Theory Aluminium Code Check Subtitle 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 i 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 nd 2 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 nd 2 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 ii Rail type KQ ........................................................................................................ 101 References ........................................................................................................................................... 102 iii 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 Addendum EN 1999-1-1:2007/A1:2009 Version info Documentation Title Release Revision Aluminium Code Check – Theoretical Background 2012.0 03/2012 1 Disclaimer This document is being furnished by SCIA for information purposes only to licensed users of 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 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 be used 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 2012 Nemetschek SCIA. All rights reserved. 2 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 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 (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 EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW 12.5) EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW EN-AW 6063 (EP) T5 (3-25) 6063 (EP,ET,ER/B) T6 (0-25) 6063 (DT) T6 (0-20) 6063 (EP,ET,ER/B) T66 (0-10) 6063 (EP) T66 (10-25) 6063 (DT) T66 (0-20) 6082 (Sheet) T4/T451 (0-12.5) 6082 (Sheet) T61/T6151 (06082 (Sheet) T6151 (12.5-100) 6082 (Sheet) T6/T651 (0-6) 6082 (Sheet) T6/T651 (6-12.5) 6082 (Sheet) T651 (12.5-100) 6082 (EP,ET,ER/B) T4 (0-25) 6082 (EP/O,EP/H) T5 (0-5) 6082 (EP/O,EP/H,ET) T6 (0-5) 6082 (EP/O,EP/H,ET) T6 (5-15) 6082 (ER/B) T6 (0-20) 6082 (ER/B) T6 (20-150) 6082 (DT) T6 (0-5) 6082 (DT) T6 (5-20) 7020 (Sheet) T6 (0-12.5) 7020 (Sheet) T651 (0-40) 7020 (EP,ET,ER/B) T6 (0-15) 7020 (EP,ET,ER/B) T6 (15-40) 7020 (DT) T6 (0-20) 8011A (Sheet) H14 (0-12.5) 8011A (Sheet) H24 (0-12.5) 8011A (Sheet) H16 (0-4) 8011A (Sheet) H26 (0-4) 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. 3 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 are included. For class 4 sections (slender sections) the effective section is calculated in each intermediary 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) 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 5.3.2 Imperfections for global analysis of frames 5.3.4 Member imperfections X X X 6 Ultimate limit states for members 6.1 Basis 6.1.3 Partial safety factors 6.1.4 Classification of cross-sections 6.1.5 Local buckling resistance 6.1.6 HAZ softening adjacent to welds X X (*) X (*) X (*) 6.2 Resistance of cross-sections 6.2.1 General 6.2.2 Section properties 6.2.3. Tension 6.2.4. Compression 6.2.5. Bending Moment 6.2.6. Shear 6.2.7. Torsion 6.2.8. Bending and shear 6.2.9. Bending and axial force 6.2.10. Bending , shear and axial force X (*) X (*) X (*) X (*) X (*) X (*) X (*) X X (*) X (*) 6.3 Buckling resistance of members 6.3.1 Members in compression 6.3.2 Members in bending 6.3.3 Members in bending and axial compression X (*) X (*) X (*) 4 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 6.7.6 Interaction X (*) 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 to define the Initial Shape. The thin-walled cross-section parts can have the following types: Fixed Part – No reduction is needed Internal cross-section part Symmetrical Outstand Unsymmetrical Outstand F I SO UO 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 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, I I RI I I RI 5 - RUO : The reinforcement is connected to only one plate with type I I RUO 6 For standard Cross-sections, the default plate type and reinforcement type are defined in the following table. Form code Shape 1 I section Initial Geometrical shape (F, none) (SO, none) (SO, none) (I, none) (SO, none) 2 RHS (F, none) (SO, none) (I, none) (F, none) (F, none) (I, none) (I, none) (F, none) 3 CHS (F, none) (I, none) (fixed value for ) 7 4 Angle section (UO, none) (F, none) 5 Channel section (UO, none) (F, none) (UO, none) (I, none) (UO, none) (F, none) 8 6 T section (UO, none) (F, none) (SO, none) 7 Full rectangular section No reduction possible 11 Full circular section No reduction possible 101 Asymmetric I section (SO, none) (SO, none) (SO, none) (F, none) (I, none) (F, none) (SO, none) (SO, none) 9 102 Rolled Z section (UO, none) (F, none) (I, none) (UO, none) (F, none) 110 General cold formed section (UO, none) (I, none) (UO, none) (I,none) (UO, none) 10 111 Cold formed angle (UO, none) (UO, none) 112 Cold formed channel (UO, none) (I, none) (UO, none) 11 113 Cold formed Z (UO, none) (I, none) (UO, none) 114 Cold formed C section (I, none) (UO, RUO) (I,none) (I,none) (UO, RUO) 115 (I, none) Cold formed Omega (I, none) (UO, RUO) (I, none) (UO, RUO) 12 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) (I,none) (UO, none) (UO, none) (I,none) (I,none) (UO, none) (UO, none) (I,none) (UO, none) (UO, 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 components 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 is independent on the magnitude of the actual forces in the cross-section, the classification is always done for each component. 