# V. CSA S16-19 - Axial Tension and Bending Interaction

Determine the capacity of a member subject to axial tension and biaxial moment per the CSA S16-19 code.

## Problem

A 3.6 m long column is subjected to a 900 kN tensile load along with concentrated moments of 50 kN·m about both major and minor axes at one end. The section is a Canadian W250x73. The steel is grade 350W.

Section Properties

• D = 253 mm
• B = 254 mm
• tw = 8.6 mm
• tf = 14.2 mm
• Ag = 9,280 mm2
• Zx = 985,000 mm3
• Zy = 463,000 mm3

Material Properties

• Fy = 350 MPa
• Fu = 450 MPa

## Calculations

Section Classification

Compression section classification per Table 1:

bel = B / 2 = 127 mm

h = D - 2×tf = 235.8 mm

• Flange classification:
$b el t = 127 14.2 = 8.94 < 200 F y = 200 350 = 10.69$
• Web classification:
$h w = 235.8 8.6 = 27.42 < 670 F y = 670 350 = 35.81$

Therefore, the section is Class 1 for compression.

Flexure section classification per Table 2:

• Flange classification about major axis:
$145 F y = 7.75 < b el t = 127 14.2 = 8.94 < 170 350 = 9.09$

Class 2

• Flange classification about minor axis:
$145 F y = 7.75 < b el t = 127 14.2 = 8.94 < 170 350 = 9.09$

Class 2

• Web classification about major axis:
$h w = 235.8 8.6 = 27.42 < 1,100 F y = 58.8$

Class 1

• Web classification about minor axis:
$h w = 235.8 8.6 = 27.42 < 1,100 F y = 58.8$

Class 1

Therefore, the section is Class 2 for flexure.

Axial Tension

Assume the gross area for both yielding and rupture limit states. Therefore, Ane = Ag = 9,280 mm2.

So, the tensile capacity = 2,923 kN.

The critical stress ratio 900 / 2,923 = 0.308.

Bending Capacity

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

Major axis:

Mpx = Zx×Fy = 985,000 × 350 (10)-6 = 344.8 kN·m

Mrx = ϕMpx = 0.9 × 344.8 = 310.3 kN·m

The ratio = 50 / 310.3 = 0.161

Minor axis:

Mpy = Zy×Fy = 463,000 × 350 (10)-6 = 162.1 kN·m

Mrx = ϕMpx = 0.9 × 162.1 = 145.8 kN·m

The ratio = 50 / 145.8 = 0.343

Combined Axial Tension and Bending

Check for axial tension and bending per Cl. 13.9.2:

$T f T r + 0.85 M fx M rx + 0.6 M fy M ry = 0.380 + 0.85 ( 0.161 ) + 0.6 ( 0.343 ) = 0.651 < 1.0$

Also, the following must be satisfied:

$M fx M rx + M fy M ry = 0.161 + 0.343 = 0.504 < 1.0$

## Comparison

Table 1. Comparison of results
Nominal flexure capacity about major axis, Mpx (kN·m) 344.8 345 negligible
Ultimate flexure capacity about major axis, Mrx (kN·m) 310.3 310.3 none
Nominal flexure capacity about minor axis, Mpy (kN·m) 162.1 162 negligible
Ultimate flexure capacity about minor axis, Mrx (kN·m) 145.8 145.8 none
Tension capacity, Tr (kN) 2,932 2,930 negligible
Critical Ratio 0.651 0.651 none

