scia - aluminium code check theory

8/19/2019 Scia - Aluminium Code Check Theory

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Subtitle

Theory

Aluminium Code Check


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Introduction.............................................................................................................................................. 1

Disclaimer ................................................................................................................................................ 2

EN 1999 Code Check ............................................................................................................................... 3

Material Properties .......................................................................................................... 3

Consulted Articles........................................................................................................... 4

Initial Shape ................................................................................................................. 5

Classification of Cross-Section .................................................................................. 13

Step 1: Calculation of stresses ............................................................................. 14

Step 2: Determination of stress gradient ........................................................... 14

Step 3: Calculation of slenderness ....................................................................... 15

Step 4: Classification of the part ........................................................................... 16

Reduced Cross-Section properties ............................................................................ 16

Calculation of Reduction factor c for Local Buckling ........................................... 17

Calculation of Reduction factor for Distortional Buckling ................................... 17

Calculation of Reduction factor HAZ for HAZ effects ............................................ 22

Calculation of Effective properties ........................................................................ 23

Section properties ...................................................................................................... 23

Tension....................................................................................................................... 24

Compression .............................................................................................................. 24

Bending moment ........................................................................................................ 24

Shear .......................................................................................................................... 25

Slender and non-slender sections ........................................................................ 25

Calculation of Shear Area ..................................................................................... 27

Torsion with warping .................................................................................................. 28

Calculation of the direct stress due to warping ..................................................... 29

Calculation of the shear stress due to warping ..................................................... 31

Standard diagrams ................................................................................................ 33

Decomposition of arbitrary torsion line ................................................................. 39

Combined shear and torsion ...................................................................................... 40

Bending, shear and axial force .................................................................................. 41

Localised welds ..................................................................................................... 41

Shear reduction ..................................................................................................... 41

Stress check for numerical sections ..................................................................... 42

Flexural buckling ........................................................................................................ 43

Calculation of Buckling ratio – General Formula .................................................. 43

Calculation of Buckling ratio – Crossing Diagonals .............................................. 45

Calculation of Buckling ratio – From Stability Analysis ......................................... 48

Torsional (-Flexural) buckling ..................................................................................... 49

Calculation of Ncr,T ................................................................................................. 49

Calculation of Ncr,TF ............................................................................................... 50

Lateral Torsional buckling .......................................................................................... 51

Calculation of Mcr – General Formula ................................................................... 51

Calculation of Moment factors for LTB ................................................................. 54

LTBII Eigenvalue solution ..................................................................................... 55

Combined bending and axial compression ................................................................ 55

Flexural buckling ................................................................................................... 56


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Lateral Torsional buckling ..................................................................................... 56

Localised welds and factors for design section .................................................... 56

Shear buckling ........................................................................................................... 59

Plate girders with stiffeners at supports ................................................................ 59

Plate girders with intermediate web stiffeners ...................................................... 61

Interaction ............................................................................................................. 64

Scaffolding ................................................................................................................. 65

Scaffolding member check for tubular members .................................................. 65

Scaffolding coupler check ..................................................................................... 67 LTBII: Lateral Torsional Buckling 2nd Order Analysis ...................................................................... 71

Introduction to LTBII ................................................................................................... 71

Eigenvalue solution Mcr .............................................................................................. 71

2nd

Order analysis ...................................................................................................... 73

Supported Sections .................................................................................................... 74

Loadings ..................................................................................................................... 75

Imperfections .............................................................................................................. 76

Initial bow imperfection v0 according to code ....................................................... 76 Manual input of Initial bow imperfections v0 and w0 ............................................ 76

LTB Restraints ........................................................................................................... 77

Diaphragms ................................................................................................................ 78

Linked Beams ............................................................................................................ 79

Limitations and Warnings ........................................................................................... 80

Eigenvalue solution Mcr ........................................................................................ 80

2nd

Order Analysis ................................................................................................. 80 Profile conditions for code check ....................................................................................................... 81

Introduction to profile characteristics ......................................................................... 81

Data for general section stability check ..................................................................... 81

Data depending on the profile shape ......................................................................... 82

I section ................................................................................................................. 82

RHS ....................................................................................................................... 83

CHS ....................................................................................................................... 84

Angle section......................................................................................................... 85

Channel section .................................................................................................... 86

T section ................................................................................................................ 87

Full rectangular section ......................................................................................... 88

Full circular section ............................................................................................... 89

Asymmetric I section ............................................................................................. 90

Z section ................................................................................................................ 91

General cold formed section ................................................................................. 92 Cold formed angle section .................................................................................... 94

Cold formed channel section ................................................................................ 95

Cold formed Z section ........................................................................................... 96

Cold formed C section .......................................................................................... 97

Cold formed Omega section ................................................................................. 98

Rail type KA .......................................................................................................... 99

Rail type KF......................................................................................................... 100


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Rail type KQ ........................................................................................................ 101 References ........................................................................................................................................... 102


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Introduction

Welcome to the Aluminium Code Check – Theoretical Background.This document provides background information on the code check according to theregulations given in:

Eurocode 9Design of aluminium structuresPart 1-1: General structural rulesEN 1999-1-1:2007

Addendum EN 1999-1-1:2007/A1:2009

Version info

Documentation Title Aluminium Code Check – Theoretical BackgroundRelease 2012.0Revision 03/2012


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Disclaimer

This document is being furnished by SCIA for information purposes only to licensed usersof SCIA software and is furnished on an "AS IS" basis, which is, without any warranties,whatsoever, expressed or implied. SCIA is not responsible for direct or indirect damage asa result of imperfections in the documentation and/or software.

Information in this document is subject to change without notice and does not represent acommitment on the part of SCIA. The software described in this document is furnishedunder a license agreement. The software may be used only in accordance with the terms ofthat license agreement. It is against the law to copy or use the software except asspecifically allowed in the license.

© Copyright 2012 Nemetschek SCIA. All rights reserved.


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EN 1999 Code Check

In the following chapters, the material properties and consulted articles are discussed.

Material Properties

The characteristic values of the material properties are based on Table 3.2a for wroughtaluminium alloys of type sheet, strip and plate and on Table 3.2b for wrought aluminiumalloys of type extruded profile, extruded tube, extruded rod/bar and drawn tube.

The following alloys are provided by default:

EN-AW 5083 (Sheet) O/H111 (0-50)EN-AW 5083 (Sheet) O/H111 (50-80)EN-AW 5083 (Sheet) H12 (0-40)EN-AW 5083 (Sheet) H22/H32 (0-40)EN-AW 5083 (Sheet) H14 (0-25)EN-AW 5083 (Sheet) H24/H34 (0-25)EN-AW 5083 (ET,EP,ER/B)O/111,F,H112 (0-200)EN-AW 5083 (DT) H12/22/32 (0-10)EN-AW 5083 (DT) H14/24/34 (0-5)EN-AW 5454 (ET,EP,ER/B)O/H111,F/H112 (0-25)EN-AW 5754 (ET,EP,ER/B)O/H111,F/H112 (0-25)EN-AW 5754 (DT) H14/H24/H34 (0-10)EN-AW 6005A (EP/O,ER/B) T6 (0-5)EN-AW 6005A (EP/O,ER/B) T6 (5-10)EN-AW 6005A (EP/O,ER/B) T6 (10-25)EN-AW 6005A (EP/H,ET) T6 (0-5)EN-AW 6005A (EP/H,ET) T6 (5-10)

EN-AW 6060 (EP,ET,ER/B) T5 (0-5)EN-AW 6060 (EP) T5 (5-25)EN-AW 6060 (ET,EP,ER/B) T6 (0-15)EN-AW 6060 (DT) T6 (0-20)EN-AW 6060 (EP,ET,ER/B) T64 (0-15)EN-AW 6060 (EP,ET,ER/B) T66 (0-3)EN-AW 6060 (EP) T66 (3-25)EN-AW 6061 (EP,ET,ER/B) T4 (0-25)EN-AW 6061 (DT) T4 (0-20)EN-AW 6061 (EP,ET,ER/B) T6 (0-25)EN-AW 6061 (DT) T6 (0-20)EN-AW 6063 (EP,ET,ER/B) T5 (0-3)

