V. AISC 360-10 Double L E.6

Verify the compressive strength of an unbraced wide-flange section column about the Y-Y axis per the LRFD and ASD methods of the AISC 360-10 code.

References

American Institute of Steel Construction. 2010 Steel Construction Manual. Fourteenth Edition. AISC: Chicago, IL.
American Institute of Steel Construction. 2011. Design Examples, Version 14.0. AISC:Chicago, IL. Example E.6, pp. E-31 - E-36

Details

From the reference example E.1C:

Determine if a 2L5×3×1/4 LLBB (3/4-in. separation) strut, ASTM A36, 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 pretensioned bolted 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: $P_{n} / Ω_{c} = \frac{71.4}{1.67} = 42.8 kips$

Results

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

Note: STAAD.Pro does not perform design per Section E6 of the specification. It is assumed that sufficient intermediate connectors of an appropriate type are present. It is left to the engineer to verify the intermediate connector design.

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\AISC 360-10 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 2010
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 2010
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-360-10-LRFD)   v1.4a
                       ********************************************
 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             0.941         3
                       60.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     75.420  L/C  :     3                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      3    Ratio     :     0.941(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(a)-3  |
|                  N/A           N/A      N/A      N/A      N/A         |
| Flexure     :  Non-Compact    20.00    15.33    25.83    T.B4.1(b)-12 |
|                  N/A           N/A      N/A      N/A      N/A         |
      STAAD SPACE                                              -- PAGE NO.    4
                       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   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      Eq. E7-1      3     0.00 |
| Min Buck    6.00E+01   8.15E+01   0.737      Eq. E7-1      3     0.00 |
| Flexural                                                              |
| Tor Buck    6.00E+01   6.38E+01   0.941      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    71.46   2.33E+01   5.61E+01   9.05E+01        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    3.88E+00   1.83E+01   7.08E+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-360-10-LRFD)   v1.4a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
|-----------------------------------------------------------------------|
| CHECK FOR FLEXURE TENS/COMP INTERACTION                               |
|                       RATIO      CRITERIA    L/C     LOC              |
| Flexure Comp          0.941      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 2010
    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-360-10-ASD)   v1.4a
                       ********************************************
 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             0.943         4
                       40.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     75.420  L/C  :     4                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      4    Ratio     :     0.943(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(a)-3  |
|                  N/A           N/A      N/A      N/A      N/A         |
| Flexure     :  Non-Compact    20.00    15.33    25.83    T.B4.1(b)-12 |
|                  N/A           N/A      N/A      N/A      N/A         |
      STAAD SPACE                                              -- PAGE NO.    7
                       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   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      Eq. E7-1      4     0.00 |
| Min Buck    4.00E+01   5.42E+01   0.738      Eq. E7-1      4     0.00 |
| Flexural                                                              |
| Tor Buck    4.00E+01   4.24E+01   0.943      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    71.46   2.33E+01   5.61E+01   9.05E+01        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    3.88E+00   1.83E+01   7.08E+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-360-10-ASD)   v1.4a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
|-----------------------------------------------------------------------|
| CHECK FOR FLEXURE TENS/COMP INTERACTION                               |
|                       RATIO      CRITERIA    L/C     LOC              |
| Flexure Comp          0.943      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                   |
|-----------------------------------------------------------------------|