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TheoryAluminium Code Check



i

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

Version info .............................................................................................................................................................1

Disclaimer.................................................................................................................................... 2EN1999 Code Check ..................................................................................................................3

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

Consulted Articles..................................................................................................................................................3

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

Step 1: Calculation of spring stiffness ..............................................................................................................18

Step 2: Calculation of Area and Second moment of area...............................................................................20

Step 3: Calculation of stiffener buckling load..................................................................................................21

Step 4: Calculation of reduction factor for distortional buckling..................................................................22

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

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

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

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

Shear ................................................................................................................................................................24

Torsion with warping........................................................................................................................................27

The direct stress due to warping is given by (Ref.[3] 7.4.3.2.3, Ref.[4]).........................28

I sections ...............................................................................................................................................................29

U sections..............................................................................................................................................................29sections..................................................................................................................................................................30

The shear stress due to warping is given by (Ref.[3] 7.4.3.2.3, Ref.[4]) ......................... 31

I sections ...............................................................................................................................................................31

U sections, sections ........................................................................................................................................32

Starting from the wM diagram, the following integral is calculated for the criticalpoints: ........................................................................................................................................32

The following 6 standard situations for St.Venant torsion, warping torque andbimoment are given in the literature (Ref.[3], Ref.[4]). ......................................................33

The value is defined as follows:.........................................................................................33

Torsion fixed ends, warping free ends, local torsional loading Mt ...............................................................34

Torsion fixed ends, warping fixed ends, local torsional loading Mt .............................................................35

Torsion fixed ends, warping free ends, distributed torsional loading mt ....................................................36

Torsion fixed ends, warping fixed ends, distributed torsional loading mt ..................................................37

One end free, other end torsion and warping fixed, local torsional loading Mt..........................................38

One end free, other end torsion and warping fixed, distributed torsional loading mt ...............................38

Decomposition for situation 1 and situation 3.................................................................................................40

Decomposition for situation 2 ............................................................................................................................40

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

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

Flexural buckling ..............................................................................................................................................42

For a non-sway structure: ...................................................................................................................................43



ii

For a sway structure: ...........................................................................................................................................43

Continuous compression diagonal, supported by continuous tension diagonal.......................................44

Continuous compression diagonal, supported by pinned tension diagonal ..............................................45

Pinned compression diagonal, supported by continuous tension diagonal...............................................45

Continuous compression diagonal, supported by continuous compression diagonal.............................46

Continuous compression diagonal, supported by pinned compression diagonal.....................................46

Pinned compression diagonal, supported by continuous compression diagonal.....................................47

Torsional (-Flexural) buckling ..........................................................................................................................48

Lateral Torsional buckling................................................................................................................................50

Diaphragms ...........................................................................................................................................................50

The factor C3 is taken out of the tables F.1.1. and F.1.2. from Ref.[14] - Annex F. .........53

Moment distribution generated by q load.........................................................................................................53

Moment distribution generated by F load.........................................................................................................53

Moment line with maximum at the start or at the end of the beam ...............................................................54

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

Members containing localized welds.................................................................................................................55

Unequal end moments and/or transverse loads..............................................................................................55

Calculation of xs....................................................................................................................................................56

Shear buckling..................................................................................................................................................57

Contribution of the web.......................................................................................................................................60

Contribution of the flanges .................................................................................................................................61

LTBII: Lateral Torsional Buckling 2nd Order Analysis......................................................64

Introduction to LTBII.........................................................................................................................................64

Eigenvalue solution Mcr ....................................................................................................................................64

2nd

Order analysis ............................................................................................................................................66

Supported Sections..........................................................................................................................................67

Loadings...........................................................................................................................................................68Imperfections....................................................................................................................................................68

Initial bow imperfection v0 according to code .................................................................................................69

Manual input of Initial bow imperfections v0 and w0......................................................................................69

LTB Restraints..................................................................................................................................................70

Diaphragms......................................................................................................................................................71

Linked Beams...................................................................................................................................................72

Limitations and Warnings ................................................................................................................................73

Eigenvalue solution Mcr......................................................................................................................................73

2nd

Order Analysis ................................................................................................................................................73

Profile conditions for code check .........................................................................................74

Introduction to profile characteristics...............................................................................................................74

Data for general section stability check...........................................................................................................74

Data depending on the profile shape ..............................................................................................................75

References ................................................................................................................................95



1

Introduction

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

Eurocode 9

Design of aluminium structures

Part 1-1: General structural rules

EN 1999-1-1:2007

Version info

Documentation Title Aluminium Code Check – Theoretical Background

Release 2008.0

Revision 01/2008



2

Disclaimer

This document is being furnished by SCIA for information purposes only to licensed users of SCIA software and isfurnished on an "AS IS" basis, which is, without any warranties, whatsoever, expressed or implied. SCIA is not

responsible for direct or indirect damage as a result of imperfections in the documentation and/or software.

Information in this document is subject to change without notice and does not represent a commitment on the part

of SCIA. The software described in this document is furnished under a license agreement. The software may beused only in accordance with the terms of that license agreement. It is against the law to copy or use the software

except as specifically allowed in the license.

© Copyright 2008 SCIA Group NV. All rights reserved.



