V. AISC 360-05 Built up I E.3

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.3, pp.E-14 - E.17
Seaburg, Paul A. and Charles J. Carter. 1997. Design Guide 9: Torsional Analysis of Structural Steel members. AISC: Chicago, IL.

Details

From reference (2):

Determine if a built-up, ASTM A572 grade 50 column with PL3/8 in.×102 in. flanges and a PL1/4 in.×7 1/4 in. web has sufficient available strength to carry a dead load of 40 kips and a live load of 120 kips in axial compression. The column unbraced length is 15 ft in both axes and the ends are pinned.

Validation

Ultimate load:

P_{u} = 1.2 (40 kips) + 1.6 (120 kips) = 240 kips

Service load:

P_{a} = 40 kips + 120 kips = 160 kips

Section Properties

Built-up section properties (ignoring fillet welds):

A_{g} = 2 (10.5) (0.375) + 7.25 (0.25) = 9.69 {in}^{2}

I_{x} = Σ A d^{2} + Σ \frac{b h^{3}}{12} = 2 [10.5 \times 0.375 {(3.81)}^{2} + \frac{10.5 {(0.25)}^{3}}{12}] + \frac{0.25 {(7.25)}^{3}}{12} = 122 {in}^{4}

I_{y} = 2 (\frac{0.375 \times {10.5}^{3}}{12}) + \frac{7.25 \times {0.25}^{3}}{12} = 72.4 {in}^{4}

r_{y} = \sqrt{\frac{I_{y}}{A}} = \sqrt{\frac{72.4}{9.69}} = 2.73 in

From the User Note in Section E4 of Reference 1:

C_{w} = \frac{I_{y} {h_{o}}^{2}}{4} = \frac{72.4 {(7.63)}^{2}}{4} = 1, 050 {in}^{6}

From Eq. 3.4 in Reference 3:

J = Σ \frac{b t^{3}}{3} = \frac{2 (10.5) {(0.375)}^{3} + 7.25 {(0.25)}^{3}}{3} = 0.407 {in}^{4}

Slenderness

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

Web slenderness:

λ = \frac{h}{t} = \frac{7.25}{0.25} = 29.0

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 not slender and there is no reduction (i.e., Q_a 1.0)

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

k_{c} = \frac{4}{\sqrt{h / t_{w}}} = \frac{4}{\sqrt{7.25 / 0.25}} = 0.743

(Table B4.1b note a)

0.35 < k_c < 0.76, OK.

Flange slenderness:

λ = \frac{b}{t} = \frac{5.25}{0.375} = 14.0

The limiting flange slenderness ratio, λ_r:

λ_{r} = 0.64 \sqrt{\frac{k_{c} E}{F_{y}}} = 0.64 \sqrt{\frac{0.743 (29, 000)}{50}} = 13.3 < λ

(Table B4.1a, case 2)

Thus, the flanges are slender.

0.64 \sqrt{\frac{k_{c} E}{F_{y}}} = 13.3 < \frac{b}{t} = 14.0 < 1.17 \sqrt{\frac{k_{c} E}{F_{y}}} = 1.17 \sqrt{\frac{0.743 (29, 000)}{50}} = 24.3

Q_{s} = 1.415 - 0.65 (\frac{b}{t}) \sqrt{\frac{F_{y}}{k_{c} E}} = 1.415 - 0.65 (14.0) \sqrt{\frac{50}{0.743 (29, 000)}} = 0.977

(Eq. E7-8)

Q = Q_{s} Q_{a} = 0.977 (1.0) = 0.977

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_{y} L}{r_{y}} = \frac{1.0 (15.0) (12)}{2.73} = 65.9

F_{e} = \frac{π^{2} E}{{(K L / r)}^{2}} = \frac{π^{2} 29, 000}{{(65.9)}^{2}} = 65.9 ksi

(Eq. E3-4)

Nominal Compressive Strength

\frac{K L}{r} = 65.9 < 4.71 \sqrt{\frac{E}{Q F_{y}}} = 4.71 \sqrt{\frac{29, 000}{0.977 (50)}} = 115

F_{c r} = Q ({0.658}^{Q F_{y} / F_{e}}) F_{y} = 0.977 [{0.658}^{0.977 (50) / (65.9)}] 50 = 35.8 ksi

(Eq E7-2)

P_{n} = F_{c r} A_{g} = 35.8 (6.69) = 347 kips

(Eq E7-1)

The available compressive strength:

LRFD: $ϕ_{c} P_{n} = 0.90 (347) = 312 kips$

ASD: $P_{n} / Ω_{c} = \frac{347}{1.67} = 208 kips$

Result Type	Reference	STAAD.Pro	Difference
ϕ_cP_n (kips) [LRFD]	312	314	negligible
P_n/Ω_c (kips) [ASD]	208	209	negligible
Slenderness ratio	65.9	65.86	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 Built up I E.STD is typically installed with the program.

