V. CSA S16-19 - Beam Shear Capacity

Check a the shear capacity of a wide flange beam per the CSA S16-19 code

References

Canadian Institute of Steel Construction. Handbook of Steel Construction. 9th Edition. 2008. Willowdale, Ont., pp.6-46 - 6-47
American Institute of Steel Construction, Steel Construction Manual. 15th Edition, American Institute of Steel Construction, 2017 p.3-213 & p.3-222
American Institute of Steel Construction, Specification for Structural Steel Buildings. AISC 360-16, American Institute of Steel Construction, 2016

Details

The beam is an 11 m long, simply-supported span. The member is subject to a uniform load of 40.033 kN/m along the length and a concentrated load of 352.27 kN at 55 mm before the right support. The member is also subjected to concentrated moments of 540 kN·m at the left support and -185 kN·m at the right support.

Assume that the web is stiffened at supports and that the member is laterally unsupported. Assume loads are factored.

The section used is a Canadian W530x82. The steel has a yield strength of 350 MPa (50 ksi) and ultimate strength of 450 MPa (65 ksi).

Section Properties

d = 528 mm
b = 209 mm
t_f = 13.3 mm
t_w = 9.5 mm
I_y = 20.3×(10)⁶ mm⁴
Z_x = 2,060 mm³
J = 518×(10)³ mm⁴
C_w = 1,340×(10)⁹ mm⁶

Material Properties

E = 205,000 MPa
G = 76,920 MPa

Calculations

Design Forces

Shear at the right end:

V_{f} = 352.27 (\frac{10.945}{11}) + \frac{1}{2} (40.033 \times 11) + \frac{185 - 540}{11} = 350.5 + 220.2 - 49.1 = 538.4 kN

Moment at point of maximum shear (right end):

M_f = 185 kN·m

Maximum moment (at left end):

M_max = 540 kN·m

Moment at quarter points of span (for calculating moment gradient coefficient), calculated per Table 3-23 in the AISC Steel Construction Manual:

M_{a} = \frac{40.033 (2.75)}{2} (11 - 2.75) + (\frac{540 - 185}{11}) 2.75 - 540 + \frac{352.27 (2.75) (0.055)}{11} = 7.7 kN·m

M_{b} = \frac{40.033 (5.5)}{2} (11 - 5.5) + (\frac{540 - 185}{11}) 5.5 - 540 + \frac{352.27 (5.5) (0.055)}{11} = 252.7 kN·m

M_{c} = \frac{40.033 (8.25)}{2} (11 - 8.25) + (\frac{540 - 185}{11}) 8.25 - 540 + \frac{352.27 (8.25) (0.055)}{11} = 194.9 kN·m

Shear Capacity

Along major axis:

Web depth, h = d - 2×t_f = 528 - 2× 13.3 = 501 mm

V_r = ϕA_wF_s

(13.4.1.1)

where

A_w	=	area of the web, = d × t_w = 528 × 9.5 = 5,016 mm²
F_s	=	the shear stress capacity.

a = 11,000 mm (distance between stiffeners)

a/h = 11,000 / 501 = 21.9 > 1.0

The shear buckling coefficient, k_v = 5.34 + 4 / (a/h)² = 5.348

\frac{h}{t_{w}} = \frac{501}{9.5} = 52.8 < 439 \sqrt{\frac{k_{v}}{F_{y}}} = 439 \sqrt{\frac{5.348}{350}} = 54.3

Therefore, F_s = 0.66F_y = 0.66 (350) = 231 MPa

V_r = 0.9 × 5,016 × 231 (10)^-3 = 1,043 kN

The ratio along the major axis = V_f / V_r = 538.4 / 1,043 = 0.516

Along minor axis: CSA S16-19 does not directly address the case of shear along the minor axis. Therefore, the method used in AISC 360-16 section G7. is used.

A_w = 2×b×t_f = 2 (209) (13.3) = 5,559 mm²

k_v = 1.2 (AISC G7)

\frac{h}{t_{w}} = \frac{b}{t_{f}} = \frac{209}{13.3} = 15.7 < 1.10 \sqrt{\frac{k_{v} E}{F_{y}}} = 1.10 \sqrt{\frac{1.2 \times 205,000}{350}} = 29.2

Therefore, C_v = 1.0.

