infograph fem nl manual

7/22/2019 InfoGraph Fem Nl Manual

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Nonlinear Structural Analysis


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The description of program functions within this documentation should not be considered a warranty of product features.All warranty and liability claims arising from the use of this documentation are excluded.

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parts thereof is subject to the written permission of InfoGraph GmbH.

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Title image: Arched Bridge Maulbeerstrasse, BaselCourtesy of DB Netz AG, Brckenmessung- u. bewertung N.BIF, Germany


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Contents

ContentsBasics 2

Area of Application 2Analysis Method 2Finite Beam Elements 3Section Analysis 4Reinforced Concrete Beams 4

Stress-strain-curves for the ultimate limit state check 5Stress-strain-curves for the serviceability check 7Torsional stiffness 9Check of the limit strains (ultimat limit state check) 9Automatic reinforcement increase (ultimate limit state check) 10Concrete creep 10

Steel Beams 10Beams of Free Material 11Area Elements 12Reinforced Concrete Area Elements 12Area Elements of Steel and Bilinear Material 15Solid Elements 16Notes on Convergence Behavior 16

Analysis Settings 17Examples 18

Crosscheck of Two Short-Term Tests 18Reinforced concrete slab 18Reinforced concrete frame 19

Calculation of the Deformation of a Ceiling Slab 20References 23

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Basics

Area of Application

The analysis option Nonlinear analysisenables the determination of the internal forces and deformations of beam and shellstructures made of reinforced concrete and steel as well as stress states of solid elements according to bilinear material lawwhile considering geometric and physical nonlinearities. Due to the considerable numerical complexity, it only makes senseto use this program module for special problems.

Specifically, the following nonlinear effects can be taken into consideration:

Equilibrium of the deformed system according to the second-order theory, if this has been activated for thecorresponding load case.

Beams and area elements made of reinforced concrete according to DIN 1045, DIN 1045-1, OENORM B4700, SIA 262and EN 1992-1-1.

Beams and area elements made of steel with bilinear stress-strain curve under consideration of the Huber-von Misesyield criterion and complete interaction with all internal forces.

Beams, area and solid elements with bilinear stress-strain curve respectively yield criterion according to Raghava andindividually definable compressive and tensile strength.

Compression-flexible beams. Beam and area element bedding with bilinear bedding curve perpendicular to and alongside the beam. Solid elements with bilinear bedding curve in the element coordinate system. The load type Prestressingcan be used with area and solid elements to simulate prestressing without bond (only load). Beams made of steel, reinforced concrete and timber in case of fire according to EN 1992-1-2, EN 1993-1-2 and

EN 1995-1-2 (see 'Structural Analysis for Fire Scenarios'chapter).

Analysis MethodThe analysis model used is based on the finite element method. To carry out a nonlinear system analysis, the standard beamelements are replaced internally with shear-flexible nonlinear beams with an increased displacement function. For thetreatment of area and volume structures nonlinear layer elements and solid elements are used. In contrast to the linear

calculation and in order to account for the nonlinear or even discontinuous properties of the structure, a finer subdivision isgenerally necessary.

Requirements

The beams are assumed to be straight. The area elements are flat. Area and beam sections are constant for each element. The dimensions of the section are small compared with the other system dimensions. Shear deformations of beams are accounted for with a shear distortion that is constant across the section, meaning the

section remains plane after the deformation but is no longer perpendicular to the beam axis.

The mathematical curvature is linearized. The load is slowly increased to its final value and does not undergo any deviation in direction as a result of the system

deformation.

Equilibrium iteration

The solution of the nonlinear system of equations is carried out based on load control according to the incrementalNewton-Raphson method. In this case, the tangential stiffness matrix is calculated as a function of the current internalforces and deformation state in every iteration cycle. Alternatively, the iteration can be carried out with a constant stiffness

matrix according to the modified Newton-Raphson method. For both methods the absorbable load factor ( 1.0) isdetermined through interval nesting. The program assumes that the actions defined in the load cases have already beenweighted with the corresponding partial safety factors.

The following figures illustrate the different iteration progressions using the example of a beam with one degree offreedom.

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Basics

Newton-Raphson method Modified Newton-Raphson method

When performing a check on the entire system according to OENORM B4700 (Section 3.4.3.3) resp. OENORMEN 1992-1-1:2007 (National specification on Section 5.8.6(3)), you need to keep in mind that the decisive actioncombination may only amount to 80% resp. 77% of the reachable system limit load. To account for this, youcan, for example, introduce an additional load factor of 1.25 resp. 1.3.

For reinforced concrete components the existing reinforcement forms the basis of the calculation. If desired, an automaticreinforcement increase can be carried out for the load bearing capacity check to achieve the required load level.

