V. AISC 360-10 Built Up I E 12

Verify the compressive strength of a built-up I section with different flange widths per the LRFD and ADS methods of the AISC 360-10 codes.

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.12, pp. E-57 - E-62
Seaburg, Paul A. and Charles J. Carter. 1997. Design Guide 9: Torsional Analysis of Structural Steel members. AISC: Chicago, IL.

Details

From reference (2):

Compute the available strength of a built-up compression member with a length of 14 ft. The ends are pinned. The outside flange is PL3/4-in.×5-in., the inside flange is PL3/4-in.×8-in., and the web is PL3/8-in.×10 1/2-in. Material is ASTM A572 Grade 50.

Validation

Material Properties

F_y = 50 ksi

F_u = 65 ksi

Section Properties

Built-up section properties (ignoring fillet welds):

A_{g} = 8.0 (0.75) + 10.5 (0.375) + 5.0 (0.75) = 13.7 {in}^{2}

\bar{y} = \frac{Σ A_{i} y_{i}}{Σ A_{i}} = \frac{6.0 (11.6) + 3.94 (6.0) + 3.75 (0.375)}{6.0 + 3.94 + 3.75} = 6.91 in

where $\bar{y}$ is measured from the bottom of the outside flange.

I_{x} = [8.0 (0.75) {(4.72)}^{2} + \frac{8.0 {(0.75)}^{3}}{12}] + [0.375 (10.5) {(0.910)}^{2} + \frac{0.375 {(10.5)}^{3}}{12}] + [5.0 (0.75) {(6.54)}^{2} + \frac{5.0 {(0.75)}^{3}}{12}] = 334 {in}^{4}

I_{y} = \frac{0.75 \times {8.0}^{3}}{12} + \frac{10.5 \times {0.375}^{3}}{12} + \frac{0.75 \times {5.0}^{3}}{12} = 39.9 {in}^{4}

r_{x} = \sqrt{\frac{I_{x}}{A}} = \sqrt{\frac{334}{13.7}} = 4.94 in

r_{y} = \sqrt{\frac{I_{y}}{A}} = \sqrt{\frac{39.9}{13.7}} = 1.71 in

From Eq. 3.4 in Reference 3:

J = Σ \frac{b t^{3}}{3} = \frac{8.0 {(0.75)}^{3} + 10.5 {(0.375)}^{3} + 5.0 {(0.75)}^{3}}{3} = 2.01 {in}^{4}

Distance between flange centroids:

h_{0} = 12.0 - \frac{0.75}{2} - \frac{0.75}{2} = 11.3 in

C_{w} = \frac{t_{f} {h_{o}}^{2}}{12} (\frac{{b_{1}}^{3} {b_{2}}^{3}}{{b_{1}}^{3} + {b_{2}}^{3}}) = \frac{0.75 {(11.3)}^{2}}{12} [\frac{{(8.0)}^{3} {(5.0)}^{3}}{{(8.0)}^{3} + {(5.0)}^{3}}] = 802 {in}^{6}

Due to symmetry about the y-axis, the shear center lies on the y axis (x₀ = 0). The distance from the centroid to the shear center can be calculated by finding the shear center from outside flange plus one-half the outside flange and then subtracting the centroid distance from that same reference point.

y_{0} = (e + \frac{t_{f}}{2}) - \bar{y} = h_{0} [\frac{{b_{1}}^{3}}{{b_{1}}^{3} + {b_{2}}^{3}}] + \frac{t_{f}}{2} - \bar{y} = 11.3 [\frac{{(8.0)}^{3}}{{(8.0)}^{3} + {(5.0)}^{3}}] + \frac{0.75}{2} - 6.91 = 2.55 in

{\bar{r}}_{0}^{2} = {x_{0}}^{2} + {y_{0}}^{2} + \frac{I_{x} + I_{y}}{A_{g}} = 0 + {(2.55)}^{2} + \frac{344 + 39.9}{13.7} = 33.8 {in}^{2}

H = 1 - \frac{{x_{0}}^{2} + {y_{0}}^{2}}{{\bar{r}}_{0}^{2}} = 1 - \frac{0 + {(2.55)}^{2}}{33.8} = 0.808

