# V. CSA S16-19 - Check for Bending and LTB

Determine the capacity of a laterally unbraced beam subject to bending moment per the CSA S16-19 code.

## Problem

A 7.5 m long, simply supported beam is subjected to a distributed load of 12 kN/m. The section is a Canadian W310x52. The steel is grade 350W.

Section Properties

• D = 318 mm
• B = 167 mm
• tw = 7.6 mm
• tf = 13.2 mm
• Zx = 841,000 mm3
• Zy = 189,000 mm3
• Iy = 10.3×106 mm4
• J = 308,000 mm4
• Cw = 238×109 mm6

Material Properties

• Fy = 350 MPa
• Fu = 450 MPa
• E = 205,000 MPa
• G = 76,920 MPa

## Calculations

Section Classification

bel = B / 2 = 83.5 mm

h = D - 2×tf = 291.6 mm

Flexure section classification per Table 2:

• Flange classification about major axis:
$b el t = 83.5 13.2 = 6.33 < 145 F y = 7.75$

Class 1

• Flange classification about minor axis:
$b el t = 83.5 13.2 = 6.33 < 145 F y = 7.75$

Class 1

• Web classification about major axis:
$h w = 291.6 7.6 = 38.37 < 1,100 F y = 58.8$

Class 1

• Web classification about minor axis:
$h w = 291.6 7.6 = 38.37 < 1,100 F y = 58.8$

Class 1

Therefore, the section is Class 1 for flexure.

Bending Capacity

For a Class 2, doubly symmetric section, the capacity is determined by Cl. 13.5a:

Major axis:

Mpx = Zx×Fy = 841,000 × 350 (10)-6 = 294.4 kN·m

Mrx = ϕMpx = 0.9 × 294.4 = 264.9 kN·m

The ratio = 84.4 / 264.9 = 0.318

Minor axis:

Mpy = Zy×Fy = 189,000 × 350 (10)-6 = 66.2 kN·m

Mrx = ϕMpx = 0.9 × 66.2 = 59.5 kN·m

Lateral Torsional Buckling

Check for lateral torsional buckling per Cl. 13.6:

$M u = ω 2 π L E I y G J + π E L 2 I y C w < 0.67 × M p$
where
 ω2 = 1.0 $E I y G J$ = 205,000 × 10.3(10)6 × 76,920 × 308,000 = 50.02 (10)21 $π E L 2 I y C w$ = $π × 205,000 7,500 2 10.3 10 6 238 10 9 = 18.08 10 21$

109.3 < 0.67 (294.4) = 197.2

Mr = ϕMu = 0.9 × 109.3 = 98.38 kN·m

Ratio = 84.4 / 98.38 = 0.858.

## Comparison

Table 1. Comparison of results
Nominal flexure capacity about major axis, Mpx (kN·m) 294.4 294 negligible
Ultimate flexure capacity about major axis, Mrx (kN·m) 264.9 264.9 none
Nominal flexure capacity about minor axis, Mpy (kN·m) 66.2 66.2 none
Ultimate flexure capacity about minor axis, Mrx (kN·m) 59.5 59.54 negligible
Ratio 0.318 0.318 none
Nominal lateral torsional bending capacity, Mu (kN) 109.3 109.3 none
Ultimate lateral torsional bending capacity, Mr (kN) 98.38 98.38 none
Critical Ratio 0.858 0.858 none

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\CSA S16-19 - Check for Bending and LTB.STD is typically installed with the program.

The following design parameters are used:
• The laterally unsupported is designated by the parameter LAT 0.
• The value of ω2 is specified as 1 using the parameter CB 1
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 10-Oct-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 7.5 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.92
ALPHA 1.2e-05
DAMP 0.03
TYPE STEEL
STRENGTH FY 350000 FU 450000 RY 1.5 RT 1.2
G 7.692e+07
END DEFINE MATERIAL
1 TABLE ST W310X52
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MY MZ
1 UNI GY -12
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
LAT 0 ALL
CB 1 ALL
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH


                       STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
***********************************************
ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
|--------------------------------------------------------------------------|
|   MEMBER NO.:      1      SECTION: ST  W310X52          (CANADIAN SECTIONS)|
|  STATUS:        PASS    CRITICAL RATIO:  0.858         LOADCASE    1     |
|  LOCATION:      3.75    CRITICAL CONDITION:            Cl. 13.8          |
|--------------------------------------------------------------------------|
|  STRENGTH CHECK SUMMARY                                                  |
|   CRITICAL RATIO:  0.858(PASS)            LOAD CASE:     1               |
|   LOCATION      :   3.75                  CONDITION: Cl. 13.8            |
|--------------------------------------------------------------------------|
|  DESIGN FORCES              LC:       1      LOC:      3.75              |
|         FX:      0.00(T)    FY:     0.00      FZ:      0.00              |
|         MX:  0.000E+00      MY: 0.000E+00     MZ: -8.438E+01             |
|--------------------------------------------------------------------------|
|  SECTION PROPERTIES (LOC:   3.75)             UNIT:      CM              |
|   AXX:      66.700     AZZ:      44.088      AYY:         23.165         |
|   IXX:      30.800     IZZ:   11900.002      IYY:       1030.000         |
|   PZZ:     841.000     PYY:     189.000                                  |
|   SZZ:     748.428     SYY:     123.353      CW :     237980.875         |
|--------------------------------------------------------------------------|
|  MATERIAL PROPERTIES                          UNIT:      N MM            |
|   FYLD:     350.000     FU:   450.000      E: 204999.984  G:  76920.000  |
|--------------------------------------------------------------------------|
|  DESIGN PARAMETERS                                                       |
|   ACTUAL MEMBER LENGTH:     7.500        LZ:    7.500    LY:    7.500    |
|   KZ:  1.000        KY:  1.000      NSF:  1.000      SLF:  1.000         |
|--------------------------------------------------------------------------|
|  SECTION CLASS                                                           |
|   COMPRESSION    :            Class 4                                    |
|   FLEXURE           MAJOR               MINOR                            |
|   SECTION        : Class 1             Class 1                           |
|   FLANGE         : Class 1             Class 1                           |
|   WEB (CRITICAL) :            Class 1                                    |
|==========================================================================|
|  SLENDERNESS                                                             |
|   ACTUAL SLENDERNESS RATIO   :     190.856          LC   :        1      |
|   ALLOWABLE SLENDERNESS RATIO:     300.000          LOC  :      0.00     |
|--------------------------------------------------------------------------|
STAAD SPACE                                              -- PAGE NO.    6
STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
***********************************************
ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
Member :     1       Contd.
|--------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL TENSION                         |
|--------------------------------------------------------------------------|
|  YIELDING                                                                |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   2.10E+03       0.000      Cl. 13.2          1     0.000    |
|  RUPTURE                                                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   2.25E+03       0.000      Cl. 13.2          1     0.000    |
|--------------------------------------------------------------------------|
|                       CHECKS FOR AXIAL COMPRESSION                       |
|--------------------------------------------------------------------------|
|  FLEX BUCKLING                                                           |
|            FORCE:    CAPACITY:   RATIO:    CRITERIA:      LC:    LOC:    |
|   MAJOR:   0.00E+00   1.56E+03     0.000  Cl. 13.3.4(a)       1    0.000 |
|   MINOR:   0.00E+00   3.07E+02     0.000  Cl. 13.3.4(a)       1    0.000 |
|  INTERMEDIATE RESULTS:                                                   |
|               Ag:(CM)     Fe:(N MM)       λ:        n:      KL/r:        |
|   MAJOR:      65.224       641.730      0.739     1.34      56.150       |
|   MINOR:      65.224        55.545      2.510     1.34     190.856       |
|--------------------------------------------------------------------------|
|  FLEXURAL TORSIONAL BUCKLING                                             |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   3.07E+02       0.000      Cl. 13.3.4        1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Ag:(CM)    65.224   Fe:(N MM)       55.545  λ:     2.510  n: 1.34      |