EN-AW 6063 (EP) T5 (3-25)EN-AW 6063 (EP,ET,ER/B) T6 (0-25)EN-AW 6063 (DT) T6 (0-20)EN-AW 6063 (EP,ET,ER/B) T66 (0-10)EN-AW 6063 (EP) T66 (10-25)EN-AW 6063 (DT) T66 (0-20)EN-AW 6082 (Sheet) T4/T451 (0-12.5)EN-AW 6082 (Sheet) T61/T6151 (0-12.5)EN-AW 6082 (Sheet) T6151 (12.5-100)EN-AW 6082 (Sheet) T6/T651 (0-6)EN-AW 6082 (Sheet) T6/T651 (6-12.5)EN-AW 6082 (Sheet) T651 (12.5-100)EN-AW 6082 (EP,ET,ER/B) T4 (0-25)EN-AW 6082 (EP/O,EP/H) T5 (0-5)EN-AW 6082 (EP/O,EP/H,ET) T6 (0-5)EN-AW 6082 (EP/O,EP/H,ET) T6 (5-15)EN-AW 6082 (ER/B) T6 (0-20)EN-AW 6082 (ER/B) T6 (20-150)EN-AW 6082 (DT) T6 (0-5)

EN-AW 6082 (DT) T6 (5-20)EN-AW 7020 (Sheet) T6 (0-12.5)EN-AW 7020 (Sheet) T651 (0-40)EN-AW 7020 (EP,ET,ER/B) T6 (0-15)EN-AW 7020 (EP,ET,ER/B) T6 (15-40)EN-AW 7020 (DT) T6 (0-20)EN-AW 8011A (Sheet) H14 (0-12.5)EN-AW 8011A (Sheet) H24 (0-12.5)EN-AW 8011A (Sheet) H16 (0-4)EN-AW 8011A (Sheet) H26 (0-4)

The default HAZ values are applied. As such, footnote 2) of Table 3.2a and footnote 4) ofTable 3.2b are not accounted for. The user can modify the HAZ values according tothese footnotes if required.


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Consulted Articles

The member elements are checked according to the regulations given in: “Eurocode 9:Design of aluminium structures - Part 1-1: General structural rules - EN 1999-1-1:2007 ”.

The cross-sections are classified according to art.6.1.4. All classes of cross-sections areincluded. For class 4 sections (slender sections) the effective section is calculated in eachintermediary point, according to Ref. [2].

The stress check is taken from art.6.2 : the section is checked for tension (art. 6.2.3),compression (art. 6.2.4), bending (art. 6.2.5 ), shear (art. 6.2.6 ), torsion (art.6.2.7 ) andcombined bending, shear and axial force (art. 6.2.8 , 6.2.9 and 6.2.10 ).

The stability check is taken from art. 6.3: the beam element is checked for buckling (art.6.3.1), lateral torsional buckling (art. 6.3.2 ), and combined bending and axial compression(art. 6.3.3).

The shear buckling is checked according to art. 6.7.4 and 6.7.6.

For I sections, U sections and cold formed sections warping can be considered.

A check for critical slenderness is also included.

A more detailed overview for the used articles is given in the following table. The articlesmarked with "X" are consulted. The articles marked with (*) have a supplementaryexplanation in the following chapters.

5.3 Imperfections

5.3.1 Basis X

5.3.2 Imperfections for global analysis of frames X

5.3.4 Member imperfections X

6 Ultimate limit states for members

6.1 Basis6.1.3 Partial safety factors X

6.1.4 Classification of cross-sections X (*)

6.1.5 Local buckling resistance X (*)

6.1.6 HAZ softening adjacent to welds X (*)

6.2 Resistance of cross-sections

6.2.1 General X (*)

6.2.2 Section properties X (*)

6.2.3. Tension X (*)

6.2.4. Compression X (*)

6.2.5. Bending Moment X (*)

6.2.6. Shear X (*)

6.2.7. Torsion X (*)

6.2.8. Bending and shear X

6.2.9. Bending and axial force X (*)

6.2.10. Bending , shear and axial force X (*)

6.3 Buckling resistance of members

6.3.1 Members in compression X (*)

6.3.2 Members in bending X (*)

6.3.3 Members in bending and axial compression X (*)


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6.5 Un-stiffened plates under in-plane loading

6.5.5 Resistance under shear X (*)

6.7 Plate girders

6.7.4 Resistance to shear X (*)

6.7.6 Interaction X (*)

Haunches and arbitrary members are not supported for the Aluminium Code Check.

Initial Shape

For a cross-section with material Aluminium, the Initial Shape can be defined.

For a General cross-section the ‘Thinwalled representation’ has to be used to be able todefine the Initial Shape.

The thin-walled cross-section parts can have the following types:

F Fixed Part – No reduction is needed

I Internal cross-section part

SO Symmetrical Outstand

UO Unsymmetrical Outstand

Parts can also be specified as reinforcement:

None Not considered as reinforcement

RI Reinforced Internal (intermediate stiffener)

RUO Reinforced Unsymmetrical Outstand (edge stiffener)

In case a part is specified as reinforcement, a reinforcement ID can be inputted. Partshaving the same reinforcement ID are considered as one reinforcement.

The following conditions apply for the use of reinforcements:

- RI: There must be a plate type I on both sides of the RI reinforcement,

RI RI

I I I I


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- RUO : The reinforcement is connected to only one plate with type I

RUOI


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For standard Cross-sections, the default plate type and reinforcement type are defined inthe following table.

Form code Shape Initial Geometrical shape

1 I section

(SO, none)(SO, none)

(SO, none) (SO, no

(F, none)

(F, none)

(I, none)

2 RHS (I, none)

(I, none)

(I, none)

(I, none)

(F, none)(F, none)

(F, none)(F, none)

3 CHS (fixed value for )


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4 Angle section

(UO, none)

(UO, none)(F, none)

5 Channel section (UO, none)

(UO, none)

(I, none)

(F, none)

(F, none)


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6 T section

(UO, none)


(F, none)

7 Full rectangularsection

No reduction possible

11 Full circular section No reduction possible101 Asymmetric I section (SO, none)(SO, none)


(F, none)

(F, none)

(I, none)


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102 Rolled Z section (UO, none)

(UO, none)

(I, none)

(F, none)

(F, none)

110 General cold formedsection

(UO, none)

(I, none)

(I,none)

(UO, none)

(UO, none)


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111 Cold formed angle

(UO, none)

(UO, none)

112 Cold formed channel (UO, none)

(I, none)

(UO, none)


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113 Cold formed Z (UO, none)

(UO, none)

(I, none)

114 Cold formed C

section

(I, none)

(I,none)

(I,none)

(UO, RUO)

(UO, RUO)

115 Cold formed Omega (I, none)

(I, none)(I, none)

(UO, RUO)(UO, RUO)


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For other predefined cross-sections, the initial geometric shape is based on the centrelineof the cross-section. For example Sheet Welded - IXw

(UO, none)(UO, none)

(UO, none)

(UO, none)

(UO, none)

(UO, none)

(UO, none)(UO, none)

(I,none)

(I,none)(I,none)

(I,none)

Classification of Cross-Section

The classification is based on art. 6.1.4.

For each intermediary section, the classification is determined and the proper checks areperformed. The classification can change for each intermediary point.

Classification for members with combined bending and axial forces is made for the loadingcomponents separately. No classification is made for the combined state of stress (see art.6.3.3 Note 1 & 2 ).

Classification is thus done for N, My and Mz separately. Since the classification isindependent on the magnitude of the actual forces in the cross-section, the classification isalways done for each component.


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Taking into account the sign of the force components and the HAZ reduction factors, thisleads to the following force components for which classification is done:

Classification for Component

Compression force N-

Tension force N+ with 0,HAZ

Tension force N+ with u,HAZ y-y axis bending My-

y-y axis bending My+

z-z axis bending Mz-

z-z axis bending Mz+

For each of these components the reduced shape is determined and the effective sectionproperties are calculated. This is outlined in the following paragraphs.

