V. AISC 360-10 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 360-10 code.

Details

A 5' × 5' portal frame consists of W12x72 columns and a steel tapered member for the beam.

The following load cases are evaluated:

A uniformly distributed load of -2.25 k/in along the beam in the global Y direction.
Lateral loads at the top of the left column of 50 kips in the global X direction and 25 kips in the global Z direction (out of plane).
A concreted torque of 0.75 in·kips at mid-span of the beam.

Material Properties

E = 29,000 ksi
Fy = 50 ksi

Validation

Design Forces

By inspection, the right end of the beam (shallowest overall depth) governs under the symmetric loads present. The following design loads are used from the STAAD.Pro analysis are then used:

M_z = 505.66 in·kips
F_y = 67.2 kips
F_x = 10.23 kips

Section Properties

The section at the right end (OD = 10 in.) has the following properties:

Inner diameter, ID = 9 in.
Area, A_g = (π/4)(OD² - ID²) = 14.92 in²
Form factor = 0.49 + 0.8 (t / OD) = 0.53
Shear area, A_y = 14.92(0.53) = 7.909 in²
Moment of inertia,
$I = \frac{π}{64} ({OD}^{4} - {ID}^{4}) = 168.8 {in}^{4}$
Elastic section modulus, $S = \frac{π}{32} (\frac{{OD}^{4} - {ID}^{4}}{OD}) = 33.76 {in}^{3}$
Radius of gyration, $r = \sqrt{\frac{I}{A_{g}}} = \sqrt{\frac{168.8}{14.92}} = 3.363 in$
Plastic section modulus, $Z = \frac{{OD}^{3} - {ID}^{3}}{6} = 45.17 {in}^{3}$
HSS Torsional constant, $C = \frac{π}{2} {(OD - t)}^{2} t = 70.88 {in}^{3}$

Section Classification

Flange in compression

λ = \frac{OD}{t} = \frac{10}{0.5} = 20

Per Table B4.1a, Case 9, $λ < λ_{r} = 0.11 \frac{E}{F_{y}} = 63.8$ , 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):

λ = \frac{OD}{t} = \frac{10}{0.5} = 20

Per Table B4.1b, Case20, $λ < λ_{r} = 0.07 \frac{E}{F_{y}} = 40.6$ , therefore flange is compact for bending.

Compression Capacity

\frac{k L}{r} = \frac{1.0 (60)}{3.363} = 17.84 < 4.71 \sqrt{\frac{E}{F_{y}}} = 4.71 \sqrt{\frac{29, 000}{50}} = 113.4

F_{e} = \frac{π^{2} E}{{(\frac{k L}{r})}^{2}} = \frac{π^{2} \times 29,000}{{(17.84)}^{2}} = 899.4 ksi

(Eq. E3-4)

\frac{F_{y}}{F_{e}} = \frac{50}{899.4} = 0.056 < 2.25

F_{crx} = {0.658}^{\frac{F_{y}}{F_{e}}} F_{y} = {0.658}^{(0.056)} 50 = 48.85 ksi

(Eq. E3-2)

P_n = A_g× F_cr = 14.92 × 48.85 = 729.0 kips

(Eq. E3-1)

The axial capacity:

The ultimate compression capacity (LRFD) = ϕ_cP_n = 0.9 × 729.0 = 656.1 kips
The allowable compression capacity (ASD) = P_n/Ω_c = 729.0 / 1.67 = 436.5 kips

Shear Capacity

The distance from the maximum shear force (ends) to zero shear force (mid-span), L_v = L / 2 = 30 in.

The critical shear stress, F_cr, is the minimum of:

F_{cr1} = \frac{1.60 E}{\sqrt{\frac{L_{v}}{OD}} {(\frac{OD}{t})}^{\frac{5}{4}}} = 633.4 ksi

F_{cr2} = \frac{0.78 E}{{(\frac{OD}{t})}^{\frac{3}{2}}} = 252.9 ksi

F_{cr3} = 0.6 F_{y} = 30 ksi

F_cr = 30 ksi

Nominal shear capacity:

V_{n} = F_{cr} \frac{A_{g}}{2} = 223.8 kips

The shear capacity:

The ultimate shear capacity (LRFD) = ϕ_cV_n = 0.9 × 223.8 = 201.5 kips
The allowable shear capacity (ASD) = V_n/Ω_c = 223.8 / 1.67 = 134.0 kips

Calculate Bending Capacity

Bending capacity in plastic yielding:

M_p = F_y × Z = 50 × 45.17 = 2,259 in·kips < 1.6F_y × S = 1.6 (50) (33.76) = 2,701 in·kips

The bending capacity for flexural yielding:

The ultimate bending capacity (LRFD) = ϕ_bM_y = 0.9 × 2,259 = 2,033 in·kips
The allowable bending capacity (ASD) = M_y/Ω_b = 2,259 / 1.67 = 1,352 in·kips

Torsional Capacity

The critical stress, F_cr, is the larger of:

F_{cr1} = \frac{1.23 E}{\sqrt{\frac{L}{OD}} {(\frac{OD}{t})}^{\frac{5}{4}}} = 344.3 ksi

F_{cr2} = \frac{1.23 E}{{(\frac{OD}{t})}^{\frac{3}{2}}} = 194.5 ksi

But shall not exceed:

F_{cr3} = 0.6 F_{y} = 30 ksi

F_cr = 30 ksi

Nominal torsional capacity:

T_{n} = F_{cr} C = 2,126 in·kips

The torsional capacity:

The ultimate torsional capacity (LRFD) = ϕ_cT_n = 0.9 × 2,126 = 1,914 kips
The allowable torsional capacity (ASD) = T_n/Ω_c = 2,126 / 1.67 = 1,273 kips

Interaction Ratio for Bending and Compression

Check the interaction ratio for Load Case 1:

\frac{P_{r}}{P_{c}} = \frac{10.23}{656.1} = 0.016 < 0.2 (LRFD)

\frac{P_{r}}{P_{c}} = \frac{10.23}{436.5} = 0.023 < 0.2 (ASD)

So use Eq. H1-1b for both methods: $\frac{P_{r}}{2 P_{c}} + (\frac{M_{r x}}{M_{c x}} + \frac{M_{r y}}{M_{c y}}) \leq 1.0$ .

\frac{0.016}{2} + (\frac{505.7}{2,033} + \frac{0}{2,033}) = 0.257 (LRFD)

\frac{0.023}{2} + (\frac{505.7}{1,352} + \frac{0}{1,352}) = 0.386 (ASD)

Results

Table 1.
Result Type		Reference	STAAD.Pro	Difference
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 welded tapered member is specified using STP 2

The remaining parameters all use their default values.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\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                   |
|-----------------------------------------------------------------------|