Steel Code Check Theory Enu

Theory

Steel Code Check

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Introduction Welcome to the Steel Code Check – Theoretical Background.

This document provides background information on the code checks according to different national and international regulations.

Version info

Documentation Title Steel Code Check – Theoretical Background

Release 2012.0

Revision 03/2012

TABLE OF CONTENTS

Introduction ....................................................................................................................... iv

Version info ....................................................................................................................... iv

EC3 – ENV 1993 .................................................................................................................. 2

EC3 – ENV code check ............................................................................................................... 2 Material properties ................................................................................................................... 2 Consulted articles .................................................................................................................... 3

Classification of sections .................................................................................................... 4 Effective cross-section properties for class 4 cross-section ............................................... 5 Section properties .............................................................................................................. 5 Bending moment ................................................................................................................ 5 Bending, shear and axial force ........................................................................................... 5 Torsion check ..................................................................................................................... 5 Built-in beams .................................................................................................................... 6 Compression members ...................................................................................................... 6 Lateral-torsional buckling ................................................................................................... 6 Use of diaphragms ............................................................................................................. 7 Shear Buckling check ........................................................................................................ 7 Shear buckling check for cold formed sections .................................................................. 7 Stability check for torsional buckling and torsional-flexural buckling .................................. 8 Bending and axial compression ....................................................................................... 10 Battened compression members ..................................................................................... 10

EC3 – ENV Fire Resistance ...................................................................................................... 12 Fire actions effect Efi ........................................................................................................ 12 Material properties ........................................................................................................... 12 Temperature analysis - Thermal actions .......................................................................... 13 Nominal temperature-time curve ...................................................................................... 13 Net heat flux ..................................................................................................................... 14 Steel Temperature ........................................................................................................... 15 Calculation model ............................................................................................................ 16 Code Check ..................................................................................................................... 16

Supported sections ................................................................................................................... 17

References ................................................................................................................................. 18

EC3 – EN 1993 .................................................................................................................. 19

EC3 – EN Code check ............................................................................................................... 19 Material properties ................................................................................................................. 19 Consulted articles .................................................................................................................. 20

Classification of sections .................................................................................................. 22 Effective cross-section properties for class 4 cross-section ............................................. 23 Section properties ............................................................................................................ 23 Bending, shear and axial force ......................................................................................... 23 Torsion check ................................................................................................................... 24 Built-in beams .................................................................................................................. 24 Compression members .................................................................................................... 24 Lateral-torsional buckling ................................................................................................. 25 Use of diaphragms ........................................................................................................... 30

Combined bending and axial compression ...................................................................... 31 Shear buckling check ....................................................................................................... 31 Battened compression members ..................................................................................... 32 Plate girders with sinusoidal corrugated webs ................................................................. 34 Moments on columns in simple construction.................................................................... 39 Scaffolding ....................................................................................................................... 41

EC3 – EN Fire Resistance ......................................................................................................... 47 Fire actions effect Efi .............................................................................................................. 47 Material properties ................................................................................................................. 48 Temperature analysis - Thermal actions ............................................................................... 48

Nominal temperature-time curve ...................................................................................... 48 Net heat flux ..................................................................................................................... 49

Steel Temperature ................................................................................................................. 50 Calculation model .................................................................................................................. 52 Code Check ........................................................................................................................... 52

EC3 – EN Cold-Formed ............................................................................................................. 53 Consulted articles .................................................................................................................. 53

Material properties ........................................................................................................... 55 Initial Shape ..................................................................................................................... 56 Geometrical Proportions .................................................................................................. 57 Effective Shape ................................................................................................................ 58 Section Checks ................................................................................................................ 64 Stability Checks ............................................................................................................... 76 Use of Diaphragms .......................................................................................................... 80 Special considerations for Purlins .................................................................................... 84

Supported sections ................................................................................................................... 91

References ................................................................................................................................. 92

DIN18800 ........................................................................................................................... 96

DIN18800 Code check ............................................................................................................... 96 Material properties ................................................................................................................. 96 Consulted articles .................................................................................................................. 97

Classification of sections ................................................................................................ 100 Net area properties ........................................................................................................ 100 Plastic interaction formula for RHS section .................................................................... 100 Plastic interaction formula for CHS section .................................................................... 104 Torsion check ................................................................................................................. 105 Built-in beams ................................................................................................................ 105 Calculation of the buckling length .................................................................................. 106 Torsional buckling .......................................................................................................... 106 Use of diaphragms ......................................................................................................... 107 LTB Check ..................................................................................................................... 108 Combined flexion for check method 2 ............................................................................ 111 Battened compression members ................................................................................... 112 Effective area properties ................................................................................................ 113 Shear buckling check ..................................................................................................... 114 Shear buckling check with buckling influence ................................................................ 114

Cold formed thin gauge members ....................................................................................... 114

Supported sections ................................................................................................................. 115

References ............................................................................................................................... 116

ONORM B 4300 ............................................................................................................... 118

ONORM B 4300 Code check ................................................................................................... 118 Material properties ............................................................................................................... 119 Consulted articles ................................................................................................................ 119


References ............................................................................................................................... 121

NEN .................................................................................................................................. 123

NEN6770/6771 Code check ..................................................................................................... 123 Material properties ............................................................................................................... 123 Consulted articles ................................................................................................................ 124

Section properties .......................................................................................................... 126 Classification of sections ................................................................................................ 126 Effective cross-section properties for class 4 cross-section ........................................... 126 Torsion check ................................................................................................................. 127 Built-in beams ................................................................................................................ 127 Buckling length ............................................................................................................... 127 Lateral-torsional buckling ............................................................................................... 127 Use of diaphragms ......................................................................................................... 128 Battened compression members ................................................................................... 129 Shear buckling check ..................................................................................................... 130 Shear buckling check with buckling influence ................................................................ 130

NEN6072 - Fire Resistance ..................................................................................................... 131 Fire actions effect ................................................................................................................ 131 Material properties ............................................................................................................... 131 Nominal temperature-time curve ......................................................................................... 132 Steel Temperature ............................................................................................................... 132 Calculation model ................................................................................................................ 135 Code Check ......................................................................................................................... 135


References ............................................................................................................................... 137

AISC – ASD : 1989 .......................................................................................................... 138

AISC - ASD Code check .......................................................................................................... 138 Classification of sections ..................................................................................................... 140 Section properties ................................................................................................................ 140 Buckling length .................................................................................................................... 140 Flexural Torsional Buckling .................................................................................................. 140 Lateral-torsional buckling ..................................................................................................... 140 Shear buckling check .......................................................................................................... 141


References ............................................................................................................................... 142

AISC – LRFD : 2001 ........................................................................................................ 144

AISC - LRFD Code check ........................................................................................................ 144 Classification of sections ..................................................................................................... 146 Section properties ................................................................................................................ 146

Buckling length .................................................................................................................... 146 Lateral-torsional buckling ..................................................................................................... 146 Use of diaphragms .............................................................................................................. 147 Shear buckling check .......................................................................................................... 147


References ............................................................................................................................... 148

ANSI/AISC 360-05:2005 .................................................................................................. 149

ANSI/AISC 360-05 Code check ............................................................................................... 149 Classification of sections ..................................................................................................... 151 Section properties ................................................................................................................ 151 Buckling length .................................................................................................................... 151 Lateral-torsional buckling ..................................................................................................... 151 Use of diaphragms .............................................................................................................. 151 Shear buckling check .......................................................................................................... 151


References ............................................................................................................................... 152

AISI NAS S100-2007 ....................................................................................................... 153

AISI NAS S100-2007 Code check ........................................................................................... 153 Consulted articles ................................................................................................................ 153

Initial Shape ................................................................................................................... 155 Dimensional limits .......................................................................................................... 156 Effective Widths ............................................................................................................. 156 Properties of Sections .................................................................................................... 157 Tension Members .......................................................................................................... 158 Flexural Members .......................................................................................................... 158 Compression Members .................................................................................................. 167 Combined Compression and Bending ........................................................................... 170 Use of diaphragms ......................................................................................................... 170 2

nd Order using Appendix 2 ............................................................................................ 174

Lapped Purlin Design ..................................................................................................... 175

References ............................................................................................................................... 176

CM66 ................................................................................................................................ 178

CM66 Code check ................................................................................................................... 178 Consulted articles ................................................................................................................ 178

Section properties .......................................................................................................... 180 Plastic coefficient ........................................................................................................... 180 Compression members .................................................................................................. 180 Factor kf ......................................................................................................................... 180 LTB Check ..................................................................................................................... 180 Use of diaphragms ......................................................................................................... 180 Combined flexion ........................................................................................................... 181 Shear buckling check ..................................................................................................... 181


References ............................................................................................................................... 182

CM66 - Additif 80 ............................................................................................................ 183

CM66 - Additif 80 Code check ................................................................................................ 183

Consulted articles ................................................................................................................ 183 Classification of sections ................................................................................................ 184 Section check ................................................................................................................. 184 Compression members .................................................................................................. 184 Lateral-torsional buckling ............................................................................................... 184 Use of diaphragms ......................................................................................................... 184


References ............................................................................................................................... 185

BS5950-1:1990 ................................................................................................................ 186

BS5950-1:1990 Code Check ................................................................................................... 186 Material properties ............................................................................................................... 186 Consulted articles ................................................................................................................ 187

Classification of sections ................................................................................................ 189 Slender cross-section .................................................................................................... 189 Section properties .......................................................................................................... 189 Bending moment ............................................................................................................ 190 Bending, shear, axial force............................................................................................. 190 Lateral torsional buckling ............................................................................................... 190 Use of diaphragms ......................................................................................................... 191 Compression member .................................................................................................... 191 Shear buckling check ..................................................................................................... 191


References ............................................................................................................................... 192

BS5950-1:2000 ................................................................................................................ 194

BS5950-1:2000 Code Check ................................................................................................... 194

SIA263 ............................................................................................................................. 195

SIA263 Code check ................................................................................................................. 195 Material properties ............................................................................................................... 195 Consulted articles ................................................................................................................ 195

Section classification ...................................................................................................... 197 Slender cross-section .................................................................................................... 197 Sections properties ........................................................................................................ 197 Lateral torsional buckling ............................................................................................... 197 Use of diaphragms ......................................................................................................... 198 Shear buckling ............................................................................................................... 198 Stability check ................................................................................................................ 198 Torsion check ................................................................................................................. 198 Built-in beams ................................................................................................................ 198

SIA263 - Fire Resistance ......................................................................................................... 198 Fire actions effect Efi ............................................................................................................ 198 Material properties ............................................................................................................... 199 Temperature analysis - Thermal actions ............................................................................. 199 Nominal temperature-time curve ......................................................................................... 199 Net heat flux ........................................................................................................................ 199 Steel Temperature ............................................................................................................... 200 Calculation model ................................................................................................................ 201 Code Check ......................................................................................................................... 201


References ............................................................................................................................... 202

GBJ 17-88 ........................................................................................................................ 203

The GBJ 17-88 code check ..................................................................................................... 203 Material properties ............................................................................................................... 203 Consulted articles ................................................................................................................ 204

Section properties .......................................................................................................... 205 Shear buckling check ..................................................................................................... 206 Buckling curves .............................................................................................................. 206 Buckling length ............................................................................................................... 206 Lateral torsional buckling ............................................................................................... 206 Local stability of compressed members ......................................................................... 206 Shear buckling check ..................................................................................................... 207


References ............................................................................................................................... 208

Korean steel code check ............................................................................................... 209

The Korean steel code check ................................................................................................. 209 Material properties ............................................................................................................... 209 Consulted articles ................................................................................................................ 209

Section classification ...................................................................................................... 210 Section properties .......................................................................................................... 211 Buckling length ............................................................................................................... 211 Lateral torsional buckling ............................................................................................... 212 Combined stresses ........................................................................................................ 213 Shear buckling check ..................................................................................................... 213


References ............................................................................................................................... 214

BSK 99 ............................................................................................................................. 216

BSK 99 Code check................................................................................................................. 216 Consulted articles ................................................................................................................ 218

Classification of sections ................................................................................................ 219 Effective cross-section properties for class 3 cross-section ........................................... 219 Section properties .......................................................................................................... 219 Section check ................................................................................................................. 219 Compression members .................................................................................................. 219 Stability check for torsional buckling and torsional-flexural buckling .............................. 220 Lateral-torsional buckling ............................................................................................... 221 Use of diaphragms ......................................................................................................... 222 Shear force ( shear buckling) ......................................................................................... 222


References ............................................................................................................................... 224

IS 800 ............................................................................................................................... 225

IS:800 Code check .................................................................................................................. 225 Material properties ............................................................................................................... 225 Consulted articles ................................................................................................................ 225

Classification of sections ................................................................................................ 226

Section properties .......................................................................................................... 227 Section check ................................................................................................................. 227 Compression members .................................................................................................. 227 Stability check for torsional buckling and torsional-flexural buckling .............................. 227 Lateral-torsional buckling ............................................................................................... 229 Use of diaphragms ......................................................................................................... 230


References ............................................................................................................................... 231

EAE code check ............................................................................................................. 232

Material properties ............................................................................................................... 232 Consulted articles ................................................................................................................ 233

Classification of sections ................................................................................................ 235 Effective cross-section properties for class 4 cross-section ........................................... 235 Section properties .......................................................................................................... 235 Torsion check ................................................................................................................. 235 Built-in beams ................................................................................................................ 235 Compression members .................................................................................................. 236 Lateral-torsional buckling ............................................................................................... 236 Use of diaphragms ......................................................................................................... 236 Combined bending and axial compression .................................................................... 237 Shear buckling check ..................................................................................................... 237


References ............................................................................................................................... 238

Calculation of buckling ratio ......................................................................................... 240

Introduction to the calculation of buckling ratio .................................................................. 240

Calculation buckling ratio – general formula ........................................................................ 240

Calculation buckling ratios for crossing diagonals ............................................................. 242 Continuous compression diagonal, supported by continuous tension diagonal ................... 242 Continuous compression diagonal, supported by pinned tension diagonal ......................... 243 Pinned compression diagonal, supported by continuous tension diagonal .......................... 243 Continuous compression diagonal, supported by continuous compression diagonal .......... 244 Continuous compression diagonal, supported by pinned compression diagonal ................. 244 Pinned compression diagonal, supported by continuous compression diagonal ................. 245

Calculation of critical Euler force for VARH elements ......................................................... 246 Definitions ............................................................................................................................ 246 Calculation of the critical Euler force ................................................................................... 246

Calculation buckling ratio for lattice tower members .......................................................... 248 Leg with symmetrical bracing .............................................................................................. 249 Leg with intermediate transverse support ............................................................................ 249 Leg with staggered bracing.................................................................................................. 250 Single Bracing ..................................................................................................................... 250 Single Bracing with SBS (Secondary Bracing System) ....................................................... 251 Cross bracing ...................................................................................................................... 251 Cross bracing with SBS ....................................................................................................... 252 K Bracing ............................................................................................................................. 253 Horizontal Bracing ............................................................................................................... 253

Horizontal Bracing with SBS ................................................................................................ 254 Discontinuous Cross bracing with horizontal member ......................................................... 255

Calculation of buckling ratio – From Stability Analysis ....................................................... 256

References ............................................................................................................................... 256

Calculation of moment factors for LTB ....................................................................... 258

Introduction to the calculation of moment factors ............................................................... 258

Calculation moment factors ................................................................................................... 258 Moment distribution generated by q load ............................................................................. 258 Moment distribution generated by F load............................................................................. 260 Moment line with maximum at the start or at the end of the beam ...................................... 261

References ............................................................................................................................... 261

LTBII: Lateral Torsional Buckling 2nd Order Analysis .............................................. 262

Introduction to LTBII ............................................................................................................... 262

Eigenvalue solution Mcr ......................................................................................................... 262

2nd

Order analysis ................................................................................................................... 264

Supported National Codes ..................................................................................................... 264

Supported Sections ................................................................................................................ 265

Loadings .................................................................................................................................. 267

Imperfections ........................................................................................................................... 267 Initial bow imperfection v0 for DIN and ONORM ................................................................. 268 Initial bow imperfection v0 for EC-EN and EAE ................................................................... 268 Initial bow imperfections v0 and w0 for other supported codes ........................................... 269

LTB Restraints ......................................................................................................................... 270

Diaphragms ............................................................................................................................. 271

Linked Beams .......................................................................................................................... 272

Limitations and Warnings ...................................................................................................... 273

References ............................................................................................................................... 274

Profile conditions for code check ................................................................................ 275

Introduction to profile characteristics ................................................................................... 275

Data for general section stability check ................................................................................ 275

Data depending On the profile shape .................................................................................... 276 I section ............................................................................................................................... 276 RHS ..................................................................................................................................... 277 CHS ..................................................................................................................................... 278 Angle section ....................................................................................................................... 279 Channel section ................................................................................................................... 280 T section .............................................................................................................................. 281 Full rectangular section........................................................................................................ 282 Full circular section .............................................................................................................. 283 Asymmetric I section ........................................................................................................... 284 Z section .............................................................................................................................. 285 General cold formed section ................................................................................................ 286 Cold formed Angle section................................................................................................... 288 Cold formed Channel section .............................................................................................. 289

Cold formed Z section ......................................................................................................... 290 Cold formed C section ......................................................................................................... 291 Cold formed Omega section ................................................................................................ 292 Cold formed C section eaves beam ..................................................................................... 293 Cold formed C Plus section ................................................................................................. 294 Cold formed ZED section..................................................................................................... 295 Cold formed ZED section asymmetric lips ........................................................................... 296 Cold formed ZED section inclined lip ................................................................................... 297 Cold formed Sigma section .................................................................................................. 298 Cold formed Sigma section stiffened ................................................................................... 299 Cold formed Sigma Plus section .......................................................................................... 300 Cold formed Sigma section eaves beam ............................................................................. 301 Cold formed Sigma Plus section eaves beam ..................................................................... 302 Cold formed ZED section both lips inclined ......................................................................... 303 Cold formed I-Plus section................................................................................................... 304 Cold formed IS-Plus section ................................................................................................ 305 Cold formed Sigma section asymmetric .............................................................................. 306 Rail type KA ......................................................................................................................... 307 Rail type KF ......................................................................................................................... 308 Rail type KQ ........................................................................................................................ 309

Warping check ................................................................................................................ 310

Stress check ............................................................................................................................ 310 Calculation of the direct stress due to warping .................................................................... 311 Calculation of the shear stress due to warping .................................................................... 314

Plastic Check ........................................................................................................................... 316

Standard diagrams for warping torque, bimoment and the St.Venant torsion .................. 319 Torsion fixed ends, warping free ends, local torsional loading Mt ........................................ 320 Torsion fixed ends, warping fixed ends, local torsional loading Mt ...................................... 321 Torsion fixed ends, warping free ends, distributed torsional loading mt .............................. 323 Torsion fixed ends, warping fixed ends, distributed torsional loading mt ............................. 324 One end free, other end torsion and warping fixed, local torsional loading Mt .................... 325 One end free, other end torsion and warping fixed, distributed torsional loading mt ........... 326

Decomposition of arbitrary torsion line ................................................................................ 327 Decomposition for situation 1 and situation 3 ...................................................................... 328 Decomposition for situation 2 .............................................................................................. 328

References ............................................................................................................................... 328

Check of numerical sections ........................................................................................ 330

Stress check ............................................................................................................................ 330

Use of diaphragms ......................................................................................................... 331

Adaptation of torsional constant ........................................................................................... 331

References ............................................................................................................................... 332

Section check for built-in beams (IFB, SFB, THQ sections) ...................................... 334

Introduction ............................................................................................................................. 334

Reduction of plastic moment capacity due to plate bending .............................................. 334

Plastic interaction formula for single bending and shear force .......................................... 337

Plastic check for plate in bending ......................................................................................... 338

Stress check for slim floor beams ......................................................................................... 339 Normal stress check ............................................................................................................ 339 Shear stress check in plate.................................................................................................. 339 Torsion check due to unbalanced loading ........................................................................... 340

References ............................................................................................................................... 342

Effective cross-section properties for lattice tower angle members........................ 343

Effective cross-section properties for compressed lattice tower angle members ............ 343

References ............................................................................................................................... 344

Scia Engineer Steel Code Check Theoretical Background

1


2

EC3 – ENV 1993

EC3 – ENV code check

The beam elements are checked according to the regulations given in

Eurocode 3

Design of steel structures

Part 1 - 1 : General rules and rules for buildings

ENV 1993-1-1:1992

Material properties

For standard steel grades, the yield strength fy and tensile strength fu are defined according to the

thickness of the element (see Ref. 1, art.3.2.2.1.)

(fy, fu in N/mm², t in mm)

t<=40 t<=40 40<t<=100 40<t<=100 100<t<=250 100<t<=250

fy fu fy fu fy fy

S235

S 235

235 360 215 340 175 320

S275

S 275

275 430 255 410 205 380

S355

S 355

355 510 335 490 275 450

S420

S 420

420 520 390 520

S460

S 460

460 550 430 550

Remark : For cold formed section, the values for fy and fu are not influenced by the previous table

Remark : For cold formed sections, the average yield strength fya can be used (by setting the proper data flag in the Cross Section input dialog).

The average yield strength is determined as follows:


3

with fyb the tensile yield strength = fy

fu the tensile ultimate strength

t the material thickness

Ag the gross cross-sectional area

k is a coefficient depending on the type of forming :

k = 0.7 for cold rolling

k = 0.5 for other methods of forming

n the number of 90° bends in the section

Consulted articles

The cross-section is classified according to Table 5.3.1. (class 1,2,3 or 4). The section is checked for tension (art. 5.4.3.), compression (art. 5.4.4.), shear (art. 5.4.6.) and the combination of bending, shear and axial force (art. 5.4.9.).

For the stability check, the beam element is checked according to art.5.5.. The following criteria are considered :

for compression : art. 5.5.1.

for lateral torsional buckling : art. 5.5.2.

for bending and axial compression : art. 5.5.4.

The shear buckling resistance is checked using the simple post-critical method from art. 5.6.3.

A more detailed overview for the used articles is given for part 5.3., 5.4., 5.5. and 5.6. in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.

5.3. Classification of cross sections

5.3.1. Basis x

5.3.2. Classification x

5.3.3. Cross-section requirements for plastic global analysis

5.3.4. Cross-section requirements when elastic global analysis is used

5.3.5. Effective cross-section properties for class 4 cross-section x (*)

5.3.6. Effects of transverse forces on webs

5.4. Resistance of cross-sections

5.4.1. General x

5.4.2. Section properties (*)

5.4.3. Tension x

5.4.4. Compression x


4

5.4.5. Bending moment x (*)

5.4.6. Shear x

5.4.7. Bending and shear x

5.4.8. Bending and axial force x

5.4.9. Bending, shear and axial force x (*)

5.4.10. Transverse forces on webs

5.5. Buckling resistance of members

5.5.1. Compression members x (*)

5.5.2. Lateral-torsional buckling x (*)

5.5.3. Bending and axial tension

5.5.4. Bending and axial compression x (*)

5.6. Shear buckling resistance

5.6.1. Basis x

5.6.2. Design methods

5.6.3. Simple post-critical method x

5.6.4. Tension field method

5.6.5. Intermediate transverse stiffeners

5.6.6. Welds

5.6.7. Interaction between shear force, bending moment and axial force x

5.9. Built-up compression members

5.9.3. Battened compression members

5.9.3.1. Application x(*)

5.9.3.2. Constructional details

5.9.3.3. Second moment of inertia x

5.9.3.4. Chord forces ar mid-length x

5.9.3.5. Buckling resistance of chords x

5.9.3.6. Moments and shear due to battening x

Classification of sections

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

For each load case/combination, the critical section classification over the member is used to perform the stability check. So, the stability section classification can change for each load case/combination.

However, for non-prismatic sections, the stability section classification is determined for each intermediary section.


5

Effective cross-section properties for class 4 cross-section

The calculation of the effective area is performed with the direct method (sigma_d = fy,k).

For each intermediary section, the classification (and if necessary, the effective area ) is determined and the proper section check is performed. The classification (and effective area) can change for each intermediary point. The most critical check is displayed on the screen.

For each load case and combination, the most critical effective area properties are saved :

Aeff is the effective area of the cross section when subject to uniform compression. Weff is the effective section modulus of the cross-section when subject only to moment about the relevant axis. eN is the shift of the relevant centroidal axis when the cross section is subject to uniform compression.

With these critical properties, the stability check is performed.

For non-prismatic elements, the effective area properties are calculated on each intermediary section, also for the stability check.

For angle sections, see chapter 'Effective cross-section properties for compressed lattice tower angle members'.

Section properties

5.4.2.2 : The net area properties are only taken into account in the Tension Check in case of lattice tower angle sections with bolted diagonal connections if the LTA functionality has been activated. For more information, reference is made to the Theoretical Background Bolted Diagonal Connections. In all other cases the net area properties are not taken into account.

5.4.2.3 : The shear lag effects are neglected .

Bending moment

5.4.5.3 : The holes for fasteners are neglected.

Bending, shear and axial force

The reduced design plastic resistance moment for the interaction of bending, shear and axial force,

is taken from Table 5.17. Ref. 2

Torsion check

For the cross section check inclusive torsion and warping, we refer to Chapter 'Warping check'.


6

Built-in beams

For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter ‘Section check for built-in beams (IFB, SFB, THQ sections)’

Compression members

5.5.1.5 For the calculation of the buckling length, we refer to chapter "Calculation of buckling ratio"

The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see chapter “Calculation of critical Euler force for VARH elements”).

The buckling curves for steel grade S420 and S460 are taken from Ref.[5], Annex D.

Lateral-torsional buckling

For I sections (symmetric and asymmetric), RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, the elastic critical moment for LTB Mcr is given by the general

formula F.2. Annex F Ref. 1. For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".

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

z2

t

z2

z2

EI

L²GI

I

Iw

L

EIMcr

with E the modulus of elasticity

G the shear modulus

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

Iw the warping constant

It the torsional constant

Iz the moment of inertia about the minor axis

See also Ref. 3, part 7 and in particular part 7.7. for channel sections.

Haunched sections (I+Ivar, Iw+Plvar, Iw+Iwvar, Iw+Ivar, I+Iwvar) and composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.

For advanced Lateral-torsional buckling analysis, see chapter ‘LTBII: Lateral Torsional Buckling 2nd Order Analysis’.


7

Use of diaphragms

See Chapter 'Adaptation of torsional constant'.

Shear Buckling check

Composed rail sections (Iw+rail, Iwn+rail, I+rail, I+2PL+rail, I+PL+rail, I+2L+rail, I+Ud+rail) are considered as equivalent asymmetric I sections.

Shear buckling check for cold formed sections

See Ref.[4] 5.8 :

The shear resistance of the web Vw,Rd shall be taken as the lesser of the shear buckling resistance Vb,Rd and the plastic shear resistance Vpl,Rd.

The shear resistance of the web should be checked if:

The shear buckling resistance Vb,Rd is given by

The plastic shear resistance Vpl,Rd is given by

with w

the relative web slenderness

fyb the basic yield strength

fy the average yield strength

sw the web length

t the web thickness

E the modulus of elasticity

fbv the shear buckling strength

M0 the partial safety factor for resistance of cross-sections where failure is caused by yielding (=1.1)

M1 the partial safety factor for resistance of cross-sections where failure is caused by buckling (=1.1)

E

f

t

s346.0

f

f83.0

ybww

_

1M

0M

y

ybw

_

1M

bvwRd,b

ftsV

3

ftsV

0M

yw

Rd,pl


8

The value for fbv is given by :

fbv

<1.40

w

ybf

48.0

1.40

Remarks:

For an arbitrary composed section, the total Vb,Rd and Vpl,Rd is taken as the sum of resistance of

each web, where the angle (teta) is larger than 45° (see figure)

The basic yield strength is taken equal to the average yield strength.

Stability check for torsional buckling and torsional-flexural buckling

See Ref.[4] 6.2.3.

The design buckling resistance Nb,Rd for torsional or torsional-flexural buckling shall be obtained using buckling curve b, and with relative slenderness given by :

w

_

²

f67.0

w

_

yb


9

with A the ratio Aeff/A (see Ref.[1] 5.5)

fyb the basic yield strength

cr the critical stress

cr,T the elastic critical stress for torsional buckling

cr,TF

the elastic critical stress for torsional-flexural buckling

G the shear modulus


IT the torsion constant of the gross section

CM the warping constant

iy the radius of gyration about yy-axis

iz the radius of gyration about zz-axis

lT the buckling length of the member for torsional buckling

y0 the position of the shear center

ly the buckling length for flexural buckling about the yy-axis


10

Bending and axial compression

When the torsional buckling and/or the torsional-flexural buckling is governing, the formula (6.12) from Ref.[4], article 6.5.2. is applied.

Battened compression members

The following section pairs are supported as battened compression member :

(1) 2I

(2) 2Uo

(3) 2Uc

Two links (battens) are used.


11

The following additional checks are performed:

- buckling resistance check around weak axis of single chord with Nf,Sd

- section check of single chord, using internal forces :

4

aVM

2

V V

N N

sG

sG

SDf,G

- section check of single batten, using the internal forces :

4

aVM

2h

aV T

s

0

s

For the calculation of Vs, the value of Ms is increased with the value of the internal force Mzz.

l

a

ho


12

EC3 – ENV Fire Resistance

Fire actions effect Efi

The design effects of actions for the fire situation Efi,d,t are taken from the results of the analysis. It is recommended to use the accidental combination rules, for calculating the internal forces used in the fire resistance check.

The accidental combination is given by

)f(AQQG dj,kj,21,k1,1kGA

with Gk characteristic values of permanent actions

Qk,1 characteristic value of the (main) variable action

Qk,j characteristic values of the other variable actions

Af(d) design values of actions from fire exposure

GA partial safety factor for permanent actions in the accidental situation

1,1 2,j combination coefficients

Material properties

The material properties are depending on the steel temperature.

Strength and deformation properties :

a

,a,E

y

,p,p

y

,y,y

EE

k

f

fk

f

fk

The variation in function of the steel temperature of the value for yield strength ky,, proportional limit

kp, and modulus of elasticity kE, is given by tables in Ref.[6], table 3.1.

For cold formed members ky, is taken from Ref.[7], table III.2.5.


13

In the simplified calculation method, the following default properties are considered to be constant during the analysis:

unit mass a 7850 kg/m³

thermal elongation l/l 14 x 10-6

(a-20)

thermal conductivity a 45 W/mK

Temperature analysis - Thermal actions

In this part, the nominal temperature-time curves and the related net heat flux are described. See Ref.[8], Section 4, and Ref.[7], II.2.2.

Nominal temperature-time curve

The following temperature-time curves can be selected :

with t time in [min]

g gas temperature in [°C]

c the coefficient of heat transfer by convection

ISO 834 curve

external fire curve


14

hydrocarbon curve

smoldering fire curve

during 21 minutes, followed by the standard ISO 834 curve

Net heat flux

r,netr,nc,netc,nd,net hhh

with hnet,d the net heat flux

hnet,c the convective heat flux

hnet,r the radiative heat flux

n,c factor depending on NAD [1.0]

n,r factor depending on NAD [1.0]

with configuration factor [1.0]

res resultant emissivity

= f m

f emissivity related to fire compartment

= [0.800]

m emissivity related to surface material

= [0.625]

r = g

gas temperature in [°C]

m surface temperature of member in [°C]

c coefficient of heat transfer by convection


15

Steel Temperature

The increase of temperature a,t in an unprotected steel member during a time interval t

thc

V/Ad,net

aa

mt,a

with Am the exposed surface area per unit length [m²/m]

V the volume of the member per unit length [m³/m]

The factor Am/V should not be taken as less than 10m-1

ca the specific heat of steel [J/kgK]

hnet,d the net heat flux per unit area [W/m²]

t the time interval [seconds]

The value should not be taken as more than 5 seconds

a the unit mass of steel [kg/m³]

The increase of temperature a,t in an insulated steel member during a time interval t

with Ap the area of fire protection material per unit length [m²/m]



cp the specific heat of fire protection material [J/kgK]

dp the thickness of the fire protection material [m]




p the unit mass of fire protection [kg/m³]

a,t the steel temperature at time t

g,t the ambient gas temperature at time t

g,t the increase of the ambient gas temperature during the time interval

p the thermal conductivity of the fire protection material [W/mK]

The value a,t 0.0

For the increase of temperature a,t in an insulated steel member with intumescent coating, we refer to the NEN specifications, Chapter 'Steel Temperature'.


16

Calculation model

The calculation can be performed in 2 domains :

- strength domain

- temperature/time domain

In the strength domain, the strength Rfi,d,t(unity check) is calculated after a given time t (e.g. strength

after 45 min). In the temperature/time domain, the critical steel temperature cr,d is computed. From this critical temperature, the fire resistance time tfi,d is calculated (the time domain).

Code Check

The section and stability checks (buckling, lateral torsional buckling) are performed according to the regulations given in 'ENV 1993-1-2:1995' and/or 'Model Code on Fire Engineering - ECCS N° 111'. The checks are performed in the resistance domain or in the temperature/time domain..

Torsional buckling and shear buckling are not considered.

For each member, the classification of the cross section, the section check and the stability check are performed.

The following checks are executed :

EC3-1-2 :

- classification of cross section : art. 4.2.2.

- resistance for tension members : art. 4.2.3.1

- resistance for compression members (class 1,2 or 3) : art. 4.2.3.2.