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 2
UNIT INCHES KIP
WIDE FLANGE
I
9.6875 8 0.25 10.5 0.375 122.496 72.361 0.406901 2 5.25 10.5 0.375
END
UNIT INCHES KIP
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29732.7
POISSON 0.3
DENSITY 0.000283
ALPHA 1.2e-05
DAMP 0.03
TYPE STEEL
STRENGTH FY 50 FU 65 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
END DEFINE MATERIAL
CONSTANTS
MATERIAL STEEL_36_KSI 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 -40
LOAD 2 LOADTYPE Live  TITLE LOAD CASE 2
JOINT LOAD
2 FY -120
UNIT INCHES KIP
LOAD COMB 3 LRFD
1 1.2 2 1.6 
LOAD COMB 4 ASD
1 1.0 2 1.0 
PERFORM ANALYSIS
LOAD LIST 3
PARAMETER 1
CODE AISC UNIFIED 2005
FU 65 ALL
FYLD 50 ALL
METHOD LRFD
TRACK 2 ALL
CHECK CODE ALL
LOAD LIST 4
PARAMETER 2
CODE AISC UNIFIED 2005
FU 65 ALL
FYLD 50 ALL
METHOD ASD
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 ST   I                        (UPT)
                           PASS   Eq. H1-1a            0.764         3
                      240.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     65.861  L/C  :     3                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      3    Ratio     :     0.764(PASS)                 |
|          Loc  :    0.00   Condition :   Eq. H1-1a                     |
|-----------------------------------------------------------------------|
| DESIGN FORCES                                                         |
| Fx:   2.400E+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:   5.250E+00     Ayy:   2.000E+00     Cw:  1.052E+03              |
| Szz:   3.062E+01     Syy:   1.378E+01                                 |
| Izz:   1.225E+02     Iyy:   7.236E+01     Ix:  4.069E-01              |
|-----------------------------------------------------------------------|
| 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 :  Slender        14.00     N/A     13.49    T.B4.1-3     |
|                Non-Slender    29.00     N/A     35.88    T.B4.1-10    |
| Flexure     :  Non-Compact    14.00     9.15    24.08    T.B4.1-1     |
|                Compact        29.00    90.55   137.27    T.B4.1-9     |
      STAAD PLANE                                              -- 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   4.36E+02   0.000      Eq. D2-1      3     0.00 |
| Rupture     0.00E+00   4.72E+02   0.000      Eq. D2-2      3     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    2.40E+02   3.57E+02   0.672      Sec. E1       3     0.00 |
| Min Buck    2.40E+02   3.14E+02   0.764      Eq. E7-1      3     0.00 |
| Flexural                                                              |
| Tor Buck    2.40E+02   3.21E+02   0.747      Eq. E7-1      3     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    9.69E+00    50.62   4.09E+01   1.12E+02   3.97E+02        |
| Min Buck    9.69E+00    65.86   3.60E+01   6.60E+01   3.49E+02        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    9.69E+00   3.68E+01   3.57E+02                            |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z     0.00E+00   1.42E+02   0.000      Eq. G2-1      3     0.00 |
| Local-Y     0.00E+00   6.00E+01   0.000      Eq. G2-1      3     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     5.25E+00     1.00     1.20    14.00   1.58E+02            |
| Local-Y     2.00E+00     1.00     0.00    29.00   6.00E+01            |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   1.50E+03   0.000      Eq. F2-1      3     0.00 |
| Minor       0.00E+00   9.35E+02   0.000      Eq. F6-1      3     0.00 |
| Intermediate   Mn        My                                           |
| Major       1.67E+03   0.00E+00                                       |
| Minor       1.04E+03   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-LATERAL TORSIONAL BUCKLING                          |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   1.36E+03   0.000      Eq. F2-2      3     0.00 |
| Intermediate   Mn        Me        Cb      Lp       Lr       Lb       |
| Major       1.51E+03   0.00E+00    1.00   115.84   354.09   180.00    |
      STAAD PLANE                                              -- PAGE NO.    5
                       STAAD.PRO CODE CHECKING - (AISC-360-05-LRFD)   v3.3a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-FLANGE LOCAL BUCKLING                               |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   1.33E+03   0.000      Eq. F3-1      3     0.00 |
| Minor       0.00E+00   7.73E+02   0.000      Eq. F6-2      3     0.00 |
| Intermediate   Mn        Fcr                                          |
| Major       1.47E+03   0.00E+00                                       |
| Minor       8.58E+02   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR FLEXURE TENS/COMP INTERACTION                               |
|                       RATIO      CRITERIA    L/C     LOC              |
| Flexure Comp          0.764      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         1.33E+03   0.00E+00   3.14E+02                   |
