1

AISC-ASD89-1

Title Design of Compression Member

Description Check the adequacy of a W10×45 section by Allowable Stress Design. A36 steel is used and service loads are 50 kips dead load and 110 kips live load.

Compression Member

Verification Example

2

Theoretical Results (AISC-ASD89) Compute slenderness ratio

( ) 1.0(30)12 83.34.32

x

x

KLr

= = ( ) 1.0(15)12 89.6

2.01y

y

KLr

= =

Compute allowable stress aF . Compare the /KL r with cC to determine whether the shoter or long column fomular applies

2 22 2 29,000 126

36cy

ECFπ π

= = =

Since the controlling /KL r of 89.6 is less than cC , the allowable stress is based on the parabolic equation for inelastic buckling. Thus, by calculation or from ASD “NUMERICAL VALUES” TABLE 3

2

2

3

3

( / )12

14.3 ksi5 3( / ) ( / )3 8 8

yc

a

c c

KL r FC

FKL r KL r

C C

−

= =+ −

Comparison of stresses

[ ]160 12.0 ksi 14.3 ksi13.3a a

g

Pf FA

= = = < =

Results by the MIDAS/Gen ------------------------------------------------------------------------------------------ MIDAS/Gen - Steel Code Checking [ AISC-ASD89 ] ===================================================== *.DEFINITION OF LOAD COMBINATIONS WITH SCALING UP FACTORS. ------------------------------------------------------------------------------------------------------------- LCB C Loadcase Name(Factor) + Loadcase Name(Factor) + Loadcase Name(Factor) ------------------------------------------------------------------------------------------------------------- 1 0 DL( 1.000) + LL( 1.000) ------------------------------------------------------------------------------------------------------------- *. UNIT SYSTEM : kip, in *. SECTION PROPERTIES : Designation = W10x45 Shape = H - Section. (Rolled)

AISC-ASD89-1

3

Depth = 10.100, Top F Width = 8.020, Bot.F Width = 8.020 Web Thick = 0.350, Top F Thick = 0.620, Bot.F Thick = 0.620 *. DESIGN PARAMETERS FOR STRENGTH EVALUATION : Ly = 3.60000e+002, Lz = 3.60000e+002, Lu = 3.60000e+002 Ky = 1.00000e+000, Kz = 5.00000e-001 *. MATERIAL PROPERTIES : Fy = 3.60000e+001, Es = 2.90000e+004, MATERIAL NAME = A36 *. FORCES AND MOMENTS AT (I) POINT : Axial Force Fxx =-1.60000e+002 =================================== [[[*]]] CHECK AXIAL STRESS. =================================== ( ). Check slenderness ratio of axial compression member (Kl/r). [ AISC-ASD89 Specification B7. ]

-. Kl/r = 89.6 < 200.0 ---> O.K. ( ). Calculate allowable compressive stress (Fa). [ AISC-ASD89 Specification E2. (E2-1) ] [ 2*(Pi^2)*Es ] -. Cc = SQRT [ ----------------- ] = 126.10 [ Fy ] -. Kl/r < Cc [ (Kl/r)^2 ] [ 1 - ----------- ]*Fy [ 2*Cc^2 ] -. Fa = ------------------------------- = 14.258 kip/in^2. 5 3*(Kl/r) (Kl/r)^3 --- + ---------- - ----------- 3 8*Cc 8*Cc^3 ( ). Calculate axial compressive stress of member (fa). -. fa = Fxx/Area = -12.030 kip/in^2. ( ). Check ratio of axial stress (fa/Fa). fa 12.030 -. ---- = ------------ = 0.844 < 1.000 ---> O.K. Fa 14.258

Verification Example

4

Comparison of Results

Reference

CHARLES G. SALMON, JOHN E. JOHNSON, Steel Structure, Example 6.11.1

Reference MIDAS/Gen

( ) /y yKL r (slenderness ratio) 89.6 89.6

cC (limit slenderness ratio) 126 126

aF (allowable compressive stress) 14.3 ksi 14.3 ksi

af (axial compressive stress) 12.0 ksi 12.0 ksi

1

AISC-ASD89-2

Title Design of Laterally Supported Beam

Description Select the lightest W or M section to carry a uniformly distributed dead load of 0.2 kip/ft superimposed (i.e., in addition to the beam weight) and 0.8 kip/ft live load. The simply supported span is 20ft. The compression flange of the beam is fully supported against lateral movement. Use Allowable Stress Design with A36 steel.

