V.CSA S16-09 - Shear Capacity Combined Stresses

Reference

Kulak, G.L. and G.Y. Grondin. 2010. Limit States Design in Structural Steel. 9th Edition. Markham, ON:CISC. Example 8.6

Problem

The W250x73 column has the following concentrated loads

C_f = 900 kN

And a uniform moment of:

M_fx = 180 kN·m

The material used is G40.21 350W steel (F_y = 350 MPa) (FYLD). The beam is braced against lateral-torsional buckling only at the ends (LAT 0, default)

Calculations

Axial Capacity

Determine the buckling load of the column:

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

{(\frac{K L}{r})}_{y} = \frac{1.0 (3, 600)}{64.6} = 55.7

The largest slenderness ratio controls:

F_{e x} = \frac{π^{2} E}{{(\frac{K L}{r})}_{x}^{2}} = \frac{π^{2} (200, 000)}{{(55.7)}^{2}} = 635.6 kN

λ = \sqrt{\frac{F_{y}}{F_{e y}}} = \sqrt{\frac{350}{635.6}} = 0.742

C_{r} = ϕ A F_{y} {(1 + λ^{2 n})}^{- 1 / n} = 0.90 \times 9, 280 \times 350 {[1 + {0.742}^{(2 \times 1.34)}]}^{- 1 / 1.34} = 2, 216 \times {(10)}^{6} N = 2, 216 kN

Moment Modifier

Uniform moment:

ω₁ = 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{1.0}{1 - \frac{900}{17, 211}} = 1.06

Bending Capacity

ω₂ = 1.0

EI_yGJ = 200,000×38.8×76.92×575= 343.2×10²¹ MPa²·mm⁸

{(\frac{π E}{L})}^{2} I_{y} C_{w} = {(\frac{π \times 200, 000}{3, 600})}^{2} 38.8 \times 553 = 653.6 \times 10^{21} {MPa}^{2} \cdot {mm}^{8}

M_{u} = \frac{ω_{2} π}{L} \sqrt{E I_{y} G J + {(\frac{π E}{L})}^{2} I_{y} C_{w}}

= \frac{1.0 \times π}{3,600} \sqrt{343.2 \times 10^{21} + 653.6 \times 10^{21}} = 871 \times 10^{6} N·mm = 871 kN·m

Plastic moment capacity:

M_p = Z_xF_y = 985×10³ × 350 = 345×10⁶ N·mm = 345 kN·m

M_u = 871 kN·m > 0.67×M_p = 0.67×345 = 231 kN·m

Therefore, the moment capacity is calculated as:

M_{r} = 1.15 ϕ M_{p} (1 - \frac{0.28 M_{p}}{M_{u}}) = 1.15 \times 0.9 \times 345 (1 - \frac{0.28 \times 345}{871}) = 317 kN·m

Check the capacity based on the strength cross-section:

M_rx = 0.9×985×10³ ×350 = 310 kN·m

This governs capacity.

Combined Stress Ratio

\frac{900}{2, 216} + \frac{0.85 \times 1.06 \times 180}{310} = 0.406 + 0.523 = 0.929 < 1.0

Comparison

Table 1. Verification Problem comparison
Criteria	Reference	STAAD.Pro	Difference
(KL/r)_x	33	32.624	negligible
(KL/r)_y	56	55.675	negligible
C_r (kN)	2,216	2,220	negligible
C_e (kN)	17,211	17,200	negligible
Stress Ratio	0.929	0.926	negligible

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

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 21-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 2.00e+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
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 MZ 180
2 FY -900 MZ -180
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN 2009
SSY 1 ALL
SSZ 1 ALL
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:  0.926(PASS)     LOAD:     1
  LOCATION (MET): 0.00  CONDITION: Cl. 13.8.2     
  SECTION: ST   W250X73                 (CANADIAN SECTIONS)
  UNIT: KN MET
  STRENGTH CHECKS:
  CRITICAL RATIO:  0.926(PASS)     LOAD CASE:     1  LOCATION (MET): 0.00  CONDITION: Cl. 13.8.2     
  DESIGN FORCES:   Fx:    900.00(C)    Fy:     0.00      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    2709.553      0.332    Cl. 13.3           1        0.000
  MINOR:         900.000    2217.120      0.406    Cl. 13.3           1        0.000
  INTERMEDIATE:   Ag:(CM)     KL/r:     Fe:(N MM)        λ          n:
  MAJOR:          92.800     32.624     1854.621      0.434        1.340
  MINOR:          92.800     55.675      636.808      0.741        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    5.360       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        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:       -1.80E+02    3.17E+02      0.567    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  1.00   0.00     0.00   8.71E+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:                0.580    Cl. 13.9.1         1       0.000
  MEMBER STRENGTH:             0.580    Cl. 13.9.2         1       0.000
  FLEXURE AND AXIAL COMPRESSION:
  C/S STRENGTH:                0.828    Cl. 13.8.2         1       0.000
  MEMBER STRENGTH:             0.926    Cl. 13.8.2         1       0.000
  LTB STRENGTH:                0.926    Cl. 13.8.2         1       0.000
  FLEX AND SHEAR:              0.422    Cl. 14.6(a)        1       0.000
  BIAXIAL FLEX:                0.580    Cl. 13.8           1       0.000
  INTERMEDIATE:
  FLEXURE AND AXIAL TENSION:
  C/S STRENGTH:   Tf:   0.00E+00  Tr:   2.92E+03  Mfz:  -1.80E+02  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: 9.00E+02 -1.80E+02  0.00E+00 2.92E+03 3.10E+02 1.46E+02 0.60 1.72E+04 5.91E+03 1.00 0.60 1.06 1.00
  MEM STR: 9.00E+02 -1.80E+02  0.00E+00 2.22E+03 3.10E+02 1.46E+02 0.85 1.72E+04 5.91E+03 0.60 1.00 1.06 0.71
  LTB STR: 9.00E+02 -1.80E+02  0.00E+00 2.22E+03 3.10E+02 1.46E+02 0.85 1.72E+04 5.91E+03 1.00 0.60 1.06 0.71
  FLEX SHEAR:  Mfz:  -1.80E+02  Mrz:   3.10E+02  Vfy:   0.00E+00  Vry:   4.52E+02
  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 --------------------