3

EN1999 Code Check

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

Material PropertiesThe characteristic values of the material properties are based on Table 3.2a for wrought aluminium alloys of type

sheet, strip and plate and on Table 3.2b for wrought aluminium alloys 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

EN-AW 5083 (Sheet) H22/H32

EN-AW 5083 (Sheet) H14

EN-AW 5083 (Sheet) H24/H34

EN-AW 5083 (ET,EP,ER/B) O/111,F,H112

EN-AW 5083 (DT) H12/22/32

EN-AW 5083 (DT) H14/24/34

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 6063 (EP,ET,ER/B) T5

EN-AW 6063 (EP) T5


EN-AW 6063 (DT) T6


EN-AW 6063 (EP) T66

EN-AW 6063 (DT) T66

EN-AW 6082 (Sheet) T4/T451

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/O,EP/H) T5

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)

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

Consulted ArticlesThe 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 are included. For class 4

sections (slender sections) the effective section is calculated in each intermediary point, according to Ref. [2].



4

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 ) and combined 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 articles marked with "X" are

consulted. The articles marked with (*) have a supplementary explanation 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 Basis

6.1.3 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



5

6.3.1 Members in compression X

(*)

6.3.2 Members in bending X

(*)6.3.3 Members in bending and axial compression X

(*)

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

(*)

Note: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 to define 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. Parts having the samereinforcement 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,



6

RI RI

I I I I

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

RUOI



7

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

Formcode 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 )



8

4 Angle section

(UO, none)

(UO, none)(F, none)

5 Channel section(UO, none)

(UO, none)

(I, none)

(F, none)

(F, none)



9

6 T section

(UO, none)


(F, none)

7 Full rectangular section No reduction possible

11 Full circular section No reduction possible

101 Asymmetric I section(SO, none)(SO, none)


(F, none)

(F, none)

(I, none)



10

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)



11

111 Cold formed angle

(UO, none)

(UO, none)

112 Cold formed channel(UO, none)

(I, none)

(UO, none)



12

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)



13

For other predefined cross-sections, the initial geometric shape is based on the centreline of 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 are performed. The

classification can change for each intermediary point.

Classification for members with combined bending and axial forces is made for the loading componentsseparately. 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 is independent on the magnitude

of the actual forces in the cross-section, the classification is always done for each component.



14

Taking into account the sign of the force components and the HAZ reduction factors, this leads 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 section properties 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 the rectangular 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

Note:Compression stress is indicated as negative.

Note:When the rectangular shape is completely under tension, i.e. beg and end are both tensile 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 stressbeg

end



15

Step 3: Calculation of slenderness

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

a) Internal part: type I

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

Stress gradient factor

Remark:

For a thin walled round tube

t

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

b) Outstand part: 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



16

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< 2 : class 2

if 2< 3 : class 3

if 3< : class 4

Note:For Table 6.2 the remark ‘with welds’ is valid for fabrication type ‘welded’. ‘Without welds’ is valid for the other fabrication types.

Note:The modified expression given in art.6.1.4.4 (4) is not supported.

The cross-section is then classified according to the highest (least favourable) class of its compression parts.

Reduced Cross-Section properties

Using the initial shape the cross-section parts are classified as specified in the previous chapter.

For calculating the reduced shape the following reduction factors are calculated:

o Local Buckling: Reduction factor c

o Distortional Buckling: Reduction factor

o HAZ effects: Reduction factor HAZ

Calculation of Reduction factor c for Local Buckling

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



18

Step 1: Calculation of spring stiffness

Spring stiffness c = cr for RI:

Spring stiffness c = cs for RUO:



19

ad p

ad

s

s

s

b

Et c

cb

Et b y

ycc

,

3

3

3

21

3

31

²)1(12

²)1(4

1

With: tad Thickness of the adjacent element

bp,ad Flat width of the adjacent element

c3 The sum of the stiffnesses from the adjacent elements

α 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 3 elements(counted as plates in initial geometry, without the reinforcement parts, e.g. channel or Z

sections)

These parameters are illustrated on the following picture:

edge stiffener

considered plate

adjacent element

t ad

bp,ad



20

Step 2: Calculation of Area and Second moment of 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 of the 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, with thickness t and effective width beff ,referred to the neutral axis a-a

beff For RI reinforcement taken as 15 t

For ROU reinforcement taken as 12 t

These parameters are illustrated on the following figures.

Ar and Ir for RI:



21

Ar and Ir for RUO:

Step 3: Calculation of stiffener buckling load

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

With: c Spring stiffness of Step 1

E Module of Young

Ir Second moment of area of Step 2

r cr r cEI N 2

,



22

Step 4: Calculation of reduction factor for distortional buckling

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 strength

c 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 the width of the adjacent

part(s).

Calculation of Reduction factor HAZ for HAZ effects

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



23

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

for 5xxx & 6xxx alloys :

120

)601(12

T

for 7xxx alloys :

120

)601(5.112

T

With: T1 Interpass temperature

n Number of heat paths

Note:The variations in numbers of heath paths 3/n is specifically intended for fillet welds. In case of a butt weld the parameter n should be set to 3 (instead of 2) to negate this effect.

The reduction factor for the HAZ is given by:

u

haz,u

haz,uf

f

o

haz,ohaz,o

f

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 previous chapter.