- resistance for beams (class 1,2) : art. 4.2.3.3.

- resistance for beams (class 3) : art.4.2.3.4.

- resistance for members (class 1,2,3) subject to bending and compression : art. 4.2.3.5.

- critical temperature : art. 4.2.4.

ECCS Model Code on Fire Engineering

- resistance for tension members : art. III.5.2.

- resistance for compression members (class 1,2 or 3) : art. III.5.3.

- resistance for beams (class 1,2) : art. III.5.4.

- resistance for beams (class 3) : art. III.5.5.

- resistance for members (class 1,2,3) subject to bending and compression : art. III.5.6.

- resistance for members (class 4) : art. III.5.7.

- critical temperature : art. III.5.8.


17

Supported sections

I Symmetric I shapes (IPE, HEA, HEB, ….)

RHS Rectangular Hollow Section

CHS Circular Hollow Section

L Angle section

U Channel section

T T section

PPL Asymmetric I shapes

Z Z section

RS Rectangular section

Cold formed section

COM Composed section in PRIMAWIN

O Solid tube

NUM Numerical section

The necessary data conditions for these sections are described in chapter "Profile conditions for code check

".

The COM and NUM sections are not read out of the profile library.

I

RHS

CHS

L

U

T

PPL

RS

Z

O

COM

NUM

Classification x x x x x x x x (1) x (1) (1) (1)

Section check class 1 x x x


Section check class 3 x x x x x x x x x x x x x

Section check class 4 x x x x x x

Stability check class 1

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

Shear buckling check x x x x

(1) sections are classified as class 3 cross section by default.


18

References

1 Eurocode 3



ENV 1993-1-1:1992, 1992

2 Essentials of Eurocode 3

Design Manual for Steel Structures in Building

ECCS - N° 65, 1991

3 R. Maquoi

ELEMENTS DE CONSTRUCTIONS METALLIQUE

Ulg , Faculté des Sciences Appliquées, 1988

[4] 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

[5] Eurocode 3


Part 1 - 1/ A1 : General rules and rules for buildings

ENV 1993-1-1:1992/A1, 1994

[6] Eurocode 3


Part 1 - 2 : General rules - Structural fire design

ENV 1993-1-2:1995, 1995

[7] Model Code on Fire Engineering

ECCS - N° 111

May 2001

[8] Eurocode 1

Basis of design and actions on structures

Part 2-2 : Actions on structures - Actions on structures exposed to fire

ENV 1991-2-2:1995


19

EC3 – EN 1993

EC3 – EN Code check

The beam elements are checked according to the regulations given in:

Eurocode 3


Part 1 - 1: General rules and rules for buildings

EN 1993-1-1:2005

Corrigendum

EN 1993-1-1:2005/AC:2006

Corrigendum

EN 1993-1-1:2005/AC:2009

Material properties


thickness of the element (see Ref. 1, table 3.1.)

Within the material properties the rules for reduction of the yield strength in function of the thickness can be edited.

Remark: For cold formed sections, the reductions of the yield strength in function of the thickness are not applied.


20

Consulted articles

The beam elements are checked according to the regulations given in "Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings - EN 1993-1-1:2005".

The cross-sections are classified according to Table 5.2. All classes of cross-sections are included. For class 4 sections (slender sections) the effective section is calculated in each intermediary point, according to prEN 1993-1-5:2003, Chapter 4.4 .

The stress check is taken from art. 6.2.: the section is checked for tension (art. 6.2.3.), compression (art. 6.2.4.), bending (art. 6.2.5.), shear (art. 6.2.6.), torsion (art.6.2.7.) and combined bending, shear and axial force (art. 6.2.8., art.6.2.9. and art.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.), combined bending and axial compression (art. 6.3.3.) and battened compression members (art. 6.4).

The shear buckling is checked according to EN 1993-1-5:2006, Chapter 5.

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

A check for critical slenderness and torsion moment is also included.

For integrated beams, the local plate bending is taken into account for the plastic moment capacity and the bending stresses in the section. The out-of-balance loading is checked.

A more detailed overview for the used articles is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation in the following chapters.

EN 1993-1-1

5.5 Classification of cross section (*)

5.5.1. Basis x

5.5.2. Classification x

6. Ultimate limit states

6.1. General 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 Uniform members in compression

x(*)

6.3.2 Uniform members in bending x

6.3.3 Uniform members in bending and axial compression x(*)


21

6.4 Uniform built-up compression members

6.4.1 General

6.4.3 Battened compression members

Annex A:Method 1:Interaction factors kij for interaction formula in 6.3.3.(4)

x

Annex B:Method 2:Interaction factors kij for interaction formula in 6.3.3.(4)

x

EN 1993-1-3

6.1.2. Axial tension x

6.1.3. Axial compression x

6.1.5. Shear force x

6.1.6. Torsional moment x

EN 1993-1-5

4.4. Plate elements without longitudinal stiffeners x

5. Resistance to shear

5.1. Basis x

5.2. Design resistance x

5.3. Contribution from webs x

5.4. Contribution from flanges x

5.5. Verification x

7.1. Interaction between shear force, bending moment and axial force

x


22



For each load case/combination, the critical section classification over the member is used to perform the stability check since stability effects are related to the whole member and not to a single cross-section.

To determine this critical classification, all sections in the Ly and Lz system lengths of the buckling

system are checked and the worst classification is used as the critical. Note that only sections on the actual member are used so in case the system length spans multiple members, only the sections of the actual member are used to determine the critical classification.

For non-prismatic sections, the stability section classification is determined for each intermediary section.

The classification according to EN 1993-1-1 Table 5.2 is done using the general formulation for

‘parts subjected to bending and compression’. In this way the beneficial effect of tension is also accounted for.

- For example, for class 1, in case of bending combined with a tensile axial force the cross-

section is less subjected to compression thus leading to an value smaller than 0,5. This in

turn leads to higher classification limits than the case for ‘part subject to bending’.

- For example, for class 1, in case of an asymmetrical section in bending, depending on the geometry of the section the cross-section is less subjected to compression thus leading to an

value smaller than 0,5. This in turn leads to higher classification limits than the case for

‘part subject to bending’.

Specifically for sheet welded Iw and Iwn cross-sections the weld size as inputted in the cross-

section is accounted for in the classification as follows:

With: Hw Web height

a Weld size


23








Section properties

The net area properties are not taken into account.

The shear lag effects are neglected.

Bending, shear and axial force

Plastic Interaction for CHS

The 2009 correction sheet to EN 1993-1-1 Ref.[11] specifies an interaction formula for CHS

sections. This formula however does not include the influence of shear. Table 5.17. Ref. 15 specifies the full interaction formula including also shear.

In addition for CHS sections the resultant shear force and resultant moment are determined.

The moment resistance is determined taking into account this resultant shear force.

The unity check then becomes the following:

Plastic Interaction for other sections

By default the plastic combined interaction check according to EN 1993-1-1 article 6.2.9.1 formula (6.41) is only executed for I-sections, full rectangular and RHS sections since Eurocode only gives formulas to reduce the plastic bending resistance of these sections in case of a normal force.

However, in case there is no normal force, no reduction of the plastic bending resistance for normal force is needed and thus the plastic interaction can be used also for other cross-sections.

This is used specifically for the following cross-sections:

Angle section (Form code 4)

U-section (Form code 5)

T-section (Form code 6)

Asymmetric I-section (Form code 101)

In case these sections are classified as class 1 or 2 and no normal force is present the combined interaction is checked according to formula (6.41) with Alpha and Beta set to 1.00.


24

Torsion check

For the cross section check including torsion and warping, we refer to Chapter 'Warping check'.

Built-in beams

For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter ‘Section check for built-in beams (IFB, SFB, THQ sections)’.

Compression members

For the calculation of the buckling length, we refer to chapter "Calculation of buckling ratio"


Buckling curves are determined according to EN 1993-1-1 Table 6.2.

In case the cross-section of the member is not listed in this table, the buckling curves are taken from the user inputted values in the cross-section properties.

For non-prismatic members with cross-sections that are not listed in Table 6.2 all generated

sections will receive the user inputted values of the buckling curves of the first section in the span.


25


For I sections (symmetric and asymmetric), and Rectangular Hollow Sections (RHS), the elastic critical moment for Lateral-Torsional Buckling Mcr is given by the general formula F.2. Annex F Ref.

10.

For any other section, the elastic critical moment for Lateral-Torsional Buckling Mcr is given by:

With E Modulus of elasticity

G Shear modulus

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

Iw Warping constant

It Torsional constant

Iz Moment of inertia about the minor axis

C1 Moment factor which by default is taken as 1,00. Within the Steel setup it can be set to use the calculated value of C1 instead of 1,00.


Circular hollow sections (CHS) are taken as non-susceptible to Lateral Torsional Buckling.

Rectangular hollow sections are classified as non-susceptible to Lateral Torsional Buckling if the following condition is fulfilled (Ref.[9] pp.119).

With: h Height of RHS section

b Width of RHS section

Relative slenderness for weak axis flexural buckling




26

Determination of the moment factors C1, C2 and C3

The coefficients C1, C2 and C3 can be calculated according to three different methods:

ENV 1993-1-1 Annex F

ECCS 119/Galea

Lopez, Yong, Serna

By default the method according to ECCS 119/Galea is applied.

The following paragraphs give information on these methods.

ENV 1993-1-1 Annex F

When this setting is chosen, the moment factors are determined according to ENV 1993-1-1 Annex F Ref.[10].

Detailed information can be found in chapter "Calculation of moment factors for LTB".

ECCS 119/Galea

When this setting is chosen, the moment factors are determined according to ECCS 119 Annex B Ref.[9].

The figures given in this reference for C1 and C2 in case of combined loading originate from Ref.[28]

which in fact also gives the tabulated values of those figures as well as an extended range.

The actual moment distribution is compared with several standard moment distributions. These standard moment distributions are moment lines generated by a distributed q load, a nodal F load, or where the moment line is maximum at the start or at the end of the beam.

The standard moment distribution which is closest to the actual moment distribution, is taken for the calculation of the factors C1 and C2.

Linear Moment

In case of a linear moment diagram the C1 coefficient is determined using formula (301) of ECCS 119

Annex B Ref.[9].

The coefficient C2 is taken as zero in this case.

Point Loading

In case of Point loading the coefficients C1 and C2 are calculated using tables 5-8 of Galea Ref.[28].

A double interpolation is used for intermediate values.

Line Loading

In case of Line loading the coefficients C1 and C2 are calculated using tables 1-4 of Galea Ref.[28].

A double interpolation is used for intermediate values.

In case k differs from 1.00 the C1 and C2 values determined from Galea Ref.[28] are overruled by

the values from ECCS 119 Annex B Ref.[9] tables 63 and 64

For all cases the factor C3 is taken from ECCS 119 Annex B Ref.[9] tables 63 and 64. The C3 value is determined based on the case of which the C1 value most closely matches the table value.


27

The table for C3 uses the value f which is taken as 0 by default.

For asymmetrical I-sections (Form code 101) f is calculated as follows:

Ifc and Ift concern the moments of inertia of the compression ( c ) and tension ( t ) flange about the

minor axis.

For this method f should be within the following range:

When this is not the case f is set to the respective limit and a warning is given.

I-section Cantilevers

ECCS 119 Annex B Ref.[9] tables 65 to 68 give values for C1, C2 and C3 for I-section cantilevers.

These coefficients are used only in case the following conditions are met:

- The member concerns a cantilever.

A cantilever is defined as a member at the end of a buckling system which has free ends for both buckling about the y-y and z-z axis. In addition, the LTB length should correspond to the full system length of the buckling system.

- The cross-section is an I-section (Form code 1) or Asymmetric I-section (Form code 101).

This method differentiates between ‘warping prevented’ and ‘warping free’ at the fixed end. This setting is taken from the buckling system.

This method uses the value f which is calculated as specified above.

For this method f should be within the following range:

When this is not the case f is set to the respective limit and a warning is given.

This method uses the coefficient which is defined as follows:

With: L System length for LTB

E Modulus of Young

G Shear modulus

Iz Inertia about the weak axis

It Torsion constant

hs Distance defined as follows:

Form Code 1: H - t

Form Code 101: H – 0,5 * tt – 0,5 * tb


28

should be within the following range:

When this is not the case is set to the respective limit and a warning is given.

In addition this method should be used in combination with k equal to 2,00 and kw equal to 1,00

When this is not the case an additional warning is given.


29

Lopez, Yong, Serna

When this setting is chosen, the moment factors are determined according to Lopez, Yong, Serna Ref.[29].

When using this method the coefficients C2 and C3 are set to zero.

The coefficient C1 is calculated as follows:

With: k1 Taken equal to kw

k2 Taken equal to kw

M1, M2, M3, M4, M5 The moments My determined on the buckling system in the given sections as shown on the above figure.

These moments are determined by dividing the beam into 10 parts (11 sections) and interpolating between these sections.

Mmax The maximal moment My along the LTB system.

This method is only supported in case both k and kw equal 0.50 or 1.00


30

Modified design rule for LTB of Channel sections

In case this setting is activated within the Steel Setup, the reduction factor for Lateral-Torsional Buckling of Channel sections is determined according to Ref.[22].

More specifically the calculation is done as follows:

This Modified design rule is applied only in case the following conditions are met:

- The section concerns a Channel section (Form Code 5)

- The General Case is used for LTB (Not the Rolled and Equivalent Welded Case) - 15 <= Lltb/h <= 40 (with Lltb the LTB length and h the cross-section height)

Correction factor kc

In case Lateral-Torsional Buckling curves for the ‘Rolled and equivalent welded’ case are used according to EN 1993-1-1 article 6.3.2.3 the correction factor kc can be determined in two ways:

By default kc is taken from Table 6.6

Alternatively, kc can be determined from C1 as follows Ref.[30]:

Use of diaphragms



31

Combined bending and axial compression

For prismatic members the value My,Ed is the maximum value of the bending moment around the strong axis in the member. The value Mz,Ed is the maximum value of the bending moment around the weak axis in the member.

For non-prismatic sections, the values My,Ed and Mz,Ed are the concurrent bending moments for each intermediary section.

When Torsional(-Flexural) buckling is governing the z value in equation (6.62) is taken

as the minimum of z and TF.

Interaction Method 1 – Annex A

For Czz the corrected formula given in correction sheet EN 1993-1-1:2005/AC:2009 Ref.[11] is used:

Interaction Method 2 – Annex B

Doubly symmetric I sections which have a reduction factor for Lateral Torsional Buckling LT equal to 1,00 are classified as non-susceptible to torsional deformations.

Circular hollow sections are classified as non-susceptible to torsional deformations.

Rectangular hollow sections are classified as non-susceptible to torsional deformations if the following condition is fulfilled (Ref.[9] pp.119).

With: h Height of RHS section

b Width of RHS section

Relative slenderness for weak axis flexural buckling

Shear buckling check



32


The following section pairs are supported as battened compression member:

(1) 2I

(2) 2Uo

(3) 2Uc

This specifically concerns hot rolled sections i.e. cold-formed pair sections are not supported.

Battened compression members are evaluated according to EN 1993-1-1 article 6.4.1 and 6.4.3.


The following additional checks are performed:

Section check of chord as beam in field between battens

This check is executed according to articles 6.4.3.1 & 6.2.9.1 using the following internal forces:

4

aVM

2

V V

N N

Ed

G

Ed

G

Edch,G

With: Nch,Ed Chord force according to formula (6.69)

VEd Shear force in the built-up member according to formula (6.70)

a Distance between battens

For I-sections a classification is made which thus supports both an elastic or plastic interaction. For U-sections always an elastic interaction is made.


33

Buckling check of chord

This concerns a weak axis buckling check of a single chord according to articles 6.4.3.1 & 6.3.1.1 using chord force Nch,Ed.

Section check of single batten

An elastic section check of a single batten is executed according to articles 6.4.3.1, 6.2.9.2 & 6.2.6 using the following forces:

4

aVM

2h

aV T

Ed

0

Ed

With: VEd Shear force in the built-up member according to formula (6.70)

a Distance between battens

h0 Distance between centroids of chords

l

a

ho


34

Plate girders with sinusoidal corrugated webs

Plate girders with sinusoidal corrugated webs (“SIN beams”) are covered in EN 1993-1-5 Annex D. The method given in this chapter is specified in Ref.[33]. Background information can be found in Ref.[16].

The check is executed for sheet welded cross-sections of type Iw c and Iwn c. The corrugations are taken

to be perpendicular to the upper flange. The dimensioning of corrugated web girders is executed for the in plane effects NEd, Vz,Ed and My,Ed.

Transformation of internal forces

For every point of the plate girder the chord forces N,og and N,ug are found by transformation. These chord forces are still parallel to the member axis while the shear force is orthogonal to the axis. The following angles are defined: - α = the slope of the lower chord against the upper chord - β = the angle between the centre line and chords.

The shear force Vz is decomposed into a corrugation-parallel component V* and an axis-parallel component N(V)*.


35

N(V)* can be added directly to the calculated normal force N. The chord forces can now be determined as

follows:

With: A,og Area of the upper flange A,ug Area of the lower flange H,steg Web height t,og Thickness of the upper flange t,ug Thickness of the lower flange

From the chord forces the chord-parallel components and the corrugation-parallel components are determined. For the upper chord this becomes:

For the lower chord the following intermediate step is used:

The actual force in the lower chord is then:

The actual component of the shear force can then be written as:


36

The chord forces Nog* and Nug* are now known. By summation of the V* and V(Nog)* and V(Nug)*

components the total shear force is obtained.

Resistance of sinusoidal corrugated web girders

The normal force and bending moment are taken by the flanges while the shear force is taken by the corrugated web.

Flanges

For the flanges the following limits are checked:

- Yielding - Local buckling - Global buckling

Yielding

NRd,yield = bf * tf * fy / M0

With: bf Flange width

tf Flange thickness

fy Yield strength

M0 Partial safety factor

Local buckling

Local buckling of the compression flange is checked according to EN 1993-1-5 article 4.4.

To avoid local buckling the slenderness is limited to 0,748. By substituting this into the formula for the slenderness the following limit is obtained for the width:

For a sinusoidal corrugated web member the total flange width thus becomes:

The resistance for local buckling can then be written out as:

NRd,local = b * tf * fy / M0


37

Global buckling

Global buckling of the compression flange (Lateral-Torsional Buckling) is checked according to EN 1993-1-1 article 6.3.2.4:

This is written out to the following resistance for the compression flange:

With: b Flange width

t Flange thickness

fy Yield strength

E Modulus of Young

Lc Length between lateral restraints (LTB length)

kc Correction factor according to EN 1993-1-1 Table 6.6

The design value can then be written out as:

NRd,global = NRk / M1

With: M1 Partial safety factor

Web

For the web the shear resistance is determined according to EN 1993-1-5 Annex D article D2.2:

Where c is taken as the lesser of the reduction factors for local buckling c,l and global buckling

c,g.

According to Ref.[34] it was found by testing and FEM that no local buckling occurs for all actually produced beams with sinusoidal corrugated webs. Therefore only the reduction factor for global

buckling c,g needs to be accounted for.


38

With: fy Yield strength

E Modulus of Young

Poisson ratio

tw Web thickness

hw Web depth

Iz Second moment of area of one corrugation of length w, calculated as:

a3 Height of a sinus wave

Taken as 40 mm for tw < 3 mm

Taken as 43 mm for tw ≥ 3 mm

w Length of the projection of a half wave

s Unfolded length of a half wave

Taken as 178 mm for tw < 3 mm

Taken as 182 mm for tw ≥ 3 mm


39

Moments on columns in simple construction

This NCCI presents a method for determining the moments on columns in simple construction due to the eccentricity of the beam-to-column joints. This method is intended for braced frames with nominally pinned joints. The method is detailed in Ref.[31] and [32].

Conditions

In case the setting is activated in the Steel Setup the additional moments will be calculated on columns in which the following conditions are satisfied:

- The column cross-section concerns an I-section (Form code 1) or RHS section (Form code 2)

- The column has structural type Column, Gable column or Secondary column

- The column is uniform i.e. does not have arbitrary sections or haunches

- Only connected beams with structural type Beam or Rafter are accounted for. In addition these

beams should have a hinge at the side where they are connected to the column.

- There can maximally be two connected beams in the same plane in the same node. These two

connected beams must have the same X-axis direction of their LCS.

Additional moments

When the above conditions are satisfied the additional moments are calculated in the following way:

With: Rb1,Ed Shear force in the considered plane in the connected beam at the

specified distance

h Profile height for an I-section

Profile height or width for an RHS-section

tw Web thickness for an I-section


40

The distribution of the additional moments to the upper and lower column sections is carried out in

proportion to their stiffness, except where the ratio of the stiffnesses (I/L) does not exceed 1.5,

when the moments may be shared equally. This is illustrated on the following picture:

With: MU Distributed moment to the upper column section

ML Distributed moment to the lower column section

IU Inertia in the considered plane of the upper column section

IL Inertia in the considered plane of the lower column section

LU System length in the considered plane of the upper column section

LL System length in the considered plane of the lower column section

These additional moments are then added to the sections in the column just above and just below the

connected beam.

The simplified procedure given in this chapter allows to account for eccentricities without specifically adding these eccentricities in the calculation model. In case however an actual member eccentricity is defined on the column member the above procedure will not be used since additional moments will already be generated during the analysis.


41

Scaffolding

The scaffolding member and coupler check are implemented according to EN 12811-1 Ref.[23].

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 Steel 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.[26] 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


42

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

Safety factor taken as M0 of EN 1993-1-1

As specified in EN 12810 Ref.[25] & 12811 Ref.[23] the scaffolding check for tubular members assumes the use of a 2

nd order analysis including imperfections.

In case these conditions are not set the default EN 1993-1-1 check should be applied instead.


43

Scaffolding coupler check

The scaffolding couplers according to EN 12811-1 Annex C Ref.[23] 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.


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




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


44

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].


Cuplock Coupler Interaction equation

Interaction 1

Interaction 2

With: Nxk Taken from the coupler properties

Myk Taken from the coupler properties

Mxk Taken from the coupler properties

N Normal force in the ledger


Mx Torsional moment about the x axis

Nv Normal force in a connecting vertical diagonal

Angle between connecting vertical diagonal and standard




45

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


46

Vz,R,d = Vzk /

With Vzk taken from the coupler properties

N Normal force in the ledger

(+) This index indicates a tensile force




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+




47

EC3 – EN Fire Resistance


Eurocode 3


Part 1 - 2 : General rules – Structural fire design

EN 1993-1-2:2005

Corrigendum

EN 1993-1-2:2005/AC:2005

Corrigendum

EN 1993-1-2:2005/AC:2009



The accidental combination is given by (see EN 1990 – Ref[5])

Eq. 6.11b Gk,j + P + Ad+ (1,l or 2,l)Qk,l+ 2,iQk,i

The choice between 1,l or 2,l is done by the user. Default is 1,l.

with Gk,j characteristic value of permanent action j

P relevant representative value of prestressing action

Qk,l characteristic value of leading variable action l

Qk,i characteristic value of accompanying variable action i

Ad design value of the accidental action

1,l 2,l combination coefficients


48

Material properties


Strength and deformation properties :

a

,a,E

y

,p,p

y

,y,y

EE

k

f

fk

f

fk

The variation in function of the steel temperature of the value for yield strength ky,, proportional limit

kp, and modulus of elasticity kE, is given by tables in ref.[6], table 3.1.

For cold formed members ky, is taken from Ref.[7]; table III.2.5.

In the simplified calculation method, the following default properties are considered to be constant during the analysis :



(a-20)



In this part, the nominal temperature-time curves and the related net heat flux are described. See Ref.[8], Section 3, and Ref.[7], II.2.2.


The following temperature-time curves can be selected :



c the coefficient of heat transfer by convection


49

ISO 834 curve

external fire curve

hydrocarbon curve

smoldering fire curve

during 21 minutes, followed by the standard ISO 834 curve

user defined temperature-time curve

Net heat flux

rnetcnetdnet hhh ,,,

with hnet,d the net heat flux

hnet,c the convective heat flux

hnet,r the radiative heat flux


50

with configuration factor [1.0]

res resultant emissivity

= f m

f emissivity related to fire compartment

= [1.00]

m emissivity related to surface material

= [0.70]

r = g

gas temperature in [°C]

m surface temperature of member in [°C]


Steel Temperature


thc

VAk dnet

aa

mshta ,,

/









ksh correction factor for the shadow effect [1.0]

The correction factor is calculated for I sections only



51














The value a,t 0.0



52

Calculation model


- strength domain




Code Check

The section and stability checks (buckling, lateral torsional buckling) are performed according to the regulations given in 'EN 1993-1-2:2005'. The checks are performed in the resistance domain or in the temperature/time domain..

Torsional buckling and shear buckling are not considered.



- classification of cross section : art. 4.2.2.

- resistance for tension members : art. 4.2.3.1

- resistance for compression members (class 1,2 or 3) : art. 4.2.3.2.

- resistance for beams (class 1,2) : art. 4.2.3.3.

- resistance for beams (class 3) : art.4.2.3.4.

- resistance for members (class 1,2,3) subject to bending and compression : art. 4.2.3.5.

- check for class 4 sections : Annex E


53

EC3 – EN Cold-Formed

The members are checked according to the regulations given in:

Eurocode 3


Part 1 - 3: Supplementary rules for cold-formed members and sheeting

EN 1993-1-3:2006

Corrigendum

EN 1993-1-3:2006/AC:2009

Eurocode 3


Part 1 - 5: Plated Structural elements

EN 1993-1-5:2006

Corrigendum

EN 1993-1-5:2006/AC:2009

Consulted articles

An 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 paragraphs.

Article Title

1 Introduction X

2 Basis of design X

3 Materials

3.1 General X

3.2 Structural Steel X(*)

5 Structural Analysis

5.1 Influence of rounded corners X(*)


54

5.2 Geometrical proportions X(*)

5.3 Structural modelling for analysis X

5.5 Local and distortional buckling

5.5.1 General

5.5.2 Plane elements without stiffeners

5.5.3 Plane elements with edge or intermediate stiffeners

5.5.3.1 General

5.5.3.2 Plane elements with edge stiffeners

5.5.3.3 Plane elements with intermediate stiffeners

X

X(*)

X(*)

X(*)

X(*)

6 Ultimate Limit States

6.1 Resistance of cross-sections

6.1.1 General

6.1.2 Axial Tension

6.1.3 Axial Compression

6.1.4 Bending moment

6.1.4.1 Elastic and elastic-plastic resistance with yielding at the compressed flange

6.1.5 Shear Force

6.1.6 Torsional Moment

6.1.7 Local Transverse Forces

6.1.8 Combined Tension and Bending

6.1.9 Combined Compression and Bending

6.1.10 Combined shear force, axial force and bending moment

6.1.11 Combined Bending moment and local load or support reaction

X

X(*)

X(*)

X(*)

X(*)

X(*)

X(*)

X(*)

X(*)

X(*)

X(*)

6.2 Buckling Resistance

6.2.1 General

6.2.2 Flexural buckling

6.2.3 Torsional buckling and torsional-flexural buckling

6.2.4 Lateral Torsional buckling of members subject to bending

6.2.5 Bending and axial compression

X

X(*)

X(*)

X(*)

X(*)

6.3 Bending and axial tension X(*)

10 Special considerations for purlins, liner trays and sheetings

10.1 Beams restrained by sheeting

10.1.1 General

10.1.2 Calculation methods

10.1.3 Design criteria

10.1.4 Design resistance

10.1.5 Rotational restraint given by sheeting

10.1.5.1 Lateral spring stiffness

X(*)

X

X

X(*)

X(*)


55

As specified in EN 1993-1-3: 1.1(3) the code does not apply to cold -formed CHS (FC 3) and RHS (FC 2) sections. For these form codes the default EN 1993-1-1 provisions apply.

Haunches, arbitrary members and cross-sections without initial shapes are not supported for the EN 1993-1-3 code check. In this case the default EN 1993-1-1 code check is executed.

The checks are executed according to the principal axis in accordance with EN 1993-1-3 art. 1.5.1(4) NOTE except where stated otherwise.

Material properties

The steel grades given within EN 1993-1-3 Table 3.1b are available in the default Material Library of

Scia Engineer.

Average Yield Strength

The average yield strength is supported according to EN 1993-1-3 art. 3.2.2.

The average yield strength is applied in the following resistance calculations:

- Axial Tension

- Axial Compression

- Bending Moment

- Torsional moment

- Flexural Buckling

- Torsional (-Flexural) Buckling

- Purlin design – Cross-section resistance

The average yield strength is calculated using Ag of the Initial shape.

Steel Core Thickness

The steel core thickness is supported according to EN 1993-1-3 art. 3.2.4.

The steel core thickness is only available for the following sections:

- Cross-section which have form code FC 111 – FC 126 & FC 129 - Cold-formed pair sections (2CFUo, 2CFUc, 2CFCo, 2CFCc, 2CFLT)

The ranges for the core thickness are set ‘for sheeting and members’.

Form codes 172 & 128 are not supported for the Steel Core Thickness.


56

Initial Shape

For a cross-section with material Steel and fabrication set to Cold-Formed, the Initial Shape can be defined.

For a General cross-section the ‘Thin-walled representation’ has to be used to be able to define the Initial Shape.

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

F Fixed Part – No reduction is needed

I Internal cross-section part

SO Symmetrical Outstand

UO Unsymmetrical Outstand

Parts can also be specified as reinforcement:

None Not considered as reinforcement

RUO Reinforced Unsymmetrical Outstand (edge stiffener)

RI Reinforced Intermediate (intermediate stiffener)

DEF Double Edge Fold (edge stiffener)

ROU and DEF reinforcement types can be set only to elements of type SO or UO.

RI types can be set only to elements of type I or UO or SO.

For general cross-sections neighbouring elements of type RI are seen as one stiffener for the

calculation of the stiffener area and inertia.

The initial shape is supported for the following cross-section types:

- Standard profile library cross-sections

- Cold formed Pair cross-sections of profile library sections

- General thin-walled sections

- General sections with thin-walled representation

- Thin-walled geometric sections


57

- All other sections which support the centerline and do not have roundings

For standard profile library cross-sections, the flat parts are taken between the roundings. The roundings are set as fixed parts.

For predefined sections without roundings, the initial shape is based on the centreline dimensions i.e. the flat parts are taken between the intersection points of the centrelines.

For standard profile library cross-sections and pair sections the Initial Shape is generated automatically. Within this automatic generation the stiffeners are handled as follows:

o For the following form codes edge stiffeners are automatically set as RUO FC 114 Cold formed C-section FC 115 Cold formed Omega section FC 116 Cold formed C-Section eaves beam FC 118 Cold formed ZED section FC 119 Cold formed ZED section asymmetric lips FC 120 Cold formed ZED section inclined lip FC 121 Cold formed Sigma section FC 124 Cold formed Sigma section eaves beam FC 126 Cold formed ZED section both lips inclined FC 129 Cold formed Sigma section asymmetric

o For the following form codes edge stiffeners are automatically set as DEF

FC 117 Cold formed C-Plus section FC 122 Cold formed Sigma section stiffened FC 123 Cold formed Sigma-Plus section FC 125 Cold formed Sigma-Plus section eaves beam FC 127 Cold formed I-Plus section FC 128 Cold formed IS-Plus section

o For the following form codes internal stiffeners are automatically set as RI

FC 121 Cold formed Sigma section FC 124 Cold formed Sigma section eaves beam FC 122 Cold formed Sigma section stiffened FC 123 Cold formed Sigma-Plus section FC 125 Cold formed Sigma-Plus section eaves beam FC 128 Cold formed IS-Plus section FC 129 Cold formed Sigma section asymmetric

Geometrical Proportions

The Geometrical proportions are checked according to EN 1993-1-3 art. 5.2(1) Table 5.1.

The limits for edge stiffeners (c) and double edge folds (d) are checked in case the correct stiffener type (RUO or DEF) has been set in the initial shape.

The limit ratio’s given in EN 1993-1-3 art. 5.2(2) are checked. Lip dimensions c and d are however

always accounted for and will not be ignored.

In addition the limit for the internal radius given in EN 1993-1-3 art. 5.1(6) is checked.

For general sections, the geometrical proportions are checked for elements I, UO and SO using their respective part lengths. Flanges including RI stiffeners are thus considered part by part and not as one whole flange.


58

Effective Shape

Influence of rounded corners

Within Scia Engineer the exact method is applied i.e. all properties and dimensions are determined including the influence of rounded corners.

The approximate procedure given in EN 1993-1-3 art. 5.1(3) and following is thus not supported.

Notional widths

The notional widths are specified in EN 1993-1-3 art. 5.1 and Figure 5.1.

The initial shape elements are taken between the roundings (i.e. internal dimensions w).

