V.CSA S16-09 - Slender Column Compression

Determine the factored compressive resistance of a slender column.

Reference

Kulak, G.L. and G.Y. Grondin. 2010. Limit States Design in Structural Steel. 9th Edition. Markham, ON:CISC. Example 4.2

Problem

A W250x73 section is used for a pedestal with a height of 11.0 m.

The material used is G40.21 350W steel (F_y = 350 MPa) (FYLD)

Calculations

The nominal yield strength is used for the actual yield strength for a Group 2 section, per Table 3 and Table 4 of Appendix A of the reference.

Local Element Buckling

Evaluate the slenderness effects of the column web:

\frac{h}{w} = \frac{225}{8.6} = 26.2 < \frac{670}{\sqrt{350}} = 35.8

Evaluate the slenderness effects of the column flanges:

\frac{b}{2 t} = \frac{254}{2 \times 14.2} = 8.9 < \frac{200}{\sqrt{350}} = 10.7

Both the column web and flanges are less than the local buckling limit, so the capacity of the column is evaluated using Eq. 4.21.

Column Capacity

Assume K = 1.0,

\frac{L}{r_{x}} = \frac{11, 000}{110} = 100

\frac{L}{r_{y}} = \frac{11, 000}{64.6} = 170

The largest slenderness ration controls, thus the slenderness factor for buckling about the minor axis:

F_{e} = \frac{π^{2} \times 200, 000}{{(1.0 \times 170)}^{2}} = 68.3 kN

λ = \sqrt{\frac{F_{y}}{F_{e}}} = \sqrt{\frac{350}{68.3}} = 2.26

When n = 1.34 (Group 2 W shape),

C_{r} = ϕ A F_{y} {(1 + λ^{2 n})}^{- 1 / n} = 0.90 \times 9, 280 \times 350 {[1 + {2.26}^{(2 \times 1.34)}]}^{- 1 / 1.34} = 0.529 \times {(10)}^{6} N = 529 kN

Comparison

Table 1. Comparison of results
Criteria	Reference	STAAD.Pro	Difference
KL/r, major	100	99.684	negligible
KL/r, minor	170	170.118	negligible
λ, critical	2.26	2.237	1.0%
C_r (kN)	529	538.161	1.7%

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\S16 2009\CSA S16-09 - Slender Column Compression.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 22-Mar-19
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 11 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 -0.001
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN 2009
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