|--------------------------------------------------------------------------|
|                             CHECKS FOR SHEAR                             |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Y AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    4.50E+01   5.02E+02       0.090    Cl. 13.4.1.1        1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)      Kv:       Ka:     Fcri:(N MM)  Fcre:(N MM)  Fs:(N MM)     |
|     24.168    5.347     0.042       326.980       653.808    231.000     |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Z AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   8.33E+02       0.000    AISC G2-1           1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)                                                                |
|     44.088                                                               |
|--------------------------------------------------------------------------|
STAAD SPACE                                              -- PAGE NO.    7
STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
***********************************************
ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
Member :     1       Contd.
|--------------------------------------------------------------------------|
|                            CHECKS FOR BENDING                            |
|--------------------------------------------------------------------------|
|  YIELDING ALONG MAJOR AXIS                                               |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|     8.44E+01  2.649E+02      0.318    Cl. 13.5(a)         1     3.750    |
|  INTERMEDIATE RESULTS: Mp:  2.94E+02   Se(CM):  0.00E+00    My:  0.00E+00|
|--------------------------------------------------------------------------|
|  YIELDING ALONG MINOR AXIS                                               |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00  5.954E+01      0.000    Cl. 13.5(a)         1     0.000     |
|  INTERMEDIATE RESULTS: Mp:  6.62E+01   Se(CM):  0.00E+00    My:  0.00E+00|
|--------------------------------------------------------------------------|
|  LATERAL TORSIONAL BUCKLING (MAJOR AXIS)                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    8.44E+01  9.838E+01      0.858   Cl. 13.6(a)(ii)      1     3.750     |
|  INTERMEDIATE RESULTS:                                                   |
|   Iyc(CM):    w 2:    w 3:    b x(CM):  Mu:       rt:   Myr:     Lu:       Lyr:  |
|  0.00E+00   1.00  0.00     0.00  1.093E+02  0.00  0.00E+00  0.00   0.00  |
|--------------------------------------------------------------------------|
|                          CHECKS FOR INTERACTION                          |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL TENSION                                               |
|                    RATIO:   CRITERIA:   LC:   LOC:     Tf:        Tr:    |
|   C/S STRENGTH   : 0.271  Cl. 13.9.2      1   3.75   0.00E+00  2.10E+03  |
|   MEMBER STRENGTH: 0.858  Cl. 13.9.3      1   0.00                       |
|                        Mfz:         Mfy:          Mrz:         Mry:      |
|   C/S STRENGTH   :    8.44E+01      0.00E+00     2.65E+02      5.95E+01  |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL COMPRESSION                                           |
|                    RATIO:   CRITERIA:   LC:   LOC:     Cf:        Cr:    |
|   C/S STRENGTH   : 0.271  Cl. 13.8.2      1   3.750 0.00E+00  2.05E+03   |
|   MEMBER STRENGTH: 0.271  Cl. 13.8.2      1   3.750 0.00E+00  3.07E+02   |
|   LTB STRENGTH   : 0.729  Cl. 13.8.2      1   3.750 0.00E+00  3.07E+02   |
|                        Mfz:      Mfy:      Mrz:      Mry:    w 1z:    w 1y:    |
|   C/S STRENGTH   :  8.44E+01  0.00E+00  2.65E+02  5.95E+01  0.60   0.60  |
|   MEMBER STRENGTH:  8.44E+01  0.00E+00  2.65E+02  5.95E+01  0.60   0.60  |
|   LTB STRENGTH   :  8.44E+01  0.00E+00  9.84E+01  5.95E+01  0.60   0.60  |
|                        b :         Cez:     Cey:         U1z:     U1y:     |
|   C/S STRENGTH   :    0.85     4.28E+03  3.70E+02     1.00     1.00      |
|   MEMBER STRENGTH:    0.85     4.28E+03  3.70E+02     1.00     1.00      |
|   LTB STRENGTH   :    0.85     4.28E+03  3.70E+02     1.00     1.00      |
|--------------------------------------------------------------------------|
|  SHEAR CAPACITY MODIFICATION  ( SHEAR AND FLEX )                         |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|     4.50E+01  5.025E+02      0.090    Cl. 14.6            1     0.000    |
|--------------------------------------------------------------------------|
|  BIAXIAL FLEXURE                                                         |
|   RATIO:  CRITERIA:  LC:   LOC:     Mfz:      Mrz:       Mfy:      Mry:  |
|   0.858  Cl. 13.8     1    3.75   8.44E+01  9.84E+01  0.00E+00  5.95E+01 |