1-1:2004 [27], BS 8110 [28], IS 456/2210 [29]/[30], ACI 318M-05 [31] and STN 73 1201.

The NEDIM design algorithm has many outstanding features distinguishing it from competing design programs on the international software market. They are outlined in this introductory chapter.

One of the outstanding features of NEDIM is its ability to deal with 2- and 3-course reinforcement nets of almost deliberate geometry, i.e. allowing for the angles closed by pairs of reinforcement directions being freely specified; however, reasonable limits are set and checked by the program. Fig. 1 shows the basic definition scheme of the reinforcement geometry : the directions of respectively 2 or 3 reinforcement courses specified for design are expressed by angles of the interval <0°, 180°) to the 1st planar axis xp.

The reinforcement geometry may be specified individually at both structural faces, concerning the direction angles and the number of reinforcement courses (2 or 3). So it is, for example, possible, to specify at one face a skew 2-course reinforce-ment net with directions, 10°/70° and, at the same time, a 3-course reinforcement net with directions 0°/60°/120° at the other face. The standard orthogonal reinforcement 0°/90°, allowed by most design programs as the only (implicitly set) reinforcement geometry specification, is in NEDIM one of an infinite variety of possible constellations, nothing more.

63

xp

yp

1 2

3

1

2

3

Fig. 1 Reinforcement geometry (upper or lower face)

Another outstanding feature of NEDIM is its extensive and reliable system of detection of non-designabilities. NEDIM distinguishes between several situations of non-designability (see Chapter Processing non-designability conditions), specifies them accurately and informs the user in the printer output file as well as in the graphic portrayal of results (by Error Code Numbers). In the practice some serious non-designability situations are regularly encountered, which are not recognizable by most commercial design programs. For example, in corners of (rectangular) floor plates, which are usually reinforced by orthogonal nets parallel to the plate edges (0°/90°), where the principal bending moments at upper and lower face are declined by about 45°/135° from the edges, i.e. acting in the plate's diagonal or perpendicularly to it, respectively, a typical stress-strain situation is encountered : All four reinforcement layers (two at upper and two at lower face) are subjected to tension! In such stress states the concrete fulfils two tasks : (a) resisting the compression zone force of the bending moment (like in 1D design), i.e. opposing the reinforcement tension force on the opposite face and (b) stiffening the reinforcement net on the actual face against distortion. The concrete medium becomes then the critical factor of the structural bearing capacity. It is possible to meet high tensions in plate corner regions by increasing the reinforcement steel amount (which is the primary task of reinforcement design). Yet it is impossible to increase the bearing capacity of the stiffening concrete (see the notion of virtual concrete strut, discussed in the subsequent text). Here the most design algorithms deliver, maybe, high statically requi-red steel amount, yet ignore at the same time the possible exhaustion of the stiffening concrete. NEDIM recognises reliably such a state of concrete failure and issues a corresponding non-designability message (error code E5). This very special design aspect will be dealt with in the next Chapter Program Theory and Algorithm (see also [18]). In NEDIM great attention is paid to the processing and reporting of compression reinforcement. The theory of Baumann [1], which was applied as basic theoretical issue to the development of NEDIM, disregards the fact that in a reinforced concrete medium the reinforcement may and must also fulfil the function of armouring the concrete being subjected to pressure. Besides the classical case of the Wall model being under pressure, also in bending the reinforcement in the pressure bending zone may be engaged in increasing the bearing capacity; this case is, as a rule, strictly regulated by the Norms. For hyperbo-lic states of stress in Walls and Shells, where there is typically one tensioned reinforcement course, the pressure forces being opposed by the stiffening concrete medium, a substantial improvement was found and implemented: NEDIM assigns, using a special combined inner forces transformation approach, a calculation pressure force to the compressed reinforcement course and only an acceptable portion of the pressure force to the stiffening concrete continuum. Thus, an efficient and consistent solution is found for all possible situations of the reinforcement net geometry. In elliptic pressure states (full pressure in all directions), both or all three, respectively, design forces are assigned to reinforcement courses, thus attaining a determined statically required reinforcement. According to the theory of Baumann [1] and other known theories of 2D design, no compression reinforcement would be designed at all; in other words: plain concrete would have to resist the load in spite of reinforced concrete!

Special Norm requirements concerning the so called minimum compression reinforcement are an interesting chapter of the structural theory of reinforced concrete. It is especially well dealt with by DIN 1045 07/1988 [5], which requires a reinforcement percentage assigned to s. c. statically required (reinforced) cross-section. Actual Norms stipulate minimum compression reinforcement (for 2D structures) quite elementary : (a) as a constant percentage of concrete gross section; (b) by a simple dependence on the normal force. Only ACI 318M-05 applies a similar approach as DIN 1045 07/1988, however, referring to the plain concrete cross-section instead of reinforced one. The principal goal of the minimum compression reinforcement is to allow for a reasonable measure of buckling stability of 2D structures from the structural point of view. Also in this field a satis-factory state of solution has been attained. The topic shall be discussed in Chapter Program Theory and Algorithm.

NEDIM enables to specify every of the load cases from the FEM analysis as design load cases. On the other side, NEDIM has no ambitions to carry out advanced superposition of load cases. This is, by declaration, a task of specialized post-proces-sor modules of the FEM solver of ESA-Prima Win and SCIA.ESA PT. However, the assignment and processing of different load factors to individual load cases and/or combinations (defaulted to 1.0), may be a part of the NEDIM input data control.


64

PROGRAM THEORY AND ALGORITHM

Introduction to the theory and algorithmization of 2D design To fulfil the task of reinforcement concrete design of 2D structures, several steps, theoretically or algorithmically, have to be performed :

o Analysis of Norm requirements or restrictions; their exact formulation by numerical values and control items.

o Set-up of a user-friendly input system enabling the engineer to understand the data at every step.

o Processing of the input data as well as the internal program data to prepare a perfect task definition.

o Transformation of inner forces of the FE solution to design inner forces.

o Reinforcement design and processing of non-designability conditions.

o Storing up of the design results in the program data base.

o Creating a user controllable printer output file containing all engineering information obtained during design.

o Performing the crack proof (generally data base bound!) additionally to the reinforcement design.

o Retrieval of the results from the data base for graphic portrayal including indication of non-designabilities.

o Retrieval of the results from the data base for printout documents.

Unlike to 1D models, the static inner forces in 2D models, as resulting from the FEM static/dynamic analysis, are, excepting very special cases, not directly applicable to reinforcement concrete design. They must be transformed to design forces. This is a very vital notion of the 2D design; it will frequently be used throughout this manual.

Both FEM inner forces and design forces are understood as generalized forces. They mean both membrane forces and bending moments, depending on the analysis/design model. However, there is no danger of misunderstanding since in each design situation it is clear what sort of inner forces is being dealt with.

Transformation of inner forces of the FEM solution to design forces Once the reinforcement design input data have been read in and analyzed and the FEM data base approached, the NEDIM design model can be created respecting all Norm rules and restrictions applicable to the analysis model : NEDIM distinguishes between the Wall, Plate and Shell analysis types. They are different not only as to the principal assumptions about the character of inner forces acting upon the cross-section but, in all Norms, also as to the requirements and restrictions these analysis types are subject to.

The first substantial step of the design procedure is the estimation of inner design forces for each item to be designed. Under the actual FEM solver there is one design item only: finite-element node. The design forces transformation procedure outlined here takes thus place at each step of the multiple design loop running over macro-elements → elements → nodes → load case combinations → individual load cases.

Inner forces of the FEM analysis can generally not be used as direct design forces. A versatile inner forces transformation procedure is indispensable to enable the 2D design.

The NEDIM transformation procedure is based on a general transformation formula devised by Baumann [1] :

ci = [sin αj sin αk + cos αj cos αk] / [sin (αj – αi) sin (αk – αi)] (1)

65

(i, j, k, i,... = 1,2,3,1,...)

In the cyclic formula (1), the subscripts i, j, k denote either the three reinforcement directions according to Fig. 1 or two reinforcement directions and the direction of the virtual concrete strut (see below), respectively. The variables in (1) have the following meaning :

i, j, k : angles between individual reinforcement/strut directions and the direction of the 1st principal moment mI (Plates) or the 1st principal membrane force nI (Walls and Shells) , respectively

: quotient (mII/mI) or (nII/nI), respectively ; it can assume negative, zero or positive value

ci : transformation coefficient associated with the transformation direction i

The design forces mi or ni, respectively, are defined by the following relations:

mi = ci mI or ni = ci nI (2)

Formula (2) yields direct design forces for Plates (mi) and Walls (ni). In Shells, an intermediate transformation step has to be performed prior to applying (2).

The combined vector of bending & membrane inner forces of the FEM analysis comprises 8 components :

{ mx, my, mxy | vx, vy | nx, ny, nxy } (3)

First it has to be transformed to reference membrane forces acting as two formally independent force systems at both structural faces. The Shell design becomes thus, formally, a “double” Wall design, executing the typical Wall design procedure twice, at both faces ± Zp quasi individually :

nx,±Zp = ± mx / z + nx / 2

ny,±Zp = ± my / z + ny / 2 (4)

nxy,±Zp = ± mxy / z + nxy / 2

The alternating moment sign in (4) refers to +Zp (+ m) and –Zp (– m) face, respectively. The reference inner force vectors {nx, ny, nxy}±Zp in (4) differ thus generally from each other. This applies equally to the corresponding principal reference forces {nI,

nII}±Zp at both faces of the Shell model.

In the following text the faces ±Zp are distinguished by index only if it is necessary in the context; generally, the design algo-rithm is identical for both faces (Plates and Shells).

In (4) z represents the inner forces lever calculated, as reference value, for the outermost reinforcement layer. The calculation of z must refer to the material properties of the concrete continuum; it is thus „material dependent“, unlike the transformation formula (1), which is based on the assumption of (any) linearly elastic material (with cracks).

The reference value of z is calculated in the cross-section perpendicular to the first principal moment mI (along with associated n in this section).

The transformation coefficients ci, cj, ck (1) represent transformation forces in a linearly elastic medium which does not make difference between tension and pressure design forces. Such a solution is generally not applicable to reinforced concrete, since concrete can oppose pressure stresses only.

Let us, for general considerations, independently of the analysis model, denote the design forces obtained by the transformation (1), (2) as {n1d, n2d, n3d}. In Walls they are transforms of the basic inner forces {nx, ny, nxy}; in Plates – of the inner moments {mx, my, mxy} and in Shells – of the reference inner forces {nx, ny, nxy}±Zp (4). Basically, the design forces {n1d, n2d, n3d} according to (1) and (2) meet the following generalized first tensor invariant condition, no matter what model they represent :

n1d + n2d + n3d = nI + nII = const (5)

where nI and nII symbolise either the principal normal forces {nI, nII} (Walls), or the principal moments {mI, mII} (Plates) or the face related principal normal forces {nI, nII}±Zp (Shells). Formula (1) yields several solutions satisfying (5). NEDIM applies a solution representing the minimum or another convenient low energy level, respectively, for 2D design.

The subscript ‘d’ with the design inner forces nid shall often be omitted in the following text because the symbols ni or mi, respectively, are used exclusively to denote a design force of the reinforced concrete medium obtained by the transformation (1); vice versa, using the symbol nd (md) without direction subscript means the representation of any of the design forces nid (mid), i=1,2,3.


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Under an elliptic state of stress, the solution for a tree-course reinforcement net, each reinforcement being active, may exist. In two-course reinforcement nets, which represent the standard use in the building practice, only two design forces can be assigned to the reinforcement. The third force of the invariant formula (5), n3d, represents then the resistance of the concrete medium. We will call this fundamental function of concrete virtual stiffening concrete strut (or concrete strut, virtual strut, stif-fening strut etc.).

Obviously, the value of n3d (5) is non-positive, since concrete is not able to resist tension. Only in special reinforcement arrangements (s.c. trajectory reinforcement) or in strictly circular states of stress, the concrete design force can vanish : n3d = 0; the virtual strut performs no mechanical work in that case and may be considered as stress-free. Of practical meaning is n3d < 0 with concrete resisting the inner forces by stiffening the reinforcement net, which would, by itself, distort under the action of tension or pressure forces in its plane.

The notion of virtual stiffening concrete strut is the fundamental topic of the 2D design, primarily distinguishing the 2D design from the 1D design (beams and columns). The graphic illustration in Fig. 2 reveals the elementary meaning of the stiffening strut.

The position of the concrete strut is, however, generally not identical with any reinforcement direction specified for design. It means that if formula (1) does not yield for a 3-course reinforcement specification all three positive design forces, at least one of the reinforcement courses is inactive (or two of them); the concrete strut does not automatically coincide with one of the reinforcement courses! The assessment of the concrete strut position is thus an important optimization task.

It is an outstanding feature of the NEDIM design algorithm, enhanced during months and years of improvements of the theoretical background and algorithm, that formula (1) can usefully be applied to all possible situations of elliptic, parabolic and hyperbolic states of stress-strain, i.e. also to elliptic pressure state, thus yielding design forces which enable optimized design.

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Fig. 2 Function of the concrete strut in a 2D reinforced concrete continuum : (a) 2D element and section geometry; (b) trajectory reinforcement with vanishing strut force; (c) general case – concrete stiffening the "distorted" steel net

Reinforcement Design 2D structures dealt with by NEDIM – Walls, Plates and Shells – are usually reinforced by two systems of steel reinforcement nets consisting of 2 or 3 reinforcement courses situated more or less close to the surfaces of both faces of the 2D structure. NEDIM sets few restrictions to the absolute position of reinforcement courses within the cross-section; the position of each reinforcement course is described by its axial concrete cover acc. to Fig. 3. However, there are relative restrictions : all con-crete covers must meet some rules to prevent ambiguousness of the geometric definition of the design task. These rules are described in the tutorial part of the NEDIM manual.


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h

u11

12

u13

u23

u22

u21

u

+Zp

-Zp

Fig. 3 Cross-section geometry (upper/lower face, cross-section height, axial covers)

There might be encountered more complex situations in a cross-section than symbolized by Fig. 3 : (1) reinforcement bars of individual layers do not need to touch each other; they might be placed at larger distances from each other within the cross sections; (2) the surfaces of bars are usually corrugated so that there is, as a rule, a greater distance between two crossing bars than expressed by their characteristic bar diameters; (3) in thick plates, e.g. foundation slabs, two or more layers in one direction (reinforcement course) or bars bundles in one layer are used, so that the axial distance (from edge) and the representative bar diameter itself are two independent quantities and qualities, which must be defined independently on input. Also, exact nominal concrete covers of outermost reinforcement bars of each reinforcement course must be defined for the crack proof. EPW and ESA PT offer an enhanced input data dialogue enabling such full-scale cross-section geometry input.

In Walls, being subjected to forces acting in their plane, the positions of reinforcement nets, by definition placed symmetrically in the cross-section, are of no static interest; however, the cross-section geometry (concrete covers and bar diameters) is meaningful for the crack proof algorithm. Thus, the Wall design branch comprises the same cross-section input dialog as the Plate and Shell models. From formal reasons (simplification of printed output), if the notion of face appears in connection with a Wall it means both faces together, since for Walls the total reinforcement is calculated under the principal assumption that Walls are armoured symmetrically with respect to both faces.

In Plates and Shells, the reinforcement covers estimate the effective static height of the reinforcement courses in the cross-section subjected (also) to bending, thus having fundamental meaning in the design process. The covers are related to the faces (Fig. 3). Thus, it is necessary to distinguish the faces clearly from each other. NEDIM had originally used the commonly understandable terms distinguishing two faces of a horizontal plate: upper and lower face. These notions had possessed their exact mathematical meaning, which had made them acceptable for Shells, too : the lower face had been the structural plane edge in direction of the positive planar (local) axis +Zp, the upper face –Zp having been opposite to it. However, at a later development stage of EPW/ ESA PT the orientation of the global co-ordinate system (X,Y,Z)glob was defined by the positive axis +Zglob pointing upwards (before – downwards). Thus, the original meaning +Zp lower face was reversed to –Zp lower face. Generally, to avoid irritations and misunderstandings with 2D members positioned deliberately in 3D space, the mathematically exact and unambiguous meaning +Zp /–Zp face is being preferred to the equivalent specification upper/lower face.

The reinforcement courses are, correspondingly to their relative position in the cross-section, called the outer(most), middle (if any) and inner(most) ones. This verbal distinguishing is in the mathematical formulation replaced by assigning them the ordinal numbers 1, 2 and 3 (if three reinforcement courses are specified at all). The same double identification may be given to other associated terms like reinforcement angles, design forces, effective static heights, inner forces levers, etc. So we can speak, e.g., about reinforcement angles , , meaning the same as when indicating 1, 2, 3.

The notions and terms of the reinforcement concrete theory are used in accordance with the general use or they strictly follow the rules postulated by the Norms implemented in NEDIM. However, for NEDIM deals with several national Norms, it is likely that a few of terms or formulations would appear somewhat unfamiliar to readers focused onto the use of one Norm branch only. It is hardly possible to create a manual text on such special topic for international use being in all respects fully conform to every country’s verbal usage. In doubts the terminology of EC 2 will be given preference.

NEDIM works internally with preferred basic and derived units of the SI system.

Design of Walls The inner forces {nx, ny, nxy} of the FEM solution are retrieved from the FEM data base for each design item (element node) and transformed by the method outlined above to the design forces {n1d, n2d, n3d} according to the formulae (1) and (2).

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Once a positive design force has been assigned to its associated reinforcement course, the corresponding statically required reinforcement amount ai is calculated from the elementary relation :

ai = ni / σsd (i = 1,2 (,3)) [m²/m] (6)

(6) has symbolic meaning only, as we do not want write down at this stage all exact calculation rules for Norms implemented in NEDIM. The symbol sd stands for effective design steel strength. Both ni and sd may be, according to the actual Norm, charged with security coefficients. We are not going to discuss the problem of 1D reinforcement design; the NEDIM algorithm strictly follows special rules stipulated by national Norms and associated Standards, as far as they are applicable to the 2D design.

Above it was emphasized that the application of the transformation formula (1) to the inner forces of the FEM model yields not yet the final result and that there are formally several solutions fitting the invariance condition (5). In a class of hyperbolic states of stress (nI > 0, nII < 0) the NEDIM algorithm finds, by means of (1), a minimum energy solution of the following quality :

n1 > 0; n2 = 0; n3 < 0 (7)

In (7) n1 > 0 is the (positive) reinforcement design force; the second reinforcement course is set inactive (or both remaining ones, if a three-course reinforcement is specified); n3 is the (negative) design force of the stiffening strut.

For a general 2-course, skew, i.e. non-orthogonal, reinforcement net (representing equivalently also 3-course reinforcement nets under hyperbolic state of stress) we will follow the explanations by means of Fig. 4.

In certain hyperbolic stress situations, the NEDIM transformation algorithm yields for the skew reinforcement specification acc. to Fig. 4a a solution of the quality described by (7). It is sketched by Fig. 4b : the reinforcement course 1 is assigned a tension design force 1n1 > 0; the reinforcement course 2 is inactive; the pressure stress acting at the cross-section of question is resisted by a relatively large pressure force of the stiffening concrete strut 1n3 < 0.

However, the heterogeneous reinforced concrete medium would hardly be practically armoured by one reinforcement course only. Even if the state of stress symbolized by (6) and Fig.3 would prevail in extensive parts of the structure, at least a 2-course reinforcement net would still be necessary to maintain the functionality of such 2D structure. Due to the minimum transversal percentage requirement, the reinforcement course No 2 deactivated (theoretically) in this case would generally be assigned a portion of the statically required reinforcement amount of the reinforcement course 1. Thus, in a practical rein-forcement design the reinforcement course 2 would also be assigned a real as value.

In addition, many of national Norms implemented in NEDIM require minimum compression reinforcement resisting pressure normal forces. In case (7) or Fig. 4b, such proof could formally not be performed since there is no data of the calculation pressure force assigned to the reinforcement : the strut pressure force is not associated to any reinforcement course!

For reasons outlined, the seek of another consistent solution fitting the transformation formula (1) yet assigning a non-zero design force to the reinforcement course 2 seems to be a logical if not indispensable algorithmic step. As a fact, NEDIM carries this step out automatically in such stress-situations and yields an alternative solution symbolized by Fig. 4c. Using vector arrows of different lengths, the stress vectors of Fig. 4b and Fig. 4c express the substantial difference of the two transformation solutions mentioned. In mathematical notation the relations are as follows :

2n1 >

1n1;

2n2 <

1n2 = 0; 0 >

2n3 >

1n3 (81)

Respecting the relations (81) NEDIM replaces automatically (i.e. by internal control) the direct solution (7) by one of two following alternative solutions:

o The consistent solution acc. to Fig. 4c is preferred to the basic solution (7) acc. to Fig. 4b (both solutions are consistent since they fit the invariance condition (5)) :

n1 = 2n1; n2 =

2n2 < 0; n3 =

2n3 < 0 (82)

o NEDIM combines the two consistent solutions (7) and (82) to an inconsistent solution by combining them according to Fig. 4d. Analytically :

n1 = 1n1; n2 =

2n2 < 0; n3 =

1n3 < 0 (83)

The choice between (82) and (83) is taken by checking the energy level of both consistent solutions. As a measure of energy level jE the sum of absolute values of the design forces jni is considered :

j E = │

j n1│+ │

j n2│+ │

j n3│ (84)

Is the energy level 1E less than 80% of 2E, i.e.

1E < 0.80 2E (85)


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the consistent solution (82) is replaced by the non-consistent “mixed” solution (83). The sense of the control by "empiric" limit 80 % is to prevent too high increase of the tension reinforcement corresponding to the first course design force 1n1 when changing to 2n1.