The notional widths bp are then calculated as follows:

For an internal element (I)

bp = w + rm * sin ( left / 2) + rm * sin ( right / 2)

For an outstand element (UO, SO)

bp = w + rm * sin ( / 2)

In addition to the notional with bp, for each element the centerline length lc is determined as follows:

For an internal element (I)

lc = bp + gr,left + gr,right

With gr,left = rm * [tan ( left / 2) - sin ( left / 2)]

gr,right = rm * [tan ( right / 2) - sin ( right / 2) ]


59

For an outstand element (UO, SO)

lc = bp + gr

With gr = rm * [tan ( / 2) - sin ( / 2)]

General procedure for one element

By default, EN 1993-1-3 specifies that the stress f (com,Ed) to be used for the effective section

calculation should be taken as fyb/M0

The reduction of an element is in general given by:

beff = p * b

With: beff Effective width

p Reduction factor

b Full width

Step 1:

For the given stress f the normal stress over the rectangular plate element of the initial

geometrical shape is calculated. These stresses are calculated based on the nominal width bp.

beg : normal stress at start point of rectangular shape – compression stress is positive

end : normal stress at end point of rectangular shape – compression stress is positive

If the rectangular shape is completely under tension, i.e. beg and end are both tensile stresses, no reduction is needed, p = 1.0


60

Step 2: Determine f1 and f2: in case

f1 = beg

f2 = end

in case

f1 = end

f2 = beg

Step 3: Calculate the stress gradient :

Step 4: If = 1 the element is under uniform compression, else the element is under stress gradient.

Depending on the stress gradient and the element type, the effective width can be calculated as specified in the following paragraphs.

EN 1993-1-3 art. 6.1.4.2 concerning the plastic reserve of the tension flange is not supported i.e. always an elastic stress distribution is used.

Internal Compression Elements

The effective width of internal compression elements is calculated according to EN 1993-1-5 art. 4.4 and Table 4.1.

This applies to elements of type I.

The notional width bp is used as

Outstand Compression Elements

The effective width of outstand compression elements is calculated according to EN 1993-1-5 art. 4.4 and Table 4.2.

This applies to elements of type UO and SO

The notional width bp is used as

When activating the checkbox “Use Lambda,p,red 4.4(4)” the reduced slenderness is

determined using com,Ed as the maximal compressive stress f1 f2 in the element.

When activating the checkbox “Use Annex E E.1(1)” the formulas given in Annex E

are used to determine the reduction factor .


61

Plane Elements with Edge Stiffeners

The procedure for determining the effective width/thickness of elements with edge stiffeners is given in EN 1993-1-3 art. 5.5.3.2 and art. 5.5.3.1.

This applies to elements of type RUO and DEF

General remarks regarding the stiffness K of the edge stiffener given in formula (5.10b) .

hw is taken as lc (centreline length) of the biggest adjacent element. Adjacent

elements are those elements connected to the flange. For typical cross-sections, there is only one adjacent element, the web.

For Sigma sections, hw is taken as the sum of the centreline lengths of the web

elements.

This concerns the following form codes: FC 121 Cold formed Sigma section FC 124 Cold formed Sigma section eaves beam FC 122 Cold formed Sigma section stiffened FC 123 Cold formed Sigma-Plus section FC 125 Cold formed Sigma-Plus section eaves beam FC 128 Cold formed IS-Plus section FC 129 Cold formed Sigma section asymmetric

General Cross-section: hw for stiffener:

o Elements connected to the stiffener are not accounted for since they are considered as flanges

o Elements connected to those flanges are all accounted for in case they are of type I and the summation is made of the lengths of these elements

o Roundings are not accounted for

General Cross-section: In case there is only one edge stiffener kf is taken as zero. (i.e. no interaction between two flanges since there is only one

flange).

General Cross-section: In case there are two edge stiffeners kf is determined by default. (i.e. interaction between the two flanges is accounted

for).

General Cross-section: In case there are more than two edge stiffeners The same logic is followed as for a single stiffener. The factor kf is thus taken as

zero.

The formula for K given in the EN 1993-1-3 is based purely on simple sections with two flanges. In case of more complex cross-sections, the only exact procedure is to perform a numerical analysis (finite strip method) to determine the critical stresses for local and distortional buckling. This is referenced as the ‘Advanced Procedure’ given in art. 5.5.1(7).

Critical stresses for local and distortional buckling obtained from a numerical analysis can be inputted in the cross-section manager.

The reduced effective area of the stiffener As,red according to art 5.5.3.2(11) is

calculated using com,Ed = fyb/M0.


62

Plane Elements with Intermediate Stiffeners

The procedure for determining the effective width/thickness of elements with intermediate stiffeners is given in EN 1993-1-3 art. 5.5.3.3 and art. 5.5.3.1.

This applies to elements of type RI

The stiffness K of the internal stiffener is determined from formula (5.11):

The reduced effective area of the stiffener As,red according to art 5.5.3.3(10) is

calculated using com,Ed = fyb/M0.

General procedure of Effective Shape calculation

The general procedure which combines the effective section calculation of plane elements without and plane elements with stiffeners is given in EN 1993-1-3 art. 5.5.2(3) and art. 5.5.3.

This procedure can be written out as follows:

Step 1: The effective width of the flanges and edge/intermediate stiffeners within the flanges are calculated based on gross section properties. This includes the optional iterative procedure for the edge/intermediate stiffeners as specified in art. 5.5.3.2(10) and art. 5.5.3.3(9).

Step 2: This partially effective shape of the previous step is used to determine the stress gradient and effective width of the web. This includes the optional iterative procedure for the intermediate stiffeners as specified in art. 5.5.3.3(9).

Step 3: The end result of the previous two steps is the effective cross-section and its properties can be calculated.

Step 4: This process can now be optionally iterated using the stress ratio based on the effective cross-section in place of the gross cross-section.

Both iteration procedures (iteration of stiffeners and iteration of the full cross-section) can be set in the Steel Setup.


63

Manufacturer provided effective section properties

In case in the Steel Setup the option ‘Use manufacturer provided effective section properties’is

activated, effective section properties from the manufacturer are taken from the Effective Section Library instead of calculated by EN 1993-1-3.

The following properties can be defined in the Effective Section Library:

Property Description

fy [MPa] Steel grade for which the effective properties have been derived

Aeff [mm^2] Effective Area for compression

eN,y [mm] Shift of centroid in y direction for compression

eN,z [mm] Shift of centroid in z direction for compression

Ieff,y My+ [mm^4] Effective moment of inertia about the y-y axis for a positive moment My

Weff,y My+ [mm^3] Effective section modulus to the extreme fiber about the y-y axis for a positive moment My

eM,z My+ [mm] Shift of centroid in z direction for a positive moment My

Ieff,y My- [mm^4] Effective moment of inertia about the y-y axis for a negative moment My

Weff,y My- [mm^3] Effective section modulus to the extreme fiber about the y-y axis for a negative moment My

eM,z My- [mm] Shift of centroid in z direction for a negative moment My

Ieff,z Mz+ [mm^4] Effective moment of inertia about the z-z axis for a positive moment Mz

Weff,z Mz+ [mm^3] Effective section modulus to the extreme fiber about the z-z axis for a positive moment Mz

eM,y Mz+ [mm] Shift of centroid in y direction for a positive moment Mz

Ieff,z Mz- [mm^4] Effective moment of inertia about the z-z axis for a negative moment Mz

Weff,z Mz- [mm^3] Effective section modulus to the extreme fiber about the z-z axis for a negative moment Mz

eM,y Mz- [mm] Shift of centroid in y direction for a negative moment Mz

In case the yield strength used for the cross-section does not match any of the yield strengths defined in the Effective Section Library the default EN 1993-1-3 calculation will be used.


64

Section Checks

Axial Tension

The Axial Tension Check is executed according to EN 1993-1-3 art. 6.1.2.

The net section resistance Fn,Rd is taken as:

With Anet taken equal to Ag since bolt holes are not accounted for.

Axial Compression

The Axial Compression Check is executed according to EN 1993-1-3 art. 6.1.3.

The choice between formula (6.2) and (6.3) is made by comparing the gross area Ag from the initial shape with the effective area Aeff of the effective shape for compression.

The gross area Ag used in the formulas is taken from the cross-section manager.

This comparison using the initial shape property is of importance for the following reasons: - Profile Library sections can have different gross properties compared to the initial shape since the gross properties come from certain sources (books, tables, …) and are mostly rounded off. - For general cross-sections the gross shape can differ from the initial shape since the initial shape concerns a thin walled representation.

Each element on which a distortional buckling reduction factor d is applied is seen as ‘stiffened’.

All other elements are seen as ‘plane’.


65

Bending Moment

The Bending Moment Check is executed according to EN 1993-1-3 art. 6.1.4.1.

The choice between formula (6.4) and (6.5) is made by comparing the elastic section modulus Wel from the initial shape with the effective section modulus Weff of the effective shape for bending.

The elastic section modulus Wel used in the formulas is taken from the cross-section manager.

Note: This comparison using the initial shape property is of importance for the following reasons: - Profile Library sections can have different gross properties compared to the initial shape since the gross properties come from certain sources (books, tables, …) and are mostly rounded off. - For general cross-sections the gross shape can differ from the initial shape since the initial shape concerns a thin walled representation.

An element of type I is seen as ‘plane’.

An element of type UO or SO is seen as ‘outstand’.

As indicated in EN 1993-1-3 art. 6.1.4.1(2) formula (6.5) is only applied in case:

There is only single bending i.e. My OR Mz

There is no torsion i.e. Mx = 0

There is no Torsional (-Flexural) buckling i.e. TF = 1,00

There is no Lateral Torsional buckling i.e. LTB = 1,00

There is no Distortional buckling i.e. all reinforcement types of the cross-section elements should be ‘none’ or, in case there are stiffeners, they should not be in compression.

The angle between the web and flange exceeds 60°.

In case formula (6.5) should be applied but the above conditions are not fulfilled, formula (6.6) is

applied.

EN 1993-1-3 art. 6.1.4.2 concerning the plastic reserve of the tension flange is not supported i.e.

always an elastic stress distribution is used.

EN 1993-1-3 art. 6.1.4.3 concerning the effects of shear lag is not supported.


66

Shear Force

The Shear Force Check is executed according to EN 1993-1-3 art. 6.1.5.

General

The shear resistance is calculated for each ‘web’ element separately and the cross-section resistance is taken as the sum of these element resistances.

Only elements with element types I, UO and SO are accounted for.

In addition, elements with stiffener type RUO or DEF are not accounted for.

Formula (6.8) is rewritten as follows for both directions:

With: i = Angle of element i related to the principal y-y axis

lc,i = Centreline length of element i

By default the Shear Check is executed ‘without stiffening at the support’ In case Local Transverse Forces data are inputted which have the checkbox ‘No Local Transverse Forces Check’ activated, the Shear Check in those sections is executed ‘with stiffening at the support’.

Elements without Internal stiffeners

The centreline length lc,i for each element i is taken from the Initial shape.

The angle i for each element i is determined as the angle related to the principal y-y axis.

The relative web slenderness for each element i is determined according to formula (6.10a).

The slant height sw,i is taken as the notional width bp,i of the element under consideration as

indicated on the following picture.


67

Sections with Internal stiffeners

Special considerations are required for cross-sections with internal stiffeners (Type RI).

The following picture illustrates a web with internal stiffener:

The internal stiffener and connected elements are seen as ‘one web’. This ‘composed’ web is seen as ‘one’ element i in the shear calculation.

For such a ‘composed’ web, the different distances are determined as follows:

The slant height sw is taken as the distance between - The starting point of the nominal width bp,i of the first element in the web. - The end point of the nominal width bp,i of the last element in the web.

The total developed slant height sd is taken as the sum of the nominal widths bp,i of all the

elements in the web.

The slant height sp concerns the notional width bp,i of the largest plane web element.

The relative web slenderness is determined according to formula (6.10b).

The inertia of the stiffener(s) Is is taken from the Initial shape


68

The centreline length lc of this composed web is calculated as follows:

In case the first or last element of the composed web has element type SO or UO:

lc = sw + gr

With gr = rm * [tan ( / 2) - sin ( / 2)]

If the first element is an outstand, gr is taken as gr at the end point of the last

element. If the last element is an outstand, gr is taken as gr at the starting point of the first

element. Reference is made to Notional widths.

In case both the first and last element of the composed web has element type I:

lc = sw + gr,first + gr,end With gr,first taken as gr at the starting point of the first element. gr,end taken as gr at the end point of the last element.

The angle of the ‘composed’ web concerns the angle of the centreline length lc relative to the

principal y-y axis.

Neighboring connected elements are seen as one ‘web’. A typical example of this is a sigma section: the web has two internal stiffeners which both are connected to the same internal element. As such they are recognized as forming one web.

Torsional Moment

The Combined Stress Check including Torsion and Warping is executed according to EN 1993-1-3 art. 6.1.6.

Regarding warping reference is made to Warping check.

The average yield strength fya in all three formulas (6.11a), (6.11b), (6.11c) will only be used in

case for all three force components separately (N, My, Mz) the average yield strength may be used (Aeff = Ag ; Weff,y = Wel,y ; Weff,z = Wel,z).

Local Transverse Forces

The local transverse forces check is executed according to EN 1993-1-3 art 6.1.7 and following.

The check is executed on the positions where there is a jump in the Vz shear force diagram.

Remarks:

The shear force diagram of both the actual member as well as adjacent members is evaluated. Adjacent members are defined as members which are in the same buckling system.

The Flange Condition depends on the definition of the initial shape. In case there is an element with reinforcement type ROU or DEF the setting is taken as ‘Stiffened ’.


69

The distances for One-flange/Two-flange and End/Interior are evaluated taking into account adjacent members. Adjacent members are defined as members which are in the same buckling system.

In case the cross-section has multiple webs, for determining the load condition the maximal web height is used.

As opposed to EN 1993-1-3 art.6.1.7.2(4), the exact inputted bearing length ss will be used

at all times i.e. the simplification of using the minimal length for both opposing loads is not supported.

As indicated on EN 1993-1-3 Figure 6.6, the local transverse force resistance is taken relative to the support, not according to the principal z-axis. Therefore FEd, is determined

according to the LCS axis system and not according to the principal axis system!

General Procedure

This paragraph specifies the general procedure to determine the local transverse web resistance which is applied for any type of cross-section except for FC 115 (Cold formed Omega).

In case the cross-section has any element with stiffener type RI, the procedure for stiffened webs is applied first.

In a first step the web height hw is determined for each ‘web’ element:

Only elements of type I are accounted for. In addition elements with stiffener types RUO and DEF are not accounted for.

For each of those elements i the centreline length lc,i is read from the Initial shape

For each of those elements i the angle i is determined as the angle of the element relative to the horizontal axis (based on Figure 6.6).

In addition, only elements with an angle i ≥ 45° are accounted for.

The web height for each element i is calculated as:

In case none of the cross-section elements fulfil the above conditions, the local transverse forces check is not supported for the cross-section.

When hw,i is determined, the local transverse resistance Rw,Rd,i for each of those elements is determined based on EN 1993-1-3 art.6.1.7.2

The final cross-section resistance is taken as the sum of the individual element resistances.

By default, the local transverse resistance Rw,Rd,i is determined using EN 1993-1-3 Figure 6.7a & 6.7b.

The following table shows the relation between the loading conditions and the cases defined in the tables.

Loading Condition Table Case

End One Flange (EOF) 6.7a a) i)

Interior One Flange (IOF) 6.7a a) ii)

End Two Flange (ETF) 6.7b b) i)

Interior Two Flange (ITF) 6.7b b) ii)


70

In case Web rotation prevented was set using Local Transverse Forces data instead of EN 1993-1-3 Figure 6.7a & 6.7b the formulas given in EN 1993-1-3 art. 6.1.7.2(4) are used.

The following table shows the relation between the loading conditions and the cases defined in this article.

Loading Condition Article Case

End One Flange (EOF) art. 6.1.7.2(4) a) i)

Interior One Flange (IOF) art. 6.1.7.2(4) a) ii)

End Two Flange (ETF) art. 6.1.7.2(4) b) i)

Interior Two Flange (ITF) art. 6.1.7.2(4) b) ii)

Omega Sections

Specifically for FC 115 (Cold formed Omega) cross-sections the special procedure for sections with two or more unstiffened webs is applied. The local transverse resistance Rw,Rd,i for each of those webs is determined according to EN 1993-1-3 art. 6.1.7.3.

Other cross-sections with two or more unstiffened webs will always be calculated according to the General Procedure, not this special procedure.

The value of in EN 1993-1-3 art. 6.1.7.3(5) is taken for ‘liner trays and hat sections’.

The following table shows the relation between the loading conditions and the categories defined in EN 1993-1-3 Figure 6.9.

Loading Condition Category

End One Flange (EOF) 1

Interior One Flange (IOF) 1

End Two Flange (ETF) 1

Interior Two Flange (ITF) 2

Figure 6.9 does not directly specify ETF. However since two flange loading is specified as category 1 and End loading is also specified as category 1, the combined condition of ETF is considered as category 1.

According to [27] to use la = 10 mm for the end support reaction force (category 1) results in a very conservative resistance. A modification is given for case 2 and 3 of Figure 6.9: la = c + Ss. By activating the setting “Use la correction in (6.18)” this modification is applied.


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Stiffened Webs

This paragraph outlines the special procedure in case of stiffened webs according to EN 1993-1-3 art. 6.1.7.4.

This method is used only in case there are one or more elements with stiffener type RI

The procedure consists of four steps.

Step 1: Creating ‘composed’ webs

In a first step, ‘composed’ webs are created using the same procedure as outlined in Sections with Internal stiffeners.

This includes the determination of the centreline length lc,i of those ‘composed’ webs.

Step 2: Evaluation of ‘composed’ webs

The special procedure outlined in EN 1993-1-3 art. 6.1.7.4 is only valid under certain conditions.

Therefore, each ‘composed’ web is evaluated to see if it meets the following requirements:

There is one or more elements with stiffener type RI

Each RI element should have element type I (i.e. it is at both sides connected to

other elements signifying it’s a fold instead of a stiffener).

Elements connected to this RI element should not have stiffener type RI. This

implies that the procedure is not applied in case of neighbouring stiffener elements i.e. elements forming ‘one’ big stiffener.

Composed webs which do not meet these requirements are further evaluated in step 3.

Composed webs which meet all requirements are further evaluated in step 4.

Example:

All four sections have ‘composed’ webs.

Section A contains two RI stiffeners which are connected. The web thus does not meet the

requirements.

Section B contains a single RI stiffener which meets all the requirements. This stiffener is thus a

‘true’ two fold stiffener so the special article applies.

Section C contains several RI stiffeners however not all match the requirements (one is an outstand,

others are connected etc). The web thus does not meet the requirements.

Section D has a composed web which contains two RI stiffeners. Both meet all the requirements

and are thus ‘true’ two fold stiffeners. The special article applies.


72

Step 3: Composed webs which do NOT meet the requirements

For composed webs which do not meet the requirements, the special article is not valid. The local transverse force resistance of these webs will be determined according to the General Procedure

In this case, the centre line length lc,i of the composed web is used in the determination of hw.

The angle i is determined as the angle of the centre line length relative to the horizontal axis.

Step 4: Composed webs which meet all requirements

For composed webs which meet all requirements, the special procedure outlined in EN 1993-1-3 art. 6.1.7.4 is applied.

The ‘system line’ of this web is taken as the centre line length lc,i.

The eccentricity e is determined at each end of an RI within the ‘composed’ web. Eccentricity emin and emax are then taken as the min and max value for the considered composed web.

In case the limit specified in formula (6.21) is not fulfilled, the special article is not applied and the

composed web is considered as a web which does not meet all requirements. For such a web the procedure outlined in step 3 is applied.

For the developed width of the loaded flange bd any RI stiffeners of element Type I are always included, independent of their angle. RI stiffeners of element Type UO or SO are always ignored.

Connected flange elements which have a relative angle > 135° are accounted for as „one‟ flange for

the determination of bd.

In case there is no connected flange, for example when using a general section, then bd is

considered as zero. Practically this implies that there is no limit for a,s.

The data is then used to determine a,s according to formula (6.22).

The Rw,Rd,i value of the composed web is then calculated as:

Rw,Rd,i = a,s * Rw,Rd,i,general

With Rw,Rd,i,general calculated according to the General Procedure

The value of hw,i for this composed web is calculated using the centre line lc,i of the composed web

as outlined in step 3.

Combined Tension and Bending

The Combined Tension and Bending Check is executed according to EN 1993-1-3 art. 6.1.8.

The bending resistances are determined using the section moduli Weff of the effective shapes for

bending.


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Combined Compression and Bending

The Combined Compression and Bending Check is executed according to EN 1993-1-3 art. 6.1.9.

Additional moments due to the shift in neutral axis are calculated at the beginning of the check and added to the internal forces.

This ensures specific bending checks are executed also in case there is no initial moment but only an additional moment.

The shifts in neutral axis eNy and eNz are read directly from the effective shape for compression.

As specified in EN 1993-1-3 art. 6.1.3(3) additional moments are only accounted for in case they

lead to an unfavourable check result.

The bending resistances are determined using the section moduli Weff of the effective shapes for

bending.

Combined Shear Force, Axial Force and Bending Moment

The Combined Shear Force, Axial Force and Bending Moment Check is executed according to EN 1993-1-3 art. 6.1.10.

In the following paragraphs formula (6.27) is written out for both directions.

Shear Vy

In case of shear Vy formula (6.27) is written out as follows:

Remarks:

Mf,Rd is taken as zero in case of Vy

(In case of weak axis bending, the ‘web’ becomes a ‘flange’. Since there is only a single ‘flange’ in that case, the moment resistance of this flange is negligible. In addition, in case of more webs like in a box section EN 1993-1-5 art. 7.1 (5) specifies Mf,Rd = 0. Therefore, as a general conservative approach for Vy the value of Mf,Rd is taken as 0.)

Shear Vz

In case of shear Vz formula (6.27) is written out as follows:


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Remarks:

According to [Ref.16] pp70 Mf,Rd is calculated as follows:

This is generalised in the following way:

a) Only elements with element types I, UO and SO are accounted for

b) Only elements which have an angle with the principal y-y axis which is 45°

are considered In case there is only one or none of such element, Mf,Rd = 0.

c) Of these elements, the one with the lowest beff is considered. The width beff

concerns the effective with of this element, read from the effective shape for bending.

d) Af = beff * t with t the thickness of the considered element.

e) Next only elements which have an angle with the principal y-y axis which is > 45°are considered.

In case there are no such elements, set Mf,Rd = 0.

f) Of these elements, the one with the highest value of lc * sin() is considered, with lc the centreline length of the element.

g) hf = lc * sin()

h) Mf,Rd is now be calculated as:

According to [Ref.16] pp70 Mpl,Rd is calculated as follows:

with Wpl read from the gross section properties.


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Combined Bending Moment and local Load/Support Reaction

The Combined Bending Moment and local Load/Support Reaction Check is executed according to EN 1993-1-3 art. 6.1.11.

In formula (6.28c) the internal force MEd is taken as the actual moment in the section considered, not

the moment at the edge of the support.


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Stability Checks

Flexural Buckling

The Flexural Buckling Check is executed according to EN 1993-1-3 art. 6.2.2 and EN 1993-1-1 art. 6.3.1.

Table 6.3 regarding the buckling curves is revised as follows:

Form code Description about axis Curve

1 I section y-y

z-z

a

b

101 Asymmetric I section y-y

z-z

a

b

114 Cold formed C section any b

116 Cold formed C-Section eaves beam any b

117 Cold formed C-Plus section any b

118 Cold formed ZED section any b

119 Cold formed ZED section asymmetric lips any b

120 Cold formed ZED section inclined lip any b

121 Cold formed Sigma section any b

122 Cold formed Sigma section stiffened any b

123 Cold formed Sigma-Plus section any b

124 Cold formed Sigma section eaves beam any b

125 Cold formed Sigma-Plus section eaves beam any b

126 Cold formed ZED section both lips inclined any b

127 Cold formed I-Plus section y-y

z-z

a

b

128 Cold formed IS-Plus section y-y

z-z

a

b

129 Cold formed Sigma section asymmetric any b

2CFCo with a = 0 y-y

z-z

a

b

2CFCc with a = 0 Closed section rule 6.2.2(3)

2CFUo with a = 0 y-y

z-z

a

b

2CFUc with a = 0 Closed section rule 6.2.2(3)

2CFLT with a = 0 any c

Any other section any c

All other sections fall in the ‘other cross-section’ case of curve c for any axis.



77

Torsional (-Flexural) Buckling

The Torsional (-Flexural) Buckling Check is executed according to EN 1993-1-3 art. 6.2.3 and EN 1993-1-1 art. 6.3.1.4.

The buckling curve for torsional (-flexural) buckling is taken as the z-z buckling curve according to the table given in Flexural Buckling.

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.[17].

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


78

Calculation of Ncr,TF

The elastic critical load Ncr,TF for torsional flexural buckling is calculated according to Ref.[17].

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

0

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

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

Ncr,T Critical axial load for torsional buckling

Lateral Torsional Buckling

The Lateral Torsional Buckling Check is executed according to EN 1993-1-3 art. 6.2.4 and EN 1993-1-1 art. 6.3.2.2.

For additional information reference is made to Lateral-torsional buckling.

For information regarding the influence of diaphragms on the Lateral Torsional Buckling Check reference is made to Use of Diaphragms.

Bending and Axial Compression

For determining the Combined Bending and Axial Compression check according to EN 1993-1-3 art. 6.2.5 EN 1993-1-3 allows two possibilities:

Use the EN 1993-1-1 interaction according to article 6.3.3

Use the alternative according to EN 1993-1-3 article 6.2.5(2)

The choice between these two methods is set in the Steel Setup.

Interaction according to EN 1993-1-1

The interaction is executed according to EN 1993-1-1 art. 6.3.3 using interaction factors from Annex A & B.

In both Method 1 (Annex A) and Method 2 (Annex B) the cold - formed sections are seen as ‘class 3 or 4’.

Alternative interaction according to EN 1993-1-3

The interaction is executed according to EN 1993-1-3 art. 6.2.5(2).

Nb,Rd is taken as the lowest value of

the flexural buckling resistance about the y-y axis

the flexural buckling resistance about the z-z axis

the torsional (-flexural) buckling resistance


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Formula (6.36) includes the strong axis bending resistance Mb,Rd. There is however no indication for a weak axis bending moment. Therefore, in case a weak axis bending moment is present, this interaction cannot be applied and the general interaction according to EN 1993-1-1 is applied.

Bending and Axial Tension

The Combined Bending and Tension Check is executed according to EN 1993-1-3 art. 6.3.

The code specifies that the same equations as for compression should be used. These interaction equations are however not fully valid in case of tension.

The purpose of the interaction check for bending and tension is to check the stresses at the compression fiber. In the AISI NAS 2007 Ref.[18] code the following formula is given in article C5:

This formula is rewritten using EC-EN notations as follows:

With: Mb,y,Rd The Lateral Torsional Buckling resistance.

Mc,z,Rd,com The moment resistance for the compression fiber in case of Mz.

Nt,Rd The Tension Resistance


80

Use of Diaphragms

The influence of a diaphragm is outlined in the following diagram.


81

First of all the lateral stiffness S of the diaphragm is determined and compared to the required stiffness Serf.

The lateral stiffness S is calculated according to Ref.19,3.5 and Ref.20,3.3.4.

L

K+K

10a.=S

s

21

4

with a The frame distance

Ls The length of diaphragm

K1 Diaphragm factor K1

K2 Diaphragm factor K2

The required stiffness Serf is determined according to EN 1993-1-3 art. 10.1.1

In case S < Serf the member is seen as Inadequately braced. In this case, when the diaphragm is

located on the compression side, the Lateral Torsional Buckling check is executed using the augmented torsional stiffness It. Reference is made to Adaptation of torsional constant.

G

lvorhCII

2

2

tid,t

with l The LTB length

G The shear modulus

vorhC The actual rotational stiffness of diaphragm

As specified in art. 10.1.1 the shear stiffness S is replaced by 0,2 S in case the diaphragm is connected every second rib only.

In case S ≥ Serf the member is seen as Fully braced. In this case, a first test is executed to evaluate if the special purlin checks according to EN 1993-1-3 Chapter 10 can be applied.


82

More specifically, this chapter is applied only in case the cross-section concerns a Z, C, or U

section:

Form code Description

5 Channel section

102 Rolled Z section

112 Cold formed channel

113 Cold formed Z

114 Cold formed C section

116 Cold formed C-Section eaves beam

117 Cold formed C-Plus section

118 Cold formed ZED section

119 Cold formed ZED section asymmetric lips

120 Cold formed ZED section inclined lip

121 Cold formed Sigma section

122 Cold formed Sigma section stiffened

123 Cold formed Sigma-Plus section

124 Cold formed Sigma section eaves beam

125 Cold formed Sigma-Plus section eaves beam

126 Cold formed ZED section both lips inclined

129 Cold formed Sigma section asymmetric

The code specifies that the chapter is also valid for hat (Omega) sections however in all further paragraphs; no specific formulas are given for Omega sections. For

example the free flange geometry is described only for Z, C and sections, not for Omega sections. Therefore, Omega sections are not supported for this special chapter.

In case the cross-section does not match any of the above, the default checks are executed. Since the member is seen as fully braced, no Lateral Torsional Buckling check needs to be executed in case the diaphragm is located on the compression side.

In case the cross-section does match the list of set form codes, a second test is executed. More specifically, the special purlin checks according to EN 1993-1-3 Chapter 10 can be applied only in

case:

The dimensional limits of article 10.1.1(1) are satisfied

The section is only loaded by N, Vz, My

Chapter 10 specifies only checks related to in plane effects N, Vz and My. Therefore, in case of other loading components, the special articles are not valid and the default checks will be applied.


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For a section which meets all requirements, the following is done:

Reduced default Checks are executed i.e. not all default checks will be executed.

Special purlin checks according to Chapter 10

More specifically, the following ‘default’ checks will be executed:

Section Check Article

Axial tension 6.1.2

Axial compression 6.1.3

Bending moment 6.1.4

Shear force 6.1.5

Torsional moment NOT

Local Transverse Forces 6.1.7

Combined tension and bending NOT

Combined compression and bending NOT

Combined shear, axial force and bending moment 6.1.10

Combined Bending and Local Transverse Force 6.1.11

Stability Check Article

Flexural buckling only for y-y 6.2.2

Torsional and Torsional-Flexural buckling NOT

Lateral-Torsional buckling NOT

Bending and axial compression NOT

Bending and axial tension NOT

The Torsional moment check will never occur in this case since the prerequisite is to have only N, Vz, My.

The combined axial and bending checks are not executed since they are replaced by the special purlin checks.

The flexural buckling check is executed for y-y buckling in accordance with EN 1993-1-3 art. 10.1.4.2(2).

Torsional buckling and Lateral-torsional buckling are prohibited by the fully braced diaphragm. The compression in the free flange is included in the special purlin checks.

The combined stability checks are not executed since they are replaced by the special purlin checks.

In contrast to art. 10.1.3.3(2) the Local Transverse Load Check and its interaction with the bending moment is executed even if the support reaction is a tensile force.


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Special considerations for Purlins

As outlined in Use of Diaphragms for a section which meets all requirements, special purlin checks according to EN 1993-1-3 Chapter 10 will be executed:

Diaphragm on the compression side

Cross-section resistance according to EN 1993-1-3 art. 10.1.4.1

In case of compression in the free flange also Stability of the free flange according to EN 1993-1-3 art. 10.1.4.2

Diaphragm on the tension side

Cross-section resistance according to EN 1993-1-3 art. 10.1.4.1

Stability of the free flange according to EN 1993-1-3 art. 10.1.4.2

Resistance of Cross-Section

The Resistance of the Cross-Section is determined according to EN 1993-1-3 art. 10.1.4.1.

Since this check concerns a separate formula for each flange (10.3a) and (10.3b) the effective section modulus Weff,y is determined for each flange separately.

The average yield strength will only be used in case for both force components separately (N, My) the average yield strength may be used (Aeff = Ag ; Weff,y = Wel,y).

Definition of the free flange geometry

The dimension h is taken as the full cross section height.

The properties of the free flange are calculated according to the z-z axis of the full cross-section.

The following table shows the supported cross-sections including the contributing web height.

Form code Description Contributing web

5 Channel section 1/5 h

102 Rolled Z section 1/5 h

112 Cold formed channel 1/5 h

113 Cold formed Z 1/5 h

114 Cold formed C section 1/5 h

116 Cold formed C-Section eaves beam 1/5 h

117 Cold formed C-Plus section 1/5 h

118 Cold formed ZED section 1/5 h

119 Cold formed ZED section asymmetric lips 1/5 h

120 Cold formed ZED section inclined lip 1/5 h

121 Cold formed Sigma section 1/6 h

122 Cold formed Sigma section stiffened 1/6 h

123 Cold formed Sigma-Plus section 1/6 h


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124 Cold formed Sigma section eaves beam 1/6 h

125 Cold formed Sigma-Plus section eaves beam 1/6 h

126 Cold formed ZED section both lips inclined 1/5 h

129 Cold formed Sigma section asymmetric 1/6 h

As the code indicates in Figure 10.2, for sigma sections the rounding which leads to the web depression is also accounted for in the height of the free flange. Therefore, to generalize this principle, within Scia Engineer the rounding between the flange and the web is always accounted for in the free flange height (for all section types).

Determination of the equivalent lateral load

The equivalent lateral load on the free flange qh,Ed is determined from the vertical load qEd on the purlin using formula (10.4).

For any given moment diagram, the equivalent vertical line load qEd is determined as the line load

which results in approximately the same bending moment diagram..

The factor kh is determined according to EN 1993-1-3 Figure 10.3.

For kh0, the general formula for Z,C or sections is applied. The formula for a simple Z-section is not supported.

For Gravity loading, the vertical loading is assumed to be positioned at the outside of the web. For Uplift loading the vertical loading is assumed to be positioned exactly in the middle of the flange width.

