V. AISC LRFD - Load Capacity of 3 Wide Flange Beams

To find the design flexural strength capacity of 3 beams of different spans using the data in Table 5-4 of AISC LRFD 3rd edition.

Reference

AISC Load Factor Resistance Design, 3rd Edition, Example 5.5, page 5-21.

Details

W21x62 section.

F_y = 50 ksi

40 ft, 30 ft, and 8 ft spans are investigated. All beams are fully braced.

Validation

Bending Capacity

The beam is fully braced so it can reach full plastic capacity:

ϕ M_{n} = 0.9 \times 50 \times 144 = 6,480 in-k

The total load capacity on the beam is then calculated by setting the ultimate capacity equal to the load, $ϕ M_{n} = M_{u} = W \times l / 8$ and then solving for the total load:

W_{1} = 8 (ϕ M_{n} / l) = 8 (\frac{6,480}{40 \times 12}) = 108 kips

W_{2} = 8 (ϕ M_{n} / l) = 8 (\frac{6,480}{30 \times 12}) = 144 kips

Shear Capacity

Check for web slenderness:

\frac{h}{t_{w}} = 46.9 < 2.45 \sqrt{E / F_{yw}} = 2.45 \sqrt{29,000 / 50} = 59.0

The nominal shear capacity of a W21x62 section:

V_{n} = 0.6 F_{yw} A_{w} = 0.6 \times 50 \times 0.4 \times 21.0 = 252 kips

ϕ V_{n} = 0.9 \times 252 = 226.8 kips

The total load capacity on the beam is then calculated by setting the ultimate capacity equal to the load, $ϕ V_{n} = V_{u} = W / 2$ and then solving for the total load:

W_{3} = 2 (ϕ V_{n}) = 2 (226.8) = 453.6 kips

This governs the total load capacity for the short span (8') case.

Using the total weight values from the reference, the following distributed loads are calculated:

Member	W (kips)	l (ft)	w (kips/ft)
1	108	40	2.7
2	144	30	4.8
3	454	8	56.75

Results

There is no direct comparison of the results from STAAD.Pro with the Reference, as the program does not directly compute the total member load. However, when the total member load is applied as a distributed load (see previous table), the Utilization ratio will then nominally be unity (1.0) for a beam loaded at its capacity.

Table 1. Comparison of results
Result Type		Reference	STAAD.Pro	Difference
Beam #1: 30 ft span	Bending capacity, ϕM_n (in·k)	6,480	6,480	none
Beam #1: 30 ft span	Utilization ratio	1.0	1.0	none
Beam #2: 20 ft span	Bending capacity, ϕM_n (in·k)	6,480	6,480	none
Beam #2: 20 ft span	Utilization ratio	1.0	1.0	none
Beam #3: 8 ft span	Shear capacity, ϕV_n (kips)	226.8	226.8	none
Beam #3: 8 ft span	Utilization ratio	1.0	1.001	negligible

STAAD Input

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

Note: The UNT parameter is set to 1.0 in. in order to approximate a fully-braced condition.

STAAD SPACE BENDING CAPACITY PER AISC LRFD 3RD ED
START JOB INFORMATION
ENGINEER DATE 23-Sep-18
END JOB INFORMATION
*
*  OBJECTIVE : TO DETERMINE THE MAXIMUM UNIFORMLY
*  DISTRIBUTED LOAD CARRYING CAPACITY OF 3 BEAMS.
*
*  EXAMPLE PROBLEM 5.5 PAGE 5-21, AISC LRFD 3RD ED.
*
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 40 0 0; 3 0 10 0; 4 30 10 0; 5 0 20 0; 6 8 20 0;
MEMBER INCIDENCES
1 1 2; 2 3 4; 3 5 6;
MEMBER PROPERTY AMERICAN
1 TO 3 TABLE ST W21X62
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 4.176e+06
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 3 5 PINNED
2 4 6 FIXED BUT MZ
*
LOAD 1
MEMBER LOAD
* ON MEMBER 1, W=108/40=2.7
1 UNI GY -2.7
* ON MEMBER 2, W=144/30
2 UNI GY -4.8
* ON MEMBER 3, W=454/8
3 UNI GY -56.75
*
PERFORM ANALYSIS
PARAMETER 1
CODE LRFD
MAIN 1 ALL
FYLD 7200 ALL
* FULLY BRACED CONDITION CAN BE ACHIEVED BY SETTING UNT TO 1.0 INCH
UNT 0.1 ALL
TRACK 1 ALL
CHECK CODE ALL
FINISH

STAAD Output

                         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   W21X62                   (AISC SECTIONS)
                           FAIL     LRFD-H1-1B-C       1.000         1
                        0.00 C          0.00        -540.00       20.00
 +---------------------------------------------------------------------+
 | DESIGN STRENGTHS FOR MEMBER     1       UNITS - KIP IN              |
 | PNC=   53.25  PNT=  823.50  MNZ= 6480.00  MNY=  942.05  VN=  226.80 |
 +---------------------------------------------------------------------+
      2  ST   W21X62                   (AISC SECTIONS)
                           PASS     LRFD-H1-1B-C       1.000         1
                        0.00 C          0.00        -540.00       15.00
 +---------------------------------------------------------------------+
 | DESIGN STRENGTHS FOR MEMBER     2       UNITS - KIP IN              |
 | PNC=   94.66  PNT=  823.50  MNZ= 6480.00  MNY=  942.05  VN=  226.80 |
 +---------------------------------------------------------------------+
 *    3  ST   W21X62                   (AISC SECTIONS)
                           FAIL     SHEAR-Y            1.001         1
                        0.00 C          0.00           0.00        0.00
 +---------------------------------------------------------------------+
 | DESIGN STRENGTHS FOR MEMBER     3       UNITS - KIP IN              |
 | PNC=  608.18  PNT=  823.50  MNZ= 6161.00  MNY=  942.05  VN=  226.80 |
 +---------------------------------------------------------------------+
   ************** END OF TABULATED RESULT OF DESIGN **************