V. AISC 360-05 Double L E.6

References

Steel Construction Manual, Version 13.0, American Institute of Steel Construction, 2005
Design Examples, Version 13.0, American Institute of Steel Construction, 2005, Example E.6, pp.E-26 - E-30

Details

From reference (2):

Determine if a 2L5×3×1/4 LLBB (3/4-in. separation) strut with a length of 8 ft and pinned ends has sufficient available strength to support a dead load of 10 kips and live load of 30 kips in axial compression. Also, calculate the required number of fully tightened or welded intermediate connectors.

Validation

Ultimate load:

P_{u} = 1.2 (10 kips) + 1.6 (30 kips) = 60 kips

Service load:

P_{a} = 10 kips + 30 kips = 40 kips

Per Table C-A-7.1 in Reference 1, K = 1.0 for pinned-pinned columns. Since the unbraced length is the same both directions, by inspection the y-y axis governs.

Slenderness

λ = \frac{b}{t} = \frac{5.0}{0.25} = 20.0

The limiting slenderness ratio, λ_r:

λ_{r} = 0.45 \sqrt{\frac{E}{F_{y}}} = 0.45 \sqrt{\frac{29, 000}{36}} = 12.8 < λ

(Table B4.1a, case 3)

Thus, the outstanding legs are slender.

Check for slender web and then determine the stiffened element (web) reduction factor.

0.45 \sqrt{\frac{E}{F_{y}}} = 12.8 < \frac{b}{t} = 20.0 < 0.91 \sqrt{\frac{E}{F_{y}}} = 0.91 \sqrt{\frac{29, 000}{36}} = 25.8

Q_{s} = 1.34 - 0.76 (\frac{b}{t}) \sqrt{\frac{F_{y}}{E}} = 1.34 - 0.76 (20.0) \sqrt{\frac{36}{29, 000}} = 0.804

(Eq. E7-11)

Q = Q_{s} Q_{a} = 0.804 (1.0) = 0.804

F_{c r} = Q ({0.658}^{Q F_{y} / F_{e}}) F_{y}

where

F_e	=	$(\frac{F_{e y} + F_{e z}}{2 H}) [1 - \sqrt{1 - \frac{4 F_{e y} F_{e z} H}{{(F_{e y} + F_{e z})}^{2}}}]$ (Eq E4-5)
K_i	=	0.5 for angles back-to-back
r_i	=	r_z = 0.652 in; the radius of gyration for a single angle
a	=	32 in spacing between connectors (using the bolts at 1/3 points from Reference 2)
${(\frac{K_{y} L}{r_{y}})}_{m}$	=	$\sqrt{{(\frac{K L}{r})}_{0}^{2} + {(\frac{K_{i} a}{r_{i}})}^{2}} = \sqrt{{(\frac{1.0 \times 96}{1.33})}^{2} + {(\frac{0.5 \times 32}{0.652})}^{2}} = 76.3$ for $\frac{a}{r_{i}} = \frac{32 in}{0.652 in} = 49.1 > 40$
F_ey	=	$\frac{π^{2} E}{{(\frac{K_{y} L}{r_{y}})}_{m}^{2}} = \frac{π^{2} (29, 000)}{{(76.3)}^{2}} = 49.2 ksi$
F_ez	=	$\frac{G J}{A_{g} {\bar{r}}_{0}^{2}} = \frac{11, 200 (0.04328) \times 2 angles}{3.88 {(2.59)}^{2}} = 37.7 ksi$

Thus, $(\frac{49.2 + 37.7}{2 \times 0.657}) [1 - \sqrt{1 - \frac{4 (49.2) (37.7) (0.657)}{{(49.2 + 37.7)}^{2}}}] = 26.8 ksi$

Nominal Compressive Strength

\frac{Q F_{y}}{2.25} = \frac{0.804 (36)}{2.25} = 12.9 ksi < 26.8 ksi

F_{c r} = Q ({0.658}^{Q F_{y} / F_{e}}) F_{y} = 0.804 [{0.658}^{0.804 (36) / (26.8)}] 36 = 18.4 ksi

(Eq E7-2)

P_{n} = F_{c r} A_{g} = 18.4 (3.88) = 71.4 kips

(Eq E7-1)