The following procedure is applied for determining the classification of a part.

Step 1: Calculation of stresses

For the given force component (N, My, Mz) the normal stress is calculated over therectangular plate part for the initial geometrical shape.

beg: normal stress at start point of rectangular shape

end: normal stress at end point of rectangular shape

Compression stress is indicated as negative.

When the rectangular shape is completely under tension, i.e. beg and end are bothtensile stresses, no classification is required.

Step 2: Determination of stress gradient

if end is the maximum compression stress

end

beg

if beg is the maximum compression stress

beg

end


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Step 3: Calculation of slenderness

Depending on the stresses and the plate type the slenderness parameter is calculated.

Internal part: ty pe I

With: b Width of the cross-section partt Thickness of the cross-section part

Stress gradient factor

Remark:

For a thin walled round tubet

D3 with D the diameter to mid-thickness of the tube

material.

Outstand p art: type SO, UO

When = 1.0 or peak compression at the toe of the plate:

peak compression at toe

When peak compression is at the root of the plate:

)1(1

80.0

)11(30.070.0

t

b

t b

)1(1

80.0

)11(30.070.0

t

b


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peak compression at root

Step 4: Classification of the part

The slenderness parameters 1, 2, 3 are determined according to Table 6.2. Using these limits, the part is classified as follows:

if 1 : class 1

if 1


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Calculation of Reduction factorc for Local Buckling

In case a cross-section part is classified as Class 4 (slender), the reduction factor c forlocal buckling is calculated according to art. 6.1.5

For a cross-section part under tension or with classification different from Class 4 the

reduction factor c is taken as 1,00.

In case a cross-section part is subject to compression and tension stresses, the reduction

factor c is applied only to the compression part as illustrated in the following figure.

compression stress

tensile stress

t

t eff

b

Calculation of Reduction factor for Distortional Buckling

To take into account distortional buckling, a simplified direct method is given in art. 6.1.4 which is only applicable for a single sided rib or lip.

In Scia Engineer a more general procedure is used according to Ref. [2] pp.66 The design of stiffened elements is based on the assumption that the stiffener itself acts asa beam on elastic foundation, where the elastic foundation is represented by a springstiffness depending on the transverse bending stiffness of adjacent parts of plane elementsand on the boundary conditions of these elements.

The following procedure is applied for calculating the reduction factor for an intermediatestiffener (RI) or edge stiffener (RUO).


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Step 1: Calculat ion of sprin g sti f fness

Spring stiffness c = cr for RI:

Spring stiffness c = cs for RUO:


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ad p

ad

s

s

s

b Et c

c

b

Et

b y

ycc

,

3

3

3

2

1

3

3

1

²)1(12

²)1(4

1

With: tad Thickness of the adjacent elementbp,ad Flat width of the adjacent elementc3 The sum of the stiffnesses from the adjacent elementsα equal to 3 in the case of bending moment load or when the cross section

is made of more than 3 elements (counted as plates in initial geometry,without the reinforcement parts)equal to 2 in the case of uniform compression in cross sections made of 3elements (counted as plates in initial geometry, without the reinforcementparts, e.g. channel or Z sections)

These parameters are illustrated on the following picture:

edge stiffener

considered plate

adjacent element

t ad

bp,ad


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Step 2: Calculat ion of Area and Second moment o f area

After calculating the spring stiffness the area Ar and Second moment of area Ir are calculated.

With: Ar the area of the effective cross section (based on teff = pc t ) composed ofthe stiffener area and half the adjacent plane elements

Ir the second moment of area of an effective cross section composed of

the (unreduced) stiffener and part of the adjacent plate elements, withthickness t and effective width beff , referred to the neutral axis a-a

beff For RI reinforcement taken as 15 tFor ROU reinforcement taken as 12 t

These parameters are illustrated on the following figures.

Ar and Ir for RI:


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Ar and Ir for RUO:

Step 3: Calculat ion of sti f fener buckl ing load

The buckling load Nr,cr of the stiffener can now be calculated as follows:

With: c Spring stiffness of Step 1E Module of YoungIr Second moment of area of Step 2

r cr r cEI N 2,


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Step 4: Calculat ion of reduction factor for distor t ional buckl ing

Using the buckling load Nr,cr and area Ar the relative slenderness c can be determined for

calculating the reduction factor :

00.11

00.1

))(0.1(50.0

60.0

20.0

220

0

2

0

0

,

c

c

c

cc

cr r

r oc

if

if

N

A f

With: f 0 0,2% proof strengthc Relative slenderness

0 Limit slenderness taken as 0,60

α Imperfection factor taken as 0,20

Reduction factor for distortional buckling

The reduction factor is then applied to the thickness of the reinforcement(s) and on half thewidth of the adjacent part(s).

Calculation of Reduction factorHAZ for HAZ effects

The extend of the Heat Affected Zone (HAZ) is determined by the distance bhaz according toart. 6.1.6 .

The value for bhaz is multiplied by the factors 2 and 3/n


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for 3xxx, 5xxx & 6xxx alloys :120

)601(12

T

for 7xxx alloys :120

)601(5.112

T

With: T1 Interpass temperaturen Number of heat paths

The variations in numbers of heath paths 3/n is specifically intended for fillet welds. Incase of a butt weld the parameter n should be set to 3 (instead of 2) to negate thiseffect.

The reduction factor for the HAZ is given by:

u

haz,u

haz,u

f

f

o

haz,o

haz,of

f

Calculation of Effective properties

For each part the final thickness reduction is determined as the minimum of .c and haz.

The section properties are then recalculated based on the reduced thicknesses.

This procedure is then repeated for each of the force components specified in the previouschapter.

Section properties

Deduction of holes, art. 6.2.2.2 is not taken into account.

Shear lag effects, art. 6.2.2.3 are not taken into account.


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Tension

The Tension check is verified using art. 6.2.3.

The value of Ag is taken as the area A calculated from the reduced shape for N+(0,HAZ)

The value of Anet is taken as the area A calculated from the reduced shape for N+(u,HAZ)

Since deduction of holes is not taken into account Aeff will be equal to Anet.

Compression

The Compression check is verified using art. 6.2.4.

Deduction of holes is not taken into account.

The value of Aeff is taken as the area A calculated from the reduced shape for N-

Bending momentThe Bending check is verified using art. 6.2.5.


The section moduli Weff ; Wel,haz; Weff,haz are taken as Wel calculated from the reduced shapefor M+ / M-

The section modulus Wpl,haz is taken as Wpl calculated from the reduced shape for M+ / M-

In case the alternative formula is used for 3,u or 3,w the critical part is determined bythe lowest value of 2 / in accordance with addendum EN 1999-1-1:2007/A1:2009.

The assumed thickness specified in art. 6.2.5.2 (2) e) is not supported.


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Shear

The Shear check is verified using art. 6.2.6 & 6.5.5.


Slender and non-slender sections

The formulas to be used in the shear check are dependent on the slenderness of the cross-section parts.

For each part i the slenderness is calculated as follows:

i

beg end

iw

wi

t

x x

t

h

With: xend End position of plate ixbeg Begin position of plate i

t Thickness of plate i

For each part i the slenderness is then compared to the limit 39

With0

250 f

and f 0 in N/mm²

39i => Non-slender plate

39i => Slender plate

I) All parts are classified as non-slender 39i

The Shear check shall be verified using art. 6.2.6.

II) One or more parts are classified as slender 39i

The Shear check shall be verified using art. 6.5.5.

For each part i the shear resistance VRd,i is calculated.


