Page 253 - Handbook of Civil Engineering Calculations, Second Edition

2.38 REINFORCED AND PRESTRESSED CONCRETE ENGINEERING AND DESIGN

ANALYSIS OF A RECTANGULAR MEMBER BY
INTERACTION DIAGRAM

A short tied member having the cross section shown in Fig. 22 is to resist an axial load
and a bending moment that induces rotation about axis N. The member is made of 4000-
lb/sq.in. (27,580-kPa) concrete, and the steel has a yield point of 50,000 lb/sq.in. (344,750
kPa). Construct the interaction diagram for this member.

Calculation Procedure:

1. Compute a and M
Consider a composite member of two
materials having equal strength in ten-
sion and compression, the member being
subjected to an axial load P and bending
moment M that induce the allowable
stress in one or both materials. Let P a
allowable axial load in absence of bend-
ing moment, as computed by dividing the
allowable ultimate load by a factor of
safety; M f allowable bending moment
FIGURE 22
in absence of axial load, as computed by
dividing the allowable ultimate moment
by a factor of safety.
Find the simultaneous allowable values of P and M by applying the interaction equation

P M
1 (44)
P a M f
Alternative forms of this equation are

P
M
M M f 1 P P a 1 (44a)
P a
M f

P a M f
P (44b)
M f P a M/
P
Equation 44 is represented by line AB in Fig. 23; it is also valid with respect to a
reinforced-concrete member for a certain range of values of P and M. This equation is not
applicable in the following instances: (a) If M is relatively small, Eq. 44 yields a value of
P in excess of that given by Eq. 41. Therefore, the interaction diagram must contain line
CD, which represents the maximum value of P.
(b) If M is relatively large, the section will crack, and the equal-strength assumption
underlying Eq. 44 becomes untenable.
Let point E represent the set of values of P and M that will cause cracking in the ex-
treme concrete fiber. And let P b axial load represented by point E; M b bending mo-
ment represented by point E; M o allowable bending moment in reinforced-concrete
member in absence of axial load, as computed by dividing the allowable ultimate moment
by a factor of safety. (M o differs from M f in that the former is based on a cracked section