Section properties

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



24

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

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.

CompressionThe 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 moment

The 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 shape for M+ / M-

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

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

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:



25

i

beg end

iw

wi

t

x x

t

h

With: xend End position of plate i .

xbeg Begin position of plate i.

t Thickness of plate i.

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

With

0

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.



26

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 uibeg end

t x x 2, cos)(

For Vz: Anet,z,i = ii HAZ uibeg end t x x 2

, sin)(

With: i The number (ID) of the plate

xend End position of plate i

xbeg Begin position of plate i

t Thickness of plate i

u,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+(u,HAZ) inthe same way as for a non-slender part.

=> VRd,i,yield

Formula (6.89) is used with a the member length or the distance between stiffeners (for I or

U-sections)=> VRd,i,buckling

=> For this slender part, the eventual VRd,i is taken as the minimum of VRd,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

Note:For a solid bar, round tube and hollow tube, all parts are taken as non-slender by default and formula (6.31) isapplied.



27

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 section

0,6 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-section shape:

n

i HAZ beg end vy t x x A1

2

,0 cos)(

n

i HAZ beg end vz t x x A1

2

,0 sin)(

With: i The number (ID) of the plate

xend End position of plate i

xbeg Begin position of plate i

t Thickness of plate i

0,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).

Note:For sections without initial shape or numerical sections, none of the above mentioned methods can be applied. Inthis case, formula (6.29) is used with Av taken as Ay or Az of the gross-section properties.

Torsion with warping

In case warping is taken into account, the combined section check is replaced by an elastic stress check including

warping stresses.



28

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,Ed Total direct stress

tot,Ed Total shear stress

M1 Partial safety factor for resistance of cross-sections

C Constant (by default 1,2)

N,Ed Direct stress due to the axial force on the relevant effective cross-section

My,Ed

Direct stress due to the bending moment around y axis on the relevant

effective cross-section

Mz,Ed

Direct stress due to the bending moment around z axis on the relevant

effective cross-section

w,Ed Direct stress due to warping on the gross cross-section

Vy,Ed Shear stress due to shear force in y direction on the gross cross-section

Vz,Ed Shear stress due to shear force in z direction on the gross cross-section

t,Ed

Shear stress due to uniform (St. Venant) torsion on the gross cross-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 are described in “Profile conditions for code

check”.

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 Bimoment

wM Unit warping

Cm Warping constant



29

I sections

For I sections, the value of wM is given in the tables (Ref. [3], Tafel 7.87, 7.88). This value is added 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

1w

With b Section width

hm Section height (see figure)

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 following figure:



31

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 warping

Cm Warping constant

t 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



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)



33

Standard diagrams

The following 6 standard situations for St.Venant torsion, warping torque and bimoment are given 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. Venant

Mxs Warping torque

Mw Bimoment

IT Torsional constant

CM Warping constant

E Modulus of elasticity

G Shear modulus



34

Torsion fixed 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 t

w

Mw for b side

)'xsinh()Lsinh(

)asinh(MM t

w



35

Torsion fixed 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 side

2DM

M tw



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(2

2

)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, distributed torsional loading mt

Mx

2

LmM

2

Lm

M

txb

t

xa

Mxp

)Lsinh(

)'xcosh()xcosh()x

2

L(

mM t

xp

Mxs

)Lsinh(

)'xcosh()xcosh(mM t

xs



37

Mw

)Lsinh(

)'xsinh()xsinh(1

mM

2

tw

Torsion fixed ends, warping fixed ends, distributed torsional loading mt

Mx

2

LmM

2

LmM

txb

txa

Mxp

)Lsinh(

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

2

L(

mM t

xp

Mxs

)Lsinh(

)'xcosh()xcosh()k 1(

mM t

xs

Mw

)Lsinh(

)'xsinh()xsinh()k 1(1

mM

2

tw

)2

Ltanh(

2L

1k



38

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

Mx

txa MM

Mxp

)Lcosh(

)'xcosh(1MM txp

Mxs

)Lcosh()'xcosh(MM txs

Mw

)Lcosh(

)'xsinh(MM t

w

One end free, other end torsion and warping fixed, distributed torsional loading mt

Mx



39

LmM txa

Mxp

)Lcosh(

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

mM t

xp

Mxs

)Lcosh(

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

mM t

xs

Mw

)Lcosh(

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

²

mM t

w

Decomposition of arbitrary torsion line

Since the SCIA•ESA PT solver does not take into account the extra DOF for warping, the determination of the

warping torque and the related bimoment, is based on some standard situations.

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



40

Decomposition for situation 1 and situation 3

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

n number of torsion lines generated by a local torsional loading Mt n

one torsion line generated by a distributed torsional loading mt one torsion line with constant torque Mt0

The values for Mxp, Mxs and Mw are taken from the previous tables for the local torsional loadings Mt n and thedistributed loading mt. The value Mt0 is added to the Mxp value.

Decomposition for situation 2

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

one torsion line generated by a local torsional loading Mtn

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 torsional loading 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 is activated, the combined

section check is replaced by an elastic stress check including warping stresses which takes into account all shear

stress effects. For more information please refer to “Torsion with warping”.

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

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



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 and calculated in the fibres of the gross cross-section.