13 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 NTension force N+ with 0,HAZ Tension force N+ with u,HAZ y-y axis bending Myy-y axis bending My+ z-z axis bending Mzz-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 Compression stress is indicated as negative. 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 beg end end if beg is the maximum compression stress beg if end is the maximum compression stress 14 Step 3: Calculation of slenderness Depending on the stresses and the plate type the slenderness parameter is calculated. Internal part: type I b t 0.70 0.30 0.80 1 With: (1 1) ( 1) b t Width of the cross-section part Thickness of the cross-section part Stress gradient factor Remark: For a thin walled round tube 3 D with D the diameter to mid-thickness of the tube t material. Outstand part: type SO, UO When = 1.0 or peak compression at the toe of the plate: b t peak compression at toe When peak compression is at the root of the plate: b t 0.70 0.30 0.80 1 (1 1) ( 1) 15 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 if 1< 2 if 2< 3 if 3< : class 1 : class 2 : class 3 : class 4 As specified in art. 6.1.4.4(3) a cross-section part is considered ‘with welds’ in case it contains a weld at any point along its width. 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 16 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 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 t eff t tensile stress 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 as a beam on elastic foundation, where the elastic foundation is represented by a spring stiffness depending on the transverse bending stiffness of adjacent parts of plane elements and on the boundary conditions of these elements. The following procedure is applied for calculating the reduction factor for an intermediate stiffener (RI) or edge stiffener (RUO). 17 Step 1: Calculation of spring stiffness Spring stiffness c = cr for RI: Spring stiffness c = cs for RUO: 18 c cs ys 1 ys 4(1 ²)b13 b12 c3 Et 3 3 Et ad c3 12(1 ²)b p ,ad With: tad bp,ad c3 α Thickness of the adjacent element Flat width of the adjacent element The sum of the stiffnesses from the adjacent elements 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: considered plate edge stiffener t ad bp,ad adjacent element 19 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 Ir beff the area of the effective cross section (based on teff = pc t ) composed of the stiffener area and half the adjacent plane elements 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 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: 20 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: N r ,cr 2 cEI r With: c E Ir Spring stiffness of Step 1 Module of Young Second moment of area of Step 2 21 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 : c f o Ar N r ,cr 0.20 0 0.60 0.50(1.0 (c 0 ) 2c ) if c 0 1.00 if c 0 With: f0 c 0 α 1 2 2c 1.00 0,2% proof strength Relative slenderness Limit slenderness taken as 0,60 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. The value for bhaz is multiplied by the factors 2 and 3/n 22 for 3xxx, 5xxx & 6xxx alloys : for 7xxx alloys : 2 1 1.5 With: T1 n 2 1 (T1 60) 120 (T1 60) 120 Interpass temperature Number of heat paths 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 f u ,haz o,haz f o,haz fu fo 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. Shear lag effects, art. 6.2.2.3 are not taken into account. 23 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 moment The 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 shape for M+ / MThe 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 by the 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. 