Page 238 - Handbook of Civil Engineering Calculations, Second Edition

REINFORCED CONCRETE 2.23

DESIGN OF REINFORCEMENT IN A
RECTANGULAR BEAM OF GIVEN SIZE

A rectangular beam of 4000-lb/sq.in. (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:

1. Calculate the moment capacity of this member
at balanced design
Record the following values: f c,allow 1800 lb/sq.in. (12,411 kPa); n 8. From Table 1,
2
2
j b 0.860; K b 324 lb/sq.in. (2234.0 kPa); M b K b bd 324(14)(23.5) 2,505,000
in.·lb (283,014.9 N·m).
2. 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) > M b . This result signi-
fies that the beam size is deficient with respect to balanced design, and the concrete will
therefore be stressed to capacity.

3. Apply the basic equations in proper sequence to obtain A s
2
2
By Eq. 24, k(3 k) 6M/( f c bd ) 6(2,640,000)/[1800(14)(23.5) ] 1.138; k 0.446.
2
By Eq. 29, p k /[2n(1 k)] 0.446 /[16(0.554)] 0.0224; A s pbd
2
2
0.0224(14)(23.5) 7.37 sq.in. (47.551 cm ).
4. Verify the result by evaluating the flexural capacity
of the member
For part b, compute A s by the exact method and then describe the approximate method
used in practice.
5. 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) < M b . This result signifies
that the beam size is excessive with respect to balanced design, and the steel will therefore
be stressed to capacity.

6. Apply the basic equations in proper sequence to obtain A s
2
By using Eq. 27, k (3 k)/(1 k) 6nM/(f s bd ) 6(8)(2,400,000)/[20,000(14)(23.5) ]
2
2
0.7448; k 0.411. By Eq. 22, j 1 0.411/3 0.863. By Eq. 25, A s M/( f s jd)
2
2,400,000/[20,000(0.863)(23.5)] 5.92 sq.in. (38.196 cm ).
7. Verify the result by evaluating the flexural capacity
of this member
The value of j obtained in step 6 differs negligibly from the value j b 0.860. Conse-
quently, in those instances where the beam size is only moderately excessive with respect
to balanced design, the practice is to consider that j j b and to solve Eq. 25 directly on
this basis. This practice is conservative, and it obviates the need for solving a cubic equa-
tion, thus saving time.