V. AISC 360-05 Tapered Tube Section
Verify the axial compression capacity, flexure capacity, and interaction ratio of a tapered tube section member per both the LRFD and ASD methods of the AISC 360-05 code.
Details
A 5' × 5' portal frame consists of W12x72 columns and a steel tapered member for the beam.
- E = 29,000 ksi
- Fy = 50 ksi
Validation
Design Forces
- Mz = 505.66 in·kips
- Fy = 67.2 kips
- Fx = 10.23 kips
Section Properties
The section at the right end (OD = 10 in.) has the following properties:
- Inner diameter, ID = 9 in.
- Area, Ag = (π/4)(OD2 - ID2) = 14.92 in2
- Form factor = 0.49 + 0.8 (t / OD) = 0.53
- Shear area, Ay = 14.92(0.53) = 7.909 in2
- Moment of inertia,
- Elastic section modulus,
- Radius of gyration,
- Plastic section modulus,
- HSS Torsional constant,
Section Classification
Flange in compression
Per Table B4.1-15a, , therefore flanges are non-slender for compression. The web is similarly non-slender for compression.
Flange in bending (use half the longer flange width as the outstanding width):
Per Table B4.1-15b, , therefore flange is compact for bending.
Compression Capacity
(Eq. E3-4) |
(Eq. E3-2) |
Pn = Ag× Fcr = 14.92 × 48.85 = 729.0 kips | (Eq. E3-1) |
- The ultimate compression capacity (LRFD) = ϕcPn = 0.9 × 729.0 = 656.1 kips
- The allowable compression capacity (ASD) = Pn/Ωc = 729.0 / 1.67 = 436.5 kips
Shear Capacity
The distance from the maximum shear force (ends) to zero shear force (mid-span), Lv = L / 2 = 30 in.
The critical shear stress, Fcr, is the minimum of:
Fcr = 30 ksi
Nominal shear capacity:
The shear capacity:- The ultimate shear capacity (LRFD) = ϕcVn = 0.9 × 223.8 = 201.5 kips
- The allowable shear capacity (ASD) = Vn/Ωc = 223.8 / 1.67 = 134.0 kips
Calculate Bending Capacity
Bending capacity in plastic yielding:
Mp = Fy × Z = 50 × 45.17 = 2,259 in·kips < 1.6Fy × S = 1.6 (50) (33.76) = 2,701 in·kips
- The ultimate bending capacity (LRFD) = ϕbMy = 0.9 × 2,259 = 2,033 in·kips
- The allowable bending capacity (ASD) = My/Ωb = 2,259 / 1.67 = 1,352 in·kips
Torsional Capacity
The critical stress, Fcr, is the larger of:
But shall not exceed:
Fcr = 30 ksi
Nominal torsional capacity:
The torsional capacity:- The ultimate torsional capacity (LRFD) = ϕcTn = 0.9 × 2,126 = 1,914 kips
- The allowable torsional capacity (ASD) = Tn/Ωc = 2,126 / 1.67 = 1,273 kips
Interaction Ratio for Bending and Compression
Check the interaction ratio for Load Case 1:
So use Eq. H1-1b for both methods: .
Results
Result Type | Reference | STAAD.Pro | Difference | Comments | |
---|---|---|---|---|---|
LRFD | Compression capacity (kips) | 656.1 | 656 | negligible | |
Shear capacity (kips) | 201.5 | 201 | negligible | ||
Bending capacity (in·kips) | 2,033 | 2,030 | negligible | ||
Torsional capacity (in·kips) | 1,914 | 1,910 | negligible | ||
Interaction ratio | 0.257 | 0.257 | none | ||
ASD | Compression capacity (kips) | 436.5 | 437 | negligible | |
Shear capacity (kips) | 134.0 | 134 | negligible | ||
Bending capacity (in·kips) | 1,352 | 1,350 | negligible | ||
Torsional capacity (in·kips) | 1,273 | 1,270 | negligible | ||
Interaction ratio | 0.386 | 0.386 | none |
STAAD.Pro Input
The following design parameters are used:
The remaining parameters all use their default values.
