(a) Section (b) Strains (c) Stresses (d) Resultant
forces
FIGURE 20Calculation Procedure:
- Compute the value of c associated with balanced design
An interaction diagram, as the term is used here, is one in which every point on the curve
represents a set of simultaneous values of the ultimate moment and allowable ultimate ax-
ial load. Let €A and eB = strain of reinforcement at A and B, respectively; ec = strain of ex-
treme fiber of concrete; FA and FB = stress in reinforcement at A and B, respectively,
lb/in^2 (kPa); FA and F 8 = resultant force in reinforcement at A and 5, respectively; Fc =
resultant force in concrete, Ib (N).
Compression will be considered positive and tension negative. For simplicity, disre-
gard the slight reduction in concrete area caused by the steel at A.
Referring to Fig. 2OZ?, compute the value of c associated with balanced design. Com-
puting Pb and Mb yields cbld = 0.0037(0.003 +fy/Es) = 87,000/(87,0OO +£); Cb9 = 10.62 in
(269.748 mm). Then eAleB = (10.62 - 2.5)7(15.5 - 10.62) > 1; therefore,/^ =£; ab =
0.85(10.62) = 9.03 in (229.362 mm); F 0 = 0.85(3000)(12^) = 276,300 Ib (1,228,982.4
N); FA = 40,000(2.00) = 80,000 Ib ((355,84O N); F 3 = - 80,000 Ib (-355,840 N); Pb =
0.70(276,300) = 193,400 Ib (860,243.2 N). Also,
r Fc(t - d) (FA - FB)(t -2d'}-\
Mb = 0.70 I -^-^ + — y -I (38)Thus, Mb = 0.70(276,300(18 - 9.03)72 + 160,000(6.5)] = 1,596,000 in-lb (180,316.1
N-m).
When c > cb9 the member fails by crushing of the concrete; when c < cb, it fails by
yielding of the reinforcement at line B.
- Compute the value of c associated with incipient yielding
of the compression steel
Compute the corresponding values of P 11 and M 14. Since eA and es are numerically equal,
the neutral axis lies at N. Thus, c = 9 in (228.6 mm); a = 0.85(9) = 7.65 in (194.31 mm);
F 0 = 30,600(7.65) = 234,100 Ib (1,041,276.8 N); FA = 80,000 Ib (355,840 N); FB =
-80,000 Ib (- 355,840 N); Pn = 0.70 (234,100) = 163,900 Ib (729,027.2 N); Mu =
0.70(234,100 x 5.18 + 160,000 x 6.5) = 1,577,000 in-lb (178,169.5 N-m). - Compute the minimum value of c at which the entire concrete
area is stressed to 085 f^
Compute the corresponding values of P 11 and Mu. Thus, a = t = 18 in (457.2 mm);