Bending and Axial Loads - Unified Theory of Concrete Structures - page 88

M a )in
Equation (3.51) is chosen to best fit the test results obtained for simply supported beams.
Equation (3.51) is also found to be applicable to cantilever beams if M a is the maximum
moment at the support. For continuous beams the ACI Code allows the effective moment of
inertia to be taken as the average of the values obtained from Equation (3.51) for the critical
positive and negative moment sections.
The effective bending rigidity E c I e , determined by Equation (3.49) and Equations (3.51)-
(3.53), is used to calculate the deflections in conjunction with Equation (3.48).

Equation (3.51) is a modification of Equation (3.50). The power of 3 for ( M cr /

3.2 Nonlinear Bending Theory
3.2.1 Bernoulli Compatibility Truss Model
Linear bending theory has been presented in Section 3.1. This linear theory utilizes the
Bernoulli compatibility truss model, which is based on Navier's three principles of equilib-
rium, Bernoulli's compatibility, and Hooke' linear constitutive laws for both concrete and
steel reinforcement. In this section, a nonlinear bending theory will be studied which also
utilizes the Bernoulli compatibility truss model. In this case, however, nonlinear consti-
tutive laws of concrete and steel will be used, in addition to equilibrium and Bernoulli's
compatibility.
The stress-strain relationship of mild steel is assumed to be linear up to yielding, followed
by a yield plateau. This elastic-perfectly plastic type of stress-strain relationship imparts a
distinct method of analysis and design for concrete members reinforced with mild steel. The
nonlinear bending theory presented in this section is, therefore, applicable only to mild steel
reinforced concrete members at ultimate load stage.
Navier's three principles will first be applied to singly reinforced rectangular beams in
Section 3.2.2, and then to doubly reinforced rectangular beams and flanged beams in Sections
3.2.3 and 3.2.4, respectively. Finally, the moment-curvature relationship of mild steel rein-
forced beams will be discussed in Section 3.2.5, using both the nonlinear bending theory and
the linear bending theory.

3.2.1.1 Equilibrium and Compatibility Conditions

A singly reinforced rectangular beam is subjected to a nominal bending moment M n at ultimate
load stage (Figure 3.9a). The moment M n will induce an ultimate curvature

φ u , which is defined
as the bending rotation per unit length of the member. The fundamental assumption relating
M n and φ u is the well-known Bernoulli compatibility hypothesis, which states that a plane

section before bending will remain plane after bending. In other words, the strains along the
depth of the cross-section will be distributed linearly, as shown in Figure 3.9(b). The maximum
compressive strain of concrete at the top surface is

ε s at the centroid of
the rebars is located at a distance d from the top surface. The neutral axis, where the strain
is zero, is located at a distance c from the top surface. The compatibility equation can then
be established in terms of c ,

ε u . The tensile strain

ε u and ε s . Using the geometric relationship of similar triangles

we have:

c
d =

ε u
ε u + ε s

(3.54)