Laterally Supported Beam

Verification Example

2

Theoretical Results (AISC-ASD89) A36 steel. Assume “compact section”since nearly all sections satisfy the width/thickness limits pλ : thus, the allowable stress bF would be 0.66b yF F= Note that rounded values (i.e., 0.66 times 36 = 23.8 ksi ; use 24 ksi) are accepted values in accordance with ASD-“NUMERICAL VALUES” TABLE

The superimposed service load (1kip/ft) bending moment is 2 2/ 8 1.0(20) / 8 50 ft-kipsM wL= = =

Required 350(12) 25 in24x

b

MSF

= = =

Try W12× 22 : 325.4 inxS = Check “compact” limits ( pλ ) of ASD-Table B5.1

4.03 4.7 10.82 2(0.425)

f

f

bt

= = < (Table 7.4.2) OK

12.31 47.3 1070.260w

dt

= = < (Table 7.4.2) OK

Note that ASD uses overall depth d whereas LRFD uses the supported height ch of the web even though the limit is the same. Check the flexural stress : 21.022(20) /8 51.1 ft-kipsM = = (including beam weight)

[ ]51.1(12) 24.1 ksi 24 ksi25.4b b

x

Mf FS

= = = ≈ = OK

Use W12× 22, 36 ksiyF =

AISC-ASD89-2

3

Results by the MIDAS/Gen ------------------------------------------------------------------------------------------ MIDAS/Gen - Steel Code Checking [ AISC-ASD89 ] ===================================================== *.DEFINITION OF LOAD COMBINATIONS WITH SCALING UP FACTORS. -------------------------------------------------------------------------------------------------------------- LCB C Loadcase Name(Factor) + Loadcase Name(Factor) + Loadcase Name(Factor) -------------------------------------------------------------------------------------------------------------- 1 0 Load( 1.000) + Self Weight( 1.000) -------------------------------------------------------------------------------------------------------------- *. UNIT SYSTEM : kip, in *. SECTION PROPERTIES : Designation = W12x22 Shape = H - Section. (Rolled) Depth = 12.310, Top F Width = 4.030, Bot.F Width = 4.030 Web Thick = 0.260, Top F Thick = 0.425, Bot.F Thick = 0.425 *. DESIGN PARAMETERS FOR STRENGTH EVALUATION : Ly = 2.40000e+002, Lz = 2.40000e+002, Lu = 0.00000e+000 Ky = 1.00000e+000, Kz = 1.00000e+000 *. MATERIAL PROPERTIES : Fy = 3.60000e+001, Es = 2.90000e+004, MATERIAL NAME = A36 *. FORCES AND MOMENTS AT (1/2) POINT : Bending Moments My = 6.13250e+002, Mz = 0.00000e+000 ================================================== [[[*]]] CHECK BENDING STRESSES ABOUT MAJOR AXIS. ================================================== ( ). Check depth-thickness ratio of web (DTR). [ AISC-ASD89 Specification B5.1 ] -. DTR = Dweb/tw = 44.077 -. DTR < 640/SQRT[Fy] ---> COMPACT SECTION ! ( ). Check width-thickness ratio of flange (BTR). [ AISC-ASD89 Specification B5.1 ] -. h/t = 44.08 < 70. ---> kc = 1.0 -. BTR = bf/2tf = 4.74 -. BTR < 65/SQRT[Fy] ---> COMPACT SECTION ! ( ). Calculate allowable bending stresses (FBCy,FBTy). [ AISC-ASD89 Specification F1.1 (F1-1) ]

Verification Example

4

-. FBCy,FBTy = 0.66*Fy = 23.760 kip/in^2. if Fy < 65 ksi. ( ). Calculate actual bending stresses of member (fbcy,fbty). -. fbcy = (My*Ccom)/Iyy = -24.196 kip/in^2. -. fbty = (My*Cten)/Iyy = 24.196 kip/in^2.

AISC-ASD89-2

5

Comparison of Results

Reference CHARLES G. SALMON, JOHN E. JOHNSON, Steel Structure, Example 7.5.1

Reference MIDAS/Gen

bF (allowable stress) 0.66 23.8 ksib bF f= = 0.66 23.8 ksib bF f= =

/ 2f fb t (width-thickness ratio of flange) 4.7 4.7

bf (flexural stress) 24.1 ksi 24.2 ksi

1

AISC-ASD89-3

Title Design for Combined Bending and Axial Load

Description Investigate the acceptability of a W16× 67 used as a beam-column under the loading shown in Figure. The total service loads are P = 350 kips and M = 60 ft-kips, and yF = 60 ksi. Use Allowable Stress Design.

