V. AISC LRFD - Wide Flange Flexural Strength 3

To determine the design flexural strength of an ASTM A992 W18X40 bent about its strong axis on a 35-ft span with a uniformly distributed load.

Reference

Hand calculation using the following reference:

AISC Load Factor Resistance Design, 3rd Edition, Example 5.1, Case (c), page 5-12.

Details

The beam is braced at the ends only and so, L_b = 420 in.

F_y = 50 ksi

Length = 35 ft

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	=	$\frac{π}{S_{x}} \sqrt{\frac{E G J A}{2}} = \frac{π}{68.4} \sqrt{\frac{29,000 (11,200) (0.810) (11.85)}{2}} = 1,813$
X₂	=	$4 \frac{C_{w}}{I_{y}} {(\frac{S_{x}}{G J})}^{2} = 4 (\frac{1,440}{19.1}) {[\frac{68.4}{11,200 (0.810)}]}^{2} = 0.0171$

L_{r} = \frac{1.27 \times 1,813}{40} \sqrt{1 + \sqrt{1 + 0.0171 {(40)}^{2}}} = 144.9 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 (68.4) (1,813) \sqrt{2}}{420 / 1.27} \sqrt{1 + \frac{{(1,813)}^{2} (0.0171)}{2 {(420 / 1.27)}^{2}}} = 675.6 in·k

The lateral-torsional buckling limit state capacity:

ϕ_bM_n = 0.9×(675.6) = 608.1 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)	608.1	607.2	negligible

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\AISC LRFD - Wide Flange Flexural Strength 3.STD is typically installed with the program.

The input parameter CB 0 directs the program to calculate the value of the modification factor.

STAAD SPACE BENDING CAPACITY PER AISC LRFD 3RD ED
START JOB INFORMATION
ENGINEER DATE 23-Sep-18
END JOB INFORMATION
*
*  EXAMPLE PROBLEM 5.1, CASE (C), PAGE 5-12, AISC LRFD 3RD ED.
*  CAPACITY (MNZ) SHOULD BE ABOUT 50.8 KIP-FT
*
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 35 0 0;
MEMBER INCIDENCES
1 1 2;
MEMBER PROPERTY AMERICAN
1 TABLE ST W18X40
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 UNI GY -0.3
PERFORM ANALYSIS
PARAMETER 1
CODE LRFD
MAIN 1 ALL
CB 0 ALL
FYLD 7200 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD Output

                       STAAD.PRO CODE CHECKING - (LRFD 3RD EDITION)   v1.0
                       ********************************************
  |--------------------------------------------------------------------------|
  |                                                    Y        PROPERTIES   |
  |*************                                       |        IN INCH UNIT |
  |            *  |=============================|   ===|===     ------------ |
  |MEMBER   1  *  |  AISC SECTIONS              |      |        AX=0.1180E+2 |
  |            *  | ST  W18X40                  |      |   --Z  AY=0.5638E+1 |
  |DESIGN CODE *  |                             |      |        AZ=0.4214E+1 |
  | LRFD 2001  *  ===============================   ===|===     PY=0.1000E+2 |
  |            *                                                PZ=0.7840E+2 |
  |            *  |&lt;---LENGTH (FT)=   35.00 --->|               RY=0.1272E+1 |
  |*************                                                RZ=0.7202E+1 |
  |                                                                          |
  |                 45.9 (KIP-FEET)                                          |
  |PARAMETER        |               L1  L1  L1                  CAPACITIES   |
  |IN KIP  INCH     |                                           IN KIP  INCH |
  |---------------  +           L1              L1              -------------|
  | KL/R-Y= 330.12  |                                           PNC=0.2310E+2|
  | KL/R-Z=  58.32  +       L1                      L1          pnc=0.0000E+0|
  | UNL   = 420.00  |                                           PNT=0.5310E+3|
  | CB    =   1.14  +   L1                              L1      pnt=0.0000E+0|
  | PHIC  =   0.85  |                                           MNZ=0.6072E+3|
  | PHIB  =   0.90  +                                           mnz=0.5512E+3|
  | FYLD  =  50.00  |L0                                     L0  MNY=0.4283E+3|
  | NSF   =   1.00  +---+---+---+---+---+---+---+---+---+---|   mny=0.0000E+0|
  | DFF =     0.00  -2.6                                        VN =0.1522E+3|
  | dff =     0.00               ABSOLUTE MZ ENVELOPE           vn =0.0000E+0|
  |                                 (WITH LOAD NO.)                          |
  |                                                                          |
  |                      MAX FORCE/ MOMENT SUMMARY  (KIP-FEET)               |
  |                      -------------------------                           |
  |                                                                          |
  |                AXIAL      SHEAR-Y    SHEAR-Z    MOMENT-Y    MOMENT-Z     |
  |                                                                          |
  |      VALUE       0.0         5.2        0.0         0.0        45.9      |
  |   LOCATION       0.0         0.0        0.0         0.0        17.5      |
  |    LOADING         0           1          0           0           1      |
  |                                                                          |
  |**************************************************************************|
  |*                                                                        *|
  |*                         DESIGN SUMMARY  (KIP-FEET)                     *|
  |*                         --------------                                 *|
  |*                                                                        *|
  |*       RESULT/     CRITICAL COND/     RATIO/         LOADING/           *|
  |         FX              MY             MZ           LOCATION             |
  |        ======================================================            |
  |          PASS        LRFD-H1-1B-C     0.908              1               |
  |          0.00 C          0.00       -45.94            17.50              |
  |*                                                                        *|
  |**************************************************************************|
  |                                                                          |
  |--------------------------------------------------------------------------|