V_r = ϕ 0.6A_wF_yC_v = 0.9 × 5,559 × 350 × 1.0 (10)^-3 = 1,051 kN

Bending Capacity

Calculate the bending capacity per Cl. 13.6.1:

M_p = Z_xF_y = 2,060 × 350 × 10^-3 = 721 kN·m

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

(13.6.1)

where

ω₂

\frac{4 M_{\max}}{\sqrt{{M_{\max}}^{2} + {4 M_{a}}^{2} + {7 M_{b}}^{2} + {4 M_{c}}^{2}}} = \frac{4 (540)}{\sqrt{{(540)}^{2} + 4 {(7.7)}^{2} + 7 {(252.7)}^{2} + 4 {(194.9)}^{2}}} = 2.29 < 2.5

M_{u} = \frac{2.29 \times π}{11,000} \sqrt{205,000 \times (20.3 \times 10^{6}) \times (76,920) \times (518 \times 10^{3}) + {(\frac{π \times 205,000}{11,000})}^{2} (20.3 \times 10^{6}) \times (1,340 \times 10^{9})}

= \frac{1}{1,530} \sqrt{165.8 {(10)}^{21} + 93.24 {(10)}^{21}} = 332.7 kN·m

M_u < 0.67M_p = 0.67 (721) = 483.1 kN·m

Therefore, M_r = ϕM_u = 0.9 ( 332.7) = 299.4 kN·m

Shear and Bending Interaction

F_s = 231 > 0.60×F_y = 210 MPa

Calculate the reduction factor for shear capacity due to combined shear and moment per Cl. 14.6:

[2.20 - 1.60 \frac{M_{f}}{M_{r}}] = [2.20 - 1.60 \frac{185}{299.4}] = 1.21

(14.6)

Therefore, no reduction in the shear capacity to combined effects of bending and shear:

V_r = 1,043 kN

Ratio = V_f / V_r = 538.4 / 1,043 = 0.516

Comparison

Table 1. Comparison of results
Criteria	Reference	STAAD.Pro	Difference
Shear buckling coefficient, k_v	5.348	5.349	negligible
Elastic shear stress capacity, F_s (MPa)	231	231	none
Factored shear capacity along major axis, V_ry (kN)	1,043	1,040	negligible
Shear ratio along major axis	0.516	0.516	none
Factored shear capacity along minor axis, V_rz (kN)	1,051	1,050	negligible
Factored shear capacity modified for shear and moment interaction, V_r (kN)	1,043	1,043	none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\CSA S16-19 - Beam Shear Capacity.STD is typically installed with the program.

The following design parameters are used:

Steel strength values are specified using FYLD 350000 and FU 450000 (the units are kN/m² or kPa).

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 18-Jan-22
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 11 0 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 W530X82
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 ENFORCED BUT FX MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
1 MZ 540
2 MZ -185
MEMBER LOAD
1 UNI GY -40.033
1 CON GY -352.27 10.945
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN 2019
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD Output

