V. AISC 360-10 - E-2

Reference

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.2, pp. E-11 - E-15

Problem

From the reference:

Verify that a built-up, ASTM A572 Grade 50 column with PL 1 in. × 8 in. flanges and a PL 4 in. × 15 in. web is sufficient to carry a dead load of 70 kips and live load of 210 kips in axial compression. The column length is 15 ft and the ends are pinned in both axes.

Validation

Ultimate load:

P_{u} = 1.2 (70 kips) + 1.6 (210 kips) = 420 kips

Service load:

P_{a} = 70 kips + 210 kips = 280 kips

Section Properties

Built-up section properties (ignoring fillet welds):

A = 2 (8.0) (1.0) + 15.0 (0.25) = 19.8 {in}^{2}

I_{x} = Σ A d^{2} + Σ \frac{b h^{3}}{12} = 2 [8 {(8.0)}^{2} + \frac{8.0 {(1.0)}^{3}}{12}] + \frac{0.25 {(15.0)}^{3}}{12} = 1, 100 {in}^{4}

I_{y} = 2 (\frac{1.0 \times {8.0}^{3}}{12}) + \frac{15.0 \times {0.25}^{3}}{12} = 85.4 {in}^{4}

r_{y} = \sqrt{\frac{I_{y}}{A}} = \sqrt{\frac{85.4}{19.8}} = 2.08 in

Elastic Flexural Buckling Stress

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.

\frac{K L}{r_{y}} = \frac{1.0 (15.0) (12)}{2.08} = 86.5

F_{e} = \frac{π^{2} E}{{(K L / r)}^{2}} = \frac{π^{2} 29, 000}{{(86.5)}^{2}} = 38.3 ksi

(Eq. E3-4)

Slenderness

Check for slender flanges and then determine the unstiffened element (flange) reduction factor.

k_{c} = \frac{4}{\sqrt{h / t_{w}}} = \frac{4}{\sqrt{15.0 / 0.25}} = 0.516

(Table B4.1b note a)

0.35 < k_c < 0.76

Flange slenderness:

λ = \frac{b}{t} = \frac{4.0}{1.0} = 4.0

The limiting flange slenderness ratio, λ_r:

λ_{r} = 0.64 \sqrt{\frac{k_{c} E}{F_{y}}} = 0.64 \sqrt{\frac{0.516 (29, 000)}{50}} = 11.1 > λ

(Table B4.1a, case 2)

Thus, the flanges are not slender and there is no reduction (Q_s = 1.0).

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

Web slenderness:

λ = \frac{h}{t} = \frac{15.0}{0.25} = 60.0

The limiting web slenderness ratio, λ_r:

λ_{r} = 1.49 \sqrt{\frac{E}{F_{y}}} = 1.49 \sqrt{\frac{29, 000}{50}} = 35.9 < λ

(Table B4.1a, case 5)

Thus, the web is slender.

Q_{a} = \frac{A_{e}}{A_{g}}

(Eq. E7-16)

where

A_e	=	the effective area based on the reduced effective width, b_e
b_e	=	$1.92 t \sqrt{\frac{E}{f}} [1 - \frac{0.34}{(b / t)} \sqrt{\frac{E}{f}}] \leq b$ (Eq. E7-17)
$f$	=	F_cr calculated based on Q = 1.0
F_cr	=	$Q ({0.658}^{Q F_{y} / F_{e}}) F_{y} = 1.0 [{0.658}^{1.0 (50) / (38.3)}] 50 = 29.0 ksi$ because $\frac{K L}{r} = 86.5 < 4.71 \sqrt{\frac{E}{Q F_{y}}} = 4.71 \sqrt{\frac{29, 000}{1.0 (50)}} = 113$

b_{e} = 1.92 (0.25) \sqrt{\frac{29, 000}{29.0}} [1 - \frac{0.34}{(15 / 0.25)} \sqrt{\frac{29, 000}{29.0}}] = 12.5 in < 15 in

A_{e} = b_{e} t_{w} + 2 b_{f} t_{f} = 12.5 (0.25) + 2 (8.0) (1.0) = 19.1 {in}^{2}

Q_{a} = \frac{19.1}{19.8} = 0.965

Q = Q_{s} Q_{a} = 1.0 \times 0.965 = 0.965

\frac{K L}{r} = 86.5 < 4.71 \sqrt{\frac{E}{Q F_{y}}} = 4.71 \sqrt{\frac{29, 000}{0.965 (50)}} = 115

F_{c r} = Q ({0.658}^{Q F_{y} / F_{e}}) F_{y} = 0.965 [{0.658}^{0.965 (50) / (38.3)}] 50 = 28.5 ksi

Nominal Compressive Strength

P_{n} = F_{c r} A_{g} = 28.5 (19.8) = 564 kips

(Eq E7-1)

The available compressive strength:

LRFD: $ϕ_{c} P_{n} = 0.90 (564) = 508 kips$

ASD: $P_{n} / Ω_{c} = \frac{564}{1.67} = 338 kips$

Comparison

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
ϕ_cP_n (kips) [LRFD]	508	513	1.0%
P_n/Ω_c (kips) [ASD]	338	341	0.9%

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\AISC 360-10 - E-2.STD is typically installed with the program.

