V. CSA S16-19 - Check for Bending and LTB
Determine the capacity of a laterally unbraced beam subject to bending moment per the CSA S16-19 code.
Problem
A 7.5 m long, simply supported beam is subjected to a distributed load of 12 kN/m. The section is a Canadian W310x52. The steel is grade 350W.Section Properties
- D = 318 mm
- B = 167 mm
- tw = 7.6 mm
- tf = 13.2 mm
- Zx = 841,000 mm3
- Zy = 189,000 mm3
- Iy = 10.3×106 mm4
- J = 308,000 mm4
- Cw = 238×109 mm6
Material Properties
- Fy = 350 MPa
- Fu = 450 MPa
- E = 205,000 MPa
- G = 76,920 MPa
Calculations
Section Classification
bel = B / 2 = 83.5 mm
h = D - 2×tf = 291.6 mm
Flexure section classification per Table 2:
- Flange classification about major axis:
Class 1
- Flange classification about minor axis:
Class 1
- Web classification about major axis:
Class 1
- Web classification about minor axis:
Class 1
Therefore, the section is Class 1 for flexure.
Bending Capacity
For a Class 2, doubly symmetric section, the capacity is determined by Cl. 13.5a:
Major axis:
Mpx = Zx×Fy = 841,000 × 350 (10)-6 = 294.4 kN·m
Mrx = ϕMpx = 0.9 × 294.4 = 264.9 kN·m
The ratio = 84.4 / 264.9 = 0.318
Minor axis:
Mpy = Zy×Fy = 189,000 × 350 (10)-6 = 66.2 kN·m
Mrx = ϕMpx = 0.9 × 66.2 = 59.5 kN·m
Lateral Torsional Buckling
Check for lateral torsional buckling per Cl. 13.6:
where= | ||
= | ||
= |
109.3 < 0.67 (294.4) = 197.2
Mr = ϕMu = 0.9 × 109.3 = 98.38 kN·m
Ratio = 84.4 / 98.38 = 0.858.
Comparison
Criteria | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Nominal flexure capacity about major axis, Mpx (kN·m) | 294.4 | 294 | negligible | |
Ultimate flexure capacity about major axis, Mrx (kN·m) | 264.9 | 264.9 | none | |
Nominal flexure capacity about minor axis, Mpy (kN·m) | 66.2 | 66.2 | none | |
Ultimate flexure capacity about minor axis, Mrx (kN·m) | 59.5 | 59.54 | negligible | |
Ratio | 0.318 | 0.318 | none | |
Nominal lateral torsional bending capacity, Mu (kN) | 109.3 | 109.3 | none | |
Ultimate lateral torsional bending capacity, Mr (kN) | 98.38 | 98.38 | none | |
Critical Ratio | 0.858 | 0.858 | none |
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\Canada\CSA S16-19 - Check for Bending and LTB.STD is typically installed with the program.
- The laterally unsupported is designated by the parameter LAT 0.
- The value of ω2 is specified as 1 using the parameter CB 1
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 10-Oct-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 7.5 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 W310X52
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE None TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -12
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN
LAT 0 ALL
CB 1 ALL
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 W310X52 (CANADIAN SECTIONS)| | STATUS: PASS CRITICAL RATIO: 0.858 LOADCASE 1 | | LOCATION: 3.75 CRITICAL CONDITION: Cl. 13.8 | |--------------------------------------------------------------------------| | STRENGTH CHECK SUMMARY | | CRITICAL RATIO: 0.858(PASS) LOAD CASE: 1 | | LOCATION : 3.75 CONDITION: Cl. 13.8 | |--------------------------------------------------------------------------| | DESIGN FORCES LC: 1 LOC: 3.75 | | FX: 0.00(T) FY: 0.00 FZ: 0.00 | | MX: 0.000E+00 MY: 0.000E+00 MZ: -8.438E+01 | |--------------------------------------------------------------------------| | SECTION PROPERTIES (LOC: 3.75) UNIT: CM | | AXX: 66.700 AZZ: 44.088 AYY: 24.168 | | IXX: 30.800 IZZ: 11900.002 IYY: 1030.000 | | PZZ: 841.000 PYY: 189.000 | | SZZ: 748.428 SYY: 123.353 CW : 237980.875 | |--------------------------------------------------------------------------| | MATERIAL PROPERTIES UNIT: N MM | | FYLD: 350.000 FU: 450.000 E: 204999.984 G: 76920.000 | |--------------------------------------------------------------------------| | DESIGN PARAMETERS | | ACTUAL MEMBER LENGTH: 7.500 LZ: 7.500 LY: 7.500 | | KZ: 1.000 KY: 1.000 NSF: 1.000 SLF: 1.000 | |--------------------------------------------------------------------------| | SECTION