The available compressive strength:

LRFD: $ϕ_{c} P_{n} = 0.90 (71.4) = 64.3 kips$

ASD: $P_{n} / Ω_{c} = \frac{71.4}{1.67} = 42.8 kips$

Result Type	Reference	STAAD.Pro	Difference	Comments
ϕ_cP_n (kips) [LRFD]	64.3	63.8	negligible
P_n/Ω_c (kips) [ASD]	42.8	42.4	negligible

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\AISC 360-05 Double L E.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 5-Dec-12
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 8 0; 2 0 0 0;
MEMBER INCIDENCES
1 1 2;
MEMBER PROPERTY AMERICAN
1 TABLE LD L50304 SP 0.0625
UNIT INCHES KIP
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 29000
POISSON 0.3
ISOTROPIC STEEL
E 29000
POISSON 0.3
DENSITY 0.000283
ALPHA 6e-06
DAMP 0.03
TYPE STEEL
STRENGTH FY 36 FU 58 RY 1.5 RT 1.2
ISOTROPIC STEEL_36_KSI
E 29000
POISSON 0.3
DENSITY 0.000283
ALPHA 6.5e-06
DAMP 0.03
TYPE STEEL
STRENGTH FY 36 FU 58 RY 1.5 RT 1.2
ISOTROPIC STEEL_50_KSI
E 29000
POISSON 0.3
DENSITY 0.000283
ALPHA 6.5e-06
DAMP 0.03
TYPE STEEL
STRENGTH FY 50 FU 62 RY 1.5 RT 1.2
END DEFINE MATERIAL
UNIT FEET KIP
CONSTANTS
MATERIAL STEEL_50_KSI ALL
SUPPORTS
2 PINNED
1 FIXED BUT FY MX MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
1 FY -10
LOAD 2 LOADTYPE None  TITLE LOAD CASE 2
JOINT LOAD
1 FY -30
LOAD COMB 3 ULTIMATE LOADS
1 1.2 2 1.6 
LOAD COMB 4 SERVICE LOADS
1 1.0 2 1.0 
PERFORM ANALYSIS
UNIT INCHES KIP
LOAD LIST 3
PARAMETER 1
CODE AISC UNIFIED 2005
METHOD LRFD
FU 58 ALL
FYLD 36 ALL
CSPACING 32 ALL
TRACK 2 ALL
CHECK CODE ALL
LOAD LIST 4
PARAMETER 2
CODE AISC UNIFIED 2005
METHOD ASD
FU 58 ALL
FYLD 36 ALL
CSPACING 32 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