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Non-slender part : Formula (6.88) is used with properties calculated from the reduced shape

for N+(u,HAZ)

For Vy: Anet,y,i = ii HAZ ibeg end t x x 2

,0 cos)(

For Vz: Anet,z,i = ii HAZ ibeg end t x x

2

,0 sin)(

With: i The number (ID) of the platexend End position of plate ixbeg Begin position of plate it Thickness of plate i

0,HAZ Haz reduction factor of plate i

Angle of plate i to the Principal y-y axis

Slender part: Formula (6.88) is used with properties calculated from the reduced

shape for N+(0,HAZ) in the same way as for a non-slender part.=> VRd,i,yield

Formula (6.89) is used with a the member length or the distancebetween stiffeners (for I or U-sections)=> VRd,i,buckling

=> For this slender part, the eventual VRd,i is taken as the minimum ofVRd,i,yield and VRd,i,buckling

For each part VRd,i is then determined.

=> The VRd of the cross-section is then taken as the sum of the resistances VRd,i of all parts.

i Rd Rd iV V

For a solid bar, round tube and hollow tube, all parts are taken as non-slender bydefault and formula (6.31) is applied.


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Calculation of Shear Area

The calculation of the shear area is dependent on the cross-section type.

The calculation is done using the reduced shape for N+(0,HAZ)

a) Solid bar and round tube

The shear area is calculated using art. 6.2.6 and formula (6.31):

evv A A

With: v 0,8 for solid section0,6 for circular section (hollow and solid)

Ae Taken as area A calculated using the reduced shape for

N+(0,HAZ)

b) All other Supported sections

For all other sections, the shear area is calculated using art. 6.2.6 and formula (6.30).

The following adaptation is used to make this formula usable for any initial cross-sectionshape:

n

i HAZ beg end vy t x x A

1

2

,0 cos)(

n

i HAZ beg end vz t x x A

1

2

,0 sin)(

With: i The number (ID) of the platexend End position of plate ixbeg Begin position of plate i

t Thickness of plate i0,HAZ HAZ reduction factor of plate i

Angle of plate i to the Principal y-y axis

Should a cross-section be defined in such a way that the shear area Av (Avy or Avz) is zero,

then Av is taken as A calculated using the reduced shape for N+(0,HAZ).

For sections without initial shape or numerical sections, none of the above mentionedmethods can be applied. In this case, formula (6.29) is used with Av taken as Ay or Azof the gross-section properties.


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Torsion with warping

In case warping is taken into account, the combined section check is replaced by an elasticstress check including warping stresses.

Ed w Ed t Ed Vz Ed Vy Ed tot

Ed w Ed Mz Ed My Ed N Ed tot

M

Ed tot Ed tot

M

Ed tot

M

Ed tot

f C

f

f

,,,,,

,,,,,

1

02

,

2

,

1

0

,

1

0

,

3

3

With f 0 0,2% proof strength

tot,E

d

Total direct stress

tot,E

d

Total shear stress

M1 Partial safety factor for resistance of cross-sections

C Constant (by default 1,2)

N,E

d

Direct stress due to the axial force on the relevant effectivecross-section

My,

Ed

Direct stress due to the bending moment around y axis on therelevant effective cross-section

Mz,

Ed

Direct stress due to the bending moment around z axis on therelevant effective cross-section

w,E

d

Direct stress due to warping on the gross cross-section

Vy,E

d

Shear stress due to shear force in y direction on the grosscross-section

Vz,E

d

Shear stress due to shear force in z direction on the grosscross-section

t,Ed Shear stress due to uniform (St. Venant) torsion on the grosscross-section

w,Ed

Shear stress due to warping on the gross cross-section

The warping effect is considered for standard I sections and U sections, and for (= “cold

formed sections”) sections. The definition of I sections, U sections and sections aredescribed in “Profile conditions for code check”.


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29

The other standard sections (RHS, CHS, Angle section, T section and rectangular sections)are considered as warping free. See also Ref.[3], Bild 7.4.40.

Calculation of the direct stress due to warping

The direct stress due to warping is given by (Ref.[3] 7.4.3.2.3, Ref.[4])

m

MwEd,w

C

wM

With Mw BimomentwM Unit warpingCm Warping constant

I sections

For I sections, the value of wM is given in the tables (Ref. [3], Tafel 7.87, 7.88). This value isadded to the profile library. The diagram of wM is given in the following figure:

The direct stress due to warping is calculated in the critical points (see circles in figure).The value for wM can be calculated by (Ref.[5] pp.135):

mM h b4

1

w

With b Section widthhm Section height (see figure)


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30

U sections

For U sections, the value of wM is given in the tables as wM1 and wM2 (Ref. [3], Tafel 7.89).These values are added to the profile library. The diagram of wM is given in the followingfigure:

The direct stress due to warping is calculated in the critical points (see circles in figure).

sections

The values for wM are calculated for the critical points according to the general approachgiven in Ref.[3] 7.4.3.2.3 and Ref.[6] Part 27.

The critical points for each part are shown as circles in the figure.


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Calculation of the shear stress due to warping

The shear stress due to warping is given by (Ref.[3] 7.4.3.2.3, Ref.[4])

s

0

M

m

xsEd,w tdsw

tC

M

With Mxs Warping torque (see "Standard diagrams")wM Unit warpingCm Warping constantt Element thickness

I sections

The shear stress due to warping is calculated in the critical points (see circles in figure)

For I sections, the integral can be calculated as follows:

A4

wt btdsw M

2/ b

0

M


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32

U sections, sections

Starting from the wM diagram, the following integral is calculated for the critical points:

s

0

M tdsw

The shear stress due to warping is calculated in these critical points (see circles in figures)


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33

Standard diagrams

The following 6 standard situations for St.Venant torsion, warping torque and bimoment aregiven in the literature (Ref.[3], Ref.[4]).

The value is defined as follows:

m

t

CE

IG

With:

Mx Total torque= Mxp + Mxs

Mxp Torque due to St. VenantMxs Warping torqueMw BimomentIT Torsional constantCM Warping constantE Modulus of elasticity

G Shear modulus


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34

Torsion fix ed ends, warping free ends, local torsional loading Mt

Mx

L

aMM

L bMM

txb

txa

Mxp for a side

)xcosh()Lsinh(

) bsinh(

L

bMM txp

Mxp for b side

)'xcosh()Lsinh(

)asinh(

L

aMM txp

Mxs for a side

)xcosh()Lsinh(

) bsinh(

MM txs

Mxs for b side

)'xcosh()Lsinh(

)asinh(MM txs

Mw for a side

)xsinh()Lsinh(

) bsinh(MM tw

Mw for b side

)'xsinh()Lsinh(

)asinh(MM tw


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35

Torsion fix ed ends, warping fixed ends, local torsional loading Mt

Mx

L

aMM

L

bMM

txb

txa

Mxp for a side

3D

L

1k 2k bMM txp

Mxp for b side

4D

L

1k a2k MM txp

Mxs for a side 3DMM txs

Mxs for b side 4DMM txs Mw for a side

1DM

M tw

Mw for b side2D

MM tw


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36

)2

tanh(2

)2

tanh(2)sinh(

)sinh()sinh(

2

)2

tanh(2

1)sinh(

)sinh()sinh(

2

)2

tanh(2

)2

tanh(2)sinh(

)sinh()sinh(

2

)2

tanh(2

1)sinh(

)sinh()sinh(

1

)sinh(

)'cosh()1)(sinh()cosh(24

)sinh(

)'cosh(1)cosh()2)(sinh(3

)sinh(

)'sinh()1)(sinh()sinh(22

)sinh(

)'sinh(1)sinh()2)(sinh(1

L L

L L

L

baba

L

L

ba

k

L L

L L

L

baba

L

L

ba

k

L

xk a xk D

L

xk xk b D

L

xk a xk D

L

xk xk b D

Torsion fixed ends, warping free ends, distr ibuted torsio nal loading mt

Mx

2

LmM

2

LmM

txb

txa

Mxp

)Lsinh(

)'xcosh()xcosh()x

2

L(

mM txp

Mxs

)Lsinh(

)'xcosh()xcosh(mM txs

Mw

)Lsinh(

)'xsinh()xsinh(1

mM

2

tw


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37

Torsion fixed ends, warping fixed ends, distr ibuted torsion al loading mt

Mx

2

LmM

2

LmM

txb

txa

Mxp

)Lsinh(

)'xcosh()xcosh()k 1()x

2

L(

mM txp

Mxs

)Lsinh(

)'xcosh()xcosh()k 1(

mM txs

Mw

)Lsinh()'xsinh()xsinh(

)k 1(1

m

M 2t

w

)2

Ltanh(

2

L

1k


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38

One end free, other end torsion and w arping fixed, local torsional loading Mt

Mx

txa MM

Mxp

)Lcosh(

)'xcosh(1MM txp

Mxs

)Lcosh(

)'xcosh(MM txs

Mw

)Lcosh(

)'xsinh(MM tw

One end free, other end torsion and w arping fixed, distr ibuted torsion alload ing mt