Finite Beam ElementsA shear-flexible beam element with 4 nodes is used. Both inner nodes are hidden to the user. Each node has the degrees of

freedom u , u , u , j , j and j .x y z x y z

Beam element with node degrees of freedom

Because independent deformation functions are used for the displacement and rotation of shear-flexible elements, alldeformations of the given beam element are approximated by third-order polynomials.

This allows an equivalent representation of the strains and curvatures, which is especially advantageous for determining thephysical nonlinearities. All internal forces are described by second-order polynomials (quadratic parabolas).

In connection to shell elements equivalent shear fixed beam elements are used to provide compatibility.

For linear calculation and constant element load in the longitudinal or lateral direction the results are precise. Other loads,

such as point loads in the element or trapezoidal loads in the element, can only be approximated. The quality of the resultsdepends on the chosen structure discretization.

Kinematics

ux(x,y,z) = ux(x) -y jz(x) +z jy(x)

uy(x,y,z) = u

y(x) -z jx(x)

u (x,y,z) = u (x) -y j (x)z z x



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x


Distortion-displacement relationships

2ux 1 uy uz 2 e = + + x 2 x x u uyxg = +xy y xu uz xg = +xz x zj

J' = xx

e = e = g = 0y z y zVirtual internal work

Y(a) = des dV + dgt dV + dJ'M dxxV V L

Normal force Shear TorsionBending

a Nodal degrees of freedom

Tangential stiffness matrix

d Y T T ds =K = d B +B dV + K +Kshear torsionT da de V=K0+Ks+Knl+Kshear+Ktorsion

with de=BdaandB =B0 +BnlIf necessary, terms from the bedding of the elements can be added.

Section AnalysisIn general, a nonlinear analysis can only be performed for polygon sections, database sections and steel sections. For allother section types and for the material types Beton and Timberan elastic material behavior is assumed. In order todetermine the nonlinear stiffnesses and internal forces, you need to perform a numeric integration of the stresses and theirderivatives across the section area. The procedure differs depending on the material.

Reinforced Concrete BeamsBased on the stress-strain curves shown further below, the resulting internal forces Nx, Myand Mzat the section can be

determined for a known strain state by integrating the normal stresses. For determining the strain and stress states at thesection the following assumptions are made:

The sections remain flat at every point in time during the deformation, even when the section is cracked as a result ofexceeding the concrete tensile strength.

The concrete and reinforcement are perfectly bonded. The tensile strength of the concrete is usually ignored, but it can be taken into account if desired. The tensile section of the stress-strain curves can be described using one of the two following curves:

With softening Bilinear



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Basics

Strain when reaching the concrete tensile strengthect

Strain after exceeding the specific fracture energy Gfectu0.7

f MNmcm -5G = G G 3.0 10f f 0 fof m cmo

MN

fcmo =10 mf Average concrete compressive strengthcm

The tension stiffening effect of the concrete is accounted for by the residual tensile stress s= cfctmor cbz. The

constant ccan be set by the user (default: 0.1).

a) Reinforced concrete section with stress-triggering strains

b) Concrete and steel stresses

c) Resultant internal forces

Depending on the chosen material type, the following stress-strain curves are applied. The tensile section of the concrete isonly activated as an option.

Stress-strain-curves for the ultimate limit state check

The following material partial safety factors are implemented. For materials according to OENORM B 4700the user-defined material safety factors for the fundamental combination are used.

Concrete acc. to DIN 1045:1988, Figure 11

Racc. to Table 12

s = -R(eb- 1/4eb) for 0 eb -2 c

Reinforcing steel acc. to DIN 1045:1988, Figure 12

E= 210000 MN/m acc. to Figure 12s

Stress-strain curves according to DIN 1045



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Stress-strain curves according to DIN 1045-1

Concrete acc. to DIN 1045-1, 9.1.5

s acc. to Equation 62 or Figure 22 for 0 e ec1c cwithfcR/gRinstead off ; gR= 1,3c

Ec0m

orElc0m

instead ofEc0

fcR

acc. to Eq. 23 or Eq. 24

Normala= 0.85

concrete:Lightweight flckinstead offckand a= 0.75concrete:

Reinforcing steel acc. to DIN 1045-1, 9.2.3 Figure 26

fyR= 1,1fyk; gR= 1.3acc. to 8.5.1(4)

k= 1.05; euk= 2.5%acc. to Table 11, Column 1 or 2

E= 200000 MN/m acc. to 9.2.4(4)s

Concrete acc. to EN 1992-1-1, 3.1.5

s acc. to Equation 3.14 for 0 e ec1or elc1c cwithfcdorflcdinstead offcmandE orElcmas specifiedcm