24 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 crosssection parts. For each part i the slenderness is calculated as follows: h xend xbeg t i i w t w i 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 250 f 0 and f0 in N/mm² i 39 => Non-slender plate i 39 => Slender plate I) All parts are classified as non-slender i 39 The Shear check shall be verified using art. 6.2.6. II) One or more parts are classified as slender i 39 The Shear check shall be verified using art. 6.5.5. For each part i the shear resistance VRd,i is calculated. 25 Non-slender part: Formula (6.88) is used with properties calculated from the reduced shape for N+(u,HAZ) For Vy: Anet,y,i = ( xend xbeg ) i 0, HAZ t i cos 2 i For Vz: Anet,z,i = ( xend xbeg ) i 0, HAZ t i sin 2 i With: Slender part: i xend xbeg t 0,HAZ The number (ID) of the plate End position of plate i Begin position of plate i Thickness of plate i Haz reduction factor of plate i Angle of plate i to the Principal y-y axis 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 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. VRd VRd i i For a solid bar, round tube and hollow tube, all parts are taken as non-slender by default and formula (6.31) is applied. 26 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): Av v Ae 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: Avy i 1 ( xend xbeg ) 0, HAZ t cos 2 n Avz i 1 ( xend xbeg ) 0, HAZ t sin 2 n With: i xend xbeg t 0,HAZ The number (ID) of the plate End position of plate i Begin position of plate i Thickness of plate i 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 mentioned methods can be applied. In this case, formula (6.29) is used with Av taken as Ay or Az of the gross-section properties. 27 Torsion with warping In case warping is taken into account, the combined section check is replaced by an elastic stress check including warping stresses. tot, Ed tot, Ed f0 M1 f0 3 M 1 2 2 tot C , Ed 3 tot , Ed tot, Ed N , Ed My, Ed f0 M1 Mz , Ed w, Ed tot, Ed Vy , Ed Vz , Ed t , Ed w, Ed With f0 0,2% proof strength Total direct stress tot,E d Total shear stress tot,E d M1 C N,E Partial safety factor for resistance of cross-sections Constant (by default 1,2) Direct stress due to the axial force on the relevant effective cross-section d My, Direct stress due to the bending moment around y axis on the relevant effective cross-section Ed Mz, Direct stress due to the bending moment around z axis on the relevant effective cross-section Ed Direct stress due to warping on the gross cross-section w,E d Vy,E Shear stress due to shear force in y direction on the gross cross-section d Vz,E Shear stress due to shear force in z direction on the gross cross-section d t,Ed Shear stress due to uniform (St. Venant) torsion on the gross cross-section Shear stress due to warping on the gross cross-section w,Ed 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”. 28 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]) w ,Ed MwwM Cm With Mw wM Cm Bimoment Unit warping 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 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): wM With 1 b hm 4 b hm Section width Section height (see figure) 29 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: 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 approach given 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. 30 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 w ,Ed M xs w M tds Cm t 0 With Mxs wM Cm t Warping torque (see "Standard diagrams") Unit warping Warping constant 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: b/2 w 0 M tds b t wM A 4 31 U sections, sections Starting from the wM diagram, the following integral is calculated for the critical points: s w M tds 0 The shear stress due to warping is calculated in these critical points (see circles in figures) 32 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: With : G It E Cm Mx Mxp Mxs Mw IT CM E G Total torque = Mxp + Mxs Torque due to St. Venant Warping torque Bimoment Torsional constant Warping constant Modulus of elasticity Shear modulus 33 Torsion fixed ends, warping free ends, local torsional loading Mt Mx Mt b L Mt a M xb L M xa Mxp for a side Mxp for b side Mxs for a side Mxs for b side Mw for a side Mw for b side b sinh(b) M xp M t cosh(x ) L sinh(L) a sinh(a ) M xp M t cosh(x ' ) L sinh(L) sinh(b) M xs M t cosh(x ) sinh(L) sinh(a ) M xs M t cosh(x ' ) sinh( L ) sinh(b) sinh(x ) sinh(L) M sinh(a ) M w t sinh(x ' ) sinh(L) Mw Mt 34 Torsion fixed ends, warping fixed ends, local torsional loading Mt Mx Mt b L Mt a M xb L M xa Mxp for a side Mxp for b side Mxs for a side Mxs for b side Mw for a side Mw for b side b k2 k1 M xp M t D3 L k 2 a k1 M xp M t D4 L M xs M t D3 M xs M t D4 M M w t D1 M M w t D2 35 (sinh(b) k 2) sinh(x ) k1 sinh(x' ) sinh(L) k 2 sinh(x) (sinh(a ) k1) sinh(x ' ) D2 