V. CSA S16-19 - Axial Tension and Bending Interaction
Determine the capacity of a member subject to axial tension and biaxial moment per the CSA S16-19 code.
Problem
A 3.6 m long column is subjected to a 900 kN tensile load along with concentrated moments of 50 kN·m about both major and minor axes at one end. The section is a Canadian W250x73. The steel is grade 350W.Section Properties
- D = 253 mm
- B = 254 mm
- tw = 8.6 mm
- tf = 14.2 mm
- Ag = 9,280 mm2
- Zx = 985,000 mm3
- Zy = 463,000 mm3
Material Properties
- Fy = 350 MPa
- Fu = 450 MPa
Calculations
Section Classification
Compression section classification per Table 1:
bel = B / 2 = 127 mm
h = D - 2×tf = 235.8 mm
- Flange classification:
- Web classification:
Therefore, the section is Class 1 for compression.
Flexure section classification per Table 2:
- Flange classification about major axis:
Class 2
- Flange classification about minor axis:
Class 2
- Web classification about major axis:
Class 1
- Web classification about minor axis:
Class 1
Therefore, the section is Class 2 for flexure.
Axial Tension
Assume the gross area for both yielding and rupture limit states. Therefore, Ane = Ag = 9,280 mm2.
So, the tensile capacity = 2,923 kN.
The critical stress ratio 900 / 2,923 = 0.308.
Bending Capacity
For a Class 2, doubly symmetric section, the capacity is determined by Cl. 13.5a:
Major axis:
Mpx = Zx×Fy = 985,000 × 350 (10)-6 = 344.8 kN·m
Mrx = ϕMpx = 0.9 × 344.8 = 310.3 kN·m
The ratio = 50 / 310.3 = 0.161
Minor axis:
Mpy = Zy×Fy = 463,000 × 350 (10)-6 = 162.1 kN·m
Mrx = ϕMpx = 0.9 × 162.1 = 145.8 kN·m
The ratio = 50 / 145.8 = 0.343
Combined Axial Tension and Bending
Check for axial tension and bending per Cl. 13.9.2:
Also, the following must be satisfied:
Comparison
Criteria | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Nominal flexure capacity about major axis, Mpx (kN·m) | 344.8 | 345 | negligible | |
Ultimate flexure capacity about major axis, Mrx (kN·m) | 310.3 | 310.3 | none | |
Nominal flexure capacity about minor axis, Mpy (kN·m) | 162.1 | 162 | negligible | |
Ultimate flexure capacity about minor axis, Mrx (kN·m) | 145.8 | 145.8 | none | |
Tension capacity, Tr (kN) | 2,932 | 2,930 | negligible | |
Critical Ratio | 0.651 | 0.651 | none |
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\Canada\CSA S16-19 - Axial Tension and Bending Interaction.STD is typically installed with the program.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 08-Oct-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 3.6 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 W250X73
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 ENFORCED BUT FY MX MZ
LOAD 1 LOADTYPE None TITLE LOAD CASE 1
JOINT LOAD
2 FY 900 MX 50 MZ 50
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT SUPPORT REACTION ALL
PARAMETER 1
CODE CANADIAN
LAT 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 W250X73 (CANADIAN SECTIONS)| | STATUS: PASS CRITICAL RATIO: 0.651 LOADCASE 1 | | LOCATION: 3.60 CRITICAL CONDITION: Cl. 13.9.2 | |--------------------------------------------------------------------------| | STRENGTH CHECK SUMMARY | | CRITICAL RATIO: 0.651(PASS) LOAD CASE: 1 | | LOCATION : 3.60 CONDITION: Cl. 13.9.2 | |--------------------------------------------------------------------------| | DESIGN FORCES LC: 1 LOC: 3.60 | | FX: 900.00(T) FY: 13.89 FZ: 13.89 | | MX: 0.000E+00 MY: 5.000E+01 MZ: -5.000E+01 | |--------------------------------------------------------------------------| | SECTION PROPERTIES (LOC: 3.60) UNIT: CM | | AXX: 92.800 AZZ: 72.136 AYY: 21.758 | | IXX: 57.500 IZZ: 11300.001 IYY: 3880.000 | | PZZ: 985.000 PYY: 463.000 | | SZZ: 893.281 SYY: 305.512 CW : 552900.312 | |--------------------------------------------------------------------------| | MATERIAL PROPERTIES UNIT: N MM | | FYLD: 350.000 FU: 450.000 E: 204999.984 G: 76920.000 | |--------------------------------------------------------------------------| | DESIGN PARAMETERS | | ACTUAL MEMBER LENGTH: 3.600 LZ: 3.600 LY: 3.600 | | KZ: 1.000 KY: 1.000 NSF: 1.000 SLF: 1.000 | |--------------------------------------------------------------------------| | SECTION CLASS | | COMPRESSION : Class 1 | | FLEXURE MAJOR MINOR | | SECTION : Class 2 Class 2 | | FLANGE : Class 2 Class 2 | | WEB (CRITICAL) : Class 1 | |==========================================================================| | SLENDERNESS | | ACTUAL SLENDERNESS RATIO : 55.675 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: | | 9.00E+02 2.92E+03 0.308 Cl. 13.2 1 0.000 | | RUPTURE | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 9.00E+02 3.13E+03 0.287 Cl. 13.2 1 0.000 | |--------------------------------------------------------------------------| | CHECKS FOR AXIAL COMPRESSION | |--------------------------------------------------------------------------| | FLEX BUCKLING | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | MAJOR: 0.00E+00 2.72E+03 0.000 Cl. 13.3 1 0.000 | | MINOR: 0.00E+00 2.23E+03 0.000 Cl. 13.3 1 0.000 | | INTERMEDIATE RESULTS: | | Ag:(CM) Fe:(N MM) λ: n: KL/r: | | MAJOR: 92.800 1900.987 0.429 1.34 32.624 | | MINOR: 92.800 652.728 0.732 1.34 55.675 | |--------------------------------------------------------------------------| | FLEXURAL TORSIONAL BUCKLING | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 0.00E+00 2.23E+03 0.000 Cl. 13.3 1 0.000 | | INTERMEDIATE RESULTS: | | Ag:(CM) 92.800 Fe:(N MM) 652.728 λ: 0.732 n: 1.34 | |--------------------------------------------------------------------------| | CHECKS FOR SHEAR | |--------------------------------------------------------------------------| | SHEAR ALONG Y AXIS | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 1.39E+01 4.52E+02 0.031 Cl. 13.4.1.1 1 0.000 | | INTERMEDIATE RESULTS: | | Aw:(CM) Kv: Ka: Fcri:(N MM) Fcre:(N MM) Fs:(N MM) | | 21.758 5.360 0.070 480.943 1414.474 231.000 | |--------------------------------------------------------------------------| | SHEAR ALONG Z AXIS | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | -1.39E+01 1.36E+03 0.010 AISC G2-1 1 0.000 | | INTERMEDIATE RESULTS: | | Aw:(CM) | | 72.136 | |--------------------------------------------------------------------------| 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: | | 5.00E+01 3.103E+02 0.161 Cl. 13.5(a) 1 3.600 | | INTERMEDIATE RESULTS: Mp: 3.45E+02 Se(CM): 0.00E+00 My: 0.00E+00| |--------------------------------------------------------------------------| | YIELDING ALONG MINOR AXIS | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 5.00E+01 1.458E+02 0.343 Cl. 13.5(a) 1 3.600 | | INTERMEDIATE RESULTS: Mp: 1.62E+02 Se(CM): 0.00E+00 My: 0.00E+00| |--------------------------------------------------------------------------| | CHECKS FOR INTERACTION | |--------------------------------------------------------------------------| | FLEXURE AND AXIAL TENSION | | RATIO: CRITERIA: LC: LOC: Tf: Tr: | | C/S STRENGTH : 0.651 Cl. 13.9.2 1 3.60 9.00E+02 2.92E+03 | | Mfz: Mfy: Mrz: Mry: | | C/S STRENGTH : 5.00E+01 5.00E+01 3.10E+02 1.46E+02 | |--------------------------------------------------------------------------| | FLEXURE AND AXIAL COMPRESSION | | RATIO: CRITERIA: LC: LOC: Cf: Cr: | | C/S STRENGTH : 0.428 Cl. 13.8.2 1 3.600 0.00E+00 2.92E+03 | | MEMBER STRENGTH: 0.428 Cl. 13.8.2 1 0.000 0.00E+00 2.23E+03 | | Mfz: Mfy: Mrz: Mry: w 1z: w 1y: | | C/S STRENGTH : 5.00E+01 5.00E+01 3.10E+02 1.46E+02 0.60 0.60 | | MEMBER STRENGTH: 5.00E+01 5.00E+01 3.10E+02 1.46E+02 0.60 0.60 | | b : Cez: Cey: U1z: U1y: | | C/S STRENGTH : 0.85 1.76E+04 6.06E+03 1.00 1.00 | | MEMBER STRENGTH: 0.85 1.76E+04 6.06E+03 1.00 1.00 | |--------------------------------------------------------------------------| | SHEAR CAPACITY MODIFICATION ( SHEAR AND FLEX ) | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 1.39E+01 4.523E+02 0.031 Cl. 14.6 1 0.000 | |--------------------------------------------------------------------------| | BIAXIAL FLEXURE | | RATIO: CRITERIA: LC: LOC: Mfz: Mrz: Mfy: Mry: | | 0.504 Cl. 13.8 1 3.60 5.00E+01 3.10E+02 5.00E+01 1.46E+02 |