Note:The interaction for mono-symmetrical sections specified in art. 6.2.9.1 (2) is not supported. For mono-symmetrical

sections the elastic stress check of art. 6.2.1 is applied.

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 least width 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.

Note:Since the extend of the HAZ is defined along the member axis, it is important to specify enough sections onaverage member in the Solver Setup when transverse welds are used.

Note: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 are reduced using a reducedyield 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

Stress check for numerical sections

For numerical sections an elastic stress check is performed according to art. 6.2.1 and formula (6.15). The

stresses are calculated in the following way:



42

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 direction

Wy Elastic section modulus around y axis

Wz Elastic section modulus around z axis

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).



43

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 Diagonalso From Stability Analysiso Manual input

These 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. These formulas are treated inreference [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

For a sway structure:

4+

x

x=l/L1

2

With: L System length

E Modulus of Young

I Moment of inertia

Ci Stiffness in node i

Mi Moment in node i

i Rotation in node i



44

21212

12

21

8+)+(

+4=x

EILC= i

i

i

ii

M=C

The values for M i and i are approximately determined by the internal forces and the deformations, calculated byload cases which generate deformation forms, having an affinity 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 are used, on the columns the

global distributed loads Qx = -10000 N/m and Qy= -10000 N/m are used.

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 bucking ratios. In such cases a more refinedapproach (from stability analysis) can be applied.

Calculation of Buckling ratio – Crossing Diagonals

For crossing diagonal elements, the buckling length perpendicular to the diagonal plane, is calculated according toRef.[10], DIN18800 Teil 2, table 15. This means that the buckling length sK is dependent on the load distribution inthe element, and it is not a purely geometrical data anymore.

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

Wit

h:

s

K

Buckling length

L Member length

L

1

Length of supporting diagonal

I Moment of inertia (in the buckling plane) of the member

I

1

Moment of inertia (in the buckling plane) of the supporting diagonal

N Compression force in m ember

N

1

Compression force in supporting diagonal

Z Tension force in supporting diagonal

E Modulus of Young

Continuous compression diagonal, supported by continuous tension diagonal



45

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.

Continuous compression diagonal, supported by pinned tension diagonal

NN

Z

Z

l/2

l1/2

l5.0s

lN

lZ75.01ls

K

1

K


Pinned compression diagonal, supported by continuous tension diagonal



46

NN

Z

Z

l/2

l1/2

)1lZ

lN(

4

lZ3)IE(

1lZ

lNl5.0s

1

2

2

1

d1

1

K


Continuous compression diagonal, supported by continuous compression diagonal

NN

N1

N1

l/2

l1/2

l5.0s

lI

l1I1

lN

lN1

ls

K

3

1

3

1

1

K


Continuous compression diagonal, supported by pinned compression diagonal



47

NN

N1

N1

l/2

l1/2

1

1

2

KlNlN

121ls

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

Pinned compression diagonal, supported by continuous compression diagonal

NN

N1

N1

l/2

l1/2

)N

lN

12(

l

lN)IE(

l5.0s

1

1

2

1

2

3

d

K

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



48

Calculation of Buckling ratio – From Stability Analysis

When member buckling data from stability are defined, the critical buckling load Ncr for a prismatic member iscalculated 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

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

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 of radiating 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-

Note:The Torsional (-Flexural) buckling check is ignored for sections complying with the rules 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).



49

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



50

Calculation of Ncr,TF

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

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

0

With: 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

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 solutiono Manual input

Note:The Lateral Torsional buckling check is ignored for circular hollow sections according to 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 the paragraph "Calculation of Moment factors for LTB".

For the other supported sections, the elastic critical moment for LTB Mcr is given by:

z2

t

z2

z2

EI

L²GI

I

Iw

L

EIMcr

With: E Modulus of elasticity

G Shear modulus

L Length of the beam between points which have lateral restraint (= lLTB)

Iw Warping constant

It Torsional constant

Iz 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.

Diaphragms



51

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

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

bCC

s

EIk C

C

1

C

1

C

1

vorhC

1

G

lvorhCII

s

s

k ,P

aa

100k ,A

a

2a

100k ,A

eff k ,M

k ,Pk ,Ak ,M

2

2

tid,t

With: l LTB length

G Shear modulus

vorh

C

Actual rotational stiffness of diaphragm

CM,k Rotational stiffness of the diaphragm

CA,k Rotational stiffness of the connection between the diaphragm and the 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 beam

ba Width of the beam flange (in mm)

C100 Rotation coefficient - see table

h Height of the beam

t Thickness of the beam flange

s Thickness of the beam web



52



53

Calculation of Moment factors for LTB

For determining the moment factors C1 and C2 for lateral torsional buckling, standard tables are used which aredefined 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 loado 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 taken for 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 distribution 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=C

2

*

2.70<E*

0.52+1.40-1.88=E*2

Moment distribution generated by F load



54

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**

eD**

(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 line with maximum at the start or at the end of the beam

M2 M1 = Beta M2

l

C2 = 0.0

2.70<1C and

0.52+1.40-1.88=1C 2

LTBII Eigenvalue solution

For calculation of Mcr using LTBII reference is made to chapter “LTBII: Lateral Torsional Buckling 2nd Order Analysis”.



55

Combined bending and axial compression

The Combined bending and axial compression check is verified using art. 6.3.3.1 & 6.3.3.2 .