sinh(L) (sinh(b) k 2) cosh(x ) k1 cosh(x' ) D3 sinh(L) k 2 cosh(x) (sinh(a ) k1) cosh(x' ) D4 sinh(L) sinh(a ) sinh(b) a b sinh(a ) sinh(b) L L 1 tanh( ) sinh(L) 2 sinh(L) 2 2 k1 L 2 L 2 tanh( ) L tanh( ) 2 2 sinh(a ) sinh(b) a b sinh(a ) sinh(b) L L 1 tanh( ) sinh(L) 2 sinh(L) 2 2 k2 L 2 L 2 tanh( ) L tanh( ) 2 2 D1 Torsion fixed ends, warping free ends, distributed torsional loading mt Mx mt L 2 m L t 2 M xa M xb Mxp Mxs M xs Mw L cosh(x ) cosh(x ' ) ( x ) sinh(L) 2 m cosh(x ) cosh(x ' ) t sinh(L) M xp Mw mt mt 2 sinh(x ) sinh(x ' ) 1 sinh(L) 36 Torsion fixed ends, warping fixed ends, distributed torsional loading mt Mx mt L 2 mt L 2 M xa M xb Mxp Mxs M xs Mw k 1 L cosh(x ) cosh(x ' ) ( x ) (1 k) sinh(L) 2 m cosh(x ) cosh(x ' ) t (1 k ) sinh(L) M xp Mw mt mt 2 sinh(x ) sinh(x ' ) 1 (1 k) sinh(L) L 2 L t anh( ) 2 37 One end free, other end torsion and warping fixed, local torsional loading Mt Mx M xa M t Mxp Mxs Mw cosh(x ' ) M xp M t 1 cosh(L) cosh(x ' ) M xs M t cosh(L) Mw Mt sinh(x ' ) cosh(L) One end free, other end torsion and warping fixed, distributed torsional loading mt Mx M xa m t L Mxp M xp mt (1 L sinh(L)) sinh(x ) x'L cosh(x ) cosh(L) 38 (1 L sinh(L)) sinh(x ) L cosh(x ) cosh(L) m (1 L sinh(L)) cosh(x ) M w t 1 L sinh(x ) ² cosh(L) Mxs M xs Mw mt Decomposition of arbitrary torsion line Since the Scia Engineer 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 39 Decomposition 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 Mt n 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 torsional loadings Mtn and the distributed 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: o one torsion line generated by a local torsional loading Mt n 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 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. 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. 40 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. The plastic 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. Since the extend of the HAZ is defined along the member axis, it is important to specify enough sections on average member in the Solver Setup when transverse welds are used. 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 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 41 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: tot tot f0 M1 f0 3 M 1 2 2 tot 3 tot C tot N My f0 M1 Mz tot Vy Vz With : f0 0,2% proof strength tot tot M1 C N My Mz Vy Vz Ax Ay Az Wy Wz Total direct stress Total shear stress Partial safety factor for resistance of cross-sections Constant (by default 1,2) Direct stress due to the axial force Direct stress due to the bending moment around y axis Direct stress due to the bending moment around z axis Shear stress due to shear force in y direction Shear stress due to shear force in z direction Sectional area Shear area in y direction Shear area in z direction Elastic section modulus around y axis Elastic section modulus around z axis 42 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 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 in reference [7], [8] and [9]. The following formulas are used for the buckling ratios (Ref[7],pp.21): For a non-sway structure: l/L = (1 2 + 5 1 + 5 2 + 24)(1 2 + 4 1 + 4 2 + 12)2 (2 1 2 + 11 1 + 5 2 + 24)(2 1 2 + 5 1 + 11 2 + 24) 43 For a sway structure: l/L = x 2 +4 1 x With: L E I Ci Mi i x= System length Modulus of Young Moment of inertia Stiffness in node i Moment in node i Rotation in node i 4 1 2 + 2 1 2 (1 + 2) + 8 1 2 CL i = i EI M i Ci = i The values for Mi and i are approximately determined by the internal forces and the deformations, calculated by load 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. 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 semirigid beam connections. For other cases, the user has to evaluate the presented bucking ratios. 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 1 and load case 2 return ky = 1,00 then ky is taken as kz. This rule is used to account for possible rotations of the cross-section. 44 Calculation of Buckling ratio – Crossing Diagonals For crossing diagonal elements, the buckling length perpendicular to the diagonal plane, is calculated according to Ref.[10], DIN18800 Teil 2, table 15. This means that the buckling length sK is dependent on the load distribution in the element, and it is not a purely geometrical data anymore. In the following paragraphs, the buckling length sK is defined, With: sK L L1 I I1 Buckling length Member length Length of supporting diagonal Moment of inertia (in the buckling plane) of the member Moment of inertia (in the buckling plane) of the supporting diagonal Compression force in member Compression force in supporting diagonal Tension force in supporting diagonal Modulus of Young N N1 Z E Continuous compression diagonal, supported by continuous tension diagonal Z l/2 N N l1/2 Z 3 Z l 4 N l1 I1 l3 1 I l13 1 sK l sK 0.5 l See Ref.