V. CSA S16-19 - Cantilever Beam with Point Load
Determine the capacity of a cantilever beam subject to a point load per the CSA S16-19 code.
Details
A 7.5 m long cantilever member is subjected to load at the free end of 5 kN. The section is a Canadian W310x52. The steel is grade 350W. The section is unbraced along the length.
Section Properties
- D = 318 mm
- B = 167 mm
- tw = 7.6 mm
- tf = 13.2 mm
- Ag = 9,280 mm2
- Zx = 841,000 mm3
- Zy = 189,000 mm3
- Iy = 1,030,00 mm4
- J = 308,000 mm4
- Cw = 238×109 mm6
Material Properties
- Fy = 350 MPa
- Fu = 450 MPa
Validation
Section Classification
bel = B / 2 = 83.5 mm
h = D - 2×tf = 291.6 mm
Cf = 0 (no compression force)
Flexure section classification per Table 2 of CSA S16-19:
- 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 1, doubly symmetric section, the capacity is determined by Cl. 13.5:
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 = 37.5 / 264.9 = 0.142
Minor axis:
Mpy = Zy×Fy = 189,000 × 350 (10)-6 = 66.2 kN·m
Mrx = ϕMpx = 0.9 × 66.2 = 59.5 kN·m
Check for lateral-torsional buckling per Cl. 13.6.h i).
where| = | 0.5 ≤ X ≤ 2.3 for a point load | |
| = | ||
| = | ||
| = |
0.67×Mp = 0.67 (294.4) = 197.2 kN·m > Mu
Mr = ϕMu = 0.9 × 90.52 = 81.5 kN·m
The ratio = 37.5 / 81.5 = 0.46
Results
| Result Type | Reference | STAAD.Pro | Difference | Comments |
|---|---|---|---|---|
| Mpx (kN·m) | 294.4 | 294 | negligible | |
| Mrx (kN·m) | 264.9 | 264.9 | none | |
| Ratio of Mfx/Mrx | 0.142 | 0.142 | none | |
| Mpy (kN·m) | 66.2 | 66.2 | none | |
| Mry (kN·m) | 59.5 | 59.54 | negligible | |
| Mu (kN·m) | 90.5 | 90.52 | negligible | |
| Mr (kN·m) | 81.5 | 81.47 | negligible | |
| ω2 | 6.07 | 6.07 | none | |
| Critical ratio | 0.46 | 0.46 | none |
STAAD.Pro Input
The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Canada\CSA S16-19 - Cantilever Beam with Point Load.STD is typically installed with the program.
The following design parameters are used:- The unbraced cantilever is specified using CAN 1.
- The load acts on the tension side of the bending member, thus LHT 1.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 02-Dec-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 FIXED
LOAD 1 LOADTYPE None TITLE LOAD CASE 1
JOINT LOAD
2 FY -5
PERFORM ANALYSIS
PARAMETER 1
CODE CANADIAN
CAN 1 ALL
LHT 1 ALL
FYLD 350000 ALL
FU 450000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH
STAAD.Pro 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: 0.460 LOADCASE 1 | | LOCATION: 0.00 CRITICAL CONDITION: Cl. 13.8 | |--------------------------------------------------------------------------| | STRENGTH CHECK SUMMARY | | CRITICAL RATIO: 0.460(PASS) LOAD CASE: 1 | | LOCATION : 0.00 CONDITION: Cl. 13.8 | |--------------------------------------------------------------------------| | DESIGN FORCES LC: 1 LOC: 0.00 | | FX: 0.00(T) FY: 5.00 FZ: 0.00 | | MX: 0.000E+00 MY: 0.000E+00 MZ: 3.750E+01 | |--------------------------------------------------------------------------| | SECTION PROPERTIES (LOC: 0.00) 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. 4 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: | | 5.00E+00 5.02E+02 0.010 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. 5 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: | | -3.75E+01 2.649E+02 0.142 Cl. 13.5(a) 1 0.000 | | 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: | | -3.75E+01 8.147E+01 0.460 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 6.07 0.00 0.00 9.052E+01 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 : 0.120 Cl. 13.9.2 1 0.00 0.00E+00 2.10E+03 | | MEMBER STRENGTH: 0.460 Cl. 13.9.3 1 0.00 | | Mfz: Mfy: Mrz: Mry: | | C/S STRENGTH : -3.75E+01 0.00E+00 2.65E+02 5.95E+01 | |--------------------------------------------------------------------------| | FLEXURE AND AXIAL COMPRESSION | | RATIO: CRITERIA: LC: LOC: Cf: Cr: | | C/S STRENGTH : 0.120 Cl. 13.8.2 1 0.000 0.00E+00 2.05E+03 | | MEMBER STRENGTH: 0.120 Cl. 13.8.2 1 0.000 0.00E+00 3.07E+02 | | LTB STRENGTH : 0.391 Cl. 13.8.2 1 0.000 0.00E+00 3.07E+02 | | Mfz: Mfy: Mrz: Mry: w 1z: w 1y: | | C/S STRENGTH : -3.75E+01 0.00E+00 2.65E+02 5.95E+01 0.60 0.60 | | MEMBER STRENGTH: -3.75E+01 0.00E+00 2.65E+02 5.95E+01 0.60 0.60 | | LTB STRENGTH : -3.75E+01 0.00E+00 8.15E+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: | | 5.00E+00 5.025E+02 0.010 Cl. 14.6 1 0.000 | |--------------------------------------------------------------------------| | BIAXIAL FLEXURE | | RATIO: CRITERIA: LC: LOC: Mfz: Mrz: Mfy: Mry: | | 0.460 Cl. 13.8 1 0.00 -3.75E+01 8.15E+01 0.00E+00 5.95E+01 | |==========================================================================| STAAD SPACE -- PAGE NO. 6