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 without localized welds thereduction factors are calculated according to art. 6.3.3.5.

Members containing localized 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 least width 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.

Note:Since the extend of the HAZ is defined along the member axis, it is important to specify enough sections onaverage member in the Solver Setup when transverse welds are used.

Note:In the calculation of xLT the buckling length l c and distance x s are taken for buckling around the z-z axis.

Unequal end moments and/or transverse loads

If the section under consideration is not located in a HAZ zone, the reduction factors x and xLT are thencalculated according to art. 6.3.3.5 .



56

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

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

Note:In the calculation of xLT the buckling length l c and distance x s are taken for buckling around the z-z axis.

Calculation of xs

The distance xs is defined as the distance from the studied section to a simple support or point 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 a denominator 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 for the calculation of xs.

Known buckling shape

The buckling shape is assumed to be known in case the buckling ratio is calculated according to the GeneralFormula specified in chapter “Calculation of Buckling ratio – General Formula”. The basic assumption is that thedeformations for the buckling load case have an affinity with the buckling shape.

Since the buckling shape (deformed structure) is known, the distance from each section to a 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 the positions where the bending moment diagram for the buckling loadcase reaches zero.

Note:Since for a known buckling shape x s can be different in each section, accurate results can be obtained by

increasing the numbers of sections on average member in the Solver Setup.

Unknown buckling shape

In case the buckling ratio is not calculated according to the General Formula specified in chapter “Calculation of

Buckling ratio – General Formula” the buckling shape is taken as unknown. This is thus the case for manual inputor if the buckling ratio is calculated from stability.



57

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

but xs ≥ 0

With: lc Buckling length

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

NEd Design value of the axial compression force

MRd Bending moment capacity

NRd Axial compression force capacity

Reduction factor for flexural buckling

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

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 a value between -1,00

and +1,00

If one of the two above stated limitations occur, the formula is not applied and instead xs is taken as half of the buckling

length for each section.

Note:The above specified formula contains the factor in the denominator of the right side of the equation. This factor was erroneously omitted in formula (6.71) of EN 1999-1-1:2007.

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 the member ends.

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

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

Note:For shear buckling only transverse stiffeners are supported. Longitudinal stiffeners are not 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 member ends. The verification isdone according to art. 6.7.4.1.

The check is executed when the following condition is met:



58

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

f 0 0,2% proof strength

The design shear resistance VRd for shear buckling consists of one part: the contribution of the 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 following table:

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:



59

For interaction see paragraph “Interaction”.



60

Plate girders with intermediate web stiffeners

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

f 0 0,2% proof strength

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

Contribution 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.



61

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

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

With: f uw Ultimate strength of the web material


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

Contribution of the flanges

First the design moment resistance of the cross-section considering only the flanges Mf,Rd is calculated.

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.



62

On each side of the web.

With: f 0f Yield strength of the flange 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”.



63

Interaction

If required, for both above cases the interaction between shear force, bending moment and axial force is checkedaccording to art. 6.7.6.1.

If the following two expressions are checked:

With:

Mf,Rd design moment resistance of the cross-section considering only

the flanges

Mpl,Rd Plastic design bending moment resistance

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

by:

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



64

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 applicationRef.[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 the respective results are

sent back to SCIA•ESA PT.

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 warpingo End and begin forces

o Loadingso Intermediate restraints (diaphragms, LTB restraints)

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

Cw_i Warping condit ion at end i (beginning of the member)

Cw_j Warping condition at end j (end of the member)

Ct_i Torsion condit ion at end i (beginning of the member)

Ct_j Torsion condit ion at end j (end of the member)

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



65

For this system, the elastic critical moment Mcr for lateral torsional buckling can be analyzed as the solution of an

eigenvalue problem:

0K K ge

Wit

h: Critical load factor

Ke Elastic linear stiffness matrix

Kg Geometrical stiffness matrix

For members with arbitrary sections, the critical moment can be obtained in each section, with: (See

Ref.[24],pp.176)

)x(MxM

MmaxM

yycr

yycr

With:

Critical load factor

Myy Bending moment around the strong axis

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

Mcr (x) Critical moment at position x

The calculated Mcr is then used in the Lateral Torsional Buckling check of SCIA•ESA PT.

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



66

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 forceso 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 freedom calculation.

The calculated torsional and warping moments (St Venant torque Mxp, Warping torque Mxs and Bimoment Mw)are then used in the Stress check of SCIA•ESA PT (See chapter “Torsion with warping”).

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

o Normal force from SCIA•ESA PT

o Maximal shear forces from SCIA•ESA PT / Frilo LTBII

o Maximal bending moments from SCIA•ESA PT / 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].



67

Supported Sections

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

FRILO LTBII CSS SCIA•ESA PT CSS Eigenvalue

analysis

2nd

Order

analysis

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 geometric

asymmetric 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 x

Thin walled geometric U x x

Thin walled Cold formed from library x x

Cold formed from graphical input x x

Double T with top

flange angle

Welded I+2L x

Sheet welded Iw+2L x

Rectangle Full rectangular from library x

Full rectangular from thin walled

geometric

x

Static values double

symmetric

all other double symmetric CSS x

Static values single

symmetric

all other single symmetric CSS x

The following picture illustrates the relation between the local coordinate system of SCIA•ESA PT and Frilo LTBII.