Mx

LmM txa

Mxp

)Lcosh(

)xsinh())Lsinh(L1()xcosh(L'x

mM txp


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39

Mxs

)Lcosh(

)xsinh())Lsinh(L1()xcosh(L

mM txs

Mw

)Lcosh(

)xcosh())Lsinh(L1()xsinh(L1

²

mM tw

Decomposition of arbitrary torsion line

Since the Scia Engineer solver does not take into account the extra DOF for warping, thedetermination of the warping torque and the related bimoment, is based on some standardsituations.

The following end conditions are considered:

warping free

warping fixed

This results in the following 3 beam situations:

situation 1 : warping free / warping free

situation 2 : warping free / warping fixed

situation 3 : warping fixed / warping fixed


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40

Decompo sit ion for situation 1 and situation 3

The arbitrary total torque line is decomposed into the following standard situations:

o n number of torsion lines generated by a local torsional loading Mtn

o one torsion line generated by a distributed torsional loading mt

o one torsion line with constant torque Mt0

The values for Mxp, Mxs and Mw are taken from the previous tables for the local torsionalloadings Mtn and the distributed loading mt. The value Mt0 is added to the Mxp value.

Decompo sit ion for situation 2

The arbitrary total torque line is decomposed into the following standard situations:

o one torsion line generated by a local torsional loading Mtn

o one torsion line generated by a distributed torsional loading mt

The values for Mxp, Mxs and Mw are taken from the previous tables for the local torsionalloading Mt and the distributed loading mt.

Combined shear and torsion

The Combined shear force and torsional moment check is verified using art. 6.2.7.3.

For I and H sections formula (6.35) is applied.

For U-sections formula (6.36) is applied without accounting for warping. In case warping isactivated, the combined section check is replaced by an elastic stress check includingwarping stresses which takes into account all shear stress effects. For more informationplease refer to “Torsion with warping”.

For all other supported sections formula (6.37) is applied.

In case of extreme torsion (unity check for torsion > 1,00) the shear resistance will bereduced to zero which will lead to extreme unity check values.


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41

Bending, shear and axial force

The combined section check is verified according to art. 6.2.8, 6.2.9 & 6.2.10

For I sections formulas (6.40) and (6.41) are applied.

For hollow and solid sections formula (6.43) is applied.

For all other supported sections an elastic stress check is performed according to art. 6.2.1 and formula (6.15). The stresses are based on the effective cross-sectional properties andcalculated in the fibres of the gross cross-section.

The plastic interaction for mono-symmetrical sections specified in art. 6.2.9.1 (2) is notsupported. For mono-symmetrical sections the elastic stress check of art. 6.2.1 isapplied.

Localised welds

In case transverse welds are inputted, the extend of the HAZ is calculated as specified in

paragraph “Calculation of Reduction factor HAZ for HAZ effects” and compared to the leastwidth of the cross-section.

The reduction factor 0 is then calculated according to art. 6.2.9.3

When the width of a member cannot be determined (Numerical section, tube …) formula(6.44) is applied.

Since the extend of the HAZ is defined along the member axis, it is important to specifyenough sections on average member in the Solver Setup when transverse welds areused.

Formula (6.44) is limited to a maximum of 1,00 in the same way as formula (6.64).

Shear reduction

Where VEd exceeds 50% of VRd the design resistances for bending and axial force arereduced using a reduced yield strength as specified in art. 6.2.8 & 6.2.10 .

For Vy the reduction factor y is calculated

For Vz the reduction factor z is calculated

The bending resistance My,Rd is reduced using z

The bending resistance Mz,Rd is reduced using y

The axial force resistance NRd is reduced by using the maximum of y and z


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Stress check for numerical sections

For numerical sections an elastic stress check is performed according to art. 6.2.1 andformula (6.15). The stresses are calculated in the following way:

Vz Vytot

Mz My N tot

M

tot tot

M

tot

M

tot

f C

f

f

1

022

1

0

1

0

3

3

With:

f 0 0,2% proof strength

tot Total direct stress

tot Total shear stress

M1 Partial safety factor for resistance of cross-sections

C Constant (by default 1,2)

N Direct stress due to the axial force

My Direct stress due to the bending moment around y axis

Mz Direct stress due to the bending moment around z axis

Vy Shear stress due to shear force in y direction

Vz Shear stress due to shear force in z direction

Ax Sectional area

Ay Shear area in y direction Az Shear area in z directionWy Elastic section modulus around y axisWz Elastic section modulus around z axis


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43

Flexural buckling

The flexural buckling check is verified using art. 6.3.1.1.

The value of Aeff is taken as the area A calculated from the reduced shape for N- however

HAZ effects are not accounted for (i.e. HAZ is taken as 1,00).

The value of AHAZ is illustrated on the following figure:

For the calculation of the buckling ratio several methods are available:

o General formula (standard method)

o Crossing Diagonals

o From Stability Analysis

o Manual inputThese methods are detailed in the following paragraphs.

Calculation of Buckling ratio – General Formula

For the calculation of the buckling ratios, some approximate formulas are used. Theseformulas are treated in reference [7], [8] and [9].

The following formulas are used for the buckling ratios (Ref[7],pp.21):

For a non-sway structure:

24)+11+5+24)(2+5+11+(2

12)2+4+4+24)(+5+5+(=l/L

21212121

21212121


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44

For a sway structure:

4+x

x=l/L1

2

With: L System lengthE Modulus of YoungI Moment of inertiaCi Stiffness in node iMi Moment in node i

i Rotation in node i

21212

12

21

8+)+(

+4=x

EI

LC=

ii

i

i

i

M=C

The values for Mi and i are approximately determined by the internal forces and thedeformations, calculated by load cases which generate deformation forms, having anaffinity with the buckling shape. (See also Ref.[11], pp.113 and Ref.[12],pp.112).

The following load cases are considered:load case 1: on the beams, the local distributed loads qy=1 N/m and qz=-100 N/m are used,on the columns the global distributed loads Qx = 10000 N/m and Qy =10000 N/m are used.load case 2: on the beams, the local distributed loads qy=-1 N/m and qz=-100 N/m areused, on the columns the global distributed loads Qx = -10000 N/m and Qy= -10000 N/mare used.

In addition, the following limitations apply (Ref[7],pp.21):- The values of ρi are limited to a minimum of 0.0001- The values of ρi are limited to a maximum of 1000- The indices are determined such that ρ1 ≥ ρ2 - Specifically for the non-sway case, if ρ1 ≥ 1000 and ρ2 ≤ 0,34 the ratio l/L is set to 0,7

The used approach gives good results for frame structures with perpendicular rigid or semi-rigid beam connections. For other cases, the user has to evaluate the presented buckingratios. In such cases a more refined approach (from stability analysis) can be applied.

The following rule applies specifically to ky: in case both the calculation for load case 1and load case 2 return ky = 1,00 then ky is taken as kz. This rule is used to account forpossible rotations of the cross-section.


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45

Calculation of Buckling ratio – Crossing Diagonals

For crossing diagonal elements, the buckling length perpendicular to the diagonal plane, iscalculated according to Ref.[10], DIN18800 Teil 2, table 15. This means that the bucklinglength sK is dependent on the load distribution in the element, and it is not a purelygeometrical data anymore.