Normal fcd= a (Eq. 3.15) with a = 1.0ccfck/gc ccconcrete:Lightweight flcd= alccflck/gc(Eq. 11.3.15)concrete: with alcc= 0.85Standard: g = 1.5acc. to Table 2.1N, Row 1c

Reinforcing steel acc. to EN 1992-1-1, 3.2.7 Figure 3.8

k= 1.05; euk= 2,5%acc. to Table C.1, Column A

E= 200000 MN/m acc. to 3.2.7(4)s

Standard: g = 1.15acc. to Table 2.1N, Row 1sStress-strain curves according to EN 1992-1-1 and OENORM B 1992-1-1

Concrete acc. to DIN EN 1992-1-1, 3.1.5

s acc. to Equation 3.14 for 0 e ec1or elc1c cwithfcR/gRinstead off ; gR= 1.3cmandE orElcmas specifiedcmfcRacc. to Eq. NA.5.12.7

NormalcR= 0.85a fckwith a = 0.85cc ccconcrete:

LightweightfcR= 0.85alccflckwith alcc= 0.75concrete:

Reinforcing steel acc. to DIN EN 1992-1-1, 3.2.7 Figure 3.8

fyR= 1.1fykacc. to NA.5.12.2; gR= 1.3

k= 1.05; euk= 2.5%acc. to NA.5.12.4 or DIN 488

E= 200000 MN/m acc. to 3.2.7(4)sStress-strain curves according to DIN EN 1992-1-1



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Basics

Concrete acc. to OENORM B 4700, 3.4.1.1 Figure 7

withf =fck+ 7.5acc. to Equation 10c andcmf

ck= 0.75f

cwkacc. to 3.4.1.1(2)

Standard: g = 1.5acc. to Table 1, Row 1c

Reinforcing steel acc. to OENORM B 4700, 3.4.1.2 Figure 9

withf =fyk

+ 10.0acc. to Equation 10dym


Standard: g = 1.15acc. to Table 1, Row 1sStress-strain curves according to OENORM B 4700

Concrete acc. to SIA 262, 4.2.1.6 and Figure 12s acc. to Equation 28 for 0 e -2 withc cfcdacc. to Eq. 2 and Eq. 26; g = 1,5acc. to 2.3.2.6cEcd=E / gcEacc. to Eq. 33cmE acc. to Eq. 10 and Eq. 11

cm

gcE= 1.2acc. to 4.2.1.17

Reinforcing steel acc. to SIA 262, 4.2.2.2, Figure 16

withfsd=fsk/g acc. to Eq. 4; g = 1.15acc. to 2.3.2.6s sk= 1.05and e

uk= 2 %acc. to Table 9, Column A

sE= 205000 MN/m acc. to Figure 16

s

Stress-strain curves according to SIA 262

Stress-strain-curves for the serviceability check

The serviceability check is based on the average strengths of the materials. The partial safety factors areassumed to be 1.0.

Concrete with linear stiffness in the compression zonewithout failure limit

Reinforcing steel acc. to DIN 1045:1988, Figure 12

E= 210000 MN/m acc. to 6.2.1s

Stress-strain curves according to DIN 1045



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Concrete acc. to DIN 1045-1, 9.1.5

s acc. to Equation 62 or Figure 22 for 0 e ec1c cwithf orflcminstead off andEc0morElc0minstead ofEc0cm c

Reinforcing steel acc. to DIN 1045-1, 9.2.3 Figure 26

k= 1.05; euk= 2.5% acc. to Table 11, Column 1 or 2

E= 200.000 MN/m acc. to 9.2.4(4)sStress-strain curves according to DIN 1045-1

Concrete acc. to EN 1992-1-1, 3.1.5

s acc. to Equation 3.14 for 0 e ec1or elc1c cwithf acc. to Table 3.1 orflcmacc. to Table 11.3.1cmandE orElcmas specifiedcm

Reinforcing steel acc. to EN 1992-1-1, 3.2.7 Figure 3.8

k= 1.05; euk= 2.5% acc. to Table C.1, Column AE= 200000 MN/m acc. to 3.2.7(4)

s

Stress-strain curves according to EN 1992-1-1, NORM B 1992-1-1 and SS EN 1992-1-1