Beam-column with Combined Bending and Axial Load

Verification Example

2

Theoretical Results (AISC-ASD89) Column effect

15(12) 732.46y

KLr

= =

/ 73 0.74797.7c

KL rC

= =

0.381(60) 22.8 ksia c yF C F= = =

350 17.8 ksi19.7a

g

PfA

= = =

17.8 0.78 0.1522.8

a

a

fF

= = >

Beam effect

76 76(10.235) 8.4 ft

60(12)f

cy

bL

F= = = (controls)

or

20,000 20,000 11.6 ft( / ) 2.40(60)12c

f y

Ld A F

= = =

12,000 12,000 27.8 ksi 0.60/ 15(12)(2.40)b y

b f

F FL d A

= = = <

15(12) 65.52.75

b

T

Lr

= =

2 2( / ) (65.5)40.0 40.0 29.9 ksi

425 425b T

bL rF = − = − =

1 20.6 0.4( / ) 0.60mC M M= − =

60(12) 6.15 ksi117bf = =

0.6(6.15) 0.1229.9

m b

b

C fF

= =

15(12) 25.96.96x

KLr

= =

' 223 ksieF =

AISC-ASD89-3

3

where the x-axis is the axis of bending. The magnification factor is

'

1 1.0 1.0 1.091 17.8 / 223 1 0.07981 /a ef F

= = =− −−

Check of ASD Formulas

For stability, Formular (H1-1)

'

1.0( ) 0.78 0.12(1.09) 0.91 1.01 /

a m b

a b a e

f C fF F f F

+ = + = <−

For yielding, Formualr (H1-2), at the braced point,

17.8 6.15 0.66 1.00.60 36 29.9

a b

y b

f fF F

+ = + = <

Results by the MIDAS/Gen ----------------------------------------------------------------------------------- MIDAS/Gen - Steel Code Checking [ AISC-ASD89 ] =================================================

*.DEFINITION OF LOAD COMBINATIONS WITH SCALING UP FACTORS. -------------------------------------------------------------------------------------------------------------- LCB C Loadcase Name(Factor) + Loadcase Name(Factor) + Loadcase Name(Factor) -------------------------------------------------------------------------------------------------------------- 1 0 Load( 1.000) -------------------------------------------------------------------------------------------------------------- *. UNIT SYSTEM : kip, in *. SECTION PROPERTIES : Designation = W16x67 Shape = H - Section. (Rolled) Depth = 16.330, Top F Width = 10.235, Bot.F Width = 10.235 Web Thick = 0.395, Top F Thick = 0.665, Bot.F Thick = 0.665 *. DESIGN PARAMETERS FOR STRENGTH EVALUATION : Ly = 1.80000e+002, Lz = 1.80000e+002, Lu = 1.80000e+002 Ky = 1.00000e+000, Kz = 1.00000e+000 *. MATERIAL PROPERTIES : Fy = 6.00000e+001, Es = 2.90000e+004, MATERIAL NAME = A572-60 *. FORCES AND MOMENTS AT (J) POINT : Axial Force Fxx =-3.50000e+002 Shear Forces Fyy = 0.00000e+000, Fzz =-4.00000e+000 Bending Moments My = 7.20000e+002, Mz = 0.00000e+000 Moments of j-node Myyj = 7.20000e+002, Mzzj = 0.00000e+000

Verification Example

4

================================ [[[*]]] CHECK AXIAL STRESS. ================================

( ). Check slenderness ratio of axial compression member (Kl/r). [ AISC-ASD89 Specification B7. ]

-. Kl/r = 73.2 < 200.0 ---> O.K. ( ). Calculate allowable compressive stress (Fa). [ AISC-ASD89 Specification E2. (E2-1) ] [ 2*(Pi^2)*Es ] -. Cc = SQRT [ -----------------] = 97.68 [ Fy ] -. Kl/r < Cc [ (Kl/r)^2 ] [ 1 - ----------- ]*Fy [ 2*Cc^2 ] -. Fa = -------------------------------- = 22.778 kip/in^2. 5 3*(Kl/r) (Kl/r)^3 --- + ---------- - ---------- 3 8*Cc 8*Cc^3 ( ). Calculate axial compressive stress of member (fa). -. fa = Fxx/Area = -17.766 kip/in^2. ( ). Check ratio of axial stress (fa/Fa). fa 17.766 -. ---- = ------------- = 0.780 < 1.000 ---> O.K. Fa 22.778

