V. CSA S16-01 - 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. Comparison of results
Criteria	Reference	STAAD.Pro	Difference
(KL/r)_x	32.7	32.624	negligible
(KL/r)_y	55.7	55.675	negligible
C_r (kN)	2,216	2,217	negligible
M_u (kN·m)	871	871.4	negligible
M_rx (kN·m)	310	310.3	negligble
Stress Ratio	0.929	0.926	negligble

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\CSA S16-01 - Shear Capacity Combined Stresses.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 26-June-20
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
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 ENFORCED BUT FY MX MZ
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 2001
SSY 1 ALL
SSZ 1 ALL
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

      STAAD SPACE                                              -- PAGE NO.    5
                       STAAD.PRO CODE CHECKING - (CAN/CSA-S16-01  ) V2.1           
                       ********************************************
 ALL UNITS ARE - KNS  MET  (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
       1 ST   W250X73                  (CANADIAN SECTIONS)
                           PASS     CSA-13.8.2C        0.926         1
                      900.00 C          0.00         180.00        3.60
      STAAD SPACE                                              -- PAGE NO.    6
 ALL UNITS ARE - KNS  MET  (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
   MEMBER PROPERTIES (UNIT = CM)
   -----------------------------
   CROSS SECTION AREA =  9.28E+01   MEMBER LENGTH =  3.60E+02
   IZ =  1.13E+04   SZ =  8.93E+02   PZ =  9.85E+02
   IY =  3.88E+03   SY =  3.06E+02   PY =  4.63E+02
   IX =  5.75E+01   CW =  5.53E+05
   MATERIAL PROPERTIES (UNIT = MPA)
   --------------------------------
   FYLD = 350.0   FU = 450.0   E =  2.00E+05   G =  7.69E+04
   SECTION CAPACITIES (UNIT - KN,M)
   ---------------------------------
   CR1 =  2.923E+03   CR2 =  2.217E+03   SECTION CLASS 2
   CRZ =  2.710E+03   CTORFLX =  2.217E+03
   TENSILE CAPACITY     =  2.923E+03   COMPRESSIVE CAPACITY =  2.217E+03
   FACTORED MOMENT RESISTANCE : MRY =  1.458E+02   MRZ =  3.103E+02
                                 MU =  8.714E+02
   FACTORED SHEAR RESISTANCE  : VRY =  4.523E+02   VRZ =  1.001E+03
   MISCELLANEOUS INFORMATION
   --------------------------
   NET SECTION FACTOR FOR TENSION =  1.000
   KL/RY =   55.675   KL/RZ =   32.624   ALLOWABLE KL/R =  200.000
   UNSUPPORTED LENGTH OF THE COMPRESSION FLANGE (M) =  3.600
   OMEGA-1 (Y-AXIS) = 1.00   OMEGA-1 (Z-AXIS) = 1.00   OMEGA-2 = 1.00
   SHEAR FORCE (KNS) : Y AXIS =  0.000E+00   Z AXIS =  0.000E+00
   SLENDERNESS RATIO OF WEB (H/W) =  2.61E+01