V.CSA S16-09 - Wide Flange Capacity Combined Stresses

Check the capacity of a wide-flange column under combined stresses.

Reference

Kulak, G.L. and G.Y. Grondin. 2010. Limit States Design in Structural Steel. 9th Edition. Markham, ON:CISC. Example 8.5

Problem

The W250x73 column has the following concentrated loads applied at the top of the column:

C_f = 900 kN

M_fx = 180 kN·m

The material used is G40.21 350W steel (F_y = 350 MPa) (FYLD). The column is assumed to be braced laterally against lateral-torsional buckling (LAT 1).

L = 3.6 m

Calculations

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.

Axial Capacity

Determine the buckling load of the column, assuming that weak axis bucking will not control due to bracing:

{(\frac{K L}{r})}_{x} = \frac{1.0 (3, 600)}{110} = 32.7

The largest slenderness ratio controls:

F_{e x} = \frac{π^{2} E}{{(\frac{K L}{r})}_{x}^{2}} = \frac{π^{2} (200, 000)}{{(32.7)}^{2}} = 1, 847 kN

λ = \sqrt{\frac{F_{y}}{F_{e y}}} = \sqrt{\frac{350}{1, 847}} = 0.435

C_{r} = ϕ A F_{y} {(1 + λ^{2 n})}^{- 1 / n} = 0.90 \times 9, 280 \times 350 {[1 + {0.435}^{(2 \times 1.34)}]}^{- 1 / 1.34} = 2, 716 \times {(10)}^{6} N = 2, 716 kN

Moment Modifier

κ = -0/180 = 0

ω₁ = 0.6 - 0.4κ = 0.6 ≤ 1.0

Calculate the elastic buckling load for the bending axis:

C_{e} = \frac{π^{2} \times 200, 000 \times 113 \times 10^{6}}{{(1.0 \times 3, 600)}^{2}} = 17, 211 kN

U_{1 x} = \frac{0.6}{1 - (\frac{900}{17, 210})} = 0.633

Bending Capacity

From a previous example, the W250x73 is a Class 2 section. It is laterally supported along its length, thus:

M_rx = 0.9×985×10³ ×350 = 310 kN·m

Combined Stress Ratio

\frac{900}{2, 716} + \frac{0.85 \times 0.633 \times 180}{310} = 0.331 + 0.312 = 0.644 < 1.0

This is the check for overall member strength. However, there is another case that actually controls in this instance: checking for cross sectional strength.

C_r = 0.9×9,280×350 = 2,923 kN

U_1x = 1.0

\frac{900}{2, 923} + \frac{0.85 \times 1.0 \times 180}{310} = 0.308 + 0.494 = 0.801 < 1.0

Comparison

Table 1. Comparison of results
Criteria	Reference	STAAD.Pro	Difference
(KL/r)_x	33	32.624	negligible
(KL/r)_y	56	55.675	negligible
C_r (kN)	2,716	2,715.935	negligible
C_e (kN)	17,211	17,600	2.3%
Stress Ratio	0.644	0.643	none
Stress Ratio (Critical)	0.801	0.801	none

Note: Both the reference and hand calculations agree with the STAAD.Pro results that the section is adequate. The reference however neglects to account for section capacity checks, which control in this case.

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 - Wide Flange Capacity Combined Stresses.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 17-Mar-19
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 ENFORCED BUT MX MY MZ
1 PINNED
2 ENFORCED BUT FY MX MZ
*2 ENFORCED BUT FX FZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
1 FY 900
*1 MZ 180
2 FY -900 MZ 180
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN 2009
FYLD 350000 ALL
LAT 1 ALL
SSZ 1 ALL
SSY 1 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:  0.801(PASS)     LOAD:     1
  LOCATION (MET): 3.60  CONDITION: Cl. 13.8.2     
  SECTION: ST   W250X73                 (CANADIAN SECTIONS)
  UNIT: KN MET
  STRENGTH CHECKS:
  CRITICAL RATIO:  0.801(PASS)     LOAD CASE:     1  LOCATION (MET): 3.60  CONDITION: Cl. 13.8.2     
  DESIGN FORCES:   Fx:    900.00(C)    Fy:    50.00      Fz:      0.00
                   Mx:   0.00E+00      My:  0.00E+00     Mz:  -1.80E+02
  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):     3.600
  PARAMETERS:                  KZ:  1.000  KY:  1.000  NSF:  1.000  SLF:  1.000
  SLENDERNESS:   ACTUAL SLENDERNESS RATIO:      55.675  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:         900.000    2715.935      0.331    Cl. 13.3           1        0.000
  MINOR:         900.000    2233.940      0.403    Cl. 13.3           1        0.000
  INTERMEDIATE:   Ag:(CM)     KL/r:     Fe:(N MM)        λ          n:
  MAJOR:          92.800     32.624     1900.987      0.429        1.340
  MINOR:          92.800     55.675      652.728      0.732        1.340
  SHEAR:          FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:          50.000     452.349      0.111    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    5.360       0.070       480.943      1414.474      231.000
  MINOR:          72.136
  UNIT: KN MET
  YIELDING:       FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  MAJOR:        1.80E+02    3.10E+02      0.580    Cl. 13.5(a)        1        3.600
  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
  INTERACTION:                RATIO:    CRITERIA:    LOAD CASE:   LOCATION(MET):
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:                0.580    Cl. 13.9.1         1       3.600
  FLEXURE AND AXIAL COMPRESSION:
  C/S STRENGTH:                0.801    Cl. 13.8.2         1       3.600
  MEMBER STRENGTH:             0.715    Cl. 13.8.2         1       0.000
  FLEX AND SHEAR:              0.472    Cl. 14.6(a)        1       3.600
  BIAXIAL FLEX:                0.580    Cl. 13.8           1       3.600
  INTERMEDIATE:
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:   Tf:   0.00E+00  Tr:   2.92E+03  Mfz:   1.80E+02  Mrz:   3.10E+02
  FLEXURE AND AXIAL COMPRESSION:
             Cf:       Mfz:       Mfy:     Cr:     Mrz:     Mry:     b :    Cez:     Cey:    w 1z:  w 1y: U1z: U1y:
  C/S STR: 9.00E+02  1.80E+02  0.00E+00 2.92E+03 3.10E+02 1.46E+02 0.60 1.76E+04 6.06E+03 0.60 0.60 1.00 1.00
  MEM STR: 9.00E+02  1.80E+02  0.00E+00 2.23E+03 3.10E+02 1.46E+02 0.85 1.76E+04 6.06E+03 0.60 0.60 0.63 0.70
  FLEX SHEAR:  Mfz:   1.80E+02  Mrz:   3.10E+02  Vfy:   5.00E+01  Vry:   4.52E+02
      STAAD SPACE                                              -- PAGE NO.    7
  BIAX FLEX:   Mfz:   1.80E+02  Mrz:   3.10E+02  Mfy:   0.00E+00  Mry:   1.46E+02
  -------------------- END OF DESIGN OUTPUT OF MEMBER      1 --------------------