Concrete acc. to DIN EN 1992-1-1, 3.1.5

s acc. to Equation 3.14 for 0 e ec1or elc1c cwithf acc. to Table 3.1 orf

lcm

acc. to Table 11.3.1cm

andE orElcmas specifiedcm

Reinforcing steel acc. to DIN EN 1992-1-1, 3.2.7 Figure 3.8

k= 1.05 ; euk= 2.5%

E= 200000 MN/msStress-strain curves according to DIN EN 1992-1-1



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Basics

Concrete acc. to OENORM B 4700, 3.4.1.1 Figure 7

withf =fck

+ 7.5acc. to Equation 10c andcm

fck

= 0.75fcwk

acc. to 3.4.1.1(2)

Reinforcing steel acc. to OENORM B 4700, 3.4.1.2 Figure 9

withf =fyk+ 10.0acc. to Equation 10dymE= 200000 MN/m acc. to Figure 9s

Stress-strain curves according to OENORM B 4700

Stress-strain curves according to SIA 262

Concrete acc. to SIA 262, 4.2.1.6 and Figure 12

s acc. to Equation 28 for 0 e -2 withc cf =f

ck+ 8 acc. to Eq. 6 instead of f

cdcm

E acc. to Eq. 10 and Eq. 11 instead ofEcdcm

Reinforcing steel acc. to SIA 262, 4.2.2.2, Figure 16

withfskinstead offsd

k= 1.05and euk= 2 % acc. to Table 9, Column As


Torsional stiffness

When calculating the torsional stiffness of the section, it is assumed that it decreases at the same rate as the bendingstiffness.

1 EI EI y,II z,II GI = GI +x,II x,I2 EI EI y,I z,I

The opposite case, meaning a reduction of the stiffnesses as a result of torsional load, cannot be analyzed. The physicalnonlinear analysis of a purely torsional load is also not possible for reinforced concrete.

Check of the limit strains (ultimat limit state check)

After completion of the equilibrium iteration the permissible limit strains for concrete and reinforcing steel are checked inthe case of frameworks (RSW, ESW). If the permissible values are exceeded the reinforcement will be increased or the loadbearing capacity will be reduced.



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Automatic reinforcement increase (ultimate limit state check)

If requested an automatic increase of reinforcement can be made for frameworks (RSW, ESW) to reach the full load-bearingcapacity. Initially based on the desired start reinforcement the reachable load level is determined. If the reached load-bearingcapacity is lower then 100%, the reinforcement will be increased and the iteration starts again. If no load-bearing capacityexists because of insufficient start reinforcement a base reinforcement can be specified within the reinforcing steel definitionand the design can be performed again.

Concrete creep

The consideration of concrete creep in nonlinear analysis is realized by modifying the underlying stress-strain curves. These

are scaled in strain-direction with the factor (1+j). Also the corresponding limit strains are multiplied by (1+j). Thefollowing figure shows the qualitative approach.

The described procedure postulates that the creep-generating stresses remain constant during the entire creep-period. Dueto stress redistributions, for example to the reinforcement, this can not be guaranteed. Therefore an insignificantoverestimation of the creep-deformations can occur.Stress relaxation during a constant strain state can also be modelled by the described procedure. It leads, however, to anoverestimation of the remaining stresses due to the non-linear stress-strain curves.The nonlinear concrete creep can be activated in the 'Load group' dialog.The plausibility of the achievable calculation results as well as the mode of action of different approaches for the concretetensile strength and the tension stiffeningare demonstrated by two examples in the 'Crosscheck of Two Short-Term Tests'section.

Steel BeamsThe section geometry is determined by the polygon boundary. In order to perform the section analysis, the programinternally generates a mesh. The implemented algorithm delivers all section properties, stresses and internal forces on thebasis of the quasi-harmonic differential equation and others.

Steel section with internal mesh generation

The shear load is described by the following boundary value problem:



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Basics

2j 2j+ = 0

2 2 (Differential equation)y zt = 0 (Boundary condition)xn

222xs (Yield criterion)+ 3t + 3t - s = 0xy xz R,d

The differential equation holds equally for Qy, QzandMx. Depending on the load, the boundary condition delivers different

boundary values of the solution function j. The solution is arrived at by integral representation of the boundary valueproblem and discretization through finite differential equation elements. The internal forces are determined by numericalintegration of the stresses across the section under consideration of the Huber-von Mises yield criterion. The interaction ofall internal forces can be considered.

At the ultimate limit state, the material safety factor gMis taken into account as follows:

Construction steel according to DIN 18800 and the general material type Stahl:The factor gMof the fundamental combination according to DIN 18800 is decisive.

Construction steel according to EN 10025-2:In accordance with EN 1993-1-1, Chapter 6.1(1), gMis assumed to be gM0= 1.0.

The serviceability check is generally calculated with a material safety factor of gM= 1.0.