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 (14.0) (12)}{1.71} = 98.2

F_{e y} = \frac{π^{2} E}{{(K L / r_{y})}^{2}} = \frac{π^{2} 29, 000}{{(98.2)}^{2}} = 29.7 ksi

(Eq. E4-8)

F_{e z} = [\frac{π^{2} E C_{w}}{{(K_{z} L)}^{2}} + G J] (\frac{1}{A_{g} {\bar{r}}_{0}^{2}}) = [\frac{π^{2} (29, 000) (802)}{{(1.0 \times 14.0 \times 12}^{2}} + 11, 200 (2.01)] (\frac{1}{13.7 \times 33.8}) = 66.2 ksi

(Eq. E4-9)

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}}}] = (\frac{29.7 + 66.2}{2 \times 0.808}) [1 - \sqrt{1 - \frac{4 (29.7) (66.2) (0.808)}{{(29.7 + 66.2)}^{2}}}] = 26.4 ksi

(Eq. E4-5)

Nominal Compressive Strength

\frac{F_{y}}{2.25} = \frac{50}{2.25} = 22.2 ksi < 26.4 ksi

F_{c r} = ({0.658}^{F_{y} / F_{e}}) F_{y} = [{0.658}^{(50) / (26.4)}] 50 = 22.6 ksi

(Eq E7-2)

P_{n} = F_{c r} A_{g} = 22.6 (13.7) = 310 kips

(Eq E7-1)

The available compressive strength:

LRFD: $ϕ_{c} P_{n} = 0.90 (310) = 279 kips$

ASD: $P_{n} / Ω_{c} = \frac{310}{1.67} = 186 kips$

Results

Result Type	Reference	STAAD.Pro	Difference	Comments
ϕ_cP_n (kips) [LRFD]	279	278	negligible
P_n/Ω_c (kips) [ASD]	186	185	negligible

