V. AISC 360-10 UPT Pipe Section

Verify the axial compression capacity, flexure capacity, and interaction ratio of a tapered I 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 UPT pipe member for the beam. The pipe outer diameter, D, is 10" and the thickness, t, is 0.25".

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 concentrated torque of 0.75 in·kips at mid-span of the beam.

Material Properties

E = 29,000 ksi
Fy = 50 ksi

Validation

Design Forces

The following design loads are used from the STAAD.Pro analysis are then used:

M_z = 599.6 in·kips, F_y = 67.5 kips

Section Properties

Inner diameter, d = 10 - 2×0.25 = 9.5 in.
Area, $A_{g} = \frac{π}{4} (D^{2} - d^{2}) = 7.66 {in}^{2}$
Shear area, A_y = A_x= A_g / 2 = 3.83 in²
Moment of inertia, $I_{x} = I_{y} = \frac{π}{64} (D^{4} - d^{4}) = 91.05 {in}^{4}$
Elastic section modulus, $S_{x x} = \frac{I_{x x}}{D / 2} = \frac{91.05}{10 / 2} = 18.21 {in}^{3}$
Radius of gyration, $r_{x} = \sqrt{\frac{I_{xx}}{A_{g}}} = \sqrt{\frac{91.05}{7.66}} = 3.448 in$
Plastic section modulus, $Z_{x} = \frac{1}{6} (D^{3} - d^{3}) = 23.77 {in}^{3}$
HSS Torsional constant, $C = \frac{π}{2} {(D - t)}^{2} t = 37.33 {in}^{3}$

Section Classification

Flange in compression

λ = \frac{D}{t} = \frac{10}{0.25} = 40

Per Table 4.1a Case 9, $λ < λ_{r} = 0.11 \frac{E}{F_{y}} = 0.11 \frac{29,000}{50} = 63.8$ , therefore flanges are non-slender for compression.

Web in compression

Per Table 4.1a, Case 9, $λ < λ_{r} = 0.11 \frac{E}{F_{y}} = 0.11 \frac{29,000}{50} = 63.8$ , therefore the web is non-slender for compression.

Flexure classification:

Per Table 4.1b, Case 20, $λ < λ_{p} = 0.07 \frac{E}{F_{y}} = 0.11 \frac{29,000}{50} = 40.6$ , therefore flange is compact for bending.

Compression Capacity

Effective slenderness ratio:

\frac{k_{x} L_{x}}{r_{x}} = \frac{1.0 (60)}{3.448} = 17.40 < 4.71 \sqrt{\frac{E}{F_{y}}} = 4.71 \sqrt{\frac{29, 000}{50}} = 113.4

Elastic buckling stress:

F_{e} = \frac{π^{2} E}{{(\frac{k_{x} L_{x}}{r_{x}})}^{2}} = \frac{π^{2} \times 29, 000}{{(17.40)}^{2}} = 945 ksi

(Eq. E3-4)

\frac{F_{y}}{F_{e}} = \frac{50}{945} = 0.053 < 2.25

Critical buckling stress:

F_{cr} = {0.658}^{\frac{F_{y}}{F_{e}}} F_{y} = {0.658}^{(0.053)} 50 = 48.91 ksi

(Eq. E3-2)

Nominal buckling strength:

P_n = A_g× F_cr = 7.66 × 48.91 = 374.5 kips

(Eq. E3-1)

The axial capacity:

The ultimate compression capacity (LRFD) = ϕ_cP_n = 0.9 × 374.5 = 337.0 kips
The allowable compression capacity (ASD) = P_n/Ω_c = 374.5 / 1.67 = 224.3 kips

Calculate Shear Capacity

Critical stress is the minimum of F_cr1, F_cr2, and F_cr3:

F_{cr1} = \frac{1.23 E}{\sqrt{\frac{L}{D}} {(\frac{D}{t})}^{(\frac{5}{4})}} = 144.8 ksi

F_{cr2} = \frac{0.78 E}{{(\frac{D}{t})}^{(\frac{3}{2})}} = 89.4 ksi

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

So, F_cr = 30 ksi.

The nominal shear strength:

V_n = F_cr × A_x = 30 × 3.83 = 114.9 kips

The axial capacity:

The ultimate shear capacity (LRFD) = ϕ_vV_n = 0.9 × 114.9 = 103.4 kips
The allowable shear capacity (ASD) = V_n/Ω_v = 114.9 / 1.67 = 68.78 kips

Calculate Bending Capacity

Bending capacity in plastic yielding (compact section):

M_y = F_y × Z_x = 50 × 23.77 = 1,189 in·kips

(Eq F8-1)

The bending capacity for flexural yielding about the Y axis:

The ultimate bending capacity (LRFD) = ϕ_bM_y = 0.9 × 1,189 = 1,070 in·kips
The allowable bending capacity (ASD) = M_y/Ω_b = 1,189 / 1.67 = 711.7 in·kips

Interaction Ratio for Bending and Compression

Check the interaction ratio for Load Case 1:

\frac{P_{r}}{P_{c}} = \frac{11.75}{337.0} = 0.034 < 0.2 (LRFD)

\frac{P_{r}}{P_{c}} = \frac{11.75}{224.3} = 0.052 < 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{0.034}{2} + (\frac{599.6}{1,070} + \frac{0}{1,070}) = 0.578 (LRFD)

\frac{0.052}{2} + (\frac{599.6}{711.7} + \frac{0}{711.7}) = 0.869 (ASD)

Calculate Torsion Capacity

Critical stress is the minimum of F_cr1, F_cr2, and F_cr3. From shear calculations, F_cr = 30 ksi.

