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
t_w = 7.6 mm
t_f = 13.2 mm
Z_x = 841,000 mm³
Z_y = 189,000 mm³
I_y = 10.3×10⁶ mm⁴

J = 308,000 mm⁴
C_w = 238×10⁹ mm⁶

Material Properties

F_y = 350 MPa
F_u = 450 MPa
E = 205,000 MPa
G = 76,920 MPa

Calculations

Section Classification

b_el = B / 2 = 83.5 mm

h = D - 2×t_f = 291.6 mm

Flexure section classification per Table 2:

Flange classification about major axis:
$\frac{b_{el}}{t} = \frac{83.5}{13.2} = 6.33 < \frac{145}{\sqrt{F_{y}}} = 7.75$

Class 1
Flange classification about minor axis:
$\frac{b_{el}}{t} = \frac{83.5}{13.2} = 6.33 < \frac{145}{\sqrt{F_{y}}} = 7.75$

Class 1
Web classification about major axis:
$\frac{h}{w} = \frac{291.6}{7.6} = 38.37 < \frac{1,100}{\sqrt{F_{y}}} = 58.8$

Class 1
Web classification about minor axis:
$\frac{h}{w} = \frac{291.6}{7.6} = 38.37 < \frac{1,100}{\sqrt{F_{y}}} = 58.8$

Class 1

Therefore, the section is Class 1 for flexure.

Bending Capacity

M_{fx} = \frac{12 {(7.5)}^{2}}{8} = 84.4 kN·m

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

Major axis:

M_px = Z_x×F_y = 841,000 × 350 (10)^-6 = 294.4 kN·m

M_rx = ϕM_px = 0.9 × 294.4 = 264.9 kN·m

The ratio = 84.4 / 264.9 = 0.318

Minor axis:

M_py = Z_y×F_y = 189,000 × 350 (10)^-6 = 66.2 kN·m

M_rx = ϕM_px = 0.9 × 66.2 = 59.5 kN·m

Lateral Torsional Buckling

Check for lateral torsional buckling per Cl. 13.6:

M_{u} = \frac{ω_{2} π}{L} \sqrt{E I_{y} G J + {(\frac{π E}{L})}^{2} I_{y} C_{w}} < 0.67 \times 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
${(\frac{π E}{L})}^{2} I_{y} C_{w}$	=	${(\frac{π \times 205,000}{7,500})}^{2} 10.3 {(10)}^{6} 238 {(10)}^{9} = 18.08 {(10)}^{21}$

\frac{(1.0) π}{7,500} \sqrt{50.02 {(10)}^{21} + 18.08 {(10)}^{21}} = 109.3 {(10)}^{6} N·mm = 109.3 kN·m

109.3 < 0.67 (294.4) = 197.2

M_r = ϕM_u = 0.9 × 109.3 = 98.38 kN·m

Ratio = 84.4 / 98.38 = 0.858.

Comparison

Table 1. Comparison of results
Criteria	Reference	STAAD.Pro	Difference
Nominal flexure capacity about major axis, M_px (kN·m)	294.4	294	negligible
Ultimate flexure capacity about major axis, M_rx (kN·m)	264.9	264.9	none
Nominal flexure capacity about minor axis, M_py (kN·m)	66.2	66.2	none
Ultimate flexure capacity about minor axis, M_rx (kN·m)	59.5	59.54	negligible
Ratio	0.318	0.318	none
Nominal lateral torsional bending capacity, M_u (kN)	109.3	109.3	none
Ultimate lateral torsional bending capacity, M_r (kN)	98.38	98.38	none
Critical Ratio	0.858	0.858	none

STAAD Input

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
MEMBER PROPERTY CANADIAN
1 TABLE ST W310X52
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -12
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN
LAT 0 ALL
CB 1 ALL
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD Output

                       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:         24.168         |
|   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 |