V. CSA S16-19 - CLASS4 UPT Cantilever Beam with UDL
Determine the capacity of a cantilever beam using a user-provided section subject to a distributed load per the CSA S16-19 code.
Details
A 7.5 m long cantilever member is subjected to uniformly distributed load of 1 kN. The section is an I shape with 300 mm × 15 mm flanges and a 420 mm × 15 mm web. The compression flange is laterally unbraced along the length.
Material Properties
- Fy = 650 MPa
Validation
Section Properties
- D = 420 + 15 + 15 = 450 mm
- Ag = 2×(300 × 15) + 420 × 15 = 15,300 mm2
- Ix =
- Iy =
- Sx =
- Sy =
- J =
- Cw =
Section Classification
bel = B / 2 = 150 mm
Cf = 0 (no compression force)
Flexure section classification per Table 2 of CSA S16-19:
- Flange classification about major axis:
Class 4
- Flange classification about minor axis:
Class 3
- Web classification about major axis:
Class 1
- Web classification about minor axis:
Class 1
Therefore, the section is Class 4 for flexure.
Bending Capacity
For a Class 4, doubly symmetric section, the capacity is determined by Cl. 13.5.c. In this case, the compression flange is Class 4 but the web is Class 3, so Cl. 13.5.c.iii applies. Therefore, the effective section modulus, Se, is used. The outstanding element length is limited to:
Therefore, beff = 2× bel = 236 mm
Major axis:
Myx = Seff,x×Fy = 1,901,000 × 650 (10)-6 = 1,236 kN·m
Mrx = ϕMyx = 0.9 × 1,236 = 1,112 kN·m
The ratio = 28.1 / 1,112 = 0.025
Minor axis:
Myy = Sy×Fy = 450,800 × 650 (10)-6 = 293.0 kN·m
Mry = ϕMyy = 0.9 × 293.0 = 263.7 kN·m
Check for lateral-torsional buckling per Cl. 13.6.h i).
where= | 0.5 ≤ X ≤ 2.5 for a distributed load | |
= | ||
= | ||
= |
0.67×My = 0.67 (1,236) = 827.8 kN·m < Mu
Mr = 1.15ϕMy[1 - 0.28 My / Mu] = 1.15 × 0.9 × 1,236 [ 1 - 0.28×(1,236 / 1,004) ] = 838.3 kN·m
The ratio = 28.1 / 838.3 = 0.034
Results
Result Type | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Myx (kN·m) | 1,236 | 1,230 | negligible | |
Mrx (kN·m) | 1,112 | 1,110 | negligible | |
Ratio of Mfx/Myx | 0.025 | 0.025 | none | |
Myy (kN·m) | 293.0 | 290.0 | negligible | |
Mry (kN·m) | 263.7 | 261.1 | negligible | |
Mu (kN·m) | 1,004 | 1,006 | negligible | |
Mr (kN·m) | 838.3 | 837.9 | negligible | |
ω2 | 15.77 | 15.77 | none | |
Critical ratio | 0.034 | 0.034 | none |
STAAD.Pro Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\Canada\CSA S16-19 - CLASS4 UPT Cantilever Beam with UDL.STD is typically installed with the program.
The following design parameters are used:- The cantilever is braced along the tension flange only specified using CAN 2.
- The load acts on the tension side of the bending member, thus LHT 1.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 3-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;
START USER TABLE
TABLE 1
UNIT METER KN
WIDE FLANGE
UPTI
0.0153 0.45 0.015 0.3 0.015 0.000518535 6.76181e-05 1.1475e-06 0.00675 -
0.006 0.3 0.015
END
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 UPTABLE 1 UPTI
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE None TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -1
PERFORM ANALYSIS
PARAMETER 1
CODE CANADIAN
CAN 2 ALL
LHT 1 ALL
FYLD 650000 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 UPTI (UPT) | | STATUS: PASS CRITICAL RATIO: 0.034 LOADCASE 1 | | LOCATION: 0.00 CRITICAL CONDITION: Cl. 13.8.4 | |--------------------------------------------------------------------------| | STRENGTH CHECK SUMMARY | | CRITICAL RATIO: 0.034(PASS) LOAD CASE: 1 | | LOCATION : 0.00 CONDITION: Cl. 13.8.4 | |--------------------------------------------------------------------------| | DESIGN FORCES LC: 1 LOC: 0.00 | | FX: 0.00(T) FY: 7.50 FZ: 0.00 | | MX: 0.000E+00 MY: 0.000E+00 MZ: 2.813E+01 | |--------------------------------------------------------------------------| | SECTION PROPERTIES (LOC: 0.00) UNIT: CM | | AXX: 153.000 AZZ: 60.000 AYY: 67.500 | | IXX: 114.750 IZZ: 51853.500 IYY: 6761.811 | | PZZ: 2619.000 PYY: 698.625 | | SZZ: 2304.600 SYY: 450.787 CW : 3198759.000 | |--------------------------------------------------------------------------| | MATERIAL PROPERTIES UNIT: N