                       STAAD.PRO CODE CHECKING - (AISC-360-05-LRFD)   v3.3a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
       1 LD   L50304                   (AISC SECTIONS)
                           PASS   Eq. H2-1             0.945         3
                       60.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     78.939  L/C  :     3                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      3    Ratio     :     0.945(PASS)                 |
|          Loc  :    0.00   Condition :   Eq. H2-1                      |
|-----------------------------------------------------------------------|
| DESIGN FORCES                                                         |
| Fx:   6.000E+01(C ) Fy:   0.000E+00     Fz:   0.000E+00               |
| Mx:   0.000E+00     My:   0.000E+00     Mz:   0.000E+00               |
|-----------------------------------------------------------------------|
| SECTION PROPERTIES  (UNIT: INCH)                                      |
| Azz:   1.500E+00     Ayy:   1.250E+00     Cw:  0.000E+00              |
| Szz:   3.055E+00     Syy:   2.075E+00                                 |
| Izz:   1.022E+01     Iyy:   7.003E+00     Ix:  8.083E-02              |
|-----------------------------------------------------------------------|
| MATERIAL PROPERTIES                                                   |
| Fyld:          36.000   Fu:          58.000                           |
|-----------------------------------------------------------------------|
| Actual Member Length:        96.000                                   |
| Design Parameters                                                     |
| Kz:    1.00 Ky:    1.00 NSF:    1.00 SLF:    1.00 CSP:   32.00        |
|-----------------------------------------------------------------------|
| SECTION CLASS  UNSTIFFENED /     λ       λp      λr        CASE       |
|                STIFFENED                                              |
| Compression :  Slender        20.00     N/A     12.77    T.B4.1-5     |
|                  N/A           N/A      N/A      N/A      N/A         |
| Flexure     :  Non-Compact    20.00    15.33    25.83    T.B4.1-6     |
|                  N/A           N/A      N/A      N/A      N/A         |
      STAAD SPACE                                              -- PAGE NO.    4
                       STAAD.PRO CODE CHECKING - (AISC-360-05-LRFD)   v3.3a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL TENSION                                               |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Yield       0.00E+00   1.26E+02   0.000      Eq. D2-1      3     0.00 |
| Rupture     0.00E+00   1.69E+02   0.000      Eq. D2-2      3     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    6.00E+01   8.72E+01   0.688      Sec. E1       3     0.00 |
| Min Buck    6.00E+01   8.09E+01   0.742      Eq. E7-1      3     0.00 |
| Flexural                                                              |
| Tor Buck    6.00E+01   6.35E+01   0.945      Eq. E7-1      3     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    3.88E+00    59.15   2.50E+01   8.18E+01   9.69E+01        |
| Min Buck    3.88E+00    72.59   2.32E+01   5.43E+01   8.99E+01        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    3.88E+00   1.82E+01   7.05E+01                            |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z     0.00E+00   2.92E+01   0.000      Eq. G2-1      3     0.00 |
| Local-Y     0.00E+00   2.43E+01   0.000      Eq. G2-1      3     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     1.50E+00     1.00     1.20    12.00   3.24E+01            |
| Local-Y     1.25E+00     1.00     1.20    20.00   2.70E+01            |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   1.58E+02   0.000      Eq. F9-1      3     0.00 |
| Minor       0.00E+00   1.08E+02   0.000      Eq. F6-1      3     0.00 |
| Intermediate   Mn        My                                           |
| Major       1.76E+02   0.00E+00                                       |
| Minor       1.20E+02   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-LATERAL TORSIONAL BUCKLING                          |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   1.04E+03   0.000      Eq. F9-4      3     0.00 |
| Intermediate   Mn        Me        Cb      Lp       Lr       Lb       |
| Major       1.16E+03   0.00E+00    1.00     0.00     0.00    96.00    |
      STAAD SPACE                                              -- PAGE NO.    5
                       STAAD.PRO CODE CHECKING - (AISC-360-05-LRFD)   v3.3a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
|-----------------------------------------------------------------------|
| CHECK FOR FLEXURE TENS/COMP INTERACTION                               |
|                       RATIO      CRITERIA    L/C     LOC              |
| Flexure Comp          0.945      Eq. H2-1      3     0.00             |
| Flexure Tens          0.000      Eq. H1-1b     3     0.00             |
| Intermediate         frbw /     frbz /     fra /                      |
|                      Fcbw       Fcbz       Fca                        |
| Flexure Comp         0.00E+00   0.00E+00   1.55E+01                   |
|                      5.18E+01   5.18E+01   1.64E+01                   |
| Intermediate         Mcx        Mrx        Pc                         |
|                      Mcy        Mry        Pr                         |
| Flexure Tens         1.58E+02   0.00E+00   1.26E+02                   |
|                      1.08E+02   0.00E+00   0.00E+00                   |
|-----------------------------------------------------------------------|
    70. LOAD LIST 4
    71. PARAMETER 2
    72. CODE AISC UNIFIED 2005
    73. METHOD ASD
    74. FU 58 ALL
    75. FYLD 36 ALL
    76. CSPACING 32 ALL
