Moment and shear strength of I shape and Channel with Fy £ 50 ksi in AISC 13th edition

Calculation of moment and shear strength of I shape and channel

Failure of I shape members and channel subjected to bending moment occurs in several different forms.

(a) Yielding

(b) Lateral-torsional buckling

(c) Flange local buckling

(d) Web local buckling.

Within the four, lateral-torsional buckling most common.  It depends on lateral bracing of compressive flange.  Yielding is depended on yield strength of steel and section modulus of the section.  Flange local buckling and web local buckling is depended on the thick to width ratio of flange and web.

First, the section is divided to compact and non-compact section.  When the section is compact, local buckling of flange and web is rule out.  Most rolled W shape beams in AISC manual with yield strength,  F_y ³ 50 ksi (are ASTM A992 steel) are compact section.  It does make the calculation easier.

When a compact section with unbraced length less or equal to the unbraced length of lateral-torsional buckling, nominal moment strength is simply yield strength of steel multiplies plastic section modulus.  When unbraced length exceeds the lateral-torsional limit, it is depended on the buckling is in-elastic or elastic.  

AISC give complicated equations to calculate nominal moment strength for elastic buckling.  Moment strength in in-elastic buckling range is linearly interpreted between yield strength and elastic lateral-torsional buckling.  The equations for calculating the limit of unbraced length is also complicated.  AISC manual provide charts and tables for practical used.

For LRFD design, a resistance factor is multiplied to nominal moment strength.  For Allowable stress design, nominal moment strength is divided by a safety factor.

How to determine unbraced length of beam

Resistance factor f_b and safety factor W_b for M_n.

f_b = 0.9

W_b = 1.67

Nominal moment strength, M_n

Nominal flexural strength of Compact I shape and Channel bend about major axis

All W, S, M, C, and MC shape with F_y £ 50 ksi have compact flange (b/t) except 

W21x48, W14x99, W14x90, W12x65, W10x12, W8x31, W8x10, W6x15, W6x9, W6x8.5, and M4x6; All current W, S, M, M, HP, C, and MC shapes have compact webs at F_y £ 65 ksi.

For compact sections, the nominal flexural strength, M_n are determined based on unsupported length L_b of the beam which  are divided in three categories:

1. L_b £ L_p (member is not subjected to lateral-torisonal buckling)

2. L_p £ L_b £ L_r (member is subjected to in-elastic lateral-torisonal buckling)

3. L_b > L_r (member is subjected to elastic lateral-torisonal buckling)

Where L_p and L_r are defined as

L_p = 1.76 r_y/ÖE/F_y         [AISC F2-5]

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            [AISC F2-6]

Where r_y is radius of gyration about minor axis

E is elastic modulus

F_y is yield strength

J = torsion constant, in⁴.

c = 1 for double symmetric I - shape.

S_x is elastic section modulus, in³.

h_o is distance between the flange centroids, in 

r_ts = (I_y/C_w)^1/2/S_x.

I_y is moment of inertia in y axis, in⁴.

C_w = warping constant, in⁶.

1. When , L_b £ L_p, M_n is equal to yielding strength.

M_n = M_p = F_yZ_x                 [AISC F2-1]

Where M_p is plastic moment, , Z_x is plastic section modulus about x axis.

2. When L_p < L_b £ L_r

M_n = C_b {M_p - (M_p - 0.75 F_yS_x)[(L_b - L_p)/(L_r-L_p)]}           [AISC F2-2]              

C_b = [12.5M_max/(2.5M_max+3M_A+4M_B+3M_C)] R_m < 3.0        [AISC F1-1]

M_max, M_A,M_B, M_C are absolute moment at maximum, 1/4, 1/2, and ¾ point of unbranced length, 

R_m is cross section monosymmetry parameter,

R_m = 1.0 for double symmetric members and single symmetric members subjected to single curvature bending

R_m = 0.5 + 2 (I_yc/I_y)2 for single symmetric members subjected to reverse curvature bending.

I_yc is moment of inertia about y axis to the compression flange, or smaller flange for reverse curvature bending.

C_b can be conservatively taken as 1.0.

3. When L_b > L_r

            M_n = F_cr S_x £ M_p       [AISC F2-3]

Where

            F_cr = [C_bp²E/(L_b/r_ts)²]Ö{1+0.78 [Jc/(S_xh_o)](L_b/r_ts)²}        [AISC F2-4]

Nominal flexural strength of non-compact W bend about their major axis

Shear strength

Nominal shear strength, resistance factor f_vand safety factor W_v 

Nominal shear strength of beams, 

V_n = 0.6 F_y A_wC_v

Where A_w (= d t_w) is area of web, and C_v is determined as follows:

1. For all current ASTM W, S, and HP shapes except W44x230, W40x149, W36x135, W33x90, W24x55, W16x26, and W12x14 for F_y = 50 ksi steel.

f_v = 1.00, (LRFD)

W_b = 1.5  (ASD)

C_v = 1.0

2. For W44x230, W40x149, W36x135, W33x90, W24x55, W16x26, and W12x14 with F_y = 50 ksi and unstiffened web.

f_v = 0.9, (LRFD)

W_b = 1.67  (ASD)

C_v = 1