The nominal shear strength:

T_n = F_cr × C = 30 × 37.33 = 1,120 in·kips

The axial capacity:

The ultimate torsional capacity (LRFD) = ϕ_TT_n = 0.9 × 1,120 = 1,008 in·kips
The allowable torsional capacity (ASD) = T_n/Ω_T = 114.9 / 1.67 = 670.6 in·kips

Results

Table 1.
Result Type		Reference	STAAD.Pro	Difference
LRFD	Compression capacity (kips)	337	337	none
	Bending capacity (in·kips)	1,070	1,070	none
	Shear capacity (kips)	103.4	103	negligible
	Interaction ratio	0.578	0.578	none
	Torsional capacity (in·kips)	1,008	1,010	negligible
ASD	Compression capacity (kips)	224.3	224.3	none
	Bending capacity (in·kips)	711.7	712	negligible
	Shear capacity (kips)	68.78	68.8	negligible
	Interaction ratio	0.869	0.869	none
	Torsional capacity (in·kips)	670.6	671	negligible

STAAD.Pro Input

The following design parameters are used:

The welded pipe member is specified using STP 2
Shear lag factor, U for tension rupture capacity is specified by SLF 0.8

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 UPT Pipe 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;
START USER TABLE
TABLE 1
UNIT INCHES KIP
PIPE
PIPE
10 9.5 0 0
END
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 3 TABLE ST W12X72
MEMBER PROPERTY
2 UPTABLE 1 PIPE
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 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 ST   PIPE                     (UPT)
                           PASS   Sec. G1              0.653         1
                       11.75 C          0.00         599.60        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     17.400  L/C  :     1                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      1    Ratio     :     0.653(PASS)                 |
|          Loc  :    0.00   Condition :   Sec. G1                       |
|-----------------------------------------------------------------------|
| DESIGN FORCES                                                         |
| Fx:   1.175E+01(C ) Fy:   6.750E+01     Fz:   0.000E+00               |
| Mx:   0.000E+00     My:   0.000E+00     Mz:   5.996E+02               |
|-----------------------------------------------------------------------|
| SECTION PROPERTIES  (UNIT: INCH)                                      |
| Azz:   3.829E+00     Ayy:   3.829E+00     Cw:  0.000E+00              |
| Szz:   1.821E+01     Syy:   1.821E+01                                 |
| Izz:   9.105E+01     Iyy:   9.105E+01     Ix:  1.821E+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    40.00     N/A     63.80    T.B4.1(a)-9  |
|                Non-Slender    40.00     N/A     63.80    T.B4.1(a)-9  |
| Flexure     :  Compact        40.00    40.60   179.80    T.B4.1(b)-20 |
|                Compact        40.00    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   3.45E+02   0.000      Eq. D2-1      1     0.00 |
| Rupture     0.00E+00   2.76E+02   0.000      Eq. D2-2      1     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    2.43E+01   3.37E+02   0.072      Eq. E3-1      2     0.00 |
| Min Buck    2.43E+01   3.37E+02   0.072      Eq. E3-1      2     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    7.66E+00    17.40   4.89E+01   9.45E+02   3.74E+02        |
| Min Buck    7.66E+00    17.40   4.89E+01   9.45E+02   3.74E+02        |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z    -6.96E-02   1.03E+02   0.001      Eq. G6-1      2     0.00 |
| Local-Y     6.75E+01   1.03E+02   0.653      Eq. G6-1      1     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     7.66E+00     0.00     0.00     0.00   1.15E+02            |
| Local-Y     7.66E+00     0.00     0.00     0.00   1.15E+02            |
|-----------------------------------------------------------------------|
| CHECK FOR TORSION                                                     |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
|             1.56E+02   1.01E+03   0.155      Eq. H3-1      2     0.00 |
| Intermediate Fcr        Tn                                            |
|             3.00E+01   1.12E+03                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major      -6.00E+02   1.07E+03   0.561      Eq. F8-1      1     0.00 |
| Minor      -2.09E+00   1.07E+03   0.002      Eq. F8-1      2     0.00 |
| Intermediate   Mn        My                                           |
| Major       1.19E+03   0.00E+00                                       |
| Minor       1.19E+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          0.578      Eq. H1-1b     1     0.00             |
| Flexure Tens          0.561      Eq. H1-1b     1     0.00             |
| Intermediate         Mcx        Mrx        Pc                         |
|                      Mcy        Mry        Pr                         |
| Flexure Comp         1.07E+03  -6.00E+02   3.37E+02                   |
|                      1.07E+03   0.00E+00   1.17E+01                   |
| Flexure Tens         1.07E+03  -6.00E+02   2.76E+02                   |
|                      1.07E+03   0.00E+00   0.00E+00                   |