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 Built Up I E 12.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 06-Feb-19
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 0 14 0;
MEMBER INCIDENCES
1 1 2;
START USER TABLE
TABLE 1
UNIT INCHES KIP
WIDE FLANGE
I
13.6875 12 0.375 8 0.75 333.426 39.8586 2.0127 4.5 6.5 5 0.75
END
UNIT FEET KIP
DEFINE MATERIAL START
ISOTROPIC STEEL
E 4.176e+06
POISSON 0.3
DENSITY 0.489024
ALPHA 6.5e-06
DAMP 0.03
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
ISOTROPIC STEEL_36_KSI
E 4.176e+06
POISSON 0.3
DENSITY 0.489024
ALPHA 6.5e-06
DAMP 0.03
TYPE STEEL
STRENGTH FY 5184 FU 8352 RY 1.5 RT 1.2
END DEFINE MATERIAL
UNIT INCHES KIP
MEMBER PROPERTY AMERICAN
1 UPTABLE 1 I
CONSTANTS
MATERIAL STEEL_36_KSI ALL
UNIT FEET KIP
SUPPORTS
1 PINNED
2 FIXED BUT FY MX MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
2 FY -20
LOAD 2 LOADTYPE None  TITLE LOAD CASE 2
JOINT LOAD
2 FY -40
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
UNIT INCHES KIP
LOAD LIST 3
PARAMETER 1
CODE AISC UNIFIED 2010
METHOD LRFD
STP 2 ALL
FU 65 ALL
FYLD 50 ALL
TRACK 2 ALL
CHECK CODE ALL
LOAD LIST 4
PARAMETER 2
CODE AISC UNIFIED 2010
METHOD ASD
STP 2 ALL
FU 65 ALL
FYLD 50 ALL
TRACK 2 ALL
CHECK CODE ALL
PRINT MEMBER PROPERTIES
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 ST   I                        (UPT)
                           PASS   Eq. H1-1a            0.316         3
                       88.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     98.449  L/C  :     3                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      3    Ratio     :     0.316(PASS)                 |
|          Loc  :    0.00   Condition :   Eq. H1-1a                     |
|-----------------------------------------------------------------------|
| DESIGN FORCES                                                         |
| Fx:   8.800E+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:   6.500E+00     Ayy:   4.500E+00     Cw:  7.947E+02              |
| Szz:   4.815E+01     Syy:   9.965E+00                                 |
| Izz:   3.334E+02     Iyy:   3.986E+01     Ix:  2.013E+00              |
|-----------------------------------------------------------------------|
| MATERIAL PROPERTIES                                                   |
| Fyld:          50.000   Fu:          65.000                           |
|-----------------------------------------------------------------------|
| Actual Member Length:       168.000                                   |
| 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     5.33     N/A     13.40    T.B4.1(a)-2  |
|                Non-Slender    28.00     N/A     35.88    T.B4.1(a)-5  |
| Flexure     :  Compact         5.33     9.15    23.84    T.B4.1(b)-11 |
|                Compact        23.17   127.82   137.27    T.B4.1(b)-16 |
      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   6.16E+02   0.000      Eq. D2-1      3     0.00 |
| Rupture     0.00E+00   6.67E+02   0.000      Eq. D2-2      3     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    8.80E+01   5.66E+02   0.156      Eq. E3-1      3     0.00 |
| Min Buck    8.80E+01   3.03E+02   0.290      Eq. E3-1      3     0.00 |
| Flexural                                                              |
| Tor Buck    8.80E+01   2.78E+02   0.316      Eq. E4-1      3     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    1.37E+01    34.04   4.59E+01   2.47E+02   6.29E+02        |
| Min Buck    1.37E+01    98.45   2.46E+01   2.95E+01   3.37E+02        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    1.37E+01   2.26E+01   3.09E+02                            |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z     0.00E+00   1.76E+02   0.000      Eq. G2-1      3     0.00 |
| Local-Y     0.00E+00   1.22E+02   0.000      Eq. G2-1      3     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     6.50E+00     1.00     1.20    10.67   1.95E+02            |
| Local-Y     4.50E+00     1.00     5.00    28.00   1.35E+02            |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   2.78E+03   0.000      Sec. F4.2     3     0.00 |
| Minor       0.00E+00   7.17E+02   0.000      Eq. F6-1      3     0.00 |
| Intermediate   Mn        My                                           |
| Major       3.09E+03   0.00E+00                                       |
| Minor       7.97E+02   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-LATERAL TORSIONAL BUCKLING                          |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   2.30E+03   0.000      Eq. F4-2      3     0.00 |
| Intermediate   Mn        Me        Cb      Lp       Lr       Lb       |
| Major       2.56E+03   0.00E+00    1.00    60.83   343.65   168.00    |
      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 BENDING-COMPRESSION FLANGE YIELDING                         |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
|             0.00E+00   2.78E+03   0.000      Eq. F4-1      3     0.00 |
| Intermediate   Mn        Rpc                                          |
|             3.09E+03   1.28E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR FLEXURE TENS/COMP INTERACTION                               |
|                       RATIO      CRITERIA    L/C     LOC              |
| Flexure Comp          0.316      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         2.30E+03   0.00E+00   2.78E+02                   |
|                      7.17E+02   0.00E+00   8.80E+01                   |
| Flexure Tens         2.30E+03   0.00E+00   6.16E+02                   |
|                      7.17E+02   0.00E+00   0.00E+00                   |
|-----------------------------------------------------------------------|
    67. LOAD LIST 4
    68. PARAMETER 2