Of-course, the non-consistent solution (83) does not bring any relief to the virtual strut (as it is the case with the alternate consistent solution (8b)). As is known, the hyperbolic stress states of this kind (7) are prone to failing virtual strut (see Chapter Processing of non-designability conditions, NSt = 5). On the other side, the solution (83) assigns the pressure force 2n2 to the second reinforcement direction, thus enabling its design, i.e. at least the estimation of minimum compression reinforcement !

Note: Actually, a more efficient optimization of this delicate design situation has been being investigated.

(a)

2 1

xp

yp

(b)

1n 3

< 0

1n 1

1n 2 = 0

Fig. 4 (Continued on next page …)

(c)

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2n 3 < 0

2n2< 0 2n

1

(d)

n3

= 1n 3

n1

= 1n1

n2

= 2n 2

Fig. 4 (…continued) Advanced NEDIM solution for a class of hyperbolic states of stress in Walls and Shells : (a) reinforcement geometry; (b) consistent solution No 1 (7); (c) consistent solution No 2 (82); (d) inconsistent solution (83) combining the consistent solutions No 1 and No 2 (83)

The preceding observations made it obvious that the virtual stiffening strut of the heterogeneous concrete-steel continuum represents a quite substantial item of the design process. While it is possible (unless the upper reinforcement percentage limit has not been exceeded) to improve the bearing capacity of the cross-section on the side of the reinforcement by augmenting its amount, the bearing limit of the concrete strut is given by the height of the cross-section and the quality of concrete only; thus its limits are predestined by the input data. The concrete strut bearing capacity condition is described by the following relation :

– n3 < Ac σcd (9)

In (9) cd represents the effective concrete design pressure strength and Ac – the concrete area of reference. In NEDIM it is generally assessed on the base of 80% of the standard design concrete pressure strength. This reduction follows the recom-mendation of Schleich and Schäfer in [2] : the compression strength of concrete is unfavourably affected by transversal tension stresses which produce cracks, parallel to the direction of pressure; this is typically the stress situation of the stiffening strut. SIA 262, §4.2.1.7 is the only Norm (implemented in NEDIM) which defines the strut reduction explicitly! For cracks parallel to the direction of strut, the reduction coefficient kc = 0.80 is stipulated, which is identical with the NEDIM default, whereas for cracks crossing the strut direction the value kc = 0.60 is specified [26] !

The cross-section area Ac in (9) is for Walls taken as the full unity rectangular cross-section h × 1.

Once a design pressure force ni, assigned to a reinforcement direction i, is known the compression reinforcement is calculated acc. to the following general formula :

ai = (– ni – Ac σcd) / σscd (i = 1,2 (,3)) [cm2/m] (10)


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In (10) scd represents the effective design compression strength of reinforcement steel differently to (6), where sd denotes the design tension strength; Ac is the gross section area. NEDIM checks if the steel attains the pressure yield strain; if not, scd reflects the actual strain level. Some Norms (ÖNORM B 4200, CSN/STN 73 1201, GBJ 10-89) stipulate different values of steel strength in tension and pressure. NEDIM follows this idea by enabling different definitions of tension and pressure steel strength to all Norm branches.

Design of Plates In the Wall model, dealt with in the preceding paragraph, the inner design forces produce constant stresses all over the cross-section; thus, there is no necessity to examine the stress distribution within the cross-section. In contrast to the Wall model, for bending in Plates it is a fundamental characteristics that the stresses are non-linearly and discontinuously distributed over the cross-section. Since all of the national Norms implemented in NEDIM exclude the tensional bearing capacity of concrete (ULS), in the tension zone („below“ the neutral axis) the only bearing material is the reinforcement steel. The resistance of concrete is exploited in the pressure bending zone only.

Fig. 5 shows symbolically one possible equilibrium situation in the reinforcement courses 1 and 2 (Fig. 5a,b) as well as in the concrete stiffening strut, i.e. in virtual course 3 (Fig. 5c). In Fig. 5, the face subscript is generally omitted, for the discussion is equally valid for both faces. The distribution of the concrete pressure stress in Fig. 5 is not related to a special Norm. However, the Norms implemented introduce different basic notions of the concrete pressure stress distribution. The assump-tion of the pressure stress distribution is in affinity to the - diagram of the concrete material, introduced by the Norms :

DIN 1045 07/1988 introduces the notion of combined parabolic (2°) and constant pressure stress distribution (s. c. “parabola-rectangle-diagram“).

ÖNORM B 4200 does not allow for fully plastified concrete in the pressure zone; the pressure stress distribution function is parabola 2° without a yield stress part. This assumption of ÖNORM B 4200 yields, for comparable material strength classes, design solutions with higher security level than DIN 1045.

EUROCODE 2 allows for all national Norm assumptions. Actually, NEDIM keeps to the parabola-constant concrete pressure stress distribution assumption. EC 2 introduces a new approach to the shear proof, which explicitly operates with the notion of the virtual (shear) strut. It also formulates a new approach to the consideration of the interaction (m/n) ↔ v. In compliance with this notion, the shear force may cause an increase of the required net reinforcement. This phenomenon was investigated by the Author of NEDIM and 1999 implemented into the EC 2 design branch as well as into all other design branches following the same (or similar) approach. For more detail on this phenomenon, named Shear Effect by the Author of NEDIM, see paragraph Advanced notes on "Shear Effect" and [17].

CSN/STN 73 1201 introduces the notion of the so called pressure block : the resisting concrete stress is assumed to develop constantly over a portion of the pressure zone only (i.e. it covers not completely the zone between the neutral axis and the compressed face) thus forming a calculation substitute for the reality approximated. Comparative tests showed that there is no substantial difference between this simplified approach and more complex pressure distribution assumptions of other Norms on the side of the design results.

SIA 162 introduces (see CSN/STN 73 1201) the notion of the pressure block and, simul-taneously, the combined parabola-constant stress distribution, analogously to DIN 1045. NEDIM uses the former assumption. It might thus be expected that the design results of CSN/STN 73 1201 and SIA 162 would differ slightly from each other for comparable material qualities. However, this proves to be true in situations with vanishing shear forces only! As a fact, SIA 162 was the first of national Norms implemented in NEDIM (and the first Norm used in practice on international scale) which formulated the impact of shear forces upon the net reinforcement at both faces(!), which was given the name Shear Effect (see above, EUROCODE 2). Because SIA 162, in difference to EC 2, DIN 1045-1 and ÖNORM B 4700, does not regularly allow for a design variant without considering the Shear Effect, the development of the shear effect algorithm of NEDIM was, as a fact, induced by SIA 162,

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rather than by EC 2. This special treatment of the Shear Effect is by SIA 162 explicitly formulated for beams, i.e. 1D structural members, only. In order to make it applicable for 2D reinforcement models, some special assumptions and algorithmic enhancements had to be made. This NEDIM genuine development was implemented 1999 [17]. SCIA holds priority not only in reporting this phenomenon but also in having developed and implemented their own 2D algorithm in a design program distributed on the international market. This phenomenon shall be discussed in more detail in the Chapter Shear Proof.

NEN 6720 operates under the assumption of linear-constant branch of the pressure distribu-tion function. NEN 6720 is a Norm of high engineering value. Especially its notion of shear and crack proof is highly sophisticated. However, NEN 6720 does not introduce the notion of the Shear Effect.

DIN 1045-1 is a Norm of the Eurocode family, developed as a successor of DIN 1045, 07/1988 [5]; it has officially been introduced into the German engineering practice in 2002. DIN 1045-1 maintains the notion of combined 2° parabolic and constant concrete pressure stress distribution of its predecessor issue. However, the material strengths are defined in the Eurocode manner. The most distinguishing features in comparison with the predecessor DIN issue are: (a) the notion of partial safety factors; (b) Eurocode-like classification of concrete; (c) substantially higher allowed ultimate steel strain; (d) the Shear Proof notion is substantially that of EC 2; however, some new elements have been introduced, not all of them being a real improvement [21]. For more details see the Paper [22] of the Author of NEDIM.

GBJ10-89 uses a similar concrete pressure distribution notion as DIN 1045-1. This norm does not substantially differ from any other European norm. A distinguishing feature of the Chinese Norm is that it pays more attention to design and detailing problems of 2D structures. Especially, the GBJ 10-89 Chapter “Deep Beams” has, compared with other European norms, a pioneer character. See the paragraph Deep Beams below and [20].

BS 8110 allows for the pressure block and the parabola-constant concrete stress distribution as basic design assumptions. However, BS 8110 has developed outside the EUROCODE family, thus interesting differences are encountered, e.g. (a) the yield concrete strain εc1 is controlled by fcu (e.g. for fcu = 25 [MPa] is εc1 = 0.98 ‰); (b) the material security factor of reinforcement steel is γm = 1.05; (c) the shear strength (global resistance to shear) is assigned the security coefficient γsh = 1.25; (d) special shear force limit has been introduced for constructive minimum shear reinforcement : SR0 ↔ SR1; (e) detailed provisions for minimum reinforcement percentages according to analysis type and steel strength; (f) "unconventional" approach to the crack control (obscure analytical formulae) etc.

ÖNORM B 4700, officially declared as “EUROCODE-like” Norm, does relatively little differ from EC 2. It stipulates for the concrete both the notion of the pressure block and the parabola-constant stress distribution analogously to DIN 1045-1. NEDIM follows the latter assumption. Reference [22] presents a comparative study ÖNORM B 4700 ↔ DIN 1045-1.

BAEL 91/99 stipulates, like SIA 162 and ÖNORM B 4700, both the assumptions of the pressure block and the parabola-constant concrete stress distribution. NEDIM follows the latter assumption, like with ÖNORM B 4700. BAEL 91/99 resembles, in the way how it deals with the problem of m/n design, the Norms of the Eurocode family. The most distinguished differences are to be found in special regions of design, like min/ max reinforcement, shear proof and crack proof, as described further below.

SIA 262 maintains the assumptions of the pressure block and the parabola-constant


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concrete stress distribution from its predecessor issue SIA 162. As a whole, there are just little differences to EC 2, e.g. (a) reduced ultimate concrete strain εcu = – 3.0‰; (b) Young's module of steel Es = 205 [GPa]; (c) simplified estimation of basic shear resistance VRd ; (d) consequent limitation of the effective concrete strength in cracked continuum (§4.2.1.7); (e) simplified semi-empirical approach to the crack control etc.

EN 1992-1-1:2004 is the EC 2 novella of the preliminary European Norm ENV 1992-1-1:1991. Extensive modifications to the original text have been taken in all sections. Espe-cially the paragraphs on shear proof, crack control and minimum reinforcement control have been expanded and diversified, e.g. (a) all "reasonable" assumptions of concrete stress-strain diagram are allowed, namely the three basic cases: pressure block, linear-constant and parabola-constant. For the block distribution, restrictions to height and stress have been introduced: λ x and η fcd ; (b) the recommended value of the strength reduction coefficient in fcd = αcc fctk,0.005 / γc is αcc = 1.0 (for the EC2 Norm family is typically αc = 0.85); (c) the strength reduction coefficient ν1 in the formula for the shear strut resistance vRd,max is more diversified and the coefficient αcw of the same formula expresses the effect of normal stress upon vRd,max on three intensity levels; (d) the crack calculation formula (direct control) resembles that of DIN 1045-1, however, the crack distance formula depends here on 4 parameters.

IS 456 is unmistakably based upon some principles of BS 8110, yet it has developed ge-nuine approaches, which distinguish it strongly from the British Standard. In particular, (a) the shear proof is based upon 3 Shear Regions, unlike BS 8110, and the crucial parameters of the shear proof are defined quite differently from BS 8110; (b) the paragraphs on mini-mum reinforcement control stipulate another approach than BS 8110. On the other hand, the crack control procedure of IS 456 is, by declaration, identical with the approach of BS 8110.

ACI 318M-05 is a comprehensive Norm on concrete. For reinforcement concrete design the Chapter 9 "Strength and Serviceability Requirements" is of fundamental importance. The most distinguishing features are: (a) ACI does not apply the notion of partial security coefficient; instead the design strengths are controlled by the state of stress, which is expressed by the strength reduction factor . Typically, = 0.90 for tension, = 0.65 for compression and = 0.75 for shear. This regulation remembers to the strain controlled secu-rity coefficient under DIN 1045 07/1988, yet this case is algorithmically more exacting since, in bending, the transition interval between "tension-controllled" and "compression-controlled" lies between εs,yield and εs,min, where εs,min corresponds to xmax – the maximum allowable bending pressure zone, and some numerical instability phenomena appear; (b) ACI allows for three basic assumptions of concrete stress distribution – rectangular, trapezoidal and parabolic, but allows for any other stress-strain diagram that "results in prediction of strength in substantial agreement with results of comprehensive tests". Since the more general assumption parabola-rectangle is a generalization of pure parabolic stress distribution, the basic three stress-strain diagrams actually maintained by NEDIM apply fully to the stipulations of ACI 318M-05; (c) for flexural members with factored axial load

n > – 0.10 fc' Ac the steel strain (at nominal strength) shall not be less than εs = 4.0‰. This is an important restriction to the design process! (d) the maximum admissible bending zone height xmax is controlled by steel strength or minimum admissible steel strain (see (c)), res-pectively; (e) the maximum usable strain at extreme concrete compression fibre shall be assumed equal to – 3.0‰ (compare with SIA 262); (f) there is a variety of stipulations con-cerning the minimum reinforcement percentage. However, they seem quite uncoordinated; (g) Crack control is not directly required; it seems that this topic is covered by a number of

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structural requirements.

In Plate models the statically required tension reinforcement of a design course is calculated by the basic formula :

ai = mi / (zi σs,eff) (i = 1,2 (,3)) [cm²/m] (11)

In (11) the special moment symbol mi for the design moment associated with the reinforcement course i is substituted for the common symbol ni for design force in order to avoid confusion. The stress symbol s,eff has a quality comparable with that introduced by (6) for Walls; it again represents the effective design steel strength for all Norms. The inner forces lever zi in (11) makes out the formal difference of (6) and (11); factually, there is no difference between them, since the quotient mi /zi

equals the steel design force Zi, which constitutes with the opposing concrete pressure zone resultant force Di the force couple representing the design bending moment mi; thus, we formally obtain (6) by substituting ni = Zi = mi /zi into (11).

(11) reveals the fundamental meaning of the inner forces lever z for the design algorithm. As a fact, by introducing the transformation formulae (3) for Shells it was made clear enough that the knowledge of the proper value of inner forces lever is indispensable for correct reinforcement design.

In NEDIM the inner forces lever z is calculated by the following procedures:

o For DIN 1045 and ÖNORM B 4200 interpolation formulae for the value of z were developed. The maximum approximation error amounts up to 2 %, however.

o For all other Norms (following the first two on the time scale) analytic integration procedures for the basic assumptions of stress block, linear-constant and parabola-constant stress function were devised; they yield exact pressure inte-grals.

The stiffening function of the concrete medium is not as transparent in Plates as in Walls. In Plates we have to do with force couples representing inner bending moments. The concrete pressure stresses are not constantly distributed over the cross-section. Thus, a direct application of the concrete strut bearing capacity limit condition (9) is not possible in Plates. NEDIM had used some approximate approaches until the best and perhaps most simply formulation of the strut bearing capacity limit was found. Instead of describing the strut control by mathematical terms, a verbal explanation of the matter relating to Fig. 5c and Fig. 5d is preferred :

o In Plates the strut design force n3 means the force couple of m3. From Fig. 5c it is obvious that m3 causes basically the same kind of stresses in its direction as the other two reinforcement design moments m1 and m2, however, with exchanged faces (i.e. m3 is of opposite sign). In this case we are not interested in analysing the situation on the opposite face; the state of stress in the stiffening strut bending pressure zone is of interest. What is the limit condition of the strut bearing capacity? What calculation value of stress integral force D3 is to be taken into account?

o The answer to this fundamental question is given by Fig. 5d : NEDIM allows for the maximum height of the bending pressure zone xmax in compliance with the design algorithm applied. If at this state of stress the equilibrium in the cross-section is not yet attained, i.e. would strengthening of the pressure zone by (pressure) rein-forcement be formally required, then this is considered by NEDIM as an unambiguous indication of the bearing capacity of the stiffening strut being exceeded. The cross-section is non-designable due to concrete failure (Non-designabilty Status – NSt=5).

Till the mid of 2007 it had not been known to the Author of NEDIM that any competing software would deal with this problem at all. Neither Norms nor theoretical publications on reinforced concrete design do not care about the state of concrete in a heterogeneous concrete-steel 2D medium. Some Norms give “standardized” recommendations as to the geometrical arrangement of reinforcement in reference to the directions of the principal moments. They are concerned with stress situations which are typical for corners of floor slabs etc. The way of how the recommendations had been formulated chal-lenged the Author of NEDIM to a critical essay in [18].


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Fig. 5 Equilibrium of design inner forces in a Plate cross-section : (a) reinforcement course 1; (b) reinforcement course

2; (c) concrete stiffening strut – course 3; (d) strain situation in the stiffening strut (bearing ability proof)

The special quality of design in plate corners is summed up by Fig. 6. This case gives a principal insight into the substance of the virtual strut in bending stress situations.

Two marginal cases are compared by following Fig. 6: o Usual reinforcement geometry (0°/90°) at both faces. This is statically the most inefficient arrangement since the

reinforcement courses decline by 45° from the principal moments mI, mII directions, which is the maximum value. All 4 reinforcement courses are under tension (!?) and the concrete pressure zones (on both faces) cannot develop in sections rectangular to the reinforcement : they are active in the directions (135°/45°) – a situation, which is completely unknown to 1D design! The reinforcement is, as a fact, under shear stress instead, as generally assumed, under pure tension. This is very a complicated situation which shall not be analyzed in detail here. The primary knowledge from Fig. 6b is that the concrete strut has to resist the double value of mI !

o The trajectory reinforcement, as demonstrated by Fig. 6c, represents the optimum reinforcement solution. The concrete strut force vanishes and the reinforcement design simplifies to two quasi 1D design cases.

The virtual strut bearing ability is a topic of acute interest. Users changing from other program systems to EPW/ ESA PT come earlier or later across the design error NSt = 5. On the SCIA hotline they then often claim : “With our old program we never have had such a problem. All the time we had been using it, no exhaustion of the concrete bearing ability was reported”. However, if the state of stress of the concrete is not monitored properly, not all of possible critical situations can be realized by the design algorithm. Programs which do not care of this crucial design phenomenon cannot report of it at all, but this disability does not make them better than software which deals precisely with the matter!

Additional arguments like: “We do not know anything about damage to structures due to insufficiency of the stiffening function of the concrete”, are of no practical impact. The structures have to be designed with a security reserve. Underestimating of the bearing capacity of concrete does not cause immediately a crash, yet generally a lesser than the required level of security required by Norms and other Standards and Regulations, which every project engineer is obliged to provide with his structure.

The most distinguishing features of the 2D reinforced concrete compared with the 1D design are summarized in the Paper [18], where some critical theses about the design approaches of EC 2, DIN 1045, DIN 1045-1 and ÖNORM B 4700 are formulated. The criticism of lacking attention to the concrete in its function as stiffening medium constitutes one of the vital points of the Paper.

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Fig. 6 Stress situation and development of the concrete strut in corner of a quadratic

plate : (a) Alternative reinforcement geometry 0°/90° or 45°/135° (both faces);

(b) Reinforcement geometry 0°/90° : state of stress in plate corner – inefficient reinforcement geometry causing maximum concrete stress;

(c) Trajectory reinforcement geometry 45°/135° : state of stress in plate corner


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Design of Shells In the design of Shells, the ideas and procedures of both the design of Walls and the design of Plates are combined. The Norm requirements and restrictions, which seldom are formulated individually for Shells, must both be considered both for Walls and Plates. Thus, the Shell design is the most complex design model dealt with by NEDIM.

From the mechanical point of view, the stress-strain situation in cross-sections of Shells may develop from a typical "Wall pattern" with constant stress distribution to a "Plate pattern" with characteristic non-linear concrete pressure stress distribution over the bending pressure zone along with a cracked region "below" the neutral axis where there the reinforcement resists the stresses from inner forces. The special situation depends, however, on the character of external load as well as on the boundary conditions of the analysis model.

NEDIM has to manage all possible stress situations arising between the Wall type and the Plate type state of stress using one unique design model to be able to produce results consistent also with quantitatively slowly yet qualitatively abruptly changing states of stress. It would be unacceptable to have a Shell design model which, on one side, yields results fully identical with a Plate solution when there is pure bending acting, i.e. the membrane forces being zero, yet produces, on the other side, obviously distorted results whenever the membrane forces differ slightly from zero. Little change in loading must imply little change in the reinforcement design results.

As a fact, all Norms were drafted focusing to the problems of 1D structural members, i.e. beams and columns. In NEDIM, many requirements and restrictions had to be given a reasonable engineering interpretation or extrapolation to fit to the special character of the 2D structures. The reinforcement at both faces consists of two mutually independent nets with 2 or 3 reinforcement courses in generally different directions. Thus, in Shells it is not possible to proceed by simply using the solutions of the reinforcement concrete design of beams.

NEDIM creates two sets of transformed design forces assigned to individual reinforcement courses and/or the stiffening concrete strut at both faces of the analysis model. The procedure goes acc. to the formulae (4). In the assessment of the inner forces lever z the Shell design procedure resembles the Plate design. By creating equivalent inner forces {nx, ny, nxy}Zp and their transforms {n1d, n2d, n3d} NEDIM follows a typical Wall design approach. Formally, we get two systems of design situations at both Shell faces which must be managed in two algorithmic steps in every cross-section by considering the situation on the other face. In this sense, the Shell design is organized like the Plate design.

