Scia - Aluminium Code Check Theory
Transcript of Scia - Aluminium Code Check Theory
-
8/19/2019 Scia - Aluminium Code Check Theory
1/110
Subtitle
Theory
Aluminium Code Check
-
8/19/2019 Scia - Aluminium Code Check Theory
2/110
-
8/19/2019 Scia - Aluminium Code Check Theory
3/110
i
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
-
8/19/2019 Scia - Aluminium Code Check Theory
4/110
ii
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
-
8/19/2019 Scia - Aluminium Code Check Theory
5/110
iii
Rail type KQ ........................................................................................................ 101 References ........................................................................................................................................... 102
-
8/19/2019 Scia - Aluminium Code Check Theory
6/110
1
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
-
8/19/2019 Scia - Aluminium Code Check Theory
7/110
2
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
8/110
3
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
9/110
4
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 (*)
-
8/19/2019 Scia - Aluminium Code Check Theory
10/110
5
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
-
8/19/2019 Scia - Aluminium Code Check Theory
11/110
6
- RUO : The reinforcement is connected to only one plate with type I
RUOI
-
8/19/2019 Scia - Aluminium Code Check Theory
12/110
7
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 )
-
8/19/2019 Scia - Aluminium Code Check Theory
13/110
8
4 Angle section
(UO, none)
(UO, none)(F, none)
5 Channel section (UO, none)
(UO, none)
(I, none)
(F, none)
(F, none)
-
8/19/2019 Scia - Aluminium Code Check Theory
14/110
9
6 T section
(UO, none)
(SO, none)(SO, none)
(F, none)
7 Full rectangularsection
No reduction possible
11 Full circular section No reduction possible101 Asymmetric I section (SO, none)(SO, none)
(SO, none)(SO, none)
(F, none)
(F, none)
(I, none)
-
8/19/2019 Scia - Aluminium Code Check Theory
15/110
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)
-
8/19/2019 Scia - Aluminium Code Check Theory
16/110
11
111 Cold formed angle
(UO, none)
(UO, none)
112 Cold formed channel (UO, none)
(I, none)
(UO, none)
-
8/19/2019 Scia - Aluminium Code Check Theory
17/110
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)
-
8/19/2019 Scia - Aluminium Code Check Theory
18/110
13
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
19/110
14
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
-
8/19/2019 Scia - Aluminium Code Check Theory
20/110
15
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
-
8/19/2019 Scia - Aluminium Code Check Theory
21/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
22/110
17
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).
-
8/19/2019 Scia - Aluminium Code Check Theory
23/110
18
Step 1: Calculat ion of sprin g sti f fness
Spring stiffness c = cr for RI:
Spring stiffness c = cs for RUO:
-
8/19/2019 Scia - Aluminium Code Check Theory
24/110
19
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
-
8/19/2019 Scia - Aluminium Code Check Theory
25/110
20
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:
-
8/19/2019 Scia - Aluminium Code Check Theory
26/110
21
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,
-
8/19/2019 Scia - Aluminium Code Check Theory
27/110
22
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
-
8/19/2019 Scia - Aluminium Code Check Theory
28/110
23
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
29/110
24
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.
Deduction of holes is not taken into account.
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
30/110
25
Shear
The Shear check is verified using art. 6.2.6 & 6.5.5.
Deduction of holes is not taken into account.
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
31/110
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 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.
-
8/19/2019 Scia - Aluminium Code Check Theory
32/110
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 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.
-
8/19/2019 Scia - Aluminium Code Check Theory
33/110
28
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”.
-
8/19/2019 Scia - Aluminium Code Check Theory
34/110
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)
-
8/19/2019 Scia - Aluminium Code Check Theory
35/110
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
36/110
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 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
-
8/19/2019 Scia - Aluminium Code Check Theory
37/110
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)
-
8/19/2019 Scia - Aluminium Code Check Theory
38/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
39/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
40/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
41/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
42/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
43/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
44/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
45/110
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
46/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
47/110
42
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
-
8/19/2019 Scia - Aluminium Code Check Theory
48/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
49/110
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
50/110
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
51/110
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
See Ref.[10], Tab. 15, case 4.
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
See Ref.[10], Tab. 15, case 5.
-
8/19/2019 Scia - Aluminium Code Check Theory
52/110
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
See Ref.[10], Tab. 15, case 2.
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).
-
8/19/2019 Scia - Aluminium Code Check Theory
53/110
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
54/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
55/110
50
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
-
8/19/2019 Scia - Aluminium Code Check Theory
56/110
51
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
57/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
58/110
53
-
8/19/2019 Scia - Aluminium Code Check Theory
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
-
8/19/2019 Scia - Aluminium Code Check Theory
60/110
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
-
8/19/2019 Scia - Aluminium Code Check Theory
61/110
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
62/110
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
63/110
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
64/110
59
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
f 0 0,2% proof strength
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:
-
8/19/2019 Scia - Aluminium Code Check Theory
65/110
60
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”.
-
8/19/2019 Scia - Aluminium Code Check Theory
66/110
61
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
f 0 0,2% proof strength
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
67/110
62
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
68/110
63
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”.
-
8/19/2019 Scia - Aluminium Code Check Theory
69/110
64
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
70/110
65
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
-
8/19/2019 Scia - Aluminium Code Check Theory
71/110
66
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
72/110
67
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.
The interaction equations are summarised as follows:
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
Vz Shear force in z direction
My Bending moment about the y axis
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
-
8/19/2019 Scia - Aluminium Code Check Theory
73/110
68
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].
The interaction equations are summarised as follows:
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
My Bending moment about the y axis
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 M0 of EN 1993-1-1 for steel couplers
Safety factor taken as M1 of EN 1999-1-1 for aluminiumcouplers
-
8/19/2019 Scia - Aluminium Code Check Theory
74/110
69
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
-
8/19/2019 Scia - Aluminium Code Check Theory
75/110
70
(+) This index indicates a tensile force
Vy Shear force in y direction
Vz Shear force in z direction
My Bending moment about the y axis
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 M0 of EN 1993-1-1 for steel couplers
Safety factor taken as M1 of EN 1999-1-1 for aluminiumcouplers
-
8/19/2019 Scia - Aluminium Code Check Theory
76/110
71
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”.
-
8/19/2019 Scia - Aluminium Code Check Theory
77/110
72
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
-
8/19/2019 Scia - Aluminium Code Check Theory
78/110
73
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].
-
8/19/2019 Scia - Aluminium Code Check Theory
79/110
74
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.
-
8/19/2019 Scia - Aluminium Code Check Theory
80/110
-
8/19/2019 Scia - Aluminium Code Check Theory
81/110
76
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