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

The following design parameters are used:
• The default value of NSF 1 is used for net section equal to the gross area.
• The default value of SLF 1 is used for a shear lag factor of 1 (i.e., no shear lag in connection)
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 08-Oct-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 3.6 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 W250X73
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 ENFORCED BUT FY MX MZ
2 FY 900 MX 50 MZ 50
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
LAT 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  W250X73          (CANADIAN SECTIONS)|
|  STATUS:        PASS    CRITICAL RATIO:  0.651         LOADCASE    1     |
|  LOCATION:      3.60    CRITICAL CONDITION:            Cl. 13.9.2        |
|--------------------------------------------------------------------------|
|  STRENGTH CHECK SUMMARY                                                  |
|   CRITICAL RATIO:  0.651(PASS)            LOAD CASE:     1               |
|   LOCATION      :   3.60                  CONDITION: Cl. 13.9.2          |
|--------------------------------------------------------------------------|
|  DESIGN FORCES              LC:       1      LOC:      3.60              |
|         FX:    900.00(T)    FY:    13.89      FZ:     13.89              |
|         MX:  0.000E+00      MY: 5.000E+01     MZ: -5.000E+01             |
|--------------------------------------------------------------------------|
|  SECTION PROPERTIES (LOC:   3.60)             UNIT:      CM              |
|   AXX:      92.800     AZZ:      72.136      AYY:         20.537         |
|   IXX:      57.500     IZZ:   11300.001      IYY:       3880.000         |
|   PZZ:     985.000     PYY:     463.000                                  |
|   SZZ:     893.281     SYY:     305.512      CW :     552900.312         |
|--------------------------------------------------------------------------|
|  MATERIAL PROPERTIES                          UNIT:      N MM            |
|   FYLD:     350.000     FU:   450.000      E: 204999.984  G:  76920.000  |
|--------------------------------------------------------------------------|
|  DESIGN PARAMETERS                                                       |
|   ACTUAL MEMBER LENGTH:     3.600        LZ:    3.600    LY:    3.600    |
|   KZ:  1.000        KY:  1.000      NSF:  1.000      SLF:  1.000         |
|--------------------------------------------------------------------------|
|  SECTION CLASS                                                           |
|   COMPRESSION    :            Class 1                                    |
|   FLEXURE           MAJOR               MINOR                            |
|   SECTION        : Class 2             Class 2                           |
|   FLANGE         : Class 2             Class 2                           |
|   WEB (CRITICAL) :            Class 1                                    |
|==========================================================================|
|  SLENDERNESS                                                             |
|   ACTUAL SLENDERNESS RATIO   :      55.675          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:       |
|    9.00E+02   2.92E+03       0.308      Cl. 13.2          1     0.000    |
|  RUPTURE                                                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    9.00E+02   3.13E+03       0.287      Cl. 13.2          1     0.000    |
|--------------------------------------------------------------------------|
|                       CHECKS FOR AXIAL COMPRESSION                       |
|--------------------------------------------------------------------------|
|  FLEX BUCKLING                                                           |
|            FORCE:    CAPACITY:   RATIO:    CRITERIA:      LC:    LOC:    |
|   MAJOR:   0.00E+00   2.72E+03     0.000  Cl. 13.3            1    0.000 |
|   MINOR:   0.00E+00   2.23E+03     0.000  Cl. 13.3            1    0.000 |
|  INTERMEDIATE RESULTS:                                                   |
|               Ag:(CM)     Fe:(N MM)       λ:        n:      KL/r:        |
|   MAJOR:      92.800      1900.987      0.429     1.34      32.624       |
|   MINOR:      92.800       652.728      0.732     1.34      55.675       |
|--------------------------------------------------------------------------|
|  FLEXURAL TORSIONAL BUCKLING                                             |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   2.23E+03       0.000      Cl. 13.3          1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Ag:(CM)    92.800   Fe:(N MM)      652.728  λ:     0.732  n: 1.34      |
|--------------------------------------------------------------------------|
|                             CHECKS FOR SHEAR                             |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Y AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    1.39E+01   4.52E+02       0.031    Cl. 13.4.1.1        1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)      Kv:       Ka:     Fcri:(N MM)  Fcre:(N MM)  Fs:(N MM)     |
|     21.758    5.360     0.070       480.943      1414.474    231.000     |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Z AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|   -1.39E+01   1.36E+03       0.010    AISC G2-1           1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)                                                                |
|     72.136                                                               |
|--------------------------------------------------------------------------|
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:       |
|     5.00E+01  3.103E+02      0.161    Cl. 13.5(a)         1     3.600    |
|  INTERMEDIATE RESULTS: Mp:  3.45E+02   Se(CM):  0.00E+00    My:  0.00E+00|
|--------------------------------------------------------------------------|
|  YIELDING ALONG MINOR AXIS                                               |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    5.00E+01  1.458E+02      0.343    Cl. 13.5(a)         1     3.600     |
|  INTERMEDIATE RESULTS: Mp:  1.62E+02   Se(CM):  0.00E+00    My:  0.00E+00|
|--------------------------------------------------------------------------|
|                          CHECKS FOR INTERACTION                          |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL TENSION                                               |
|                    RATIO:   CRITERIA:   LC:   LOC:     Tf:        Tr:    |
|   C/S STRENGTH   : 0.651  Cl. 13.9.2      1   3.60   9.00E+02  2.92E+03  |
|                        Mfz:         Mfy:          Mrz:         Mry:      |
|   C/S STRENGTH   :    5.00E+01      5.00E+01     3.10E+02      1.46E+02  |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL COMPRESSION                                           |
|                    RATIO:   CRITERIA:   LC:   LOC:     Cf:        Cr:    |
|   C/S STRENGTH   : 0.428  Cl. 13.8.2      1   3.600 0.00E+00  2.92E+03   |
|   MEMBER STRENGTH: 0.428  Cl. 13.8.2      1   0.000 0.00E+00  2.23E+03   |
|                        Mfz:      Mfy:      Mrz:      Mry:    w 1z:    w 1y:    |
|   C/S STRENGTH   :  5.00E+01  5.00E+01  3.10E+02  1.46E+02  0.60   0.60  |
|   MEMBER STRENGTH:  5.00E+01  5.00E+01  3.10E+02  1.46E+02  0.60   0.60  |
|                        b :         Cez:     Cey:         U1z:     U1y:     |
|   C/S STRENGTH   :    0.85     1.76E+04  6.06E+03     1.00     1.00      |
|   MEMBER STRENGTH:    0.85     1.76E+04  6.06E+03     1.00     1.00      |
|--------------------------------------------------------------------------|
|  SHEAR CAPACITY MODIFICATION  ( SHEAR AND FLEX )                         |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|     1.39E+01  4.523E+02      0.031    Cl. 14.6            1     0.000    |
|--------------------------------------------------------------------------|
|  BIAXIAL FLEXURE                                                         |
|   RATIO:  CRITERIA:  LC:   LOC:     Mfz:      Mrz:       Mfy:      Mry:  |
|   0.504  Cl. 13.8     1    3.60   5.00E+01  3.10E+02  5.00E+01  1.46E+02 |