DESIGN OF REINFORCEMENT IN A
RECTANGULAR BEAM OF GIVEN SIZE
A rectangular beam of 4000-lb/in
2
(27,580-kPa) concrete has a width of 14 in (355.6 mm)
and an effective depth of 23.5 in (596.9 mm). Determine the area of reinforcement if the
beam is to resist a bending moment of (a) 220 ft-kips (298.3 kN-m); (b) 200 ft-kips (271.2
kN'm).
Calculation Procedure:- Calculate the moment capacity of this member
at balanced design
Record the following values:/callow = 1800 lb/in
2
(12,411 kPa); n = 8. From Table I 9 jb =
0.860; Kb = 324 lb/in
2
(2234.6 kPa); M^^bd
2
= 324(14)(23.5)
2
= 2,505,000 in-lb
(283,014.9 N-m). - Determine which material will be stressed to capacity under
the stipulated moment
For part a, M= 220,000(12) = 2,640,000 in-lb (3,579,840 N-m) > Mb. This result signifies
that the beam size is deficient with respect to balanced design, and the concrete will there-
fore be stressed to capacity. - Apply the basic equations in proper sequence to obtain A 3
By Eq. 24, *(3 - k) = 6M/(fcbf) = 6(2,640,000)/[1800(14)(23.5)
2
] = 1.138; A:= 0.446. By
Eq. 29, p = P/[2n(l - k)] = 0.446
2
/[ 16(0.554)] = 0.0224; As = pbd = 0.0224(14)(23.5) =
7.37 in^2 (47.55 lcm^2 ). - Verify the result by evaluating the flexural capacity
of the member
For part b, compute As by the exact method and then describe the approximate method
used in practice. - Determine which material will be stressed to capacity under
the stipulated moment
Here M= 200,000(12) = 2,400,000 in-lb (3,254,400 N-m) < Mb. This result signifies that
the beam size is excessive with respect to balanced design, and the steel will therefore be
stressed to capacity. - Apply the basic equations in proper sequence to obtain A 8
By using Eq. 27, #(3 - K)/(I -k) = 6nMI(fJbf) = 6(8)(2,400,000)/[20,000(14)(23.5)^2 ] =
0.7448; K = 0.411. By Eq. 22, j = 1 - 0.411/3 = 0.863. By Eq. 25, A 5 = Ml(fjd) =
2,400,000/[20,000(0.863)(23.5)] = 5.92 in^2 (38. 196 cm^2 ). - Verify the result by evaluating the flexural capacity
of this member
The value of/ obtained in step 6 differs negligibly from the value jb = 0.860. Consequent-
ly, in those instances where the beam size is only moderately excessive with respect to
balanced design, the practice is to consider that7 =jb and to solve Eq. 25 directly on this
basis. This practice is conservative, and it obviates the need for solving a cubic equation,
thus saving time.