In the following paragraphs, the buckling length sK is defined,

With: sK Buckling lengthL Member lengthL1 Length of supporting diagonalI Moment of inertia (in the buckling plane) of the memberI1 Moment of inertia (in the buckling plane) of the supporting

diagonalN Compression force in memberN1 Compression force in supporting diagonalZ Tension force in supporting diagonalE Modulus of Young

Cont inuous com press ion diagonal , supported by cont inuous tens ion

diagonal

NN

Z

Z

l/2

l1/2

l5.0s

lI

l1I1

lN4

lZ31

ls

K

3

1

3

1

K

See Ref.[10], Tab. 15, case 1.


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46

Continuo us comp ression diagonal, supp orted by pinned tension diagonal

NN

Z

Z

l/2

l1/2

l5.0s

lN

lZ75.01ls

K

1

K


Pinned compressio n diagonal, supp orted by continuous tension diagonal

NN

Z

Z

l/2

l1/2

)1lZ

lN(

4

lZ3)IE(

1lZ

lN

l5.0s

1

2

21

d1

1

K



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47

Cont inuous com press ion d iagonal , supported by cont inuous com press ion

diagonal

NN

N1

N1

l/2

l1/2

l5.0s

lI

l1I1

lN

lN1

ls

K

3

1

3

1

1

K


Cont inuous com press ion diagonal , supported by pinned compress ion

diagonal

NN

N1

N1

l/2

l1/2

1

1

2

KlN

lN

121ls

See Ref.[10], Tab. 15, case 3 (2).


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48

Pinned compress ion diagonal , supported by cont inuous com press ion

diagonal

NN

N1

N1

l/2

l1/2

)N

lN

12(

l

lN)IE(

l5.0s

1

12

1

2

3

d

K

See Ref.[10], Tab. 15, case 3 (3).

Calculation of Buckling ratio – From Stability Analysis

When member buckling data from stability are defined, the critical buckling load Ncr for aprismatic member is calculated as follows:

Ed cr N N

Using Euler’s formula, the buckling ratio k can then be determined:

With: Critical load factor for the selected stability combination

NEd Design loading in the member

E Modulus of Young

I Moment of inertia

s Member length

In case of a non-prismatic member, the moment of inertia is taken in the middle of theelement.


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49

Torsional (-Flexural) buckling

The Torsional and Torsional-Flexural buckling check is verified using art. 6.3.1.1.

If the section contains only Plate Types F, SO, UO it is regarded as ‘ Composed entirely ofradiating outstands’. In this case Aeff is taken as A calculated from the reduced shape for

N+(0,HAZ).

In all other cases, the section is regarded as ‘General’. In this case Aeff is taken as A calculated from the reduced shape for N-

The Torsional (-Flexural) buckling check is ignored for sections complying with therules given in art. 6.3.1.4 (1).

The value of the elastic critical load Ncr is taken as the smallest of Ncr,T (Torsional buckling)and Ncr,TF (Torsional-Flexural buckling).

Calculation of Ncr,T

The elastic critical load Ncr,T for torsional buckling is calculated according to Ref.[13].

With: E Modulus of Young

G Shear modulus

It Torsion constant

Iw Warping constant

lT Buckling length for the torsional buckling mode

y0 and z0 Coordinates of the shear center with respect to the centroid

iy radius of gyration about the strong axis

iz radius of gyration about the weak axis


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Calculation of Ncr,TF

The elastic critical load Ncr,TF for torsional flexural buckling is calculated according toRef.[13].

Ncr,TF is taken as the smallest root of the following cubic equation in N:

0With: Ncr,y Critical axial load for flexural buckling about the y-y axis

Ncr,z Critical axial load for flexural buckling about the z-z axis

Ncr,T Critical axial load for torsional buckling


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Lateral Torsional buckling

The Lateral Torsional buckling check is verified using art. 6.3.2.1.

For the calculation of the elastic critical moment Mcr the following methods are available:

o General formula (standard method)

o LTBII Eigenvalue solution

o Manual input

The Lateral Torsional buckling check is ignored for circular hollow sections accordingto art. 6.3.3 (1).

Calculation of Mcr – General Formula

For I sections (symmetric and asymmetric) and RHS (Rectangular Hollow Section) sections

the elastic critical moment for LTB Mcr is given by the general formula F.2. Annex F Ref.14. For the calculation of the moment factors C1, C2 and C3 reference is made to theparagraph "Calculation of Moment factors for LTB".For the other supported sections, the elastic critical moment for LTB Mcr is given by:

z2

z2

z2

EI

L²GI

I

Iw

L

EIMcr

With: E Modulus of elasticityG Shear modulusL Length of the beam between points which have lateral restraint (=

lLTB)

Iw Warping constantIt Torsional constantIz Moment of inertia about the minor axis

See also Ref. 15, part 7 and in particular part 7.7 for channel sections.Composed rail sections are considered as equivalent asymmetric I sections.


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52

Diaphragms

When diaphragms (steel sheeting) are used, the torsional constant It is adapted forsymmetric/asymmetric I sections, channel sections, Z sections, cold formed U, C , Zsections.

See Ref.[16], Chapter 10.1.5., Ref.17,3.5 and Ref.18,3.3.4.

The torsional constant It is adapted with the stiffness of the diaphragms:

12

³sI

)th(

IE3C

200 b125if 100

bC25.1C

125 bif 100

b

CC

s

EIk C

C

1

C

1

C

1

vorhC

1

G

lvorhCII

s

sk ,P

aa

100k ,A

a

2

a

100k ,A

eff k ,M

k ,Pk ,Ak ,M

2

2

tid,t

With: l LTB lengthG Shear modulus

vorhC Actual rotational stiffness of diaphragm

CM,k Rotational stiffness of the diaphragm

C A,k Rotational stiffness of the connection between the diaphragm andthe beam

CP,k Rotational stiffness due to the distortion of the beam

k Numerical coefficient= 2 for single or two spans of the diaphragm= 4 for 3 or more spans of the diaphragm

EIeff Bending stiffness per unit width of the diaphragms Spacing of the beamba Width of the beam flange (in mm)C100 Rotation coefficient - see tableh Height of the beamt Thickness of the beam flanges Thickness of the beam web


58/110

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59/110

54

Calculation of Moment factors for LTB

For determining the moment factors C1 and C2 for lateral torsional buckling, standard tablesare used which are defined in Ref.[19] Art.12.25.3 table 9.1.,10 and 11.

The current moment distribution is compared with several standard moment distributions.

These standard moment distributions are:o Moment line generated by a distributed q load

o Moment line generated by a concentrated F load

o Moment line which has a maximum at the start or at the end of the beam

The standard moment distribution which is closest to the current moment distribution is takenfor the calculation of the factors C1 and C2. These values are based on Ref.[14].

The factor C3 is taken out of the tables F.1.1. and F.1.2. from Ref.[14] - Annex F.

Moment distr ibution generated by q load

if M2 < 0

C1 = A* (1.45 B

* + 1) 1.13 + B

* (-0.71 A

* + 1) E

*

C2 = 0.45 A* [1 + C* eD*

(½ + ½)]

if M2 > 0

C1 = 1.13 A* + B

* E

*

C2 = 0.45A*

With:l+q|M2|8

lq=A

2

2*

l+q|M2|8

|M2|8=B

2

*

)ql

|M2|-72(=D

2

2

*

ql

|M2|94=C2

*

2.70


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55

Moment distr ibution g enerated by F load

F

M2 M1 = Beta M2

l

M2 < 0

C1 = A** (2.75 B

** + 1) 1.35 + B

** (-1.62 A

** + 1) E

**

C2 = 0.55 A** [1 + C

** e

D** (½ + ½)]

M2 > 0

C1 = 1.35 A** + B

** E

**

C2 = 0.55 A**

With: +Fl|M2|4

Fl=A **

+Fl|M2|4

|M2|4=**B

Fl

|M2|38=C **

)Fl

|M2|-32(=D

2**

The values for E** can be taken as E

* from the previous paragraph.