Fig. 7c symbolizes the Shell design : there is a design force nd (subscript i = 1,2,3 is omitted) assigned to a reinforcement course at the upper face (the same procedure applies to the lower face). The symbol nd,opp is used for the virtual design force at the opposite face acting in the same direction as at the actual face; it is unimportant if there is a congruent reinforcement course parallel to that at the actual face or not. The normal force in this cross-section is in Fig. 7c denoted as nvirt (virtual normal force). The virtual bending moment mvirt is defined complementary to nvirt. Thus, the virtual normal force eccentricity (241), (242) can be estimated.

Fig. 7c demonstrates also the fact that the design at a Shell face is typically Wall design; however, the design force nd is not applied to the total cross-section area as in Walls (Fig. 7a), yet to some portion of it : Ac,eff = heff × 1.0. NEDIM assigns Ac,eff basically in accordance with the suggestions of Baumann in [5].

Fig. 7 Comparison of design situations in three NEDIM design models :

(a) Wall : total cross-section under tension/compression design force nd; (b) Plate : design bending moment md acting over the effective height d; (c) Shell : combined action of bending moments/membrane forces expressed by nd : design normal force at active

face; nd,opp : design normal force at passive face; nvirt : total virtual normal force in cross-section; mvirt : virtual bending moment conjugated with nvirt

In the area assignment formula

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Ac,eff = kA Ac (12)

the value of the coefficient kA varies in the range <0.35, 0.42> in stress situations with neutral axis within the cross-section. Principally, this approach may be compared with the approximation of the stress distribution in the bending pressure zone by the pressure block (see above). Recent theoretical and algorithmic enhancements of NEDIM made it possible to distinguish efficiently between bending-like and membrane-like stress situations in Shells, thus enabling to apply the full cross-section, i.e. kA = 0.50, to the virtual strut proof when the strut cross-section is over-pressed.

The proof of the virtual strut resistance is formally governed by (9), like for Walls. However, instead of the total cross-section area Ac, the effective one-face area Ac,eff (12) is to substitute into (9).


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SHEAR PROOF

Introduction to Shear Proof All Norms implemented in NEDIM introduce own mechanical model of how slender structures like Plates and Shells resist the shear force and how they can be strengthen to increase their shear resistance. As a fact, the Shear Proof is still more diver-sified than the longitudinal reinforcement design. However, this diversity becomes more relative by near consideration : there are basic shear proof principles shared by most of the implemented Norms.

The shear stress is associated with bending state of stress in Plates and Shells. Walls are not involved in this design algorithm; the membrane shear, represented by the inner force nxy is principally another kind of inner force and has nothing in common with the following discussion.

The shear force components vx and vy at the point of consideration are transformed to the design shear force vd according to the following „geometric sum“ formula :

vd = (vx² + vy²) [kN/m] (13)

Fig. 8 Notion of the design shear force : – direction of the principal design force vd ; j – directions of supplement design forces vjd associated with mI, nI

The symbol vd in (13) denotes the principal design shear force associated with the direction = arctan (vy / vx) (Fig. 8). In most Norms the overall state of stress plays a role in the shear proof, thus the actual bending moment and normal force become factors of the shear proof process. The logical consequence of considering the action (m / n) is to effect-uate the shear proof in the directions j of the principal inner forces mI and nI (Fig. 8), too. To do this, the notion of supplement design shear forces vjd is introduced:

vjd = vx cos j + vy sin j j = j (mI, nI) (j = 1,2,3) (14)

All Norms define certain stress levels which, when attained/exceeded, indicate either the shear reinforcement being required to ensure the cross-section resistance to shear, or signalize the structure being non-designable to shear. In compliance with the terminology of DIN 1045 07/1988, the lowest limit restricts the so called Shear Region 1 (abbreviated as SR1). Within SR1, no shear reinforcement is required. In the Shear Region 2 (SR2) the shear reinforcement is calculated on the base of the design value vd (13). When the (upper) limit of SR2 is attained in a 2D continuum, the shear bearing ability of the cross-section is considered to be exhausted. This limit is the threshold to Shear Region 3. If SR3 is indicated in the output protocol of the 2D design it means non-designability of the cross-section under scrutiny.

Not all of the Norm proof procedures implemented in NEDIM are discussed here in full detail; their theoretical fundamentals may be studied in the corresponding Norm texts and accompanying publications. A theoretical summary of the 2D shear proof is presented in the Paper [17], which comprises some unconventional insights into the computerized 2D shear proof tech-niques. Generally, only selected characteristic details of the shear proof procedure shall be given focus to in this Chapter. Since there are principles common to all Norms, they shall not be repeatedly discussed in all paragraphs on individual Norm branches, which had continually been being added to this manual text during their implementation into NEDIM since 1992.

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The program users are thus encouraged to read all Norm paragraphs of this manual to get sure that no useful information has been ignored.

Some Norms stipulate minimum shear reinforcement. This requirement is regulated by Norm tables combining e.g. the control by the concrete and steel strength classes. Analogously, CSN/STN 73 1201 stipulate an additional, lower shear stress limit, which, if attained, implies the need of the so called structural shear reinforcement in corresponding parts of the structure designed. A comparable intermediate shear limit is stipulated also by BS 8110. To incorporate the corresponding shear level into the proof procedure, the notion of Shear Region 0 was introduced for CSN/STN and BS 8110! In these Norm branches, SR0 means "No required shear reinforcement"; SR1 – "Minimum shear reinforcement"; SR2 – "Statically required shear reinforcement"; SR3 – "Non-designable".

To enable better understanding of the design results, the most important characteristics of the Norm oriented shear proof procedures applied in NEDIM are summarized here. Prior to all individual specifications it is to declare that the assumption of vertical stirrups, i.e. stirrups which are orthogonal to the structural member normal axis, is common to all Norm branches implemented in NEDIM.

Shear Proof according to DIN 1045 07/1988 DIN 1045 introduces the notion of the so called truss model analogy (“Fachwerkanalogie” of “Fachwerkmodel”) by E. Mörsch of the shear stress resistance mechanism. The Shear Region limits (see the introductory paragraph for definition of the notion Shear Region) are expressed in terms of allowable shear stresses as functions of the concrete strength class. Depending on the continuity of the tension reinforcement in individual spans, one of two sets of shear stress limits applies.

Shear Proof according to ÖNORM B 4200 ÖNORM B 4200 stipulates a shear proof approach similar to DIN 1045. Differently to DIN 1045, the continuity of net reinforcement is no design factor ; on the other side, the amount of net reinforcement (both tension and compression reinforcement!) at both faces is declared a bearing capacity increasing factor. Since ÖNORM B 4200 formulates this approach for 1D members only, NEDIM implies the following interpretation : the geometric sums of individual active reinforcement direction amounts (analogously to (13)), evaluated at both faces sepa-rately, are added up and used as effective values of longitudinal reinforcement.

Shear Proof according to EUROCODE 2 EUROCODE 2 introduces a progressive shear proof method (compared with DIN 1045 88/7 and other Norms of the pre-Eurocode era) based upon the concept of Strut inclination. The notion “Strut” is a symbolic representation of the concrete medium resistance to principal pressure stresses, as described in [17]; the strut inclination is then the assumed direction of the principal pressure stress field reactivated in the concrete by the shear (transversal) force v.

The basic shear resistance of the concrete cross-section without transversal shear reinfor-cement (stirrups) is determined by a sophisticated formula considering continuity and percentage of longitudinal reinforcement, height of the cross-section and the magnitude of the normal force (both compression and tension). This EC2 formula (4.18), written in terms of NEDIM for 2D design, assumes the following form :

vRd1 = (τrd k (1.2 + 40 ρ1) – 0.15 σcp) d [MN/m] (141)

with τRd – basic value of design shear strength according to Table 4.8 ; ρ1 – effective longi-tudinal reinforcement percentage according to the following restriction :

ρ1 = min (100 asl / d , 2. ) [%] (142)

For (142) EC2 strictly specifies tension reinforcement asl, and so it is handled in NEDIM. However, if there is no tension reinforcement (i.e. over-pressed cross-section), the active compression reinforcement is dealt with instead, in order to prevent inconsistent design results


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in cross-sections differing, perhaps, only by normal force slightly alternating around zero. Furthermore, (141) uses the term σcp = nd / d < 0 – medium normal stress in cross-section due to the isolated effect of the normal force nd (compare with NEN 6720 where there the full cross-section height h is referred to). Differently to EC2, formula (4.18) the term 0.15σcp in (142) appears with negative sign since the tension normal force is considered to be positive! NEDIM controls, however, the basic shear strength not to become negative with high normal tension.

The coefficient k in (142) is represents the continuity of longitudinal (tension) reinforcement as well as the static height of cross-section :

k = max (1.6 – d , 1.) [–] (143)

with d – the cross-section static height in [m]; k assumes the minimum value of 1.0 if more than 50% of the longitudinal (or “field”) reinforcement is discontinuous.

The cross-section remains in SR1 as long as the basic shear resistance vRd1 fits the following condition :

vRd1 ≥ vd [MN/m] (144)

For higher load intensity than (144), shear reinforcement is required. Two alternative shear reinforcement design methods are stipulated by EC 2; both are implemented in NEDIM) :

o The s. c. “Standard method” EC 2, §4.3.2.4.3 is based on the concept of constant strut inclination. This approach allows in the calculation of the statically required shear reinforcement for basic concrete resistance vcd (s. c. “Abzugswert” by Mörsch), which reduces the effective value of the reinforcement design force vwd. On the other hand, the strut inclination angle is applied constantly as θ = 45° :

asw = (vd – vcd) / (0.9 d fywd) [m²/m²] (145)

where vcd is, basically, vRd1 (141); compare with EC 2, formulae (4.22) and (4.23)). It is noteworthy a fact that the fundamental formulae (159) for NEN 6720 and (173) for BAEL 91/99 (see below), respectively, are principally identical, excepting, however, different symbols used.

o The advanced method EC 2, §4.3.2.4.4 applies the concept of variable strut inclination. The statically required shear reinforcement asw is calculated from the elementary EC 2 formula (4.27); with NEDIM symbolism :

asw = vd tan θ / (z fywd) [m²/m²] (146)

which is principally identical for all Norms of the Eurocode family. It does not contain any reduction component considering the basic concrete resistance, like vcd in (145). How-ever, the limited resistance capacity of the virtual concrete strut is paid respect to by the EC 2 condition (4.27) :

asw ≤ 0.5 ν fcd / fywd [m²/m²] (147)

which means, that the cross-section resistance cannot be increased deliberately by increasing the stirrup reinforcement ! The control factor ν = f (fck) is defined by the EC 2 formula (4.21).

Note. EC 2 introduced the symbol θ for the virtual strut inclination. However, it seems that the process of European unification has obviously not yet reached the reinforced con-crete symbolism, since there are at least 3 different symbols for the strut inclination in use : θ, α (SIA 162) and β (ÖNORM B 4700). In general considerations of this manual, the EC 2 symbol θ is preferred.

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The resistance to shear force of a reinforced cross-section is exhausted when the bearing capacity vRd2 of the virtual shear-compression strut is less than the shear impact vd ; the cross-section transits to SR3 :

vRd2 = 0.5 z ν fcd sin (2θ) [MN/m] (148)

In (148) ν considers the reduced concrete strut strength, as analogy to the reduced strength of the bending-membrane stiffening strut (see Design of Walls above). A further reduction of the strut resistance vRd2 follows from compression normal force. EC2, §4.3.2.2(4), formula (4.15) introduces the effective value of vRd2,eff which then replaces vRd2 of (148).

Shear Proof according to CSN 73 1201 and STN 73 1201 CSN/STN 73 1201 stipulate shear proof in oblique section (compare with GBJ 10-89), which is based on the concrete tension strength merely than on allowable shear stress. In addition to most other Norms, CSN/STN 73 1201 require minimum shear reinforcement in special design situations. For this reason, the notion of Shear Regions, introduced by DIN 1045 and applied to other Norms as well, has for CSN/STN 73 1201 (and also BS 8110) been extended by the formal introduction of SR0; this is equivalent to the standard meaning of SR1, used for other Norms. For CSN/STN, in SR1 the minimum shear reinforcement is estimated. SR2/3 have then equal meaning as with other Norms.

The basic shear resistance of the concrete cross-section without transversal shear reinforce-ment is determined by §5.3.3, formula (137); in terms of NEDIM :

qbu = h κq γb Rbtd / 3. [MN/m] (14a)

with h – cross-section height, Rbtd – design tension strength of concrete, γb – effectivity factor of concrete (§5.1.2) and κq – shear strength factor; κq = f (h) → {1.25, 1.5, 1.6} (§5.3.5.2). The value of qbu serves as criteria of determining the Shear Region (§5.3.2.2), formulae (133) – (135).

The cross-section is in SR0 if the design shear force does not exceed the value of qbu :

qd ≤ qbu [MN/m] (14b)

SR1, i.e. required minimum shear reinforcement, corresponds to the shear intensity :

qbu < qd ≤ 2.5 qbu [MN/m] (14c)

For SR1 the minimum stirrup shear reinforcement is specified by §5.3.6, formula (146) ; in terms of NEDIM :

asv = (κq / 2.)² Rbtd / Rssd [m²/m²] (14d)

with Rbtd – design tension strength of concrete (14a), Rssd – design tension strength of the stirrup reinforcement steel and κq – shear strength factor (14a).

SR2 is limited by the condition :

qd > 2.5 qbu [MN/m] (14e)

The statically required shear reinforcement in SR2 is specified by §5.3.4.1, formula (138) ; in terms of NEDIM :


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asv = (qd – qbu) / (γs Rssd c) [m²/m²] (14f)

with Rssd – design tension strength of the stirrup reinforcement steel (14d), γs – effectivity factor of stirrup steel (§5.1.2) and c – length of oblique section projection onto the member neutral axis, which is stipulated by §5.3.5.1, formulae (140) and (141); in terms of NEDIM :

c0 = 1.2 γb Rbtd he² / (qd – qbu) [m] (14g)

c = max (min (0.18 Rbd h /(κq Rbtd), c0), zb) [m] (14h)

with he – effective static height, zb – inner forces lever (used in shear proof) and c0 – auxiliary variable (14g) → (14h); other variables used in (14g), (14h) – see preceding formulae and explanations.

The limit to SR3 is described as the so called reliability condition by §5.3.2.1, formula (131); in terms of NEDIM :

qd ≤ h min (γb Rbd,18.) / 3. [MN/m] (14i)

with Rbd – design compression strength of concrete, γb – effectivity factor (§2.1.2.1). The upper limit of 18. [MPa] on γb Rbd is stipulated as additional condition by §5.3.2.1, formula (132). If (14i) is not met, the cross-section does not possess sufficient reliability to the shear impact in the sense of CSN/STN (exhaustion of the virtual concrete strut); undesignability to shear is reported.

Shear Proof according to SIA 162 SIA 162 applies the variable shear strut inclination method (see mode (b) of EC2, above) as standard shear proof approach. However, the special requirement of §3 24 203 (see the para-graph Shear Effect), which is hardly to control in 2D structures, made it necessary for NEDIM to introduce the alternative mode of the so called variable-fixed shear strut incli-nation available for user's control. This special approach applies the basic value of = 45° constantly throughout the whole calculation; however, it is not the same approach as stipulated by the EC 2 “Standard method” (145).

Shear Proof according to NEN 6720 NEN 6720 stipulates a sophisticated, progressive method of shear proof, which resembles a combination of the ideas of EC 2 and SIA 162. Especially, it also applies the notion of vari-able shear strut inclination. The notion of Shear Regions (see above) is well applicable to the NEN 6720 proof procedure as a classification base.

The NEN 6720 design branch is one of the most crucial NEDIM design algorithms. Originally, it was coded according to the NEN 6720:1995 edition. The Norm novella A3:2004 (and other corrections) has been, however, considered by additional algorithm enhancements. Especially, the shear proof stipulations, as described by the paragraph 8.2 “Dwarskracht” have thus become quite labyrinthine. In order to make the NEDIM design procedure transparent to the users, it is here described, adapted to the ESA PT 2D design terminology and symbolism, in quite a detail: Shear Region 1 (SR1). The notion of Shear Region introduced by DIN 1045 07/1988 (see above) has been adapted to the NEN 6720 shear proof procedure. The cross-section lies in SR1 if the design shear stress d (related to the cross-section width b0 = 1.0 [m]) is not less than the basic shear stress limit 1 (152) :

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τ1 ≥ τd = vd / d [MPa] (151)

It means that the resistance of the cross-section to shear force is effectuated without shear reinforcement (stirrups). The basic shear bearing ability is defined as :

τ1 = 0.4 fb max (kh kλ 3ω0, 1.) [MPa] (152)

In (152) τ1 is, consequently, the SR1 limit; it is calculated from fb, as stipulated by the Norm Table 3. From (152) it follows that τ1 attains at least the value τ1 = 0.4 fb. The parameters kλ , kh and 3ω0 in (151) have the following meaning :

kh = max (1.6 – h, 1.) [–] (153)

with h – the cross-section height in [m]. The parameter kλ accounts for increased shear resistance to concentrate loads acting near to margin supports. It is defined by a formula (§8.2.3.1), which implicitly comprises the so called shear force slenderness λv = Md,max / (d Vd,max) . However, to apply this typical 1D formula, ESA PT does not contain any necessary geometric information for 2D members; anyway, the application of the above λv formula to 2D structures would generally be indefinite ! Thus, NEDIM applies the parameter kλ as a constant :

kλ = 1. [–] (154)

This is, according to the definition in §8.2.3.1, the lower limit of kλ, thus providing a secure proof result. Compare the meaning of kλ with the definition of the parameter β in EC2, §4.3.2.2(5) and (9), formula (4.17) !

The most important parameter of (152) is the longitudinal reinforcement percentage, defined as

ω0 = min (100 as,eff / d, 2.) [%] (155)

(also related to unit cross-section width b = 1.0), appearing in cubic root. In (155) as,eff represents the effective amount of longitudinal net reinforcement under tension ! To calculate this special value, NEDIM applies the routine SUMREI(3,…). It yields for as,eff the geometric sum obeying the cos² rule :

as,eff = Σj =1,3 (ast,j cos² (αj – β)) Zp [m²/m] (156)

(156) is written in algorithmic symbolism : (a) Zp means, that the summations Σj =1,3 (...) are carried out separately for upper and lower face, then applying the square root operator to the final sum of + Zp and – Zp; (b) ast,j symbolizes that only active reinforcement courses under tension (t) are considered. The reinforcement percentage ω0 (154) is limited to [2%] ! The cosine function in (156) is the reinforcement course effectiveness factor, ensuring that reinforcement parallel to the design shear force direction β is engaged by 100%, whereas reinforcement course perpendicular to direction β disappears effectively from (156). The square power of the cosine function considers the tensor character of the longitudinal stresses which ast,j is associated with.

Shear Region 2 (SR2). The shear bearing capacity associated with stirrup reinforcement (α =

90°) is represented by the shear stress limit s :

τs = vs / d = asv z fs / (d tan θ) [MPa] (157)

with vs – the partial shear force acting upon the stirrups; asv – effective area of stirrups; z – the inner forces lever; θ – actual (chosen) inclination angle of the virtual concrete strut. The statically required amount of asv follows from the condition of equilibrium of active and resisting inner forces :

τu ≡ τ1 + τs = τd [MPa] (158)

Thus :

asv = (τd – τ1) d tan θ / (z fs) [m²/m²] (159)

It is noteworthy a fact that the shear reinforcement formula (159) for NEN 6720 and the cor-responding formula (145) of the so called “Standard method” of EC 2 (see above) comprise both the shear impact force (or stress) reduced by the basic concrete resistance capacity.

Shear Region 3 (SR3). According to (159), the resistance to shear can be increased within quite wide limits by increasing the stirrup reinforcement asv. However, if the capacity of the


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virtual strut is exhausted, the cross-section collapsed due to concrete insufficiency, no matter how strong the shear reinforcement has been provided. The limit value of SR3 is the shear limit 2 defined by the following relation :

τ2 = 0.2 f'b kn kθ [MPa] (15A)

The transition to SR3, i.e. non-designability to shear force, is described by the relation :

τ2 < τd [MPa] (15B)

For perpendicular stirrups, the factor kθ in (15A) is defined as :

kθ = 2 / (tan θ + cot θ) = sin 2θ [–] (15C)

The factor kn in (15A) represents the unfavourable effect of the compression normal force nd upon the strut resistance. It is described by the formula :

kn = min [ (1 + σnd / f’b) 5. / 3., 1.] [–] (15D)

(compare with EC2, §4.3. 2.2(4), (4.15)) with σnd = nd / h < 0 – medium normal stress in cross-section due to the isolated effect of the normal force nd (note the difference to NEN 6720 symbolism: σnd ≡ – σ 'bmd ). (15D) is valid both for σnd < 0, i.e. for compression, and for σnd ≥ 0, i.e. tension; in the latter case the second term of (15D) applies, i.e. kn = 1.0 for tension normal forces nd (or pure bending, nd = 0).

The notion of virtual shear strut with variable inclination, as stipulated by NEN 6720, has consequently been implemen-ted in NEDIM. A comparison of the formulae (159) and (15A,C) reveals the “double-edged” character of the strut

inclination : (a) higher values of cause increased demand of shear reinforcement; (b) lower values of diminuish the strut resistance. Since both dependances are described by continuous functions, there exists an optimum value of θopt, which yields the minimum required shear reinforcement amount, yet still preventing the transition to SR3 (15B). The θopt value is found iteratively : the iteration process starts with the minimum allowed strut angle min = 30° (§8.2.4) and, incre-menting by 1°, equilibrium at the lowest amount of required shear reinforcement is searched, meeting the shear resistance requirement u ≡ 1 + s = 2 = d. The ESA PT input control allows, as a practical proof alternative, for user's choice of the constant inclination = 45° (compare with the so called variable-fixed shear strut inclination method of the SIA 162 branch).

