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

F_y = 50 ksi

Length = 40 ft

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

Validation

The inelastic limiting unbraced length is given by:

L_{r} = \frac{r_{y} X_{1}}{F_{L}} \sqrt{1 + \sqrt{1 + X_{2} {F_{L}}^{2}}}

(F1-6)

where

F_L	=	F_yf - F_r = 50 ksi - 10 ksi = 40 ksi
X₁	=	$\frac{π}{S_{x}} \sqrt{\frac{E G J A}{2}} = \frac{π}{93.0} \sqrt{\frac{29,000 (11,200) (0.803) (14.1)}{2}} = 1,449$
X₂	=	$4 \frac{C_{w}}{I_{y}} {(\frac{S_{x}}{G J})}^{2} = 4 (\frac{3,950}{38.7}) {[\frac{93.0}{11,200 (0.803)}]}^{2} = 0.0437$

L_{r} = \frac{1.66 \times 1,449}{40} \sqrt{1 + \sqrt{1 + 0.0437 {(40)}^{2}}} = 184.6 in.

L_b > L_r, thus the section is subject to elastic lateral-torsional buckling.

The modification factor, C_b is given by:

C_{b} = \frac{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} = \frac{w l^{2}}{8}

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

M_{A} = M_{C} = \frac{w (l / 4)}{2} (l - \frac{l}{4}) = \frac{3 w l^{2}}{32}

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

C_{b} = \frac{12.5 (1/8)}{2.5 (1/8) + 3 (3/32) + 4 (1/8) + 3 (3/32)} = \frac{12.5 (1/8)}{6.5 (1/8) + 6 (3/32)} = 1.14

The critical buckling moment is given by:

M_{cr} = \frac{C_{b} S_{x} X_{1} \sqrt{2}}{L_{b} / r_{y}} \sqrt{1 + \frac{{X_{1}}^{2} X_{2}}{2 {(L_{b} / r_{y})}^{2}}}

(F1-13)

M_{cr} = \frac{1.14 (93.0) (1,449) \sqrt{2}}{480 / 1.66} \sqrt{1 + \frac{{(1,449)}^{2} (0.0437)}{2 {(480 / 1.66)}^{2}}} = 935.0 in·k

The lateral-torsional buckling limit state capacity:

ϕ_bM_n = 0.9×(935.0) = 841.5 in·k

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference	Comments
Modification factor, C_b	1.14	1.14	none
Design flexural strength, ϕ_bM_n (in·kips)	841.5	834.4	<1%

STAAD.Pro Input

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
*
LOAD 1
MEMBER LOAD
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 Output

                       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|
  |                                 (WITH LOAD NO.)                          |
  |                                                                          |
  |                      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      |
  |    LOADING         0           1          0           0           1      |
  |                                                                          |
  |**************************************************************************|
  |*                                                                        *|
  |*                         DESIGN SUMMARY  (KIP-FEET)                     *|
  |*                         --------------                                 *|
  |*                                                                        *|
  |*       RESULT/     CRITICAL COND/     RATIO/         LOADING/           *|
  |         FX              MY             MZ           LOCATION             |
  |        ======================================================            |
  |          FAIL        LRFD-H1-1B-C     1.917              1               |
  |          0.00 C          0.00      -133.33            13.33              |
  |*                                                                        *|
  |**************************************************************************|
  |                                                                          |
  |--------------------------------------------------------------------------|