Tip: You can copy and paste this content directly into a .std file to run in STAAD.Pro.

STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 31-Oct-12
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 0 15 0;
MEMBER INCIDENCES
1 1 2;
START USER TABLE
*TABLE 1
*UNIT INCHES KIP
*ISECTION
*I1
*17 0.25 17 8 1 8 1 0 0 0
TABLE 2
UNIT INCHES KIP
WIDE FLANGE
I
19.75 17 0.25 8 1 1095.65 85.3529 5.41146 4.25 10.6667 8 1
END
UNIT inch KIP
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29732.7
POISSON 0.3
DENSITY 0.000283
ALPHA 1.2e-005
DAMP 0.03
TYPE STEEL
STRENGTH FY 50 FU 65 RY 1.5 RT 1.2
END DEFINE MATERIAL
UNIT INCHES KIP
CONSTANTS
MATERIAL STEEL ALL
MEMBER PROPERTY AMERICAN
1 UPTABLE 2 I
UNIT FEET KIP
SUPPORTS
1 PINNED
2 FIXED BUT FY MX MY MZ
LOAD 1 LOADTYPE Dead  TITLE LOAD CASE 1
JOINT LOAD
2 FY -70
LOAD 2 LOADTYPE Live  TITLE LOAD CASE 2
JOINT LOAD
2 FY -210
UNIT INCHES KIP
LOAD COMB 3 COMBINATION LOAD CASE 3
1 1.2 2 1.6
LOAD COMB 4 COMBINATION LOAD CASE 4
1 1.0 2 1.0
PERFORM ANALYSIS
LOAD LIST 3
PARAMETER 1
CODE AISC UNIFIED 2010
FU 65 ALL
FYLD 50 ALL
METHOD LRFD
TRACK 2 ALL
CHECK CODE ALL
LOAD LIST 4
PARAMETER 2
CODE AISC UNIFIED 2010
FU 65 ALL
FYLD 50 ALL
METHOD ASD
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD 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 ST   I                        (UPT)
                           PASS   Eq. H1-1a            0.819         3
                      420.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     86.586  L/C  :     3                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      3    Ratio     :     0.819(PASS)                 |
|          Loc  :    0.00   Condition :   Eq. H1-1a                     |
|-----------------------------------------------------------------------|
| DESIGN FORCES                                                         |
| Fx:   4.200E+02(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:   1.067E+01     Ayy:   4.250E+00     Cw:  5.461E+03              |
| Szz:   1.289E+02     Syy:   2.134E+01                                 |
| Izz:   1.096E+03     Iyy:   8.535E+01     Ix:  5.411E+00              |
|-----------------------------------------------------------------------|
| MATERIAL PROPERTIES                                                   |
| Fyld:          50.000   Fu:          65.000                           |
|-----------------------------------------------------------------------|
| Actual Member Length:       180.000                                   |
| Design Parameters                                                     |
| Kz:    1.00 Ky:    1.00 NSF:    1.00 SLF:    1.00 CSP:   12.00        |
|-----------------------------------------------------------------------|
| SECTION CLASS  UNSTIFFENED /     λ       λp      λr        CASE       |
|                STIFFENED                                              |
| Compression :  Non-Slender     4.00     N/A     13.66    T.B4.1(a)-1  |
|                Slender        60.00     N/A     36.33    T.B4.1(a)-5  |
| Flexure     :  Compact         4.00     9.27    24.39    T.B4.1(b)-10 |
|                Compact        60.00    91.69   139.00    T.B4.1(b)-15 |
      STAAD PLANE                                              -- 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   8.89E+02   0.000      Eq. D2-1      3     0.00 |
| Rupture     0.00E+00   9.63E+02   0.000      Eq. D2-2      3     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    4.20E+02   8.03E+02   0.523      Eq. E7-1      3     0.00 |
| Min Buck    4.20E+02   5.13E+02   0.819      Eq. E7-1      3     0.00 |
| Flexural                                                              |
| Tor Buck    4.20E+02   6.84E+02   0.614      Eq. E7-1      3     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    1.86E+01    24.17   4.52E+01   5.02E+02   8.93E+02        |
| Min Buck    1.91E+01    86.59   2.89E+01   3.91E+01   5.70E+02        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    1.88E+01   3.85E+01   7.60E+02                            |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z     0.00E+00   2.88E+02   0.000      Eq. G2-1      3     0.00 |
| Local-Y     0.00E+00   1.15E+02   0.000      Eq. G2-1      3     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     1.07E+01     1.00     1.20     4.00   3.20E+02            |
| Local-Y     4.25E+00     1.00     5.00    60.00   1.27E+02            |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   6.39E+03   0.000      Eq. F2-1      3     0.00 |
| Minor       0.00E+00   1.45E+03   0.000      Eq. F6-1      3     0.00 |
| Intermediate   Mn        My                                           |
| Major       7.10E+03   0.00E+00                                       |
| Minor       1.61E+03   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-LATERAL TORSIONAL BUCKLING                          |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   5.44E+03   0.000      Eq. F2-2      3     0.00 |
| Intermediate   Mn        Me        Cb      Lp       Lr       Lb       |
| Major       6.04E+03   0.00E+00    1.00    89.22   311.02   180.00    |
      STAAD PLANE                                              -- 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          0.819      Eq. H1-1a     3     0.00             |
| Flexure Tens          0.000      Eq. H1-1b     3     0.00             |
