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
- tf = 13.3 mm
- tw = 9.5 mm
- Iy = 20.3×(10)6 mm4
- Zx = 2,060 mm3
- J = 518×(10)3 mm4
- Cw = 1,340×(10)9 mm6
Material Properties
- E = 205,000 MPa
- G = 76,920 MPa
Calculations
Design Forces
Shear at the right end:
Moment at point of maximum shear (right end):
Mf = 185 kN·m
Maximum moment (at left end):
Mmax = 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:
Shear Capacity
Along major axis:
Web depth, h = d - 2×tf = 528 - 2× 13.3 = 501 mm
Vr = ϕAwFs | (13.4.1.1) |
= | ||
= |
a = 11,000 mm (distance between stiffeners)
a/h = 11,000 / 501 = 21.9 > 1.0
The shear buckling coefficient, kv = 5.34 + 4 / (a/h)2 = 5.348
Therefore, Fs = 0.66Fy = 0.66 (350) = 231 MPa
Vr = 0.9 × 5,016 × 231 (10)-3 = 1,043 kN
The ratio along the major axis = Vf / Vr = 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.
Aw = 2×b×tf = 2 (209) (13.3) = 5,559 mm2
kv = 1.2 (AISC G7)
Therefore, Cv = 1.0.
Vr = ϕ 0.6AwFyCv = 0.9 × 5,559 × 350 × 1.0 (10)-3 = 1,051 kN
Bending Capacity
Calculate the bending capacity per Cl. 13.6.1:
Mp = ZxFy = 2,060 × 350 × 10-3 = 721 kN·m
(13.6.1) |
= |
Mu < 0.67Mp = 0.67 (721) = 483.1 kN·m
Therefore, Mr = ϕMu = 0.9 ( 332.7) = 299.4 kN·m
Shear and Bending Interaction
Fs = 231 > 0.60×Fy = 210 MPa
Calculate the reduction factor for shear capacity due to combined shear and moment per Cl. 14.6:
(14.6) |
Therefore, no reduction in the shear capacity to combined effects of bending and shear:
Vr = 1,043 kN
Ratio = Vf / Vr = 538.4 / 1,043 = 0.516
Comparison
Criteria | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Shear buckling coefficient, kv | 5.348 | 5.349 | negligible | |
Elastic shear stress capacity, Fs (MPa) | 231 | 231 | none | |
Factored shear capacity along major axis, Vry (kN) | 1,043 | 1,040 | negligible | |
Shear ratio along major axis | 0.516 | 0.516 | none | |
Factored shear capacity along minor axis, Vrz (kN) | 1,051 | 1,050 | negligible | |
Factored shear capacity modified for shear and moment interaction, Vr (kN) | 1,043 | 1,043 | none |
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\Canada\CSA S16-19 - Beam Shear Capacity.STD is typically installed with the program.
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