Moment l ine with maximum at the start or at the end of the beam

M2 M1 = Beta M2

l

C2 = 0.0

2.70


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56

Flexural buckling

For I sections formulas (6.59) and (6.60) are applied.

For solid sections formula (6.60) is applied for bending about either axis.

For hollow sections formula (6.62) is applied.

For all other supported sections formula (6.59) is applied for bending about either axis.

Lateral Torsional buckling

For all sections except circular hollow sections formula (6.63) is applied.For circular hollow sections the check is ignored according to art. 6.3.3(1).

In case a cross-section is subject to torsional (-flexural) buckling, the reduction factor z is

taken as the minimum value of z for flexural buckling and TF for torsional (-flexural)buckling.

Localised welds and factors for design section

The HAZ-softening factors are calculated according to art. 6.3.3.3. For sections withoutlocalized welds the reduction factors are calculated according to art. 6.3.3.5.

Members containing lo cal ized welds

In case transverse welds are inputted, the extend of the HAZ is calculated as specified in

chapter “Calculation of Reduction factor HAZ for HAZ effects” and compared to the leastwidth of the cross-section.

The reduction factors 0, x, xLT are then calculated according to art. 6.3.3.3

When the width of a member cannot be determined (Numerical section, tube …) formula(6.64) is applied.

The calculation of the distance xs is discussed further in this chapter.

Since the extend of the HAZ is defined along the member axis, it is important to specifyenough sections on average member in the Solver Setup when transverse welds areused.

In the calculation of xLT the buckling length lc and distance xs are taken for bucklingaround the z-z axis.


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57

Unequal end m oments and/or transverse loads

If the section under consideration is not located in a HAZ zone, the reduction factors x and

xLT are then calculated according to art. 6.3.3.5 .

In this case 0 is taken equal to 1,00.

For the calculation of the distance xs reference is made to the following paragraph.

In the calculation of xLT the buckling length lc and distance xs are taken for bucklingaround the z-z axis.

Calculat ion of x s

The distance xs is defined as the distance from the studied section to a simple support orpoint of contra flexure of the deflection curve for elastic buckling of axial force only.

By default xs is taken as half of the buckling length for each section. This leads to adenominator of 1,00 in the formulas of the reduction factors following Ref.[20] and [21].

Depending on how the buckling shape is defined, a more refined approach can be used forthe calculation of xs.

Known buckling shapeThe buckling shape is assumed to be known in case the buckling ratio is calculatedaccording to the Gener al Formula specified in chapter “Calculation of Buckling ratio – General Formula”. The basic assumption is that the deformations for the buckling load casehave an affinity with the buckling shape.

Since the buckling shape (deformed structure) is known, the distance from each section toa simple support or point of contra flexure can be calculated. As such xs will be different in

each section. A simple support or point of contra flexure are in this case taken as thepositions where the bending moment diagram for the buckling load case reaches zero.

Since for a known buckling shape xs can be different in each section, accurate resultscan be obtained by increasing the numbers of sections on average member in theSolver Setup.


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58

Unknown buckling shapeIn case the buckling ratio is not calculated according to the General Formula specified inchapter “Calculation of Buckling ratio – General Formula” the buckling shape is taken asunknown. This is thus the case for manual input or if the buckling ratio is calculated fromstability.

When the buckling shape is unknown, xs can be calculated according to formula (6.71):

but xs ≥ 0With: lc Buckling length

MEd,1 and MEd,2 Design values of the end moments at the system length of themember

NEd Design value of the axial compression force

MRd Bending moment capacity

NRd Axial compression force capacity

Reduction factor for flexural buckling

The above specified formula contains the factor in the denominator of the right side ofthe equation in accordance with addendum EN 1999-1-1:2007/A1:2009.

Since the formula returns only one value for xs, this value will be used in each section of themember.

The application of the formula is however limited:

o The formula is only valid in case the member has a linear moment diagram.

o

Since the left side of the equation concerns a cosine, the right side has to return avalue between -1,00 and +1,00If one of the two above stated limitations occur, the formula is not applied and instead xs is takenas half of the buckling length for each section.


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Shear buckling

The shear buckling check is verified using art. 6.7.4 & 6.7.6. Distinction is made between two separate cases:

o No stiffeners are inputted on the member or stiffeners are inputted only at themember ends.

o Any other input of stiffeners (at intermediate positions, at the ends and intermediatepositions …).

The first case is verified according to art. 6.7.4.1. The second case is verified according toart. 6.7.4.2.

For shear buckling only transverse stiffeners are supported. Longitudinal stiffeners arenot supported.

In all cases rigid end posts are assumed.

Plate girders with stiffeners at supports

No stiffeners are inputted on the member or stiffeners are inputted only at the memberends. The verification is done according to art. 6.7.4.1.

The check is executed when the following condition is met:

0

37,2

f

E

t

h

w

w

With: hw Web height

tw Web thickness

Factor for shear buckling resistance in the plastic range

E Modulus of Young


The design shear resistance VRd for shear buckling consists of one part: the contribution ofthe web Vw,Rd.

The slenderness w is calculated as follows:

E

f

t

h

w

ww

035,0

Using the slenderness w the factor for shear buckling v is obtained from the followingtable:


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In this table, the value of is taken as follows:

With: f uw Ultimate strength of the web material

f 0w Yield strength of the web material

The contribution of the web Vw,Rd can then be calculated as follows:

For interaction see paragraph “Interaction”.


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Plate girders with intermediate web stiffeners

Any other input of stiffeners (at intermediate positions, at the ends and intermediatepositions …). The verification is done according to art. 6.7.4.2 .

The check is executed when the following condition is met:

With: hw Web height

tw Web thickness

Factor for shear buckling resistance in the plastic range

k Shear buckling coefficient for the web panel

E Modulus of Young


The design shear resistance VRd for shear buckling consists of two parts: the contribution ofthe web Vw,Rd and the contribution of the flanges Vf,Rd.

Contributio n of the web

Using the distance a between the stiffeners and the height of the web hw the shear buckling

coefficient k

can be calculated:

The value k

can now be used to calculate the slenderness w.


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Using the slenderness w the factor for shear buckling v is obtained from the followingtable:

In this table, the value of is taken as follows:

With: f uw Ultimate strength of the web materialf 0w Yield strength of the web material

The contribution of the web Vw,Rd can then be calculated as follows:

Contributio n of the flangesFirst the design moment resistance of the cross-section considering only the flanges Mf,Rd iscalculated.

When then Vf,Rd = 0

When then Vf,Rd is calculated as follows:

With: bf and tf the width and thickness of the flange leading to the lowest resistance.

On each side of the web.


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With: f 0f Yield strength of the flange materialf 0w Yield strength of the web material

If an axial force NEd is present, the value of Mf,Rd is be reduced by the following factor:

With: Af1 and Af2 the areas of the top and bottom flanges.

The design shear resistance VRd is then calculated as follows:

For interaction see paragraph “Interaction”.


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Interaction

If required, for both above cases the interaction between shear force, bending moment andaxial force is checked according to art. 6.7.6.1.

If the following two expressions are checked:

With:Mf,Rd design moment resistance of the cross-sectionconsidering only the flangesMpl,Rd Plastic design bending moment resistance

If an axial force NEd is also applied, then Mpl,Rd is replaced by the reduced plastic momentresistance MN,Rd given by:

With: Af1 and Af2 the areas of the top and bottom flanges.


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Scaffolding

The scaffolding member and coupler check are implemented according to EN 12811-1Ref.[31].The following paragraphs give detailed information on these checks.

Scaffolding member check for tubular members

The check is executed specifically for circular hollow sections (Form code 3) andNumerical sections in case the proper setting is activated in the Aluminium Setup.

The check is executed according to Equation 9 given in EN 12811-1 article 10.3.3.2.However, the EN 12811-1 only gives an interaction equation in case of a low shear force.Since the EN 12811-1 is based entirely on DIN 4420-1 Teil 1 Ref.[34] the interactionformulas according to Tabelle 7 of DIN 4420-1 Teil 1 are applied in case of a large shearforce.