For Gravity loading the general formula including the shear center distance e is used.

For Uplift loading the general formula including the shear center distance f is used. In case of a symmetrical Z section this distance will become a.

The load qh,Ed is given a positive sign in case it follows the same convention as shown in the code.

The load is given a negative sign in case it points in the other direction.

Determination of the lateral bending moment

Table 10.1 provides the formulas to determine Mfz,Ed for specific positions within the beams: at the

ends (e) and at the position of the maximal moment (m).

Within Scia Engineer however, the check is executed in different sections. Therefore, the values of Mfz,Ed need to be known in each section.

To this end, as indicated in the code in EN 1993-1-3 art. 10.1.4.1(7), the general equations have

been derived using the theory of beams on an elastic Winkler foundation.


86

The differential equation for the displacement of a beam on elastic foundation loaded by a line load is written out as follows Ref.[21]:

With: E Section modulus

I Bending stiffness

L Member length, taken as La

q Line load, taken as qh,Ed

K Foundation stiffness, taken as lateral spring stiffness K

A,B,C,D Integration constants

The integration constants are determined depending on the boundary conditions for the cases given in Table 10.1.

Using the beam equation with the second derivative of the displacement the equation for the bending moment Mfz,Ed is obtained and leads to the following solutions:


87

Solution for a beam on elastic Winkler foundation with Hinged end conditions

Solution for a beam on elastic Winkler foundation with Hinged-Fixed end conditions

Solution for a beam on elastic Winkler foundation with Fixed end conditions


88

The determination of a hinged or fixed end for Mfz,Ed is done as follows:

A single span member is always considered to have hinged ends. A single span member is defined as a member with only one part in the buckling system for Ly.

An LTB restraint is always considered as a fixed end.

For multi-span members, the ends of the buckling system for Ly are considered as hinged. The internal points of the buckling system for Ly are considered as fixed.

As specified in EN 1993-1-3 art. 10.1.4.1(5) in case the free flange is in tension Mfz,Ed is taken equal

to zero. To determine if the free flange is in tension or compression the following stress is calculated:

(My,Ed / Weff,y,free flange) + (Ned / Aeff)

In case this stress results in tension, the free flange is considered to be in tension.

In case this stress results in compression, the free flange is considered to be in compression.

The sign of Mfz,Ed determines the tension/compression side of the free flange and thus determines which Wfz is used in the check.

The limit of R 40 given in art. 10.1.4.1(6) does not apply since the general Winkler theory is used instead of table 10.1.

Determination of the distance between anti-sag bars

The code defines anti-sag bars as bars which provide lateral rigid support to the free flange. Within Scia Engineer, LTB restraints are thus seen as anti-sag bars.

In case LTB restraints are defined at the free flange, the length La is taken as the length between these restraints. In case there are no LTB restraints defined at the free flange, La is read from the

buckling system.

Determination of the lateral spring stiffness

The lateral spring stiffness K is determined according to EN 1993-1-3 art. 10.1.5.1(4).

The developed height of the purlin web hd is taken as the total developed slant height sd used in the

Shear Check, as described in Shear Force.

The rotational restraint CD is taken as vorhC, the rotational stiffness of the diaphragm, as described

in Adaptation of torsional constant.

The dimension bmod depends on the direction of the equivalent horizontal load qh,Ed and the type of

cross-section. According to the code this depends if the load brings the purlin into contact with the sheeting at the purlin web or at the tip of the purlin flange.


89

This is clarified in the following picture:

The distance a i.e. position of the fastener is taken as 0,5 b. The fastener is thus assumed to be

positioned in the middle of the flange.

Buckling Resistance of the Free Flange

The Buckling Resistance of the Free Flange is determined according to EN 1993-1-3 art. 10.1.4.2.

To determine if the free flange is in tension or compression the following stress is calculated:

(My,Ed / Weff,y,free flange) + (Ned / Aeff)

In case this stress results in tension, the free flange is considered to be in tension.

In case this stress results in compression, the free flange is considered to be in compression.

For a free flange in tension the buckling resistance does not need to be checked.

For determining the buckling length lfz of the free flange a difference is made between gravity

loading (downward –z loading) and uplift loading (upward +z loading).

Gravity Loading

In case of downward –z loading the buckling length of the free flange is determined according to formula (10.9).

The i factors are determined according to EN 1993-1-3 Table 10.2a.

Art. 10.1.4.2(4) is not supported.


90

Uplift Loading

In case of upward +z loading the buckling length of the free flange is determined according to formula (10.9).

The i factors are determined according to EN 1993-1-3 Table 10.2b.

The method according to art. 10.1.4.2(6) & (7) is not supported.

General Notes

For both loading types, Tables 10.2a & b differentiate between ‘simple span’, ‘end span’ and ‘intermediate span’. This is based on the Ly system length.

In case the member under consideration has only one part for Ly then it is considered as ‘simple

span’.

When the member has more parts for Ly it is considered as multi-span. For a multi-span, sections

located in the first or last part of the system length are considered as ‘end span’. Sections located in the other parts are considered as ‘intermediate span’.

Table 10.2a does not specify ‘simple span’. The values for a ‘simple span’ are taken equal as an ‘end span’.

The ‘number of anti-sag’ bars used in Tables 10.2a & b concerns the number of LTB restraints

defined on the actual member. Only LTB restraints at the side of the free flange are accounted for in this ‘number’.

EN 1993-1-3 art. 10.1.4.2(5) specifies a method for the buckling length in case of a ‘relatively large

axial force’. Within Scia Engineer this is quantified using a limit value, which is set in the Steel Setup.

In case the axial load is considered as large, the method described in EN 1993-1-3 art. 10.1.4.2(5)

is applied.

This procedure applies to both gravity and uplift loading using Table 10.2a and 10.2b respectively.


91

Supported sections




L Angle section

U Channel section

T T section


Z Z section


Cold formed section


O Solid tube



".


I

RHS

CHS

L

U

T

PPL

RS

Z

O

COM

NUM







x x x


x x x




x x x x x x


(1) Sections are classified as class 3 cross section by default.


92

References

1 Eurocode 3



EN 1993-1-1:2005

[2] Eurocode 3


Part 1-3: General rules

Supplementary rules for cold-formed members and sheeting

EN 1993-1-3:2006

3 Eurocode 3


Part 1.5 : Plated structural elements

EN 1993-1-5 : 2006

4 R. Maquoi



[5] EN 1990

Eurocode – Basis of structural design

EN 1990:2002 E

[6] Eurocode 3



EN 1993-1-2:2005


ECCS - N° 111

May 2001

[8] Eurocode 1

Actions on structures

Part 1-2 : General Actions - Actions on structures exposed to fire

prEN 1991-1-2:2002

[9] Rules for Member Stability in EN 1993-1-1

Background documentation and design guidelines

ECCS - N° 119

2006


93

[10] Eurocode 3



ENV 1993-1-1:1992/A1, 1994

[11] Eurocode 3



EN 1993-1-1:2005/AC:2009 Corrigendum

[12] Eurocode 3




[13] Eurocode 3


Part 1-3: General rules

Supplementary rules for cold-formed members and sheeting


[14] Eurocode 3


Part 1.5 : Plated structural elements

EN 1993-1-5 : 2006/AC:2009 Corrigendum

[15] Essentials of Eurocode 3


ECCS - N° 65, 1991

[16] Commentary and Worked Examples to EN 1993-1-5 “Plated Structural Elements”

Johansson B., Maquoi R., Sedlacek G., Müller C., Beg D.,

JRC - ECCS, 2007.

[17] SN001a-EN-EU

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

Access Steel, 2006

www.access-steel.com

[18] AISI S100-2007

North American Specification for the Design of Cold-Formed Steel Structural Members

2007 edition

[19] E. Kahlmeyer

Stahlbau nach DIN 18 800 (11.90)

Werner-Verlag, Düsseldorf

http://www.access-steel.com/


94

[20] Beuth-Kommentare

Stahlbauten

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

Beuth Verlag, Berlin-Köln 1993

[21] D. Vandepitte

Berekening van Constructies

Boekdeel 1 pp522

www.berekeningvanconstructies.be

[22] Design rule for Lateral Torsional Buckling of Channel Sections

A-2007.9 O-2007.21

Karin de Louw

2007

[23] EN 12811-1

Temporary works equipment

Part 1: Scaffolds – performance requirements and general design

2004

[24] EN 12810-1

Façade scaffolds made of prefabricated components

Part 1: Products specifications

2004

[25] EN 12810-2

Façade scaffolds made of prefabricated components

Part 2: Particular methods of structural design

2004

[26] DIN 4420 Teil 1

Arbeits- und Schutzgerüste

Allgemeine Regelungen, Sicherheitstechnische Anforderungen, Prüfungen

Dezember 1990

[27] Corrections and amendments to EN 1993-1-3

Meeting of ECCS-TWG 7.5

T. Höglund

2010

[28] Déversement élastique d’une poutre à section bi-symétrique soumise à des moments d’extrémité et une charge répartie ou concentrée.

Y. Galéa

CTICM, Construction Métallique, n° 2-2002.

http://www.berekeningvanconstructies.be/


95

[29] Lateral-Torsional buckling of steel beams:

A general expression for the moment gradient factor.

A. López, D. J. Yong, M. A. Serna

Stability and Ductility of Steel Structures, 2006.

[30] SC001a-EN-EU

Code commentary: Collection No. 1

Access-Steel, 2007.

[31] SN005a-EN-EU

Determination of moments on columns in simple construction

Access-Steel, 2005.

[32] Steel Building Design

Medium Rise Braced Frames

SCI PUBLICATION P365.

[33] Target specification Dimensioning Profiles

ZEMAN & CO. GmbH

Wien, 2006.

[34] New proposals for EN 1993-1-5, Annex D:

Plate girders with corrugated webs.

H. Pasternak, J. Robra, G. Kubieniec

IABSE-FIB Conference, Dubrovnik, 2010.

[35] Zulassung Nr. Z-8.22-208

Modulsystem "CUPLOK"

Deutsches Institut für Bautechnik, 2006.

[36] Zulassung Nr. Z-8.22-64

Modulsystem "Layher-Allround"

Deutsches Institut für Bautechnik, 2008.


96

DIN18800

DIN18800 Code check


DIN 18800 Teil 1

Stahlbauten

Bemessung und Konstruktion

DK 693.814.014.2, November 1990

DIN 18800 Teil 2

Stahlbauten

Stabilitätsfälle, Knicken von Stäben und Stabwerken

DK 693.814.074.5, November 1990

DIN 18800 Teil 3

Stahlbauten

Stabilitätsfälle, Plattenbeulen

DK 693.814.073.1, November 1990

Material properties


thickness of the element (see Ref. 1, Tab.1)

The standard steel grades are :


t<=40 t<=40 40<t<=80 40<t<=80

fy fu fy fu

S235

S 235

St 37-2

240 360 215 360

S275

S 275

280 430 255 430

S355

S 355

St 52-3

360 510 325 510


97

t<=40 t<=40 40<t<=100 40<t<=100

fy fu fy fu

S420

S 420

420 520 390 520

S460

S 460

460 550 430 550

Consulted articles

For the section check, the cross section is classified according to DIN18800 Teil I, Table 12,13,14,15 and 18.. Depending on this classification, the section is checked as slender section, EL/EL (elastic/elastic), as EL/PL (elastic/plastic) or as PL/PL (plastic/plastic).

For the EL/EL check, DIN18800 Teil I, Element (746), (747), (748), (749), (750) are used.

The EL/PL check takes the rules from DIN18800 Teil I, Element (756), (757) and Table (16) ,(17). The PL/PL check is done according to DIN18800 Teil I, Element (758), Table (16),(17).

The slender cross section is checked according to DIN18800 Teil 2, Element (715).

For the stability check, the beam element is checked according to DIN18800 Teil 2 for buckling, lateral torsional buckling and bending and compression. The following criteria are used :

compression : Element (304),(306)

lateral torsional buckling : Element (311),(309)

bending and axial compression : Element (313),(321),(322)

bending (LTB) and compression : Element (320),(323)

For slender sections, the following criteria are used :

calculation of effective area : Element (705),(706),(708),(709),(712),(713)

buckling check : Element (715),(716),(718),(719)

LTB check : Element (725),(726),(728),(729)

For the shear buckling check, the beam element is checked according to DIN18800 Teil 3. The following criteria are used : Element (113), (504), (602),(603)

A more detailed overview for the used articles is given for the relevant parts following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.


98

Teil 1

7.5. Verfahren beim Tragsicherheitsnachweis Nachweise (*)

7.5.1. Abgrenzungskriterien und Detailregelungen (*)

7.5.2. Nachweis nach dem Verfahren Elastisch-Elastisch

(745)………………………………………………………………………………

(746) ………………………………………………………………………………

(747) ………………………………………………………………………………

(748) ………………………………………………………………………………

(749) ………………………………………………………………………………

(750) ………………………………………………………………………………

x

x

x

x

x

x

x

Nachweis nach dem Verfahren Elastisch-Plastisch

(753) ………………………………………………………………………………

(756) ………………………………………………………………………………

(757) ………………………………………………………………………………

x

x

x

x

Nachweis nach dem Verfahren Plastisch-Plastisch

(758) ………………………………………………………………………………

x

x

Teil 2

3.2. Planmässig mittiger Druck

3.2.1. Biegeknicken

(304) ………………………………………………………………………………

x

x

x (*)

3.2.2. Biegedrillknicken

(306) ………………………………………………………………………………

x

x (*)

3.3. Einachsige Biegung ohne Normalkraft

3.3.1. Allgemeines

(307) ………………………………………………………………………………

x

x

x

3.3.2. Behinderung der Verformung

(309) ………………………………………………………………………………

x

x (*)

3.3.3. Nachweis des Druckgurtes als Druckstab


(311) ………………………………………………………………………………

x

x (*)

3.4. Einachsige Biegung mit Normalkraft

3.4.1. Stäbe mit geringer Normalkraft

(312) ………………………………………………………………………………

x

x

x

3.4.2. Biegeknicken

(314) ………………………………………………………………………………

x

x


(320) ………………………………………………………………………………

x

x

3.5. Zweiachsige Biegung mit oder ohne Normalkraft

3.5.1. Biegeknicken

(321) ………………………………………………………………………………

(322) ………………………………………………………………………………

x

x

x

x(*)


(323) ………………………………………………………………………………

x

x


99

4. Mehrteilige, einfeldrige Stäbes

4.1. Allgemeines

4.2. Häufig verwendete Formelzeichnen

(404) ………………………………………………………………………………

4.3. Ausweichen rechtwinklig zur stofffreien Achse

(405) ………………………………………………………………………………

(406)……………………………………………………………………………….

(408)……………………………………………………………………………….

(409)……………………………………………………………………………….

x(*)

x

x

x

x

x

7. Planmässig gerade Stäbe mit ebenen dünnwandigen Quenschnittsteilen

7.1. Allgemeines

(701) ………………………………………………………………………………

(702) ………………………………………………………………………………

(704) ………………………………………………………………………………

x

x

x

x

x

7.2. Berechnungsgrundlage

(705) ………………………………………………………………………………

(706) ………………………………………………………………………………

(707) ………………………………………………………………………………

(708) ………………………………………………………………………………

(709) ………………………………………………………………………………

x

x

x

x

x

x

7.3. Wirksame Breite beim Verfahren Elastisch-Elastisch

(711) ………………………………………………………………………………

(712) ………………………………………………………………………………

(713) ………………………………………………………………………………

x

x

x (*)

x

7.4. Wirksame Breite beim Verfahren Elastisch-Plastisch

7.5. Biegeknicken

7.5.1. Spannungsnachweis beim Verfahren Elastisch-Elastisch

(715) ………………………………………………………………………………

x

x

x

7.5.2. Vereinfachte Nachweise

(716) ………………………………………………………………………………

(718) ………………………………………………………………………………

(719) ………………………………………………………………………………

(721) ………………………………………………………………………………

x

x

x

x

x

7.6. Biegedrillknicken

(722) ………………………………………………………………………………

(723) ………………………………………………………………………………

(725) ………………………………………………………………………………

(726) ………………………………………………………………………………

(728) ………………………………………………………………………………

(729) ………………………………………………………………………………

x

x

x

x

x

x

x

Teil 3

5. Nachweise

(504) ………………………………………………………………………………

(*)

x


100

6. Abminderungsfaktoren

(601) ………………………………………………………………………………

(602) ………………………………………………………………………………

x

x

x





Net area properties

The net area properties are not taken into account .

The holes for fasteners are neglected.

Plastic interaction formula for RHS section

b

s/2

h

AG

AS/2


101

For RHS section, classified as Plastic-Plastic or Elastic-Plastic, the plastic interaction formula according to Ref.[13], can be selected.

Used variable :

A sectional area

AS = s h

AG = (A-AS)/2.0

Wel,y elastic section modulus around y axis

Wel,z elastic section modulus around z axis

fy,d yield strength

y,d shear strength

Vz,pl,Rd = AS y,d

Vy,pl,Rd = 2AG y,d

NSd normal force

My,Sd bending moment around y axis

Mz,Sd bending moment around z axis

Vy,Sd shear force in y direction

Vz,Sd shear force in z direction

MT,Sd torsional moment

2

Rd,pl,z

Sd,T

Sd,z

z

z

Rd,pl,z

Sd,T

Sd,z

V

b

MV

1else

0.14

1

V

b

MV

if


102

2

Rd,pl,y

Sd,T

Sd,y

y

y

Rd,pl,y

Sd,T

Sd,y

V

h

MV

1else

0.14

1

V

h

MV

if

Ar= zAS + 2yAG

r

Sz

A

A

Npl,Rd = Ar fy,d

ydy,elRd,plRd,pl,y fW25.1,hN

4

2minM

ydz,elRd,plRd,pl,z fW25.1,bN

4

1minM

Rd,pl

Sd

N

Nn

Rd,pl,y

Sd,y

yM

Mm

Rd,pl,z

Sd,z

zM

Mm

The following interaction formula are checked :


103


104

Plastic interaction formula for CHS section

For CHS section, classified as Plastic-Plastic or Elastic-Plastic, the plastic interaction formula according to Ref.[14], Tafel 6.74, is used :

selQ,plQ,pl

srQ,pl

r

2

pl

v

pl

v

pl

v

spl

2

z

2

yv

2

z

2

yv

plQ

vQ,pl

v

W25.1,Nd

minM

AN

dtA

Q

Q1:

4

1

Q

Q

1:4

1

Q

Q

3

dt2Q

MMM

QQQ

1

2N

Ncos

1

M

M

with Qy,Qz internal shear force

Nv internal normal force

My,Mz internal bending moments

s yield strength

d,t dimensions from CHS

Wel elastic section modulus


105

t

d

Torsion check


The stability check (DIN 18800 T2, formula 28 & 30) for doubly symmetric I section becomes (Ref.[9], pp. 259) :

)30(0.1kM

MMk

M

M

N

N

)28(0.1kM

MMk

M

M

N

N

z

d,z,pl

w,zz

y

d,y,plM

y

d,plz

z

d,z,pl

w,zz

y

d,y,pl

y

d,pl

with Mz,w

h

M2 w

Mw bimoment (see chapter 'Standard diagrams for warping torque, bimoment and the St.Venant torsion')

kz = 1.50 In case there is no compression force kz is taken as 1.00 (Ref.[9], pp. 270).

Built-in beams

For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter ‘Section check for built-in beams (IFB, SFB, THQ sections)’


106

Calculation of the buckling length

For the calculation of the buckling length, we refer to chapter "Calculation of buckling ratio".

The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see “Calculation of critical Euler force for VARH elements”).


Torsional buckling

The slenderness for torsional buckling vi is given by (see Ref.6 , 7.5):

with l0 the torsional buckling length, refers to the input value for the system length lyz

lz the system length for buckling around zz-axis

Remark : the z-axis refers to the axis which goes through the shear force centre.

z refers to the buckling ratio around the zz-axis

Remark : the z-axis refers to the axis which goes through the shear force centre.

0 refers to end warping and is input by the value kxy

zM the shear center

iy the radius of gyration around major axis

iz the radius of gyration around minor axis

ip² = iy² + iz²

iM² = ip² + zM²


Iz the moment of inertia around minor axis


With this slenderness vi and the buckling curve c, the reduction factor is calculated.


107

Use of diaphragms

(see also Ref.7,3.5 and Ref.8,3.3.4.)

The shear stiffness S for diaphragm is calculated as follows:

L

K+K

10a.=S

s

21

4

with a the frame distance

Ls the length of diaphragm

K1 factor K1

K2 factor K2

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

G

lvorhCII

2

2

tid,t

with l the LTB length

G the shear modulus

vorh

C

the actual rotational stiffness of diaphragm


108

LTB Check

For aysmmetric I sections, RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, the elastic critical moment for LTB Mcr is given by the general formula F.2. Annex

F Ref. 4. For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".

Depending on the input of the basic data, Mcr for symmetric I sections is given by the general

formula F.2. Annex F Ref. 4, by the DIN formula (19), or by formula according to Ref.[11] "Roik, Carl, Lindner, Biegetorsionsprobleme gerader dünnwandiger Stäbe, Verlag von Wilhelm Ernst & Sohn, 1972".

DIN formula (19) :

with l,l0 the LTB length

z refers to rotational end-restraint ‘in plan’ (about the z-z local axis).

0 refers to end warping

zp the point of load application


Iz the moment of inertia around minor axis


A the sectional area


vi the slenderness for torsional buckling ( see above)

the moment factor ( equivalent for factor C1)


109

Roik, Carl & Lindner

z

tw

p

2

pzcry,ki

I

I²l039.0Ic

²

z5²c

²

z5

²l

²EIMM

with moment factor according to Roik, Carl, Lindner

E modulus of elasticity

I

z moment of inertia around weak axis zz

l system length for LTB

z

p application point for loading, negative value is on top and has negative influence

I

w warping constant

I

t torsional constant

The factor is supported for the following cases (described in Ref.[11], tables 5.13, 5.14, 5.15, 5.18, 5.19, 5.20, 5.21, 5.22, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28, 5.29, 5.30, 5.33) :

Linear moment distribution :

Moment line according to distributed loading


110

Moment line according to concentrated loading


111


z

2

t

2

z

2

z

2

EI

GIL

I

Iw

L

EIMcr


G the shear modulus







For full rectangular sections the value of n according to DIN 18800-2 tabelle 9 is taken as 1,5 according to Ref.[8] pp 175.


Combined flexion for check method 2

The value My is the maximum value of the bending moment around the strong axis in the member. The value Mz is the maximum value of the bending moment around the weak axis in the member.

For non-prismatic sections, the values My and Mz are the concurrent bending moments for each intermediary section.


112



(1) 2I

(2) 2Uo

(3) 2Uc


The following additional checks are performed :

- buckling resistance check around weak axis of single chord with NG

- section check of single chord, using internal forces (Ref.[7], pp.88-95) :

4

amaxVM

2

maxV V

W

A)

l

asin(Mmax

2

N N

y

G

y

G

*

z

GzG

- section check of single batten, using the internal forces (Ref.[7], pp.88-95) :

2

TeM

2h

amaxV T

y

y

For the calculation of maxVy, the value of Mz is increased with the value of the internal force Mzz.


113

l

a

hy

e

Effective area properties

The calculation of the effective area is performed with the direct method (sigma_d = fy,k) according to the El-El procedure (DIN18800 T2, 7.3.).


For each load case and combination, the most critical effective area properties are saved. The most critical effective area properties are the effective area properties on the position where the appropriate moment of inertia is the minimum.




114



Shear buckling check with buckling influence

The influence of the buckling effect into the shear buckling control, is neglected when there is a

bending moment present.

It means that k=1 if <0.9. See also Ref.[3], Element 503.

Cold formed thin gauge members

The following table includes a list of DASt-Richtlinie 016 (Ref.[12]) elements which are implemented in Scia Engineer by using the related DIN18800 T2 (Ref.[2]) element.

Supported elements from

DASt - Richtlinie 016

Covered by DIN 18800 T2 elements

Remarks

3.7.1. Grenzzustand der Tragfähigkeit

328 Tab.26

329 712

330 712

333 Tab.27

335 706

4.3.1. Biegemomententragfähigkeit

404 715

4.4. Biegedrillknicken biegebeanspruchter Bauteile

4.4.3. Allgemeiner Nachweis

421 311

422 311

423 725, 726

4.5. Druckbeanspruchte einteilige Stäbe

4.5.1. Allgemeines

429 708-710

430 708-710

431 708-710

432 708-710


115

433 708-710

434 708-710

4.5.2. Planmäig mittiger Druck

435 716 AD

ef is not used

436 manual input / input in profile library for KSL

437 723

438 723

4.5.3. Einachsige Biegung mit Druck

440 707

441 718

442 728

4.5.3. Zweiachsige Biegung mit Druck

443 707

444 721 AD

ef is not used

445 729

Supported sections


RHS Rectangular Hollow Section (RHS)

CHS Circular Hollow Section (CHS)

L Angle section

U Channel section

T T section



Cold formed section


O Solid tube



".



116

I

RHS CHS L U T PPL RS O COM NUM

Classification x x x x x x x x x (1) (1) (1)

Section check PL-PL x x

Section check EL-PL x x

Section check EL-EL x x x x x x x x x x x x

Section check slender section

x x x x x x

Stability check x x x x x x x x x x x x

Stability check slender section

x x x x x x


(1) sections are classified as EL-EL cross section by default.

References

1 DIN 18800 Teil 1

Stahlbauten


DK 693.814.014.2, November 1990

2 DIN 18800 Teil 2

Stahlbauten


DK 693.814.074.5, November 1990

3 DIN 18800 Teil 3

Stahlbauten


DK 693.814.073.1, November 1990

[4] Eurocode 3



ENV 1993-1-1:1992, 1992

[5] R. Maquoi



[6] G. Hünersen, E. Fritzsche

Stahlbau in Beispielen

Berechnungspraxis nach DIN 18 800 Teil 1 bis Teil 3 (11.90)

Werner-Verlag, Düsseldorf 1991


117

[7] E. Kahlmeyer




Stahlbauten



[9] Stahlbau Kalender 1999

DSTV

Ernst & Sohn, 1999

[10] Eurocode 3



ENV 1993-1-1:1992/A1, 1994

[11] Roik, Carl, Lindner

Biegetorsionsprobleme gerader dünnwandiger Stäbe

Verlag von Wilhelm Ernst & Sohn

1972

[12] DASt-Richtlinie 016

Bemessung und konstruktive Gestaltung von Tragwerken aus dünnwandigen kaltgeformted Bauteilen

Stahlbau-Verlagsgesellschaft - 1992

[13] H. Rubin,

Interaktionsbeziehungen für doppeltsymmetrische I- und Kasten-Querschnitte bei zweiachsiger Biegung und Normalkraft

Der Stahlbau 5/1978, 6/1978

[14] Stahl im Hochbau

14. Auflage, Band I / Teil 2

1986, Verlag Stahleisen mbH, Düsseldorf


118

ONORM B 4300

ONORM B 4300 Code check


ÖNORM B 4300-1

Stahlbau

Berechnung und Konstruktion der Tragwerke

Bemessung nach Grenzzuständen

DK 624.014.2.046, März 1994

ÖNORM B 4300-2

Stahlbau

Knicken von Stäben und Stabwerken

Bedingungen für die gemeinsame Anwendung von DIN 18 800 Teil 2 und ÖNORM B 4300-1

DK 624.014.2.075.2, April 1994

ÖNORM B 4300-3

Plattenbeulen


DK 624.014.2.075.4, April 1994

DIN 18800 Teil 1

Stahlbauten


DK 693.814.014.2, November 1990

DIN 18800 Teil 2

Stahlbauten


DK 693.814.074.5, November 1990

DIN 18800 Teil 3

Stahlbauten


DK 693.814.073.1, November 1990


119

Material properties


thickness of the element (see Ref. 1, 2.1. and Ref. 4, Tab.1)

The standard steel grades are:


t<=40 t<=40 40<t<=80 40<t<=80

fy fu fy fu

St 360

S235

S 235

240 360 215 360

St 430

S275

S 275

280 430 255 430

St 510

S355

S 355

360 510 325 510

t<=40 t<=40 40<t<=100 40<t<=100

fy fu fy fu

S420

S 420

420 520 390 520

S460

S 460

460 550 430 550

Consulted articles

For the section check, the cross section is classified according to ONORM B 4300-1 Tab.3,4,5 and to DIN18800 Teil I, Table 15,18. Depending on this classification, the section is checked as slender section, EL/EL (elastic/elastic), as EL/PL (elastic/plastic) or as PL/PL (plastic/plastic).

For the EL/EL check, ONORM B 4300-1 Art. 5.2. is used. (The 7% increase of the moment of

inertia is taken into account for rolled I section - see Ref. 1, Art. 5.2.5.4.).

The EL/PL check takes the rules from DIN18800 Teil I, Element (756), (757) and Table (16) ,(17). The PL/PL check is done according to DIN18800 Teil I, Element (758), Table (16),(17).

The slender cross section is checked according to DIN18800 Teil 2, Element (715).


120

For the stability check, the beam element is checked according to DIN18800 Teil 2 for buckling, lateral torsional buckling and bending and compression. The following criteria are used :

compression : Element (304),(306)

lateral torsional buckling : Element (311),(309)

bending and axial compression : Element (313),(321),(322)

bending (LTB) and compression : Element (320),(323)

For slender sections, the following criteria are used :

calculation of effective area : Element (705),(706),(708),(709),(712),(713)

buckling check : Element (715),(716),(718),(719)

LTB check : Element (725),(726),(728),(729)

For the shear buckling check, the beam element is checked according to DIN18800 Teil 3. The following criteria are used : Element (113), (504), (602),(603)

A more detailed overview for the used articles is given in "DIN18800 Code check".

Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube

NUM Numerical sections


".



121

I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM


Section check PL-PL x

Section check EL-PL x

Section check EL-EL x x x x x x x x x x x x

Section check slender section

x x x x x x

Stability check x x x x x x x x x x x x

Stability check slender section

x x x x x x


(1) sections are classified as EL-EL cross section by default.

References

1 ÖNORM B 4300-1

Stahlbau

Berechnung und Konstruktion der Tragwerke

Bemessung nach Grenzzuständen

DK 624.014.2.046, März 1994

2 ÖNORM B 4300-2

Stahlbau

Knicken von Stäben und Stabwerken


DK 624.014.2.075.2, April 1994

3 ÖNORM B 4300-3

Plattenbeulen


DK 624.014.2.075.4, April 1994

[4] DIN 18800 Teil 1

Stahlbauten


DK 693.814.014.2, November 1990

[5] DIN 18800 Teil 2

Stahlbauten


DK 693.814.074.5, November 1990


122

[6] DIN 18800 Teil 3

Stahlbauten


DK 693.814.073.1, November 1990


123

NEN

NEN6770/6771 Code check


Staalconstructies TGB 1990

Basiseisen en basisrekenregels voor overwegend statisch belaste constructies

NEN 6770, december 1991

Staalconstructies TGB 1990

Stabiliteit

NEN 6771, december 1991-januari 2000

Material properties


thickness of the element (see Ref. 1, art.9.1.2.1.1.)

The standard steel grades are :


t<=40 t<=40 40<t<=100 40<t<=100 100<t<=250 100<t<=250

fy fu fy fu fy fy

S235

S 235

235 360 215 340 175 320

S275

S 275

275 430 255 410 205 380

S355

S 355

355 510 335 490 275 450

S420

S 420

420 520 390 520

S460

S 460

460 550 430 550

Remark : For cold formed section, the values for fy and fu are not influenced by the previous table.


124

Consulted articles

The cross section is classified according to NEN 6771 Table 1. (Class 1, 2, 3 or 4).

The section is checked on following criteria:

Tension: NEN 6770 Art. 11.2.1., NEN 6771 Art. 11.2.1.

Compression: NEN 6770 Art. 11.2.2., NEN 6771 Art. 11.2.2.

Shear: NEN 6770 Art. 11.2.4., NEN 6771 Art. 11.2.4.

Bending, shear and axial force: NEN 6770 Art. 11.3., NEN 6771 Art. 11.3.

For the stability check, the element is checked on following criteria:

Compression: NEN 6771 Art.12.1.1.1/ 12.1.2./12.1.3.

Lateral torsional buckling : NEN 6771 Art.12.2.

Bending and axial compression: NEN 6771 Art.12.3.

Shear buckling : NEN 6771 Art.13.8. / 13.9.

A more detailed overview for the used articles is given for NEN6770 part 11,12 and NEN6771 part 10,11,12,13. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.