MM | | FYLD: 650.000 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 4 Class 3 | | FLANGE : Class 4 Class 3 | | WEB (CRITICAL) : Class 1 | |==========================================================================| | SLENDERNESS | | ACTUAL SLENDERNESS RATIO : 112.817 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 8.95E+03 0.000 Cl. 13.2 1 0.000 | | RUPTURE | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | -0.00E+00 5.16E+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 5.81E+03 0.000 Cl. 13.3.4(a) 1 0.000 | | MINOR: 0.00E+00 1.67E+03 0.000 Cl. 13.3.4(a) 1 0.000 | | INTERMEDIATE RESULTS: | | Ag:(CM) Fe:(N MM) λ: n: KL/r: | | MAJOR: 129.731 1219.039 0.730 1.34 40.740 | | MINOR: 129.731 158.965 2.022 1.34 112.817 | |--------------------------------------------------------------------------| | FLEXURAL TORSIONAL BUCKLING | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 0.00E+00 1.67E+03 0.000 Cl. 13.3.4 1 0.000 | | INTERMEDIATE RESULTS: | | Ag:(CM) 129.731 Fe:(N MM) 158.965 λ: 2.022 n: 1.34 | |--------------------------------------------------------------------------| | CHECKS FOR SHEAR | |--------------------------------------------------------------------------| | SHEAR ALONG Y AXIS | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 7.50E+00 2.61E+03 0.003 Cl. 13.4.1.1 1 0.000 | | INTERMEDIATE RESULTS: | | Aw:(CM) Kv: Ka: Fcri:(N MM) Fcre:(N MM) Fs:(N MM) | | 67.500 5.353 0.056 610.909 1228.900 429.000 | |--------------------------------------------------------------------------| | SHEAR ALONG Z AXIS | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | -0.00E+00 2.11E+03 0.000 AISC G2-1 1 0.000 | | INTERMEDIATE RESULTS: | | Aw:(CM) | | 60.000 | |--------------------------------------------------------------------------| 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: | | -2.81E+01 1.110E+03 0.025 Cl. 13.5(c) 1 0.000 | | INTERMEDIATE RESULTS: Mp: 0.00E+00 Se(CM): 1.90E+03 My: 1.23E+03| |--------------------------------------------------------------------------| | YIELDING ALONG MINOR AXIS | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 0.00E+00 2.611E+02 0.000 Cl. 13.5(c) 1 0.000 | | INTERMEDIATE RESULTS: Mp: 0.00E+00 Se(CM): 4.46E+02 My: 2.90E+02| |--------------------------------------------------------------------------| | LATERAL TORSIONAL BUCKLING (MAJOR AXIS) | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | -2.81E+01 8.379E+02 0.034 Cl. 13.6(b)(ii) 1 0.000 | | INTERMEDIATE RESULTS: | | Iyc(CM): w 2: w 3: b x(CM): Mu: rt: Myr: Lu: Lyr: | | 0.00E+00 15.77 0.00 0.00 1.006E+03 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 : 0.025 Cl. 13.9.1 1 0.00 -0.00E+00 5.16E+03 | | MEMBER STRENGTH: 0.034 Cl. 13.9.3 1 0.00 | | Mfz: Mfy: Mrz: Mry: | | C/S STRENGTH : -2.81E+01 0.00E+00 1.11E+03 2.61E+02 | |--------------------------------------------------------------------------| | FLEXURE AND AXIAL COMPRESSION | | RATIO: CRITERIA: LC: LOC: Cf: Cr: | | C/S STRENGTH : 0.025 Cl. 13.8.4 1 0.000 0.00E+00 7.59E+03 | | MEMBER STRENGTH: 0.025 Cl. 13.8.4 1 0.000 0.00E+00 1.67E+03 | | LTB STRENGTH : 0.034 Cl. 13.8.4 1 0.000 0.00E+00 1.67E+03 | | Mfz: Mfy: Mrz: Mry: w 1z: w 1y: | | C/S STRENGTH : -2.81E+01 0.00E+00 1.11E+03 2.61E+02 0.60 0.60 | | MEMBER STRENGTH: -2.81E+01 0.00E+00 1.11E+03 2.61E+02 0.60 0.60 | | LTB STRENGTH : -2.81E+01 0.00E+00 8.38E+02 2.61E+02 0.60 0.60 | | b : Cez: Cey: U1z: U1y: | | C/S STRENGTH : N/A 1.87E+04 2.43E+03 1.00 1.00 | | MEMBER STRENGTH: N/A 1.87E+04 2.43E+03 1.00 1.00 | | LTB STRENGTH : N/A 1.87E+04 2.43E+03 1.00 1.00 | |--------------------------------------------------------------------------| | SHEAR CAPACITY MODIFICATION ( SHEAR AND FLEX ) | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 7.50E+00 2.606E+03 0.003 Cl. 14.6 1 0.000 | |--------------------------------------------------------------------------| | BIAXIAL FLEXURE | | RATIO: CRITERIA: LC: LOC: Mfz: Mrz: Mfy: Mry: | | 0.034 Cl. 13.8 1 0.00 -2.81E+01 8.38E+02 0.00E+00 2.61E+02 | |==========================================================================| STAAD SPACE -- PAGE NO. 6