On the basis of DIN 18800, the interaction between normal and shear stresses is considered only when one of the followingthree conditions is met:

Qy 0.25Qpl,y,d

Qz 0.33Qpl,z,d

Mx 0.20Mpl,x,d

Limiting the bending moments Mpl,y,d and Mpl,z,d according to DIN 18800 to 1.25 times the corresponding elastic limit

moment does not occur.

Stability failure due to cross-section buckling or lateral torsional buckling is not taken into account.

Beams of Free MaterialFor polygon sections of the general material type Frei, the following bilinear stress-strain curve with user-defined tensilestrength (fy,tension ) and compressive strength (fy,compression ) is used.

Stress-strain curve for the general material type Frei

Due to the differing strengths in the compressive and tensile section, the Huber-von Mises yield criterion is not used here.The interaction of normal and shear stresses is also not accounted for.



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Area ElementsSo-called layer elements are used to enable the integration of nonlinear stresses across the area section. This is done bydetermining the stresses in each layer according to the plain stress theory, two-dimensional stress state, under considerationof the physical nonlinearities. The integration of the stresses to internal forces is carried out with the help of the trapezoidalrule.

n-1 n-1nx = t

sxo + sxn + 2 sxi mx = t sxo zo + sxn zn + 2 sxi zi i=1 i=1

n-1 n-1ny = t syo + syn + 2 syi my = t syo zo + syn zn + 2 syi zi

i=1 i=1

n-1 n-1nxy = t

txyo + txyn + 2 txyi mxy = t txyo zo + txyn zn + 2 txyi zi i=1 i=1

Reinforced Concrete Area Elements

The biaxial concrete behavior is realized with the help of the concept of equivalent one-axial strains (Finite Elemente imStahlbetonbau (Finite Elements in Reinforced Concrete Construction), Stempniewski and Eibl, Betonkalender 1993). Theexisting strain state is first transformed into the principal direction in each corresponding layer and then reformed to theequivalent one-axial relationship of the form

0 d e11 0 d e22

G(E1,E2 ) d e12

E 0 0 d e 1 1u1

= 0 E2 0 d e2 u 1- n 0 0 G(E ,E ) d e 1 2 12

with

de = de1

2 n+E

Ede1u 11 22

de2

1 n=E

Ede + de2 u 11 22

1G(E ,E ) = [E +E1 2

4 (1- n) 1 2]212 EE n

.

The shear modulus G is considered to be direction independent in this approach. The poisson's ratio is assumed to beconstant.

The required tangential stiffnesses and stresses are represented using the Saenz curve

s

s

s

12

22

11

d

d

d

n

n

n-= 221

211

001

1EEE

EEE



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Basics

E e0 ius =i 2E e e 0 iu iu1+ - 2 + E e e s iBi iBi with i= 1, 2

under consideration of the biaxial stiffnesses according to Kupfer/Hilsdorf/Rsch.Only when checking the serviceability according to DIN 1045 it is assumed that the compressive area exhibits a linear elastic material behavior.Input parameters for the stress-strain relationship used are: Initial stiffness E0of the one-axial stress-strain curve

Strain eiBiwhen the biaxial concrete compressive strength is reached

Secant stiffnessEwhen the biaxial concrete compressive strength is reacheds

Concrete tensile strengthfct

Factor cfor the residual concrete tensile strength (tension stiffening)

Biaxial concrete compressive strengthfcBi

Biaxial failure curve according to Kupfer/Hilsdorf/Rsch

Standardized equivalent one-axial stress-strain curve for concrete



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Stress-strain curve for the serviceability check according to DIN 1045

The consideration of the concrete creep occurs analog to the procedure for beams through modification of the underlyingstress-strain curves and can be activated in the 'Load group' dialog.

Interaction with the shear stresses from the lateral forces is not considered here.During the specification of thecrack direction the so-called 'rotating crack model' is assumed, meaning that the crack direction can change during thenonlinear iteration as a function of the strain state. Conversely, a fixed crack direction after the initial crack formation can

lead to an overestimation of the load-bearing capacity.The Finite Element module currently does not check the limit strain nor does it automatically increasereinforcement to improve the load-bearing capacity due to the significant numerical complexity.

For clarification of the implemented stress-strain relationships for concrete, the curves measured in the biaxial test accordingto Kupfer/Hilsdorf/Rsch are compared with those calculated using the approach described above.

Biaxial test according to Kupfer/Hilsdorf/Rsch

Calculation results for C25/30;f = 25 + 8 = 33 MN/mcm



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Basics

The existing reinforcing steelis modeled as a 'blurred' orthogonal reinforcement mesh. The corresponding strain state istransformed into the reinforcement directions (p, q) of the respective reinforcing steel layer. The stress-strain curve of theselected reinforced concrete standard is used to determine the stress.