NEN6770

11.Toetsing van de doorsnede

11.1. Algemeen

x

x

11.2. Enkelvoudige krachten en momenten

11.2.1. Axiale trek

x

x

11.2.2. Axiale druk x

11.2.3. Buiging

11.2.4. Afschuiving x

11.2.5. Torsie x

11.3. Combinaties van krachten en momenten

11.3.1. Enkele buiging met normaalkracht en afschuiving

x

x

11.3.2. Dubbele buiging met normaalkracht en afschuiving x

11.4. Vloeicriterium x

11.5. De invloed van de boutgaten (*)

NEN6771

10.2.4. Doorsneden x (*)

11.Toetsing van de doorsnede

11.1. Algemeen

x

x

11.2. Enkelvoudige krachten en momenten

11.2.1. Axiale trek

x

x

11.2.2. Axiale druk x


125

11.2.3. Buiging

11.2.4. Afschuiving x

11.2.5. Torsie

11.3. Combinaties van krachten en momenten x

12. Toetsing van de stabiliteit

12.1. Op druk belaste staven

12.1.1. Knikstabiliteit

x

x

x (*)

12.1.2. Torsiestabiliteit x

12.1.3. Torsieknikstabiliteit x

12.1.4. Verend gesteunde staven

12.1.5. Staven in vakwerken

12.1.6. Samengestelde staven

12.1.6.1 Algemeen

12.1.6.2. Benodigde grootheden

12.1.6.3. Toetsing van het middenveld van de samengestelde staaf

12.1.6.4. Toetsing van de eindvelden van de samengestelde staaf

12.1.6.4.2 Staven met raamwerkverband

x(*)

x

x

x

x

x

12.2. Op buiging belaste staven(kipstabiliteit)

12.2.1. Toepassingsgebied

xx

x

12.2.2. Toetsingsregel x

12.2.3. Ongesteunde lengte

12.2.4. Opleggingen en zijdelingse steunen

12.2.5. Het theoretisch elastische kipmoment x (*)

12.3. Op druk en buiging belaste staven

12.3.1. Knikstabiliteit

x

x

12.3.2. Torsiestabilteit x

12.3.3. Torsieknikstabiliteit x

12.4. Op trek en buiging belaste staven

13. Toetsing van de plooistabiliteit

13.1. Algemeen

x

x

13.2. Geometrie van het verstijfde en onverstijfde plaatveld x

13.3. Geometrie van de verstijvingen

13.4. Belasting in het vlak van het plaatveld

13.4.1. Normaalspanning in langsrichting

x

x

13.4.2. Schuifspanningen x

13.4.3. Normaalspanningen in dwarsrichting

13.4.4. Platen in en loodrecht op hun vlak belast

13.5. Belasting op verstijvingen

13.6. Ideële kritieke plooispanning van een onverstijfd plaatveld x

13.7. De plooispanning van een onverstijfd plaatveld

13.7.1. Bepaling van de relatieve slankheid van het plaatveld

x

x


126

13.7.2. De plooispanning voor een onverstijfd plaatveld met als opleggingen dwarsverstijving(en) en/of randen

x

13.7.3. De plooispanning voor een onverstijfd plaatveld met ten minste een langsverstijving als oplegging

13.8. Eisen waaraan plaatvelden en verstijvingen moeten voldoen

13.8.1. Onverstijfd plaatveld

x

x

13.8.2. Dwarsverstijvingen

13.8.3. Langsverstijvingen

13.8.4. Stijfheidseisen te stellen aan langs- en dwarsverstijvingen

13.8.5. Doorsnedecontrole voor langs- en dwarsverstijvingen

13.9. Interactie tussen plooi en knik

13.9.1. Algemeen

x (*)

x

13.9.2. Constructies opgebouwd uit plaatvelden al of niet verstijfd met dwarsverstijvingen

x

13.9.3. Constructies opgebouwd uit plaatvelden verstijfd met langsverstijvingen en/of niet verstijfd met dwarsverstijvingen

13.9.4. Berekeningen van de dwarsverstijvingen

Section properties

The influence of the bore hole is neglected.












127


For angle sections, see chapter 'Effective cross-section properties for compressed lattice tower angle members'.

Torsion check


Built-in beams

For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter ‘Section check for built-in beams (IFB, SFB, THQ sections)

Buckling length

For the calculation of the buckling length, we refer tochapter "Calculation of buckling ratio".




For symmetric I sections and RHS (Rectangular Hollow Section) sections, the elastic critical moment

for LTB Mcr is given by the formula of Ref 2, part 12.2.5.. When the factor > 5000, the elastic

critical moment for LTB Mcr is given by the general formula in EC3, Annex F, F.2. Ref 3. For asymmetric I sections, the elastic critical moment for LTB Mcr is given by the general formula in

EC3, Annex F, F.2. Ref 3.

For the calculation of the moment factors C1, C2 and C3 we refer to Ref.[7], tables 9 (case 1), 10 and 11.


z

2

t

2

z

2

z

2

EI

GIL

I

Iw

L

EIMcr


G the shear modulus






128




Use of diaphragms



129



(1) 2I

(2) 2Uo

(3) 2Uc


The following additional checks are performed :

- buckling resistance check around weak axis of single chord with Nf,s;d

- section check of single chord, using internal forces :

4

aQM

2

Q V

N N

f;s;d

G

f;s;d

G

f;s;dG

- section check of single batten, using the internal forces :

4

aQM

2h

aQ V

ds;f;

ds;k;

0

ds;f;

ds;k;

For the calculation of Qf;s;d, the value of My;s;d is increased with the value of the internal force Mzz.


130

l

a

ho



Shear buckling check with buckling influence

The influence of the buckling effect into the shear buckling control, is neglected when there is a

bending moment present, i.e. if <0.9.


131

NEN6072 - Fire Resistance For more info, reference is made to to Ref.[8], Ref.[9].

Fire actions effect

The design effects of actions for the fire situation are taken from the results of the analysis. It is recommended to use the special combination rules according to Ref.[10], NEN6702 6.2.2., for calculating the internal forces used in the fire resistance check.

This special combination is given by

rep;aa;frep;iiq;frepg;f FQG

with Grep characteristic values of permanent actions

Qi characteristic value of the variable action

Fa;rep design values of special action (from fire exposure)

f;g partial safety factor for permanent actions in the special combination

=1.0

f;q partial safety factor for variable actions in the special combination

=1.0

f;a partial safety factor for special actions in the special combination

=1.0

I the 'momentaaan' factor for the variable action

Material properties

The yield strength is depending on the steel temperature :

d;yd;;y ff

The variation in function of the steel temperature of the value for yield strength is given by :

- =1.0 when a 400° C

- when 400°C < a 1200° C


132

with

2.39

482a

a steel temperature in °C

fy;d design value for yield strength at room temperature

fy;;d design value for yield strength at increased temperature

The following default properties are considered to be constant during the analysis :



(a-20)



The standard temperature-time (ISO 834) curve is used :

)1t8(log34520 10g



Steel Temperature

The increase of temperature a in an unprotected steel member during a time interval t



P = Am/V

t gas temperature in [°C]


133

a steel temperature [°C]




r resultant emissivity

= 0.5


= 25 W/(m²K)

The increase of temperature a in an insulated (non intumescent coating) steel member during a time

interval t



Pi = Ap/V


ci the specific heat of fire protection material [J/kgK]

di the thickness of the fire protection material [m]




i the unit mass of fire protection [kg/m³]

a the steel temperature at time t

t the ambient gas temperature at time t

t the increase of the ambient gas temperature during the time interval

i;d;ef the thermal conductivity of the fire protection material [W/mK]


134

The increase of temperature a in an insulated (intumescent coating) steel member during a time interval

t



Pi = Ap/V


Kd;ef coefficient of heat transfer of the intumescent coating




a the steel temperature at time t

t the ambient gas temperature at time t

i;d;ef the thermal conductivity of the fire protection material [W/mK]


135

Calculation model


- strength domain


In the strength domain, the strength (unity check) is calculated after a given time t (e.g. strength

after 45 min). In the temperature/time domain, the critical steel temperature a,cr is computed. From this critical temperature, the fire resistance time is calculated (the time domain).

The critical steel temperature a,cr is given by :

with degree of utilization at time t=0

correction factor

= 1.00 for tension elements

= 1.00 for beams, statically determined, 4 side exposure

= 0.70 for beams, statically determined, 3 side exposure

= 0.85 for beams, statically undetermined, 4 side exposure

= 0.60 for beams, statically undetermined, 3 side exposure

= 1.20 for compression elements (inclusive the buckling check)

= 1.20 for compression and bending elements (inclusive the buckling and LTB check)

Code Check

The section and stability checks (buckling, lateral torsional buckling) are performed according to the regulations given in NEN6770/6771, adapted with the yield strength for the increased temperature and the correction factor. The checks are performed in the resistance domain or in the temperature/time domain. Shear buckling is not considered.


136

Supported sections




L Angle section

U Channel section

T T section


Z Z section


Cold formed section


O Solid tube



". The COM and NUM sections are not read out of the profile library.

I

RHS

CHS

L

U

T

PPL

RS

Z

O

COM

NUM







x x x


x x x




x x x x x x




137

References 1 Staalconstructies TGB 1990



2 Staalconstructies TGB 1990

Stabiliteit


3 Eurocode 3



ENV 1993-1-1:1992, 1992

[4] R. Maquoi



[5] Eurocode 3



ENV 1993-1-1:1992/A1, 1994

[6] ENV 1993-1-3:1996




CEN 1996

[7] Staalconstructies TGB 1990

Stabiliteit

NEN 6771, januari 2000

[8] NEN 6072

Rekenkundige bepaling van de brandwerendheid van bouwdelen

Staalconstructies

December 1991

[9] NEN 6072/A2 - Wijzigingsblad

Rekenkundige bepaling van de brandwerendheid van bouwdelen

Staalconstructies

December 2001

[10] NEN 6702

Belastingen en vervormingen TGB 1990

December 1991


138

AISC – ASD : 1989

AISC - ASD Code check


Manual of Steel Construction

Allowable Stress Design

Part 5 : Specification and Codes

AISC, Ninth Edition, 1989

The cross section is classified according to Table B5.1. (compact, non compact, or slender section).

The member is checked on following criteria:

tension : D1

compression : E2, E3

flexural members : F1,F2,F3,F4

plate girders : G2

combined forces : H1,H2

A more detailed overview for the used articles of the relevant parts is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.

B. DESIGN REQUIREMENTS

B1. Gross Area x

B2. Net Area (*)

B3. Effective Area

B4. Stability

B5. Local Buckling

1.Classification of Steel Sections

2.Slender Compression Elements

(*)

x

x

B6. Rotational Restraint at Points of Support

B7. Limiting Slenderness Ratios x

B8. Simple Spans

B9. End Restraint

B10. Proportions of Beams and Girders

B11. Proportioning of Crane Girders


139

D. TENSION MEMBERS

D1. Allowable Stress x (*)

D2. Built-up members

D3. Pin-Connected Members

E. COLUMN AND OTHER COMPRESSION MEMBERS

E1. Effective Length and Slenderness Ratio x (*)

E2. Allowable Stress x

E3. Flexural-torsional Buckling x (*)

E4. Built-up Members

E5. Pin-Connected Compression Members

E6. Column Web Shear

F. BEAMS AND OTHER FLEXURAL MEMBERS (*)

F1. Allowable Stress : Strong Axis Bending of I-Shaped Members and Channels

1.Members with Compact Sections

2.Members with Non-Compact Sections

3.Members with Compact or Non-Compact Sections with Unbraded Length Greater then Lc

x

x

x

x

F2. Allowable Stress : Weak Axis Bending of I-Shaped Members, Solid Bars and Rectangular Plates



x

x

x

F3. Allowable Stress : Bending of Box Members, Rectangular Tubes and Circular Tubes



x

x

x

F4. Allowable Shear Stress x

F5. Transverse Stiffeners

F6. Built-up Members

F7. Web-tapered Members

G. PLATE GIRDERS

G1. Web Slenderness Limitations

G2. Allowable Bending Stress x

G3. Allowable Shear Stress with Tension Field Action

G4. Transverse Stiffeners

G5. Combined Shear and Tension Stress

H. COMBINED STRESSES

H1. Axial Compression and Bending x

H2. Axial Tension and Bending x


140

APPENDIX B. DESIGN REQUIREMENTS

B5. Local Buckling x


For each intermediary section, the classification is determined..

For each load case/combination, the critical section classification over the member is used to perform the code check. However, for non-prismatic sections, the section classification is determined for each intermediary section.

Section properties

The influence of the bore hole is neglected, i.e. only the gross area is used.

Buckling length

For the calculation of the buckling length, we refer to "Calculation of buckling ratio".


Flexural Torsional Buckling

The slenderness ratio for flexural torsional buckling (KL/r)e is given by

Fe

E

r

KL

e

See Ref. 1, Commentary Chapter E1.

The calculation of Fe is given in Ref. 2, Appendix E.


For I sections and channel sections, the allowable LTB stress is given in F1.

For RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) , the allowable LTB stress is given in F3.

For angle sections with symmetrical legs, the allowable LTB stress is given in Ref. 1, pp.309-314, “Specification for allowable stress - Design of single-angle members”.


141


z

2

t

2

z

2

z

2

EI

GIL

I

Iw

L

EIMcr


G the shear modulus





See also Ref. 4, part 7.

With this moment Mcr, the critical LTB stress LTB is calculated :

y

crLTB

I

M

with Iy the moment of inertia about the major axis

The slenderness ratio for LTB LTB, is given by

LTBLTB

E

The allowable LTB stress is calculated using the slenderness LTB with the formulas given in

Ref.1, E2.

See also Ref. 5, Bijlage E.





142

Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube




I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM

Classification x x x x x x x x x (1)

(1) (1)

Compact section x x x x x

Non-compact section

x x x x x x x x x x x x

Slender section x x x x x x


x x x

(1) sections are classified as non-compact section by default.

References

1 Manual of Steel Construction

Allowable Stress Design

AISC, Ninth Edition, 1989


Load & Resistance Factor Design

AISC, First Edition, 1986


143


Load & Resistance Factor Design

AISC, Volume I, Second Edition, 1995

[4] R. Maquoi



[5] NBN B 51-001

Stalen Bouwconstructies

BIN, 5e uitg. April 1977


144

AISC – LRFD : 2001

AISC - LRFD Code check


AISC – Manual of steel construction

Load and Resistance Factor Design

Part 16 Specifications and Codes

Third Edition

2001


The member is checked on following criteria :

tension : D1

compression : E2, E3, Appendix E3

flexural members : F1,Appendix F1, Appendix F2

plate girders : Appendix G2, Appendix G3, Appendix G5

combined forces : H1,H2



B1. Gross Area x

B2. Net Area (*)

B3. Effective Area for Tension Members

B4. Stability

B5. Local Buckling

1.Classification of Steel Sections

2.Slender Compression Elements

3.Slender-Element Compression Sections

(*)

x

x

x

B6. Bracing at Support

B7. Limiting Slenderness Ratios x

B8. Simple Spans


145

B9. End Restraint

B10. Proportions of Beams and Girders

D. TENSION MEMBERS

D1. Design Tensile Strength x (*)

D2. Built-up members

D3. Pin-Connected Members and Eyebars

E. COLUMN AND OTHER COMPRESSION MEMBERS

E1. Effective Length and Slenderness Limitations

1.Effective Length

2.Design by Plastic Analysis

x

x (*)

E2. Design Compressive Strength for Flexural Buckling x

E3. Design Compressive Strength for Flexural-Torsional Buckling x

E4. Built-up Members

E5. Pin-Connected Compression Members

F. BEAMS AND OTHER FLEXURAL MEMBERS (*)

F1. Design for Flexure

1.Yielding

2.Lateral-Torsional Buckling

x

x

x

F2. Design for Shear x


F4. Beams and Girders with Web Openings

G. PLATE GIRDERS x

H. MEMBERS UNDER COMBINED FORCES AND TORSION

H1. Symmetric Members Subject to Bending and Axial Force x

H2. Unsymmetric Members and Members under Torsion and Combined Torsion, Flexure, Shear and/or Axial Force

x

H3. Alternative Interaction Equation for Members under Combined Stress

APPENDIX B. DESIGN REQUIREMENTS

B5. Local Buckling x

APPENDIX E. COLUMN AND OTHER COMPRESSION MEMBERS

E3. Design Compressive Strength for Flexural-Torsional Buckling x

APPENDIX F. BEAMS AND OTHER FLEXURAL MEMBERS

F1. Design for Flexure x

F2. Design for Shear x


146


APPENDIX G. PLATE GIRDERS

G1. Limitations

G2. Design Flexural Strength x(*)

G3. Design Shear Strength with Tension Field Action x(*)

G4. Transverse Stiffeners

G5. Flexure-Shear Interaction x(*)




Section properties


Buckling length




For I sections, channel sections, RHS (Rectangular Hollow Section) sections, T sections, rectangular sections, and asymmetric I sections, the critical LTB moment is given in F1 and Appendix F1.

For angle sections with symmetrical legs, the critical LTB moment is given in Ref. 1, pp.281-288, “Specification for Load and Resistance Factor Design of Single-Angle members”.


z

2

t

2

z

2

z

2

EI

GIL

I

Iw

L

EIMcr


G the shear modulus






147

See also Ref. 2, part 7.


Use of diaphragms




Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube



".


I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM



Non-compact section




x x x


148

(1) sections are classified as non-compact section by default.

References

1 AISC – Manual of steel construction

Load and Resistance Factor Design

Third Edition

2001

2 R. Maquoi




149

ANSI/AISC 360-05:2005

ANSI/AISC 360-05 Code check


ANSI/AISC 360-05

Specifications for Structural Steel Buildings

2005

The steel code check can be executed according to either ASD or LRFD provisions.


The member is checked on following criteria:

tension : Chapter D

compression : Chapter E

flexural members :Chapter F

shear : Chapter G

combined forces : Chapter H



B2. Loads and Load Combination x

B3. Design Basis

1.Required Strength

2.Limit States

3.Design for Strength using LRFD

4.Design for Strength using ASD

x

x

B4. Classification of Sections for Local Buckling x

D. DESIGN OF MEMBERS FOR TENSION

D1. Slenderness Limitation x

D2. Tensile Strength x

D3. Area Determination x(*)


150

E. DESIGN OF MEMBERS FOR COMPRESSION

E1. General Provisions x

E2. Slenderness Limitations and Effective Length x(*)

E3. Compressive Strength for Flexural Buckling of members without Slender Elements

x

E4. Compressive Strength for Torsional and Flexural-Torsional Buckling of members without Slender Elements

x

E7. Members with Slender Elements x

F. DESIGN FOR MEMBERS FOR FLEXURE

F1. General Provisions x

F2. Doubly Symmetric Compact I-Shaped Members and Channels Bent about their Major Axis

x

F3. Doubly Symmetric I-Shaped Members with Compact Webs and Noncompact or Slender Flanges Bent about Their Major Axis

x

F4. Other I-Shaped Members with Compact or Noncompact Webs Bent about Their Major Axis

x

F5. Doubly Symmetric and Singly Symmetric I-Shaped Members with Slender Webs Bent about Their Major Axis

x

F6. I-Shaped Members and Channels Bent about Their Minor Axis x

F7. Square and Rectangular HSS and Box-Shaped Members x

F8. Round HSS x

F9. Tees and Double Angle Loaded in Plane of Symmetry x

F10. Single Angle x

F11. Reactangular Bars and Rounds x

F12. Unsymmetrical Shapes

G. DESIGN OF MEMBERS FOR SHEAR

G1. General Provisions x

G2. Members with Unstiffened or Stiffened Webs x

G4. Single Angles x

G5. Rectangular HSS and Box Members x

G6. Round HSS x

G7. Weak Axis Shear in Singly and Doubly Symmetric Shapes x

H. DESIGN OF MEMBERS FOR COMBINED FORCES AND TORSION

H1. Doubly and Singly Symmetric Members Subject to Flexure and Axial Force x

H2. Unsymmetric and Other Members Subject to Flexure and Axial Force x

H3. Members Under Torsion and Combined Torsion and Combined Stress x


151




Section properties


Buckling length





Use of diaphragms





152

Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube




I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM



Non-compact section




x x x x x x

(1) Sections are classified as non-compact section by default.

References

1 ANSI/AISC 360-05

Specifications for Structural Steel Buildings

2005


153

AISI NAS S100-2007

AISI NAS S100-2007 Code check


AISI S100-2007


2007 edition

AISI S100-07-E1

Errata to North American Specification for the Design of Cold-Formed Steel Structural Members

2007 edition February 20, 2008

Amended September 25, 2008

Amended June 4, 2009

AISI S100-07/S1-09

Supplement No. 1 to the North American Specification for the Design of Cold-Formed Steel Structural Members, 2007 edition August, 2009

AISI S100-07/S2-10

Supplement No. 2 to the North American Specification for the Design of Cold-Formed Steel Structural Members, 2007 edition February, 2010

The steel code check is supported for the United States provisions and can be executed according to either ASD or LRFD principles. The Canadian LSD method is not supported.

Consulted articles

An 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 paragraphs.

Article Title

A General Provisions

A4 Allowable Strength Design X

A5 Load and resistance Factor Design X

B Elements

B1 Dimensional Limits and Considerations X(*)


154

B2

Effective Widths of Stiffened Elements

B2.1 Uniformly Compressed Stiffened Elements

B2.3 Webs and Other Stiffened Elements under Stress Gradient

X(*)

X(*)

B3 Effective Widths of Unstiffened Elements

B3.1 uniformly Compressed Unstiffened Elements

B3.2 Unstiffened Elements and Edge Stiffeners with Stress Gradient

X(*)

X(*)

B4 Effective Width of Uniformly Compressed Elements with a Simple Lip Edge Stiffener

X(*)

C Members

C1 Properties of Sections X(*)

C2 Tension Members X(*)

C3 Flexural members

C3.1 Bending

C3.1.1 Nominal Section Strength

C3.1.2 Lateral-Torsional Buckling Strength

C3.1.3 Flexural Strength of Closed Cylindrical Tubular Members

C3.1.4 Distortional Buckling Strength

X(*)

X(*)

X(*)

X(*)

C3.2 Shear

C3.2.1 Shear Strength of Webs without Holes

C3.3 Combined Bending and Shear

X(*)

X(*)

C3.4 Web Crippling

C3.4.1 Web Crippling Strength of Webs without Holes

C3.5 Combined Bending and Web Crippling

X(*)

X(*)

C3.6 Combined Bending and Torsional Loading X(*)

C4 Concentrically Loaded Compression Members

C4.1 Nominal Strength for Yielding, Flexural, Flexural-Torsional and Torsional Buckling

C4.2 Distortional Buckling Strength

X(*)

X(*)

C5 Combined Axial Load and Bending

C5.1 Combined Tensile Axial Load and Bending

C5.2 Combined Compressive Axial Load and Bending

X

X(*)

D Structural Assemblies and Systems

D6 Metal Roof and Wall Systems

D6.1 Purlins, Girts and Other Members

D6.1.1 Flexural Members Having One Flange Through-Fastened to Deck of Sheeting

D6.1.3 Compression Members Having One Flange Through-Fastened to Deck of Sheeting

X(*)

X(*)


155

Appendix 2 Second-Order Analysis

2.1 General requirements

X(*)

Haunches, arbitrary members and cross-sections without initial shapes are not supported for the AISI NAS S100-2007 code check. In this case the default AISC 2005 code check is executed.

Initial Shape

For a cross-section with material Steel and fabrication set to Cold-Formed, the Initial Shape can be defined.

For a General cross-section the ‘Thin-walled representation’ has to be used to be able to define the Initial Shape.

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

F Fixed Part – No reduction is needed

I Internal cross-section part

SO Symmetrical Outstand

UO Unsymmetrical Outstand

Parts can also be specified as reinforcement:

None Not considered as reinforcement

RUO Reinforced Unsymmetrical Outstand (edge stiffener)

ROU reinforcement types can be set only to elements of type SO or UO.

The initial shape is supported for the following cross-section types:

- Standard profile library cross-sections

- Cold formed Pair cross-sections of profile library sections

- General thin-walled sections

- General sections with thin-walled representation

- Thin-walled geometric sections

- All other sections which support the centerline and do not have roundings

For standard profile library cross-sections, the flat parts are taken between the roundings. The roundings are set as fixed parts.

For predefined sections without roundings, the initial shape is based on the centreline dimensions i.e. the flat parts are taken between the intersection points of the centrelines.


156

Dimensional limits

Dimensional limits are supported according to article B1.1 and B1.2.

Article B1.1 (a) (1) for a simple lip is checked for an internal element (I) connected to a stiffener

(RUO).

Article B1.1 (a) (2) is checked for an internal element (I).

Article B1.1 (a) (3) is checked for an outstand element (UO or SO).

Articles B1.1 (b) concerning flange curling and (c) concerning shear lag effects are not supported.

Article B1.2 (a) is checked for web elements under stress gradient. Webs are defined as elements

perpendicular (tolerance +/-45°) to the axis of bending.

Effective Widths

Uniformly Compressed Stiffened elements

The effective width of Uniformly Compressed Stiffened elements is calculated according to article B2.1 (a) Strength Determination.

More specifically, this concerns elements of type I with stress gradient = 1

Serviceability Determination is not supported.

Webs and Other Stiffened Elements under Stress Gradient

The effective width of Webs and Other Stiffened elements under stress gradient is calculated according to article B2.3 (a) Strength Determination.

More specifically, this concerns elements of type I with stress gradient ≠ 1


Uniformly Compressed Unstiffened elements

The effective width of Uniformly Compressed Unstiffened elements is calculated according to article B3.1 (a) Strength Determination.

More specifically, this concerns elements of type SO or UO (with or without reinforcement type

RUO) with stress gradient = 1



157

Unstiffened elements and Edge Stiffeners with Stress Gradient

The effective width of Unstiffened elements and Edge Stiffeners with Stress Gradient is calculated according to article B3.2 (a) Strength Determination.

More specifically, this concerns elements of type SO or UO (with or without reinforcement type

RUO) with stress gradient ≠ 1

The alternative methods for unstiffened C-sections are not supported.


Effective width of Uniformly Compressed elements with a Simple Lip Edge Stiffener

The effective width of Uniformly Compressed elements with a Simple Lip Edge Stiffener is calculated according to article B4 (a) Strength Determination.

More specifically, this concerns elements of type I with stress gradient = 1 which are connected to a fixed element (rounding) which in turn is connected to an element of type UO or SO with reinforcement type RUO.


Effective section properties can never be bigger than gross section properties (for example in case of manually inputted gross section properties which have been rounded down).

Properties of Sections

Deductions for holes, openings and cut-outs are not supported.


158

Tension Members

The tensile strength is determined according to article C2.

For yielding in the gross section:

For rupture in the net section:

With: Fy Yield strength

Fu Tensile strength

Ag Gross area of cross-section

An Net area of cross-section

Since deductions for holes, openings … are not supported An = Ag.

Flexural Members

Nominal Section Strength

The nominal section strength is determined according to article C3.1.1. More specifically Procedure I - Based on Initiation of Yielding is applied.

Lateral Torsional Buckling Open Section

The Lateral Torsional Buckling strength for open sections is determined according to article C3.1.2.1 (a).

For diaphragms reference is made to “Use of diaphragms”. The simplified formulas of article C3.1.2.1 (b) are not supported.

Doubly symmetric sections

For Doubly symmetric sections formula (C3.1.2.1-4) is used for either axis.

This applies to the following form codes:

1 (Symmetric I shape)

7 (Rectangular section)

11 (Solid tube)

In addition this applies to the cold formed pair sections 2CFUo, 2CFUc, 2CFCo, 2CFCc


159

Formula (C3.1.2.1-4) is rewritten as follows:

Remarks:

For x-x bending the LTB length is used instead of the effective length KyLy.

For y-y bending Kx is taken as the buckling ratio about the x-axis and Lx the system

length for buckling about the x-axis.

The equation for r0 is expanded to allow any type of cross-section:

Cb for x-x bending is calculated according to formula (C3.1.2.1-6)

Cb for y-y bending is taken as unity.


160

Point symmetric sections

For Point symmetric sections formula (C3.1.2.1-5) is used for either axis.


102 (Z section)

113 (Cold formed Z section)

118 (Cold formed ZED section)

119 (Cold formed ZED section asymmetrical lips)

120 (Cold formed ZED section inclined lip)

126 (Cold formed ZED section both lips inclined)

Formula (C3.1.2.1-5) is rewritten as follows:

The same remarks are valid as for doubly symmetric sections.

Singly symmetric sections

For Singly symmetric sections formula (C3.1.2.1-4) is used for bending about the x-x axis and formula (C3.1.2.1-10) for bending about the y-y axis.


5 (Channel section)

112 (Cold formed Channel section)

114 (Cold formed C section)

117 (Cold formed C-Plus section)

121 (Cold formed Sigma section)

122 (Cold formed Sigma section stiffened)

123 (Cold formed Sigma-Plus section)

Formulas (C3.1.2.1-4) and (C3.1.2.1-10) are written as follows:


161

The same remarks are valid as for doubly symmetric sections.

The parameter j is calculated using the formula for C-sections given in Ref. [4].

Other section types

All other cross-sections which are not covered by the previous paragraphs are considered to be doubly symmetric, except for the following form codes:

2 (Rectangular Hollow Section)

3 (Circular Hollow Section)

Lateral Torsional Buckling Box Section

The Lateral Torsional Buckling strength for box sections is determined according to article C3.1.2.2.

This applies to the following form code:


In addition this applies to the cold formed pair sections 2CFUc and 2CFCc with distance a = 0 mm

Formulas (C3.1.2.2-1) and (C3.1.2.2-2) are rewritten as follows:

The same remarks are valid as for open doubly symmetric sections.

Flexural Strength Closed Cylindrical Tubular members

The Flexural Strength of Closed Cylindrical Tubular members is determined according to article C3.1.3.



In case the diameter to thickness ratio D/t exceeds the limit 0,441 E/Fy the check is not executed and a warning is issued on the output.


162

Distortional Buckling Strength

For both bending axis the distortional buckling strength is determined according to article C3.1.4. More specifically the general Procedure (a) is followed using formula (C3.1.4-6).

The check is executed in case the following conditions are met:

The cross-section has at least one element with reinforcement type RUO

For the given bending moment in the section, at least one of these elements is in compression

More specifically this implies that, if the stiffener is in compression distortional buckling can occur (even if the flange itself is partially in tension). This is in accordance with the distortional buckling shapes for weak axis bending of typical C-sections obtained using numerical analysis Ref. [9].

Remarks:

The unbraced length Lm is taken as the LTB length and this for both bending axis.

In case a diaphragm is positioned on the compression side and the diaphragm provides full bracing, the member is regarded as continuously restrained and Lm = Lcr.

The rotational stiffness of the restraining element is by default taken as zero.

In case a diaphragm is located on the compression side, is taken as the

rotational stiffness vorhC of the diaphragm.

For diaphragms reference is made to Use of diaphragms.

For calculating the compression flange properties, the default Scia Engineer axis convention is used (x-y axis system located at the centroid of the flange, with the x-axis measured positive to the right from the centroid and the y-axis positive up from the centroid)

The elastic section modulus of the full unreduced section relative to the extreme fiber in first yield Sfy is taken as Sfy,x for x-x bending and Sfy,y for y-y bending.

In determining the stress gradient in the web, pure symmetrical bending is assumed. This implies that for x-x bending this parameter equals 2 and for y-y bending this parameter equals zero.

The distance b0 for a standard profile library section is taken as the width property.

For a general section this is taken as the summation of the Internal (I) parts of the flange.

The distance h0 for a standard profile library section is taken as the height property.

For a Sigma section (Form Code 121 – 125) this is taken as the (full) height property. For a general section this is taken as the maximal height of the ‘web’ elements. Web elements are defined as elements with an angle > 45° to the horizontal axis.

When there is no ‘web’ element (i.e. CHS section ), distortional buckling is not checked.

Flanges are defined as elements with angle < 45° to the horizontal axis. Connected flange elements which have a relative angle > 135° are accounted for as ‘one’ flange for distortional buckling.


163

For cross-sections with roundings, the flange/web junction is taken to be at the intersection between the flange/web rounding and the flat part of the flange.

The thickness t is taken as the smallest thickness of the cross-section elements.

For Omega sections (Form Code 115) the top flange is not seen as flange for distortional buckling.

Shear

The shear strength is determined according to article C3.2.1.

In the calculation of Aw only elements with element types I, UO and SO are accounted for. In addition, elements with reinforcement type ROU are not accounted for.

For each element i the shear area Aw,i is calculated as follows:

With: i = The number (ID) of the element.

xend = End position of element i .

xbeg = Begin position of element i.

t = Thickness of element i.

= Angle of element i to the horizontal x-x axis

In addition, for each element i the nominal shear stress Fv,i is calculated.

The shear strength of the element then becomes Vn,i = Aw,i * Fv,i

The nominal shear strength Vn for the cross section is taken as the sum of the Vn,i of the related

elements.

Transverse stiffeners are not supported, therefore the shear buckling coefficient kv is taken as 5,34.

AISI NAS S100-2007 does not give provisions to calculate the shear resistance for circular hollow sections (Form Code 3). Therefore the default AISC 2005 provisions are used in this case.

Combined Bending and Shear

The combined bending and shear check is determined according to articles C3.3.1 and C3.3.2.

Transverse stiffeners are not supported; therefore the equations for unreinforced webs are used.


164

Web Crippling Strength

The web crippling strength is determined according to article C3.4.1.

More specifically the general equation (C3.4.1-1) is applied.

The alternative given in equation (C3.4.1-2) is not supported.

The web crippling check is executed on the positions where there is a jump in the Vy shear force

diagram.

Remarks:

The shear force diagram of both the actual member as well as adjacent members is evaluated. Adjacent members are defined as members which are in the same buckling system.

The angle between the plane of the web and the plane of the bearing surface is taken as 90°.

The Flange Conditions depend on the definition of the initial shape. In case there is an element with reinforcement type ROU the setting is taken as ‘Stiffened or Partially Stiffened

Flanges’.

