 # V. AISC LRFD - Wide Flange Compression Capacity 4

To determine the design bending strength of a non-compact American wide flange section bent about its strong axis.

## Reference

AISC Load Factor Resistance Design, 3rd Edition, Example 5.2, case (c), page 5-16.

## Details

W21x48 shape

 Fy = 50 ksi

 Length = 40 ft

 Lb = 480 in. (braced at end points only)

## Validation

The inelastic limiting unbraced length is given by:

 $L r = r y X 1 F L 1 + 1 + X 2 F L 2$ (F1-6)
where
 FL = Fyf - Fr = 50 ksi - 10 ksi = 40 ksi X1 = $π S x E G J A 2 = π 93.0 29,000 ( 11,200 ) ( 0.803 ) ( 14.1 ) 2 = 1,449$ X2 = $4 C w I y ( S x G J ) 2 = 4 ( 3,950 38.7 ) [ 93.0 11,200 ( 0.803 ) ] 2 = 0.0437$

Lb > Lr, thus the section is subject to elastic lateral-torsional buckling.

The modification factor, Cb is given by:
 $C b = 12.5 M max 2.5 M max + 3 M A + 4 M B + 3 M C$ (F1-3)

Since the moment diagram is symmetric about the mid-span point of the beam, the maximum moment is the same as the mid-span moment and the quarter-point moments are equal.

$M max = M B = w l 2 8$

The quarter-point moments of a uniformly loaded beam are likewise equal:

$M A = M C = w ( l / 4 ) 2 ( l - l 4 ) = 3 w l 2 32$

The $w l 2$ terms factor out and the constants simply as follows:

$C b = 12.5 ( 1/8 ) 2.5 ( 1/8 ) + 3 ( 3/32 ) + 4 ( 1/8 ) + 3 ( 3/32 ) = 12.5 ( 1/8 ) 6.5 ( 1/8 ) + 6 ( 3/32 ) = 1.14$

The critical buckling moment is given by:

 $M cr = C b S x X 1 2 L b / r y 1 + X 1 2 X 2 2 ( L b / r y ) 2$ (F1-13)

The lateral-torsional buckling limit state capacity:

ϕbMn = 0.9×(935.0) = 841.5 in·k

## Results

Table 1. Comparison of results
Modification factor, Cb 1.14 1.14 none
Design flexural strength, ϕbMn (in·kips) 841.5 834.4 <1%

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\AISC LRFD - Wide Flange Compression Capacity 4.STD is typically installed with the program.

STAAD SPACE BENDING CAPACITY PER AISC LRFD 3RD ED
START JOB INFORMATION
ENGINEER DATE 23-Sep-18
END JOB INFORMATION
*
*  EXAMPLE PROBLEM 5.2, CASE (C), PAGE 5-16, AISC LRFD 3RD ED.
*  CAPACITY (MNZ) SHOULD BE ABOUT 70.2 KIP-FT
*
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 40 0 0;
*
MEMBER INCIDENCES
1 1 2;
*
MEMBER PROPERTY AMERICAN
1 TABLE ST W21X48
*
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 4.176e+06
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
*
SUPPORTS
1 PINNED
2 FIXED BUT MZ
*
1 CON GY -10 13.333
1 CON GY -10 26.667
*
PERFORM ANALYSIS
*
PARAMETER 1
CODE LRFD
MAIN 1 ALL
FYLD 7200 ALL
CB 0 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH


                       STAAD.PRO CODE CHECKING - (LRFD 3RD EDITION)   v1.0
********************************************
|--------------------------------------------------------------------------|
|                                                    Y        PROPERTIES   |
|*************                                       |        IN INCH UNIT |
|            *  |=============================|   ===|===     ------------ |
|MEMBER   1  *  |  AISC SECTIONS              |      |        AX=0.1410E+2 |
|            *  | ST  W21X48                  |      |   --Z  AY=0.7210E+1 |
|DESIGN CODE *  |                             |      |        AZ=0.4667E+1 |
| LRFD 2001  *  ===============================   ===|===     PY=0.1490E+2 |
|            *                                                PZ=0.1070E+3 |
|            *  |&lt;---LENGTH (FT)=   40.00 --->|               RY=0.1657E+1 |
|*************                                                RZ=0.8247E+1 |
|                                                                          |
|                133.3 (KIP-FEET)                                          |
|PARAMETER        |               L1  L1  L1                  CAPACITIES   |
|IN KIP  INCH     |                                           IN KIP  INCH |
|---------------  +           L1              L1              -------------|
| KL/R-Y= 289.73  |                                           PNC=0.3584E+2|
| KL/R-Z=  58.20  +       L1                      L1          pnc=0.0000E+0|
| UNL   = 480.00  |                                           PNT=0.6345E+3|
| CB    =   1.14  +                                           pnt=0.0000E+0|
| PHIC  =   0.85  |   L1                              L1      MNZ=0.8344E+3|
| PHIB  =   0.90  +                                           mnz=0.1600E+4|
| FYLD  =  50.00  |L0                                     L0  MNY=0.6367E+3|
| NSF   =   1.00  +---+---+---+---+---+---+---+---+---+---|   mny=0.0000E+0|
| DFF =     0.00  -7.4                                        VN =0.1947E+3|
| dff =     0.00               ABSOLUTE MZ ENVELOPE           vn =0.1000E+2|
|                                                                          |
|                      MAX FORCE/ MOMENT SUMMARY  (KIP-FEET)               |
|                      -------------------------                           |
|                                                                          |
|                AXIAL      SHEAR-Y    SHEAR-Z    MOMENT-Y    MOMENT-Z     |
|                                                                          |
|      VALUE       0.0        10.0        0.0         0.0       133.3      |
|   LOCATION       0.0         0.0        0.0         0.0        13.3      |
|                                                                          |
|**************************************************************************|
|*                                                                        *|
|*                         DESIGN SUMMARY  (KIP-FEET)                     *|
|*                         --------------                                 *|
|*                                                                        *|