[10], Tab. 15, case 1. 45 Continuous compression diagonal, supported by pinned tension diagonal Z l/2 N N l1/2 Z sK l 1 0.75 Zl N l1 sK 0.5 l See Ref.[10], Tab. 15, case 4. Pinned compression diagonal, supported by continuous tension diagonal Z l/2 N N l1/2 Z sK 0.5 l N l1 1 Z l 3 Z l12 N l1 (E I1)d ( 1) 42 Z l See Ref.[10], Tab. 15, case 5. 46 Continuous compression diagonal, supported by continuous compression diagonal N1 l/2 N N l1/2 N1 N1 l N l1 l I1 l3 1 I l13 1 sK sK 0.5 l See Ref.[10], Tab. 15, case 2. Continuous compression diagonal, supported by pinned compression diagonal N1 l/2 N N l1/2 N1 sK 2 N1 l l 1 12 N l1 See Ref.[10], Tab. 15, case 3 (2). 47 Pinned compression diagonal, supported by continuous compression diagonal N1 l/2 N N l1/2 N1 sK 0.5 l (E I)d N l1 N l3 2 ( ) 2 l1 12 N1 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 a prismatic member is calculated as follows: Ncr N Ed 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 the element. 48 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- 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). 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 49 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 50 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 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: Mcr With: 2 EI z L2 E G L Iw It Iz Iw L²GI t 2 EI z Iz Modulus of elasticity Shear modulus Length of the beam between points which have lateral restraint (= lLTB) Warping constant Torsional constant 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. 51 Diaphragms When diaphragms (steel sheeting) are used, the torsional constant It is adapted for symmetric/asymmetric I sections, 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: l2 2G 1 1 C A ,k C P ,k I t ,id I t vorhC 1 1 vorhC C M ,k C M ,k k C A ,k EI eff s b C100 a 100 2 b C A ,k 1.25 C100 a 100 3 E Is C P ,k (h t) s³ Is 12 With: l G vorhC CM,k CA,k CP,k k EIeff s ba C100 h t s if b a 125 if 125 b a 200 LTB length Shear modulus Actual rotational stiffness of diaphragm Rotational stiffness of the diaphragm Rotational stiffness of the connection between the diaphragm and the beam Rotational stiffness due to the distortion of the beam Numerical coefficient = 2 for single or two spans of the diaphragm = 4 for 3 or more spans of the diaphragm Bending stiffness per unit width of the diaphragm Spacing of the beam Width of the beam flange (in mm) Rotation coefficient - see table Height of the beam Thickness of the beam flange 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 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 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 D* C2 = 0.45 A* [1 + C* e (½ + ½)] if M2 > 0 * * * C1 = 1.13 A + B E * C2 = 0.45A With: A* = q l2 8 | M2 | +q l2 B* = 8 | M2 | 8 | M2 | +q l2 D * = -72( C* = 94 | M2 | E* = 1.88 - 1.40 + 0.52 ql2 E* < 2.70 | M2 | ql2 )2 2 54 Moment distribution generated by F load F M1 = Beta M2 M2 l M2 < 0 ** ** ** ** ** C1 = A (2.75 B + 1) 1.35 + B (-1.62 A + 1) E ** ** D** C2 = 0.55 A [1 + C e (½ + ½)] M2 > 0 ** ** ** C1 = 1.35 A + B E ** C2 = 0.55 A A ** = With: Fl 4 | M2 | +Fl B** = 4 | M2 | 4 | M2 | +Fl 38 | M2 | Fl | M2 | 2 = -32( ) Fl C ** = D ** ** * 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 M1 = Beta M2 M2 l C2 = 0.0 C1 = 1.88 - 1.40 + 0.52 2 and C1 < 2.70 LTBII Eigenvalue solution For calculation of Mcr using LTBII reference is made to chapter “LTBII: Lateral Torsional Buckling 2nd Order Analysis”. Combined bending and axial compression The Combined bending and axial compression check is verified using art. 6.3.3.1 & 6.3.3.2. 55 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 the reduction 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. Since the extend of the HAZ is defined along the member axis, it is important to specify enough sections on average member in the Solver Setup when transverse welds are used. In the calculation of xLT the buckling length lc and distance xs are taken for buckling around the z-z axis. 56 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 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 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 General Formula specified in chapter “Calculation of Buckling ratio – General Formula”. The basic assumption is that the deformations 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 load case reaches zero. Since for a known buckling shape xs can be different in each section, accurate results can be obtained by increasing the numbers of sections on average member in the Solver Setup. 