The distances for One-flange/Two-flange and End/Interior are evaluated taking into account adjacent members. Adjacent members are defined as members which are in the same buckling system.

The following paragraphs specify the supported cross-section types.

Built-Up Sections

For built-up sections table C3.4.1-1 is used.

This applies to cold formed pair sections 2CFUo and 2CFCo with distance a = 0 mm and the

following form codes:

127 (Cold formed I-Plus section)

128 (Cold formed IS-Plus section)

Since these pair sections consist of two webs the resistance of the full section is obtained by adding the values of each web.

Single Web Channel and C-Sections

For single web channel and C-sections table C3.4.1-2 is used.


5 (Channel section)



116 (Cold formed C section eaves beam)


In addition this applies to the following pair sections:

2CFUc and 2CFCc

2CFUo and 2CFCo with distance a > 0 mm


165

Since the pair sections consist of two webs the resistance of the full section is obtained by adding the values of each web.

Single Web Z-Sections

For single web Z-sections table C3.4.1-3 is used.


102 (Z section)






Single Hat Sections

For single hat sections table C3.4.1-4 is used.


115 (Cold formed Omega section)

Since these sections consist of two webs the resistance of the full section is obtained by adding the values of each web.

Other Sections

For any other cross-section types as those listed in the previous paragraphs no web crippling check is executed.

In addition table C3.4.1-5 is not supported.


166

Combined Bending and Web Crippling

The combined bending and web crippling check is determined according to articles C3.5.1 and C3.5.2.

Requirement (a) is applied to the following form codes/sections:

5 (Channel section)



116 (Cold formed C section eaves beam)


102 (Z section)






115 (Cold formed Omega section)

2CFUc and 2CFCc

2CFUo and 2CFCo with distance a > 0 mm

Requirement (b) is applied to the following form codes/sections:

2CFUo and 2CFCo with distance a = 0 mm

Requirement (c) is applied to the following form codes/sections in case the check is executed within

a lapped zone:

102 (Z section)






Remarks:

The exception given for requirement (a) is not supported.

In case a lapped Z section does not meet the limits for requirement (c) the provisions of requirement (a) are applied instead.

For requirement (c) it is assumed that conditions (1), (2), (3) & (4) are fulfilled.


167

Combined Bending and Torsion

Combined bending and torsion loading is evaluated according to article C3.6.

In each fiber of the cross-section the bending stresses Sigma Mx and Sigma My are calculated.

These stresses are based on the effective cross-sectional properties and calculated in the fibers of the gross cross-section. In addition, in each fiber the shear stress due to torsion Tau t is calculated based on gross section

properties.

Using these stresses, the R factor is calculated according to equation (C3.6-1) using the following

expressions:

f bending = Sigma Mx + Sigma My

f torsion = Tau t

f bending + f torsion = (composed stress)

The critical fiber is taken as the fiber with the biggest composed stress.

The increase of the R factor in case of C-sections is not supported.

For diaphragms reference is made to “Use of diaphragms”.

Compression Members

Nominal axial strength

The nominal axial strength is determined according to article C4.1 using Fn = Fy.

Flexural Buckling

The stress Fe for flexural buckling is determined according to article C4.1.1.

For the calculation of the effective length factor, reference is made to “Calculation buckling ratio – general formula”.

In case an LTB restraint of type ‘Both’ is inputted, it specifies that both the top and bottom flange are held into position. As such, this point is seen as a fixed point for weak axis buckling. This implies that the system length Ly is taken between the LTB restraints of type ‘Both’ and the member ends. In addition the effective length factor ky is set to 1,00.



168

Torsional (-Flexural) Buckling

The stress Fe for torsional (-flexural) buckling is determined according to the general method given

in Ref. [7]. Doubly symmetric and hollow sections are taken as not subject to torsional (-flexural) buckling.


1 (Symmetric I shape)



For any other section the stress Fe is taken as the smallest of Sigma,t and Sigma,TF

Sigma,t = Ncr,T / Ag

Sigma,TF = Ncr,TF / Ag

With: Ncr,T Critical axial load for torsional buckling

Ncr,TF Critical axial load for torsional-flexural buckling

Ag Gross section area

Determination of Ncr,T

The elastic critical load Ncr,T for torsional buckling is calculated according to Ref.[7].

With: E Modulus of Young

G Shear modulus

J Torsion constant

Cw Warping constant

lT Buckling length for the torsional buckling mode

x0 and y0 Coordinates of the shear center with respect to the centroid

rx radius of gyration about the x-x axis

ry radius of gyration about the y-y axis


169

Determination of Ncr,TF

The elastic critical load Ncr,TF for torsional flexural buckling is calculated according to Ref.[7].

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

0

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

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

Ncr,T Critical axial load for torsional buckling

The smallest value of Fe (flexural, torsional and torsional-flexural buckling) is used for calculating Fn according to article C4.1.


Closed Cylindrical Tubular sections

The axial strength for closed cylindrical tubular sections is determined according to article C4.1.5.



In case the diameter to thickness ratio D/t exceeds the limit 0,441 E/Fy the check is not executed and a warning is issued on the output.

Distortional Buckling Strength

The distortional buckling strength is determined according to article C4.2. More specifically the general Procedure (a) is followed using formula (C4.2-6).

The check is executed in case the cross-section has at least one element with reinforcement type RUO.

Remarks:

The same remarks are valid as for distortional buckling of flexural members.

The elastic distortional buckling stress Fd is determined for each flange separately. The minimal Fd is taken as the limiting value of the cross-section.

Because of this separate determination, a diaphragm on only one flange is accounted for in the Fd calculation of that specific flange.

In addition, this procedure allows stiffened flanges of unequal dimensions.



170

Combined Compression and Bending

The combined compression and bending check is executed according to article C5.2.

The shifts ex and ey of the neutral axis are determined for the required compressive axial strength.

The additional moments due to these shifts are then calculated by multiplying the required compressive axial strength with these respective shifts.

The special provisions for angle sections apply for the following form codes:

4 (Angle section) 111 (Cold formed Angle section)

In case of 2nd

order analysis, reference is made to “2nd Order using Appendix 2”.

Use of diaphragms

Diaphragms are used specifically in conjunction with article D6.1 concerning purlin and girt design.

The lateral stiffness S for a diaphragm is calculated as follows in case the bolt pitch of the diaphragm

is set as ‘br’: (Ref.11,3.5 and Ref.12,3.3.4.):

With a Frame distance

Ls Diaphragm length

K1 Diaphragm stiffness factor K1

K2 Diaphragm stiffness factor K2

For a bolt pitch of ‘2br’ the shear stiffness S is replaced by 0,2 S (Ref.11 p22).

For the rotational stiffness vorhC of a diaphragm reference is made to “Adaptation of torsional

constant”.

The available lateral strength S is compared to the required lateral strength Serf Ref.[8]:

With E Modulus of Young

CW Warping constant of the purlin

L LTB length of the purlin

G Shear modulus

J Torsion constant of the purlin

Iy Second moment or area about the y-y axis of the purlin

h Height of the purlin


171

In case the available lateral strength S is higher than or equal to the required strength Serf, the diaphragm is providing sufficient stiffness and the purlin is seen as fully braced.

In case the available lateral strength S is lower than the required strength Serf, the diaphragm is not providing sufficient stiffness and the purlin is seen as inadequately braced.

The influence of a diaphragm on different checks (bending, compression and torsion) is outlined in the following overview.

Bending

Diaphragm on the compression flange

The lateral stiffness S is calculated and compared to the required stiffness Serf.

In case S ≥ Serf the member is taken as fully braced.

As a result no LTB check is required for bending about the x-x axis.

Distortional buckling still needs to be checked. For distortional buckling is taken as vorhC.

See Ref.[2] pp 47 “Since the distortional buckling has an intermediate buckling half wavelength; the distortional buckling still needs to be considered even for braced members.”

In case S < Serf the member is seen as inadequately braced.

As a result the LTB check for bending about the x-x axis is executed using the augmented torsional stiffness J.

Reference is made to “Adaptation of torsional constant”.

Distortional buckling still needs to be checked. For distortional buckling is taken as vorhC.

Diaphragm on the tension flange


In case S ≥ Serf the member is taken as fully braced on the tension flange.

In this case article D6.1.1 is applied.

As a result no LTB check is required for bending about the x-x axis.

In addition, no distortional buckling check is required.

In case S < Serf or in case the limits of article D6.1.1 are not met, the member is seen as

inadequately braced.

As a result the LTB check for bending about the x-x axis is executed by default, without an increased torsional stiffness J.

In addition distortional buckling is checked taking as zero.


172

Compression

Diaphragm on one flange


In case S ≥ Serf the member is taken as fully braced.

In this case article D6.1.3 is applied.

As a result no distortional buckling check is required.

In case S < Serf or in case the limits of article D6.1.3 are not met, the member is seen as

inadequately braced .

As a result the default compression checks are executed.

In addition distortional buckling will be checked taking as zero.

Diaphragm on both flanges

In this case the specifications of the previous step apply using the largest lateral stiffness S of both

diaphragms.

Torsion

Diaphragm on any flange


In case S ≥ Serf the member is taken as fully braced against torsion.

In this case the reduction due to torsion is not applied.

In case S < Serf, the member is taken as inadequately braced.

As a result the reduction for torsion is determined by default.


173

Flexural members having one flange through-fastened to sheeting

The nominal flexural strength is determined according to article D6.1.1.

This article is only applied in case the following conditions are met:

The member is in bending about the x-x axis

The diaphragm is located on the tension flange

The diagram is through fastened

The lateral stiffness S ≥ Serf

The conditions for article D6.1.1 are met

Remarks:

The article is only valid for C and Z sections with edge stiffeners (i.e. elements with reinforcement type ROU).

This applies to the following form codes: 114 (Cold formed C-section) 116 (Cold formed C-section eaves beam) 117 (Cold formed C-Plus section) 118 (Cold formed ZED section) 119 (Cold formed ZED section asymmetric lips) 120 (Cold formed ZED section inclined lip) 126 (Cold formed ZED section both lips inclined)

For determining the R factor a difference is made between simple span and continuous spans. This difference is based on the system length Lx. When the member under consideration has only one part for Lx it is taken as simple span. When the member has more parts for Lx it is taken as continuous span.

The article is not applied for cantilevers. A cantilever is defined as a member at the end of a buckling system which has free ends for both buckling about the x-x and y-y axis.

In addition, the article is not applied for continuous beams in the region between inflection points adjacent to a support.

It is assumed that conditions (8), (9), (10), (11), (12) & (13) are fulfilled.

The correction factor r for compressed insulation is not supported.

Compression members with one flange through-fastened to sheeting

The compressive strength is determined according to article D6.1.3.

This article is only applied in case the following conditions are met:

The member is in compression

The diaphragm is located on one or both flanges

The diagram is through fastened

The lateral stiffness S ≥ Serf

The conditions for article D6.1.3 are met


174

Remarks:

The article is only valid for C and Z sections with edge stiffeners (i.e. elements with reinforcement type ROU). This applies to the following form codes: 114 (Cold formed C-section) 116 (Cold formed C-section eaves beam) 117 (Cold formed C-Plus section) 118 (Cold formed ZED section) 119 (Cold formed ZED section asymmetric lips) 120 (Cold formed ZED section inclined lip) 126 (Cold formed ZED section both lips inclined)

The fastener distance x is taken as 0,5.

It is assumed that conditions (7) & (8) are fulfilled.

2nd Order using Appendix 2

In case the proper setting is activated in the steel setup, the provisions according to article 2.1 of Appendix 2 are applied.

More specifically, when the check is executed for a non-linear combination the following changes are applied:

Effective length factor Kx is set to 1,00

Effective length factor Ky is set to 1,00

x for article C5.2 is taken as 1,00

y for article C5.2 is taken as 1,00

Cmx for article C5.2 is taken as 1,00

Cmy for article C5.2 is taken as 1,00

Article 2.2 of Appendix 2 is not supported.


175

Lapped Purlin Design

For the analysis, the purlin line is considered prismatic i.e. the increased stiffness due to the doubled cross-section within the lap is ignored Ref.[5].

Since the lap length is defined along the member axis, it is important to specify a sufficient ‘number of sections on average member’ in the Solver Setup when using overlaps.

Combined Strength

The strength within the lapped zones is taken as the sum of the strengths of the individual members Ref.[4].

The use of the combined strength of the individual members is applied for the following checks:

- Nominal Bending Check

- Shear Check

- Combined Bending and Shear Check

- Web crippling Check

- Combined Bending and Web Crippling Check

- Bending – Distortional Buckling Check

For distortional buckling, the distortional buckling stress Fd is calculated for the critical flange i.e. the flange resulting in the lowest Fd value.

The following equations are then used: Mcrd = (Sfsection 1 + Sfsection 2) * Fd My = (Sfysection 1 + Sfysection 2) * Fy

Special considerations for Lateral Torsional Buckling

Within a lapped zone, at the bottom flange the LTB check depends on the Bottom flange fully braced setting within the Overlap data.

In case this setting is activated it implies the bottom flange within the lapped zone is fully fixed and thus no LTB occurs. This has the following implications:

- Within the lapped zone, in case the bottom flange is in compression, no LTB check is executed.

- Outside of the lapped zone the LTB length is taken to the end of the lap.


176

Diaphragm on the tension flange

In case the following conditions are met:

- Diaphragm on the top flange which provides full bracing - Setting Bottom flange fully braced activated in the overlap data

- The top flange is in tension

By default it would imply article D6.1.1 should be applied however this article is only valid in case the

compression flange is free. Since in this case the compression flange is fully braced this article is not applied and the nominal bending strength is used.

References

[1] AISI S100-2007


2007 edition

[2] AISI S100-2007-C

Commentary on North American Specification for the Design of Cold-Formed Steel Structural Members

2007 edition

[3] AISI S100-07-E1

Errata to North American Specification for the Design of Cold-Formed Steel Structural Members

2007 edition February 20, 2008

Amended September 25, 2008

Amended June 4, 2009

[4] AISI SG03-2

Cold-Formed Steel Design Manual

2002 edition

[5] G. J. Hancock, T. M. Murray, D. S. Ellifritt

Cold-Formed Steel Structures to the AISI Specification

Marcel Dekker, Inc., 2001

[6] A Gerhsi, R. Landolfo, F.M. Mazzolani

Design of Metallic cold formed thin-walled members

Spon Press, London, UK, 2002

[7] SN001a-EN-EU

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

Access Steel, 2006

www.access-steel.com

http://www.access-steel.com/


177

[8] EN 1993-1-3:2006

Eurocode 3 - Design of steel structures

Part 1-3: General rules - Supplementary rules for cold-formed members and sheeting

CEN, 2006

[9] Schafer, B.W., Ádány, S.

Buckling analysis of cold-formed steel members using CUFSM: conventional and constrained finite strip methods.

Eighteenth International Specialty Conference on Cold-Formed Steel Structures,

Orlando, FL. October 2006.

[10] J. Schikowski

Stabilisierung von Hallenbauten unter besonderer Berücksichtigung der Scheibenwirkung von Trapez- und Sandwichelementdeckungen, 1999 http://www.jschik.de/

[11] E. Kahlmeyer




Stahlbauten



[13] AISI S100-07/S1-09

Supplement No. 1 to the North American Specification for the Design of Cold-Formed Steel Structural Members, 2007 edition August, 2009

[14] AISI S100-07/S2-10

Supplement No. 2 to the North American Specification for the Design of Cold-Formed Steel Structural Members, 2007 edition February, 2010

http://www.jschik.de/


178

CM66

CM66 Code check


Règles de calcul des constrcutions en acier

ITBTP / CTICM

Régles CM Decembre 1966

Editions Eyrolles 1982

Consulted articles

The cross-section is checked for tension (art. 3,1), bending (art. 3,2.) and shear (art. 3,3.).

For the stability check, the following criteria are considered:

for compression : art. 3,4.

for compression and bending : art. 3,5

for lateral torsional buckling : art. 3,6.

for double bending and axial compression : art. 3,7.

for shear buckling : art 5,212

A more detailed overview for the used articles is given for the relevant parts in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.

3 Règles générales concernant les calculs de résistance et de déformation

3,0 Données numériques x

3,1 Pièces soumises à traction simple x (*)

3,2 Pièces soumises à flexion simple ou déviée x

3,21 Flexion simple x(*)

3,22 Flexion déviée

3,3 Effet de l’effort tranchant dans les pièces fléchies x


179

3,4 Pièces soumises à la compression – flambement simple

3,40 Généralités x(*)

3,41 Pièces comprimées a parois pleines x

3,42 Pièces composées a treilis

3,43 Pièces composées a traverses de liaison

3,44 Conditions spéciales imposées aux éléments comprimés a parois minces x

3,5 Pièces soumises à compression avec flexion dans le plan de flambement

3,50 Principe x

3,51 Coefficient d’amplification des contraintes de flexion x (*)

3,52 Vérfication des pièces a parois pleines

x

3,53 Vérification des pièces composées à treilis

3,54 Vérification des pièces composées à traverses de liaison

3,6 Déversement en flexion simple

3,60 Généralités x

3,61 Pièces symétriquement chargées et appuyées

3,611 Poutres à äme pleine x(*)

3,612 Poutres à treilis

3,62 Cas des piéces soumises à deux moments différents au droit des appuis x(*)

3,63 Cas des poutrelles en console parfaitement encastrées

3,64 Coeffcients utilisés pour la détermination de kd

3,641 Coefficient D x

3,642 Coefficient C x(*)

3,643 Coefficient B x(*)

3,7 Flexion composée

3,70 Domaine d’application x

3,71 Notations x

3,72 Principe des vérifications x

3,73 Formules enveloppes pour les pièces à parois pleines x (*)

3,8 Flambement dans les systémes hyperstatiques

3,9 Déformations x

5 Règles spéciales à certains éléments

5,212 Poutres composées à âme pleine – âmes x


180

Section properties


Plastic coefficient

The plastic coefficients are calculated according to the Ref.[1], 13,212 (Valeurs du coefficient ψ d’adaptation plastique).

Compression members



Factor kf

The factor kf is calculated using the formula given in Ref[1], 3,516

3;1

lM

A172.125.0

k

2

med

M

f

If Mmed ≈ 0.0, the formula 3,513 is used : 3.1

25.0k f

LTB Check

The LTB check is performed for symmetric I sections. For other cross sections the factor kd=1.0.

For the calculation of the coefficient C, we refer to "Calculation of moment factors for LTB".

The coefficient B is calculated by interpolating the table for B given in Ref[1] 3,643, and using the calculated C value with table for C given in Ref[1] 3,642.


Use of diaphragms



181

Combined flexion

The values fx is the maximum value of the bending stress in the member for the bending around

the strong axis. The value fy is the maximum value of the bending stress in the member for the bending around the weak axis.

For non-prismatic sections the values fx and fy are the local (i.e. in each intermediary section) bending stresses.



Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube




I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM

Section check x x x x x x x x x x x x

Buckling check x x x x x x x x x x x x

Slender section buckling check

x x x x x x x x

LTB Check x



182

References

1 Règles de calcul des constrcutions en acier

ITBTP / CTICM




183

CM66 - Additif 80

CM66 - Additif 80 Code check

The beam elements are checked according to the regulations given in Additif 80

Consulted articles

The cross-section is classified according to art. 5,12. (classification 'plastic' or 'elastic').

The section is checked for tension and compression (art. 4,2), bending (art 4,3), shear force (art. 4,4), the combination of bending and axial force (art. 4,5 and art 4.6).

For the stability check, the following criteria are considered:

for lateral torsional buckling : art. 5,2.

for compression : art. 5,31.

for compression and bending : art. 5,32


4 Resistance des sections

4,1 Règle générale (*)

4,2 Effort normale x

4,3 Moment de flexion x

4,4 Effort tranchant x

4,5 Moment de flexion et effort normal x

4,6 Momens de flexion, effort normal et effort tranchant x

5 Stabilité des éléments

5,1 Conditions de non voilement local x (*)

5,2 Résistance au déversement des poutre fléchies

5,21 Règles de contreventement latéral au voisinage des sections plastifiées

5,22 Moment ultime de déversement en flexion simple x (*)

5,23 Dimensionnement des entretoises

5,24 Résistance au déversement en flexion déviée x

5,3 Résistance au flambement

5,31 Eléments simplement comprimés x

5,32 Eléments comprimés et fléchis x

5,33 Longueur de flambement (*)


184





Section check

If the sections are not according to the conditions specified in art. 5,1, the sections are checked according to the regulations given in Ref.[2].

If a torsional moment is present, the sections are checked according to the regulations given in Ref.[2].

Compression members




For the calculation of the moment factors C1 and C2, we refer to "Calculation of moment factors for LTB", using the EC3 values.


Use of diaphragms



185

Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube




I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM

Classification Add 80

x x

Plastic section check Add 80

x x

Buck:ling check Add 80

x x

LTB check Add 80 x x

Compression + bending Add 80

x x

References

[1] Additif 80

2 Règles de calcul des constrcutions en acier

ITBTP / CTICM




186

BS5950-1:1990

BS5950-1:1990 Code Check


British Standard BS 5950

Structural use of steelwork in building

Part1. Code of practice for design in simple

and continuous construction:hot rolled section

British Standard distribution BS5950 Part1 1990 revised in 1992

Material properties

For standard steel grades, the yield strength py is defined according to the thickness of the element (see Table 6 Art.3.1.1.). The standard steel grades are :

Grade 43 : yield strength defined between 245 and 275 N/mm²



(pY in N/mm², t in mm)

Steel grade

Thickness limits

PY

Grade 43

t16 mm

275 N/Mm²

t40 mm

265 N/mm²

t63 mm

255 N/mm²

t100 mm

245 N/mm²

t16 mm

355 N/mm²


187

Grade 50

t40 mm

345 N/mm²

t63 mm

340 N/mm²

t100 mm

325 N/mm²

Grade 55

t16 mm

450 N/mm²

t25 mm

430 N/mm²

t40 mm

415 N/mm²

t63 mm

400 N/mm²

Remark: For cold-formed section, values for Py are not influenced by the previous table.

Remark : The reduction rules from previous table are only valid when the used material is defined as material for the selected code.

Consulted articles

According to Art. 3.5. and table 7, cross sections are classified in 4 types:

Plastic

Compact

Semi-compact

Slender

A reduction factor is applied to the design strength of the material in use for slender sections by following the rules described in Art. 3.6 and in Table 8. Partial safety factor of design strength is included in py value.

The section is checked for bending (Art.4.2.), tension (Art.4.6.), compression (Art.4.7.), shear (Art.4.2.3.), combined moment and axial force (Art. 4.8.) and biaxial moments (Art.4.9.). For the stability check, the beam element is checked for lateral torsional buckling, shear buckling, compression and bending with axial compression. Articles used for this stability check are the following:

for lateral torsional buckling : Art. 4.3.

shear buckling : Art. 4.4.5.

for compression : Art. 4.7.

for bending and axial compression : Art. 4.8.

A more detailed overview of used articles is given in the following table.


188

Part. 3 Section properties

3.5. Limiting proportions of cross sections Art. 3.5.1.

Art. 3.5.2.

Art. 3.5.4.

Table 7

Fig.3

3.6. Slender cross section Art. 3.6.1.

Art. 3.6.2.-3.6.3.

Art. 3.6.4.

Table 8

Part. 4 Design of structural elements

4.2. Member in bending Art. 4.2.1.3. (a) (c)

Shear capacity Art. 4.2.3.

Moment capacity with low shear Art. 4.2.5.

Moment capacity with high shear Art. 4.2.6.

4.3. Lateral torsional buckling

Member in bending Art. 4.3.7.

LTB factor

General Art. 4.3.7.1.

Equivalent uniform moment Art. 4.3.7.2.

Buckling Resistance Art. 4.3.7.3.

Bending strength pb Art. 4.3.7.4.

Equivalent slenderness LT, , , u, v

Art. 4.3.7.5.

Appendix B.

Factors m, n Art. 4.3.7.6.

Equal flanged rolled section Art. 4.3.7.7.

Buckling resistance moment for single angle Art.4.3.8.

4.4. Plate Girders

General Art. 4.4.1.

Dimensions of webs and flanges Art. 4.4.2.2. Art. 4.4.2.3.

Moment capacity Art. 4.4.4.

Section with slender webs Art. 4.4.4.2. (a)

Shear buckling resistance of thin webs Art. 4.4.5.1.

Design without using tension field action Art. 4.4.5.3. and Appendix H.1.

4.6. Axially loaded tension members


189

Tension capacity Art. 4.6.1.

Effective Area of simple tension members Art. 4.6.3.1. Art. 4.6.3.3.

4.7. Compression member

Slenderness Art. 4.7.3.2.

Compression resistance Art. 4.7.4.

Compressive strength Art. 4.7.5. Appendix C

4.8. Axially loaded members with moments

Tension members with moments Art. 4.8.2. + EC3 5.4.9.&Annex F

Compression members with moments Art. 4.8.3.

Local capacity check Art. 4.8.3.2.

Buckling check with exact approach Art. 4.8.3.3.2.

4.9. Members with biaxial moments

See 4.8.


For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point. For each load case/combination, the critical section classification over the member is used to perform the stability check.

So, the stability section classification can change for each load case/combination. However, for non-prismatic sections, the stability section classification is determined for each intermediary section.

Slender cross-section

Slender sections are particularly sensitive to local buckling. British Standard code (Art. 3.6.) defines stress reduction factor to prevent this phenomenon. For webs subject to moments and axial load and for circular hollow sections, the design strength py should be assumed such that the limiting proportions for semi-compact section are met. For other sections, where a slender outstand is in compression, the design strength should be reduced by the factor given in Table 8.

Section properties

The net area of a section is taken as its gross section neglecting the deduction due to fastener holes: Art. 3.3. Shear area of a cross-section is calculated by using Art. 4.2.3.


190

Bending moment

Before any calculation of members in bending, it's necessary to determine the shear capacity. For plastic and compact section with high shear load, moment capacity is calculated with the plastic modulus only for I and PLL sections (Art. 4.2.6. and 4.8.). For other cross-section, with plastic or compact section classification, characterised or not by a low shear load, we assumed that the moment capacity is calculated by using the same approach than for semi-compact section: the elastic modulus (elastic calculation).

Bending, shear, axial force

For plastic and compact sections, BS5950 Art. 4.8.2. & 4.8.3.2. (b) prescribes a detailed approach to determine the unity check of axially loaded members with moments. The detailed relationship allows a greater economy for plastic and compact section . In this expression, we use a reduced moment capacity Mr respectively about the major and the minor axis. Those values are determined by using EC3 Art.5.4.9. (see Ref.[5]). For semi-compact and slender section, the simplified approach is applied following Art. 4.8.2.and Art. 4.8.3.2. (a).

Lateral torsional buckling

For I sections (symmetric and asymmetric PPL), rectangular sections (solid and hollow), T sections, channel sections and angle section, the critical lateral torsional buckling moment is given by the general formula Art. 4.3.7. and Annex B2&3. For other sections, we follow conservative recommendation described in Art. 4.3.7.5. and calculation proposed in EC3 to determine the elastic critical moment Mcr EC3 Annex F1.1. Formula (F.1.) see Ref [5].

The condition to be satisfied in all the cases is that

with

Mb=Sxpb

and

(m is an equivalent uniform moment factor)

pb is the bending strength and is related to the equivalent slenderness :

in which n is an equivalent slenderness factor.

For beam without loading point between points of lateral restraint, n=1 and m depends on the ratio of the end moments at the points of restraint.


191

For beam loaded between point of lateral restraint, m=1 and n depend on the ratio of the end moments at the points of restraint and on the ratio of the larger moment to the mid-span free moment.

There are thus two methods for dealing with lateral torsional buckling namely:

'm approach' i.e. the 'equivalent uniform moment method' with n=1

'n approach' i.e. the 'equivalent slenderness method' with m=1

In any given situation, only one method will be admissible, taking into account that it is always conservative to use m=n=1. Since the publication of BS5950 Part 1 1990, doubt has been cast on the correctness of using n factors less than 1 in combination with an effective length LLTB less than

the length of the member L in the calculation of LTB. However, as a interim measure, pending

clarification ina future version of BS5950, it is recommended that LTB is taken as the smaller of the two following values:

By using the settings of BS5950, the user can define which method correspond to his situation or define his choice as the conservative method m=n=1.


Use of diaphragms


Compression member

For member submitted to compression, we applied the recommendations given in BS 5950 and Appendix C to determine the compressive strength.




192

Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube




I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM

Classification x x x x x x x (1) x (1) (1) (1)

Section check class 1 x x x x x x x x


Section check class 3 x x x x x x x x x x x x


Stability check class 1 x x x x x x x x


Stability check class 3 x x x x x x x x x x x x


Shear buckling check x x x

(1)sections are classified as class 3 cross section by default

References

[1] British Standard BS5950 Part 1 : 1990+Revised text 1992

Structural use of steel work in building

Part1 Code of practice for design in simple and continuous construction: hot rolled sections


193

[2] Plastic design to BS5950

J.M. Davies & B.A. Brown

The steel Construction institute

[3] Steelwork design

Guide to BS5950: Part 1: 1990

Volume 2 Worked examples (revised edition)

[4] Essentials of Eurocode 3


ECCS - N° 65, 1991

[5] Eurocode 3



ENV 1993-1-1:1992

[6] R. Maquoi




194

BS5950-1:2000

BS5950-1:2000 Code Check

The background to this code check can be found within the document “BS 5950-1:2000 steel code check Theory”


195

SIA263

SIA263 Code check


SIA263

Construction en acier

SIA263:2003

Material properties

The most common steel grades are used in SIA263. Their mechanical properties are described in table 1 SIA263. The following table gives the yield strength for each type of grade commonly used in function of the nominal web thickness:

t<=40 t<=40 40<t<=100 40<t<=100

fy fu fy fu

S235

S 235

235 360 215 340

S275

S 275

275 430 255 410

S355

S 355

355 510 335 490

S460

S 460

460 550 430 530

Consulted articles

The classification described in SIA263 is based on the calculation method. The calculation method in SIA263 distinguish the method used respectively to determine the internal forces and to perform the section and the stability check.

By facility, we can obviously make a parallel between the calculation method of SIA263 and the section classification proposed in EC3.


196

According to SIA263 Table 5a-5b , cross sections are classified in 4 types:

PP (plastic-plastic) or class 1

EP (elastic-plastic) or class 2

EE (elastic-elastic) or class 3

EER (elastic-elastic reduced) or class 4

The first letter of the classification denomination is related to the method used to calculate internal forces in the structure. The second letter indicates if we perform the section and the stability check with a elastic or a plastic approach. Finally, we must note that the steel code SIA263 is essentially oriented for symmetrical and bisymmetrical profile like I profiles. In the present modulus, others profiles are calculated by using a classic elastic approach (EE classification) and EC3 prescriptions.

The section is checked for tension, compression, shear, combination of bending and axial forces. For the stability check, the beam element is checked for lateral torsional buckling, shear buckling, compression and bending with axial compression. A more detailed overview for the used articles is given in the following table :

4 Analyse structurale et dimensionnement

4.1 Généralités x

4.2 Bases de l'analyse structurale et du dimensionnement

4.3 Modélisation

4.3.1 Classification des sections

x

4.4 Résistance des sections

4.4.1 Effort normal

x

4.4.2 Flexion x

4.4.3 Effort tranchant x

4.4.4 Flexion et effort tranchant x

4.4.5 Flexion et effort normal x

4.4.6 Sollicitations multiaxiales x

4.5 Stabilité

4.5.1 Flambage

x

4.5.2 Déversement des poutres fléchies x

4.5.3 Flexion et compression x

4.5.4 Voilement des éléments plans comprimés x

4.5.5 Voilement des éléments plans cisaillés x

4.8 Situtation de projet incendie

4.8.1 PRINCIPES

x

4.8.2 Propriétés de l'acier en cas d'incendie x

4.8.5 Méthode de calcul simplifiée x

5 Eléments de construction

5.1 POUTRES ET POTEAUX DES CLASSES DE SECTION 1 ET 2

x

5.3 Eléments comprimés à section composée

5.3.1 Barres étrésillonées ( à travers de liaison)

x

5.4 Poutres composées à âme pleine


197

5.4.1 Résistance à la flexion x

5.4.2 Résistance à l'effort tranchant x

5.4.3 Interaction entre flexion et effort tranchant x

Annexe B Moment critique de déversement élastique Mcr x

Annexe C Echauffement des éléments de construction en cas d'incendie x

Section classification

For each intermediary section, the classification is determined and the proper section check is performed. The classification can change for each intermediary point. For each load case/combination, the critical section classification over the member is used to perform the stability check.

So, the stability section classification can change for each load case/combination. However, for non-prismatic sections, the stability section classification is determined for each intermediary section.

Slender cross-section

The design of a section that not satisfies the table 5 of SIA263 is always performed by using a reduced area. This classification correspond to the EER method. The determination of a reduced area is based on the effective width of each compression element in the current section (Art. 4.5.4). The using of a reduced area implies the recalculation of the shear centre position, the inertia and the elastic modulus.

Sections properties

The holes due to fastener are neglected in the area of a section


For double symmetric I profile, we don't have to perform any lateral torsional buckling check if

NEd/Npl,Rd0.15 and the conditions provided in Table 6 SIA263 are satisfied. For any other case, a LTB check must be perform.