|-----------------------------------------------------------------------|
    57. PARAMETER 2
    58. CODE AISC UNIFIED 2010
    59. SLF 0.8 MEMB 2
    60. FYLD 50 ALL
    61. FU 60 ALL
    62. METHOD ASD
    63. STP 2 MEMB 2
    64. TRACK 2 MEMB 2
    65. 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 ST   PIPE                     (UPT)
                           PASS   Sec. G1              0.981         1
                       11.75 C          0.00         599.60        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     17.400  L/C  :     1                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      1    Ratio     :     0.981(PASS)                 |
|          Loc  :    0.00   Condition :   Sec. G1                       |
|-----------------------------------------------------------------------|
| DESIGN FORCES                                                         |
| Fx:   1.175E+01(C ) Fy:   6.750E+01     Fz:   0.000E+00               |
| Mx:   0.000E+00     My:   0.000E+00     Mz:   5.996E+02               |
|-----------------------------------------------------------------------|
| SECTION PROPERTIES  (UNIT: INCH)                                      |
| Azz:   3.829E+00     Ayy:   3.829E+00     Cw:  0.000E+00              |
| Szz:   1.821E+01     Syy:   1.821E+01                                 |
| Izz:   9.105E+01     Iyy:   9.105E+01     Ix:  1.821E+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    40.00     N/A     63.80    T.B4.1(a)-9  |
|                Non-Slender    40.00     N/A     63.80    T.B4.1(a)-9  |
| Flexure     :  Compact        40.00    40.60   179.80    T.B4.1(b)-20 |
|                Compact        40.00    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   2.29E+02   0.000      Eq. D2-1      1     0.00 |
| Rupture     0.00E+00   1.84E+02   0.000      Eq. D2-2      1     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    2.43E+01   2.24E+02   0.108      Eq. E3-1      2     0.00 |
| Min Buck    2.43E+01   2.24E+02   0.108      Eq. E3-1      2     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    7.66E+00    17.40   4.89E+01   9.45E+02   3.74E+02        |
| Min Buck    7.66E+00    17.40   4.89E+01   9.45E+02   3.74E+02        |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z    -6.96E-02   6.88E+01   0.001      Eq. G6-1      2     0.00 |
| Local-Y     6.75E+01   6.88E+01   0.981      Eq. G6-1      1     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     7.66E+00     0.00     0.00     0.00   1.15E+02            |
| Local-Y     7.66E+00     0.00     0.00     0.00   1.15E+02            |
|-----------------------------------------------------------------------|
| CHECK FOR TORSION                                                     |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
|             1.56E+02   6.71E+02   0.233      Eq. H3-1      2     0.00 |
| Intermediate Fcr        Tn                                            |
|             3.00E+01   1.12E+03                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major      -6.00E+02   7.12E+02   0.843      Eq. F8-1      1     0.00 |
| Minor      -2.09E+00   7.12E+02   0.003      Eq. F8-1      2     0.00 |
| Intermediate   Mn        My                                           |
| Major       1.19E+03   0.00E+00                                       |
| Minor       1.19E+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          0.869      Eq. H1-1b     1     0.00             |
| Flexure Tens          0.843      Eq. H1-1b     1     0.00             |
| Intermediate         Mcx        Mrx        Pc                         |
|                      Mcy        Mry        Pr                         |
| Flexure Comp         7.12E+02   3.58E+02   2.24E+02                   |
|                      7.12E+02  -2.09E+00   2.43E+01                   |
| Flexure Tens         7.12E+02   3.58E+02   1.84E+02                   |
|                      7.12E+02  -2.09E+00   0.00E+00                   |
|-----------------------------------------------------------------------|
| CHECK FOR FLEXURE TENS/COMP TORSION SHEAR INTERACTION                 |
|                       RATIO      CRITERIA    L/C     LOC              |
| Flx Tor Comp Shr      0.775      Eq. H3-6      2     0.00             |
| Flx Tor Tens Shr      0.667      Eq. H3-6      2     0.00             |
| Intermediate         Mcx /      Mcy /      Mrx /      Mry /           |
|                      Vcx /      Vcy /      Vrx /      Vry /           |
|                      Tc         Tr         Pc         Pr              |
| Flx Tor Comp Shr     7.12E+02   7.12E+02   3.58E+02  -2.09E+00        |
|                      6.88E+01   6.88E+01  -6.96E-02  -1.18E+01        |
|                      6.71E+02   1.56E+02   2.24E+02   2.43E+01        |
| Flx Tor Tens Shr     7.12E+02   7.12E+02   3.58E+02  -2.09E+00        |
|                      6.88E+01   6.88E+01  -6.96E-02  -1.18E+01        |
|                      6.71E+02   1.56E+02   1.84E+02   0.00E+00        |
|-----------------------------------------------------------------------|