 # V. AISC ASD - 2D Frame Validation

Determine the allowable stresses (per 1989 AISC code) for the members of the structure as shown in the figure. Also, perform a code check for these members based on the results of the analysis.

## Reference

Attached step by step hand calculation as per 1989 AISC ASD code. Ninth Edition.

## Problem ### Steel braced frame

Members 1, 2 = W12X26, Members 3, 4 = W14X43

Members 5, 6, 7 = W16X36, Memb 8= L40404, Memb 9 = L50506

F = 15 kip (wind load)

Steel: Fy = 36ksi

## Hand Calculation

Manual / Code refers to AISC Manual of Steel Construction, Allowable Stress Design, ninth edition.

## Member 1

Size W12x26, L = 10 ft., a = 7.65 in2, Sz = 33.39 in3

Per Section F2 is the lower value for the yielding and lateral-torsional buckling limit states:

Mn = Mp = FyZx = 36ksi × 37.2 in3 = 1,339 in·k = 111.3 ft·k

$L p = 1.76 r y E F y = 1.76 ( 1.51 in ) 29 , 000 ksi 36 ksi = 75.4 ⁢ in$

From clause F1-2, page 5-45 of Manual, Lc = 6.85 ft.

From observation Load case 1 will govern,

Fx = 25.0 kip (compression), Mz = 56.5 ft·kip

Area of compressive flange = 6.49*0.38 = 2.466 in2

Allowable bending stress, per Clause F1-8 (page 5-47 of Manual)

fb = 12. · 1000 · 1.0/[10 · 12 · (12.22/2.466)] = 20.1817 ksi

(kl/r)y = 120/1.5038 = 79.8, so fa = 15.38 ksi

Stresses per Table C-36 (page 3-16 of manual)

fa = 25./7.65 = 3.268

fb = 56.5 · 12/33.39 = 20.31 ksi

(kl/r)z = 120/5.1639 = 23.238, so F'ez = = 276.54 ksi

Equation H1-1, page 5-54 of Manual

3.268/15.38 + 0.85 · 20.31 / [(1-3.268/276.54) · 20.1817]  =  1.078

Equation H1-2, page 5-54 of Manual.

3.268/(0.6 · 36) +20.31/20.1817 = 1.1576

Therefore, equation H1-2 governs and ratio = 1.1576

## Member 2

Size W12x26, L = 5 ft., a = 7.65 in2, Sz = 33.39 in3

From observation load case 1 will govern,

Fx = 8.71 kip (compression), Mz = 56.50 ft·kip

Since L is less than Lc = 6.85ft, per Clauses F1-1 & F1-2 (page 5-45 of Manual)

fb = 0.66 · 36 = 23.76 ksi

(kl/r)y = 60/1.5038 = 39.90, so fa = 19.19 ksi

Per Table C-36 (page 3-16 of Manual)

fa = 8.71/7.65 = 1.1385, fb = 56.5 · 12/33.39 = 20.31 ksi

Since fa/Fa less than 0.15, apply equation H1-3 (page 5-54 of Manual)

1.1385/19.19 + 20.31/23.76 = .0593 + .8548 = 0.9141

## Member 3

Size W14X43, L = 11ft., a = 12.6 in2, Sz = 62.7 in3

From observation load case 3 will govern,

Fx = 25.5 kip (compression), Mz = 112.173 ft·kip

Referring to clause F1-2, page 5-45 of Manual.

Lc = 8.4 ft. Therefore

fb = 0.6 · 36 = 21.6 ksi

(kl/r)y = 132/1.8941 = 69.69,

so fa = 16.46 ksi per Table C-36 (page 3-16 of Manual)

fa = 25.5/12.6 = 2.024

fb = 112.173 · 12/62.66 = 21.48 ksi

since fa/Fa less than 0.15, use equation H1-3, page 5-54 of Manual

2.024/16.46 + 21.48/21.6 = 0.123 + 0.994 = 1.117

## Member 4

Size W14x43, L = 4ft, a = 12.6 in2, Sz = 62.7 in3

From observation, load case 3 will govern,

Fx = 8.75 kip (tension), Mz = 112.173 ft·kip

Since L is less than Lc = 8.4 ft

fb = 0.66 · 36 = 23.76 ksi

Clause F1-1 (page 5-45 of Manual)

fa = 8.75/12.6 = 0.694

fb = 112.73x12/62.66 = 21.48 ksi

Combined tension and bending, use equation H2-1, page 5-55 of Manual.