Area Elements of Steel and Bilinear MaterialWhen dealing with bilinear material properties, the yield criterion according to Raghava is used.

F = J2+ 1/3 f

ydf - (f

yd+f ) s = 0yz yz m

withs = 1/3 ( s + s + s )m x y z

J2

= 1/2 (s +s +s )+ t + t + tx y z xy yz zx

s = s - sx x m

s = s - sy y m

s = s - sz z m

Starting from the strain state, a layer is iterated on the yield surface with the help of a 'backward Euler return' that, inconjunction with the Newton-Raphson method, ensures quadratic convergence. (Non-linear Finite Element Analysis of Solids and Structures, M.A. Crisfield, Publisher John Wiley & Sons).The yield surface mentioned above is illustrated below for the 2D stress state, as used for layer elements, for the principalstresses s1and s2as well as for the components sx, syand txy.In this examplef

yd= -20 MN/m andf = 2 MN/m.yz

sy -30-20

-100

1010

55

00

-5-5

-10-10

txysx

Yield surface for the 2D stress state according to Raghava



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Solid ElementsFor solid elements of steel, the material type Freior standardized type of concrete the Raghava yield criterion is used (seeArea Elements of Steel and Bilinear Material). All other material types are assumed to be elastic.

The compressive strength of the standardized concrete types is fck. The concrete tensile strength is cfctm if the analysis

settings with softeningor bilinearare selected or zero in all other cases. At the moment a softening of the material in thetensile area is not supported for solid elements.

The failure criteria of concrete are not represented correctly with the yield criterion pictured below.

Yield surface according to Raghava for principal stresses s1, s2and s3withfyd= -20 MN/m andf = 2 MN/myz

For materials with identical tensile and compressive strength the yield criterion of Raghava is equivalent to the Huber-v.Mises yield criterion.

Yield surface according to Raghava for principal stresses s1, s2and s3with fyd= -235 MN/m andf = 235 MN/myz

Notes on Convergence BehaviorThe implemented Newton-Raphson method with tangential stiffness matrix results in a stable convergence behavior given aconsistent relationship between the stress-strain relation and its derivative. As previously mentioned, this is especially true ofbilinear materials. Reinforced concrete, however, typically displays a poor convergence due to its more complex materialproperties. This is caused by crack formation, not continuously differentiable stress-strain relationships, two-componentmaterials, etc.

Especially for checks of the serviceability and determination of the tensile stiffness with softening, markedly worse

convergence can result due to the negative tangential stiffness in the softening area. If this results in a singular globalstiffness matrix, it is possible to assume a bilinear function in the tensile section or to perform a calculation without tensilestrength.



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Analysis Settings

Analysis SettingsThe following settings are made on the Ultimate Limit state and the Serviceabilitytabs in the Settings for the nonlinearanalysis of the menu itemAnalysis- Settings.

With the nonlinear system analysis, load cases are calculated under consideration of physical and geometrical nonlinearities,whereby the latter only becomes active if the second-order theory is activated in the load case. The load bearing capacityand serviceability check as well as the stability check for fire differ according to the load case that is to be checked, thematerial safety, the different stress-strain-curves and the consideration of the concrete tensile strength.

Consider the following load cases

The load cases from the left list box are calculated.

Start reinforcement

The nonlinear system analysis is carried out on reinforced concrete sections based on the reinforcement selected here. Thisresults from a reinforced concrete design carried out in advance. The starting reinforcement Nullcorresponds to the basereinforcement of the reinforcing steel layers. When performing a check for fire scenarios, special conditions apply asexplained in the "Structural Analysis for Fire Scenarios'chapter.

Automatic reinforcement increase (RSW, ESW)

For the ultimate limit state check a reinforcement increase is carried out for reinforced concrete sections to achieve therequired load-bearing safety.

Concrete tensile strength; Factor c

This option defines the behavior in the tensile zone for the nonlinear internal forces calculation for all reinforced concretesections. By default the ultimate limit state check is performed without considering the concrete tensile strength.

With softening Bilinear

Max. iterations per load step

Maximum number of iteration steps, in order to reach convergence within one load step. If the error threshold is exceeded,the iteration is interrupted and an expectable reduced load level is determined on the basis of load.

Layers per area element

Number of integration levels of an area element. Members subject to bending should be calculated with 10 layers. Structuremostly subject to normal forces can be adequately analyzed with 2 layers.

Modified Newton method

The iteration is done according to the 'Modified Newton Raphson method'. If the switch is not set, then the 'NewtonRaphson method'is used.