|                      7.73E+02   0.00E+00   2.40E+02                   |
| Flexure Tens         1.33E+03   0.00E+00   4.36E+02                   |
|                      7.73E+02   0.00E+00   0.00E+00                   |
|-----------------------------------------------------------------------|
    65. LOAD LIST 4
    66. PARAMETER 2
    67. CODE AISC UNIFIED 2005
    68. FU 65 ALL
    69. FYLD 50 ALL
    70. METHOD ASD
    71. TRACK 2 ALL
    72. CHECK CODE ALL
 STEEL DESIGN    
      STAAD PLANE                                              -- 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 ST   I                        (UPT)
                           PASS   Eq. H1-1a            0.765         4
                      160.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     65.861  L/C  :     4                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      4    Ratio     :     0.765(PASS)                 |
|          Loc  :    0.00   Condition :   Eq. H1-1a                     |
|-----------------------------------------------------------------------|
| DESIGN FORCES                                                         |
| Fx:   1.600E+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:   5.250E+00     Ayy:   2.000E+00     Cw:  1.052E+03              |
| Szz:   3.062E+01     Syy:   1.378E+01                                 |
| Izz:   1.225E+02     Iyy:   7.236E+01     Ix:  4.069E-01              |
|-----------------------------------------------------------------------|
| 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 :  Slender        14.00     N/A     13.49    T.B4.1-3     |
|                Non-Slender    29.00     N/A     35.88    T.B4.1-10    |
| Flexure     :  Non-Compact    14.00     9.15    24.08    T.B4.1-1     |
|                Compact        29.00    90.55   137.27    T.B4.1-9     |
      STAAD PLANE                                              -- 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   2.90E+02   0.000      Eq. D2-1      4     0.00 |
| Rupture     0.00E+00   3.15E+02   0.000      Eq. D2-2      4     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    1.60E+02   2.38E+02   0.674      Sec. E1       4     0.00 |
| Min Buck    1.60E+02   2.09E+02   0.765      Eq. E7-1      4     0.00 |
| Flexural                                                              |
| Tor Buck    1.60E+02   2.14E+02   0.749      Eq. E7-1      4     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    9.69E+00    50.62   4.09E+01   1.12E+02   3.97E+02        |
| Min Buck    9.69E+00    65.86   3.60E+01   6.60E+01   3.49E+02        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    9.69E+00   3.68E+01   3.57E+02                            |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z     0.00E+00   9.43E+01   0.000      Eq. G2-1      4     0.00 |
| Local-Y     0.00E+00   4.00E+01   0.000      Eq. G2-1      4     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     5.25E+00     1.00     1.20    14.00   1.58E+02            |
| Local-Y     2.00E+00     1.00     0.00    29.00   6.00E+01            |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   9.97E+02   0.000      Eq. F2-1      4     0.00 |
| Minor       0.00E+00   6.22E+02   0.000      Eq. F6-1      4     0.00 |
| Intermediate   Mn        My                                           |
| Major       1.67E+03   0.00E+00                                       |
| Minor       1.04E+03   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-LATERAL TORSIONAL BUCKLING                          |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   9.02E+02   0.000      Eq. F2-2      4     0.00 |
| Intermediate   Mn        Me        Cb      Lp       Lr       Lb       |
| Major       1.51E+03   0.00E+00    1.00   115.84   354.09   180.00    |
      STAAD PLANE                                              -- PAGE NO.    8
                       STAAD.PRO CODE CHECKING - ( AISC-360-05-ASD)   v3.3a
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-FLANGE LOCAL BUCKLING                               |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   8.82E+02   0.000      Eq. F3-1      4     0.00 |
| Minor       0.00E+00   5.14E+02   0.000      Eq. F6-2      4     0.00 |
| Intermediate   Mn        Fcr                                          |
| Major       1.47E+03   0.00E+00                                       |
| Minor       8.58E+02   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR FLEXURE TENS/COMP INTERACTION                               |
|                       RATIO      CRITERIA    L/C     LOC              |
| Flexure Comp          0.765      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         8.82E+02   0.00E+00   2.09E+02                   |
|                      5.14E+02   0.00E+00   1.60E+02                   |
| Flexure Tens         8.82E+02   0.00E+00   2.90E+02                   |
|                      5.14E+02   0.00E+00   0.00E+00                   |
|-----------------------------------------------------------------------|