Handbook of Civil Engineering Calculations

The basic equations for the working-stress de-

sign of a rectangular beam reinforced solely in ten-

sion are

k

-Jw

(21)

J = 1 - f (22)

M = Qd = Vifckjbd

2

(23) ™™J "•

Stress 3

^

result

M=y*fck(3-k)bd

2

(24)

M=Tjd=fsAJd (25)

M=fspjbd^2 (26)

/^P(3-yt)W^2

M

=^V-

(27)

, = f (28)

>-2*lb (29)

k = [2pn + (pw)^2 ]^0 -^5 - PW (30)

For a given set of values offc,fs, and «, Mis directly proportional to the beam proper-
ty bd^2. Let K denote the constant of proportionality. Then

M = Kbd^2 (31)

where

K=V2fckj=fspj (32)

The allowable flexural stress in the concrete and the value of w, which are functions of
the ultimate strength/,', are given in the ACI Code, as is the allowable flexural stress in
the steel. In all instances in the following procedures, the assumption is that the reinforce-
ment is intermediate-grade steel having an allowable stress of 20,000 lb/in^2 (137,900
kPa).
Consider that the load on a beam is gradually increased until a limiting stress is in-
duced. A beam that is so proportioned that the steel and concrete simultaneously attain
their limiting stress is said to be in balanced design. For each set of values offc
f
and^,
there is a corresponding set of values of K 9 k, j, and p associated with balanced design.
These values are recorded in Table 1.