# 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

Cf = 900 kN

And a uniform moment of:

Mfx = 180 kN·m

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

## Axial Capacity

Determine the buckling load of the column:

$( K L r ) x = 1.0 ( 3 , 600 ) 110 = 32.7$
$( K L r ) y = 1.0 ( 3 , 600 ) 64.6 = 55.7$

The largest slenderness ratio controls:

$λ = F y F e y = 350 635.6 = 0.742$

## Moment Modifier

Uniform moment:

ω1 = 1.0

Calculate the elastic buckling load for the bending axis:

$C e = π 2 × 200 , 000 × 113 × 10 6 ( 1.0 × 3 , 600 ) 2 = 17 , 211 kN$
$U 1 x = 1.0 1 − 900 17 , 211 = 1.06$

## Bending Capacity

ω2 = 1.0

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

$M u = ω 2 π L E I y G J + ( π E L ) 2 I y C w$

Plastic moment capacity:

Mp = ZxFy = 985×103 × 350 = 345×106 N·mm = 345 kN·m

Mu = 871 kN·m > 0.67×Mp = 0.67×345 = 231 kN·m

Therefore, the moment capacity is calculated as:

Check the capacity based on the strength cross-section:

Mrx = 0.9×985×103 ×350 = 310 kN·m

This governs capacity.

## Combined Stress Ratio

$900 2 , 216 + 0.85 × 1.06 × 180 310 = 0.406 + 0.523 = 0.929 < 1.0$

## Comparison

Table 1. Comparison of results
(KL/r)x 32.7 32.624 negligible
(KL/r)y 55.7 55.675 negligible
Cr (kN) 2,216 2,217 negligible
Mu (kN·m) 871 871.4 negligible
Mrx (kN·m) 310 310.3 negligble
Stress Ratio 0.929 0.926 negligble

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
1 TABLE ST W250X73
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 ENFORCED BUT FY MX MZ
1 FY 900 MZ 180
2 FY -900 MZ -180
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
SSY 1 ALL
SSZ 1 ALL
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH


      STAAD SPACE                                              -- PAGE NO.    5
STAAD.PRO CODE CHECKING - (CAN/CSA-S16-01  ) V2.1
********************************************
ALL UNITS ARE - KNS  MET  (UNLESS OTHERWISE Noted)
FX            MY             MZ       LOCATION
=======================================================================
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)
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