Special attention is required for U sections due to the inversion of the y and z-axis.



68

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

Loadings

The following load impulses are supported:

o Point force in node (if the node is part of the exported beam)

o Point force on beamo Line force in beam

o Moment in node (if the node is part of the exported beam)o Moment on beamo Line moment in beam (only for Mx in LCS)

The supported load impulses and their eccentricities are transformed into the local LCS of the exported member.

The dead load is replaced by an equivalent line force on the beam.

Load eccentricities are replaced by torsional moments.

The forces in local x-direction are ignored, except for the torsional moments.

Note:In Frilo LTBII a distinction is made between the centroid and the shear center of a cross-section. Load impulseswhich do not pass through the shear center will cause additional torsional moments.

Imperfections

In the 2nd

Order LTB analysis the bow imperfections v0 (in local y direction) and w0 (in local z direction) can betaken into account.



69

v0

y, v0

zy

Initial bow imperfection v0 according to codeFor EC-EN the imperfections can be calculated according to the code. The code indicates that for a 2

ndOrder

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•ESA PT.

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

00 ek v

Wit

h

k Factor by default taken as 0,5

e0 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.



70

LTB Restraints

LTB restraints are transformed into 'Supports' (Ref.[23] p22), with horizontal elastic restraint Cy:

Cy = 1e15 kN/m

The position of the restraint z(Cy) is depending on the position of the LTB restraint (top/bottom).

The use of an elastic restraint allows the positioning of the restraint since this is not possible for a fixed restraint.

(Ref.[23] p23)

Specifically for U-sections, an elastic restraint Cz is used with position y(Cz) due to the rotation of U-sections in the

Frilo LTBII solver (see paragraph “Supported Sections”).



71

Diaphragms

Diaphragms are transformed into 'Elastic Foundations' of type ‘elastic restraint’ (Ref.[23] p25). Both a horizontalrestraint Cy and a rotational restraint C are used.

The elastic restraint Cy [kN/m^2] is calculated as follows (Ref.[23] p52 and Ref.26 p40):

2

L

S Cy

Wit

h:

S Shear stiffness of the diaphragm

L Diaphragm length along the member

The above formula for Cy is valid in case the bolt pitch of the diaphragm is set as ‘br’. For a bolt pitch of ‘2br’ the

shear stiffness S is replaced by 0,2 S (Ref.26 p22).

The shear stiffness S for a diaphragm is calculated as follows (Ref. 28,3.5 and Ref.29,3.3.4.):

L

K +K

10a.=S

s

21

4

With: a Frame distance

Ls Length of the diaphragm

K1 Factor K1 of the diaphragm

K2 Factor K2 of the diaphragm

The position of the restraint z(Cy) is depending on the position of the diaphragm.

Specifically for U-sections, an elastic restraint Cz is used with position y(Cz) due to the rotation of U-sections in the

Frilo LTBII solver (see paragraph “Supported Sections”).



72

The rotational restraint C [kNm/m] is taken as vorhC (see paragraph “Calculation of Mcr – General Formula”).

Linked Beams

Linked beams are transformed into 'Supports' (Ref.[23] p22), with elastic restraint.

The direction of the restraint is dependent on the direction of the linked beam:

If the linked beam has an angle less than 45° with the local y-axis of the beam under consideration, the restraint is

set as Cy. In all other cases the restraint is set as Cz.

The position of the restraint z(Cy) or y(Cz) is depending on the application point of the linked beam (top/bottom).

The position is only taken into account in case of a flexible restraint (Ref.[23] p23).

The end forces of the linked beam are transformed to point loads on the considered 1D member,o in z -direction for linked beams considered as y-restraint

o in y- direction for linked beams considered as z-restraint

Specifically for U-sections, if the linked beam has an angle less than 45° with the local y-axis of the beam under

consideration, the restraint is set as Cz. In all other cases the restraint is set as Cy. This is due to the rotation of U-sections in the Frilo LTBII solver (see paragraph “Supported Sections”).



73

Limitations and Warnings

The FRILO LTBII solver is used with following limitations:

o Only straight members are supported

o LTBII analysis is done for the whole 1D member, not for a part of the member, not for more memberstogether

o When a LTB system length is inputted which differs from the member length, a warning will be given.o Intermediate lateral restraints should be defined through LTB restraints, diaphragms and linked beams.

During the analysis, the Frilo LTBII solver may return a warning message. The most important causes of the

warning message are listed here.

Eigenvalue solution Mcr - Lateral Torsional Buckling is not governing – relative slenderness < 0,4

Due to the low relative slenderness, no LTB check needs to be performed. In this case it is not required to use the

Frilo LTBII solver.

- Design Torsion! Simplified analysis of lateral torsional buckling is not possible.

Due to the torsion in the member it is advised to execute a 2 nd

order analysis instead of an eigenvalue calculation.

- Bending of U-section about y-axis!

The program calculates the minimum bifurcation load only.

2nd

Order Analysis

- Load is greater then minimum bifurcation load (Error at elastic calculation – system is instable in II.Order )

The loading on the member is too big, a 2 nd

order calculation cannot be executed.