                       STAAD.PRO CODE CHECKING - S16-09 (v1.1)
                     *******************************************
 ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
  -------------------- START OF DESIGN OUTPUT OF MEMBER      1 --------------------
  MEMBER NO:     1    CRITICAL RATIO:  0.000(PASS)     LOAD:     1
  LOCATION (MET): 0.00  CONDITION: Cl. 13.8.2     
  SECTION: ST   W250X73                 (CANADIAN SECTIONS)
  UNIT: KN MET
  STRENGTH CHECKS:
  CRITICAL RATIO:  0.000(PASS)     LOAD CASE:     1  LOCATION (MET): 0.00  CONDITION: Cl. 13.8.2     
  DESIGN FORCES:   Fx:      0.00(C)    Fy:     0.00      Fz:      0.00
                   Mx:   0.00E+00      My:  0.00E+00     Mz:   0.00E+00
  UNIT: CM
  SECTION PROPERTIES:       AZZ:      72.136 AYY:      20.537 CW:     552900.312
                            SZZ:     893.281 SYY:     305.512
                            IZZ:   11300.001 IYY:    3880.000
  UNIT: NEW MM
  MATERIAL PROPERTIES:      FYLD:      350.000   FU:      450.000
  ACTUAL MEMBER LENGTH(MET):    11.000
  PARAMETERS:                  KZ:  1.000  KY:  1.000  NSF:  1.000  SLF:  1.000
  SLENDERNESS:   ACTUAL SLENDERNESS RATIO:     170.118  LOAD:     1   LOC.(MET):   0.000
                 ALLOWABLE SLENDERNESS RATIO:  200.000
  SECTION CLASS:
  COMPRESSION:     Class 1     
  FLEXURE:         Class 2     
  FLANGE:          Class 2     
  WEB:             Class 1     
  UNIT: KN MET
  TENSION:        FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  YIELDING:        0.000    2923.200      0.000    Cl. 13.2           1        0.000
  RUPTURE:         0.000    3132.000      0.000    Cl. 13.2           1        0.000
      STAAD SPACE                                              -- PAGE NO.    6
                       STAAD.PRO CODE CHECKING - (          S16-09)   v1.1
                       ********************************************
 ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
  COMPRESSION:    FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:           0.001    1266.716      0.000    Cl. 13.3           1        0.000
  MINOR:           0.001     538.161      0.000    Cl. 13.3           1        0.000
  INTERMEDIATE:   Ag:(CM)     KL/r:     Fe:(N MM)        λ          n:
  MAJOR:          92.800     99.684      203.610      1.311        1.340
  MINOR:          92.800    170.118       69.912      2.237        1.340
  SHEAR:          FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:           0.000     452.349      0.000    Cl. 13.4.1.1       1        0.000
  MINOR:           0.000    1363.370      0.000    AISC G2-1          1        0.000
  INTERMEDIATE:   Aw:(CM)    Kv:          Ka:       Fcri:(N MM)  Fcre:(N MM)  Fs:(N MM)
  MAJOR:          20.537    5.342       0.023       480.151      1409.819      231.000
  MINOR:          72.136
  UNIT: KN MET
  YIELDING:       FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:        0.00E+00    3.10E+02      0.000    Cl. 13.5(a)        1        0.000
  MINOR:        0.00E+00    1.46E+02      0.000    Cl. 13.5(a)        1        0.000
  INTERMEDIATE:   Mp:        Se:         My:
  MAJOR:        3.45E+02    0.00E+00    0.00E+00
  MINOR:        1.62E+02    0.00E+00    0.00E+00
  UNIT: KN MET
  LAT TOR BUCK:   FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:        0.00E+00    2.83E+02      0.000    Cl. 13.6(a)(i)     1        0.000
  INTERMEDIATE:  Iyc(CM):   w 2:     w 3:      b x:     Mu:     rt:     Myr:     Lu:     Lyr:
  MAJOR:        0.00E+00  2.50   0.00     0.00   4.66E+02  0.00   0.00E+00  0.00   0.00
  UNIT: KN MET
  INTERACTION:                RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:                0.000    Cl. 13.9.1         1       0.000
  FLEXURE AND AXIAL COMPRESSION:
  C/S STRENGTH:                0.000    Cl. 13.8.2         1       0.000
  MEMBER STRENGTH:             0.000    Cl. 13.8.2         1       0.000
  LTB STRENGTH:                0.000    Cl. 13.8.2         1       0.000
  FLEX AND SHEAR:              0.000    Cl. 14.6(a)        1       0.000
  BIAXIAL FLEX:                0.000    Cl. 13.8           1       0.000
  INTERMEDIATE:
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:   Tf:   0.00E+00  Tr:   2.92E+03  Mfz:   0.00E+00  Mrz:   3.10E+02
      STAAD SPACE                                              -- PAGE NO.    7
                       STAAD.PRO CODE CHECKING - (          S16-09)   v1.1
                       ********************************************
 ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
  FLEXURE AND AXIAL COMPRESSION:
             Cf:       Mfz:       Mfy:     Cr:     Mrz:     Mry:     b :    Cez:     Cey:    w 1z:  w 1y: U1z: U1y:
  C/S STR: 1.00E-03  0.00E+00  0.00E+00 2.92E+03 3.10E+02 1.46E+02 0.85 1.89E+03 6.49E+02 0.60 0.60 1.00 1.00
  MEM STR: 1.00E-03  0.00E+00  0.00E+00 5.38E+02 3.10E+02 1.46E+02 0.85 1.89E+03 6.49E+02 0.60 0.60 1.00 1.00
  LTB STR: 1.00E-03  0.00E+00  0.00E+00 5.38E+02 2.83E+02 1.46E+02 0.85 1.89E+03 6.49E+02 0.60 0.60 1.00 1.00
  FLEX SHEAR:  Mfz:   0.00E+00  Mrz:   2.83E+02  Vfy:   0.00E+00  Vry:   4.52E+02
  BIAX FLEX:   Mfz:   0.00E+00  Mrz:   2.83E+02  Mfy:   0.00E+00  Mry:   1.46E+02
  -------------------- END OF DESIGN OUTPUT OF MEMBER      1 --------------------