Calculations described in Annex B for I,U and PPL can be applied to T sections only if the flange is subjected to compression. Otherwise, as for section not supported by SIA263 in the LTB check, we use prescriptions given in EC3 Annex F. Those rules allow us to determine a elastic critical moment for lateral torsional buckling for symmetrical (formula F.2 EC3) and non symmetrical (formula F.1. EC3) sections around the minor axis.

In the case of I, U, PPL and, T only with compression in flange, characterised by a reduced area or not, we have to determined before any calculation irc, defined as the radius of gyration of a section comprising the compression flange plus 1/3 of the compression web area, taken about an axis in the plane of the web.




198

Use of diaphragms


Shear buckling


Stability check

For double symmetric I profile PP or EP, SIA263 provides specific formula to perform the stability check of member submitted to biaxial moment. For other sections, non symmetric or from EE and EER classification, a general formula is provided to design member under mono-axial sollicitations.

Torsion check


Built-in beams

For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter ‘Section check for built-in beams (IFB, SFB, THQ sections).

SIA263 - Fire Resistance



The accidental combination is given by

Gk + Pk + Ad+ 2,iQk,i

with Gk characteristic values of permanent actions

Qk,i characteristic value of the variable action i

Ad design values of accidental action from fire exposure

2,j combination coefficients

Pk characteristic value of prestressing action


199

Material properties


Strength and deformation properties:

20,a,E,a

20,y,y,y

EkE

fkf

The variation in function of the steel temperature of the value for yield strength ky, and modulus of

elasticity kE, is given by tables in ref.[1], Figure 15.

In the simplified calculation method, the following default properties are considered to be constant during the analysis :


(a-20)



In this part, the nominal temperature-time curves and the related net heat flux are described. For more info, EC3 Chapter 'Temperature analysis - Thermal actions'


See EC3 Chapter 'Nominal temperature-time curve'.

Net heat flux

See EC3 Chapter 'Net heat flux'


200

Steel Temperature

See Ref.[1], Annexe C.


thc

V/Ad,net

aa

mt,a























The value a,t 0.0


201


Calculation model


- strength domain




Code Check

The section and stability checks (buckling, lateral torsional buckling) are performed according to the regulations given in Ref.[1], 4.8.5.



- classification of cross section : art. 4.8.5.2.

- resistance for tension members : art. 4.8.5.4.

- resistance for compression members (class 1,2 or 3) : art. 4.8.5.5..

- resistance for beams (class 1,2,3) : art. 4.8.5.6., art. 4.8.5.7., art. 4.8.5.8.

- resistance for members (class 4) : art. 4.8.5.9.

Supported sections




L Angle section

U Channel section

T T section


Z Z section


202


Cold formed section

COM Composed section

O Solid tube




I

RHS

CHS

L

U

T

PPL

RS

Z

O

COM

NUM


Section check PP x x(2) x(3)

Section check EP x x(2) x(3)

Section check EE x x x x x x x x x x x x x

Section check EER x x x x x x

Stability check PP x x x x x x x x x x x x x

Stability check EP x x x x x x x x x x x x x

Stability check EE x x x x x x x x x x x x x

Stability check EER x x x x x x


LTB x x(4) x(4) x(4) x(4) x(4) x x(4) x(4) x(4) x(4) x(4) x(4)


(2) check according to EN 1993-1-1

(3) check according to ENV 1993-1-1

(4) general formula for Mcr

References [1] SIA263

Construction en acier

SIA263:2003

[2] SIA263/1

Construction en acier / Spécification complémentaires

SIA263/1:2003


203

GBJ 17-88

The GBJ 17-88 code check

The beam elements are checked according to the regulations given in :

National standard of the People’s Republic of China

Code for design of steel structures

GBJ 17-88

Beijing 1995

Material properties

The used steel grades are:

Grade3

16Mn

16Mnq

15Mn

15Mnq

For Steel3, the following groups are defined according to the element thickness (in mm):

Group Diameter or thickness of bars Thickness of L-, I- and U sections

Thickness of Plates

1 <=40 <=15 <=20

2 >40-100 >15-20 >20-40

3 >20 >40-80

The design values are (in N/mm²)

Steel Group Thickness f fv fce fy

Steel3 1

2

3

215

200

190

125

115

110

320

320

320

235

235

235

16Mn

16Mnq

<=16

17-25

26-36

315

300

290

185

175

170

445

425

410

345

345

345

15Mn

15Mnq

<=16

17-25

26-36

350

335

320

205

195

185

450

435

415

390

390

390


204

with f the resistance design value for tension, compression, bending (N/mm²)

fv the resistance design value for shear (N/mm²)

fce the bearing resistance (N/mm²)

fy the yield strength (N/mm²)

Remark : The reduction rules from previous table are only valid when the used material is defined as material for the selected code. If they are not defined as GBJ material, the following rule is used

f = 0.91 x yield strength

fv = 0.58 x yield strength

Consulted articles

The section and elements are checked according to part 4 and 5. When plastic design is allowed, part 9 is supported.

A more detailed overview for the used articles of the relevant parts is given in the following table. The chapters marked with “x” are consulted.

4. Calculation of flexural members

4.1.Strength

4.1.1.

4.1.2.

x (*)

x

4.2.Overall stability (*)

4.2.1.

4.2.2.

4.2.3.

4.2.4.

x

x

x

x

4.3.Local stability (*)

4.3.1.

4.3.2.

4.3.3.

4.3.9.

x

x

x

x

5.Calculation of axially loaded members and members subjected to combined axial load and bending

5.1.Axially loaded members

5.1.1.

5.1.2.

x(*)

x(*)

5.2.Members subjected ot combined axial load and bending


205

5.2.1.

5.2.2.

5.2.5.

x(*)

x

x

5.3.Effective length and allowable slenderness ratio (*)

5.4.Local stability of compression members

5.4.1.

5.4.2.

5.4.3.

5.4.4.

5.4.5.

x

x

x

x

x (*)

9.Plastic design

9.1.General requirements

9.1.3.

9.1.4.

x

x

9.2.Calculation of members (*)

9.2.1.

9.2.2.

9.2.3.

9.2.4.

x

x

x

x

9.3.Allowable slenderness and detailing requirements

Appendix 1 Overall stability factor of beams

A1.1.Simply supported beam of uniform welded I section x

A1.2.Simply supported beam of rolled I section x

A1.3.Simply supported beam of rolled channel section x

A1.4.Cantilever beams of doubly symmetric I section x

A1.5.Approximate calculation of overall stability factors x

Appendix 2 Calculation of local stability of girder web

A2.1.Web plate strengthened with transverse stiffeners x(*)

A2.2.Web strengthened with transverse and longitudinal stiffeners

A2.2.Web strengthened with transverse, longitudinal and short stiffeners

Appendix 3 Stability factor of axially loaded compression members x

Section properties

The influence of the net section is neglected, i.e. only the gross area is used.


206


The local compressive stress c, is considered as 0.0.

Buckling curves

For welded I and PPL sections the default value for the buckling curve about the weak axis is “b”. This can be changed to “c” on users request.

Buckling length


The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see "Calculation of critical Euler force for VARH elements").


The LTB check is supported for the following sections : I section, U section, RHS section, T section, PPL section.

For the other section type, the factor b = 1.0.


Local stability of compressed members

For each intermediary section, the ratio’s are determined. The section classification and the effective area properties are determined for each intermediary section for performing the section check.

For each load case/combination, the critical section classification and the effective area properties over the member are used to perform the stability check. However, for non-prismatic sections, the section classification and the effective area properties are determined for each intermediary section to perform the stability check.

When the web ratio ( dept /thickness) does not conform to the requirements, the web is reduced for calculating of the section check and stability check. A width of 20 tw sqrt(235/fy) on each side of the web is taken into account.


207

yw

f

235t20d



Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube



".



208

I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM

Plastic (single bending) x x

Compact section (with ) x x x x x x

Non-compact section x x x x x x x x x x


Normal buckling x x x x x x x x x x x x

LTB x x x x x

Shear buckling x x x

Plastic stability check (single bending)

x x

References

[1] Chinese Steel Code

GBJ 17-88

(Chinese version)

.[2] National standard of the People’s Republic of China

Code for design of steel structures

GBJ 17-88

Beijing 1995


209

Korean steel code check

The Korean steel code check

Material properties

The following design values are used :

Steel fy

t<=40 mm

fy

t>40 mm

SS41

SPS41

SPSR41

240 220

SS50 280 260

SS55 380 380

with fy the yield strength (N/mm²)

The following steel characteristics are valid :

modulus of elasticity 210000 N/mm²

shear modulus 81000 N/mm²

coefficient of linear thermal expansion 12 x 10-6

density 7850 kg/m³

Consulted articles

The section and elements are checked according to part 2 and 3. The shear buckling check is perfromed using article 7.5.2. The classiffication of sections is based on the rules of part 4.

A more detailed overview for the used articles of the relevant parts is given in the following table. The chapters marked with “x” are consulted.

TEXT

2.Allowable stress

2.1.Structural material x

2.1.1.Allowable tensile stress x


210

2.1.2.Allowable shear stress x

2.1.3.Allowable compressive stress x

2.1.4.Allowable bending stress

a)

b)

c)

(*)

x

x

x

2.1.5.Allowable bearing stress

3.Load and stresses

3.3.Combined stresses (*)

3.3.1.Compression force and bending moment x

3.3.2.Tensile force and bending moment x (*)

3.3.3.Shear force and tensile stress

4.Width-Thickness ratio of plates (*)

4.1.1.Cantilever plate x

4.1.2.Two side fixed plate x

4.1.3.Effective area x

4.2.CHS section and thickness ratio x

5. Tensile member

6.Compressive member

6.1.Slenderness ratio x

6.2.Buckling length x(*)

7.Beam element

7.5.Stiffener

7.5.2.Buckling verification of the web

a)

x

Section classification


For each load case/combination, the critical section classification and the effective area properties over the member are used to perform the code check. However, for non-prismatic sections, the section classification and the effective area properties are determined for each intermediary section.

When the element properties don’t satisfy the limiting values for the ratios, the section is classified as slender. The section have to be reduced for the calculation of the stresses. For outstand


211

compression elements, the part that is situated on the fixed side, remains. The length of the part b’ is calculated by the equation in which the ratio b’/t is equal on the limiting ratio.

For internal compression elements, the remaining parts are symmetrically divided to the end of the elements. The length of the part d’ is calculated by the equation in which the ratio d’/t is equal on the limiting ratio.

The reduced section properties are calculated for I, U, PPL, RHS and Cold formed sections-types.

The slenderness ratios (for buckling and LTB) are calculated with the full section properties.

Section properties


Buckling length

For the calculation of the buckling length, we refer to "Calculation of buckling ratio"

The buckling properties for a VARH element are calculated by using the critical Euler force for this member(see "Calculation of critical Euler force for VARH elements") .


212


For I sections, PPL sections, U sections RHS and CHS sections, the formulas from 2.1.4 are used.


z

2

t

2

z

2

z

2

EI

GIL

I

Iw

L

EIMcr

with L LTB length

E modulus of elasticity

G shear modulus

Iw warping constant

It torsion constant

Iz moment of inertia about minor axis

With this moment Mcr, the critical LTB stress LTB is calculated :

y

crLTB

I

M

with Iy moment of inertia about major axis

The slenderness ratio for LTB LTB, is given by

LTB

LTB

E

The allowable LTB stress is calculated using the slenderness LTB with the formulas given in 2.1.3.



213

Combined stresses

For compression and bending, the following formulas are used:

1f

tt

1f

c

f

c

f

t

cbybx

by

by

bx

bx

c

c

For tension and bending, the following formulas are used :

1f

tt

1f

c

f

c

f

t

bybxt

by

by

bx

bx

bx

t

with c normal compression stress

t normal tension stress

cb bending compression stress

tb bending tension stress

cbx bending compression stress around x axis

tbx bending tension stress around x axis

cby bending compression stress around y axis

tby bending tension stress around y axis

ft allowable tension stress

fc allowable compression stress

fb allowable bending stress

fbx allowable bending stress around x axis

fby allowable bending stress around y axis




214

Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube




I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM

Slender sections

x x x x x

Allowable stresses


Shear buckling x x x

References

[1] Korean Standard

(Korean Version)

1983

[2] Extracts Korean Standard

(Internal English Version)

Translated by Karam Kim - 19.03.1998

[3] Regulations of Structural Standard of

Building Architecture

(internal english document)


215


216

BSK 99

BSK 99 Code check


BSK 99

StalKonstruktioner

Boverket, Byggavdelningen, 2000

Material properties

For standard steel grades, the characteristic yield strength fyk and tensile strength fuk are defined

according to the thickness of the element (see Ref. 1, tab.2:21a and tab.2:21b)

The standard steel grades are:

Steel

Name

Type E-modulus (N/mm

2)

Poisson

Unit mass (kg /m

3)

Extensibility (m/m K)

Ultimate tensile strength (N/mm

2)

Yield strength (N/mm

2)

S235

S 235

Steel 210000 0.3 7850 12*10-6

340 235

S275

S 275

Steel 210000 0.3 7850 12*10-6

410 275

S355

S 355

Steel 210000 0.3 7850 12*10-6

490 355

S420

S 420

Steel 210000 0.3 7850 12*10-6

500 420

S460

S 460

Steel 210000 0.3 7850 12*10-6

530 460

S500

S 500

Steel 210000 0.3 7850 12*10-6

590 500

S550

S 550

Steel 210000 0.3 7850 12*10-6

640 550

S620

S

Steel 210000 0.3 7850 12*10-6

700 620


217

620

S690

S 690

Steel 210000 0.3 7850 12*10-6

770 690

(fyk, fuk in N/mm², t in mm)

Steel grade Thickness fuk fyk

S235, S 235 0 < t <= 16 340 235

16 < t <= 40 340 225

40 < t <= 63 340 215

63 < t <= 80 340 215

80 < t <=100

340 215

S275, S 275 0 < t <= 16 410 275

16 < t <= 40 410 265

40 < t <= 63 410 255

63 < t <= 80 410 245

80 < t <=100

410 235

S355, S 355 0 < t <= 16 490 355

16 < t <= 40 490 345

40 < t <= 63 490 335

63 < t <= 80 490 325

80 < t <=100

490 315

S420, S 420 0 < t <= 16 500 420

16 < t <= 40 500 400

40 < t <= 63 500 390

S460, S 460 0 < t <= 16 530 460

16 < t <= 40 530 440

40 < t <= 63 530 430

S500, S 500 0 < t <= 50 550 500

50 < t <= 100

550 480

S550, S 550 0 < t <= 50 640 550

50 < t <= 100

640 550

S620, S 620 0 < t <= 50 700 620

50 < t <= 100

700 580

S690, S 690 0 < t <= 50 770 690

50 < t <= 100

760 650


218

Remark : For cold formed section, the values for fy and fu are not influenced by the previous table.

Remark : The reduction rules from previous table are only valid when the used material is defined as material for the selected code.

Consulted articles

The cross-section is classified according to Table 6:211a . (class 1,2 or 3).

The section is checked for tension (art. 6:22), compression (6:23), bending (6:24), shear force (art. 6:26), torsion (art. 6:27), the combination of bending and axial force (art. 6:25).

A more detailed overview for the used articles is given for part 6:2 in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation in the following chapters.

6:2.Calculation of the capacity of construction elements

6:21.Limiting values of slenderness for cross section parts x

6:211.Classes of cross sections x (*)

6:212.Design methods for the different section classes x (*)

6:22.Tensile force x

6:23.Compression force x

6:231. Initial curvature, initial inclination and load eccentricity

6:232.Loss of restraint x (*)

6:233.Reduction factor for flexural buckling x

6:24.Bending moment x

6:241.Cross section classes x (*)

6:242.Shape factors in flexure x (*)

6:243.Bending moment x

6:244.Lateral torsional buckling x (*)

6:2441.Lateral bracing of beam x

6:2442.Reduction factor for LTB x

6:25. Bending and axial force

6:251.Section check x

6:252.Flexural buckling x

6:253.Flexural-torsional buckling x

6:26.Shear force and concentrated load

6:261.Shear force x(*)

6:262.Web crippling under concentrated force

6:263.Local compression

6:27.Torsional moment x

6:271.Pure torsion x

6:272.Warping

6:273.Torsional moment, shear force and bending moment x


219


For each intermediary section, the classification is determined and the proper section check is performed using the actual internal forces. The classification can change for each intermediary point.


The calculation of the effective area properties is performed according to the rules given in [5], part :23 and :24.


Aeff is the effective area of the cross section when subject to uniform compression. Weff is the effective section modulus of the cross-section when subject only to moment about the relevant axis. With these properties, the section and stability check is performed.

Section properties

6:22 ; 6:243 ; 6:251 ; 6:261 : The net area properties are not taken into account .

Section check

- Double symmetric I sections (I) use the formula (6:251a) and (6:251b)

- Solid sections (O, RS) and hollow sections (RHS, CHS) use the formula (6:251c)

- For single bending, the sections U, PPL, T use formula (6:251a). For double bending the biaxial state of stress is consulted.

- All other cases use the biaxial state of stress.

The (bi)axial stress check is given by formula (3:412a) and (3:412c):

yd

22

x

ydx

f3

f

with =1.1

Compression members

6:232 : For the calculation of the buckling length, we refer to "Calculation of buckling ratio". The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see "Calculation of critical Euler force for VARH elements") .

For class 3 sections, the rules given in [5], part :34 are used, including the calculating of Idef.


220


See [5], part :37.

The design buckling resistance for torsional or torsional-flexural buckling shall be obtained using the

following reduction factor c and slenderness c :

with fyk the basic yield strength



cr,TF the elastic critical stress for torsional-flexural buckling

G the shear modulus










221

The calculation of cr based on [6], part 6.2.3.(5).


Alternatively to the regulations given in 6:2442. for bisymmetric sections, the elastic critical moment for LTB Mcr for I sections (symmetric and asymmetric), RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, can be calculated using the formula given by the

general formula F.2. Annex F Ref. 3.

For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".


z2

t

z2

z2

EI

L²GI

I

Iw

L

EIMcr


G the shear modulus






222


For class 3 section, Izdef according to [5], part :44 is used.


Use of diaphragms


Shear force ( shear buckling)

The shear buckling check is using the values for v from table 6:261 in column 2.

The value for w is (according to [5], part :26, (18:26d)) taken as below :

2

w

w

2

w

w

k

yk

w

ww

a

b34.500.4k1

b

aif

a

b00.434.5k1

b

aif

E

f

t

b

k

81.0

with Ek the modulus of elasticity

fyk the yield strength

a the field length

bw the field height

tw the web thickness


223

a

bw


Supported sections




L Angle section

U Channel section

T T section



Cold formed section


O Solid tube




I

RHS

CHS

L

U

T

PPL

RS

O

COM

NUM


Section check

double bending


Class 3 support x x x x x x

Buck:ling check x x x x x x x x x x x x

LTB check x x x x x x x x x x x x

Compression + x


224

bending

double bending

Compression + bending

single bending

x x x x x x x x

Compression + LTB

double bending

x

Shear buckling x x x x

Torsional check x


References

[1] BSK 99

StalKonstruktioner

Boverket, Byggavdelningen, 2000

[2] Swedish Regulations for Steel Structures

BSK

SBI Swedish Institute of Steel Construction, Publication 118, 1989

[3] Eurocode 3



ENV 1993-1-1:1992, 1992

4 R. Maquoi



[5] Torsten Höglund

K18, Dimensionering av Stalkonstruktioner

Utdrag ur Handboken Bygg, kapitel K18 och K19

C E Fritzes AB, Stockholm

[6] ENV 1993-1-3:1996




CEN 1996


225

IS 800

IS:800 Code check


IS 800 Draft version (for 3rd

Revision)

Material properties

The following steel grades are supported :

Grade/ Classification Yield stress(Mpa) Ultimate tensile stress(Mpa)

A/Fe410WA 250(<20mm), 240(20mm to 40mm), 230(>40mm) 410

B/Fe410WB 250(<20mm), 240(20mm to 40mm), 230(>40mm) 410

C/Fe410WC 250(<20mm), 240(20mm to 40mm), 230(>40mm) 410

Fe440 300(<16mm), 290(16mm to 40mm), 280(>41mm to 63mm)

440

Fe440B 300(<16mm), 290(16mm to 40mm), 280(>41mm to 63mm)

440


490


490


540


540

The string in the column ‘Grade/Classification’ is used to determine the proper yield stress reduction.

Consulted articles

The cross-section is classified according to Table 3.1.

The section is checked for tension (Section 6), compression (Section 7), bending (Section 8) and the combination of forces (Section 9).


226


3.7. Classification of Cross Section x(*)

6.1. Tension members x

6.2. Design strength due to Yielding of Gross section

7.1. Design Strength x

8.2. Design strength in bending x

8.2.1. Laterally supported beam

8.2.1.1. Section with slender webs x

8.2.1.2. When factored shear force < 0.6 Vd x

8.2.1.3. When factored shear force > 0.6 Vd x

8.2.2. Laterally unsupported beam x

8.2.2.1. Elastic Lateral Torsional Buckling moment x

8.4. Shear x

8.4.1. The nominal plastic shear resistance x

8.4.2. Resistance to shear buckling x

9.1. General x

9.2. Combined Shear and bending x

9.3. Combined Axial Force and Bending Moment x

Appendix F x

Remarks

- the design of slender compression elements is outside the scope of this implementation

- the shear buckling check is only using the Simple Post Critical Method




However, for non-prismatic sections, the stability section classification is determined for each intermediary section


227

The cross sections are classified as

- class 1 : plastic

- class 2 : compact

- class 3 : semi-compact

- class 4 : slender section

The class 4 (slender) section check is not supported. For this sections a class 3 (semi-compact) section check is performed.

Section properties


Section check

In the case of high shear for class 3 section, the allowable normal stress is reduced with a factor (1-

). When torsional shear stress is present, the VonMisis criterium is checked.

Compression members

For the calculation of the buckling length, we refer to "Calculation of buckling ratio". The buckling properties for a VARH element are calculated by using the critical Euler force for this member (see "Calculation of critical Euler force for VARH elements") .


The design buckling resistance Nb,Rd for torsional or torsional-flexural buckling shall be obtained using buckling for buckling around the weak axis, and with relative slenderness given by :


228

with fyb the basic yield strength



cr,TF the elastic critical stress for torsional-flexural buckling

G the shear modulus










229


The elastic critical moment for LTB Mcr for I sections (symmetric and asymmetric), RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, can be calculated using the formula given by Annex F.

For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".


z2

t

z2

z2

EI

L²GI

I

Iw

L

EIMcr


G the shear modulus








230

Use of diaphragms


Supported sections

The following standard sections are defined :




L Angle section

U Channel section

T T section


Z Z section


Cold formed section

COM Composed section ( sheet welded, section pairs, …)

O Solid tube




In the following matrix is shown which sections are supported for the different analysis parts in the Indian steel Code check :

I

RHS

CHS

L

U

T

PPL

RS

Z

O

COM

NUM

Section Classification x x x x x x x x (1) x (1) (1) (1)




Section check class 4

Stability check class 1 x x x

Stability check class 2 x x x

Stability check class 3 x x x x x x x x x x x x x





231

References

[1] IS:800

2005


232

EAE code check


Instrucción EAE

Documento 0 de la Instrucción de Acero Estructural

Comisión Permanente de Estructuras de Acero

November 2004

Material properties

For standard steel grades, the yield strength fy and tensile strength fu are defined according to

Capítulo VI of Ref. 1.

Steel Grade fy (N/mm²)

fu (N/mm²)

S 235 235 360

S 275 275 430

S 355 355 510

S 275 N/NL 275 390

S 355 N/NL 355 490

S 420 N/NL 420 540

S 460 N/NL 460 570

S 275 M/ML 275 380

S 355 M/ML 355 470

S 420 M/ML 420 520

S 460 M/ML 460 550

S 460 Q/QL/QL1

460 570

S 235 W 235 360

S 355 W 355 510

S 235 H 235 360

S 275 H 275 430

S 355 H 355 510

S 275 NH/NLH 275 370

S 355 NH/NLH 355 470

S 460 NH/NLH 460 550

S 275 MH/MLH 275 360

S 355 MH/MLH 355 470

S 420 MH/MLH 420 500

S 460 MH/MLH 460 530


233

The name of the steel grade (e.g. 'S 355 W') is used to identify the steel grade.

Remark : For cold formed section, the values for fy and fu are not influenced by the previous table

Remark : For cold formed sections, the average yield strength fya can be used (by setting the proper data flag in the Cross Section input dialog) according to Ref.[4].

The average yield strength is determined as follows :

with fyb the tensile yield strength = fy

fu the tensile ultimate strength

t the material thickness

Ag the gross cross-sectional area

k is a coefficient depending on the type of forming :

k = 0.7 for cold rolling

k = 0.5 for other methods of forming

n the number of 90° bends in the section

Consulted articles

The beam elements are checked according to the regulations given in " Instrucción EAE, Documento 0 de la Instrucción de Acero Estructural, Comisión Permanente de Estructuras de Acero, November 2004".

The cross-sections are classified according to Artículo 20 of Capítulo V. All classes of cross-sections are included. For class 4 sections (slender sections) the effective section is calculated in each intermediary point, according to Artículo 20 of Capítulo V.

The member check is executed according to Capítulo IX. The stress check is taken from art. 34.: the section is checked for tension (art. 34.2.), compression (art. 34.3.), bending (art. 34.4.), shear (art. 34.5.), torsion (art. 34.6.) and combined bending, shear and axial force (art. 34.7.1., art. 34.7.2. and art. 34.7.3.).

The stability check is taken from art. 35.: the beam element is checked for buckling (art. 35.1.), lateral torsional buckling (art. 35.2.), and combined bending and axial compression (art. 35.3.).

The shear buckling is checked according to prEN 1993-1-5:2003, Chapter 5.

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

A check for critical slenderness and torsion moment is also included.

For integrated beams, the local plate bending is taken into account for the plastic moment capacity and the bending stresses in the section. The out-of-balance loading is checked.


234

A more detailed overview for the used articles is given in the following table. The chapters marked with “x” are consulted. The chapters marked with (*) have a supplementary explanation the following chapters.

Instrucción EAE

20. Clasificación de las secciones transversales (*)

20.2. Clasificación de las secciones transversales metálicas x

20.3. Criterios de asignación de Clase en secciones metálicas no rigidizadas x

20.7. Características de la sección reducida en secciones transversales esbeltas

x

34. Estado límite de resistencia de las secciones

34.1. Principios generales del cálculo x

34.1.2. Características de las secciones transversales x(*)

34.2. Esfuerzo axil de tracción x

34.3. Esfuerzo axil de compresión x

34.4. Momento flector x

34.5. Esfuerzo cortante x

34.6. Torsión x(*)

34.7. Interacción de esfuerzos

34.7.1. Flexión y cortante x

34.7.2. Flexión y esfuerzo axil x

34.7.3. Flexión, cortante y esfuerzo axil x

35. Estado límite de inestabilidad

35.1. Elementos sometidos a compresión x(*)

35.2. Elementos sometidos a flexión x

35.3. Elementos sometidos a compresión y flexión x(*)

35.5. Abolladura del alma a cortante x

35.7. Interacción

35.7.1. Cortante, flexión y esfuerzo axil x

For cold formed sections EN 1993-1-3 is applied.

6.1.2. Axial tension x

6.1.3. Axial compression x

6.1.5. Shear force x

6.1.6. Torsional moment x


235












Section properties


The shear lag effects are neglected .

Torsion check


Built-in beams

For built-in beam sections (IFB, SFB, THQ sections), proper section checks are performed, taking into account the local plate bending. See Chapter ‘Section check for built-in beams (IFB, SFB, THQ sections)’.


236

Compression members




For I sections (symmetric and asymmetric), RHS (Rectangular Hollow Section) sections and CHS (Circular Hollow Section) sections, the elastic critical moment for LTB Mcr is given by the general

formula F.2. Annex F Ref. 5. For the calculation of the moment factors C1, C2 and C3 we refer to "Calculation of moment factors for LTB".


z2

t

z2

z2

EI

L²GI

I

Iw

L

EIMcr


G the shear modulus








Use of diaphragms



237

Combined bending and axial compression

For prismatic members the value My,Ed is the maximum value of the bending moment around the strong axis in the member. The value Mz,Ed is the maximum value of the bending moment around the weak axis in the member.

For non-prismatic sections, the values My,Ed and Mz,Ed are the concurrent bending moments for each intermediary section.

Interaction Method Calculation of Czz

By default for Czz the formula given in Ref.[1] is used:

In this formula however the position of the factor eLT is incorrect. For exact analysis the formula according to Ref.[9] can be used:



Supported sections




L Angle section

U Channel section

T T section


Z Z section


Cold formed section


O Solid tube




238


I

RHS

CHS

L

U

T

PPL

RS

Z

O

COM

NUM







x x x


x x x




x x x x x x



References

1 Instrucción EAE

Documento 0 de la Instrucción de Acero Estructural

Comisión Permanente de Estructuras de Acero

November 2004

2 Essentials of Eurocode 3


ECCS - N° 65, 1991

3 R. Maquoi



[4] ENV 1993-1-3:1996




CEN 1996

[5] Eurocode 3



ENV 1993-1-1:1992/A1, 1994


239

[6] Eurocode 3



ENV 1993-1-2:1995, 1995


ECCS - N° 111

May 2001

[8] Eurocode 1

Basis of design and actions on structures

Part 2-2 : Actions on structures - Actions on structures exposed to fire

ENV 1991-2-2:1995


240

Calculation of buckling ratio

Introduction to the calculation of buckling ratio

For the calculation of buckling ratio, several methods can be applied.

The general method is described in chapter "Calculation buckling ratio – general formula".

For crossing diagonals, the buckling ratio is explained in chapter "Calculation buckling ratios for crossing diagonals".

For VARH elements, the critical Euler force is calculated according to the method given in chapter "Calculation of critical Euler force for VARH elements".

For lattice tower members, see the chapter "Calculation buckling ratio for lattice tower members".

When using member buckling data the buckling ratio can be calculated from a stability analysis. See chapter Calculation of buckling ratio – From Stability Analysis.

Calculation buckling ratio – general formula

For the calculation of the buckling ratios, some approximate formulas are used. These formulas are treated in reference [1], [2] and [3].

The following formulas are used for the buckling ratios (Ref[1],pp.21) :

for a non sway structure :

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

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

21212121

21212121

for a sway structure :

4+x

x=l/L1

2

with L the system length

E the modulus of Young

I the moment of inertia

Ci the stiffness in node i

Mi the moment in node i

Fi the rotation in node i


241

21212

12

21

8+)+(

+4=x

EI

LC= i

i

i

ii

M=C

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 form. (See also Ref.[5], pp.113 and Ref.[6],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[1],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 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.


242

Calculation buckling ratios for crossing diagonals

For crossing diagonal elements, the buckling length perpendicular to the diagonal plane, is calculated according to Ref.[4], 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 chapters, the buckling length sK is defined,

with sK buckling length

l member length

l1 length of supporting diagonal

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

I1 moment of inertia (in the buckling plane) of the supporting diagonal

N compression force in member

N1 compression force in supporting diagonal

Z tension force in supporting diagonal

E elastic modulus

Continuous compression diagonal, supported by continuous tension diagonal

NN

Z

Z

l/2

l1/2

l5.0s

lI

l1I1

lN4

lZ31

ls

K

3

1

3

1

K

See Ref.[4], Tab. 15, case 1.


243

Continuous compression diagonal, supported by pinned tension diagonal

NN

Z

Z

l/2

l1/2

l5.0s

lN

lZ75.01ls

K

1

K


Pinned compression diagonal, supported by continuous tension diagonal

NN

Z

Z

l/2

l1/2

)1lZ

lN(

4

lZ3)IE(

1lZ

lN

l5.0s

1

2

2

1

d1

1

K



244

Continuous compression diagonal, supported by continuous compression diagonal

NN

N1

N1

l/2

l1/2

l5.0s

lI

l1I1

lN

lN1

ls

K

3

1

3

1

1

K


Continuous compression diagonal, supported by pinned compression diagonal

NN

N1

N1

l/2

l1/2

1

1

2

KlN

lN

121ls

See Ref.[4], Tab. 15, case 3 (2).


245

Pinned compression diagonal, supported by continuous compression diagonal

NN

N1

N1

l/2

l1/2

)N

lN

12(

l

lN)IE(

l5.0s

1

1

2

1

2

3

d

K

See Ref.[4], Tab. 15, case 3 (3).


246

Calculation of critical Euler force for VARH elements

Definitions

A VARH element is defined as follows :

The member has the properties of a symmetric I secion (formcode=1), where only the height is linear variable along the member. The system length for buckling around the local yy axis (strong axis), is equal to member length.

For this non-prismatic section, the critical Euler force is given in Ref[7].

