V. CSA S16-19 - Short Column Compression

Determine the capacity of a short column per the CSA S16-19 code.

Problem

A Canadian W250x73 section is used for a pedestal with a height of 1.1 m. The axial load is 1,000 kN in compression. 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

I_x = 113×10⁶ mm⁴

I_y = 38.8×10⁶ mm⁴

J = 575,000 mm⁴
C_w = 553×10⁹ 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

Compression section classification per Table 1

Flange classification:
$\frac{b_{el}}{t} = \frac{127}{14.2} = 8.94 < \frac{200}{\sqrt{F_{y}}} = 10.69$
Web classification about major axis:
$\frac{h}{w} = \frac{235.8}{8.6} = 27.4 < \frac{670}{\sqrt{F_{y}}} = 35.8$

Therefore, the section is Class 1 for compression.

Column Capacity

r_{x} = \sqrt{\frac{I_{x}}{A}} = \sqrt{\frac{113 \times {(10)}^{6}}{9,280}} = 110.3 mm

r_{y} = \sqrt{\frac{I_{y}}{A}} = \sqrt{\frac{38.8 \times {(10)}^{6}}{9,280}} = 64.7 mm

Assume K = 1.0,

\frac{K L}{r_{x}} = \frac{1.0 \times 1,100}{110.3} = 9.97

\frac{K L}{r_{y}} = \frac{1.0 \times 1,100}{64.7} = 17.0

Elastic buckling stress:

F_{ex} = \frac{π^{2} E}{{(\frac{K L_{x}}{r_{x}})}^{2}} = \frac{π^{2} \times 205,000}{{(9.97)}^{2}} = 20,360 MPa

F_{ey} = \frac{π^{2} E}{{(\frac{K L_{y}}{r_{y}})}^{2}} = \frac{π^{2} \times 205,000}{{(17.0)}^{2}} = 6,991 MPa

λ_{x} = \sqrt{\frac{F_{y}}{F_{ex}}} = \sqrt{\frac{350}{20,360}} = 0.131

λ_{y} = \sqrt{\frac{F_{y}}{F_{ey}}} = \sqrt{\frac{350}{6,991}} = 0.224

Per Cl. 13.3.1, n = 1.34 (Group 2 W shape):

C_{rx} = ϕ A F_{y} {(1 + {λ_{x}}^{2 n})}^{- 1 / n} = 0.90 \times 9,280 \times 350 {[1 + {0.131}^{(2 \times 1.34)}]}^{- 1 / 1.34} {(10)}^{-3} = 2,914 kN

C_{ry} = ϕ A F_{y} {(1 + {λ_{y}}^{2 n})}^{- 1 / n} = 0.90 \times 9,280 \times 350 {[1 + {0.224}^{(2 \times 1.34)}]}^{- 1 / 1.34} {(10)}^{-3} = 2,884 kN

Flexural Torsional Buckling

{r_{o}}^{2} = {r_{x}}^{2} + {r_{y}}^{2} + X^{2} + Y^{2} = {(110.3)}^{2} + {(64.7)}^{2} + 0 + 0 = 16,360 {mm}^{2}

F_{ez} = [\frac{π^{2} E C_{w}}{{(K_{z} L_{z})}^{2}} + G J] \frac{1}{A \times {r_{o}}^{2}}

= [\frac{π^{2} \times 205,000 \times 553 {(10)}^{9}}{{(1,100)}^{2}} + 76,920 \times 575,000] \frac{1}{9,280 \times 16,360} = 6,382 MPa

Take the minimum of F_ex, F_ey, and F_ez as F_FTB:

F_FTB = F_ez = 6,382 MPa

λ_{FTB} = \sqrt{\frac{F_{y}}{F_{e}}} = \sqrt{\frac{350}{6,382}} = 0.234

C_{r,FTB} = ϕ A F_{y} {(1 + {λ_{z}}^{2 n})}^{- 1 / n} = 0.90 \times 9,280 \times 350 {[1 + {0.234}^{(2 \times 1.34)}]}^{- 1 / 1.34} {(10)}^{-3} = 2,879 kN