0.694/(0.6 · 36) + 21.48/23.76 = 0.032 + 0.904 = 0.936

## Member 5

Size W16x36, L = 5ft, a = 10.6 in2, Sz = 56.49 in3

From observation, load case 3 will govern.

Fx = 14.02 kip (compression), Mz =57.04 ft·kip

Since L is less than Lc = 7.37 (per Clause F1-2, page 5-45 of Manual)

fb = 0.66 · 36 = 23.76 ksi

(kl/r)y = 60./1.52 = 39.47

so fa = 19.23 ksi per Table C-36 (page 3-16 of Manual)

fa = 14.02/10.6 = 1.32

fb = 57.04 · 12/56.5 = 12.12 ksi

Since fa/Fa less than 0.15, use equation H1-3, page 5-54 of Manual

1.32/19.23 + 12.12/23.76 = 0.069 + 0.510 = 0.579

## Member 6

Size W16x36, L = 16ft, a = 10.6 in2, Sz = 56.49 in3

From observation, load case 1 will govern. Forces at midspan are

Fx = 5.65 kip (compression), Mz = 71.25 ft·kip

From Chapter F of the AISC ASD 9Pth P ed. specs., with Cb = 1.0,

(102,000Cb/Fy)1/2 =53.229

(510,000Cb/Fy)1/2 =119.02

L/rT = 192/1.79 = 107.26

53.229 < 107.26 < 119.02

Therefore Fb (as per F1-6, page 5-47 of Manual)

[(2/3) – 36 · 107.26 · 107.26/(1,530,000)] · 36 = 14.25 ksi

(Kl/r)y = 192/1.5203 = 126.29

so fa = 9.36 per Table C-36 (page 3-16 of Manual)

fa = 5.65/10.6 = 0.533

fb = 71.25x12/56.49 = 15.14 ksi

Since fa/Fa less than 0.15 use formula [H1-3, page 5-54 of Manual]

0.533/9.36 + 15.14/14.25 = 0.057 + 1.062 = 1.119

## Member 7

Size W16x36, L =4ft, a = 10.6in2, Sz =56.49in3

Lc = 7.37ft (Clause F1-2 page 5-45 of Manual)

From observation load case 3 will govern, Fx = 24.06 kip (tension), Mz = 62.96 ft·kip

From Clause F1-1, the allowable compressive stress is

fb = 0.66 Fy = 23.76 ksi

Since section is in tension, per Clause F1-5 (page5-45 of Manual)

fb = 0.60 · 36 = 21.60 Ksi

Choosing the larger of above 2 values, fb = 23.76 Ksi

fa = 24.06/10.6 = 2.2698

fb = 62.96 · 12/56.49 = 13.37

Since combined tension and bending, use equation H 2-1, page 5-55 of the AISC ASD 9Pth P ed. specs.

2.2698/(0.6 · 36) + 13.37(23.76) = 0.105 + 0.5627 = 0.6677

## Member 8

Size L4x4x1/4, L = 7.071 ft, a = 1.94 in2

From observation load case 1 will govern, Fx = 23.04 kip (Comp.)

Fa is computed as per page 5-310 of the AISC ASD 9Pth P ed.specs.

Qs = 1.34 – 0.00447 · (4/0.25)·(36)1/2 = 0.9108

Qa = 1.0

Q = Qs · Qa = 0.9108

Cc= 92.0·π2·E/(Q·Fy)]1/2 = [2.0·π2 · 29000/(0.9108 · 36)]1/2 = 132.1241

Kl/r = 7.071 · 12/0.795 = 106.73 < Cc.