- You want to calculate the structural safety with Elastic-Plastic method. This analytical procedure cannot be used

for this cross-section. It is recommended to use the Elastic-Elastic method.

Plastic calculation is not possible, use imperfection according to code elastic instead of plastic.

For more information, reference is made to Ref.[22] and [23].



74

Profile conditions for code check

Introduction to profile characteristics

The standard profile sections have fixed sections properties and dimensions, which have to be present in theprofile library.

The section properties are described in chapter “Data for general section stability check".

The required dimension properties are described in chapter "Data depending on the profile shape”.

Data for general section stability check

The following properties have to be present in the profile library for the execution of the section and the stability check:

Description Property number

Iy moment of inertie yy 8

W

y

elastic section modulus yy 10

Sy statical moment of area yy 6

Iz moment of inertia zz 9

W

z

elastic section modulus zz 11

Sz statical moment of area zz 7

It* torsional constant 14

Wt

*

torsional resistance 13

A0 sectional area 1

Iyz centrifugal moment 12

iy radius of gyration yy 2

iz radius of gyration zz 3

M

py

plastic moment yy 30

M

pz

plastic moment zz 31

fa

b

fabrication code

0=rolled section (default value)

1=welded section

2=cold formed section

105

The fabrication code is not obligatory.

When the section is made out of 1 plate, the properties marked with (*) can be calculated by the calculation routine

in the profile library. When this is not the case, these properties have to be input by the user in the profile library.

The plastic moments are calculated with a yield strength of 240 N/mm².



75

Data depending on the profile shape

I section

Formcode 1

PSS Type .

I.

Propert

y

Descriptio

n

49 H

48 B

44 t

47 s

66 R

74 W

140 wm1

61 R1

146

109 1

B

s

w

t

R

R1

a

H



76

RHS

Formcode 2

PSS Type .

M

.

Property Description

49 H

48 B

67 s

66 R

109 2

B

s

H

R



77

CHS

Formcode 3

PSS Type .R

O.


64 D

65 s

109 3

D

w



78

Angle section

Formcode 4

PSS Type .

L.


49 H

48 B

44 t

61 R166 R

74 W1

75 W2

76 W3

109 4

B

R

R1

w1

w2

t

w3

w1

w2



79

Channel section

Formcode 5

PSS Type .

U

.


49 H

48 B

44 t

47 s

66 R

68

41

61 R1

146

109 5

B

s

H

t

R

R1

a



80

T section

Formcode 6

PSS Type .

T

.


49 H

48 B

44 t

47 s

66 R

61 R1

62 R2

146 1

147 2

109 6

B

s

t

R

a1

H

a2

R1

R2



82

Full circular section

Formcode 11

PSS Type .R

U.


64 D

109 11

D



83

Asymmetric I section

Formcode 1

0

1

PSS Type


49 H

48

44

47 s42 Bt

43 Bb

45 tt

46 tb

66 R

109 101

R

H

Bt

Bb

tt

tb



84

Z section

Formcode 102

PSS Type .

Z.


49 H

48 B

44 t

47 s

67 R

61 R1

109 102

B

s

t

H

RR1



85

General cold formed section

Each section is considered as a composition of rectangular parts. Each part represents a plate unit which is

considered as element for defining the effective width. The start and end parts are considered as un-stiffenedelements, the intermediate parts are considered as stiffened parts.

This way of definition of the section assumes that the area is concentrated at its centre line. The rounding in the

corners is ignored.

Description Property number Valu

e

form code 109 110

Dy* 22

Dz* 23

CM* 26

buckling curve around yy axis 106 (1)

buckling curve around zz axis 107 (1)

buckling curve for LTB 108 (1)

(1) The values for the buckling curves are defined as follows :

1 = buckling curve a

2 = buckling curve b

3 = buckling curve c

4 = buckling curve d

The conditions are that the section is an open profile. Only the geometry commands O, L, N, A may be used in the

geometry description.

When the section is made out of 1 plate, the properties marked with (*) can be calculated by the calculation routine

in the profile library. The properties from the reduced section can be calculated by the code check.



86

When the section is made out of more than 1 plate, the properties marked with (*) can NOT be calculated by the

calculation routine in the profile library. The properties from the reduced section can be calculated, except for the

marked properties. These properties have to be input by the user in the profile library.

Formcode 1

10

PSS Type


44 s

61 r

48 B

142 sp

143 e268 H

109 110

Remark:

r is rounding, special for KLS section (Voest Alpine)

sp is number of shear planes

B

H

e2

s

Cold formed angle section

Formcode 1

1

1

PSS Type



87


44 s

61 r

48 B

68 H

109 111

B

s

H

r

Cold formed channel section

Formcode 1

1

2

PSS Type



88


44 s

61 r

48 B

49 H

109 112

B

s

H

r

Cold formed Z section

Formcode 1

1

3

PSS Type



89


44 s

61 r

48 B

49 H

109 113

B

s

H

R

Cold formed C section

Formcode 1

14

PSS Type




90

44 s

61 r

48 B

49 H

68 c

109 114

B

s

H

r

c

Cold formed Omega section

Formcode 115

PSS Type


44 s



91

61 r

48 B

49 H

42 c

109 115

B

s

H

c

R

Rail type KA

Formcode 15

0

PSS Type .K

A.