    69. CODE AISC UNIFIED 2010
    70. METHOD ASD
    71. STP 2 ALL
    72. FU 65 ALL
    73. FYLD 50 ALL
    74. TRACK 2 ALL
    75. 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 ST   I                        (UPT)
                           PASS   Eq. H1-1a            0.324         4
                       60.00 C          0.00           0.00        0.00
|-----------------------------------------------------------------------|
| SLENDERNESS                                                           |
| Actual Slenderness Ratio    :     98.449  L/C  :     4                |
| Allowable Slenderness Ratio :    200.000  LOC  :     0.00             |
|-----------------------------------------------------------------------|
| STRENGTH CHECKS                                                       |
| Critical L/C  :      4    Ratio     :     0.324(PASS)                 |
|          Loc  :    0.00   Condition :   Eq. H1-1a                     |
|-----------------------------------------------------------------------|
| 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:   6.500E+00     Ayy:   4.500E+00     Cw:  7.947E+02              |
| Szz:   4.815E+01     Syy:   9.965E+00                                 |
| Izz:   3.334E+02     Iyy:   3.986E+01     Ix:  2.013E+00              |
|-----------------------------------------------------------------------|
| MATERIAL PROPERTIES                                                   |
| Fyld:          50.000   Fu:          65.000                           |
|-----------------------------------------------------------------------|
| Actual Member Length:       168.000                                   |
| 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     5.33     N/A     13.40    T.B4.1(a)-2  |
|                Non-Slender    28.00     N/A     35.88    T.B4.1(a)-5  |
| Flexure     :  Compact         5.33     9.15    23.84    T.B4.1(b)-11 |
|                Compact        23.17   127.82   137.27    T.B4.1(b)-16 |
      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   4.10E+02   0.000      Eq. D2-1      4     0.00 |
| Rupture     0.00E+00   4.45E+02   0.000      Eq. D2-2      4     0.00 |
|-----------------------------------------------------------------------|
| CHECK FOR AXIAL COMPRESSION                                           |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Maj Buck    6.00E+01   3.77E+02   0.159      Eq. E3-1      4     0.00 |
| Min Buck    6.00E+01   2.02E+02   0.297      Eq. E3-1      4     0.00 |
| Flexural                                                              |
| Tor Buck    6.00E+01   1.85E+02   0.324      Eq. E4-1      4     0.00 |
| Intermediate                                                          |
| Results     Eff Area     KL/r      Fcr        Fe         Pn           |
| Maj Buck    1.37E+01    34.04   4.59E+01   2.47E+02   6.29E+02        |
| Min Buck    1.37E+01    98.45   2.46E+01   2.95E+01   3.37E+02        |
| Flexural       Ag       Fcr        Pn                                 |
| Tor Buck    1.37E+01   2.26E+01   3.09E+02                            |
|-----------------------------------------------------------------------|
| CHECK FOR SHEAR                                                       |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Local-Z     0.00E+00   1.17E+02   0.000      Eq. G2-1      4     0.00 |
| Local-Y     0.00E+00   8.08E+01   0.000      Eq. G2-1      4     0.00 |
| Intermediate                                                          |
| Results        Aw        Cv       Kv      h/tw      Vn                |
| Local-Z     6.50E+00     1.00     1.20    10.67   1.95E+02            |
| Local-Y     4.50E+00     1.00     5.00    28.00   1.35E+02            |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-YIELDING                                            |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   1.85E+03   0.000      Sec. F4.2     4     0.00 |
| Minor       0.00E+00   4.77E+02   0.000      Eq. F6-1      4     0.00 |
| Intermediate   Mn        My                                           |
| Major       3.09E+03   0.00E+00                                       |
| Minor       7.97E+02   0.00E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR BENDING-LATERAL TORSIONAL BUCKLING                          |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
| Major       0.00E+00   1.53E+03   0.000      Eq. F4-2      4     0.00 |
| Intermediate   Mn        Me        Cb      Lp       Lr       Lb       |
| Major       2.56E+03   0.00E+00    1.00    60.83   343.65   168.00    |
      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 BENDING-COMPRESSION FLANGE YIELDING                         |
|                                                                       |
|              FORCE     CAPACITY   RATIO      CRITERIA     L/C    LOC  |
|             0.00E+00   1.85E+03   0.000      Eq. F4-1      4     0.00 |
| Intermediate   Mn        Rpc                                          |
|             3.09E+03   1.28E+00                                       |
|-----------------------------------------------------------------------|
| CHECK FOR FLEXURE TENS/COMP INTERACTION                               |
|                       RATIO      CRITERIA    L/C     LOC              |
| Flexure Comp          0.324      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         1.53E+03   0.00E+00   1.85E+02                   |
|                      4.77E+02   0.00E+00   6.00E+01                   |
| Flexure Tens         1.53E+03   0.00E+00   4.10E+02                   |
|                      4.77E+02   0.00E+00   0.00E+00                   |
|-----------------------------------------------------------------------|