V. CSA S16-19 - Axial Comp and Bending Interaction

Check a wide flange column member subject to both axial compression and bending per the CSA S16-19 code.

References

Canadian Institute of Steel Construction. Handbook of Steel Construction. 9th Edition. 2008. Willowdale, Ont., pp.6-52 - 6-53

Details

The column is simply supported and has a span of 3.6 m. The member is subjected to an axial load of 900 kN along with moments in both strong and weak directions of 50 kN·m at the roller end. The section used is a Canadian W250x73. The steel has a yield strength of 350 MPa (50 ksi) and ultimate strength of 450 MPa (65 ksi).

Section Properties

From the Handbook of Steel Construction:

D = 253 mm
B = 254 mm
t_w = 8.6 mm
t_f = 14.2 mm
A = 9,280 mm²
I_x = 113×(10)⁶ mm⁴
I_y = 38.8×(10)⁶ mm⁴
S_x = 891×(10)³ mm³
S_y = 306×(10)³ mm³
Z_x = 985×(10)³ mm³
Z_y = 463×(10)³ mm³
r_x = 110 mm
r_y = 64.6 mm
C_w = 553×(10)⁹ mm⁶
J = 575×(10)³ mm⁴

Material Properties

E = 205,000 MPa
G = 76,920 MPa

Calculations

Section Classification

b_el = B/2 = 127 mm

h = D - 2×t_f = 224.6 mm

Compression section classification as per Table 1:

Flange classification:

b_{el} / t = \frac{127}{14.2} = 8.94 < \frac{200}{\sqrt{F_{y}}} = 10.69

Web classification:

h / w = \frac{224.6}{8.6} = 26.11 < \frac{670}{\sqrt{F_{y}}} = 35.81

Section is Class 1 for compression.

Flexural section classification as per Table 2

Flange classification (major and minor axes):

\frac{145}{\sqrt{F_{y}}} = 7.75 < b_{el} / t = \frac{127}{14.2} = 8.94 < \frac{170}{\sqrt{F_{y}}} = 9.09

Web classification (major and minor axes):

h / w = \frac{224.6}{8.6} = 26.11 < \frac{525}{\sqrt{F_{y}}} = 28.06

\frac{M_{fy}}{S_{y}} > \frac{0.9 M_{fx}}{S_{x}}

\frac{M_{fy}}{S_{y}} = \frac{50}{306 {(10)}^{3}} = 0.163 {(10)}^{-3} > \frac{M_{fx}}{S_{x}} = \frac{50}{891 {(10)}^{3}} = 0.051 {(10)}^{-3}

Section is Class 2 for flexure.

Axial Compression Capacity

Check for axial compression per Cl. 13.3.1.1:

C_{r} = \frac{ϕ A F_{y}}{{(1 + λ^{2 n})}^{\frac{1}{n}}}

(13.3.1.1)

where

n	=	1.34 for I sections
λ	=	$\sqrt{\frac{F_{y}}{F_{e}}}$

F_{ex} = \frac{π^{2} E}{{(K L_{x} / r_{x})}^{2}} = \frac{π^{2} \times 205,000}{{(1.0 \times 3,600 / 110)}^{2}} = 1,890 MPa

λ_{x} = \sqrt{\frac{350}{1,890}} = 0.430

C_{rx} = \frac{0.9 \times 9,280 \times 350}{{[1 + {0.430}^{(2 \times 1.34)}]}^{\frac{1}{1.34}}} 10^{-3} = 2,715 kN

F_{ey} = \frac{π^{2} E}{{(K L_{y} / r_{y})}^{2}} = \frac{π^{2} \times 205,000}{{(1.0 \times 3,600 / 64.6)}^{2}} = 651 MPa

λ_{y} = \sqrt{\frac{350}{651}} = 0.733

C_{ry} = \frac{0.9 \times 9,280 \times 350}{{[1 + {0.733}^{(2 \times 1.34)}]}^{\frac{1}{1.34}}} 10^{-3} = 2,233 kN

Check for flexural-torsional buckling per Cl. 13.3.1.2:

F_{ez} = [\frac{π^{2} E C_{w}}{{(K L_{z})}^{2}} + G J] \frac{1}{A {\overset{⎯}{r}}_{0}^{2}}

(13.3.1.2)

where

{\overset{⎯}{r}}_{0}^{2}

{x_{0}}^{2} + {y_{0}}^{2} + {r_{x}}^{2} + {r_{y}}^{2}

; since the section is doubly-symmetric, x₀ = y₀ = 0

{\overset{⎯}{r}}_{0}^{2} = {(110)}^{2} + {(64.6)}^{2} = 16,273 {mm}^{2}

F_{ez} = [\frac{π^{2} \times 205,000 \times 553 \times {(10)}^{9}}{{(1.0 \times 3,600)}^{2}} + 76,920 \times 575 \times {(10)}^{3}] \frac{1}{9,280 \times 16,273} = 865 MPa

For flexural torsional buckling, F_e is the least of F_ex, F_ey, and F_ez. Therefore, F_e = 651 MPa.

λ = 0.733

C_r = C_ry = 2,233 kN

The critical ratio = C_f / C_r = 900 / 2,233 = 0.403

Bending Capacity

Check the bending capacity for a Class 2 section as per Cl. 13.5.a:

About major axis:

M_px = Z_x×F_y = 985 × (10)³ × 350 × 10^-6 = 344.8 kN·m

M_rx = ϕM_px = 0.9 (344.8) = 310.3 kN·m

Ratio = M_fx / M_rx = 50 / 310.3 = 0.161

About minor axis:

M_py = Z_y×F_y = 463 × (10)³ × 350 × 10^-6 = 162.1 kN·m

M_ry = ϕM_py = 0.9 (162.1) = 145.8 kN·m

Ratio = M_fy / M_ry = 50 / 145.8 = 0.343

Combined Axial Compression & Bending

Check cross-section strength for a Class 2 section per Cl. 13.8.2-a:

\frac{C_{f}}{C_{r}} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} + \frac{β U_{1y} M_{fy}}{M_{ry}} \leq 1.0

(13.8.2)

where

C_r	=	C_r as per Cl. 13.3 with the value of λ = 0 (Cl. 13.8.2-a-i), which reduces to: $= ϕ A F_{y} = 0.9 \times 9,280 \times 350 \times {(10)}^{-3} = 2,923 kN$
β	=	0.6 + 0.4 ×λ_y ≤ 0.85 0.6 + 0.4 (0.733) = 0.893; thus limit β = 0.85

About both major and minor axes, the moment at the "start" joint, M_A = 0 and the moment at the "end" joint, M_B = 50 kN·m. Thus, K = M_A / M_B = 0.

ω₁ = 0.6 - 0.4×K = 0.6

(13.8.6)

U_{1} = \frac{ω_{1}}{1 - \frac{C_{f}}{C_{e}}} \geq 1.0

(13.8.5 & 13.8.2-a-iii)

where

C_e

\frac{π^{2} E I}{L^{2}}

C_{ez} = \frac{π^{2} E I_{z}}{L^{2}} = \frac{π^{2} 205,000 \times 113 \times {(10)}^{3}}{{(3,600)}^{2}} = 17,640 kN

C_{ey} = \frac{π^{2} E I_{y}}{L^{2}} = \frac{π^{2} 205,000 \times 38.8 \times {(10)}^{3}}{{(3,600)}^{2}} = 6,057 kN

U_{1z} = \frac{0.6}{1 - \frac{900}{17,640}} 0.632 < 1.0

U_{1y} = \frac{0.6}{1 - \frac{900}{6,057}} 0.705 < 1.0

Thus, U₁ = 1.0 about both major and minor axes.

\frac{900}{2,923} + \frac{0.85 \times 1.0 \times (50)}{310.3} + \frac{0.85 \times 1.0 \times (50)}{145.8} = 0.736 \leq 1.0

Check overall member strength for a Class 2 section per Cl. 13.8.2-b:

C_r used is taken from the axial compression calculations (K = 1).
Use U_1z and U_1y = 1.0 for an unbraced frame.