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\US\AISC\AISC 360-10 Tapered Tube Section.STD is typically installed with the program.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 13-Apr-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT INCHES KIP
JOINT COORDINATES
1 0 0 0; 2 0 60 0; 3 60 60 0; 4 60 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29000
POISSON 0.3
DENSITY 0.000283
ALPHA 6.5e-06
DAMP 0.03
G 11200
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 TABLE ST W12X72
3 TABLE ST W12X72
MEMBER PROPERTY
2 PRIS ROUND STA 12 END 10 THI 0.5
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
4 FIXED
LOAD 1 LOADTYPE None TITLE LOAD CASE 1
MEMBER LOAD
2 UNI GY -2.25
LOAD 2 LOADTYPE None TITLE LOAD CASE 2
JOINT LOAD
2 FX 50 FZ 25
LOAD 3 LOADTYPE None TITLE LOAD CASE 3
MEMBER LOAD
2 CMOM GX 0.75
PERFORM ANALYSIS
PRINT ANALYSIS RESULTS
PARAMETER 1
CODE AISC UNIFIED 2010
SLF 0.8 MEMB 2
FYLD 50 ALL
FU 60 ALL
METHOD LRFD
STP 2 MEMB 2
TRACK 2 MEMB 2
CHECK CODE MEMB 2
PARAMETER 2
CODE AISC UNIFIED 2010
SLF 0.8 MEMB 2
FYLD 50 ALL
FU 60 ALL
METHOD ASD
STP 2 MEMB 2
TRACK 2 MEMB 2
CHECK CODE MEMB 2
FINISH
STAAD.Pro Output
STAAD.PRO CODE CHECKING - (AISC-360-10-LRFD) v1.4a ******************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted) MEMBER TABLE RESULT/ CRITICAL COND/ RATIO/ LOADING/ FX MY MZ LOCATION ======================================================================= 2 PRISMAT (AISC SECTIONS) PASS Sec. G1 0.334 1 10.23 C 0.00 505.66 60.00 |-----------------------------------------------------------------------| | SLENDERNESS | | Actual Slenderness Ratio : 17.839 L/C : 1 | | Allowable Slenderness Ratio : 200.000 LOC : 60.00 | |-----------------------------------------------------------------------| | STRENGTH CHECKS | | Critical L/C : 1 Ratio : 0.334(PASS) | | Loc : 60.00 Condition : Sec. G1 | |-----------------------------------------------------------------------| | DESIGN FORCES | | Fx: 1.023E+01(C ) Fy: -6.719E+01 Fz: 0.000E+00 | | Mx: 0.000E+00 My: 0.000E+00 Mz: 5.057E+02 | |-----------------------------------------------------------------------| | SECTION PROPERTIES (UNIT: INCH) | | Azz: 7.909E+00 Ayy: 7.909E+00 Cw: 0.000E+00 | | Szz: 3.376E+01 Syy: 3.376E+01 | | Izz: 1.688E+02 Iyy: 1.688E+02 Ix: 3.376E+02 | |-----------------------------------------------------------------------| | MATERIAL PROPERTIES | | Fyld: 50.000 Fu: 60.000 | |-----------------------------------------------------------------------| | Actual Member Length: 60.000 | | Design Parameters | | Kz: 1.00 Ky: 1.00 NSF: 1.00 SLF: 0.80 CSP: 12.00 | |-----------------------------------------------------------------------| | SECTION CLASS UNSTIFFENED / λ λp λr CASE | | STIFFENED | | Compression : Non-Slender 20.00 N/A 63.80 T.B4.1(a)-9 | | Non-Slender 20.00 N/A 63.80 T.B4.1(a)-9 | | Flexure : Compact 20.00 40.60 179.80 T.B4.1(b)-20 | | Compact 20.00 40.60 179.80 T.B4.1(b)-20 | STAAD SPACE -- PAGE NO. 7 STAAD.PRO CODE CHECKING - (AISC-360-10-LRFD) v1.4a ******************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted) |-----------------------------------------------------------------------| | CHECK FOR AXIAL TENSION | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | Yield 0.00E+00 8.13E+02 0.000 Eq. D2-1 1 0.00 | | Rupture 0.00E+00 6.50E+02 0.000 Eq. D2-2 1 0.00 | |-----------------------------------------------------------------------| | CHECK FOR AXIAL COMPRESSION | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | Maj Buck 2.39E+01 6.56E+02 0.036 Eq. E3-1 2 60.00 | | Min Buck 2.39E+01 6.56E+02 0.036 Eq. E3-1 2 60.00 | | Intermediate | | Results Eff Area KL/r Fcr Fe Pn | | Maj Buck 1.49E+01 17.84 4.89E+01 8.99E+02 7.29E+02 | | Min Buck 1.49E+01 17.84 4.89E+01 8.99E+02 7.29E+02 | |-----------------------------------------------------------------------| | CHECK FOR SHEAR | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | Local-Z -5.39E-02 2.01E+02 0.000 Eq. G6-1 2 60.00 | | Local-Y -6.72E+01 2.01E+02 0.334 Eq. G6-1 1 60.00 | | Intermediate | | Results Aw Cv Kv h/tw Vn | | Local-Z 1.49E+01 0.00 0.00 0.00 2.24E+02 | | Local-Y 1.49E+01 0.00 0.00 0.00 2.24E+02 | |-----------------------------------------------------------------------| | CHECK FOR TORSION | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | 2.38E+02 1.91E+03 0.125 Eq. H3-1 2 60.00 | | Intermediate Fcr Tn | | 3.00E+01 2.13E+03 | |-----------------------------------------------------------------------| | CHECK FOR BENDING-YIELDING | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | Major -5.06E+02 2.03E+03 0.249 Eq. F8-1 1 60.00 | | Minor 1.62E+00 2.03E+03 0.001 Eq. F8-1 2 60.00 | | Intermediate Mn My | | Major 2.26E+03 0.00E+00 | | Minor 2.26E+03 0.00E+00 | STAAD SPACE -- PAGE NO. 8 STAAD.PRO CODE CHECKING - (AISC-360-10-LRFD) v1.4a ******************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted) |-----------------------------------------------------------------------| | CHECK FOR FLEXURE TENS/COMP INTERACTION | | RATIO CRITERIA L/C LOC | | Flexure Comp 0.257 Eq. H1-1b 1 60.00 | | Flexure Tens 0.249 Eq. H1-1b 1 60.00 | | Intermediate Mcx Mrx Pc | | Mcy Mry Pr | | Flexure Comp 2.03E+03 -5.06E+02 6.56E+02 | | 2.03E+03 0.00E+00 1.02E+01 | | Flexure Tens 2.03E+03 -5.06E+02 0.00E+00 | | 2.03E+03 0.00E+00 0.00E+00 | |-----------------------------------------------------------------------| 52. PARAMETER 2 53. CODE AISC UNIFIED 2010 54. SLF 0.8 MEMB 2 55. FYLD 50 ALL 56. FU 60 ALL 57. METHOD ASD 58. STP 2 MEMB 2 59. TRACK 2 MEMB 2 60. CHECK CODE MEMB 2 STEEL DESIGN WARNING: For member# 2 the profile has been selected from HSS Rect/Round (non A1085)/Pipe tables of AISC database. Weld type is considered to be Elect-resist-weld. Thickness is already reduced from the table. Further reductions will not be done. STAAD SPACE -- PAGE NO. 9 STAAD.PRO CODE CHECKING - ( AISC-360-10-ASD) v1.4a ******************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted) MEMBER TABLE RESULT/ CRITICAL COND/ RATIO/ LOADING/ FX MY MZ LOCATION ======================================================================= 2 PRISMAT (AISC SECTIONS) PASS Sec. G1 0.501 1 10.23 C 0.00 505.66 60.00 |-----------------------------------------------------------------------| | SLENDERNESS | | Actual Slenderness Ratio : 17.839 L/C : 1 | | Allowable Slenderness Ratio : 200.000 LOC : 60.00 | |-----------------------------------------------------------------------| | STRENGTH CHECKS | | Critical L/C : 1 Ratio : 0.501(PASS) | | Loc : 60.00 Condition : Sec. G1 | |-----------------------------------------------------------------------| | DESIGN FORCES | | Fx: 1.023E+01(C ) Fy: -6.719E+01 Fz: 0.000E+00 | | Mx: 0.000E+00 My: 