ANALYSIS OFA 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/in^2 (27,580-kPa) concrete, and the steel has a yield point of 50,000 lb/in^2 (344,750
kPa). Construct the interaction diagram for this member.Calculation Procedure:- Compute a and M
Consider a composite member of two materi-
als having equal strength in tension and com-
pression, the member being subjected to an
axial load P and bending moment M that in-
duce the allowable stress in one or both ma-
terials. Let P 0 = allowable axial load in ab-
sence of bending moment, as computed by
dividing the allowable ultimate load by a fac-
tor of safety; Mf = allowable bending mo-
FIGURE (^22) ment jn absence of axjai joad 5 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 equa-
tion
£+if=>
Alternate forms of this equation are
*-",(!-£) ^('-f) <
44
«)
p= PaMf
^jjrP ^
Equation 44 is represented by line AB in Fig. 23; it is also valid with respect to a rein-
forced-concrete member for a certain range of values of P and M. This equation is not ap-
plicable in the following instances: (a) If Mis 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 Pb = axial load represented by point E; Mb = bending mo-
ment represented by point E\ M 0 = 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 0 differs from Mf in that the former is based on a cracked section