57 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 input or if the buckling ratio is calculated from stability. When the buckling shape is unknown, xs can be calculated according to formula (6.71): With: but xs ≥ 0 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 The above specified formula contains the factor in the denominator of the right side of the 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 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. 58 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. 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 is done according to art. 6.7.4.1. The check is executed when the following condition is met: hw 2,37 tw With: E f0 hw Web height tw Web thickness Factor for shear buckling resistance in the plastic range E Modulus of Young f0 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: w 0,35 hw tw f0 E Using the slenderness w the factor for shear buckling v is obtained from the following table: 59 In this table, the value of is taken as follows: With: fuw Ultimate strength of the web material f0w Yield strength of the web material The contribution of the web Vw,Rd can then be calculated as follows: 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 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 f0 0,2% proof strength The design shear resistance VRd for shear buckling consists of two parts: the contribution of the web Vw,Rd and the contribution 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: fuw Ultimate strength of the web material f0w Yield 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. On each side of the web. 62 With: f0f Yield strength of the flange material f0w 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”. 63 Interaction If required, for both above cases the interaction between shear force, bending moment and axial 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-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 Scaffolding The scaffolding member and coupler check are implemented according to EN 12811-1 Ref.[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) and Numerical 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 interaction formulas according to Tabelle 7 of DIN 4420-1 Teil 1 are applied in case of a large shear force. The interaction equations are summarised as follows: Conditions Interaction for tubular member and and and and 65 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 f0 in case the section is not located in a HAZ zone and f0,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 tubular nd members assumes the use of a 2 order analysis including imperfections. In case these conditions are not set the default EN 1999-1-1 check should be applied instead. 66 Scaffolding coupler check The scaffolding couplers according to EN 12811-1 Annex C Ref.[31] are provided by default 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 force Vy 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 67 Manufacturer couplers In addition to the scaffolding couplers listed above, specific manufacturer couplers are provided within Scia Engineer. The interaction checks of these couplers are executed according to the respective validation reports. Cuplock The 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 aluminium couplers 68 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 69 (+) 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 aluminium couplers 70 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 2 Order calculation including torsional and warping effects nd For both methods, the member under consideration is sent to the Frilo LTBII solver and the respective 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 Cw_j Ct_i Ct_j Warping condition at end i (beginning of the member) Warping condition at end j (end of the member) Torsion condition at end i (beginning of the member) Torsion condition at end j (end of the member) To take into account loading and stiffness of linked beams, see paragraph “Linked Beams”. 71 For this system, the elastic critical moment Mcr for lateral torsional buckling can be analyzed as the solution of an eigenvalue problem: Ke Kg 0 With: Ke Kg Critical load factor Elastic linear stiffness matrix Geometrical stiffness matrix For members with arbitrary sections, the critical moment can be obtained in each section, with: (See Ref.