Additional considerations on SR1. The basic shear stress limit 1 (152) of SR1 depends, analogously to the SR3 limit 2 (15A), upon the normal force nd , which is, basically, the integral component of the membrane stress state in the direction of the design shear force vd (see (13), (14) and Fig.8). Compression normal forces are favourable, i.e. increase 1 ; tension normal forces are unfavourable. Using the ESA/NEDIM symbolism, the dependence on nd is expressed by the formula :

τ'1 = τ1 + τn [MPa] (15E)

with '1 – the effective SR1 limit; n = f (nd) (compare with EC2, §4.3.2.3(1), (4.18)) :

τn = – 0.15 σnd > 0. for nd < 0. [MPa] (15F)

τn = – 0.5 σnd ≤ 0. for nd ≥ 0. [MPa] (15G)

with σnd = nd / h – medium normal stress in cross-section due to the isolated effect of the normal force nd (note the difference to NEN 6720 symbolism: σnd ≡ – σ 'bmd). To prevent that 1 becomes negative for high tension normal forces, (15E) is applied in the consistent form :

τ'1 = max (τ1 + τn , 0.) [MPa] (15H)

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There is another favourable effect of compression normal force which manifests itself in case of low bending stresses. If in the cross-section the bending stress is limited :

σbd,max < 0.25 fbr [MPa] (15I)

then NEN 6720, §8.2.3.3 allows to consider the first principal stress σI instead of the shear stress 1 (compression is negative!) :

σI = (τd² + σbd² / 4) + σbd / 2 [MPa] (15J)

In (15J) the symbols d and σbd represent variable shear and bending stress (concrete), respec-tively, in the section parallel to the neutral plane where the principal stress σI attains its maximum value. NEDIM searches for such section iteratively. Note that (15J) has general validity for both tension and compression normal forces. NEN 6720, §8.2.3.3 (“Toelich-ting”) focuses to compression stresses σ 'bd only; such constraint is redundant since (15J) fits consistently to normal tension, too.

If the condition (51I) is met and σI (15J) does not exceed the concrete tension strength :

σI ≤ fb [MPa] (15K)

then NEN 6720 does not require any further shear proof ! The cross-section resistance to shear is taken as granted, without shear reinforcement.

Shear Proof according to DIN 1045-1 07.2001 DIN 1045-1 was implemented into NEDIM with EPW 3.0. The DIN 1045-1 design branch follows the EC 2 thread in dealing with the problem of variable strut inclination method and considering of the Shear Effect. There are, however, some interesting differences to EC 2 and ÖNORM B 4700, which have excessively been dealt with in [22]. The most distinguishing feature from other Norms is that the lower limit of the strut inclination θ is determined by DIN 1045-1, §10.3.4(3), formula (73) :

θmin = arccot ((1.2 – 1.4 σc,d / fcd) / (1. – vRd,c / vEd)) [-] (16)

which, nevertheless, is limited by θinf = 18.4° ≤ θmin. For meaning of symbols in (16) see [13]. The NEDIM θ - iteration starts with θmin (16) and runs until θmax = 45°.

Shear Proof according to GBJ 10-89 GBJ 10-89 was implemented into NEDIM with EPW 3.1. It presents, as a whole, some interesting reinforcement concrete design ideas. As to the shear proof, its method resembles that of CSN/STN 73 1201 (stress proof in an oblique section). The notions of the variable strut inclination and of the Shear Effect are, however, not implemented into the GBJ 10-89 proof approach. On the other part, there are some advanced ideas as to the consideration of the type and the position of loads on the structure – compare with NEN 6720, coefficient λc. Typically, these specifications, focusing to 1D members, are hardly to follow in the ESA PT 2D design. Thus, corresponding upper limit provisions have been active in the NEDIM algo-rithm to ensure that the security requirements are not underestimated in extreme load situati-ons.

Shear Proof according to BS 8110 BS 8110 defines the ultimate shear stress vc, i.e. the basic shear resistance of cross-section


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without shear reinforcement, by the generator formula of Table 3.8; in terms of NEDIM :

vc = 0.79 ρeff1/3 max ((400 / d)1/4, 1. ) / γm [MPa] (161)

with ρeff – effective longitudinal reinforcement percentage according to the following re-striction :

ρeff = min (100 as / d, 3.) [%] (162)

BS 8110, Table 2.2 assesses the partial safety factor for concrete shear strength without shear reinforcement in (161) by a very special value of γm = 1.25 ! With (161), the Norm Table 3.8 is of no practical meaning for NEDIM; perhaps, for numerical check of (161).

The design stress v = vd / d is checked against the shear resistance vc to specify the Shear Region : (a) SR0: v < vc ; (b) SR1 : vc ≤ v ≤ vc + 0.4 ; (c) SR2 : vc + 0.4 < v < min (0.8 √ fcu , 5.) [MPa]. For SR1 the minimum shear reinforcement formula is specified in Table 3.16 ; in terms of NEDIM :

asv = 0.4 / (0.95 fyv,eff) [m²/m²] (163)

with fyv,eff = min ( fyv, 460.) [MPa] – effective strength of stirrup reinforcement (for limitation of fyv to 460 [MPa] – see §3.5.5.1). Note that the amount of 0.4 [MPa] in (163) represents the scope of the SR1 interval (Pt (b) above). The required shear reinforcement amount in SR2 is specified by a formula in Table 3.16 ; in terms of NEDIM :

asv = (v – vc) / (0.95 fyv,eff) [m²/m²] (164)

In (164) the difference term (v – vc) reveals the reduction effect of the basic shear resistance ("Abzugswert" by Mörsch). This approach of BS 8110 resembles the so called “Standard method” of EC 2, §4.3.2.4.3 (see above).

The cross-section becomes non-designable, i.e. transits to SR3, if the design shear stress v

exceeds the lesser value of 0.8 √ fcu and 5. [MPa] (Table 3.16) :

v > min (0.8 √ fcu, 5.) [MPa] (165)

For M40 is 0.8 √ fcu = 5.06 [MPa]; thus, the paired value of 5.0 [MPa] in (165) is the effective strength limit for concrete grades M40 and higher. A comparison with IS 456, Table 20 shows that the corresponding maximum effective strength for M40 is τc,max = 4.0 [MPa]; this makes 80% of the BS 8110 limit. However, since IS 456 allows for 2D members only half the value of τc,max as effective strength, the conclusion is justified that IS 456 is much more conservative than BS 8110 (40%), concerning the ultimate shear resistance !

Shear Proof according to ÖNORM B 4700 ÖNORM B 4700 was implemented into NEDIM with EPW 3.40. The overall design method of ÖNORM B 4700 is based on EC 2. This NEDIM design branch follows the same prin-ciples of dealing with the variable strut method and the Shear Effect like EC 2 or DIN 1045-1, respectively.

Shear Proof according to BAEL 91/99 BAEL 91/99 was introduced into NEDIM with EPW 3.50. The shear proof method applies, like most of the Norms implemented, the concept of the truss analogy (“Fachwerkmodell”) devised by Mörsch. However, there are some features strongly distinguishing the BAEL

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approach from other Norm branches. They will become obvious when reviewing the characteristic items of the BAEL shear proof, which is here described, adapted to the NEDIM 2D design terminology and symbolism, in detail: Shear strut inclination. Differently from other Norms, BAEL allows for the constant inclination angle 45° only. It resembles the constant strut inclination variant of EC 2 (compare (173) and (145)).

Inner forces lever z. BAEL strictly stipulates the use of the constant lever z = 0.9 d for shear proof. Since the most important BAEL shear proof formulae refer explicitly to this relation (i.e. using the term 0.9 d for z) NEDIM keeps to this rule, ignoring the more precise value of zcalc submitted by the preceding m/n design phase.

Shear Region 1 (SR1). The notion of Shear Region "borrowed" from the terminology of DIN 1045 07/1988 can very well be adapted to the BAEL shear proof procedure. The basic shear bearing ability ("Abzugswert" by Mörsch) expressed by the shear stress limit 0 :

τ0 = 0.3 k min (ftj , 3.3) [MPa] (171)

is precisely the SR1 shear stress limit! The values which it assumes for the BAEL concrete strength classes (Table A.6.1,21 [24]) are comparable with those of DIN 1045 07/1988. The coefficient k in (171) articulates the effect of tension/pressure nor-mal force acting in the cross-section upon the stress limit 0. More than that : also the status of cracking (see below) and the state of the joint between old and new part of composed cross-sections can be expressed by assigning the appropriate value to the factor k, which can assume both positive and negative values.

Shear Region 2 (SR2). The representative mean ultimate shear stress u is calculated by the following formula (simplified for 2D structures by setting, as usual, the cross-section width to b0 = 1) :

τu = vd / d [MPa] (172)

The required amount of shear reinforcement asv is then calculated (assumed orthogonal stir-rups, = 90°), by the following formula :

asv = γs (τu – τ0) / (0.9 d fe) [m²/m²] (173)

with s – material security coefficient for steel (1.15) and fe – steel strength [MPa]. It is noteworthy a fact that the shear reinforcement formula (173) for BAEL 91/99 and the cor-responding formula (145) of the s. c. “Standard method” of EC 2 are principally identical, excepting, however, different symbols used.

Two different upper limits are set for u [24], controlled by three shear modes, which are specified in relation to the status of cracking : (a) cracking is considered generally as non-critical (“peu préjudiciable”) for the functionality of structure; (b) cracking is considered as critical (“préjudiciable”); (c) cracking is considered as very critical (“très préjudiciable”). These characteristics, distinguishing BAEL 91/99 from other Norms, are controlled on program input! On the other part, there are no requirements on the continuity of reinforce-ment, as with some Norms of the EC family.

τu ≤ min(0.2 fcj / γb , 5.) [MPa] (cracking status (a)) (174)

τu ≤ min(0.15 fcj / γb , 4.) [MPa] (cracking status (b) & (c)) (175)

In over-pressed sections (i.e. zero axis is outside the section area) the shear proof can be considered as delivered if the shear stress u does not exceed the following limit value :


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τu ≤ min (0.06 fcj / γb , 1.5) [MPa] (176)

with b – the material security coefficient of concrete (b = 1.5) and fcj – the actual concrete strength [MPa].

Shear Region 3 (SR3). The cross-section is declared as non-designable for shear if the upper shear limit (174) or (175), respectively, is not met by u.

Shear Proof according to SIA 262 SIA 262 principally follows EC 2. The basic shear resistance vRd is estimated by

vRd = kd τ0 d [MN/m] with kd = 1 / (1 + kv d) (181)

In the formula for kd (181), the coefficient kv depends on the expected deformations; for NEDIM the application of the maximum value.

kv = 3.0 (182)

is only plausible. The virtual shear strut inclination can freely be chosen within the interval (no control by normal force like SIA 162) :

25° ≤ α ≤ 45° (183)

NEDIM starts the iteration process with 25°, unless the fixed strut option has been chosen (see the paragraph on SIA 162).

Shear Proof according to EN 1992-1-1:2004 EN 1992-1-1:2004 is a novella of the preliminary European Norm ENV 1992-1-1: 1991 (EC 2). Compared with EC 2 the basic shear resistance vRd,c is defined by an improved formula containing special coefficients CRd,c, vmin and k1 (bw = 1) :

vRd,c = [CRd,c k (100 ρI fck)1/3 + k1 σcp] × d [MN/m] (191)

The value of vRd,c is restricted by the lower limit formula

vRd,c = [vmin + k1 σcp] × d [MN/m] (192)

The values of CRd,c, vmin and k1 may be found in the National Annexes. The recommended values are CRd,c = 0.18 / γc and k1 = 0.15 (NEDIM defaults), while vmin is, by default, calculated from :

vmin = 0.035 k √ (k fck) [MN/m] (193)

The formula for shear strut resistance vRd,max contains a new coefficient αcw (bw = 1) :

vRd,max = 0.5 αcw z ν1 fcd sin (2θ) [MN/m] (194)

Here αcw takes account of the state of stress in the virtual shear strut by distinguishing three intensity levels of axial compression, thus unifying the estimation of the effective value of vRd,max by one formula. The strength reduction coefficient ν1 is more diversified than corres-ponding coefficient ν in EC2, formula (4.21) – see (148) above.

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Shear Proof according to IS 456 IS 456 defines the design shear strength of concrete τc without shear reinforcement by Table 19; that is a tabled functional dependence of τc on concrete grade and longitudinal reinforcement percentage. NEDIM employs, however, an exact analytical formula for τc, which generates the values of Table 19 ; in terms of NEDIM :

τc = 0.85 √ (0.8 fck) [√ (1. + 5 β) – 1.] / (6 β) [MPa] (201)

Notice the analogy to EC2, §4.3.2.3, formula (4.18) → (145), term vcd and BS 8110 → (161), term vc. The auxiliary parameter β in (201) is defined by the formula :

β = 0.8 fck / (6.89 ρt) [-] (202)

with ρt – percentage of longitudinal tension reinforcement in cross-section, yet not less than 0.15% (basic value of minimum tension reinforcement) and not more than 3.0% :

ρt = max (min (100 ast / d, 3.) , 0.15) [%] (203)

Following §40.2.1.1, in 2D shear proof, the tabled values τc are augmented by the factor k, depending on the cross-section depth d ; thus, the effective value τc,eff is used instead :

τc,eff = k τc [MPa] (204)

The function k = f (d) is also tabled (§40.2.1.1), assigning to k values from the interval <1.0 ;

1.3> : the lowest value, k = 1.0, corresponds to d ≥ 300 [mm], whereas the maximum, k = 1.3, is assigned to d ≤ 150 [mm]. The dependence k = f (d) is a linear function. NEDIM employs the algorithmic formula (d is inserted in [m]) :

k = max (min (1.6 – 2d, 1.3) , 1.) [-] (205)

instead of linear interpolation of the tabled values (note the factor 2 with d ; compare with (143) and (153)).

The design stress τv = vu / d is checked against τc,eff (201) to determine the Shear Region. In SR1, i.e. if the effective design shear strength of concrete τc,eff is less than the shear impact τv

:

τc,eff ≤ τv [MPa] (206)

no shear reinforcement is required. The cross-section resists the shear impact by the con-crete strength fck in combination with the so called “dowel effect” of the longitudinal rein-forcement, expressed by the factor β (202).

In SR2, i.e. if the condition (206) is not satisfied, transversal shear reinforcement (vertical stirrups are assumed by NEDIM) is required to ensure the cross-section resistance to the shear impact vu (§40.4); in terms of NEDIM :

asv = αs (τv – τc,eff) / fyv,eff = αs τvs / fyv,eff [m²/m²] (207)

with fyv,eff = min ( fyvk, 415.) and αs = 1.15 – partial safety factor of steel material (see §26.5.1.5). Design formula (207) demonstrates the fact that the basic concrete shear strength τc,eff reduces the design value of shear stress τv to effectively τvs = τv – τc,eff (s. c. “Abzugswert” by Mörsch), like EC2 (145), NEN 6720 (159) and BS 8110 (164).


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Following §40.2.3.1, the cross-section becomes non-designable, i.e. transits to SR3, if the design shear stress τv exceeds half the value (specially for 2D) of τc,max of Table 20 (§40.2.3.1) :

τv > τc,max / 2 [MPa] (208)

The maximum shear stress τc,max, i.e. the ultimate shear resistance of a 1D cross-section, is a function of concrete strength. For 2D members, half the value of τc,max is applicable, as expressed by (208). Table 20 assigns to τc,max values from the interval <2.5, 4.0>, depending on concrete grades M15 to M40; the lowest value τc,max = 2.5 corresponds to M15 (and lower strengths), whereas the highest value τc,max = 4.0 [MPa] belongs to M40 (and higher strengths). NEDIM applies a linear interpolation routine to Table 20 to assign correspond-ing τc,max value to any concrete grade.

Shear Proof according to ACI 318M-05 ACI 318M-05 assigns to the shear proof the strength reduction factor of = 0.75, i.e. the nominal shear resistance is factorized to 75%. The basic shear proof formula assumes the form :

vn = (vc + vs ) ≥ vu [MN/m] (211)

with – security coefficient for shear, vn – total nominal shear strength, vc – nominal shear strength provided by concrete, vs – nominal shear strength provided by reinforcement and vu – factored shear force in the cross-section considered. The shear strength vc may be calculated either by the simplified or the (more) detailed method. Here, only the simplified formula including the axial compression effect is presented (bw = 1) :

vc = 0.17 [1 – nu / (14 Ac)] √ fc' d [MN/m] (212)

with nu – axial pressure force and Ac – gross concrete section and fc' – specified compressive strength of concrete. Formula (212) implies primarily nu ≤ 0 (pressure axial force increasing shear strength!). For tension axial forces, there is just a statement saying that in case of "significant" axial tension vc → 0. For a coded algorithm it is obligatory to have strictly specified what means "significant". NEDIM deals with this uncertainty mathematically consistently : allowing (212) also for nu > 0 (tension axial force decreases the shear strength!) an algorithmically applicable formula is constituted :

vc = max { 0.17 [1 - nu / (14 Ac)] √ fc' d , 0. } [MN/m] (213)

which covers all values of vc 0. Thus, the applied specification of the notion "significant tension force" means such values of nu > 0 which turn vc (212) in a non-positive value, i.e. nu > 14 Ac.

The detailed method specifies two different formulae for nu > 0 and nu < 0, thus avoiding in-terpretations as with (213), although the unspecified term "significant tension" is still used. NEDIM offers both proof methods with equal preference.

Shear Effect The Shear Effect phenomenon, originally introduced by SIA 162, §3 24 206 into the engineering practice, is a vital concept of the 2D reinforcement design. Its character makes it a link between the (m/n) design and the shear proof. The state of stress in a Plate or Shell cross-section due to the shear force can be compared, in some sense, with the situation in a Wall design model under the effect of membrane shear forces! There the concrete has to withstand membrane shear by both or two of three, respectively, courses of the reinforcement net, which are stiffened against “lost of shape” by the virtual strut, i.e. by pure concrete resistance. The analogy between the membrane state of stress and the state of bending shear in a Plate or Shell

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cross-section has been described in [17]. Typically, one resisting reinforcement course is represented by the shear reinfor-cement (stirrups); the other reinforcement course is represented by the longitudinal reinforcement. However, as shown in [17], in high cross-sections (typically in deep 1D members) the horizontal reinforcement associated with the shear force has to be arranged along the cross-section height (at bar-web faces).

Due to the Shear Effect even the bending pressure zone may be over-tensioned, so that tension reinforcement appears as final result where there no tension reinforcement would be expected [17]. Also, in regions near to end supports, where bending moments approach zero while the shear stresses attain extreme values, the difference between the design results disregarding the interaction m/n ↔ v and the results comprising the Shear Effect increments of the longitudinal reinforcement may be surprisingly high! Similar observation are regularly made in the design of flat slabs supported by columns (singularity regions [19])

The concept of Shear Effect [17], is basically the same throughout all Norms, since it is based on general mechanical considerations. Since SIA 162 was the first Norm to introduce explicitly this concept into the practice, it is explained here by referring to the stipulations and symbolism of SIA 162 [11]. Additionally, special notes are supplied on other Norm branches.

SIA 162 introduces the concept of Shear Effect in paragraph §3 24 207. The total Shear Effect force ft(v) is calculated by the following formula :

ft(v) = vR cot α [MN/m] (221)

where vR is the required cross-section shear resistance value and – the strut inclination angle. The required stirrup reinforcement amount asv is then defined by

asv = vR tan α / (fy z) [m²/m²] (222)

(compare with (159), (163) , (173)), where z is the inner forces lever from the (m/n) design. We recognize from (221), (222) that the shear reinforcement amount and the Shear Effect force component acting upon the net reinforcement are indirectly pro-portional (tan = 1/cot ). The strut inclination may be chosen, according to §3 24 203, within a quite wide range; in Plates : 25° < < 65°. Since the minimization of the shear reinforcement is the primary goal of the design, the NEDIM design algorithm starts an iteration loop with the lower limit value of min = 25° and, increasing it by 1°, seeks an equilibrium solution establishing the required strut resistance at minimum inclination. From this solution of , the corresponding Shear Effect force component is derived according to (221).

If the constant strut inclination is active as input control or the cross-section is over-tensioned (automatic control), respectively, no iterative estimation of the strut inclination is started : the central value of o = 45° is set. In such a case ft(v) = vR, i.e. the design value of the Shear Effect force equals the shear resistance force vR = γR vd !

The NEDIM algorithm proceeds in 2 algorithmic steps: (1) 1st step is standard design and shear proof as described above; (2) in the 2nd step, the total Shear Effect force is assigned by halves to the upper/ lower reinforcement nets, where it is merged with the bending and membrane forces (mx, my, mxy, nx, ny, nxy), following a genuine NEDIM approach. We abstain from de-scribing this transformation procedure, characteristic for the high performance of NEDIM; for detail, please, refer to [17]. Repeated reinforcement design consistently respecting the Shear Effect increment forces is carried out.

In cases of low shear stress, i.e. when a cross-section pertains to Shear Region 1, no shear reinforcement is required to ensure the cross-section shear resistance.

The concept of Shear Effect is principally associated with SR2, i.e. with the statically required stirrup reinforcement. In SR1 the shear resistance mechanism is assumed to be basically linear-elastic state of stress, where the principal tension stress is resisted by the concrete itself (remarkably, in contradiction to the common assumption that concrete does not withstand tension stress). Thus, in SR1, principally no Shear Effect is to be considered. This fact contradicts, however, to the generally applied constructive approach to assign a Shift to the tension-force diagram of the longitudinal reinforcement both in SR2 and SR1. To provide for diversified control by the user, NEDIM is equipped with advanced possibilities. There are 3 control options for user’s control :

o Shear Effect is not considered at all. For SIA 162 this is, as a fact, an illegal control situation, since there are no other alternatives, like Shift of reinforcement, stipulated. This control option offers also the possibility to carry out “benchmark” test and comparative design calculations freed from the Shear Effect.

o Shear Effect is considered in Shear Region 2 only. This is the standard control for most Norm branches.

o Shear Effect is considered both in Shear Regions 1 + 2. This is non-standard control for most Norm branches (see preceding note).


