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
t_w = 8.6 mm
t_f = 14.2 mm
A_g = 9,280 mm²
Z_x = 985,000 mm³
Z_y = 463,000 mm³

Material Properties

F_y = 350 MPa
F_u = 450 MPa

Calculations

Section Classification

Compression section classification per Table 1:

b_el = B / 2 = 127 mm

h = D - 2×t_f = 235.8 mm

Flange classification:
$\frac{b_{el}}{t} = \frac{127}{14.2} = 8.94 < \frac{200}{\sqrt{F_{y}}} = \frac{200}{\sqrt{350}} = 10.69$
Web classification:
$\frac{h}{w} = \frac{235.8}{8.6} = 27.42 < \frac{670}{\sqrt{F_{y}}} = \frac{670}{\sqrt{350}} = 35.81$

Therefore, the section is Class 1 for compression.

Flexure section classification per Table 2:

Flange classification about major axis:
$\frac{145}{\sqrt{F_{y}}} = 7.75 < \frac{b_{el}}{t} = \frac{127}{14.2} = 8.94 < \frac{170}{\sqrt{350}} = 9.09$

Class 2
Flange classification about minor axis:
$\frac{145}{\sqrt{F_{y}}} = 7.75 < \frac{b_{el}}{t} = \frac{127}{14.2} = 8.94 < \frac{170}{\sqrt{350}} = 9.09$

Class 2
Web classification about major axis:
$\frac{h}{w} = \frac{235.8}{8.6} = 27.42 < \frac{1,100}{\sqrt{F_{y}}} = 58.8$

Class 1
Web classification about minor axis:
$\frac{h}{w} = \frac{235.8}{8.6} = 27.42 < \frac{1,100}{\sqrt{F_{y}}} = 58.8$

Class 1

Therefore, the section is Class 2 for flexure.

Axial Tension

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

T_{r} = {}_{\min}{\begin{matrix} ϕ A_{g} F_{y} = 0.9 \times 350 \times 9,280 {(10)}^{-3} = 2,923 kN \\ ϕ_{u} A_{ne} F_{u} = 0.75 \times 450 \times 9,280 {(10)}^{-3} = 3,132 kN \end{matrix}

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:

M_px = Z_x×F_y = 985,000 × 350 (10)^-6 = 344.8 kN·m

M_rx = ϕM_px = 0.9 × 344.8 = 310.3 kN·m

The ratio = 50 / 310.3 = 0.161

Minor axis:

M_py = Z_y×F_y = 463,000 × 350 (10)^-6 = 162.1 kN·m

M_rx = ϕM_px = 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:

\frac{T_{f}}{T_{r}} + 0.85 \frac{M_{fx}}{M_{rx}} + 0.6 \frac{M_{fy}}{M_{ry}} = 0.380 + 0.85 (0.161) + 0.6 (0.343) = 0.651 < 1.0

Also, the following must be satisfied:

\frac{M_{fx}}{M_{rx}} + \frac{M_{fy}}{M_{ry}} = 0.161 + 0.343 = 0.504 < 1.0

Comparison

Table 1. Comparison of results
Criteria	Reference	STAAD.Pro	Difference
Nominal flexure capacity about major axis, M_px (kN·m)	344.8	345	negligible
Ultimate flexure capacity about major axis, M_rx (kN·m)	310.3	310.3	none
Nominal flexure capacity about minor axis, M_py (kN·m)	162.1	162	negligible
Ultimate flexure capacity about minor axis, M_rx (kN·m)	145.8	145.8	none
Tension capacity, T_r (kN)	2,932	2,930	negligible
Critical Ratio	0.651	0.651	none

STAAD Input

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
MEMBER PROPERTY CANADIAN
1 TABLE ST W250X73
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 ENFORCED BUT FY MX MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
2 FY 900 MX 50 MZ 50
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN
LAT 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  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:         21.758         |
|   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 |