148 h1

149 h2

150 h3



92

151 b1

152 b2

153 b3

154 k

155 f1

156 f2

157 f3

61 r1

62 r2

63 r3

158 r4

159 r5

160 a

109 150

r1

r2

r4

r3

r5

b3

k

b2

b1

f3

f2

f1

h1

h3h2

Rail type KF

Formcode 151

PSS Type .K

F.


48 b

154 k

49 h

153 b3



93

155 f1

157 f3

148 h1

149 h2

61 r1

62 r2

63 r3

109 151

r1

r2r2

r2 r2

r3

k

bb3

f3

f1

h

h1 h2

Rail type KQ

Formcode 15

2

PSS Type .K

Q.


48 b

154 k

49 h

153 b3

155 f1



95

References

1

Eurocode 9

Design of aluminum structures

Part 1 - 1 : General structural rules

EN 1999-1-1:2007

[

2

]

TALAT Lecture 2301

Design of members

European Aluminium Association

T. Höglund, 1999.

[

3]

Stahl im Hochbau

14. Auglage Band I/ Teil 2

Verlag Stahleisen mbH, Düsseldorf 1986

[

4

]

Kaltprofile

3. Auflage

Verlag Stahleisen mbH, Düsseldorf 1982

[

5

]

Dietrich von Berg

Krane und Kranbahnen – Berechnung Konstruktion Ausführung

B.G. Teubner, Stuttgart 1988

[

6

]

C. Petersen

Stahlbau : Grundlagen der Berechnung und baulichen Ausbildung von Stahlbauten

Friedr. Vieweg & Sohn, Braunschweig 1988

[

7

]

Handleiding moduul STACO VGI

Staalbouwkundig Genootschap

Staalcentrum Nederland

5684/82

[

8

]

Newmark N.M. A simple approximate formula for effective end-fixity of columns

J.Aero.Sc. Vol.16 Feb.1949 pp.116

[9

]

Stabiliteit voor de staalconstructeur

uitgave Staalbouwkundig Genootschap

[

1

0]

DIN18800 Teil 2

Stahlbauten : Stabilitätsfälle, Knicken von Stäben und Stabwerken

November 1990

[

1

1]

Rapportnr. BI-87-20/63.4.3360

Controleregels voor lijnvormige constructie-elementen

IBBC Maart 1987



96

[

12

]

Staalconstructies TGB 1990

Basiseisen en basisrekenregels voor overwegend statisch belaste constructies

NEN 6770, december 1991

[

13

]

SN001a-EN-EU

NCCI: Critical axial load for torsional and flexural torsional buckling modes

Access Steel, 2006

www.access-steel.com

[

14]

Eurocode 3

Design of steel structures

Part 1 - 1 : General rules and rules for buildings

ENV 1993-1-1:1992

[15

]

R. MaquoiELEMENTS DE CONSTRUCTIONS METALLIQUE

Ulg , Faculté des Sciences Appliquées, 1988

[

16

]

ENV 1993-1-3:1996

Eurocode 3 : Design of steel structures

Part 1-3 : General rules

Supplementary rules for cold formed thin gauge members and sheeting

CEN 1996

[

17]

E. Kahlmeyer

Stahlbau nach DIN 18 800 (11.90)

Werner-Verlag, Düsseldorf

[

18

]

Beuth-Kommentare

Stahlbauten

Erläuterungen zu DIN 18 800 Teil 1 bis Teil 4, 1.Auflage

Beuth Verlag, Berlin-Köln 1993

[

19

]

Staalconstructies TGB 1990

Stabiliteit

NEN 6771 - 1991

[

2

0]

A Gerhsi, R. Landolfo, F.M. Mazzolani (2002)

Design of Metallic cold formed thin-walled members

Spon Press, London, UK

[

21

]

G. Valtinat (2003)

Aluminium im Konstruktiven Ingenieurbau

Ernst & Sohn, Berlin, Germany

[

22

]

Frilo LTBII software

Friedrich + Lochner Lateral Torsional Buckling 2nd

Order AnalysisBiegetorsionstheorie II.Ordnung (BTII)

http://www.frilo.de

[ Friedrich + Lochner LTBII Manual



23

]

BTII HandbuchRevision 1/2006

[

24

]

J. Meister

Nachweispraxis Biegeknicken und Biegedrillknicken

Ernst & Sohn, 2002

[

2

5]

Eurocode 3

Design of steel structures

Part 1 - 1 : General rules and rules for buildings

EN 1993-1-1:2005

[

2

6]

J. Schikowski

Stabilisierung von Hallenbauten unter besonderer Berücksichtigung der Scheibenwirkung

von Trapez- und Sandwichelementdeckungen, 1999

http://www.jschik.de/

[27

]

DIN 18800 Teil 2

Stahlbauten

Stabilitätsfälle, Knicken von Stäben und Stabwerken

November 1990

[

28

]

E. Kahlmeyer

Stahlbau nach DIN 18 800 (11.90)

Werner-Verlag, Düsseldorf

[

29

]

Beuth-Kommentare

Stahlbauten

Erläuterungen zu DIN 18 800 Teil 1 bis Teil 4, 1.Auflage

Beuth Verlag, Berlin-Köln 1993

aluminium code check theory enu

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