[24],pp.176) M cr max M yy M cr x M yy ( x ) With: Myy Myy(x) Mcr(x) Critical load factor Bending moment around the strong axis Bending moment around the strong axis at position 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]. 72 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 Loadings o 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 2 calculation. nd Order 7 degrees of freedom 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 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 2 it is no more required to execute a Lateral Torsional Buckling Check. nd Order stress check, For more background information, reference is made to Ref.[23]. 73 Supported Sections The following table shows which cross-section types are supported for which type of analysis: Scia Engineer CSS Double T I section from library Thin walled geometric I Sheet welded Iw IPY from library Thin walled geometric asymmetric I Haunched sections Welded I+Tl Sheet welded Iwn HAT Section IFBA, IFBB U section from library Thin walled geometric U Cold formed from library Cold formed from graphical input Welded I+2L x x x x x x x x x x x x x x x x x x x x x x x x x x Sheet welded Iw+2L Full rectangular from library Full rectangular from thin walled geometric all other double symmetric CSS x x x all other single symmetric CSS x Double T unequal U cross section Thin walled Double T with top flange angle Rectangle Static values double symmetric Static values single symmetric Eigenvalue analysis nd FRILO LTBII CSS 2 Order analysis x x The following picture illustrates the relation between the local coordinate system of Scia Engineer and Frilo LTBII. Special attention is required for U sections due to the inversion of the y and z-axis. 74 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 beam o Line force in beam o Moment in node (if the node is part of the exported beam) o Moment on beam o 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. In Frilo LTBII a distinction is made between the centroid and the shear center of a cross-section. Load impulses which do not pass through the shear center will cause additional torsional moments. 75 Imperfections nd In the 2 Order LTB analysis the bow imperfections v0 (in local y direction) and w0 (in local z direction) can be taken into account. v0 y z y, v0 Initial bow imperfection v0 according to code For EC-EN the imperfections can be calculated according to the code. The code indicates nd that for a 2 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) v0 k e0 With k e0 Factor taken from National Annex 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. 76 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”). 77 Diaphragms Diaphragms are transformed into 'Elastic Foundations' of type ‘elastic restraint’ (Ref.[23] p25). Both a horizontal restraint 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): Cy S L With: 2 S L Shear stiffness of the diaphragm 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.): S= a. 104 K2 K1 + Ls With: a Ls K1 K2 Frame distance Length of the diaphragm Factor K1 of the diaphragm 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”). The rotational restraint C [kNm/m] is taken as vorhC (see paragraph “Calculation of Mcr – General Formula”). 78 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 yaxis 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”). 79 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 members together 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 eigenvalue calculation. nd order analysis instead of an - 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 ElasticElastic 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]. 80 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 the profile 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: Iy Description moment of inertia yy Property number 8 Wy elastic section modulus yy 10 Sy statical moment of area yy 6 Iz moment of inertia zz 9 Wz 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 Mpy plastic moment yy 30 Mpz plastic moment zz 31 fab fabrication code 0=rolled section (default value) 1=welded section 2=cold formed section 105 The fabrication code is not obligatory. 