CLASS | | COMPRESSION : Class 4 | | FLEXURE MAJOR MINOR | | SECTION : Class 1 Class 1 | | FLANGE : Class 1 Class 1 | | WEB (CRITICAL) : Class 1 | |==========================================================================| | SLENDERNESS | | ACTUAL SLENDERNESS RATIO : 190.856 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 2.10E+03 0.000 Cl. 13.2 1 0.000 | | RUPTURE | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | -0.00E+00 2.25E+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 1.56E+03 0.000 Cl. 13.3.4(a) 1 0.000 | | MINOR: 0.00E+00 3.07E+02 0.000 Cl. 13.3.4(a) 1 0.000 | | INTERMEDIATE RESULTS: | | Ag:(CM) Fe:(N MM) λ: n: KL/r: | | MAJOR: 65.224 641.730 0.739 1.34 56.150 | | MINOR: 65.224 55.545 2.510 1.34 190.856 | |--------------------------------------------------------------------------| | FLEXURAL TORSIONAL BUCKLING | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 0.00E+00 3.07E+02 0.000 Cl. 13.3.4 1 0.000 | | INTERMEDIATE RESULTS: | | Ag:(CM) 65.224 Fe:(N MM) 55.545 λ: 2.510 n: 1.34 | |--------------------------------------------------------------------------| | CHECKS FOR SHEAR | |--------------------------------------------------------------------------| | SHEAR ALONG Y AXIS | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 4.50E+01 5.02E+02 0.090 Cl. 13.4.1.1 1 0.000 | | INTERMEDIATE RESULTS: | | Aw:(CM) Kv: Ka: Fcri:(N MM) Fcre:(N MM) Fs:(N MM) | | 24.168 5.347 0.042 326.980 653.808 231.000 | |--------------------------------------------------------------------------| | SHEAR ALONG Z AXIS | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | -0.00E+00 8.33E+02 0.000 AISC G2-1 1 0.000 | | INTERMEDIATE RESULTS: | | Aw:(CM) | | 44.088 | |--------------------------------------------------------------------------| 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: | | 8.44E+01 2.649E+02 0.318 Cl. 13.5(a) 1 3.750 | | INTERMEDIATE RESULTS: Mp: 2.94E+02 Se(CM): 0.00E+00 My: 0.00E+00| |--------------------------------------------------------------------------| | YIELDING ALONG MINOR AXIS | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 0.00E+00 5.954E+01 0.000 Cl. 13.5(a) 1 0.000 | | INTERMEDIATE RESULTS: Mp: 6.62E+01 Se(CM): 0.00E+00 My: 0.00E+00| |--------------------------------------------------------------------------| | LATERAL TORSIONAL BUCKLING (MAJOR AXIS) | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 8.44E+01 9.838E+01 0.858 Cl. 13.6(a)(ii) 1 3.750 | | INTERMEDIATE RESULTS: | | Iyc(CM): w 2: w 3: b x(CM): Mu: rt: Myr: Lu: Lyr: | | 0.00E+00 1.00 0.00 0.00 1.093E+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 : 0.271 Cl. 13.9.2 1 3.75 -0.00E+00 2.10E+03 | | MEMBER STRENGTH: 0.858 Cl. 13.9.3 1 0.00 | | Mfz: Mfy: Mrz: Mry: | | C/S STRENGTH : 8.44E+01 0.00E+00 2.65E+02 5.95E+01 | |--------------------------------------------------------------------------| | FLEXURE AND AXIAL COMPRESSION | | RATIO: CRITERIA: LC: LOC: Cf: Cr: | | C/S STRENGTH : 0.271 Cl. 13.8.2 1 3.750 0.00E+00 2.05E+03 | | MEMBER STRENGTH: 0.271 Cl. 13.8.2 1 3.750 0.00E+00 1.56E+03 | | LTB STRENGTH : 0.729 Cl. 13.8.2 1 3.750 0.00E+00 3.07E+02 | | Mfz: Mfy: Mrz: Mry: w 1z: w 1y: | | C/S STRENGTH : 8.44E+01 0.00E+00 2.65E+02 5.95E+01 0.60 0.60 | | MEMBER STRENGTH: 8.44E+01 0.00E+00 2.65E+02 5.95E+01 0.60 0.60 | | LTB STRENGTH : 8.44E+01 0.00E+00 9.84E+01 5.95E+01 0.60 0.60 | | b : Cez: Cey: U1z: U1y: | | C/S STRENGTH : 0.85 4.28E+03 3.70E+02 1.00 1.00 | | MEMBER STRENGTH: 0.85 4.28E+03 3.70E+02 1.00 1.00 | | LTB STRENGTH : 0.85 4.28E+03 3.70E+02 1.00 1.00 | |--------------------------------------------------------------------------| | SHEAR CAPACITY MODIFICATION ( SHEAR AND FLEX ) | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 4.50E+01 5.025E+02 0.090 Cl. 14.6 1 0.000 | |--------------------------------------------------------------------------| | BIAXIAL FLEXURE | | RATIO: CRITERIA: LC: LOC: Mfz: Mrz: Mfy: Mry: | | 0.858 Cl. 13.8 1 3.75 8.44E+01 9.84E+01 0.00E+00 5.95E+01 |