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There is another factor to be considered. SCIA 162, §3 24 203 presents a closing sentence causing some confusion. It reads : "The strut inclination, once chosen, ought to be considered constant over the whole length of the shear region". It is not clearly defined what is meant by “shear region” here (the notion of Shear Region used by NEDIM has another meaning – see above), yet it is evident that this requirement is concerned with 1D members (beams), where geometric relations are better controllable than in 2D members. This sentence, whatever it may mean, cannot be considered by the NEDIM design (EC 2, DIN 1045-1 and ÖNORM B 4700 do not stipulate such a requirement). The same Article, §3 24 203, determines special strut inclination control for Shells by considering the magnitude of tension or pressure normal forces; this is effectuated by intro-ducing a central inclination value 0. Other Norms do not stipulate this condition at all, or they do it in implicit way, respectively (see DIN 1045-1).

DIN 1045 07/1988, ÖNORM B 4200 and CSN/STN 73 1201 are classical Norms of the pre-Eurocode era. They do not stipulate any Shear Effect rules for direct calculation like SIA 162 or EC 2. Instead, they specify the so called moment/reinforcement Shift, e.g. DIN 1045, §18.7.2, Table 25 “Versatzmaß”. Principally, the Shift concerns both SR1 and SR2. However, NEDIM does not deal with this phenomenon for the 3 Norms named !

EUROCODE 2 introduced the Shear Effect principle according to §4.3.2.4.4(5); as usual, in 1D formulation. In early NEDIM versions (before 1997), the explicit calculation of the impact of shear forces upon the net reinforcement was not implemented; it was to be dealt with on the base of reinforcement Shift, as described by §4.3.2.4.4(6), which allows to apply constructive measures to the main net reinforcement. Actually, NEDIM deals with the 2D Shear Effect on the base of the theory described in [17].

NEN 6720, does not explicitly introduce the concept of Shear Effect by stipulating formulae like (221), (222). However, the formulations of paragraph §8.1.1, where the notion “Shift of the moment line” is linked to the strut inclination θ, lead effectively to the Shear Effect approach, as described in the SIA 162 paragraph above, which can consistently be applied to the NEN 6720 design branch. Interestingly, §8.1.1 explicitly regulates the application of Shear Effect to SR1 (like DIN 1045 7/88) ! In case of low bending stress, as described by the relation (15I) and with σI fitting the condition (15K), the Shear Effect need not to be considered in this domain of SR1. In all other stress situations of SR1 as well as in the whole SR2 the Shear Effect procedure, described by (221) and (222) is in effect : (a) SR1: α ≡ θ = 45°; (b) SR2: α ≡ θ – variable inclination.

DIN 1045-1, §10.3.4(9), Fig.30 stipulates the Shear Effect approach in accordance with EC 2. The alternative constructive approach of reinforcement Shift is referred to by §13.2.2(3). NEDIM applies the concept [17] and enables user control in 3 variants, as described above.

GBJ-10/89 does not stipulate the Shear Effect approach. Actually, NEDIM does not enable the activation of the Shear Effect procedure as alternative non-standard design option.

BS 8110 does not stipulate the Shear Effect approach. Nevertheless, NEDIM enables the activation of this procedure according to EC 2, to §4.3.2.4.4(5) as alternative non-standard design option.

ÖNORM B 4700, §3.4.4.2(15), Fig.19 stipulates the Shear Effect approach in agreement with EC 2. The alternative constructive approach of reinforcement Shift is referred to by §3.4.4.2(16).

BAEL 91/99 does not stipulate the Shear Effect approach. Actually, NEDIM does not enable the activation of this procedure as alternative non-standard design option.

SIA 262, §4.3.3.4.9 (42) stipulates the Shear Effect approach in agreement with EC 2, i.e. for SR2. Like with SIA 162, there is not (explicitly) allowed for the reinforcement Shift . Nevertheless, NEDIM alternatively enables both controls, on user’s account.

IS 456 does not stipulate the Shear Effect approach. Nevertheless, NEDIM enables the activation of this procedure according to EC 2, to §4.3.2.4.4(5), as alternative non-standard design option.

EN 1992-1-1:2004, §6.2.3(7) stipulates the Shear Effect approach as advanced EC 2 procedure. The alternative constructive approach of reinforcement Shift is referred to by §6.2.2(5); however, there are explicitly mentioned „flexure cracks“ only. Article §9.2.1.3(2), which is, consequtively, referred to, recommends explicitly the reinforcement Shift for SR1, too. Thus, NEDIM extends the Shear Effect procedure to SR1, where the constant value θ = 45° is applied (compare with the NEN 6720 approach).

ACI 318M-05 does not stipulate the Shear Effect approach. Nevertheless, NEDIM enables the activation of this procedure according to EC 2, §4.3.2.4.4(5) as alternative non-standard design option.

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Dealing with singularities in Shear Proof The SCIA hotline deals, from time to time, with cases which indicate that the Shear Effect results in a high increase of net reinforcement, hardly acceptable by the engineer’s experience. A close analysis regularly brought to daylight that this reinforcement increase happened along line supports of plates with extreme shear force gradients. This phenomenon has been analyzed in [19] : it is a defect of the Mindlin’s 2D FEM model, which causes the redirection of the well known Kirchhoff’s edge reactions to the 1st finite-element row along the line edge. The issue is demonstrated by Fig. 9, an EPW screen copy of vx results of a rectangular plate with 2 FE mesh variants – (10x10) and (20x20) – to accentuate the singular character of the FEM solution, which depends on the mesh coarseness (vx(10×10),max = 45.3 ~ vx(20×20),max = 70.7 [kN/m]), in comparison with regular maximum vx values (15.4 ~ 17.0 [kN/m]) .

In [19] practical hints were given how to deal with this problem. However, it is primarily no concern of NEDIM, but of the FEM theory and FEM solver.

Conclusion. The 2D design results cannot achieve a higher quality than the results of the FEM analysis. Defects of the FEM solution propagate into the 2D design, affecting the design results. Thus, when unacceptable design results appear, the inner forces of the static analysis ought to be subject to scrutiny prior to checking (and criticising) the 2D design results.

Fig. 9 Mindlin’s shear force edge defect of a quadratic plate (2 plate halves with different FE mesh displayed, with regular course of vx at midspan y = L/2) :

(a) vx singularity on edge y = L with FE mesh (10×10) – upper half of plate

(b) vx singularity on edge y = 0 with FE mesh (20×20) – lower half of plate


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REINFORCEMENT AMOUNT CONTROL

Introduction to reinforcement amount control The statically required reinforcement amount (maintaining equilibrium of external forces and internal resistances) is, generally, not the final design result. There are, dependent on the Norm, some restrictions to be applied to the pure statically required reinforcement. The minimum reinforcement limits assure certain lower amount of the reinforcement to be built in. Usually, every Norm stipulates also a maximum reinforcement limit. The sense of this upper limit is : (a) to ensure good workability of the concrete material during the casting process; (b) to make sure that the steel-concrete medium still works like reinforcement concrete in the sense of the generally applied theories.

The upper/lower limits on reinforcement are, in most Norms, specified relatively as percentages of either the gross section area of concrete or the s.c. effective cross-section area. A modern approach to estimating minimum tension reinforcement, as practised by DIN 1045-1, BAEL 91/99 and SIA 262, focuses on the transient stress situation of the s.c. first crack by stipulating the robustness reinforcement which is aimed at preventing brittle fracture of cross-section when the tensile stresses resisted by concrete in non-cracked state (I) is abruptly redistributed to the reinforcement due to the opening of the first crack.

Maximum reinforcement A typical upper limit set upon the reinforcement amount by the Norms of interest lies within the range <5 %, 9 %> of the gross section of concrete. Most Norms apply the value of 8 %. ÖNORM B 4200, Vol. 9, Table 9 introduces the upper limit in the range <3 %, 6 %> depending on the concrete and steel strength classes.

CSN/STN 73 1201 deals with this restriction in a more complex manner. Not only the overall upper limit of 8 % is checked, also the fulfilment of the limit condition 4 % at both faces separately and, moreover, the condition of 2 % of two congruent rein-forcement courses at both faces simultaneously. The Norms of the Eurocode family stipulate for Walls the limit condition 4 % for the vertical direction; NEDIM checks this condition, however, in all directions.

If the actual upper limit is exceeded by the required “static” reinforcement, the Non-designability Condition NST = 4 is issued (see Chapter Processing of non-designability conditions).

Minimum transversal reinforcement It is a good structural practice to provide at least two reinforcement courses even if (theoretically) only in one of the specified reinforcement directions a reinforcement is required; this is typically the case of parabolic and a class of hyperbolic states of stress. Even if both (or all three) reinforcement courses at one face are active (i.e. calculated as non-zero quantities), it may be required that at least some portion of the “principal” reinforcement is maintained in the transversal direction(s). In NEDIM, this requirement is formulated by the minimum transversal reinforcement percentage requirement.

This restriction ensures that the "secondary" reinforcement courses are provided at least a defined portion of the "principal" reinforcement, no matter what attribute (tension, pressure) they possess.

ÖNORM B 4200, Vol. 9, §3.4.2 and ÖNORM B 4700, §3.4.9. 5(1) & §6.6.1 contain most explicit formulations of this dependence : at least 20 % of the "principal" reinforcement course amount in each other course at one face are required. ÖNORM B 4200, §3.4.2 accomplishes the 20 % rule by the requirement that at least 50% of the values of table 7 are provided.

EUROCODE 2, §5.4.7.3 recommends for the horizontal reinforcement courses of walls at least 50% of the reinforcement amount of the vertical reinforcement (assumed as compressed). To prevent confusion, NEDIM does not control this special requirement ! It has should be controlled by the user on input by setting 50 % as general transversal reinforcement restriction.

NEN 6720, §9.9.3.1 stipulates the value of 20 % as well.

DIN 1045 07/1988, §20.1.6.3(1) and DIN 1045-1, §13.3.2(2) introduce the value of 20 % for s.c. “one-axis spanned plates”, i.e. not binding for all analysis types.

GBJ 10-89, §7.1.5 requires 10 % minimum for s.c. distribution reinforcement of plates. NEDIM extrapolates this stipulation to all three design models as 1st input default.

BAEL 91/99 : 25 % is assigned as 1st input default to all three design models.

SIA 262, §5.5.3.2 requires for plates minimum of 20 % for transversal reinforcement; NEDIM relates this stipulation to Plates and Shells as 1st input default. For walls, §5.5. 4.11 requires that horizontal reinforcement amounts at least to 25 % of vertical reinforcement. NEDIM assigns this value to Walls as 1st input default.

EN 1992-1-1:2004, §9.3.1.1(2) introduce the value of 20% for s.c. “one-way plates”.

CSN/STN 731201, BS 8110 and IS 456 do not take explicitly care of this parameter.

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IS 456 : 0 % is assigned as first input default upon a request of the Indian support.

NEDIM presets generally 20 % as first input default for Norm branches, or to design models, respectively, where there is not stipulated (or known) any special requirement. The input default may be redefined by the user, i.e. also to 0 % (e.g. for tests).

Compression reinforcement in general In reinforced concrete the compression reinforcement is of another quality than the tension reinforcement. The special character of the compression reinforcement requires a specific theoretical and algorithmic dealing.

In preceding paragraphs it was pointed out that the compression reinforcement calculation has characteristic features, compared with the tension reinforcement estimation, and that it implies special approach by the design algorithm. Baumann [1] avoids completely discussions on this topic. Thanks to advanced theoretical and algorithmic development, NEDIM is able to deal with pressure forces and compression reinforcement effectively.

Once the design pressure force associated to the actual reinforcement course is known, (10) yields a rule for calculation of the required compression reinforcement amount in Walls and Shells. Of-course, the minimum compression reinforcement rules discussed below may cause an augmentation of this statically minimum value!

In the Plate design model, the compression reinforcement control is more complex. The upper and lower faces are not strictly separated from each other, since the inner design forces {Zi, Di} (see Fig.4), resisting the design moment mi, act at both faces. If the stress integral of the bending pressure zone resists the design moment mi itself there is no need of compression reinforcement. In the opposite case, the reinforcement at that face must be activated to support the concrete in its bearing function. For the NEDIM algorithm the following rules are binding:

o The reinforcement courses at both faces must have pair-wise identical directions (con-gruent reinforcement). It means also that at both faces simultaneously 2 or 3 rein-forcement courses, respectively, are specified. However, the congruent reinforcement courses may be assigned different ordinal numbers in the cross-section and reinforcement geometry specification.

o The compression reinforcement is calculated for both/all reinforcement courses at one face, i.e. not only for that reinforcement courses which are directly assigned statically required compression reinforcement amount to strengthen the pressure zone.

o The statically required compression reinforcement is marked by a trailing asterisk in the output document if it constitutes the maximum value of all design load cases. If another LC yields a higher reinforcement amount resisting tension or, respectively, the constructive reinforcement percentage assigns a higher reinforcement value (transversal reinforcement), the marking by typographic asterisk is overridden by another symbol ('s' for "structural reinforcement") or becomes simply blank (for the standard case – tension).

The compression reinforcement of the bending pressure zone is, however, restricted by upper limits. As elementary restriction the maximum reinforcement percentage limit of the whole cross-section (see above) is active; yet this restriction is seldom the cause of non-designability. DIN 1045 07/1988, §17.2.3 restricts the bending pressure zone reinforcement amount a maximum of 1 % of the gross section. SIA 162, §3 24 16 limits the force assigned to the compression reinforcement to 50 % of the pres-sure inner force, by which the concrete resists the bending moment in the pressure zone (i.e. not more than 33.3 % of the total pressure inner force). Is this limit exceeded by the statically required amount of compression reinforcement, the non-design-ability Flag NSt = 2 is set (see Chapter Processing of non-designability conditions).

This “50% restriction” of SIA 162 has been applied as first input default to all NEDIM Norm branches which do not stipulate any restriction upon the compression reinforcement in bending pressure zone, thus, excepting DIN 1045 07/1988.

Minimum compression reinforcement The primary goal of providing minimum compression reinforcement to structural members subject to prevailing compression axial forces is to give slender members a structurally based minimum security against buckling; generally, it is a good practice, even if there is, prom static point of view, no reinforcement required.

According to most Norms the minimum compression reinforcement is calculated as percentage of gross section, i.e. (cross-section width : bc = 1.0) :

asc,min = c h [m²/m] (231)

with c – minimum compression reinforcement ratio and h – gross section height. (231) makes it clear that large cross-sections are assigned absolutely more minimum compression reinforcement than slender ones. In bulky cross-sections high


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absolute minimum compression reinforcement amounts are provided even if the pressure stresses are insignificant. Thus, (231) ignores the real state of stress of the cross-section !

DIN 1045 07/1988 introduces the notion of the statically required cross-section. It is the minimum part of concrete cross-section reinforced according to the proportionality rule (231) which would still resist the normal force. If the gross section has a greater area than statically required it is assigned a lower reinforcement percentage than by (231). According to the symbolism of DIN 1045 07/1988 the minimum compression reinforcement formula assumes the form :

asc,min = – γ n / (ßR /μc + ßs) [m²/m] (232)

with βR – concrete strength, βs – steel (compression) strength, n – design (nd) or virtual (nvirt) compression normal force (see Fig. 7a,c); n is factored by the security coefficient s = 2.10. Paragraph §25.5.5.2(2) stipulates for 2D structures the minimum percentage of 0.50%, i.e. c = 0.005.

Fig. 10 enables an insight into the very different approaches of DIN 1045 07/1988 and the EUROCODE family Norms, respectively, here compared with the novella DIN 1045-1 (for 2D structures). The modern concept follows the elementary, less efficient rule (231).

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Fig. 10 Comparison of the minimum compression rules : scheme of the 1st crack DIN 1045 07/1988 – sophisticated, efficient reinforcement rule according to (232)

DIN 1045-1 (Novella 2001/5) – elementary rule according to (231)

ÖNORM B 4200, Vol. 9, §8.2 follows (232) by issuing the minimum percentage acc. to Table 8. The value c = 0,005 is preset as 1st input default.

EUROCODE 2, §5.4.1.2.1(2) estimates the minimum compression reinforcement by a two-term condition, which also takes respect to the magnitude of the normal force. The EC2 formula (5.13) reads in NEDIM symbolism (bc = 1.0) :


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asc,min = max (0.15 nvirt / fyd , 0.003) × h [m²/m] (233)

The value c = 0,003 may be modified by the pertinent National Document. The paragraph §5.4.1.2.1(2) has been given preference to §5.4.7.1(1), concerning specially walls by stipulating c = 0.004 (i.e. the simple approach acc. to (232)). However, the user may apply his control by setting this value on input (see “Closing notes” on this paragraph below).

CSN/STN 73 1201, §3.1.4.2, (29) & §3.1.4.4, (37), (38) follow the elementary relation (231), however, subjecting the minimum reinforcement ratio c to the “slenderness condition”

c = le / h × 10

-

4 (234)

with le – effective (buckling) length of member and h – cross-section height, c being restricted to the interval [0.0005, 0.0025], i.e. percentage [0.05%, 0.25%]. A significant difference to other Norms is that c is related to one face, thus the total minimum reinforcement in cross-section is to be doubled (see “Closing notes” on this paragraph).

SIA 162, §4 44 3 follows (232) with percentage 0.60%. NEDIM presets the value c = 0,006 (twice that of EC 2, however, for statically required cross-section !) as 1st input default.

NEN 6720 does not care about this sort of minimum reinforcement at all, thus allowing, as a fact, for plain concrete in compression members. NEDIM makes it, however, possible for NEN 6720 users to follow the elementary approach (231); NEDIM presets c = 0 as 1st input default to (231)!

DIN 1045-1, §13.7.1(3) estimates basic percentage of 0.15%, i.e. c = 0,0015, which is to be doubled for “slender walls” acc. to §8.6.3 or for higher axial stresses. The latter condition is, in NEDIM terms (bc = 1.0) :

nvirt = 0.3 fcd h [MN/m] (235)

If nvirt meets (235) the percentage is set 0.30% (of gross section), i.e. NEDIM presets c = 0,003 as 1st input default to (231).

GBJ 10-89, §6.1.15, Table 6.1.15 estimates the percentage 0.40%, i.e. c = 0,004, for all concrete classes up to C60. NEDIM assigns this value as 1st input default to (231).

BS 8110, Part 1, §3.12.5.3 estimates the percentage 0.40%, i.e. c = 0,004 (Table 3.25, steel grade independent). NEDIM assigns this value as 1st input default to (231).

ÖNORM B 4700, §3.4.9.3 estimates the percentage 0.28%, i.e. NEDIM presets c = 0,0028 1st input default to (231).

BAEL 91/99, §A.8.1,21 estimates the percentage 0.20%, i.e. NEDIM presets c = 0,002 as 1st input default to (231).

SIA 262, §5.5.4.3 maintains the approach of the predecessor edition SIA 162, i.e. follows (232) with the percentage 0.60% related to the statically required cross-section. NEDIM presets the value c = 0,006 as 1st input default.

EN 1992-1-1:2004, §9.6.2(1)+(2) estimates the percentage 0.20% (declared as modifiable by NAD), i.e. NEDIM presets c = 0,002 as 1st input default to (231).

IS 456 does not care about minimum compression reinforcement. NEDIM follows, however, the elementary approach (231) : c = 0 is preset as 1st input default to (231).

ACI 318M-05, §10.9 estimates the percentage 1.0%, i.e. c = 0,01. This very high value, compared with European Norms, may be explained by the status which has “seismic risk” in US building industry. However, §10.8.4 allows to base the minimum compression reinforcement calculation on a reduced effective area (bw = 1.0) :

eff Ac = max (stat req A, h / 2) [m²] (235)

In (235) stat req A means the statically required cross-section discussed with DIN 1045 07/1988. ACI stipulates thus a similar

approach as expressed by (232). However, here the value of stat req A is limited by 50% of the gross section ! Thus, the effective pressure reinforcement percentage may attain the minimum value of 0.5% !

For the Wall design model the considerations on this topic are complete, since the normal force nd in formulae (23#) is identical with the design force assigned. In Shells the situation is more complex. The normal pressure force acts in combination with bending moment. It ought to be defined a limit state of stress to which minimum compression reinforcement is assigned. This is made by distinguishing three types of state of stress : (a) prevailing axial compression; (b) prevailing bending; (c) prevailing axial tension. NEDIM designs min asc to type (a) only !

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Mechanically, the limit is expressed by the critical eccentricity ecr of the virtual normal force nvirt (see paragraph Design of Shells, Fig. 7c) acting upon the cross-section. The value of ecr is stipulated by ÖNORM B 4200/4700 and by DIN 1045/1045-1 when assigning the attribute "member subject to prevailing flexion" to a class of members. The limit condition is :

e = abs (m / n) > εcr h = ecr [m] (241)

with h – total cross-section height and m, n – bending moment and axial force; for NEDIM are mvirt, nvirt to substitute. ÖNORM B 4700, §3.4.3(3) stipulates εcr = 2.0, DIN 1045-1, §3.1.18 sets εcr = 3.5. The lesser value εcr = 2.0 is applied as 1st input default to all Norm branches, except DIN 1045/1045-1, to which the value εcr = 3.5 applies.

Is the eccentricity condition (241) not met, i.e.

e = abs (m / n) ≤ ecr [m] (242)

the actual state of stress is declared either prevailing axial tension – if nvirt > 0 or prevailing axial compression – if nvirt < 0, respectively. The latter constitutes the case (a), as classified above, i.e. subject to minimum compression reinforcement design.