\frac{900}{2,233} + \frac{0.85 \times 1.0 \times (50)}{310.3} + \frac{0.85 \times 1.0 \times (50)}{145.8} = 0.831 \leq 1.0

Check moment-only capacity for biaxial bending per Cl. 13.8.2-d:

\frac{M_{fz}}{M_{rz}} + \frac{M_{fy}}{M_{ry}} = \frac{50}{310.3} + \frac{50}{145.8} = 0.504 \leq 1.0

Comparison

Table 1. Comparison of results
Criteria	Reference	STAAD.Pro	Difference
Elastic buckling stress about the major axis, F_ex (MPa)	1,890	1,900.987	negligible
Elastic buckling stress about the minor axis, F_ey (MPa)	651	652.728	negligible
Non-dimensional slenderness parameter about major axis, λ_x	0.430	0.429	negligible
Non-dimensional slenderness parameter about minor axis, λ_y	0.733	0.732	negligible
Factored axial compressive resistance about major axis, C_rx (kN)	2,715	2,720	negligible
Factored axial compressive resistance about major axis, C_ry (kN)	2,233	2,230	negligible
Flexural torsional buckling stress, F_ez (MPa)	651	652.728	negligible
Factored axial compressive resistance in flexural torsional buckling, C_r (kN)	2,233	2,230	negligible
Factored moment resistance about the major axis, M_rx (kN·m)	310.3	310.3	none
Factored moment resistance about the minor axis, M_ry (kN·m)	145.8	145.8	none
Factored axial compressive stress used for cross-section strength, C_r (kN)	2,923	2,920	negligible
Elastic buckling strength about major axis, C_ez (kN)	17,640	17,600	negligible
Elastic buckling strength about minor axis, C_ey (kN)	6,057	6,060	negligible
β	0.85	0.85	none
Cross-sectional strength ratio	0.736	0.736	none
Factored axial compressive stress used for overall member strength, C_r (kN)	2,233	2,230	negligible
Overall member strength ratio	0.831	0.831	none
Biaxial flexure ratio	0.504	0.504	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 Comp and Bending Interaction.STD is typically installed with the program.

The following design parameters are used:

Full lateral support is specified using LAT 1.
Steel strength values are specified using FYLD 350000 and FU 450000 (the units are kN/m² or kPa).

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 17-Jan-22
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:  0.831         LOADCASE    1     |
|  LOCATION:      3.60    CRITICAL CONDITION:            Cl. 13.8.2        |
|--------------------------------------------------------------------------|
|  STRENGTH CHECK SUMMARY                                                  |
|   CRITICAL RATIO:  0.831(PASS)            LOAD CASE:     1               |
|   LOCATION      :   3.60                  CONDITION: Cl. 13.8.2          |
|--------------------------------------------------------------------------|
|  DESIGN FORCES              LC:       1      LOC:      3.60              |
|         FX:    900.00(C)    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 2                                    |
|==========================================================================|
|  SLENDERNESS                                                             |
|   ACTUAL SLENDERNESS RATIO   :      55.675          LC   :        1      |
|   ALLOWABLE SLENDERNESS RATIO:     200.000          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   2.92E+03       0.000      Cl. 13.2          1     0.000    |
|  RUPTURE                                                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   3.13E+03       0.000      Cl. 13.2          1     0.000    |
|--------------------------------------------------------------------------|
|                       CHECKS FOR AXIAL COMPRESSION                       |
|--------------------------------------------------------------------------|
|  FLEX BUCKLING                                                           |
|            FORCE:    CAPACITY:   RATIO:    CRITERIA:      LC:    LOC:    |
|   MAJOR:   9.00E+02   2.72E+03     0.331  Cl. 13.3            1    0.000 |
|   MINOR:   9.00E+02   2.23E+03     0.403  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:       |
|    9.00E+02   2.23E+03       0.403      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   : 0.343  Cl. 13.9.2      1   3.60   0.00E+00  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   : 0.736  Cl. 13.8.2      1   3.600 9.00E+02  2.92E+03   |
|   MEMBER STRENGTH: 0.831  Cl. 13.8.2      1   0.000 9.00E+02  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 |
|==========================================================================|
      STAAD SPACE                                              -- PAGE NO.    8