Hence, fa = 11.6027 ksi (computed per equation 4-1)

Actual compressive stress

fa = 23.04/1.94 = 11.876 ksi

Therefore, Ratio =

fa/fa = 11.876/11.602 = 1.024

## Member 9

Size L5x5x3/8, L = 5.657 ft, a = 3.61 in2

From observation, load case 1 governs, Fx = 48.44 kip (Comp.)

Fa is computed as per page 5-310 of the AISC ASD 9Pth P ed. specs.

Qs = 1.34 – 0.00447 · (5/0.375)·(36)1/2 = 0.9824

Qa = 1.0

Q = Qs · Qa = 0.9824

Cc= 92.0·π2·E/(Q·Fy)]1/2 = [2.0·π2 · 29000/(0.9824 · 36)]1/2 = 127.2238

(Kl/r)min = 5.657 · 12/0.99 = 68.57 < Cc.

Hence, fa = 16.301 ksi (computed per equation 4-1)

Actual compressive stress

fa = 48.44/3.61 = 13.418 ksi

Therefore Ratio =

fa/fa = 13.418/16.301 = 0.823

## Comparison

Governing stress ratio at member no. Hand Calculation STAAD.Pro Difference
1 1.158 1.154 negligible
2 0.914 0.913 negligible
3 1.117 1.121 negligible
4 0.936 0.939 negligible
5 0.579 0.580 negligible
6 1.119 1.107 1.1%
7 0.668 0.670 negligible
8 1.024 1.045 2.1%
9 0.823 0.815 1.0%

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\AISC ASD - 2D Frame validation.STD is typically installed with the program.

STAAD PLANE VERIFICATION PROBLEM NO 13
START JOB INFORMATION
ENGINEER DATE 22-Sep-18
END JOB INFORMATION
*
*  THIS DESIGN EXAMPLE IS VERIFIED BY HAND CALCULATION
*  FOLLOWING AISC-89 CODE.
*
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 25 0 0; 3 0 10 0; 4 25 11 0; 5 0 15 0; 6 25 15 0; 7 5 15 0;
8 21 15 0;
MEMBER INCIDENCES
1 1 3; 2 3 5; 3 2 4; 4 4 6; 5 5 7; 6 7 8; 7 8 6; 8 3 7; 9 4 8;
MEMBER PROPERTY AMERICAN
1 2 TABLE ST W12X26
3 4 TABLE ST W14X43
5 TO 7 TABLE ST W16X36
8 TABLE ST L40404
9 TABLE ST L50506
MEMBER TRUSS
8 9
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 4.176e+06
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 2 PINNED
5 TO 7 UNI Y -2
5 FX 15
1 0.75 2 0.75
PERFORM ANALYSIS
PRINT MEMBER FORCES
PARAMETER 1
CODE AISC
TRACK 1 ALL
CHECK CODE ALL
FINISH