In Shells the situation is, again, more complex than in Walls (compare in Fig. 7a,c). The design force nd, assigned to the reinforcement direction at actual face, participates in the direct reinforcement design according to (6), if nd > 0 or (10), if nd < 0, respectively. However, the sign of nd has no impact upon the minimum compression reinforcement control ! If nvirt < 0 and (242) is true the minimum compression reinforcement is evaluated acc. to one of formulae (23#). If it is higher than the reinfor-cement acc. to (6) or (10), respectively, it becomes the new statically required reinforcement amount, and the corresponding reinforcement attribute is set. Is this value the highest of all design load cases, the minimum reinforcement attribute is main-tained. A trailing 'm' as symbol for minimum compression reinforcement is attached in the output document to the reinfor-cement amount value.

Closing notes

o If the input parameter c in (231) or (232) is set 0% on input, no calculation of the minimum compression reinforcement takes part, whatever Norm is being dealt with; pure statically required reinforcement is then estimated.

o In all Norms the minimum reinforcement ratio c is, by definition, assigned to the gross section, excepting CSN/STN 73 1201, where it is related to one face. Thus, for Plate and Shell the resulting min asc is assigned 50/50 to both faces in all Norm branches except CSN/STN, where there min asc has to be doubled.

o NEDIM assigns additionally, i.e. as non-standard provision, the control according to the basic proportionality relation (231). The user is thus enabled to control the minimum compression reinforcement by its own value of c, either alternatively or ad-ditively to the standard Norm control.

Minimum tension reinforcement Some of the Norms implemented in NEDIM stipulate no requirements upon the minimum amount of the (statically required) tension reinforcement, since it is, generally, controlled by the serviceability proofs. However, Norms which do not comprise any special serviceability proof requirements control explicitly the minimum amount of tension reinforcement. SIA 162 prescribes general constructive rules. ÖNORM B 4200 introduces a control in relation to the concrete and steel strength classes.

Generally, the minimum tension reinforcement amount is calculated acc. to the following elementary relation (which is an analogy to (231) for tension):

ast,min = μt h [m²/m] (251)

with t – minimum tension reinforcement ratio and h – gross section height. As guideline for t required by the Norms the value of t ≈ 0.001, i.e. percentage 0,1% of gross section area may be considered.

DIN 1045 07/1988 does not specify any rules for minimum tension reinforcement in the meaning of the proportionality assignment (251).

ÖNORM B 4200, Vol. 9, §6.5.1 follows principally (251), however, substituting the effective static height d (i.e. axial distance of reinforcement bar to opposite face) rather than the gross section height h :

ast,min = μt d [m²/m] (252)


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Direct consequence of this approach may be different values of min as,t calculated for individual reinforcement courses, which regularly imply different static heights. To avoid confusion with users expecting equal values for all courses at one face, NEDIM applies the static height of the outermost reinforcement course for all courses. Table 7 stipulates the values of t depending on steel and concrete strength classes.

EUROCODE 2, §5.4.2.1.1(1), (5.14) presents a double-term limit condition. Additionally to the condition (251) a parametric term is introduced, referring to fyk (bt = 1.0) :

ast,min = max (0.60 / fyk , 0.0015) × d [m²/m] (253)

In (253) d denotes, like with ÖNORM B 4200 (252) , the reference static height of the outermost reinforcement course at actual face rather than h in (251).

CSN/STN 73 1201, §3.1.4.2, (29) & §3.1.4.3, (35) follow the elementary relation (251), however, specifying the minimum reinforcement ratio t as function of concrete design tension strength Rbtd and the steel design tension strength Rsd :

t = Rbtd / (3 × Rsd) (254)

SIA 162, §3 33 41 requires a minimum tension reinforcement as provision against the brittle fracture of concrete in case of 1st crack's appearance (see SIA 262 below). This control has not been implemented into the SIA 162 design branch. Instead, the standard proportionality control by (251) has been active, t = 0 being the 1st input default; the user may define his own control by setting t > 0.

NEN 6720 does not specify explicit requirements upon the minimum tension reinforcement according to (251); the approach applied is more sophisticated [4] : low tension reinforcement amounts are checked against some linear stress conditions and, if they are not met, the statically required tension reinforcement has to be augmented by 25%. This procedure has the character of an elementary crack control.

DIN 1045-1, §13.1.1(1) introduces a new notion of minimum tension reinforcement : robustness reinforcement. Its declared task is to resist the transition inner forces arising abruptly in reinforced concrete cross-section when the 1st crack appears. More specifically: the robustness reinforcement is aimed at preventing brittle fracture, i.e. sudden failure without warning. The reinforcement steel may be, in this exceptional case, exploited up to the characteristic strength, i.e. σs ≤ fyk (compare with SIA 262, where no more than fsd,. i.e. the design strength, is allowed).

Article §13.1.1(1) restricts this control to "members subject to prevailing flexion". This class of stress situations meets the condition (241), i.e. bending with relatively small normal force nvirt 0.

The Authors of DIN 1045-1 preferred another interpretation of the notion brittle fracture than the natural approach described by BAEL 91/99 and SIA 262 (see below, (263)). It is stipulated in Heft 525 DAfSt [32] by the relation (using NEDIM symbolism) :

mr = (fctm + nvirt / h) × h² / 6 [MNm/m] (261)

with fctm – mean concrete tension strength and h – cross-section height. The idea of this quite unusual approach is illustrated by Fig. 11. The crack moment mr applied to the estimation of the robustness reinforcement is a virtual moment which, if

applied to the cross-section (without any axial force!), produces the maximum tensile strength σr = (fctm + nvirt / h) in the outermost fibres.

Fig. 11 DIN 1045-1 scheme of the 1st-crack moment mr (non-developed crack) :

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(a) cross-section subject to design inner forces (m, n); (b) pure bending ;

(c) n > 0 : augmenting mr ; (d) n < 0 : diminishing mr (down to zero)

Since this minimum tension reinforcement control is the only one stipulated by DIN 1045-1 in ULS (besides the crack control in SLS) it is indispensable for NEDIM to introduce additionally the control by the elementary proportionality relation (251). Thus, the user may define his own control setting t > 0 to (251), replacing or superposing the Norm control value, respectively; as a fact, for Walls the user control by t > 0 is the only minimum tension reinforcement control to be activated for DIN 1045-1.

GBJ 10-89, §6.1.15, Table 6.1.15 estimates the percentage 0.15%, i.e t = 0,0015, for all concrete classes up to C35, or 0.20%, i.e. t = 0,0020, for higher concrete classes, respectively. These are 1st input defaults set automatically by NEDIM. The user may redefine the t values on input.

BS 8110, Part 1, §3.12.5.3, Table 3.25 estimates the percentage 0.24%, i.e. t = 0,0024 for steel grade 250, or 0.13%, i.e. t = 0,0013, for steel grades 460 & 500. NEDIM distinguishes the control for fy ≤ 460 and fy > 460 [MPa]. These are 1st input defaults set automatically by NEDIM. The user may redefine the t values on input.

ÖNORM B 4700, §3.4.9.4(1), (61) pursues the double-term approach of EC 2, yet referring to fyd instead of fyk (bt = 1.0) :

ast,min = max (1.22 / fyd , 0.0028) × ht [m²/m] (262)

ht denotes (for rectangular cross-section, i.e. also for 2D members) the half value of the cross-section height : ht = h / 2 (i.e. the height of tension zone in case of pure bending); t = 0.0028 (i.e. 0.28%).

BAEL 91/99, §A.4.2,1 requires a minimum tension reinforcement as provision against the brittle fracture of concrete in case of 1st crack's appearance. Article §B.6.4 requires, additionally, the minimum ratio of t = 0.001 (251), which superposes the result calculated acc. to §A.4.2,1. BAEL stipulates this control globally; it is thus not restricted to "prevailingly flexural members" like DIN 1045-1. NEDIM seeks the state of 1st crack by simultaneously augmenting both inner forces {m, n} (if non-zero), main-taining the normal force eccentricity, i.e. the ratio m/n, until the 1st crack appears due to exhaustion of the concrete resistance to tension, which, in this very special case, is not supposed to be zero from the begin on (as in pure ULS design) :

m / n = mr / nr = e = const »» m → mr & n → nr (263)

In (263) {m, n} symbolize (Fig. 7) either md (Plate : nd ≡ 0 ~ e → ), or nd (Wall : md ≡ 0) or {mvirt, nvirt} (Shells). To avoid certain numerical instability effects with vanishing m or n, some practical control provisions have been implemented into the NEDIM algorithm. The user may redefine the �t value acc. to §B.6.4 and also suppress the brittle fracture control by §A.4.2.1.

Three typical states of stress at 1st crack occurrence (an "undeveloped crack" is supposed to open due to some not explicitly specified (local) overstressing, e.g. also due to undefined temperature load, creep, shrinkage and other causes) are shown in Fig. 12, where the affinity of the explicitly defined load {m, n} and the envisaged 1stcrack combination {mr, nr} is demonstrated as basic assumption of the brittle fracture control. In the explanation on §A.4.2.1 [24, Page 29], an example of pure axial load is given, which leads to the simple formula (using here �t instead of ρ) :

μt = ftj / fe → ast,min = μt h [m²/m] (264)

NEDIM calculates, however, consistent values of ast,min for all states of stress between "pure bending" and "pure axial tension" (264).


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Fig. 12 BAEL 91/99 & SIA 162 scheme of the 1st-crack inner forces combination (mr, nr) : (a) pure bending; (b) cross-section subject to non-zero design inner forces {m, n}; (c) prevailing tension → disconnection crack

SIA 262, §5.5.3.2 & §4.4.2.3.9, Table 16/1 & §4.4.2.3.10, Fig.31 requires a minimum tension reinforcement as provision against the brittle fracture of concrete in case of 1st crack’s appearance. SIA 262 stipulates this control globally, for all design models and stress situations. The calculation process follows (263), i.e. principally the same rules as BAEL 91/99. However, SIA 162 does not simultaneously stipulate the control by t (251), thus t = 0 is 1st input default. Since NEDIM V04.5.2.0, the user may, nevertheless, define his own control by (251), setting t > 0, the result superposing or replacing the Norm value, respectively.

EN 1992-1-1:2004, §9.2.1.1(1), formula (9.1) stipulates, like EC 2, a double-term limit condition. Additionally to the condition (251) a parametric term is introduced, referring to fctm and fyk (bt = 1.0) :

ast,min = max ( 0.26 fctm / fyk , 0.0013 ) × d [m²/m] (265)

In (265) d denotes, like with EC 2 (253), the reference static height of the outermost reinforcement course at actual face rather than h in (251).

IS 456, §26.5.2.1 estimates the percentage 0.15%, i.e. t = 0,0015 for steel class Fe 250, or 0.12%, i.e. t = 0,0012, for steel classes Fe 415 & Fe 500. NEDIM distinguishes the control by fy ≤ 415 and fy > 415 [MPa]. These are 1st input defaults set au-tomatically by NEDIM. The user may redefine the t values on input.

ACI 318M-05, §10.5.1 stipulates direct calculation of ast,min analytically, referring to fc', fy and d. The ACI formula (10-3),

rewritten for t (bw = 1.0):

t = max ( 0.25 √fc' / fy , 1.4 / fy ) (266)

NEDIM presets the actual value of t calculated from (266) – for both faces individually – as the 1st input default. The user may redefine the t values on input.

Overall minimum reinforcement Any of the Norms dealt with by NEDIM do not raise requirements upon the amount of the overall (base) reinforcement.

NEDIM keeps this option open by presetting the first input default of this parameter equal to 0%.

There is, however, another similar control of the s.c. basic reinforcement in NEDIM : the user may define (member-wise) the desired minimum provided reinforcement in absolute values, e.g. envisaging standard webs like Q384 (DIN) etc.

Minimum shear reinforcement All Norms introduce the notion of minimum shear reinforcement for 1D members. Actually, only EC 2, §5.4.2.2(5) stipulates minimum shear reinforcement for 2D structures, referring to Table 5.5 for 1D members and requiring 60% of the table values for 2D structures as min asv ! The minimum shear reinforcement for 2D structures applies in Shear Region 2 only, i.e. if shear reinforcement is required at all.

CSN/STN 73 1201 and BS 8110 introduce the notion of structural minimum reinforcement, see paragraph Shear Proof methods above. Yet this is, as a fact, a basic level of statically required shear reinforcement and is calculated by the program, i.e. not controllable by the user

For EC 2, the 60% values of Table 5.5 are preset as 1st input default sv ; they may be redefined by the user, i.e. also set to 0.

Minimum reinforcement of Deep Beams The notion of Deep Beams (in its substance, not verbally!) has not been explicitly introduced by all Norms implemented in NEDIM. In this generalized sense only CSN/ STN 73 1201, SIA 162, NEN 6720 and SIA 262 do not deal with this notion.

Basically, Deep Beam is a notion of a wall-like structure being loaded (prevailingly) in its plane, being supported by more or less concentrated supports and spanning one or more fields. From the point of view of the building mechanics it can be considered as a simply or continuous beam being characterized by large structural height in relation to its spans. The relation l/h ≤ 2 might generally be recognized as the threshold between a 1D beam and a 2D wall. For NEDIM, however, this distinguishing is of no concern. If a Wall or a Shell design model is, by input specification, assigned the attribute Deep Beam, it is dealt with as structure of this type. NEDIM defaults Walls to possess this attribute; Shells are defaulted to the opposite attribute.

All Norms which introduce this notion stipulate a base structural reinforcement being controlled by two conditions :

o An absolute amount of reinforcement is assigned to all reinforcement courses, making no difference of their direction in the structure. Typically, such settings fall within the interval [0.5, 2.0] cm²/m per course.

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o A relative amount of reinforcement is stipulated, too. All Norms, excepting BAEL 91/99, make the base reinforcement dependent on the cross-section height, thus following a relation like (231). Typically for all Norms, the minimum percentage lies within the interval [0.05, 0.15] %.

These design conditions are, generally, accomplished by constructive rules upon the reinforcement nets. Typically, rectangular, quadratic meshes with bar distance no more than 300 [mm] (and less) are required.

DIN 1045, §23.3(4) stipulates a double-term limit condition : (a) DB = 0.0005, i.e. 0.05% of gross section (note that it is a third of the of EC 2 value); (b) absolute amount of 1.5 [cm²/m] – for each reinforcement course. Expressed by formula (bDB = 1.0) :

min aDB = max ( 0.0005 h, 0.00015) [m²/m] (271)

EUROCODE 2, §5.4.5(2) requires DB = 0.0015, i.e. 0.15% of gross section for both reinforcement directions, at both faces.

DIN 1045-1, §13.6(2) stipulates a double-term limit condition : (a) DB = 0.00075, i.e. 0.075% of gross section (note that this is half the value of EC 2); (b) absolute amount of 1.5 [cm²/m] – both for each reinforcement course. Analytically (bDB = 1.0) :

min aDB = max ( 0.00075 h, 0.00015) [m²/m] (272)

GBJ 10-89, §7.6, Table 7.6.13 requires for "vertical distribution" and "longitudinal tension" bars the minimum reinforcement ratio of DB,v = 0.0020, i.e. 0.20% of gross section, for steel Grade I; for higher steel grades, DB,v = 0.0015, i.e. 0.15% is stipulated. For horizontal reinforcement respectively DB,h = 0.0025 or DB,h = 0.0020 is required. NEDIM assigns to both/all reinforcement courses the higher value of DB,h → DB = 0.0025 or DB = 0.0020 if this control is active.

BS 8110, §3.12.7.4 does not, as a fact, introduce the notion Deep Beam, yet it requires special ‘minimum horizontal’ reinforcement ratio of h = 0.0030, i.e. 0.30 % of gross section, for fy = 250 [MPa]; alternatively, for fy = 460 [MPa], h = 0.0025, i.e. 0.25 % is stipulated. BS 8110 maintains this requirement for vertical reinforcement ratio up to 2% (compare with §13.2.7.5). For low ratios of the vertical reinforcement v, NEDIM controls the horizontal reinforcement ratio h to become no higher than v. If no transversal reinforcement control is activated by the user, the minimum reinforcement acc. to §3.12.7.4 is not calculated ! Because of the character of this reinforcement control it was incorporated into the Deep Beam control group.

ÖNORM B 4700, §11.3(2) stipulates, like DIN 1045-1, a double-term limit condition : (a) DB = 0.001, i.e. 0.1% of gross

section; (b) an absolute amount of 1.5 [cm²/m] – both for each reinforcement course. Expressed by formula (bDB = 1.0) :

min aDB = max ( 0.001 h, 0.00015) [m²/m] (273)

BAEL 91/99, §E.5.4,211 stipulates the minimum reinforcement ratio by the formula

DB = fe (274)

With fe – design steel strength (BAEL uses the symbol ρ instead of ) i.e. indirectly proportional to the value of steel strength fe

[MPa]. On the other side, BAEL does not require any absolute amount of reinforcement for Deep Beams. However, §A.8.1,21 contains a statement concerning compression reinforcement in wall structures generally, which assigns the total amount of 4 [cm²/m] to any reinforcement direction under pressure. For NEDIM this requirement has been implemented in a generalized manner as the minimum structural reinforcement condition of Deep Beams. Thus, the amount of 4 [cm²/m] is checked against the amount of (274) and the maximum value of both is assigned to the Wall or Shell design model; provided that the option Deep Beam was activated on input!

EN 1992-1-1:2004, §9.7 stipulates exactly the same conditions as ÖNORM B 4700 : (a) DB = 0.001, i.e. 0.1 % of gross section; (b) an absolute amount of 1.5 [cm²/m], see (273).

IS 456 → IS 2210, §12.2.5 does not, as a fact, introduce the notion Deep Beam, yet it requires special "minimum compression" reinforcement of 2.5 [cm²/m] for over-pressed Shell cross-sections. Nevertheless, because of its character this reinforcement control has been incorporated into the Deep Beam control group. However, this control is under NEDIM active in case of congruent reinforcement +Zp/-Zp only !

ACI 315M-05, §11.8.4 requires a minimum vertical (‘shear’?!) reinforcement of DB,v = 0.0025 according to (251) for Deep Beams, i.e. 0.25% of gross section. For horizontal reinforcement (parallel to the main horizontal "bending reinforcement") §11.8.5 stipulates DB,h = 0.0015, i.e. 0.15% of gross section. NEDIM assigns to both/all reinforcement courses the higher

value of DB,v → DB = 0.0025 if this control is active. Additionally, there is a special requirement for ‘deformed’ horizontal bars of walls acc. to §14.3.3, assigning DB,h = 0.0020, i.e. 0.20% of gross section for ≤ 16 and fy ≥ 420, or DB,h = 0.0025, i.e. 0.25% to other deformed bars; respectively. This condition, however, can hardly be organized automatically. However, it is comprised in the above NEDIM control according to §11.8.4; thus, it was omitted from the NEDIM control.

Closing note : The Author of NEDIM published a critical discussion [20] on a Paper by Dajun who reports on investigations done in preparation of the chapter Deep Beams of the Chinese Norm GBJ 10-89 [14]. In that GBJ chapter a semi-empirical, quasi 1D approach to the design of Deep Beams is stipulated. In [20] it was shown that a general 2D solution by EPW or ESA PT, respectively, is generally safer and more economic than the quasi-empiric solutions according to the GBJ 10-89 formulae !


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SERVICEABILITY PROOFS

Introduction to serviceability proofs The most important serviceability proof is the Crack Proof. As first implementation in NEDIM the crack proof according to NEN 6720 was introduced 1997.

The contemporary theories of crack development in a concrete-steel compound medium have become very complex. Practical engineers, confronted with a plenitude of contradictory ideas and formulae, may feel doubts on the reliability of such calculations. However, it should be understand that all crack theories have probabilistic a nature. They try more or less successfully to give analytical explanation to empirical data of the crack behaviour of real structures. In 2D structures even the fundamental question, in what direction the main (first) cracks arise, has not been decided uniquely :

o perpendicular to the direction of 1st principal forces nI or mI (Fig. 13a,b);

o perpendicular to the reinforcement courses (Fig. 13c);

o parallel to the virtual stiffening concrete strut;

o erratic (combined) crack patterns etc. This assumption comes, no doubt, most closely to the reality; on the other hand, it is obviously least productive in stimulating efficient, simply crack control methods.

The NEDIM crack proof algorithm follows formally the assumption of Fig. 13c. However, it appears contradictory to the Baumann transformation theory [1], which prefers the assumption of crack parallel to the virtual stiffening concrete strut. Nevertheless, the NEDIM approach to the crack proof may be defended by following considerations:

o The design forces ndim, assigned to the reinforcement courses, attain, as a rule, values comparable with the governing principal forces nI,II (mI,II), since the strut force n3d is negative (n3d denotes, strictly speaking, the strut force in 2-course nets; as a fact, in 3-course reinforcement nets the strut force need not to be assigned the subscript ‘3’). Only in three-course nets the relation max |nd| < nI,II may become true under elliptic states of stress. However, such states of stress are the less critical for the cross-section resistance, and the cracks may the more tend to erratic patterns distributed over all three courses.

o The NEDIM crack proof algorithm is able, as will be demonstrated below, to distinguish qualitatively between different states of stress of the structure. Thus the formal “linearization” of the crack proof process does not ignore the 2D character of the reinforcement concrete medium.

o All Norms stipulate crack control formulae primarily for a 1D state of stress. The NEDIM calculation assumption of cracks developing perpendicular to reinforcement courses enables to organise the crack proof in quasi 1D steps running over individual reinforcement courses, in the same manner like with the ULS design.

