V. CSA S16-19 - Axial Comp and Bending Interaction
Check a wide flange column member subject to both axial compression and bending per the CSA S16-19 code.
References
Canadian Institute of Steel Construction. Handbook of Steel Construction. 9th Edition. 2008. Willowdale, Ont., pp.6-52 - 6-53
Details
The column is simply supported and has a span of 3.6 m. The member is subjected to an axial load of 900 kN along with moments in both strong and weak directions of 50 kN·m at the roller end. The section used is a Canadian W250x73. The steel has a yield strength of 350 MPa (50 ksi) and ultimate strength of 450 MPa (65 ksi).
Section Properties
From the Handbook of Steel Construction:
- D = 253 mm
- B = 254 mm
- tw = 8.6 mm
- tf = 14.2 mm
- A = 9,280 mm2
- Ix = 113×(10)6 mm4
- Iy = 38.8×(10)6 mm4
- Sx = 891×(10)3 mm3
- Sy = 306×(10)3 mm3
- Zx = 985×(10)3 mm3
- Zy = 463×(10)3 mm3
- rx = 110 mm
- ry = 64.6 mm
- Cw = 553×(10)9 mm6
- J = 575×(10)3 mm4
Material Properties
- E = 205,000 MPa
- G = 76,920 MPa
Calculations
Section Classification
bel = B/2 = 127 mm
h = D - 2×tf = 224.6 mm
Compression section classification as per Table 1:
Flange classification:
Web classification:
Section is Class 1 for compression.
Flexural section classification as per Table 2
Flange classification (major and minor axes):
Web classification (major and minor axes):
Section is Class 2 for flexure.
Axial Compression Capacity
Check for axial compression per Cl. 13.3.1.1:
(13.3.1.1) |
= | ||
= |
Check for flexural-torsional buckling per Cl. 13.3.1.2:
(13.3.1.2) |
= |
For flexural torsional buckling, Fe is the least of Fex, Fey, and Fez. Therefore, Fe = 651 MPa.
λ = 0.733
Cr = Cry = 2,233 kN
The critical ratio = Cf / Cr = 900 / 2,233 = 0.403
Bending Capacity
Check the bending capacity for a Class 2 section as per Cl. 13.5.a:
About major axis:
Mpx = Zx×Fy = 985 × (10)3 × 350 × 10-6 = 344.8 kN·m
Mrx = ϕMpx = 0.9 (344.8) = 310.3 kN·m
Ratio = Mfx / Mrx = 50 / 310.3 = 0.161
About minor axis:
Mpy = Zy×Fy = 463 × (10)3 × 350 × 10-6 = 162.1 kN·m
Mry = ϕMpy = 0.9 (162.1) = 145.8 kN·m
Ratio = Mfy / Mry = 50 / 145.8 = 0.343
Combined Axial Compression & Bending
Check cross-section strength for a Class 2 section per Cl. 13.8.2-a:
(13.8.2) |
= | ||
= | 0.6 + 0.4 (0.733) = 0.893; thus limit β = 0.85 |
About both major and minor axes, the moment at the "start" joint, MA = 0 and the moment at the "end" joint, MB = 50 kN·m. Thus, K = MA / MB = 0.
ω1 = 0.6 - 0.4×K = 0.6 | (13.8.6) |
(13.8.5 & 13.8.2-a-iii) |
= |
Thus, U1 = 1.0 about both major and minor axes.
Check overall member strength for a Class 2 section per Cl. 13.8.2-b:
- Cr used is taken from the axial compression calculations (K = 1).
- Use U1z and U1y = 1.0 for an unbraced frame.
Check moment-only capacity for biaxial bending per Cl. 13.8.2-d:
Comparison
Criteria | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Elastic buckling stress about the major axis, Fex (MPa) | 1,890 | 1,900.987 | negligible | |
Elastic buckling stress about the minor axis, Fey (MPa) | 651 | 652.728 | negligible | |
Non-dimensional slenderness parameter about major axis, λx | 0.430 | 0.429 | negligible | |
Non-dimensional slenderness parameter about minor axis, λy | 0.733 | 0.732 | negligible | |
Factored axial compressive resistance about major axis, Crx (kN) | 2,715 | 2,720 | negligible | |
Factored axial compressive resistance about major axis, Cry (kN) | 2,233 | 2,230 | negligible | |
Flexural torsional buckling stress, Fez (MPa) | 651 | 652.728 | negligible | |
Factored axial compressive resistance in flexural torsional buckling, Cr (kN) | 2,233 | 2,230 | negligible | |
Factored moment resistance about the major axis, Mrx (kN·m) | 310.3 | 310.3 | none | |
Factored moment resistance about the minor axis, Mry (kN·m) | 145.8 | 145.8 | none | |
Factored axial compressive stress used for cross-section strength, Cr (kN) | 2,923 | 2,920 | negligible | |
Elastic buckling strength about major axis, Cez (kN) | 17,640 | 17,600 | negligible | |
Elastic buckling strength about minor axis, Cey (kN) | 6,057 | 6,060 | negligible | |
β | 0.85 | 0.85 | none | |
Cross-sectional strength ratio | 0.736 | 0.736 | none | |
Factored axial compressive stress used for overall member strength, Cr (kN) | 2,233 | 2,230 | negligible | |
Overall member strength ratio | 0.831 | 0.831 | none | |
Biaxial flexure ratio | 0.504 | 0.504 | none |
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\Canada\CSA S16-19 - Axial Comp and Bending Interaction.STD is typically installed with the program.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 17-Jan-22
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: 0.831 LOADCASE 1 | | LOCATION: 3.60 CRITICAL CONDITION: Cl. 13.8.2 | |--------------------------------------------------------------------------| | STRENGTH CHECK SUMMARY | | CRITICAL RATIO: 0.831(PASS) LOAD CASE: 1 | | LOCATION : 3.60 CONDITION: Cl. 13.8.2 | |--------------------------------------------------------------------------| | DESIGN FORCES LC: 1 LOC: 3.60 | | FX: 900.00(C) 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 2 | |==========================================================================| | SLENDERNESS | | ACTUAL SLENDERNESS RATIO : 55.675 LC : 1 | | ALLOWABLE SLENDERNESS RATIO: 200.000 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 2.92E+03 0.000 Cl. 13.2 1 0.000 | | RUPTURE | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | 0.00E+00 3.13E+03 0.000 Cl. 13.2 1 0.000 | |--------------------------------------------------------------------------| | CHECKS FOR AXIAL COMPRESSION | |--------------------------------------------------------------------------| | FLEX BUCKLING | | FORCE: CAPACITY: RATIO: CRITERIA: LC: LOC: | | MAJOR: 9.00E+02 2.72E+03 0.331 Cl. 13.3 1 0.000 | | MINOR: 9.00E+02 2.23E+03 0.403 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: | | 9.00E+02 2.23E+03 0.403 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 : 0.343 Cl. 13.9.2 1 3.60 0.00E+00 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 : 0.736 Cl. 13.8.2 1 3.600 9.00E+02 2.92E+03 | | MEMBER STRENGTH: 0.831 Cl. 13.8.2 1 0.000 9.00E+02 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 | |==========================================================================| STAAD SPACE -- PAGE NO. 8