The interaction equations are summarised as follows:

Conditions Interaction for tubular member

and

and

and

and


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With: M

V

Npld

Vpld

Mpld

A Area of the cross-section

Wel Elastic section modulus

Wpl Plastic section modulus

N Normal force

Vy Shear force in y direction

Vz Shear force in z direction

My Bending moment about the y axis

Mz Bending moment about the z axis

fy Yield strength of the material taken as f 0 incase the section is not located in a HAZzone and f 0,HAZ otherwise.

Safety factor taken as M1 of EN 1999-1-1

As specified in EN 12810 Ref.[33] & 12811 Ref.[31] the scaffolding check for tubularmembers assumes the use of a 2

nd order analysis including imperfections.

In case these conditions are not set the default EN 1999-1-1 check should be appliedinstead.


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Scaffolding coupler check

The scaffolding couplers according to EN 12811-1 Annex C Ref.[31] are provided bydefault within Scia Engineer.The interaction check of the couplers is executed according to EN 12811-1 article 10.3.3.5.


Coupler type Interaction equation

Right angle coupler

Friction sleeve

With: Fsk Characteristic Slipping force

Taken as Nxk and Vzk of the coupler properties

2Fsk = Nxk + Vzk

Fpk Characteristic Pull-apart force

Taken as Vyk of the coupler properties

MBk Characteristic Bending moment

Taken as Myk of the coupler properties

N Normal forceVy Shear force in y direction



Safety factor taken as M0 of EN 1993-1-1 for steel couplers

Safety factor taken as M1 of EN 1999-1-1 for aluminium couplers


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Manufacturer couplers

In addition to the scaffolding couplers listed above, specific manufacturer couplers areprovided within Scia Engineer.The interaction checks of these couplers are executed according to the respectivevalidation reports.

CuplockThe cuplock coupler which connects a ledger and a standard is described in Zulassung Nr.Z-8.22-208 Ref.[35].


Cuplock Coupler Interaction equation

Interaction 1

Interaction 2

With: Nxk Taken from the coupler properties

Myk Taken from the coupler properties

Mxk Taken from the coupler properties

N Normal force in the ledger


Mx Torsional moment about the x axis

Nv Normal force in a connecting vertical diagonal

Angle between connecting vertical diagonal and standard


Safety factor taken as M1 of EN 1999-1-1 for aluminiumcouplers


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Layher Variante II & K2000+The Layher coupler which connects a ledger and a standard is described in Zulassung Nr.Z-8.22-64 Ref.[36]. Both Variante II and Variante K2000+ are provided.

Layher Coupler Interaction equation

Interaction 1 Variante II:

Variante K2000+:

Interaction 2

With: NR,d = Nxk /

With Nxk taken from the coupler properties

My,R,d = Myk /

With Myk taken from the coupler properties

MT,R,d = Mxk /

With Mxk taken from the coupler properties

Vz,R,d = Vzk /

With Vzk taken from the coupler properties

N Normal force in the ledger


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(+) This index indicates a tensile force

Vy Shear force in y direction



Mx Torsional moment about the x axis

Nv Normal force in a connecting vertical diagonal

Angle between connecting vertical diagonal and standard

e = 2,75 cm for Variante II

= 3,30 cm for Variante K2000+

eD = 5,7 cm for Variante II and Variante K2000+

= 1,26 cm for Variante II

= 1,41 cm for Variante K2000+


Safety factor taken as M1 of EN 1999-1-1 for aluminiumcouplers


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LTBII: Lateral Torsional Buckling 2nd Order Analysis

Introduction to LTBII

For a detailed Lateral Torsional Buckling analysis, a link was made to the Friedrich +Lochner LTBII application Ref.[22].

The Frilo LTBII solver can be used in 2 separate ways:

o Calculation of Mcr through eigenvalue solution

o 2nd

Order calculation including torsional and warping effects

For both methods, the member under consideration is sent to the Frilo LTBII solver and therespective results are sent back to Scia Engineer.

A detailed overview of both methods is given in the following paragraphs.

Eigenvalue solution Mcr

The single element is taken out of the structure and considered as a single beam, with:

o Appropriate end conditions for torsion and warping

o End and begin forces

o Loadings

o Intermediate restraints (diaphragms, LTB restraints)

The end conditions for warping and torsion are defined as follows:

Cw_i Warping condition at end i (beginning of the member)Cw_j Warping condition at end j (end of the member)Ct_i Torsion condition at end i (beginning of the member)Ct_j Torsion condition at end j (end of the member)

To take into account loading and stiffness of linked beams , see paragraph “Linked Beams”.


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For this system, the elastic critical moment Mcr for lateral torsional buckling can be analyzedas the solution of an eigenvalue problem:

With: Critical load factor

Ke Elastic linear stiffness matrixKg Geometrical stiffness matrix

For members with arbitrary sections, the critical moment can be obtained in each section,with: (See Ref.[24],pp.176)

With: Critical load factor

Myy Bending moment around the strong axisMyy(x) Bending moment around the strong axis at position xMcr (x) Critical moment at position x

The calculated Mcr is then used in the Lateral Torsional Buckling check of Scia Engineer.

For more background information, reference is made to Ref.[23].

0K K ge

)x(MxM

MmaxM

yycr

yycr


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2nd Order analysis

The single element is taken out of the structure and considered as a single beam, with:

o Appropriate end conditions for torsion and warping

o End and begin forces

o Loadingso Intermediate restraints (diaphragms, LTB restraints)

o Imperfections

To take into account loading and stiffness of linked beams, see paragraph “Linked Beams”.

For this system, the internal forces are calculated using a 2nd

Order 7 degrees of freedomcalculation.

The calculated torsional and warping moments (St Venant torque Mxp, Warping torque Mxsand Bimoment Mw) are then used in the Stress check of Scia Engineer (See chapter“Torsion with warping”).

Specifically for this stress check, the following internal forces are used:

o Normal force from Scia Engineer

o Maximal shear forces from Scia Engineer / Frilo LTBII

o Maximal bending moments from Scia Engineer / Frilo LTBII

Since Lateral Torsional Buckling has been taken into account in this 2nd

Order stress check,it is no more required to execute a Lateral Torsional Buckling Check.

For more background information, reference is made to Ref.[23].


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Supported Sections

The following table shows which cross-section types are supported for which type ofanalysis:

FRILO LTBII CSS Scia Engineer CSS Eigenvalueanalysis

2n

Orderanalysis

Double T I section from library x x

Thin walled geometric I x x

Sheet welded Iw x x

Double T unequal IPY from library x x

Thin walled geometricasymmetric I

x x

Haunched sections x x

Welded I+Tl x x

Sheet welded Iwn x x

HAT Section IFBA, IFBB x x

U cross section U section from library x xThin walled geometric U x x

Thin walled Cold formed from library x x

Cold formed from graphicalinput

x x

Double T with top flangeangle

Welded I+2L x

Sheet welded Iw+2L x

Rectangle Full rectangular from library x

Full rectangular from thin walledgeometric

x

Static values doublesymmetric

all other double symmetric CSS x

Static values singlesymmetric

all other single symmetric CSS x

The following picture illustrates the relation between the local coordinate system of SciaEngineer and Frilo LTBII. Special attention is required for U sections due to the inversion ofthe y and z-axis.


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Imperfections

In the 2nd

Order LTB analysis the bow imperfections v0 (in local y direction) and w0 (in localz direction) can be taken into account.

Initial bow imperfection v0 according to code

For EC-EN the imperfections can be calculated according to the code. The code indicatesthat for a 2

nd Order calculation which takes into account LTB, only the imperfection v0

needs to be considered.

The sign of the imperfection according to code depends on the sign of Mz in Scia Engineer.

The imperfection is calculated according to Ref.[1] art. 5.3.4(3)

00 ek v

With k Factor taken from National Annexe0 Bow imperfection of the weak axis

Manual input of Initial bow imperfections v0 and w0

In case the user specifies manual input, both the imperfections v0 and w0 can be inputted.

v0

y, v0

z

y

scia - aluminium code check theory

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