Handbook of Civil Engineering Calculations

2.10 REINFORCED AND PRESTRESSED CONCRETE ENGINEERING AND DESIGN
Calculation Procedure:
1. Compute the values of q b, q max, and p max for a singly
reinforced beam
As the following calculations will show, it is necessary to reinforce the beam both in tension
and in compression. In Fig. 6, let A s area of tension reinforcement, sq.in. (cm 2 ); A s
area of compression reinforcement, sq.in. (cm 2 ); d distance from compression face of
concrete to centroid of compression reinforcement, in. (mm); f s stress in tension steel,
lb/sq.in. (kPa); f sstress in compression steel, lb/sq.in. (kPa); s strain in compression
steel; p A s/(bd); pA s/(bd); q pf y /f c; M u ultimate moment to be resisted by member,
in.·lb (N·m); M u1 ultimate-moment capacity of member if reinforced solely in
tension; M u2 increase in ultimate-moment capacity resulting from use of compression
reinforcement; C u1 resultant force in concrete, lb (N); C u2 resultant force in compression
steel, lb (N).
If f f y, the tension reinforcement may be resolved into two parts having areas of A s
A s and A s. The first part, acting in combination with the concrete, develops the moment
M u1. The second part, acting in combination with the compression reinforcement, develops
the moment M s2.
To ensure that failure will result from yielding of the tension steel rather than crushing
of the concrete, the ACI Code limits p p to a maximum value of 0.75p b, where p b has
the same significance as for a singly reinforced beam. Thus the Code, in effect, permits
setting f sf y if inception of yielding in the compression steel will precede or coincide
with failure of the concrete at balanced-design ultimate moment. This, however, introduces
an inconsistency, for the limit imposed on p p precludes balanced design.
By Eq. 9, q b 0.85(0.80)(87/137) 0.432; q max 0.75(0.432) 0.324; p max
0.324(5/50) 0.0324.
2. Compute M u1, M u2, and C u2
Thus, M u 690,000(12) 8,280,000 in.·lb (935,474.4 N·m). Since two rows of tension
bars are probably required, d 24 3.5 20.5 in. (520.7 mm). By Eq. 6, M u1
0.90(14)(20.5) 2 (5000) (0.324)(0.809) 6,940,000 in.·lb (784,081.2 N·m); M u2
8,280,000 6,940,000 1,340,000 in.·lb (151,393.2 N·m); C u2 M u2/(d d)
1,340,000/(20.5 2.5) 74,400 lb (330,931.2 N).
FIGURE 6. Doubly reinforced rectangular beam.