=================================================== [[[*]]] CHECK BENDING STRESSES ABOUT MAJOR AXIS. ===================================================

( ). Check laterally unbraced length of compression flange (Lu). [ AISC-ASD89 Specification F1.1 (F1-2) ] -. Lcr1 = (76*bf)/SQRT[Fy] = 100.42 in. -. Lcr2 = 20000/((d/Af)*Fy) = 138.93 in. -. Lcr = MIN[ Lcr1, Lcr2 ] = 100.42 in. -. Lu = 180.00 in. > Lcr ( ). Calculate bending coefficient (Cb). [ AISC-ASD89 Specification F1.3 ] -. Cb = 1.000 (User defined or default value) ( ). Calculate radius of gyration (rT) -. Azz = Bf*tf + tw*(Ccom-tf)/3 = 7.7938 -. Izz = tf*Bf^3/12 + {(Ccom-tf)/3}*tw^3/12 = 59.4289 -. rT = SQRT[Izz/Azz] = 2.761

AISC-ASD89-3

5

( ). Check ratio of Lu-rT (Lu/rT). [ AISC-ASD89 Specification F1.3 ] -. CRrog2 = SQRT[ (510000*Cb)/Fy ] = 92.195 -. Lu/rT = 65.185 < CRrog2 ( ). Calculate allowable compressive bending stresses (FBC). [ AISC-ASD89 Specification F1.3 (F1-6,F1-8) ] 12000*Cb -. FBCi = -------------- = 27.786 kip/in^2. (Lu*d)/Af 2 Fy*(Lu/rT)^2 -. FBCj = [ --- - ------------------ ]*Fy = 30.002 kip/in^2. 3 1530000*Cb -. FBC = MAX( FBCi, FBCj ) = 30.002 kip/in^2. ==================================== [[[*]]] CHECK COMBINED STRESSES. ==================================== ( ). Check interaction ratio of combined stresses (Axial compression + bending). [ AISC-ASD89 Specification H1. (H1-1, H1-2) ] -. fa/Fa > 0.15 -. Single Curvature Bending. -. Cmy = 0.6 - 0.4*(My1/My2) = 0.600 -. Single Curvature Bending. -. Cmz = 0.6 - 0.4*(Mz1/Mz2) = 1.000 -. Cmz > 1.0 ---> Cmz = 1.0 12*(Pi^2)*Es -. F'ey = ----------------- = 223.267 kip/in^2. 23*(Kl/r)^2 Cmy -. SFy = --------------- = 0.652 1. - fa/F'ey *. Check interaction ratio of combined stress at member end point. fa Cmy*fbcy Cmz*fbcz -. Rmax1 = ---- + ----------------------- + ---------------------- Fa (1-fa/F'ey)*FBCy (1-fa/F'ez)*FBCz fa SFy*fbcy SFz*fbcz = ---- + ------------- + ------------- (H1-1) Fa FBCy FBCz = 0.914 < 1.000 ---> O.K. fa fbcy fbcz -. Rmax2 = ----------- + --------- + -------- (H1-2) 0.6*Fy FBCy FBCz = 0.699 < 1.000 ---> O.K.

Verification Example

6

Comparison of Results

Reference

CHARLES G. SALMON, JOHN E. JOHNSON, Steel Structure, Example 12.14.1

Reference MIDAS/Gen

( ) /y yKL r (slenderness ratio) 73 73

aF (allowable compressive stress) 22.8 ksi 22.8 ksi

af (axial compressive stress) 17.8 ksi 17.8 ksi

bF (allowable compressive bending stress) 29.9 ksi 30.0 ksi

mC (equivalent moment correction factor) 0.6 0.6 '

eF (Euler stress devided by a factor of a safety) 223 ksi 223 ksi

Interaction ratio of combined stresses (for stability) (for yielding)

0.91

0.70

0.91

0.70

'

1.0( )1 /

a m b

a b a e

f C fF F f F

+−

0.60a b

y b

f fF F

+