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Examples

Crosscheck of Two Short-Term TestsIn the following section two simple structures made of reinforced concrete (taken from Krtzig/Meschke 2001) are analyzed.The goal is to demonstrate the plausibility of the achieved calculation results as well as the mode of action of differentapproaches for determining concrete tensile strength and tension stiffening.

Reinforced concrete slab

The system described below was analyzed experimentally by Jofriet & McNeice in 1971. For the crosscheck a system with10x10 shell elements was used as illustrated below.

Element system with supports and load. Dimensions in [cm]

The load-displacement curve determined in the test for the slab middle is contrasted with the results of the static calculationin the following diagram. The material parameters were set according to the specifications of the authors.

Load-displacement curves from the crosscheck and experiment (Jofriet/McNeice)

In order to demonstrate the mode of action of the methods implemented in the program for the concrete tensile stresses,

three variants were calculated. The curve accounting for the concrete tensile strength with softening (c=0.2) exhibits theclosest agreement with the test results, as expected. The behavior at the beginning of crack formation as well as close toload-bearing capacity display a very large level of agreement. Only during the crossover to state II is the stiffness of the slabsomewhat overestimated.

The curve for the bilinear behavior in the concrete tensile area was intentionally calculated with the same value ( c=0.2). Thestiffness of the slab is thus, as expected, underestimated at the beginning of crack formation. The load-bearing safety is,



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Examples

however, hardly influenced by this. This means that the calculation is on the safe side.The curve for the 'naked state II' is for the most part determined by the characteristic curve of the reinforcing steel and thus has exhibits nearly linear behavior in the area under examination. Here as well a comparable load-bearing safety results interms of the plasticity theory; however, with significantly larger deformations.

Reinforced concrete frame

The following reinforced concrete frame was analyzed by Ernst, Smith, Riveland & Pierce in 1973. The static system withmaterial parameters is shown below.

System with sections, load and dimensions [mm]

Material parameters of the experiment:

Concrete: f = 40.82 [MPa]cm

Reinforcing steel: 9.53:fy= 472.3 [MPa],Es= 209 [GPa] (cold-drawn)

12.7:fy= 455.0 [MPa],Es= 200 [GPa] (hot-rolled)

Material parameters of the crosscheck:Concrete: C 30/37(EN 1992-1-1:2004),f = 38 [MPa],fctm= 2.9 [MPa]cmReinforcing steel: 9.53:fyk= 472.,3 [MPa], 12.7:fyk= 455.0 [MPa]

Es= 200 [GPa], (ft/fy)k= 1.05 (bilinear)

The following figure shows stress-strain-curves of the reinforcing steel used for the experiment and the crosscheck withinthe strain area up to 2 .

ss[MN/m] 9.53 (Calculation) 12.7 (Calculation)600

400

200

0

12.7 (Experiment) 9.53 (Experiment)

0 5 10 15 20 25 e[]

Stress-strain-curves of reinforcement steel from experiment and calculation

The crosscheck was performed according to the second-order theory. Softening in the concrete tensile area was consideredfor the first calculation variant whereas no concrete tensile strength was assumed for the second variant. According to theexperiment, cracking occurs at a limit load of P = 5 kN corresponding to a concrete tensile strength of about 2.2 MPa. Incompliance with the standard, the crosscheck was based on a tensile strength of 2.9 MPa resulting in a slightly raisedcracking level. The attaining of the system limit load is indicated by stability failure. The failure is caused by reinforcementflow at a load of approx. 35-40 kN and formation of a plastic hinge in the frame center. The slightly lower failure load is



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due to different stress-strain-curves for concreted and reinforcing steel decisive for the experiment and the crosscheck. Asexpected, the calculation without tensile strength results in a lower bound for the load-displacement curve of the framecenter.

Load-displacement curves from the crosscheck and experiment (Ernst et al. 1973)

Calculation of the Deformation of a Ceiling SlabThis example shows a ceiling slab of a building construction which is based on the chapter 'Grundlagen der Bemessungnach DIN 1045-1 in Beispielen', Betonkalender 2001. Deformations in the serviceability and ultimate limit state shall becalculated for the depictured slab. As normal forces also occur in the slab during a nonlinear structural analysis, shellelements are to be used, and the horizontal supports should preferably be free of restraint. Because the longitudinalreinforcement has an essential influence on the deformations within a nonlinear structural analysis, a realistic reinforcementarrangement is to be ensured. For that reason firstly a linear-elastic load case calculation with subsequent design according

to EN 1992-1-1 is carried out.

Loads

Load case 1, permanent loads

Load case 2 to 9, area loads

Element system with dimensions [m]



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Examples

Design according to EN 1992-1-1

The slab made of C30/37 is designed for the internal forces resulting from the 'permanent and temporary situation' usingthe base reinforcement listed below. Only in the area around the columns and the corners an increase of reinforcementresults, which approximately corresponds to the inserted reinforcement from the example in the literature.