    77. TRACK 2 ALL
    78. CHECK CODE ALL
 STEEL DESIGN    
      STAAD SPACE                                              -- PAGE NO.    6
                       STAAD.PRO CODE CHECKING - ( AISC-360-05-ASD)   v3.3a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
       1 LD   L50304                   (AISC SECTIONS)
                           PASS   Eq. H2-1             0.947         4
                       40.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     78.939  L/C  :     4                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      4    Ratio     :     0.947(PASS)                 |
|          Loc  :    0.00   Condition :   Eq. H2-1                      |
|-----------------------------------------------------------------------|
| DESIGN FORCES                                                         |
| Fx:   4.000E+01(C ) Fy:   0.000E+00     Fz:   0.000E+00               |
| Mx:   0.000E+00     My:   0.000E+00     Mz:   0.000E+00               |
|-----------------------------------------------------------------------|
| SECTION PROPERTIES  (UNIT: INCH)                                      |
| Azz:   1.500E+00     Ayy:   1.250E+00     Cw:  0.000E+00              |
| Szz:   3.055E+00     Syy:   2.075E+00                                 |
| Izz:   1.022E+01     Iyy:   7.003E+00     Ix:  8.083E-02              |
|-----------------------------------------------------------------------|
| MATERIAL PROPERTIES                                                   |
| Fyld:          36.000   Fu:          58.000                           |
|-----------------------------------------------------------------------|
| Actual Member Length:        96.000                                   |
| Design Parameters                                                     |
| Kz:    1.00 Ky:    1.00 NSF:    1.00 SLF:    1.00 CSP:   32.00        |
|-----------------------------------------------------------------------|
| SECTION CLASS  UNSTIFFENED /     λ       λp      λr        CASE       |
|                STIFFENED                                              |
| Compression :  Slender        20.00     N/A     12.77    T.B4.1-5     |
|                  N/A           N/A      N/A      N/A      N/A         |
| Flexure     :  Non-Compact    20.00    15.33    25.83    T.B4.1-6     |
|                  N/A           N/A      N/A      N/A      N/A         |
      STAAD SPACE                                              -- PAGE NO.    7
                       STAAD.PRO CODE CHECKING - ( AISC-360-05-ASD)   v3.3a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL TENSION                                               |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Yield       0.00E+00   8.36E+01   0.000      Eq. D2-1      4     0.00 |
| Rupture     0.00E+00   1.13E+02   0.000      Eq. D2-2      4     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    4.00E+01   5.80E+01   0.689      Sec. E1       4     0.00 |
| Min Buck    4.00E+01   5.38E+01   0.743      Eq. E7-1      4     0.00 |
| Flexural                                                              |
| Tor Buck    4.00E+01   4.22E+01   0.947      Eq. E7-1      4     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    3.88E+00    59.15   2.50E+01   8.18E+01   9.69E+01        |
| Min Buck    3.88E+00    72.59   2.32E+01   5.43E+01   8.99E+01        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    3.88E+00   1.82E+01   7.05E+01                            |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z     0.00E+00   1.94E+01   0.000      Eq. G2-1      4     0.00 |
| Local-Y     0.00E+00   1.62E+01   0.000      Eq. G2-1      4     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     1.50E+00     1.00     1.20    12.00   3.24E+01            |
| Local-Y     1.25E+00     1.00     1.20    20.00   2.70E+01            |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   1.05E+02   0.000      Eq. F9-1      4     0.00 |
| Minor       0.00E+00   7.16E+01   0.000      Eq. F6-1      4     0.00 |
| Intermediate   Mn        My                                           |
| Major       1.76E+02   0.00E+00                                       |
| Minor       1.20E+02   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-LATERAL TORSIONAL BUCKLING                          |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   6.93E+02   0.000      Eq. F9-4      4     0.00 |
| Intermediate   Mn        Me        Cb      Lp       Lr       Lb       |
| Major       1.16E+03   0.00E+00    1.00     0.00     0.00    96.00    |
      STAAD SPACE                                              -- PAGE NO.    8
                       STAAD.PRO CODE CHECKING - ( AISC-360-05-ASD)   v3.3a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
|-----------------------------------------------------------------------|
| CHECK FOR FLEXURE TENS/COMP INTERACTION                               |
|                       RATIO      CRITERIA    L/C     LOC              |
| Flexure Comp          0.947      Eq. H2-1      4     0.00             |
| Flexure Tens          0.000      Eq. H1-1b     4     0.00             |
| Intermediate         frbw /     frbz /     fra /                      |
|                      Fcbw       Fcbz       Fca                        |
| Flexure Comp         0.00E+00   0.00E+00   1.03E+01                   |
|                      3.45E+01   3.45E+01   1.09E+01                   |
| Intermediate         Mcx        Mrx        Pc                         |
|                      Mcy        Mry        Pr                         |
| Flexure Tens         1.05E+02   0.00E+00   8.36E+01                   |
|                      7.16E+01   0.00E+00   0.00E+00                   |
|-----------------------------------------------------------------------|