0.000E+00 Mz: 5.057E+02 | |-----------------------------------------------------------------------| | SECTION PROPERTIES (UNIT: INCH) | | Azz: 7.909E+00 Ayy: 7.909E+00 Cw: 0.000E+00 | | Szz: 3.376E+01 Syy: 3.376E+01 | | Izz: 1.688E+02 Iyy: 1.688E+02 Ix: 3.376E+02 | |-----------------------------------------------------------------------| | MATERIAL PROPERTIES | | Fyld: 50.000 Fu: 60.000 | |-----------------------------------------------------------------------| | Actual Member Length: 60.000 | | Design Parameters | | Kz: 1.00 Ky: 1.00 NSF: 1.00 SLF: 0.80 CSP: 12.00 | |-----------------------------------------------------------------------| | SECTION CLASS UNSTIFFENED / λ λp λr CASE | | STIFFENED | | Compression : Non-Slender 20.00 N/A 63.80 T.B4.1(a)-9 | | Non-Slender 20.00 N/A 63.80 T.B4.1(a)-9 | | Flexure : Compact 20.00 40.60 179.80 T.B4.1(b)-20 | | Compact 20.00 40.60 179.80 T.B4.1(b)-20 | STAAD SPACE -- PAGE NO. 10 STAAD.PRO CODE CHECKING - ( AISC-360-10-ASD) v1.4a ******************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted) |-----------------------------------------------------------------------| | CHECK FOR AXIAL TENSION | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | Yield 0.00E+00 5.41E+02 0.000 Eq. D2-1 1 0.00 | | Rupture 0.00E+00 4.34E+02 0.000 Eq. D2-2 1 0.00 | |-----------------------------------------------------------------------| | CHECK FOR AXIAL COMPRESSION | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | Maj Buck 2.39E+01 4.37E+02 0.055 Eq. E3-1 2 60.00 | | Min Buck 2.39E+01 4.37E+02 0.055 Eq. E3-1 2 60.00 | | Intermediate | | Results Eff Area KL/r Fcr Fe Pn | | Maj Buck 1.49E+01 17.84 4.89E+01 8.99E+02 7.29E+02 | | Min Buck 1.49E+01 17.84 4.89E+01 8.99E+02 7.29E+02 | |-----------------------------------------------------------------------| | CHECK FOR SHEAR | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | Local-Z -5.39E-02 1.34E+02 0.000 Eq. G6-1 2 60.00 | | Local-Y -6.72E+01 1.34E+02 0.501 Eq. G6-1 1 60.00 | | Intermediate | | Results Aw Cv Kv h/tw Vn | | Local-Z 1.49E+01 0.00 0.00 0.00 2.24E+02 | | Local-Y 1.49E+01 0.00 0.00 0.00 2.24E+02 | |-----------------------------------------------------------------------| | CHECK FOR TORSION | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | 2.38E+02 1.27E+03 0.187 Eq. H3-1 2 60.00 | | Intermediate Fcr Tn | | 3.00E+01 2.13E+03 | |-----------------------------------------------------------------------| | CHECK FOR BENDING-YIELDING | | | | FORCE CAPACITY RATIO CRITERIA L/C LOC | | Major -5.06E+02 1.35E+03 0.374 Eq. F8-1 1 60.00 | | Minor 1.62E+00 1.35E+03 0.001 Eq. F8-1 2 60.00 | | Intermediate Mn My | | Major 2.26E+03 0.00E+00 | | Minor 2.26E+03 0.00E+00 | STAAD SPACE -- PAGE NO. 11 STAAD.PRO CODE CHECKING - ( AISC-360-10-ASD) v1.4a ******************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted) |-----------------------------------------------------------------------| | CHECK FOR FLEXURE TENS/COMP INTERACTION | | RATIO CRITERIA L/C LOC | | Flexure Comp 0.386 Eq. H1-1b 1 60.00 | | Flexure Tens 0.374 Eq. H1-1b 1 60.00 | | Intermediate Mcx Mrx Pc | | Mcy Mry Pr | | Flexure Comp 1.35E+03 -5.06E+02 4.37E+02 | | 1.35E+03 0.00E+00 1.02E+01 | | Flexure Tens 1.35E+03 -5.06E+02 0.00E+00 | | 1.35E+03 0.00E+00 0.00E+00 | |-----------------------------------------------------------------------|