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Fig. 13 Assumptions about crack propagation in 2D continuum : (a) Cracks perpendicular to the direction of principal tension (trajectory reinforcement); (b) Cracks perpendicular to the direction of principal tension, yet non-perpendicular to reinforcement courses; (c) Cracks perpendicular to

reinforcement courses


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The basic Problem of the 2D crack proof is obvious from the only formula dedicated by the Norms of the Eurocode family, here exemplary EN 1992-1-1:2004, formula (7.15)), to 2D structures:

sr,max = 1 / {cos φ / sr,max,1 + sin φ / sr,max,2 } [mm] (281)

In (281) the symbols sr,max denote the maximum allowable or calculated, respectively, crack distances, which play a distinguished role in most Norm proof theories (besides the crack width wmax). The indices 1, 2 in (281) refer to 1st and 2nd reinforcement course, here assuming orthogonality.

(281) and the following discussion refers to Fig. 13a,b. It relates the crack distances sr,max,1 und sr,max,2 to the direction of principal tension. The relation is, however, contradictory, what is obvious from Fig. 13b : the principal direction divides

symmetrically the right angle between the reinforcement directions 1 and 2, i.e. φ = 45°. From (281) follows thus sr,max = sr,1 / √2 ≈ 0,707 sr,1 (where sr,1 ≡ sr,max,1 = sr,max,2).

The announced contradiction consists in the fact that this most ineffective reinforcement geometry [18], which causes about 200% of req as in ULS compared with the corresponding trajectory reinforcement is assigned a significantly lower design crack distance sr,max (about 71%). This conclusion is unacceptable, obviously defective.

Thus, the NEDIM approach, as symbolized by Fig. 13c, proves, also from this point of view, to be the most realistic in a 2D reinforced concrete continuum.

The proof methods are based on similar assumptions of crack propagation mechanism :

o High tension stress in a reinforcement bar causes high steel strain. The adhesion between concrete and the reinforcement bar is disturbed, and cracks arise in the concrete continuum. The higher is the ratio of steel stress and the adhesion resistance, the wider become the cracks along the reinforcement bar. Thus, the higher the representative reinforcement diameter , the higher the ratio of the steel stress and the adhesion resistance, since the cross-section area of a bar grows with the square of whereas the surface of (unit length) of bar depends linearly on .

o Cracks arise not only close to the reinforcement bars yet merely between them. Thus, the transversal distance s of reinforcement bars may also become a crucial factor of the cracks width development. However, some Norms, like ÖNORM B 4700, do not introduce the distance s as independent factor of the crack proof at all.

To limit or reduce, respectively, crack widths (as a fact, not the number of cracks but the representative crack width is of interest for the crack proof) the following measures have to be taken :

o Specification of as small reinforcement diameters as possible.

o Reduction of the representative (transversal) reinforcement bar distance s. However, there is a dependence between and s : with given and provided as, s is determined by

s = 0.25 × π ×� ² / as [mm] (282)

o Augmenting the statically required reinforcement amount. Due to this provision the steel stress in the serviceability state is lowered, thus the crack widths are reduced as direct consequence. This steel amount control (augmenting of reinforcement amount from the ULS design) is the basic concern of the NEDIM crack proof algorithm.

Practically, NEDIM follows a two-step thread : (a) ULS design, yielding statically required reinforcement amount; (b) SLS design, referring to the characteristic bar diameter k and/or a characteristic bar distance sk as specified by the user on input. NEDIM carries out the crack proof according to the Norm proof approach and increases the statically required reinforcement amount where it is needed to meet completely the crack proof requirements.

NEDIM, however, allows for merging of load cases for the ultimate and serviceability states within a calculation process in order to enable the crack proof procedure outlined above. In the following paragraphs it is shown that different attributes may be assigned to the load cases, in accordance with the individual stipulations of the Norms.

In Chapter Program Theory and Algorithm the notion of the virtual cross-section design force nvirt was introduced (Fig. 7c). The effect of this algorithmic enhancement is, along with that discussed with the shear proof and the minimum compression reinforcement, a consistent description of the state of stress in the cross-section, especially in case of non-congruent reinforcement at both faces. Since most Norms consider the stress distribution pattern (bending ↔ centric tension) as important a factor of the crack development, the knowledge of nvirt is indispensable to reliable crack proof design. Upon the analysis types dealt with by NEDIM it has the following impact:

o Walls : the general inner forces vector (3) degenerates to

{ nx, ny, nxy } (291)

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There is no use of nvirt, since nvirt nd in this model. All design forces are membrane forces with zero eccentricity, causing either tension or pressure in the cross-section.

o Plates : the general inner forces vector (3) degenerates to

{ mx, my, mxy, vx, vy } (292)

Thus, instead of the design forces nd, design moments md and shear force vd are active in Plate design. There is effectively no (virtual) normal force nvirt in pure flexural members, even if hyperbolic cases like in Fig. 6b suggest that such an interpretation of the rather complicated type of stress state may be discussible : both reinforcement courses at upper/lower face appear to be under tension, thus the conclusion seems to be justified that there is a normal force action upon the cross-section. However, with Fig. 6b it was explained that, in such hyperbolic cases, the prevailing stress is shear, not tension, and that also the reinforcement is, effectively, subject to shear rather than to tension; the representative stress pattern in the design section is thus the shear stress triangle (Fig. 6b). As a fact, it was made an attempt in NEDIM to deal with such states of stress as with “prevailing tension”. This had, however, serious consequences to the crack and shear proof results : especially under ÖNORM B 4700, unacceptable crack reinforcement increments to statically required reinforcement were casually obtained.

o Shells : the general inner forces vector (3) applies to Shell design, rewritten here :

{ mx, my, mxy, vx, vy, nx, ny, nxy } (293)

Although the two-step reinforcement design (running separately for both faces) assigns a half cross-section to each reinforcement course, the crack proof must take into consideration the total cross-section, even if there is no congruent reinforcement at opposite (actually inactive) face. The information needed is delivered by the virtual normal force nvirt and the complementary virtual bending moment mvirt.

All possible states of stress have to be correctly interpreted and managed by the NEDIM crack proof algorithm. As symbolized by Fig. 14, for the crack proof procedure it is not enough to determine tensile stresses at the actual face, yet also the stress pattern over the cross-section is of eminent importance; especially, the s. c. “disconnection cracks” are of interest.

Fig. 14 Typical stress patterns considered by NEDIM’s crack proof procedure :


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(a) bending crack – neutral axis within cross-section;

(b) disconnection crack due to tension force with low eccentricity;

(c) over-pressed cross-section – no crack proof

Crack Proof according to DIN 1045 07/1988 The crack proof algorithm of NEDIM follows the basic specifications of DIN 1045 07/1988 [5] as well as the enhancements described in Heft 400 [23], which is in the German engineering practice regarded as authorized Norm complement.

According to the basic notion of §17.6 [5] the primary goal of the crack proof is the control of the limit bar diameter ds and/or the maximum reinforcement bar distance s. The control is governed by the Tables 14 and 15, which define the functions ds = f1 (s) and s = f2 (s) for selected pivot values of ds and s. If the steel stress corresponding to the provided reinforcement as,ULS (retrieved from the data base) is higher than allowed by Table 14 or 15, respectively, one of the following measures is necessary : (a) lowering the input value ds,inp; (b) lowering the input value sinp; (c) increasing the provided reinforcement to

as,SLS > as,ULS. The NEDIM algorithm runs, however, acc. to (c) since ds,inp or sinp, respectively, are fixed values, assessed by the user.

NEDIM distinguishes in the DIN 1045 branch four attributes of load cases :

o Load cases or their combinations representing external loads, assigned the attribute “ULS”. They yield the statically required reinforcement, automatically saved in the data base to be retrieved by the crack proof procedure as a base for crack control. However, they can simultaneously be specified and used as load cases of the types specified below; then, generally provided with another LC factor.

o Load cases/combinations for crack proof causing so called force induced deformations. These serviceability load cases are assigned the attribute “SLS/ external forces”.

o Load cases/combinations causing both force and strain imposed deformations. The effect of yielding of supports and other external causes, which cannot be directly identified with external loads, is assumed. These are serviceability load cases are assigned the attribute “SLS/ externally imposed deformations”.

o Load cases/combinations causing both force and strain imposed deformations. The effect of temperature variation, shrinkage, creep and other internal causes, which cannot be identified with forces at all, is assumed. These are serviceability load cases are assigned the attribute “SLS/ internally imposed deformations”.

The minimum tension reinforcement stipulated by §17.6.2(3), formula (19) [5] is of another kind than discussed in Chapter Reinforcement amount control :

z k0 × βbz / s (30)

with μz –reinforcement ratio related to tension zone Abz; k0 – special parameter respecting the cross-section stress pattern; σs – steel tension stress according to the crack control described above (i.e. corresponding to the reinforcement amount calculated, depending by Table 14 on the bar diameter); βbz – calculation concrete tension strength. The minimum tension reinforcement may be, following §17.6.2(2), omitted in special situations.

The procedure just described is the elementary crack proof directly stipulated by [5]. The crack control guaranteed by this method means that the mean crack width is “implicitly” limited to the value wcal = 0.25 [mm]. This elementary method is sometimes called crack limiting.

However, Heft 400 DAfSt [23] provides an enhanced theory on crack propagation and control, using formulae which are basically identical with the EC 2 approach. It has been implemented into the DIN 1045 07/1988 branch as innovation of the original approach described by (30).

Crack Proof according to EUROCODE 2 According to the basic notion of EC 2 [9], §4.4.2 two possible crack proof strategies are at choice :

o §4.4.2.3 : crack limiting without direct calculation. This method is almost identical to the elementary crack limiting method stipulated by DIN 1045 [5] (see above). By meeting the requirements of §4.4.2.3 the mean crack width will be limited to the value wk = 0.30 [mm].

o §4.4.2.4 : method of calculating mean crack width wk by formula (4.80) :

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wk = β srm εsm (31)

with wk – calculation value of the crack width; β – security factor distinguishing force induced cracks (β = 1.7) and cracks induced by imposed deformations (β = 1.3); srm – mean crack distance in case of fully developed crack pattern; sm – mean steel strain, considering tension stiffening between the cracks. (31) represents a sophisticated procedure taking several factors into account. The procedure by Heft 400 [23], as supplement of DIN 1045 [5], is almost identical to that described EC 2, formula (4.80), i.e. (31) above.

The enhanced procedure acc. to (31) enables to control the mean crack width wk in the structure by varying bar diameter or bar distance s. However, the NEDIM procedure, with given input values of inp or sinp, is aimed at controlling the statically re-quired reinforcement as,ULS : if required by crack proof, as,ULS is augmented in order to lower the steel stress s, which is the

crucial factor affecting the value of mean strain sm in (31). This procedure is called crack reduction, since generally cracks wk < 0.30 [mm] are aspired to.

NEDIM controls the crack proof procedure of the EC 2 branch by distinguishing four different load case attributes acc. to the same principles as described in the paragraph on the DIN 1045 07/1988 branch.


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Crack Proof according to NEN 6720 The crack proof algorithm of NEDIM follows the specifications of NEN 6720 as well as the theoretical fundamentals described in [3], [12].

According to the method of NEN 6720, the primary goal of the crack limitation is the assessment of the characteristic average bar diameter k and/or the maximum allowable reinforcement bar distance sk. NEDIM performs both proof variants simultaneously.

NEN 6720 describes two states of cracks propagation: developed and undeveloped cracks. Developed means that over the whole span no more cracks will occur; only the crack width will increase. Undeveloped means that the cracks are considered either not to exist at all or not to be fully developed. For undeveloped crack patterns the initiation of cracks is an important factor, while for developed crack patterns their widening is of interest. Both the initiation and the widening of cracks are checked by limiting the diameter of reinforcement bars as well as by limiting their distance s.

Clauses 8.7.2 and 8.7.3 [12] provide two checks for developed and undeveloped cracks. In case of developed cracks only one of the requirements A, B is to be met; undeveloped cracks are controlled by meeting both checks simultaneously:

8.7.2. (developed – req. A): 1

1c,

sk k

σ

ξ k (321)

8.7.2. (developed – req. B): ec,2

sk kk

ξ kms

1.3 1 σ

2100 (322)

8.7.3. (undeveloped): 50)( 2

ssr

ck3k σσ

f' ξ k and

ssrk σσ

ξ k

1 (323)

For explanation of parameters in formulae (32#) see [3], [12]. Here we will discuss the para-meter ke only. It represents the stress-strain pattern characteristic for the cross-section of interest taking on the values between 1.0 for “pure” bending and 0.5 for over-tensioned cross-sections. Plates are structures with declared “pure” bending.

It is the parameter ke which is exclusively responsible for expressing this aspect in (322). With respect to Fig. 14 it means concretely : the situation of Fig. 14b corresponds to the value of ke

= 0.5; Fig. 14c corresponds to ke = 1.0. Thus, in case of equal magnitudes of the principal mo-ment mI in the plate corner of Fig. 14, (322) yields for the statically inefficient reinforcement geometry 0°/90° half the value of maximum allowable reinforcement bar distance s than in case of the statically optimum reinforcement geometry 45°/ 135°!

It is interesting to realise that the parameter ke is active in (322) only, thus having effect upon s only; the calculation of the bar diameter is not affected by the stress-strain pattern, unlike the ÖNORM B 4700, which controls by the analogous parameter k the limit bar diameter dgr (compare with (35)).

NEDIM determines the steel stresses s and sr in formulae (32#) following an iterative process in which both steel and concrete obey a linear law, the concrete, however, being disabled in tension.

The crack proof theory [12] distinguishes three types of load cases for the crack proof according to NEN 6720 (for details see Crack Proof according to DIN 1045) :

o Load cases representing external loads. In the NEDIM input system they are assigned the attribute “ULS”.

o Load cases representing the force imposed deformations. These serviceability load cases are assigned the attribute “SLS/ external forces”.

o Load cases representing both force and strain imposed deformations. Deformations indu-

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ced by LC combinations of this type cannot be assigned to external forces only. The effect of temperature variation, shrinkage, creep, yielding of supports etc. is involved, too. Servi-ceability load cases of this type are assigned the attribute “SLS/ externally and internally imposed deformations”.

The NEN 6720 crack proof branch of NEDIM has been equipped with 4 different proof branches. All of them are useful means of crack proof analysis :

1. Non-controlled Crack Proof : The crack proof calculation is not controlled by any user-specified restrictions. For each reinforcement course NEDIM calculates the characteristic average bar diameter k and the maximum allowable reinforcement bar distance sk at any design point to fit the crack proof requirements. By examination of the results (graphic portrayal) the user gains an overview over the development of the characteristic values of k and sk over the part of model analyzed. In special cases, when the crack proof requirements are completely met for the input values of diameters or distances, the crack control for these regions is already effectuated.

2. Controlled Crack Proof : The crack proof calculation is controlled by the maximum bar diameter specified by the user. NEDIM calculates the characteristic average bar diameter k at any design point and compares this value with the input value of . If k calculated is less than the input diameter inp the reinforcement augmentation process described above is started. As a result, a higher reinforcement amount fitting the crack proof for the input diameter inp replaces the original statically required reinforcement amount in the data base. This proof variant enables the engineer to specify a constant bar diameter inp envisaged for some parts of the structure. The results of this NEDIM crack proof variant ensure that the crack proof requirements will be met everywhere, when the reinforcement saved in the data base is applied (superposition of the ULS and SLS design states).

3. sControlled Crack Proof : The crack proof calculation is controlled by the maximum bar distance s specified by the user. NEDIM calculates the maximum allowable rein-forcement bar distance sk in any design point and compares it with the input value of sinp. If sk calculated is less than the input diameter sinp the reinforcement amount augmentation process is started. As a result, a higher reinforcement amount fitting the crack proof for the input bar distance sinp superposes the statically required reinforcement amount saved in the data base. This proof variant enables the engineer to specify a constant bar distance sinp envisaged as structural provision for parts of structure checked. The results of this NEDIM crack proof variant ensure that the crack proof requirements will be met everywhere, when using the reinforcement saved in the data base, augmented, if necessary, by the crack proof procedure.

4. Optimized Crack Proof : This is algorithmically the most exacting variant of the NEN 6720 crack proof algorithm. The calculation is controlled both by the maximum bar diameter and the maximum bar distance s specified by the user. NEDIM combines the procedures described under Pt. 2 and 3 above. Following variants may be encountered in course of the calculations : (a) If k calculated is greater than or equal the input diameter inp the crack proof requirements have been met. There is no need of augmenting the reinforcement amount; (b) If sk calculated is greater or equal than the input distance sinp the crack proof requirements have been met; (c) if neither inp nor sinp specified meet the crack proof requirements the procedure described under Pt. 2 and 3 is started to find the best fit of one of both conditions ( or s) by augmenting the statically required reinforcement from the ULS design. “Best fit” means to seek that one of the or s conditions which implies


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lesser reinforcement augmentation. This variant yields, generally, the lowest total reinforcement augmentation amount of all three variants described by Pt. 2, 3 and 4; this is why it is called optimized.

Crack Proof according to DIN 1045-1 07.2001 DIN 1045-1 stipulates, like EC 2, two approaches to the crack control :

o Empiric approach, §11.2.3 : limitation of the crack without direct calculation. Analogously to the ideas of EUROCODE 2, the functional dependences ds = f1 (s,wk) and s = f2 (s, wk) are defined by Tables 28 and 29. While EC 2 relates its table values to the implicitly assumed crack width wk = 0.30 [mm], DIN 1045-1 distinguishes between 3 basic cases : wk = {0.20, 0.30, 0.40} [mm]. Thus, DIN 1045-1 presents a real enhancement for hand calculations on this elementary proof level. However, for the automated crack proof by NEDIM this enhancement is of no advantage, compared with the genuine analytic approach, since a 3-step interpolation (compare with the ÖNORM B 4700 algorithm, described below) would be necessary. For these reasons it was decided not to include this procedure into the NEDIM processing.

o Analytic approach, §11.2.4 : crack limitation by direct calculation has been implemented into NEDIM. It is an analogy of the crack reduction method introduced by Eurocode 2 (see above). However, the DIN 1045-1 crack formula differs from (31) :

wk = sr,max (εsm – εcm) (331)

with wk – calculation value of the crack width; sr,max – maximum crack distance in case of fully developed crack pattern; sm – mean steel strain, considering tension stiffening between the cracks; cm – mean concrete strain between cracks. The calculation of the strain difference (sm –cm) is described by DIN 1045-1 formula (136). The maximum crack distance sr,max is described by DIN 1045-1 formula (137); in NEDIM terms :

sr,max = min (1 / ρeff, σs / fct,eff) / 3.6 × ds (332)

with ρeff – efficient reinforcement ratio; ds – calculation bar diameter.

NEDIM controls the crack proof procedure of the DIN 1045-1 branch by distinguishing two load case attributes :

o SLS load cases due to external loads combined with externally imposed deformations, if occuring. They are assigned the attribute “SLS/ external”.

o SLS load cases due to internally imposed deformations, i.e. hydration heat. They are as-signed the attribute “SLS/ internal”.

Crack Proof according to GBJ 10-89 The crack proof algorithm of NEDIM follows the specifications of GBJ 10-89 as well as the theoretical fundamentals described in [15].

The GBJ 10-89 crack proof method focuses upon the dealing with the bar diameter d only; bar distances do not play any direct role.

GBJ 10-89 introduces the explicit crack width control as the central proof item. The scope of crack widths dealt with is the interval [0.10, 0.40] [mm]. The maximum allowable crack width value wmax desired to be maintained throughout the structure or its part is specified on input.

The GBJ 10-89 algorithm, following principally the same threads as described, e.g. in the NEN 6720 crack proof paragraph, augments, if necessary, the statically required reinforcement, retrieved from the data base, to fit the crack proof requirements associated with the maximum allowable crack width value wmax given by the formula :

wk = acr ψ (2.7 c + 0.1 d /�ρte) ν × σss /Es (34)

In (34) acr is an analogy to the stress-strain pattern parameters ke (NEN 6720) and k (ÖNORM B 4700). GBJ 10-89 assigns the values acr = {2.7, 2.4, 2.1} – corresponding to axial tension, eccentric tension and pure bending, respectively. NEDIM interpolates, however, between acr = 2.7 and acr = 2.1 to distinguish smoothly between different states of stress from “pure” bending to centric tensioned cross-section. For explanation of the other factors in (34) see [14].

In the actual version of NEDIM, for both faces of a 2D member different control values of winp may be defined on input. This is, however, a common input approach to all crack proof branches implemented in NEDIM.

Crack Proof according to ÖNORM B 4700 The crack proof specifications of ÖNORM B 4700, relatively detailed elaborated, present an interesting individual approach to the problem of crack control in comparison with EC2 or DIN 1045-1. Acc. to ÖNORM B 4700 the crack development is controlled by the limit reinforcement bar diameter dgr rather than by bar distance s. However, it has to be understood (con-cerning both ÖNORM B 4700 and GBJ 10-89 and other crack proof branches of NEDIM) that explicitly focusing to bar dia-

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meters dgr as leading proof factor does not mean the distance of bars being of no impact upon the crack control. The bar distances s in a reinforcement net correspond to the reinforcement amount provided, as shown by (282). The difference between the methods of DIN 1045-1, EUROCODE 2 and NEN 6720 on one part and GBJ 10-89, ÖNORM B 4700 on the other part consists just in the manner of how the bar distances are dealt with : the formerly mentioned Norm methods make them explicit control parameters; the latter Norms use bar distances as implicit quantities of the crack proof.

ÖNORM B 4700 introduces the notion of crack width as a central proof item. The crack widths to be dealt with are assigned to the interval <0.15, 0.30> [mm]. The maximum allowable crack width value wk desired to be maintained throughout the structure or its part is defined on input. Higher or lesser values of wk are obviously outside the scope covered by the ÖNORM B 4700 stipulations. If there appears wk < 0.15 or wk > 0.30 on input, it is adapted to the nearest limit value (wk = 0.15 or wk = 0.30, respectively).