Calculation of the critical Euler force

For a VARH element (form node i to node j), we can define:

L beam length

Ii, Ij moment of inertia at end i and j

Ai, Aj

sectional area at end i and j

E modulus of Young

Ncr critical Euler force

Ri, Rj

beam stiffness at end i and j

The stiffness R and R' is given by:

EI

LR=R

EI

LR=R

M=R

ijj

iii

I

I=

i

j

The critical Euler force is given by

L

EI=N 2

i2cr

To calculate , the next steps are followed :


247

1. Calculate L, Ii, Ij, Ri, Rj, R'i, R'j, ξ

2. We suppose that

2

1>

1-

3. Calculate a, b, c and d as follows

)]lncotg(+2

1(

1)-(+[1

1=d

]1-

)ln(sin-[1

1=c=b

)]lncotg(-2

11)(-(+[1

1=a

4

1-

)1-(=

2

2

2

2

2

4. For a beam in non-sway system, we solve

0=RRbc)-(ad+Rd+Ra+1 jiji

For a beam in sway system, we solve

0=bc))-(ad-d+c-b-(aRR+-)d-(1R+)a-(1R2

ji22

j2

i

5. When a solution is found, we check if

2

1>

1-

6. If not, then recalculate a,b,c en d as follows :


248

])-(

))+2

1(-)-

2

11)((-(

+[11

=d

])-(

1)-(2-[1

1=c=b

]-

))+2

1(-)-

2

11)((-(

+[11

=a

-

-

2

-2

-

2

and resolve the proper equation of 4.

Calculation buckling ratio for lattice tower members When the national code EC-ENV is selected, the following buckling configuration can be selected.

For each configuration, the critical slenderness to be considered, is defined.

The values are taken from Ref.[8].

y

y

z z

v

v

We define :

iyy radius of gyration around yy axis

izz radius of gyration around zz axis

ivv radius of gyration around vv axis

With the option 'Bracing members are sufficiently supported', the effective slenderness may be reduced as follows:

- for vv-axis : vv7.035.0

- for yy-axis : yy7.050.0

The buckling curve 'b' is used.


249

Leg with symmetrical bracing

vvi

L

Leg with intermediate transverse support

yyi

L


250

Leg with staggered bracing

vv

yy

i

52.1)2a,1amax(

i

L

Single Bracing

vvi

L


251

Single Bracing with SBS (Secondary Bracing System)

yy

2

vv

1

i

L

i

L

Cross bracing

yy

com

com

y

E

E

com

com

com

com

1

1

1b

1

com

sup

1b

2bcomb

'

2

zz

'

2

yy

'

2

vv

1

i

L

f

E

58.070.0K

L

L

K

1125.0K

0.15.0K

1125.0

F

F

K

1138.075.0K

LKLKL

i

L,

i

L

i

L


252

with Lcom Length of compressed member (L2 from figure)

Fcom Force in compressed member (L2 from figure)

Fsup Force in supporting member (member crossing member L2)

E Modulus of Young

fy Yield strength

Cross bracing with SBS

3bcomb

'

3

zz

'

3

yy

'

3

zz

2

yy

2

vv

1

LKLKL

i

L,

i

L

i

L,

i

L

i

L

with Lcom Length of compressed member (L3 from figure)

Fcom Force in compressed member (L3 from figure)

Fsup Force in supporting member (member crossing member L3)

Kb See Chapter 'Cross bracing'


253

K Bracing

zz

3

yy

3

zz

2

yy

2

vv

1

i

L,

i

L

i

L,

i

L

i

L

Horizontal Bracing

L

1R0

P

PR

73.0R316.0R085.0k

i

Lk

1

2

2

vv

with P1 Compression load

P2 Tensile load


254

Horizontal Bracing with SBS

L

1R0

P

PR

73.0R316.0R085.0k

i

Lk

1

2

2

yy

with P1 Compression load

P2 Tensile load


255

Discontinuous Cross bracing with horizontal member

N1 N2

N1N2

F F

a

a

cos)2N1N(,FmaxF

i

a,

i

a2

Sd

vvyy

with F normal force to check

FSd actual compression force in horizontal member

N1 tensile force in diagonal

N2 compression force in diagonal


256

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:

Edcr NN

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.

References

[1] Handleiding moduul STACO VGI

Staalbouwkundig Genootschap

Staalcentrum Nederland

5684/82

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

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

[3] Stabiliteit voor de staalconstructeur

uitgave Staalbouwkundig Genootschap

[4] DIN18800 Teil 2

Stahlbauten : Stabilitätsfälle, Knicken von Stäben und Stabwerken

November 1990

[5] Rapportnr. BI-87-20/63.4.3360

Controleregels voor lijnvormige constructie-elementen

IBBC Maart 1987


257




[7] Y. Galéa

Flambement des poteaux à inertie variable

Construction Métallique 1-1981

[8] NEN-EN 50341-3-15

Overhead electrical lines exceeding AC 45 kV - Part 3: Set of National Normative Aspects

Number 15: National Normative Aspects (NNA) for The Netherlands


258

Calculation of moment factors for LTB

Introduction to the calculation of moment factors

For determining the moment factors C1 and C2 for lateral torsional buckling (LTB), we use the standard tables which are defined in Ref.[1] 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 moment lines generated by a distributed q load, a nodal F load, or where the moment line is 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.

The factor C3 is taken out of the tables F.1.1. and F.1.2. from Ref.[2] - Annex F.

Calculation moment factors

Moment distribution generated by q load

For ENV 1993, IS800 and CM66

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*


259

For DIN18800 and ONORM4300

if M2 < 0

C1 = A* (1.45 B

* + 1) 1.12 + B

* (-0.71 A

* + 1) E

*

C2 = 0.45 A* [1 + C

* e

D* (½ + ½)]

if M2 > 0

C1 = 1.12 A* + B

* E

*

C2 = 0.45A*

with : l+q|M2|8

lq=A

2

2*

ql

|M2|94=C

2

*

l+q|M2|8

|M2|8=B

2

* )

ql

|M2|-72(=D

2

2

*

For DIN18800 / ONORM 4300

0.77-1.77=E*

For ENV 1993 and IS800

2.70<E*

0.52+1.40-1.88=E*2

For NEN6770/6771, SIA263

E*=1.75-1.05*+0.30*² and E*<2.3

For CM66

2.70<E*

0.52+1.40-1.88=E*2


260

Moment distribution generated by F load

FM2 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 chapter "Moment distribution generated by q load".


261

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

M2 M1 = Beta M2

l

C2 = 0.0

For DIN18800 / ONORM 4300

0.77-1.77=1C

For ENV 1993 / IS800

2.70<1C and

0.52+1.40-1.88=1C2

For CM66

For NEN6770/6771, SIA263 Code

E*=1.75-1.05*+0.30*² and E*<2.3

References


Stabiliteit

NEN 6771 - 1991

[2] Eurocode 3 : Design of steel structures

Part 1-1 : General rules and rules for buildings

ENV 1993-1-1:1992


262

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.[1].

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

1) Calculation of Mcr through eigenvalue solution

2) 2nd

Order calculation including torsional and warping effects

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

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

Eigenvalue solution Mcr

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

- Appropriate end conditions for torsion and warping

- End and begin forces

- Loadings

- 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 chapter “Linked Beams”.

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


263

With

Critical load factor

Ke Elastic linear stiffness matrix

Kg Geometrical stiffness matrix

For members with arbitrary sections, the critical moment can be obtained in each section, with: (See Ref.[3],pp.176)

With

Critical load factor

Myy Bending moment around the strong axis

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

Mcr(x) Critical moment at position x

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

For more background information, reference is made to Ref[2].

0KK ge

)x(MxM

MmaxM

yycr

yycr


264

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 chapter “Linked Beams”.

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

Order 7 degrees of freedom calculation.

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

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[2].

Supported National Codes The following codes are supported for the analysis of Mcr.

- EC3 - ENV

- EC3 - EN

- DIN18800

- ONORM

- NEN

- SIA

- IS

- EAE


265

For the following national codes, the 2nd Order analysis approach is supported.

- EC3 - ENV

- EC3 - EN

- DIN18800

- ONORM

- NEN

- SIA

- EAE

Supported Sections

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

FRILO LTBII CSS Scia Engineer CSS Eigenvalue

analysis

2nd

Order

analysis

Double T I section from library x x

Thin walled geometric I x x

Sheet welded Iw x x

Double T unequal IPY from library x x

Thin walled geometric asymmetric I x x

Haunched sections x x

Welded I+Tl x x

Sheet welded Iwn x x

HAT Section IFBA, IFBB x x

U cross section U section from library x x

Thin walled geometric U x x

Thin walled Cold formed from library x x

Cold formed from graphical input x x

Double T with top flange angle

Welded I+2L x

Sheet welded Iw+2L x

Rectangle Full rectangular from library x

Full rectangular from thin walled geometric

x

Static values double symmetric

all other double symmetric CSS x

Static values single symmetric

all other single symmetric CSS x

Remark: Haunched sections are replaced by equivalent asymmetric I sections, by ignoring the middle flanges.


266

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.

For more information, reference is made to Ref[2]


267

Loadings

The following load impulses are supported:

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

- Point force on beam

- Line force in beam

- Moment in node (if the node is part of the exported beam)

- Moment on beam

- 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.

Imperfections

In the 2nd

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

v0

y, v0

z

y


268

For DIN, ONORM, EC-EN and EAE the imperfections can be calculated according to the code. The codes indicate that 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.

Initial bow imperfection v0 for DIN and ONORM

The imperfection is calculated according to Ref.[6] article 2.2

For prismatic uniform members:

Resistance check Section Bucking curve v0

EE

(Elastic)

any a0 L/1050

any a L/900

any b L/750

any c L/600

any d L/450

EP

PP

(Plastic)

I section a0 L/700

I section a L/600

I section b L/500

I section c L/400

I section d L/300

For non-uniform members, the bow imperfection is considered at the centre of the buckling system length L.

Initial bow imperfection v0 for EC-EN and EAE

The imperfection is calculated according to Ref.[4] article 5.3.4(3)

00 ekv

With k Factor taken from the National Annex of EC-EN

Factor taken as 0,5 for EAE

e0 Bow imperfection of the weak axis


269

The value of e0 is taken from following table:

Buckling curve

eo /L – elastic analysis

eo/L – plastic analysis

a0 1/350 1/300

a 1/300 1/250

b 1/250 1/200

c 1/200 1/150

d 1/150 1/100

With L Member system length

Initial bow imperfections v0 and w0 for other supported codes

For all other supported codes (EC-ENV, NEN and SIA) as well as DIN, ONORM, EC-EN and EAE the user can manually input the imperfections v0 and w0.


270

LTB Restraints LTB restraints are transformed into 'Supports' (Ref.[2] 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.[2] 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 Chapter “Supported Sections”)


271

Diaphragms Diaphragms are transformed into 'Elastic Foundations' of type ‘elastic restraint’ (Ref.[2] 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.[2] p52 and Ref.5 p40):

2

LSCy

With

S Shear stiffness of the diaphragm

L Diaphragm length along the member

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

pitch of ‘2br’ the shear stiffness S is replaced by 0,2 S (Ref.5 p22).

The shear stiffness S for a diaphragm is calculated as follows (Ref.7,3.5 and Ref.8,3.3.4.):

L

K+K

10a.=S

s

21

4

With a Frame distance

Ls Length of the diaphragm

K1 Factor K1 of the diaphragm

K2 Factor K2 of the diaphragm

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

Specifically for U-sections, an elastic restraint Cz is used with position y(Cz) due to the rotation of U-sections in the FriLo LTBII solver. (see Chapter “Supported sections”)


272

The rotational restraint C [kNm/m] is taken as vorhC (see Chapter “Adaptation of Torsional Constant”)

Linked Beams

Linked beams are transformed into 'Supports' (Ref.[2] 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 then 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.[2] p23).

The end forces of the linked beam are transformed to point loads on the considered 1D member,

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

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

Specifically for U-sections, if the linked beam has an angle less then 45° with the local y-axis of the beam under consideration, the restraint is set as Cz. In all other cases the restraint is set as Cy. This is due to the rotation of U-sections in the FriLo LTBII solver. (see Chapter “Supported Sections”)


273

Limitations and Warnings

The FRILO LTB 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. 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 2nd

order analysis instead of an eigenvalue calculation.

- Bending of U-section about y-axis!

The program calculates the minimum bifurcation load only.

2nd

Order Analysis

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

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

order calculation cannot be executed.

- You want to calculate the structural safety with Elastic-Plastic method. This analytical procedure cannot be used for this cross-section. It is recommended to use the Elastic-Elastic method.

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

For more information, reference is made to Ref[1] and [2].


274

References

[1] FriLo LTBII software Friedrich + Lochner Lateral Torsional Buckling 2

nd Order Analysis

Biegetorsionstheorie II.Ordnung (BTII)

http://www.frilo.de

[2] Friedrich + Lochner LTBII Manual BTII Handbuch Revision 1/2006

[3] J. Meister

Nachweispraxis Biegeknicken und Biegedrillknicken

Ernst & Sohn, 2002

[4] Eurocode 3



EN 1993-1-1:2005

[5] J. Schikowski

Stabilisierung von Hallenbauten unter besonderer Berücksichtigung der Scheibenwirkung von Trapez- und Sandwichelementdeckungen, 1999 http://www.jschik.de/

[6] DIN 18800 Teil 2

Stahlbauten


November 1990

[7] E. Kahlmeyer




Stahlbauten



http://www.frilo.de/

http://www.jschik.de/


275

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 :

Description Property number

Iy moment of inertia yy 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.

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.


276

Data depending On the profile shape

I section

Formcode 1

PSS Type .I.


49 H

48 B

44 t

47 s

66 R

74 W

140 wm1

61 R1

146

109 1

B

s

w

t

R

R1

a

H


277

RHS

Formcode 2

PSS Type .M.


49 H

48 B

67 s

66 R

109 2

B

s

H

R


278

CHS

Formcode 3

PSS Type .RO.


64 D

65 s

109 3

D

w


279

Angle section

Formcode 4

PSS Type .L.


49 H

48 B

44 t

61 R1

66 R

74 W1

75 W2

76 W3

109 4

B

R

R1

w1

w2

t

w3

w1

w2


280

Channel section

Formcode 5

PSS Type .U.


49 H

48 B

44 t

47 s

66 R

68

41

61 R1

146

109 5

B

s

H

t

R

R1

a


281

T section

Formcode 6

PSS Type .T.


49 H

48 B

44 t

47 s

66 R

61 R1

62 R2

146 1

147 2

109 6

B

s

t

R

a1

H

a2

R1

R2


282

Full rectangular section

Formcode 7

PSS Type .B.


48 B

67 H

109 7

B

H


283

Full circular section

Formcode 11

PSS Type .RU.


64 D

109 11

D


284

Asymmetric I section

Formcode 101

PSS Type


49 H

48

44

47 s

42 Bt

43 Bb

45 tt

46 tb

66 R

109 101

R

H

Bt

Bb

tt

tb


285

Z section

Formcode 102

PSS Type .Z.


49 H

48 B

44 t

47 s

67 R

61 R1

109 102

B

s

t

H

RR1


286

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 unstiffened 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 Property number Value

form code 109 110

Dy* 22

Dz* 23

CM* 26

buckling curve around yy axis 106 (1)

buckling curve around zz axis 107 (1)

buckling curve for LTB 108 (1)

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

1 = buckling curve a

2 = buckling curve b

3 = buckling curve c

4 = buckling curve d

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

When the section is made out of 1 plate, the properties marked with (*) can be calculated by the calculation routine in the profile library. The properties from the reduced section can be calculated by the code check.

When the section is made out of more than 1 plate, the properties marked with (*) can NOT be calculated by the calculation routine in the profile library. The properties from the reduced section can be calculated, except for the marked properties. These properties have to be input by the user in the profile library.


287

Formcode 110

PSS Type


44 s

61 r

48 B

142 sp

143 e2

68 H

109 110

Remark:

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

sp is number of shear planes

B

H

e2

s


288

Cold formed Angle section

Formcode 111

PSS Type


44 s

61 r

48 B

68 H

109 111

B

s

H

r


289

Cold formed Channel section

Formcode 112

PSS Type


44 s

61 r

48 B

49 H

109 112

B

s

H

r


290

Cold formed Z section

Formcode 113

PSS Type


44 s

61 r

48 B

49 H

109 113

B

s

H

R


291

Cold formed C section

Formcode 114

PSS Type


44 s

61 r

48 B

49 H

68 c

109 114

B

s

H

r

c


292

Cold formed Omega section

Formcode 115

PSS Type


44 s

61 r

48 B

49 H

42 c

109 115

B

s

H

c

R


293

Cold formed C section eaves beam

Formcode 116

PSS Type


49 H

44 t

48 B

61 r1

68 R

163 A

109 116


294

Cold formed C Plus section

Formcode 117

PSS Type


49 H

44 t

48 B

61 r1

68 R

164 PL

167 APL

109 117


295

Cold formed ZED section

Formcode 118

PSS Type


49 H

44 t

42 Bt

43 Bb

61 r1

68 R

109 118


296

Cold formed ZED section asymmetric lips

Formcode 119

PSS Type


49 H

44 t

42 Bt

43 Bb

61 r1

68 R

164 PL

109 119


297

Cold formed ZED section inclined lip

Formcode 120

PSS Type


49 H

44 t

42 Bt

43 Bb

61 r1

68 R

164 PL

165 AL

109 120


298

Cold formed Sigma section

Formcode 121

PSS Type


48 B

166 B1

49 H

50 H1

51 H2

68 R

44 t

61 r1

109 121


299

Cold formed Sigma section stiffened

Formcode 122

PSS Type


48 B

166 B1

49 H

50 H1

51 H2

68 R

44 t

61 r1

164 PL

109 122


300

Cold formed Sigma Plus section

Formcode 123

PSS Type


48 B

166 B1

49 H

50 H1

51 H2

68 R

44 t

61 r1

164 PL

167 APL

109 123


301

Cold formed Sigma section eaves beam

Formcode 124

PSS Type


48 B

166 B1

49 H

50 H1

51 H2

68 R

44 t

61 r1

163 A

109 124


302

Cold formed Sigma Plus section eaves beam

Formcode 125

PSS Type


48 B

166 B1

49 H

50 H1

51 H2

68 R

44 t

61 r1

163 A

164 PL

167 APL

109 125


303

Cold formed ZED section both lips inclined

Formcode 126

PSS Type


49 H

44 t

42 Bt

43 Bb

61 r1

68 R

164 PL

165 AL

109 126


304

Cold formed I-Plus section

Formcode 127

PSS Type


49 H

44 t

48 B

61 r1

68 R

164 PL

167 APL

168 a

109 127


305

Cold formed IS-Plus section

Formcode 128

PSS Type


48 B

166 B1

49 H

50 H1

51 H2

68 R

44 t

61 r1

164 PL

167 APL

168 a

109 128


306

Cold formed Sigma section asymmetric

Formcode 129

PSS Type


42 Bt

43 Bb

166 B1

49 H

50 H1

51 H2

68 R

44 t

61 r1

164 PL

109 129


307

Rail type KA

Formcode 150

PSS Type .KA.


148 h1

149 h2

150 h3

151 b1

152 b2

153 b3

154 k

155 f1

156 f2

157 f3

61 r1

62 r2

63 r3

158 r4

159 r5

160 a

109 150

r1

r2

r4

r3

r5

b3

k

b2

b1

f3

f2

f1

h1

h3h2


308

Rail type KF

Formcode 151

PSS Type .KF.


48 b

154 k

49 h

153 b3

155 f1

157 f3

148 h1

149 h2

61 r1

62 r2

63 r3

109 151

r1

r2r2

r2 r2

r3

k

bb3

f3

f1

h

h1 h2


309

Rail type KQ

Formcode 152

PSS Type .KQ.


48 b

154 k

49 h

153 b3

155 f1

149 h2

150 h3

61 r1

109 152

b

k

b3

r1

h3

h2

f1


310

Warping check

Stress check

In cross sections subject to torsion, the following is checked:

Ed,wEd,tEd,VzEd,VyEd,tot

Ed,wEd,MzEd,MyEd,NEd,tot

M

y2

Ed,tot

2

Ed,tot

0M

y

Ed,tot

M

y

Ed,tot

f1.13

3

f

f

with fy the yield strength

tot,Ed the total direct stress

tot,Ed the total shear stress

M = M0 (class 1,2 and 3 section)

= M1 (class 4 section)

M0 the partial safety factor for resistance of cross-sections where failure is caused by yielding

M1 the partial safety factor for resistance of cross-sections where failure is caused by buckling

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

My,Ed the direct stress due to the bending moment around y axis on the relevant effective cross-section

Mz,Ed the direct stress due to the bending moment around z axis on the relevant effective cross-section

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

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

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

t,Ed the shear stress due to uniform (St. Venant) torsion on the gross cross-section

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


311

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

sections”) sections. The definition of I sections and U sections, and sections are described in ‘Profile conditions for code check’.

The other standard sections ( RHS, CHS, Angle section, T section and rectangular sections) are considered as warping free. See also Ref.[2], Bild 7.4.40.

Calculation of the direct stress due to warping

The direct stress due to warping is given by (Ref.[2] 7.4.3.2.3, Ref.[3])

m

MwEd,w

C

wM

with Mw the bimoment

wM the unit warping

Cm the warping constant


312

I sections

For I sections, the value of wM is given in the tables (Ref. [2], Tafel 7.87, 7.88). This value is added to the profile library. The diagram of wM is given in the following figure:

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

The value for wM can be calculated by (Ref.[5] pp.135) :

mM hb4

1w

with b the section width

hm the section height (see figure)


313

U sections

For U sections, the value of wM is given in the tables as wM1 and wM2 (Ref. [2], 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.[2] 7.4.3.2.3 and Ref.[8] Part 27.

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


314

Calculation of the shear stress due to warping

The shear stress due to warping is given by (Ref.[2] 7.4.3.2.3, Ref.[3])

s

0

M

m

xsEd,w tdsw

tC

M

with Mxs the warping torque (see "Standard diagrams for warping torque, bimoment and the St.Venant torsion")

wM the unit warping

Cm the warping constant

t the element thickness

I sections

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

For I sections, we have the following :

A4

wtbtdsw M

2/b

0

M


315

U sections, sections

Starting from the wM diagram, we calculate the value

s

0

M tdsw

for the critical points.

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


316

Plastic Check For doubly symmetric I sections of class 1 and class 2 (plastic check), the interaction formula given in Ref.[10] is used.

b

tw

tf

h Hy y

z

z

Used variables

Section Properties

A sectional area

b width

H heigth of section

tf flange thickness

tw web thickness

h = H - tf

Aw = 1.05 (h+tf) tw for rolled section

Aw = h tw for welded sections

ff tb2A

A

A ff

fw 1

Wz,pl plastic section modulus around z axis

Wy,pl plastic section modulus around y axis

Material Properties


317

fy,d yield strength

y,d shear strength

Internal forces

NSd normal force

My,Sd bending moment around y axis

Mz,Sd bending moment around z axis

Mw,Sd bimoment

Vy,Sd shear force in y direction

Vz,Sd shear force in z direction

Mxp,Sd torque due to St. Venant

Mxs,Sd warping torque

Plastic capacities

Npl,Rd = A fy,d

Mz,pl,Rd = Wz,pl fy,d

Vz,pl,Rd = Aw y,d

d,y

2

w2

fRd,pl,xp2

thbtM

My,pl,Rd = Wy,pl fy,d

2

hMM Rd,pl,zRd,pl,w

Vy,pl,Rd = Af y,d

2

hVM Rd,pl,yRd,pl,xs

Rd,pl

Sd

N

Nn

Rd,pl,y

Sd,y

yM

Mm

Rd,pl,z

Sd,z

zM

Mm

Rd,pl,w

Sd,w

wM

Mm


318

Rd,pl,y

Sd,y

yV

Vv

Rd,pl,z

Sd,z

zV

Vv

Rd,pl,xp

Sd,xp

xpM

Mm

Rd,pl,xs

Sd,xs

xsM

Mm

Shear force reduction

Sign

p=sign ( Mz,Sd x Mw,Sd)

ww

swwss

np1s4


319

Unity checks:

Remark: the values between must be > 0.

Standard diagrams for warping torque, bimoment and the St.Venant torsion

The following 6 standard situations are given in the literature (Ref.[2], Ref.[3]).

The value is defined as follows :

m

t

CE

IG

with Mx the total torque

= Mxp + Mxs

Mxp the torque due to St. Venant

Mxs the warping torque

Mw the bimoment

IT the torsional constant



G the shear modulus


320

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

Mx

L

aMM

L

bMM

txb

txa

Mxp for a side

)xcosh(

)Lsinh(

)bsinh(

L

bMM txp

Mxp for b side

)'xcosh(

)Lsinh(

)asinh(

L

aMM txp

Mxs for a side

)xcosh(

)Lsinh(

)bsinh(MM txs

Mxs for b side

)'xcosh(

)Lsinh(

)asinh(MM txs

Mw for a side

)xsinh(

)Lsinh(

)bsinh(MM t

w

Mw for b side

)'xsinh(

)Lsinh(

)asinh(MM t

w


321

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

Mx

L

aMM

L

bMM

txb

txa

Mxp for a side

3D

L

1k2kbMM txp

Mxp for b side

4D

L

1ka2kMM txp

Mxs for a side 3DMM txs

Mxs for b side 4DMM txs

Mw for a side 1D

MM t

w

Mw for b side 2D

MM t

w


322


323

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

Mx

2

LmM

2

LmM

txb

txa

Mxp

)Lsinh(

)'xcosh()xcosh()x

2

L(

mM t

xp

Mxs

)Lsinh(

)'xcosh()xcosh(mM t

xs

Mw

)Lsinh(

)'xsinh()xsinh(1

mM

2

tw


324

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

Mx

2

LmM

2

LmM

txb

txa

Mxp

)Lsinh(

)'xcosh()xcosh()k1()x

2

L(

mM t

xp

Mxs

)Lsinh(

)'xcosh()xcosh()k1(

mM t

xs

Mw

)Lsinh(

)'xsinh()xsinh()k1(1

mM

2

tw

)2

Ltanh(

2

L

1k


325

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

Mx

txa MM

Mxp

)Lcosh(

)'xcosh(1MM txp

Mxs

)Lcosh(

)'xcosh(MM txs

Mw

)Lcosh(

)'xsinh(MM t

w


326

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

Mx

LmM txa

Mxp

)Lcosh(

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

mM t

xp

Mxs

)Lcosh(

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

mM t

xs

Mw

)Lcosh(

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

²

mM t

w


327

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


328

Decomposition for situation 1 and situation 3

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

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

one torsion line generated by a distributed torsional loading mt

one torsion line with constant torque Mt0

The values for Mxp, Mxs and Mw are taken from the previous tables for the local torsional loadings 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:

one torsion line generated by a local torsional loading Mtn

one torsion line generated by a distributed torsional loading mt

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

References

[1] ENV 1993-1-3:1996


Part 1-3 : General rules – Supplementary rules for cold formed thin gauge members and sheeting

CEN 1996

[2] Stahl im Hochbau

14. Auglage Band I/ Teil 2

Verlag Stahleisen mbH, Düsseldorf 1986

[3] Kaltprofile

3. Auflage

Verlag Stahleisen mbH, Düsseldorf 1982

[4] Roik, Carl, Lindner

Biegetorsionsprobleme gerader dünnwandiger Stäbe

Verlag von Wilhem ernst & Sohn, Berlin 1972

[5] Dietrich von Berg

Krane und Kranbahnen – Berechnung Konstruktion Ausführung

B.G. Teubner, Stuttgart 1988


329

[6] DASt-Richtlinie 016

Bemessung und konstruktive Gestaltung von Tragwerken aus dünnwandigen kaltgeformten Bauteilen

Stahlbau-Verlagsgesellschaft, Köln 1992

[7] Esa Prima Win

Steel Code Check Manual

SCIA

EPW 3.10

[8] C. Petersen

Stahlbau : Grundlagen der Berechnung und baulichen Ausbildung von Stahlbauten

Friedr. Vieweg & Sohn, Braunschweig 1988

[9] Eurocode 3



ENV 1993-1-1:1992, 1992

[10] I. Vayas,

Interaktion der plastischen Grenzschnittgrössen doppelsymmetrischer I-Querschnitte

Stahlbau 69 (2000), Heft 9


330

Check of numerical sections

Stress check

The stress calculation for a numerical section is as follows:

with vm the VonMises stress, the composed stress

tot the total normal stress

tot the total shear stress

N the normal stress due to the normal force N

My the normal stress due to the bending moment Myy around y axis

Mz the norma stress due to the bending moment Mzz around z axis

Vy the shear stress due to shear force Vy in y direction

Vz the shear stress due to shear force Vz in z direction

Ax the sectional area

Ay the shear area in y direction

Az the shear area in z direction

Wy the elastic section modulus around y axis

Wz the elastic section modulus around z axis

z

zVz

y

y

Vy

z

zzMz

y

yy

My

x

N

VzVytot

MzMyNtot

2tot

2totvm

A

V

A

V

W

M

W

M

A

N

3


331

Use of diaphragms

Adaptation of torsional constant

See Ref.[1], Chapter 10.1.5., Ref.2,3.5 and Ref.3,3.3.4..

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.

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

12

³sI

)th(

IE3C

200b125if100

bC25.1C

125bif100

bCC

s

EIkC

C

1

C

1

C

1

vorhC

1

G

lvorhCII

s

sk,P

aa

100k,A

a

2

a100k,A

effk,M

k,Pk,Ak,M

2

2

tid,t

with l the LTB length

G the shear modulus

vorhC the actual rotational stiffness of diaphragm

CM,k the rotational stiffness of the diaphragm

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

CP,k the 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 of per unit width of the diaphragm

s spacing of the beam

ba the width of the beam flange (in mm)


332

C100 rotation coefficient - see table

h beam height

t thickness beam flange

s thickness beam web

References

[1] ENV 1993-1-3:1996




CEN 1996

[2] E. Kahlmeyer




Stahlbauten


333




334

Section check for built-in beams (IFB, SFB, THQ sections)

Introduction

For the national codes EC-ENV, EC-EN, NEN6770/6771, DIN18800 and SIA263, special checks are performed for built-in beams, according to Ref.[1].

Reduction of plastic moment capacity due to plate bending

bu

e1

e2=bo

bo

tu

0.5 q0.5 q


335

bu

e1

e2=bo

bo

tu

0.5 q0.5 q

to

bu

e1

bo

tu

0.5 q0.5 q

e2=0

to

When the lower plate is loaded by q-load (uniform distributed load), the effective area of the loaded plate (flange) for the calculation of the plastic capacity is reduced as follows :


336

for THQ and IFB beams :

for SFB beam :

oouueff AAA

with e1, e2, tu, bu see the figures above

q load on flange, plate (as N/m)

fy yield strength

M partial safety factor

see formula

u =

o analog to u, but with

bu=bo

e1=bo

tu=to

e2=tw


337

Plastic interaction formula for single bending and shear force

The following plastic interaction formula can be used, when single bending around yy-axis My,Sd, in combination with shear force Vz,Sd, is acting :

y,pl

fm

Rd,z,pl

Sd,z

m

v

Rd,y,pl

Sd,y

W2

hA

0.1V

V

A

A

M

M

with My,Sd, Vz,Sd internal forces

Mpl,y,Rd plastic bending capacity around yy axis

Vpl,z,Rd plastic shear capacity in z direction

Av shear area (see figure)

Am = A - | Ao,x - Au,x | (see figure)

hf = h+tu/2-to/2 (see figure)

Wpl,y plastic section modulus around yy axis - reduced if necessary


338

Plastic check for plate in bending

The following condition for the plate in bending must be verified:

with e1, e2, tu see figures

q load on flange, plate (as N/m)

= qmax+qmin

(Ksi)

q

qq minmax

fy yield strength

M partial safety factor

0.5 q (1+Ksi)0.5 q (1-Ksi)


339

Stress check for slim floor beams

Normal stress check

At the edges of the bottom plate, the following composed stress check is performed:

Shear stress check in plate

In the middle of the bottom plate, transverse shear stress is checked:

u

minmax

M

y2

x

t

)q,q(

2

3

f²3


340

Torsion check due to unbalanced loading

for IFB and SFB beams:

12

bEtEI

GI

EIh2L

L

Ltanh

2

QeLM

htb

M

2

3

L

L

L

Ltanh

12

QeLM

I

tM

3

f

3

ooo

t

ofk

k

kmax,w

foo

max,w

max,w

k

k

max,t

t

omax,t

max,t

M

y

max,wmax,t

with to, bo see figures


It torsional constant for complete section

E modulus of Young

G shear modulus

L system length for Lyz

Q,e see figure


341

Q

e

for THQ beams :

2

V

b

e1

4

qL Rd,z,pl

f

with e, bf see figure


q load on flanges, plate (as N/m)

= qmax+qmin

(Ksi)

q

qq minmax


342

q maxq min

bf

ee

References

[1] Multi-Storey Buildings in Steel

Design Guide for Slim Floors with Built-in Beams

ECCS N° 83 - 1995


343

Effective cross-section properties for lattice tower angle members

Effective cross-section properties for compressed lattice tower angle members The effective cross-section properties shall be based on the effective width beff of the leg. See Ref.[1], Chapter J.2.3.

b

The effective width shall be obtained from the nominal width of the leg, assuming uniform stress distribution:

bb

f

235

43.0K

K4.28

t

b

eff

y

c

c

p

p

p


344

For a rolled angle:

2

p

p

p

p

p

98.0213.1

91.02213.191.0

0.191.0

For a cold formed angle:

2

p

p

p

p

p

98.0213.1

3

404.05

213.1809.0

0.1809.0

with t the thickness

b the nominal width

fy the yield strength in Mpa

References

[1] EN 50341-1:2001

Overhead electrical lines exceeding AC 45 kV Part 1: General requirements

Documents

Steel Code Check Theory Enu