81 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. Data depending on the profile shape I section Formcode PSS Type Property 49 48 44 47 66 74 140 61 146 109 1 .I. Description H B t s R W wm1 R1 1 B t R1 s H R w a 82 RHS Formcode PSS Type 2 .M. Property 49 48 67 66 Description H B s R 109 2 R H s B 83 CHS Formcode PSS Type 3 .RO. Property 64 65 Description D s 109 3 D w 84 Angle section Formcode PSS Type 4 .L. Property 49 48 44 61 66 Description H B t R1 R 74 75 76 W1 W2 W3 109 4 R1 t w2 w1 w2 R B w1 w3 85 Channel section Formcode PSS Type 5 .U. Property 49 48 44 47 66 Description H B t s R 68 41 61 146 R1 109 5 B R1 H s a R t 86 T section Formcode PSS Type 6 .T. Property 49 48 44 47 66 61 62 146 147 Description H B t s R R1 R2 1 2 109 6 B R2 s a2 H t R R1 a1 87 Full rectangular section Formcode PSS Type 7 .B. Property 48 67 Description B H 109 7 H B 88 Full circular section Formcode PSS Type 11 .RU. Property 64 Description D 109 11 D 89 Asymmetric I section Formcode PSS Type 101 Property 49 48 44 47 42 43 45 46 66 Description H s Bt Bb tt tb R 109 101 Bt tt H tb R Bb 90 Z section Formcode PSS Type 102 .Z. Property 49 48 44 47 67 61 Description H B t s R R1 109 102 B H s R1 R t 91 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-stiffened elements, 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 form code Dy* Dz* CM* buckling curve around yy axis buckling curve around zz axis buckling curve for LTB Property number 109 22 23 26 106 107 108 Value 110 (1) (1) (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. 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 92 reduced section can be calculated, except for the marked properties. These properties have to be input by the user in the profile library. Formcode PSS Type 110 Property 44 61 48 142 143 68 Description s r B sp e2 H 109 110 Remark: r is rounding, special for KLS section (Voest Alpine) sp is number of shear planes e2 s H B 93 Cold formed angle section Formcode PSS Type 111 Property 44 61 48 Description s r B 68 H 109 111 H s r B 94 Cold formed channel section Formcode PSS Type 112 Property 44 61 48 Description s r B 49 H 109 112 H s r B 95 Cold formed Z section Formcode PSS Type 113 Property 44 61 48 Description s r B 49 H 109 113 B H s R 96 Cold formed C section Formcode PSS Type 114 Property 44 61 48 Description s r B 49 68 H c 109 114 c H s r B 97 Cold formed Omega section Formcode PSS Type 115 Property 44 61 48 Description s r B 49 42 H c 109 115 c H s R B 98 Rail type KA Formcode PSS Type 150 .KA. Property 148 149 150 151 152 153 154 155 156 157 61 62 63 158 159 160 Description h1 h2 h3 b1 b2 b3 k f1 f2 f3 r1 r2 r3 r4 r5 a 109 150 k r1 h2 h3 b3 r3 r2 h1 r4 f2 f3 f1 r5 b2 b1 99 Rail type KF Formcode PSS Type 151 .KF. Property 48 154 49 153 155 157 148 149 61 62 63 Description b k h b3 f1 f3 h1 h2 r1 r2 r3 109 151 k r1 h1 h2 r2 h r2 f3 r2 r2 f1 r3 b3 b 100 Rail type KQ Formcode PSS Type 152 .KQ. Property 48 154 49 153 155 149 150 61 Description b k h b3 f1 h2 h3 r1 109 152 k r1 h3 h2 f1 b3 b 101 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 [10] DIN18800 Teil 2 Stahlbauten : Stabilitätsfälle, Knicken von Stäben und Stabwerken November 1990 [11] Rapportnr. BI-87-20/63.4.3360 Controleregels voor lijnvormige constructie-elementen IBBC Maart 1987 [12] Staalconstructies TGB 1990 Basiseisen en basisrekenregels voor overwegend statisch belaste constructies NEN 6770, december 1991 102 [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. Maquoi ELEMENTS 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 Staalconstructies TGB 1990 Stabiliteit NEN 6771 - 1991 [19] [20] 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 nd Friedrich + Lochner Lateral Torsional Buckling 2 Order Analysis Biegetorsionstheorie II.Ordnung (BTII) http://www.frilo.de [23] Friedrich + Lochner LTBII Manual BTII Handbuch Revision 1/2006 [24] J. Meister Nachweispraxis Biegeknicken und Biegedrillknicken Ernst & Sohn, 2002 103 [25] Eurocode 3 Design of steel structures Part 1 - 1 : General rules and rules for buildings EN 1993-1-1:2005 [26] 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 [30] EN 1999-1-1:2007/A1:2009 Addendum to EN 1999-1-1:2007 Eurocode 9: Design of Aluminium structures Part 1-1: General structural rules CEN, 2009 [31] EN 12811-1 Temporary works equipment Part 1: Scaffolds – performance requirements and general design 2004 [32] EN 12810-1 Façade scaffolds made of prefabricated components Part 1: Products specifications 2004 [33] EN 12810-2 Façade scaffolds made of prefabricated components Part 2: Particular methods of structural design 2004 [34] DIN 4420 Teil 1 Arbeits- und Schutzgerüste Allgemeine Regelungen, Sicherheitstechnische Anforderungen, Prüfungen Dezember 1990 [35] Zulassung Nr. Z-8.22-208 Modulsystem "CUPLOK" Deutsches Institut für Bautechnik, 2006. 104 [36] Zulassung Nr. Z-8.22-64 Modulsystem "Layher-Allround" Deutsches Institut für Bautechnik, 2008. 105
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