                         STAAD.Pro CODE CHECKING - (AISC 9TH EDITION)   v1.0
***********************
ALL UNITS ARE - KIP  FEET (UNLESS OTHERWISE Noted)
FX            MY             MZ       LOCATION
=======================================================================
*    1  ST   W12X26                   (AISC SECTIONS)
FAIL     AISC- H1-2         1.154         1
25.00 C          0.00          56.50       10.00
-----------------------------------------------------------------------
| MEM=     1, UNIT KIP-INCH, L= 120.0 AX=   7.65 SZ=    33.4 SY=     5.3|
| KL/R-Y=  79.8  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 20.21 |
| FTZ= 21.60  FCY= 27.00  FTY= 27.00  FC= 15.38  FT= 21.60  FV= 14.40   |
-----------------------------------------------------------------------
2  ST   W12X26                   (AISC SECTIONS)
PASS     AISC- H1-3         0.913         1
8.72 C          0.00          56.50        0.00
-----------------------------------------------------------------------
| MEM=     2, UNIT KIP-INCH, L=  60.0 AX=   7.65 SZ=    33.4 SY=     5.3|
| KL/R-Y=  39.9  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 23.76 |
| FTZ= 23.76  FCY= 27.00  FTY= 27.00  FC= 19.20  FT= 21.60  FV= 14.40   |
-----------------------------------------------------------------------
*    3  ST   W14X43                   (AISC SECTIONS)
FAIL     AISC- H1-3         1.121         3
25.50 C          0.00        -112.20       11.00
-----------------------------------------------------------------------
| MEM=     3, UNIT KIP-INCH, L= 132.0 AX=  12.60 SZ=    62.5 SY=    11.3|
| KL/R-Y=  69.7  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 21.60 |
| FTZ= 21.60  FCY= 27.00  FTY= 27.00  FC= 16.46  FT= 21.60  FV= 14.40   |
-----------------------------------------------------------------------
4  ST   W14X43                   (AISC SECTIONS)
PASS     AISC- H2-1         0.939         3
8.83 T          0.00        -112.20        0.00
-----------------------------------------------------------------------
| MEM=     4, UNIT KIP-INCH, L=  48.0 AX=  12.60 SZ=    62.5 SY=    11.3|
| KL/R-Y=  25.3  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 23.76 |
| FTZ= 23.76  FCY= 27.00  FTY= 27.00  FC= 19.85  FT= 21.60  FV= 14.40   |
-----------------------------------------------------------------------
5  ST   W16X36                   (AISC SECTIONS)
PASS     AISC- H1-3         0.580         3
14.02 C          0.00         -57.00        5.00
VERIFICATION PROBLEM NO 13                               -- PAGE NO.    6
ALL UNITS ARE - KIP  FEET (UNLESS OTHERWISE Noted)
FX            MY             MZ       LOCATION
=======================================================================
-----------------------------------------------------------------------
| MEM=     5, UNIT KIP-INCH, L=  60.0 AX=  10.60 SZ=    56.4 SY=     7.0|
| KL/R-Y=  39.5  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 23.76 |
| FTZ= 23.76  FCY= 27.00  FTY= 27.00  FC= 19.23  FT= 21.60  FV= 14.40   |
-----------------------------------------------------------------------
*    6  ST   W16X36                   (AISC SECTIONS)
FAIL     AISC- H1-3         1.107         1
5.65 C          0.00         -71.25        8.00
-----------------------------------------------------------------------
| MEM=     6, UNIT KIP-INCH, L= 192.0 AX=  10.60 SZ=    56.4 SY=     7.0|
| KL/R-Y= 126.3  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 14.45 |
| FTZ= 21.60  FCY= 27.00  FTY= 27.00  FC=  9.36  FT= 21.60  FV= 14.40   |
-----------------------------------------------------------------------
7  ST   W16X36                   (AISC SECTIONS)
PASS     AISC- H2-1         0.670         3
24.13 T          0.00          63.00        0.00
-----------------------------------------------------------------------
| MEM=     7, UNIT KIP-INCH, L=  48.0 AX=  10.60 SZ=    56.4 SY=     7.0|
| KL/R-Y=  31.6  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 23.76 |
| FTZ= 23.76  FCY= 27.00  FTY= 27.00  FC= 18.88  FT= 21.60  FV= 14.40   |
-----------------------------------------------------------------------
*    8  ST   L40404                   (AISC SECTIONS)
FAIL     AISC- H1-1         1.045         1
23.03 C          0.00           0.00        0.00
-----------------------------------------------------------------------
| MEM=     8, UNIT KIP-INCH, L=  84.9 AX=   1.93 SZ=     0.8 SY=     1.7|
| KL/R- = 108.4  CB=  0.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ=  0.00 |
| FTZ=  0.00  FCY=  0.00  FTY=  0.00  FC= 11.42  FT= 21.60  FV=  0.00   |
-----------------------------------------------------------------------
9  ST   L50506                   (AISC SECTIONS)
PASS     AISC- H1-1         0.815         3
48.55 C          0.00           0.00        0.00
VERIFICATION PROBLEM NO 13                               -- PAGE NO.    7
ALL UNITS ARE - KIP  FEET (UNLESS OTHERWISE Noted)