Steel beam design with AISC 13th Ed.

Example 1:

Situation: A structural steel beam is supporting a roof as s

The beam is simply supported at each end.

Design Code: AISC 13th edition

Roof live load: W_L = 20 psf

Roof dead load: W_D = 20 psf

Length of beam: L = 42 ft

Unsupported length (Joist spacing): L_b = 5 ft

Tributary width: TriB = 35 ft

Material: ASTM A36, yield strength, F_y = 50 ksi

Requirements:  Select a W24 beam

Solution:  

LRFD design:

Factored load on beam: W_u= (1.2W_D+1.6W_L) TriB = 1960  lb/ft.

Maximum factored moment: M_u = W_u L²/8 = 432.2 kip-ft  

ASD design:

Factored load on beam: W= (W_D+W_L) TriB = 1400  lb/ft.

Maximum factored moment: M = W L²/8 = 308.7 kip-ft

Maximum unsupported length, L_b = 5 ft

Try W24x55, From AISC steel Table, 

d = 23.57 in, b_f = 7 in, t_f =0.5 in, A_f = b_f t_f=3.5 in², t_w = 5/16 in, ry = 1.34 in

Section modulus, S_x = 114 in³, Zx = 134 in³.

Moment of inertia, I_x = 1350 in⁴.  

Unbraced length, L_p = 1.76 r_y/ÖE/F_y  = 4.7 ft < L_b = 5 ft  

For W24x55, c = 1, r_ts = 1.71 in, J = 1.18 in⁴, h_o = 23.1 in

L_r = 1.95 r_ts(E/0.7F_y)(Jc/S_xh_o)^1/2 {1+[1+6.76(0.7F_yS_xh_o/EJc)²]^1/2}^1/2  = 13.9 ft

Note: Lp and Lr can also be obtained from AISC manual 13th ed. Table 4-1.

Calculate nominal moment strength: 

M_p = F_yZ_x = (50)(134) =558.3 kip-ft, Cb = 1

M_n = C_b {M_p - (M_p - 0.75 F_yS_x)[(L_b - L_p)/(L_r-L_p)]} = 552.4 ft kip

LRFD design

f_b = 0.9, f_bM_n = 497.2 ft-kip > Mu = 432 ft-kip  O.K.

ASD design,

W_b = 1.67, M_n/ W_b= 330.8 ft-kip > M = 308.7 ft-kip O.K.

Design using AISC chart on Table 3-10

W24x55 for an unbraced length of 6 ft

LRFD design

f_bM_n = 4472 ft-kip > Mu = 432 ft-kip  O.K.

ASD design,

M_n/ W_b= 315 ft-kip > M = 308.7 ft-kip O.K.

Check deflection: Elastic modulus, E = 29000 ksi

Service load, W = (W_D+W_L) TriB = 1400 lb/ft

Total deflection, D = 5 W L⁴/(384 E I_x ) = 2.5 in

Deflection ratio, L/D = 1/201   < 1/ 180 O.K

Live load deflection, D_L = D (20/40) = 1.25 in,

Live load deflection ratio, D_L /L = 1/402 < 1/360 O.K.

Check shear force,

LRFD design

Factor shear force, V_u = W_uL/2 = 41.2 kips  

Cv = 1, f_v = 0.9

Shear strength, fV_n = (f_v0.6 F_y t_wd)Cv = 198.9 kip  

ASD design

Shear force, V = W L/2 = 29.4 kips  

Cv = 1, W_v = 1.67

Shear strength, V_n / W_v= (0.6 F_y t_wd)Cv / W_v = 132.3 kip