V. AISC LRFD - Non Compact Wide Flange 2

To determine the design bending strength of a non-compact American wide flange section bent about its strong axis. The deflection at mid-span due to two 10 kip concentrated loads acting at the third points along the beam is also to be calculated.

Reference

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

Details

W21x48 shape

F_y = 50 ksi

Length = 40 ft

L_b = 60 in.

Validation

Evaluate section for lateral-torsional buckling:

The plastic limiting unbraced length is given by:

L_{p} = 1.76 r_{y} \sqrt{\frac{E}{F_{y}}} = 1.76 (1.66) \sqrt{\frac{29,000}{50}} = 70.4 in.

L_b < L_p; thus, the member is plastic.

Check flange slenderness:

λ_{p} = 0.38 \sqrt{E / F_{y}} = 0.38 \sqrt{\frac{29,000}{50}} = 9.152 < \frac{b}{2 t_{f}} = 9.47

F_L = 50 ksi - 10 ksi = 40 ksi

λ_{r} = 0.83 \sqrt{E / F_{L}} = 0.83 \sqrt{\frac{29,000}{40}} = 22.35 > \frac{b}{2 t_{f}} = 9.47

M_r = F_LS_x = 40 × 93.0 = 3,720 in·k

(F1-7)

Flanges are non-compact. The nominal moment capacity for the limit state of flange local buckling is:

M_{n} = M_{p} - (M_{p} - M_{r}) (\frac{λ - λ_{p}}{λ_{r} - λ_{p}})

(A-F1-3)

M_p = F_yZ_x = 50 × 107 = 5,350 in·k

M_{n} = 5,350 - (5,350 - 3,720) (\frac{9.47 - 9.152}{22.35 - 9.152}) = 5,311 in·k

The flange local buckling limit state capacity:

ϕ_bM_n = 0.9×(5,311) = 4,780 in·k

The ultimate moment is:

M_{u} = P_{u} (l / 3) = 10 (\frac{30 \times 12}{3}) = 1,600 in-k

The capacity ratio is: 1,600 / 4,780 = 0.335

The deflection at mid-span for a beam loaded at third points:

Δ_{\max} = \frac{P l^{3}}{28 E I} = \frac{10 \times {(40 \times 12)}^{3}}{28 \times 29,000 \times 959} = 1.42 in.

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
Design flexural strength, ϕ_bM_n (in·kips)	4,780	4,780.24	negligible
Capacity ratio	0.335	0.335	none
Deflection (in)	1.42	1.4114	negligible

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\AISC LRFD - Non Compact Wide Flange 2.STD is typically installed with the program.

The UNT 5 parameter specifies an unbraced length of 5'.

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 (A), PAGE 5-16, AISC LRFD 3RD ED.
*  CAPACITY (MNZ) SHOULD BE ABOUT 398 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
*
SECTION 0.5 ALL
PRINT SECTION DISPL
*
PARAMETER 1
CODE LRFD
MAIN 1 ALL
FYLD 7200 ALL
UNT 5 ALL
TRACK 1 ALL
CHECK CODE ALL
FINISH

STAAD Output

     MEMBER SECTION DISPLACEMENTS
     ----------------------------
     UNITS ARE - INCH
     MEMB  LOAD     GLOBAL X,Y,Z DISPL FROM START TO END JOINTS AT 1/12TH PTS
       1     1       0.0000    0.0000    0.0000    0.0000   -0.3644    0.0000
                     0.0000   -0.7057    0.0000    0.0000   -1.0010    0.0000
                     0.0000   -1.2273    0.0000    0.0000   -1.3654    0.0000
                     0.0000   -1.4114    0.0000    0.0000   -1.3654    0.0000
                     0.0000   -1.2273    0.0000    0.0000   -1.0010    0.0000
                     0.0000   -0.7057    0.0000    0.0000   -0.3644    0.0000
                     0.0000    0.0000    0.0000
     MAX LOCAL  DISP =    1.41141   AT     240.00  LOAD    1   L/DISP=    340
         ************ END OF SECT DISPL RESULTS ***********
    40. *
    41. PARAMETER 1
    42. CODE LRFD
    43. MAIN 1 ALL
    44. FYLD 7200 ALL
    45. UNT 5 ALL
    46. TRACK 1 ALL
    47. CHECK CODE ALL
 STEEL DESIGN    
      BENDING CAPACITY PER AISC LRFD 3RD ED                    -- PAGE NO.    4
                         STAAD.Pro CODE CHECKING - (LRFD 3RD EDITION)   v1.0
                         ***********************
 ALL UNITS ARE - KIP  FEET (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
      1  ST   W21X48                   (AISC SECTIONS)
                           PASS     LRFD-H1-1B-C       0.335         1
                        0.00 C          0.00        -133.33       13.33
 +---------------------------------------------------------------------+
 | DESIGN STRENGTHS FOR MEMBER     1       UNITS - KIP IN              |
 | PNC=   35.84  PNT=  634.50  MNZ= 4780.24  MNY=  636.75  VN=  194.67 |
 +---------------------------------------------------------------------+