                       STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
                     ***********************************************
   ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
|--------------------------------------------------------------------------|
|   MEMBER NO.:      1      SECTION: ST  W530X82          (CANADIAN SECTIONS)|
|  STATUS:        FAIL    CRITICAL RATIO:  1.803         LOADCASE    1     |
|  LOCATION:      0.00    CRITICAL CONDITION:            Cl. 13.8          |
|--------------------------------------------------------------------------|
|  STRENGTH CHECK SUMMARY                                                  |
|   CRITICAL RATIO:  1.803(FAIL)            LOAD CASE:     1               |
|   LOCATION      :   0.00                  CONDITION: Cl. 13.8            |
|--------------------------------------------------------------------------|
|  DESIGN FORCES              LC:       1      LOC:      0.00              |
|         FX:      0.00(T)    FY:   254.22      FZ:      0.00              |
|         MX:  0.000E+00      MY: 0.000E+00     MZ:  5.400E+02             |
|--------------------------------------------------------------------------|
|  SECTION PROPERTIES (LOC:   0.00)             UNIT:      CM              |
|   AXX:     105.000     AZZ:      55.594      AYY:         50.160         |
|   IXX:      51.800     IZZ:   47700.008      IYY:       2030.000         |
|   PZZ:    2060.000     PYY:     303.000                                  |
|   SZZ:    1806.818     SYY:     194.258      CW :    1340255.500         |
|--------------------------------------------------------------------------|
|  MATERIAL PROPERTIES                          UNIT:      N MM            |
|   FYLD:     350.000     FU:   450.000      E: 204999.984  G:  76920.000  |
|--------------------------------------------------------------------------|
|  DESIGN PARAMETERS                                                       |
|   ACTUAL MEMBER LENGTH:    11.000        LZ:   11.000    LY:   11.000    |
|   KZ:  1.000        KY:  1.000      NSF:  1.000      SLF:  1.000         |
|--------------------------------------------------------------------------|
|  SECTION CLASS                                                           |
|   COMPRESSION    :            Class 4                                    |
|   FLEXURE           MAJOR               MINOR                            |
|   SECTION        : Class 2             Class 2                           |
|   FLANGE         : Class 2             Class 2                           |
|   WEB (CRITICAL) :            Class 1                                    |
|==========================================================================|
|  SLENDERNESS                                                             |
|   ACTUAL SLENDERNESS RATIO   :     250.172          LC   :        1      |
|   ALLOWABLE SLENDERNESS RATIO:     300.000          LOC  :      0.00     |
|--------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    6
                       STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
                     ***********************************************
   ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
   Member :     1       Contd.
|--------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL TENSION                         |
|--------------------------------------------------------------------------|
|  YIELDING                                                                |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   3.31E+03       0.000      Cl. 13.2          1     0.000    |
|  RUPTURE                                                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   3.54E+03       0.000      Cl. 13.2          1     0.000    |
|--------------------------------------------------------------------------|
|                       CHECKS FOR AXIAL COMPRESSION                       |
|--------------------------------------------------------------------------|
|  FLEX BUCKLING                                                           |
|            FORCE:    CAPACITY:   RATIO:    CRITERIA:      LC:    LOC:    |
|   MAJOR:   0.00E+00   2.25E+03     0.000  Cl. 13.3.4(a)       1    0.000 |
|   MINOR:   0.00E+00   2.53E+02     0.000  Cl. 13.3.4(a)       1    0.000 |
|  INTERMEDIATE RESULTS:                                                   |
|               Ag:(CM)     Fe:(N MM)       λ:        n:      KL/r:        |
|   MAJOR:      89.688         759.622       0.679    1.34      51.609     |
|   MINOR:      89.688          32.328       3.290    1.34     250.172     |
|--------------------------------------------------------------------------|
|  FLEXURAL TORSIONAL BUCKLING                                             |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   2.53E+02       0.000      Cl. 13.3.4        1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Ag:(CM)    89.688   Fe:(N MM)       32.328  λ:     3.290  n: 1.34      |
|--------------------------------------------------------------------------|
