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WORKING STRESS DESIGN EQUATIONS - RECTANGULAR SECTIONS





The design assumptions, coefficients and cross section geometry used in the derivation of the flexural design equations for rectangular sections are illustrated in the figure above with equations provided below.

In the design of reinforced rectangular sections, the first step is to locate the neutral axis. This can be accomplished by determining the coefficient, k, which is the ratio of the depth of the compressive stress block to the total depth from the compression face to the reinforcing steel, d.

k is derived by equating the moment of the transformed steel area about the centroidal axis of the cross section to the moment of the compression area about the centroidal axis as follows:

b(kd)[kd/2] = nA_s (d - kd)

OR

b(kd)² /2 - nA_s (d - kd) = 0

The steel ratio, p, is determined by:
p = A_s/bd

Substituting phd for A_s
b(kd)2/2 - npbd(d - kd) = 0

Dividing through by bd²
k²/2 - pn(1 - k) = 0

From which;
k = [ (np² + 2np ]^0.5 -np

The coefficient j, which is the ratio of the distance between the resultant compressive force and the centroid of the tensile force to the distance d, is determined by
j = 1 - k/3

The balanced steel ratio in the working stress design method, p , is defined as the reinforcing ratio where the steel and the masonry reach their maximum allowable stresses for the same applied moment. p is determined as follows:
p_e = n / [2(F_s/F_m) (n + F_s/F_m)]
Where:
Fs = The allowable tensile stress in the reinforcing steel, psi.
Fm = The allowable flexural compressive stress in the masonry, psi.

The working stresses for the steel and the masonry are computed as follows:

If p < p_e , the steel stress, f_s , will reach its allowable stress before the masonry and the following equation will control.
f_s = M/(A_sjd)

If p > p_e , the masonry stress, f_m , will reach its allowable stress before the steel and the following equation will control.
f_m = 2M/(kjbd²)

The resisting moment for the reinforcement, M_s , can be determined by substituting the allowable steel stress, F_s , for the computed steel stress in the above equation and solving for the moment.
M_rs = F_sA_sjd/12

The resisting moment for masonry, M_m , can be determined by substituting the allowable masonry stress, F_m , for the computed masonry stress in the above equation and solving for the moment.
M_m = F_mkjbd²/(2(12))

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