Reinforcement for area elements

No. Lay. Qual. d1x[m]

d2x[m]

asx[cm/m]

d1y[m]

d2y[m]

asy[cm/m]

asfix

1 1 500M 0.045 5.670 0.035 5.6502 500M 0.035 7.850 0.025 11.340

2 1 500M 0.045 5.670 0.035 5.6502 500M 0.035 7.850 0.025 5.030

as Base reinforcementd1 Distance from the upper edged2 Distance from the lower edge

The z axis of the element system points to the lower edge

Bending reinforcement from design

Bending reinforcement upper layer asy.1

[cm/m] Bending reinforcement lower layer asy.2

[cm/m]

Deformations and stresses

Subsequently the different deformation results are compared. For better comparison all situations are calculated with fullload and a partial safety factor g= 1.In all cases a load factor of 1.0 is reached.The load cases (21 and 22) in the serviceability limit state are calculated with softening in the tensile zone of the concreteand a residual tensile strength of cf

ctm= 0.12.9 = 0.29 MN/m.

In load case 22 the nonlinear creeping of the concrete is additionally taken into account with the creep coefficient j= 2.5The load case 23 for the ultimate limit state is calculated without consideration of the concrete tensile strength.Load cases max u

z[mm]

10 Full load with elastic material behavior 10.8

11 Creeping with elastic material behavior 27.1

12 Total (load case 10 + load case 11) 37.921 Nonlinear analysis, serviceability limit state (j= 0 with softening) 42.722 Nonlinear analysis, serviceability limit state j= 2.5 with softening) 53.023 Nonlinear analysis, ultimate limit state (j= 0 without concrete tensile strength) 64.8



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Load case 22: Color gradient of the deformations u [mm]z

Load case 22: Concrete stresses [MN/m] in y-direction at the bottom

Load case 22: Steel stresses [MN/m] in the lower steel layer in y-direction



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References

ReferencesDuddeck, H.; Ahrens, H.

Statik der Stabwerke (Statics of Frameworks), Betonkalender 1985.Ernst & Sohn Berlin 1985.

Ernst, G.C.; Smith, G.M.; Riveland, A.R.; Pierce, D.N.Basic reinforced concrete frame performance under vertical and lateral loads.

ACI Material Journal 70(28), pp. 261-269.American Concrete Institute, Farmingten Hills 1973.

Hirschfeld, K.Baustatik Theorie und Beispiele (Structural Analysis Theory and Examples).Springer Verlag, Berlin 1969.

Jofriet, J.C.; McNeice, M.Finite element analysis of reinforced concrete slabs.Journal of the Structural Division (ASCE) 97(ST3), pp. 785-806.American Society of Civil Engineers, New York 1971.

Kindmann, R.Traglastermittlung ebener Stabwerke mit rumlicher Beanspruchung

(Limit Load Determination of 2D Frameworks with 3D Loads).Institut fr Konstruktiven Ingenieurbau,Ruhr-Universitt Bochum, Mitteilung Nr. 813, Bochum 1981.

Knig, G.; Weigler, H.Schub und Torsion bei elastischen prismatischen Balken (Shear and Torsion for Elastic Prismatic Beams).Ernst & Sohn Verlag, Berlin 1980.

Krtzig, W.B.; Meschke, G.Modelle zur Berechnung des Stahlbetonverhaltens und von Verbundphnomenen unter Schdigungsaspekten(Models for Calculating the Reinforced Concrete Behavior and Bonding Phenomena under Damage Aspects).Ruhr-Universitt Bochum, SFB 398, Bochum 2001.

Link, M.

Finite Elemente in der Statik und Dynamik (Finite Elements in Statics and Dynamics).Teubner Verlag, Stuttgart 1984.

Petersen, Ch.Statik und Stabilitt der Baukonstruktionen (Statics and Stability of Constuctions).Vieweg Verlag, Braunschweig 1980.

Quast, U.Nichtlineare Stabwerksstatik mit dem Weggrenverfahren (Non-linear Frame Analysis with the Displacement-Method).Beton- und Stahlbetonbau 100.Ernst & Sohn Verlag, Berlin 2005.

Schwarz, H. R.Methode der finiten Elemente (Method of Finite Elements).

Teubner Studienbcher.Teubner Verlag, Stuttgart 1984.

Stempniewski, L.; Eibl, J.Finite Elemente im Stahlbetonbau (Finite Elements in Reinforced Concrete Construction).Betonkalender 1993.Ernst & Sohn Verlag, Berlin 1993.



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