| Intermediate         Mcx        Mrx        Pc                         |
|                      Mcy        Mry        Pr                         |
| Flexure Comp         5.44E+03   0.00E+00   5.13E+02                   |
|                      1.45E+03   0.00E+00   4.20E+02                   |
| Flexure Tens         5.44E+03   0.00E+00   8.89E+02                   |
|                      1.45E+03   0.00E+00   0.00E+00                   |
|-----------------------------------------------------------------------|
    63. LOAD LIST 4
    64. PARAMETER 2
    65. CODE AISC UNIFIED 2010
    66. FU 65 ALL
    67. FYLD 50 ALL
    68. METHOD ASD
    69. TRACK 2 ALL
    70. CHECK CODE ALL
 STEEL DESIGN    
      STAAD PLANE                                              -- 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 ST   I                        (UPT)
                           PASS   Eq. H1-1a            0.820         4
                      280.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     86.586  L/C  :     4                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      4    Ratio     :     0.820(PASS)                 |
|          Loc  :    0.00   Condition :   Eq. H1-1a                     |
|-----------------------------------------------------------------------|
| DESIGN FORCES                                                         |
| Fx:   2.800E+02(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:   1.067E+01     Ayy:   4.250E+00     Cw:  5.461E+03              |
| Szz:   1.289E+02     Syy:   2.134E+01                                 |
| Izz:   1.096E+03     Iyy:   8.535E+01     Ix:  5.411E+00              |
|-----------------------------------------------------------------------|
| MATERIAL PROPERTIES                                                   |
| Fyld:          50.000   Fu:          65.000                           |
|-----------------------------------------------------------------------|
| Actual Member Length:       180.000                                   |
| Design Parameters                                                     |
| Kz:    1.00 Ky:    1.00 NSF:    1.00 SLF:    1.00 CSP:   12.00        |
|-----------------------------------------------------------------------|
| SECTION CLASS  UNSTIFFENED /     λ       λp      λr        CASE       |
|                STIFFENED                                              |
| Compression :  Non-Slender     4.00     N/A     13.66    T.B4.1(a)-1  |
|                Slender        60.00     N/A     36.33    T.B4.1(a)-5  |
| Flexure     :  Compact         4.00     9.27    24.39    T.B4.1(b)-10 |
|                Compact        60.00    91.69   139.00    T.B4.1(b)-15 |
      STAAD PLANE                                              -- 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   5.91E+02   0.000      Eq. D2-1      4     0.00 |
| Rupture     0.00E+00   6.42E+02   0.000      Eq. D2-2      4     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    2.80E+02   5.35E+02   0.524      Eq. E7-1      4     0.00 |
| Min Buck    2.80E+02   3.41E+02   0.820      Eq. E7-1      4     0.00 |
| Flexural                                                              |
| Tor Buck    2.80E+02   4.55E+02   0.615      Eq. E7-1      4     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    1.86E+01    24.17   4.52E+01   5.02E+02   8.93E+02        |
| Min Buck    1.91E+01    86.59   2.89E+01   3.91E+01   5.70E+02        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    1.88E+01   3.85E+01   7.60E+02                            |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z     0.00E+00   1.92E+02   0.000      Eq. G2-1      4     0.00 |
| Local-Y     0.00E+00   7.63E+01   0.000      Eq. G2-1      4     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     1.07E+01     1.00     1.20     4.00   3.20E+02            |
| Local-Y     4.25E+00     1.00     5.00    60.00   1.27E+02            |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   4.25E+03   0.000      Eq. F2-1      4     0.00 |
| Minor       0.00E+00   9.65E+02   0.000      Eq. F6-1      4     0.00 |
| Intermediate   Mn        My                                           |
| Major       7.10E+03   0.00E+00                                       |
| Minor       1.61E+03   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-LATERAL TORSIONAL BUCKLING                          |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   3.62E+03   0.000      Eq. F2-2      4     0.00 |
| Intermediate   Mn        Me        Cb      Lp       Lr       Lb       |
| Major       6.04E+03   0.00E+00    1.00    89.22   311.02   180.00    |
      STAAD PLANE                                              -- 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          0.820      Eq. H1-1a     4     0.00             |
| Flexure Tens          0.000      Eq. H1-1b     4     0.00             |
| Intermediate         Mcx        Mrx        Pc                         |
|                      Mcy        Mry        Pr                         |
| Flexure Comp         3.62E+03   0.00E+00   3.41E+02                   |
|                      9.65E+02   0.00E+00   2.80E+02                   |
| Flexure Tens         3.62E+03   0.00E+00   5.91E+02                   |
|                      9.65E+02   0.00E+00   0.00E+00                   |
|-----------------------------------------------------------------------|