# V. CSA S16-14 - 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. Verification Problem comparison
(KL/r)x 33 32.624 negligible
(KL/r)y 56 55.675 negligible
Cr (kN) 2,216 2,220 negligible
Ce (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.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\Verification Models\09 Steel Design\Canada\CSA S16-14 - Shear Capacity Combined Stresses.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 17-Sep-13
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
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
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.PRO CODE CHECKING - S16-14 (v1.0)
*******************************************
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 : 0.00  CONDITION: Cl. 13.8.2
UNIT: KN   MET
STRENGTH CHECKS:
CRITICAL RATIO:  0.926(PASS)     LOAD CASE:     1  LOCATION : 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 (N MM):      350.000   FU (N MM):      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.:   0.000
ALLOWABLE SLENDERNESS RATIO:  200.000
SECTION CLASS:
COMPRESSION:     Class 1
FLEXURE      MAJOR          MINOR
SECTION:    Class 2        Class 2
FLANGE:     Class 2        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 - (          S16-14)   v1.0
********************************************
ALL UNITS ARE - KN   MET  (UNLESS OTHERWISE Noted)
COMPRESSION:    FORCE:    CAPACITY:    RATIO:    CRITERIA:    LOAD CASE:   LOCATION:
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:
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 - (          S16-14)   v1.0
********************************************
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 --------------------
***NOTE:OMEGA1 AND OMEGA2 ARE CALCULATED USING C/S MOMENT OF ANALYTICAL MEMBERS