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
1
Verification Example
Theoretical Results (AISC-ASD89) Compute slenderness ratio ( KL ) x 1.0(30)12 = = 83.3 r x 4.32
( KL ) y r y
=
1.0(15)12 2.01
= 89.6
Compute allowable stress F a . Compare the KL / r with C c to determine whether the shoter or long column fomular applies 2π 2 E
C c =
F y
=
2π 2 29, 000 36
= 126
Since the controlling KL / r of 89.6 is less than C c , the allowable stress is based on the parabolic equation for inelastic buckling. Thus, by calculation or from ASD “NUMERICAL VALUES” TABLE 3
F a =
( KL / r ) 2 1 − 2C 2 F y c 5 3
+
3( KL / r ) 8C c
−
( KL / r )
3
= 14.3 ksi
8C c3
Comparison of stresses P 160 = = 12.0 ksi < [ F a = 14.3 ksi] f a = A g 13.3
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 DL( 1.000) + LL( 1.000) ------------------------------------------------------------------------------------------------------------*. UNIT SYSTEM : kip, in *. SECTION PROPERTIES : Designation = W10x45 Shape = H - Section. (Rolled) 2
AISC-ASD89-1
Depth = Web Thick =
10.100, Top F Width = 0.350, Top F Thick =
8.020, Bot.F Width 0.620, Bot.F Thick
= =
8.020 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
3
Verification Example
Comparison of Results Reference
MIDAS/Gen
( KL) y / r y (slenderness ratio)
89.6
89.6
C c (limit slenderness ratio)
126
126
F a (allowable compressive stress)
14.3 ksi
14.3 ksi
f a (axial compressive stress)
12.0 ksi
12.0 ksi
Reference CHARLES G. SALMON, JOHN E. JOHNSON, Steel Structure, Example 6.11.1 4
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
1
Verification Example
Theoretical Results (AISC-ASD89) A36 steel. Assume “compact section”since nearly all sections satisfy the width/thickness limits λ : thus, the allowable stress F b would be Fb
=
0.66F y
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 M = wL / 8 = 1.0(20) / 8 = 50 ft-kips Required S x
=
M
=
F b
Try W12 × 22 : S x
=
50(12) 24
=
25 in 3
25.4 in 3
Check “compact” limits ( λ p ) of ASD-Table B5.1 b f 2t f d t w
=
=
4.03 2(0.425)
12.31 0.260
=
=
4.7 < 10.8
47.3 < 107
(Table 7.4.2)
OK
(Table 7.4.2)
OK
Note that ASD uses overall depth d whereas LRFD uses the supported height hc of the web even though the limit is the same. Check the flexural stress : M = 1.022(20) 2 /8 = 51.1 ft-kips f b
=
M S x
=
51.1(12) 25.4
Use W12 × 22, F y
=
=
(including beam weight)
24.1 ksi ≈ [ F b
36 ksi
2
=
24 ksi ]
OK
AISC-ASD89-2
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 Web Thick = 0.260, Top F Thick = 0.425, Bot.F Thick
= =
4.030 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) ] 3
Verification Example
-. 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.
4
AISC-ASD89-2
Comparison of Results Reference F b (allowable stress)
Fb
b f / 2t f (width-thickness ratio of flange) f b (flexural stress)
=
0.66 f b
=
23.8 ksi
MIDAS/Gen Fb
=
0.66 f b
23.8 ksi
4.7
4.7
24.1 ksi
24.2 ksi
Reference CHARLES G. SALMON, JOHN E. JOHNSON, Steel Structure, Example 7.5.1
5
=
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 F y = 60 ksi. Use Allowable Stress Design.
Beam-column with Combined Bending and Axial Load
1
Verification Example
Theoretical Results (AISC-ASD89) Column effect KL 15(12) =
r y
KL / r C c Fa
=
f a
=
f a
=
73 97.7 =
P
350
A g
=
17.8 22.8
Beam effect 76b f Lc = F y
73
=
Cc F y
=
F a
=
2.46
0.747
0.381(60) = 22.8 ksi
=
19.7
17.8 ksi
0.78 > 0.15
=
76(10.235)
=
60(12)
=
8.4 ft
(controls)
or Lc
=
Fb
=
Lb
f b
12, 000
=
Lb d / Af
=
Cm
=
(d / A f ) F y
=
r T F b
20, 000
15(12)
=
2.75 40.0 −
20, 000
=
2.40(60)12 12, 000
15(12)(2.40)
=
11.6 ft
27.8 ksi < 0.60F y
65.5
( Lb / r T ) 2 425
=
40.0 −
(65.5) 2 425
0.6 − 0.4( M 1 / M 2 ) = 0.60
=
60(12)
= 6.15 ksi 117 Cm f b 0.6(6.15) = = 0.12 F b 29.9 =
KL r x '
F e
=
=
15(12) 6.96
=
25.9
223 ksi 2
=
29.9 ksi
AISC-ASD89-3
where the x-axis is the axis of bending. The magnification factor is 1 1.0 1.0 = = = 1.09 ' 1 − f a / F e 1 − 17.8 / 223 1 − 0.0798 Check of ASD Formulas For stability, Formular (H1-1) f a Cm f b 1.0 + ( ) = 0.78 + 0.12(1.09) = 0.91 < 1.0 ' Fa Fb 1 − f a / F e For yielding, Formualr (H1-2), at the braced point, f a f b 17.8 6.15 + = + = 0.66 < 1.0 0.60 F y F b 36 29.9
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 = Web Thick = 0.395, Top F Thick = 0.665, Bot.F Thick =
10.235 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 3
Verification Example
================================ [[[*]]] 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 -. Izz = tf*Bf^3/12 + {(Ccom-tf)/3}*tw^3/12 -. rT = SQRT[Izz/Azz] 4
= = =
7.7938 59.4289 2.761
AISC-ASD89-3
( ). Check ratio of Lu-rT (Lu/rT). [ AISC-ASD89 Specification F1.3 ] -. CRrog2 = SQRT[ (510000*Cb)/Fy ] = -. Lu/rT = 65.185 < CRrog2
92.195
( ). 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 = = -. Rmax2 = =
fa SFy*fbcy SFz*fbcz ---- + ------------- + ------------Fa FBCy FBCz 0.914 < 1.000 ---> O.K. fa fbcy fbcz ----------- + --------- + -------0.6*Fy FBCy FBCz 0.699 < 1.000 ---> O.K.
5
(H1-1)
(H1-2)
Verification Example
Comparison of Results Reference
MIDAS/Gen
73
73
F a (allowable compressive stress)
22.8 ksi
22.8 ksi
f a (axial compressive stress)
17.8 ksi
17.8 ksi
F b (allowable compressive bending stress)
29.9 ksi
30.0 ksi
C m (equivalent moment correction factor)
0.6
0.6
223 ksi
223 ksi
0.91
0.91
0.70
0.70
( KL) y / r y (slenderness ratio)
'
F e (Euler stress devided by a factor of a safety) Interaction ratio of combined stresses f a Fa
+
Cm f b Fb
f a 0.60 F y
+
1.0 ( ) (for stability) 1 − f a / F e' f b
F b
(for yielding)
Reference CHARLES G. SALMON, JOHN E. JOHNSON, Steel Structure, Example 12.14.1 6