|                             CHECKS FOR SHEAR                             |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Y AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|   -5.38E+02   1.04E+03       0.516    Cl. 13.4.1.1        1    11.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)      Kv:       Ka:     Fcri:(N MM)  Fcre:(N MM)  Fs:(N MM)     |
|     50.160    5.349     0.048       237.748       345.654    231.000     |
|--------------------------------------------------------------------------|
|  SHEAR ALONG Z AXIS                                                      |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00   1.05E+03       0.000    AISC G2-1           1     0.000    |
|  INTERMEDIATE RESULTS:                                                   |
|   Aw:(CM)                                                                |
|     55.594                                                               |
|--------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    7
                       STAAD.PRO CODE CHECKING - CSA S16-19 (v1.0)
                     ***********************************************
   ALL UNITS ARE IN KN   MET   UNLESS OTHERWISE Noted
   Member :     1       Contd.
|--------------------------------------------------------------------------|
|                            CHECKS FOR BENDING                            |
|--------------------------------------------------------------------------|
|  YIELDING ALONG MAJOR AXIS                                               |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    -5.40E+02  6.489E+02      0.832    Cl. 13.5(a)         1     0.000    |
|  INTERMEDIATE RESULTS: Mp:  7.21E+02   Se(CM):  0.00E+00    My:  0.00E+00|
|--------------------------------------------------------------------------|
|  YIELDING ALONG MINOR AXIS                                               |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    0.00E+00  9.545E+01      0.000    Cl. 13.5(a)         1     0.000     |
|  INTERMEDIATE RESULTS: Mp:  1.06E+02   Se(CM):  0.00E+00    My:  0.00E+00|
|--------------------------------------------------------------------------|
|  LATERAL TORSIONAL BUCKLING (MAJOR AXIS)                                 |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|   -5.40E+02  2.994E+02      1.803   Cl. 13.6(a)(ii)      1     0.000     |
|  INTERMEDIATE RESULTS:                                                   |
|   Iyc(CM):    w 2:    w 3:    b x(CM):  Mu:       rt:   Myr:     Lu:       Lyr:  |
|  0.00E+00   2.29  0.00     0.00  3.327E+02  0.00  0.00E+00  0.00   0.00  |
|--------------------------------------------------------------------------|
|                          CHECKS FOR INTERACTION                          |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL TENSION                                               |
|                    RATIO:   CRITERIA:   LC:   LOC:     Tf:        Tr:    |
|   C/S STRENGTH   : 0.707  Cl. 13.9.2      1   0.00   0.00E+00  3.31E+03  |
|   MEMBER STRENGTH: 1.803  Cl. 13.9.3      1   0.00                       |
|                        Mfz:         Mfy:          Mrz:         Mry:      |
|   C/S STRENGTH   :   -5.40E+02      0.00E+00     6.49E+02      9.54E+01  |
|--------------------------------------------------------------------------|
|  FLEXURE AND AXIAL COMPRESSION                                           |
|                    RATIO:   CRITERIA:   LC:   LOC:     Cf:        Cr:    |
|   C/S STRENGTH   : 0.707  Cl. 13.8.2      1   0.000 0.00E+00  2.83E+03   |
|   MEMBER STRENGTH: 0.707  Cl. 13.8.2      1   0.000 0.00E+00  2.25E+03   |
|   LTB STRENGTH   : 1.533  Cl. 13.8.2      1   0.000 0.00E+00  2.53E+02   |
|                        Mfz:      Mfy:      Mrz:      Mry:    w 1z:    w 1y:    |
|   C/S STRENGTH   : -5.40E+02  0.00E+00  6.49E+02  9.54E+01  0.74   0.60  |
|   MEMBER STRENGTH: -5.40E+02  0.00E+00  6.49E+02  9.54E+01  0.60   0.74  |
|   LTB STRENGTH   : -5.40E+02  0.00E+00  2.99E+02  9.54E+01  0.74   0.60  |
|                        b :         Cez:     Cey:         U1z:     U1y:     |
|   C/S STRENGTH   :    0.85     7.98E+03  3.39E+02     1.00     1.00      |
|   MEMBER STRENGTH:    0.85     7.98E+03  3.39E+02     1.00     1.00      |
|   LTB STRENGTH   :    0.85     7.98E+03  3.39E+02     1.00     1.00      |
|--------------------------------------------------------------------------|
|  SHEAR CAPACITY MODIFICATION  ( SHEAR AND FLEX )                         |
|   FORCE:      CAPACITY:     RATIO:     CRITERIA:      LC:     LOC:       |
|    -5.38E+02  1.043E+03      0.516    Cl. 14.6            1    11.000    |
|--------------------------------------------------------------------------|
|  BIAXIAL FLEXURE                                                         |
|   RATIO:  CRITERIA:  LC:   LOC:     Mfz:      Mrz:       Mfy:      Mry:  |
|   1.803  Cl. 13.8     1    0.00  -5.40E+02  2.99E+02  0.00E+00  9.54E+01 |
|==========================================================================|
      STAAD SPACE                                              -- PAGE NO.    8