A characteristic feature of ÖNORM B 4700 is the absence of analytic formulae for wk. Tables 8a,b to 10 describe the functions dsr = f1(tr) or dsr = f2(tr,sD), respectively, of the calculation bar diameter dsr; f1(tr) – for strain imposed deformations and

f2(tr,sD) – force imposed deformations (compare with the paragraph on NEN 6720). NEDIM applies a double interpolation process based upon the Norm Tables 8a or 9 (for wk = 0.15) and 8b or 10 (for wk = 0.30) to establish the limit bar diameter value dgr respectively as function of the reinforcement amount percentage tr alone or of tr and the characteristic steel tension sD.

The relation between the limit reinforcement bar diameter dsr and the calculation reinforcement bar diameter dgr is described by the following formula:

dgr = dsr k (35)

with parameter k representing the characteristic stress-strain pattern of the cross-section, attaining values from the interval k

[0.5, 1.0]. Compared with the crack proof acc. to NEN 6720, k is a direct analogy to the parameter ke of (322); however, it is calculated in another way than ke.

ÖNORM B 4700 distinguishes several situations of exploitation concerning the crack proof procedure and the use of Norm Tables 8a,b or 9 and 10, respectively. NEDIM deals with such situations by assigning attributes to corresponding load cases :

o Load cases representing external loads. They are assigned the attribute “ULS” when applied to the design of the statically required reinforcement, which is saved in the data base to be retrieved by the crack proof procedure. However, they can simultaneously be specified with the attribute “SLS” and used as load cases of the types specified below; then, most probably, provided with another LC factor.

o Serviceability load cases representing the force imposed deformations for crack proof. For dealing with load cases of this type the functional relation dsr = f2 (tr,sD) governed by the Tables 9 and 10 is of fundamental importance. An iteration process equilibrates the required reinforcement amount by checking tr against dgr, which is related to dsr by (35); dsr is specified on input as principal control quantity along with the crack width wk. Load cases of this group are assigned the attribute “SLS/ external forces”.

o Serviceability load cases representing strain imposed deformations acc. to §4.2.2(1), (2), (3) and (4). Load cases of this type are governed by the functional relation dsr = f1 (tr) defined by the Norm Tables 8a and 8b. The cause of imposed deformations has to be expressed by means of load cases. The LC are assigned the attribute “SLS/ external strain”.

o Serviceability load cases representing the strain imposed deformations acc. to §4.2.2(5). They are special LC dealing with non-linearly distributed, self-induced stresses within the cross-section. These situations of exploitation allow for a reduction of the required reinforcement amount percentage tr within the interval <60%, 80%>. On the whole, the crack iteration process follows the same rules as described by the preceding point. Load cases of this group are assigned the attribute “SLS/internal strain”.

o Serviceability Load Cases representing the strain imposed deformations acc. to the §4.2.2(6). They are special crack proof serviceability LC dealing with cracks in an early stage of concrete hardening. These situations of exploitation allow, according to §4.2.2(6), for a reduction of the required reinforcement amount percentage tr to 70% of


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the standard value. By this it is assumed that the average tension strength fctm,t attains 50% of fctm (after 28 days). For other percentages linear interpolation between these limits is allowed. NEDIM respects these special rules. For fctm,t the corresponding percentage of fctm is expected on input (active for this type of LC only), rather than its absolute value. LC of this group are assigned the attribute early stage strain.

o Serviceability load cases representing the strain imposed deformations due to hydration heat flow acc. to §4.2.2(6). These situations of exploitation allow for a reduction of the required reinforcement amount percentage tr to 70% of the standard reinforcement value in case of normally hardening cements (e.g. PZ 275, PZ 375). For other cements a special proof is required). LC of this group are assigned the attribute “SLS/ hydration strain”.

Crack Proof according to BAEL 91/99 Concerning the basic assumptions on the crack development in a 2D reinforced concrete medium, NEDIM maintains the general notion formulated introductory to this chapter also for BAEL 91/99, since BAEL does not contain any special hints as to the organization of this design part called “Limit State of Crack Opening” when dealing with the 2D structures.

The crack proof specifications of BAEL 91/99, §A.4.5,3 present themselves, in comparison with the procedures stipulated by the EUROCODE family, NEN 6720 etc., as very a simple approach. This statement concerns, however, exclusively the practical formulae of the BAEL 91/99 crack theory, not the scientific fundamentals of it! As basic control parameter, the status of cracking, introduced already in the paragraph Shear Proof methods, redirects the crack control calculations acc. to BAEL 91/99 into three branches:

o Cracking is considered generally as non-critical (“peu préjudiciable”) for the functionality of structure. In such cases no crack proof is necessary in the sense of NEDIM. Thus, specifying this status equals to abstaining from the crack proof at all.

o Cracking is considered as critical (“préjudiciable”). This might be the standard case of crack proof. The approach of BAEL 91/99 consists of proving that the steel tension corresponding to the serviceability load case combination specified does not exceed the limit value given by the following formula (351) :

lim s = ξ = min { max [110 √(η ftj), ½ fe] , ⅔ fe} [MPa] (351)

o Cracking is considered as very critical (“très préjudiciable”). This status of cracking implies a stronger tension limit (thus decreasing the characteristic crack width) by requiring :

lim s = 0.8 ξ [MPa] (352)

with ξ from (351).

In (351) the factor η [1.0, 1.6] is, as a fact, a steel surface characteristics, thus expressing the adhesion ability of steel bars to the concrete. It is interesting to realise that in the crack formula (351), differently from all other crack control approaches applied by the Norms implemented in NEDIM, neither the steel bar diameter nor the bar distance are active as factors of the crack proof ! Equally interesting is to note that also the shear proof (see above) of BAEL 91/99 is controlled by the status of cracking. It might be concluded, that BAEL 91/99 pays more attention to shear induced cracks than other Norms !?

Crack Proof according to SIA 262 SCIA 262, §4.4.2 declares the crack control by providing special minimum reinforcement in order to limit the opening of cracks to an acceptable value. In paragraph Minimum tension reinforcement is was pointed to the fact that §4.4.2.3.9, Table 16 & §4.4. 2.3.10, Fig.31 require a minimum tension reinforcement as provision against the brittle fracture of concrete in case of 1st crack’s appearance. While that first step of SIA 262 crack control is carried out for steel stress σs,adm = fsd (case A, see Table 1), i.e. for design steel strength (compare with DIN 1045, allowing for fyk, i.e. characteristic steel strength), the second step of (full) crack proof, discussed here, restricts the admissible steel strength as function of bar distance, i.e. σs,adm = f (s). Table 16 of SIA 262 is reproduced here as Table 1 to give insight into the crack control philosophy of SIA 262 :

Table 1. Adm. steel strength σs,adm as function of bar distance s and proof objective

Proof objective Exigencies

Normal Accrued Elevated

1 Preventing brittle fracture when fctd is attained A A A

2 Limiting the opening of cracks under force impact or due to imposed deformations (when fctd is attained)

A*)

B*)

C*)

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3 Limiting the opening of cracks under quasi permanent load cases (ac. to SIA 260)

– – C

4 (4) Limiting the opening of cracks under frequent load cases (ac. to SIA 260)

– fsd – 80 fsd – 80

*) the codes A, B and C point to σs,adm = f (s) graphical function definition by SIA 262, Fig. 31

The functional relation σs,adm = f (s), coded A, B, C in Tab. 1, is defined graphically by SIA 262, Fig. 31. NEDIM uses a 3° parabola interpolation representation of Fig. 31 functional graphs to allow for analytic evaluation by the crack proof algorithm.

NEDIM effectuates the crack proof procedure of the SIA 262 branch by distinguishing four load case attributes :

o ULS load cases applied to the brittle fracture control (Tab. 1, Row 1). They are assigned the attribute "ULS".

o SLS load cases applied to the crack control due to imposed or restrained deformations (Tab. 1, Row 2). They are assigned the attribute "SLS/ deformation".

o SLS load cases applied to the crack control due to quasi permanent loads (Tab. 3, Row 4). They are assigned the attribute "quasi permanent".

o SLS load cases applied to the crack control due to frequent loads (Tab. 1, Row 4). They are assigned the attribute "frequent".

Closing note. The 1st step of crack control acc. to SIA 262, as summarized by Table 1, is the brittle fracture control. This is, however, considered as a standard procedure of minimum tension reinforcement estimation. Thus, the brittle fracture control is carried out primarily by the load cases attributed "ULS". The "genuine" crack proof, described as "limiting the opening of cracks", is effectuated by the SLS design run.

Crack Proof according to EN 1992-1-1:2004 The crack proof stipulations of EN 1992-1-1:2004 are obviously the result of continued development of Eurocode 2 (ENV 1992-1-1:1992) combined with DIN 1045-1. Here, two approaches to the crack control are stipulated as well : empiric approach, §7.3.3 and analytic approach, §7.3.4. For more details on the former one – see paragraph on DIN 1045-1. NEDIM applies the analytic approach.

Analytic approach, §7.3.4 : the basic formula for maximum crack width wk is identical with the crack formula (331) according to DIN 1045-1 :

wk = sr,max (εsm – εcm) (361)

with wk – calculation value of the crack width; sr,max–maximum crack distance in case of fully developed crack pattern; sm–mean steel strain, taking into account the effects of tension stiffening between the cracks; cm–mean concrete strain between cracks. The calculation of the difference (sm–cm) is described by DIN 1045-1, formula (136). The maximum crack distance sr,max is described by formula (7.11), resembling merely the EC2 formula (4.82); in NEDIM terms:

sr,max = k3 c + k1 k2 k4 / ρeff (362)

with c – concrete cover of the longitudinal reinforcement; k1, k2 – parameters from EC2; k3, k4 – coefficients which may be found in the NAD; recommended values are : k3 = 3.4 and k4 = 0.425.

Where the spacing of bonded reinforcement exceeds 5 (c + /2), i.e. 5 times the axial concrete cover, the alternative upper distance bound acc. to formula (7.14) applies :

sr,max = 1.3 (h – x) (363)

NEDIM controls the crack proof procedure of the DIN 1045-1 branch by applying load cases of a unique class : SLS load cases due to external loads combined with imposed deformations, if occuring. They are assigned the attribute "SLS"

Crack Proof according to BS 8110 and IS 456 IS 456 follows the same thread of crack proof methodology as BS 8110. The calculation procedure is described in BS 8110, Part 2, Chapter 3.8; it applies, by declaration, to cracks caused by loads causing prevailingly bending (flexural cracks), however, it is admitted to extend it to members subjected dominantly to axial tension.


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BS 8110 does not consider the bar diameter as a primary factor affecting the crack width. The bar distance s is considered to affect the crack width at least indirectly ("proximity of reinforcing bars to the point considered …"). Some of the symbols used in the crack proof formulae are, regrettably, obscure since (a) general cross-sections (1D) are envisaged; (b) no illustrating scheme is presented. Thus, some interpretation effort was necessary in order to get suitable crack formulae for 2D crack proof.

The design surface crack width formula (12) reads, with NEDIM terms :

wmax = 3 (acr εm) / (1. + 2 (acr – c) / (h – x)) (371)

with εm – average strain at the level where the cracking is being considered; acr – distance from the point considered to the surface of the nearest longitudinal bar; c – minimum cover to the tension steel bar; h – cross-section height; x – depth of the neutral axis.

The design value of acr is, related to a standard 2D member cross-section, calculated according the following NEDIM formula, which assumes the "level of cracking" on the tensioned surface, at half distance s/2 between two adjacent bars :

acr = – / 2 + (( / 2)² + (c + ) c + (s / 2)²) (372)

with – bar diameter; s – bar distance; other symbols are the same as in (371). The average strain εm fomula BS 8110, (13), "interpreted" for 2D (bt = 1 and a' ≡ h) reads :

εm = ε1 – (h – x)² / (3 Es as (d – x)) (373)

with ε1 – average strain at the level considered, calculated ignoring the tension stiffening effect; Es – modulus of elasticity of reinforcement steel [MPa]; as – (provided) area of tension reinforcement; d – effective static height of reinforcement course considered.

The calculation formula for ε1 has been derived from the declared meaning as :

ε1 = (h – x) / (d – x) × εs (374)

with εs – steel strain calculated in cracked section under the assumption of linear elasticity of steel and concrete (“Linear State II”).

The extrapolation of the application scope of (373) to over-tensioned cross-sections, as mentioned above, is effectuated by means of replacing the term (h - x)² in (373) by (a) h² if the neutral axis just coincides with the opposite face edge; (b) 2h² – for axial tension (§3.8.3, formula (13)). NEDIM interpolates between these margin values by inserting the term αh², with α according to the elementary formula :

α = max (min (1 + ε2 / ε1, 2.), 1.) (374)

with ε1 > 0. – strain at actual face, ε2 – strain at opposite face.

In (373), negative value of εm indicates that the cross-section is non-cracked. In assessing the strains, the modulus of elasticity of the concrete should be taken as half the instantaneous value. In case of abnormally high shrinkage (εφ > 0.0006), εm should be increased by adding 50 % of the expected shrinkage strain. Both recommendations are obeyed by NEDIM :

o The condition Ec,eff = 0.5 Ec is active as default to all load cases engaged in the crack proof . However, NEDIM allows for setting an individual value of Ec,eff as individual attribute to each load case.

o The creep effect upon the deformations may automatically be considered by defining a corresponding load case (combina-tion) of type "long-term". Additionally an increase factor kε, as indicated above, may be assigned as another load case attribute value, e.g. kε = 1.50, in order to define an increased value of εm, i.e. εm' = kε × εm, as provision for creep effects.

NEDIM controls the crack proof procedure of the BS 8110/ IS 456 branches by applying load cases of a unique class : SLS load cases due to external loads combined with imposed deformations, if occuring. They are assigned the attribute “SLS”

119

PROCESSING OF NON-DESIGNABILITY CONDITIONS

Processing of non-designability conditions A reliable system of reporting non-designabilities is one powerful feature of NEDIM. There are two kinds of irregularities: errors and warnings. The maximum number of possible errors within a design run may be restricted. At the present stage of NEDIM development, the upper error limit has formally been set infinite.

Warnings and errors in the sense of this Chapter are not program errors.

NEDIM defines eight non-designability and a warning condition. They are represented by their Non-designability STatus NST specified by the following Table :

Indication of the Non-designability Status (NSt)

Table 2. NEDIM design errors classification

NSt Indication of non-designability status (NSt >0)

<0 Control indication: Element/Node has not been subjected to design as yet. If there are some design results found in the corresponding Data Base record, nst<0 is an illegal value (probably caused a program malfunction).

0 Normal design result of element or element node, respectively.

>0 An error condition encountered – Verbal description of 8 specified error cases follows :

1 Compression reinforcement in bending (Plates only!) would be necessary, yet is prohibited (as a rule, by user’s specification).

2 Compression reinforcement in bending would be required, yet is non-designable (too heavily exploited cross-section).

3

Plate model only: Compression reinforcement in bending would be necessary, yet non-congruent reinforcement geometry at upper/lower faces encountered. Recommendation : if congruent reinforcement geometry impossible, design as Shell (in Shells, independent reinforcement geometry is acceptable in all design situations)

4 Maximum reinforcement ratio exceeded.

5 Concrete strength exhausted (generally inefficient reinforcement geometry).

6 General non-designability (reserved for special design cases).

7 Shear non-designability.

8 Multiple non-designability encountered. As a rule, a combination of one of the cases 1 to 6 with the shear non-designability case 7


120

REFERENCES This reference list comprises all references made throughout the above text to : (a) national Norm codes; (2) general publications; (3) publications of the Author of this Handbook; (4) internal SCIA publications.

Table 3. List of references

[1] Baumann, Th. : "Zur Frage der Netzbewehrung von Flächentragwerken". In : Der Bauingenieur 47 (1972), P. 36 ff. Ernst & Sohn, Berlin 1975

[2] Schlaich, J. and Schäfer, K. : "Konstruieren im Stahlbetonbau". In : Der Betonkalender 1993, Part 2, P. 327 ff. Ernst & Sohn, Berlin 1993

[3] Willemse, A. H. D. : "Cracking limitation according to NEN 6720". Internal Paper SCIA international, Belguim-Netherlands, Oosterbeek 1997

[4] Willemse, A. H. D. : "Concrete calculation for plates according to NEN 6720". Internal Paper SCIA internat-ional, Belguim-Netherlands, Oosterbeek 1997

[5] DIN 1045 07/1988: "Beton und Stahlbeton, Bemessung und Ausführung", Ausgabe Juli 1988. In : "Betonkalender 1997", Part II, P. 197 ff. Ernst & Sohn, Berlin 1997

[6] DIN 1045 12/78 : "Beton und Stahlbeton, Bemessung und Ausführung", Ausgabe Dezember 1978. In : "Betonkalender 1982", Part II, P. 239 ff. Ernst & Sohn, Berlin 1982

[7] ÖNORM B 4200, Part 8 : "Stahlbetontragwerke. Berechnung und Ausführung I", Ausgabe April 1970. Österreichisches Normungsinstitut (ON), Wien 1970

[8] ÖNORM B 4200, Part 9 : "Stahlbetontragwerke. Berechnung und Ausführung II", Ausgabe August 1971. Österreichisches Normungsinstitut, Wien 1971

[9] Eurocode 2, Teil 1, Deutsche Fassung ENV 1992-1-1 : 1991, Juni 1992. In : "Betonkalender 1995", Part II, P. 252 ff. Ernst & Sohn, Berlin 1995

[10] CSN 73 1201 : "Design of reinforcement concrete structures". Czechoslovak State Norm, State Normalization Institute, Praha 1986 (Czech)

[11] SIA-Norm 162 : "Betonbauten". In : "Betonkalender 1990", Teil II, Seite 373 ff. Ernst & Sohn, Berlin 1990

[12] NEN 6720, "Betonvoorschriften TGB 1990. Constructieve eisen en rekenmethoden", VBC 1995, 2 editie, Nederlands Normalisatie Instituut, Delft 1995

[13] DIN 1045-1 (07.2001) : "Tragwerke aus Beton, Stahlbeton und Spannbeton". In : Avak, R. und Goris, A. (Hrsg.) : "Stahlbetonbau aktuell 2007", Bauwerk Verlag, Berlin 2007

[14] GBJ10-89 : "Code for design of concrete structures", January 1, 1990. National Standard of the People´s Republic of China (English), Ministry of Construction of the P. R. China, New World Press, Beijing 1994

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[15] Oogink, H. : "Internal development specifications on the Chinese Code for design of concrete structures". SCIA n.v., Oosterbeek (NL) 1999

[16] ÖNORM B 4700 : "Stahlbetontragwerke. Eurocode-nahe Berechnung, Bemessung und konstruktive Durchbildung", Ausgabe 2000-09-01. Österreichisches Normungsinstitut, Wien 2000

[17] Hobst, Ed. : "Schubbemessung von Flächentragwerken nach SIA 162, Eurocode 2, ÖNORM B 4700 und E DIN 1045-1". In : "Beton- und Stahlbeton 95", 2000, Heft 6, Ernst & Sohn, Berlin 2000

[18] Hobst, Ed. : Kritische Hinweise zur Biegebemessung von Platten nach DIN 1045, Eurocode 2, DIN 1045-1 und ÖNORM B 4700. In : "Bautechnik 77", 2000, Heft 10, Ernst & Sohn, Berlin 2000

[19] Hobst, Ed. : "Randbedingungen und Singularitäten – wie genau ist die Finite-Elemente-Methode?", In : "Beton- und Stahlbeton 95", 2000, Heft 10, Ernst & Sohn, Berlin 2000

[20] Hobst, Ed. : "Bemerkungen zum Artikel 'Ding Dajun : Bemessung von Stahlbetonscheiben'". In : "Beton- und Stahlbeton 96", 2001, Heft 12, Ernst & Sohn, Berlin 2001

[21] Hobst, Ed.: "Bemerkungen zum Artikel 'Uzunoglu, T.: Notizen aus der Praxis' (Diskussionsforum)". In: "Beton- und Stahlbetonbau 96", 2001, Heft 5, S. 379-382, Ernst & Sohn, Berlin 2001

[22] Hobst, Ed.: "Bemessung von Flächentragwerken nach DIN 1045-1 und ÖNORM B 4700". In: "Beton- und Stahlbeton 98", 2003, Heft 3, Ernst & Sohn, Berlin 2003

[23] Schießl, P : "Grundlagen der Neuregelung zur Beschränkung der Rissbreite". In: "Heft 400 der DAfSt-Schriftenreihe", Beuth-Verlag, Berlin-Köln 1989

[24] Régles BAEL 91, 3e édition, Eyrolles, Paris, 2e tirage 2000 [25] "Béton armé", BAEL 91 modifié 99 et DTU associés, 2e édition, Eyrolles, Paris 2000

[26] SIA-Norm 262:2003 : "Betonbau", Schweizerischer Ingenieur- und Architektenverein, Zürich 2003 (German) & "Construction en béton", Société suisse des ingénieurs et des architectes, Zurich 2003 (French)

[27] Eurocode 2 : Design of concrete structures – Part 1-1. general rules and rules for buildings (EN 1992-1-1:2004), European Standard, 16 April 2004

[28] British Standard BS 8110 : Structural use of concrete, Part 1 : Code of practice for design and construction, BSI 1997; Part 2 : Code of practice for special circumstances, BSI 1985

[29] IS 456:2000 : Plain and reinforced concrete – Code of practice, Indian Standard

[30] IS 2210:1988/1998 : Criteria for design of reinforced concrete shell structures and folded plates, Indian Standard 1998

[31] ACI 318M-05 : Building Code Requirements for Structural Concrete, ACI Standard 2005

[32] Deutscher Ausschuss für Stahlbeton: Erläuterungen zu DIN 1045-1, Heft 525, Page 113 ff, Beuth Verlag, Berlin 2003

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