Ratio = 1,000 / 2,879 = 0.347

Comparison

Table 1. Comparison of results
Criteria	Reference	STAAD.Pro	Difference
λ_x	0.131	0.131	none
F_ex (MPa)	20,360	20,360.98	negligible
C_rx (kN)	2,914	2,910	negligible
λ_y	0.224	0.224	none
F_ey (MPa)	6,991	6,991,204	negligible
C_ry (kN)	2,884	2,880	negligible
λ_FTB	0.234	0.234	none
F_FTB (MPa)	6,382	6,381.727	negligible
C_FTB (kN)	2,879	2,880	negligible
Critical ratio	0.347	0.347	none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\CSA S16-19 - Short Column Compression.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 20-Sep-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 1.1 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 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
2 FY -1000
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN 2019
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:  0.347         LOADCASE    1     |
|  LOCATION:      0.00    CRITICAL CONDITION:            Cl. 13.8.2        |
|--------------------------------------------------------------------------|
|  STRENGTH CHECK SUMMARY                                                  |
|   CRITICAL RATIO:  0.347(PASS)            LOAD CASE:     1               |
|   LOCATION      :   0.00                  CONDITION: Cl. 13.8.2          |
|--------------------------------------------------------------------------|
|  DESIGN FORCES              LC:       1      LOC:      0.00              |
|         FX:   1000.00(C)    FY:     0.00      FZ:      0.00              |
|         MX:  0.000E+00      MY: 0.000E+00     MZ:  0.000E+00             |
|--------------------------------------------------------------------------|
|  SECTION PROPERTIES (LOC:   0.00)             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:     1.100        LZ:    1.100    LY:    1.100    |
|   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   :      17.012          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:   1.00E+03   2.91E+03     0.343  Cl. 13.3            1    0.000 |
|   MINOR:   1.00E+03   2.88E+03     0.347  Cl. 13.3            1    0.000 |
|  INTERMEDIATE RESULTS:                                                   |
|               Ag:(CM)     Fe:(N MM)       λ:        n:      KL/r:        |
|   MAJOR:      92.800       20360.980       0.131    1.34       9.968     |
|   MINOR:      92.800        6991.204       0.224    1.34      17.012     |
|--------------------------------------------------------------------------|
|  FLEXURAL TORSIONAL BUCKLING                                             |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    1.00E+03   2.88E+03       0.347      Cl. 13.3          1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Ag:(CM)    92.800   Fe:(N MM)     6381.727  &#955;:     0.234  n: 1.34      |
|--------------------------------------------------------------------------|
|                             CHECKS FOR SHEAR                             |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Y AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   4.52E+02       0.000    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    5.552     0.224       489.474      1465.103    231.000     |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Z AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   1.36E+03       0.000    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:       |
|     0.00E+00  3.103E+02      0.000    Cl. 13.5(a)         1     0.000    |
|  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:       |
|    0.00E+00  1.458E+02      0.000    Cl. 13.5(a)         1     0.000     |
|  INTERMEDIATE RESULTS: Mp:  1.62E+02   Se(CM):  0.00E+00    My:  0.00E+00|
|--------------------------------------------------------------------------|
|  LATERAL TORSIONAL BUCKLING (MAJOR AXIS)                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00  3.103E+02      0.000   Cl. 13.6(a)(i)       1     0.000     |
|  INTERMEDIATE RESULTS:                                                   |
|   Iyc(CM):    w 2:    w 3:    b x(CM):  Mu:       rt:   Myr:     Lu:       Lyr:  |
|  0.00E+00   2.50  0.00     0.00  1.982E+04  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   : 0.000  Cl. 13.9.2      1   0.00   0.00E+00  2.92E+03  |
|                        Mfz:         Mfy:          Mrz:         Mry:      |
|   C/S STRENGTH   :    0.00E+00      0.00E+00     3.10E+02      1.46E+02  |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL COMPRESSION                                           |
|                    RATIO:   CRITERIA:   LC:   LOC:     Cf:        Cr:    |
|   C/S STRENGTH   : 0.342  Cl. 13.8.2      1   0.000 1.00E+03  2.92E+03   |
|   MEMBER STRENGTH: 0.347  Cl. 13.8.2      1   0.000 1.00E+03  2.88E+03   |
|   LTB STRENGTH   : 0.347  Cl. 13.8.2      1   0.000 1.00E+03  2.88E+03   |
|                        Mfz:      Mfy:      Mrz:      Mry:    w 1z:    w 1y:    |
|   C/S STRENGTH   :  0.00E+00  0.00E+00  3.10E+02  1.46E+02  0.60   0.60  |
|   MEMBER STRENGTH:  0.00E+00  0.00E+00  3.10E+02  1.46E+02  0.60   0.60  |
|   LTB STRENGTH   :  0.00E+00  0.00E+00  3.10E+02  1.46E+02  0.60   0.60  |
|                        b :         Cez:     Cey:         U1z:     U1y:     |
|   C/S STRENGTH   :    0.69     1.89E+05  6.49E+04     1.00     1.00      |
|   MEMBER STRENGTH:    0.69     1.89E+05  6.49E+04     1.00     1.00      |
|   LTB STRENGTH   :    0.69     1.89E+05  6.49E+04     1.00     1.00      |
|--------------------------------------------------------------------------|
|  SHEAR CAPACITY MODIFICATION  ( SHEAR AND FLEX )                         |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|     0.00E+00  4.523E+02      0.000    Cl. 14.6            1     0.000    |
|--------------------------------------------------------------------------|
|  BIAXIAL FLEXURE                                                         |
|   RATIO:  CRITERIA:  LC:   LOC:     Mfz:      Mrz:       Mfy:      Mry:  |
|   0.000  Cl. 13.8     1    0.00   0.00E+00  3.10E+02  0.00E+00  1.46E+02 |
|==========================================================================|
      STAAD SPACE                                              -- PAGE NO.    8