Bending and Axial Loads - Unified Theory of Concrete Structures - page 71

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Bending and Axial Loads

3.1 Linear Bending Theory
3.1.1 Bernoulli Compatibility Truss Model
3.1.1.1 Basic Principles

The analysis of a prismatic member subjected to bending, or flexure, is illustrated in Figure 3.1.
On a rectangular cross-section with width b and height h (Figure 3.1a), a bending moment
M creates compression stresses in the top part of the cross-section and tensile stresses in the
bottom part. Since concrete is weak in tension, the bottom part of the cross-section will crack
and the tensile stresses will be picked up by the reinforcing bars (rebar, in short) indicated
by the area A s . The basic concept of reinforced concrete is to utilize the high compressive
strength of concrete to resist the compression at the top, and the high tensile strength of steel
reinforcement to resist tension at the bottom. This action is similar to a truss, where the top
and bottom chords resist the compressive and tensile forces, respectively.
The linear bending theory described in this section is based on Bernoulli compatibility truss
model as stated in Section 1.3.1.3. This theory is rational and rigorous because it satisfies
Navier's three principles of the mechanics of deformable bodies. First, the stresses in the
concrete and the reinforcement, Figure 3.1(c), satisfy the equilibrium condition. Second,
the linear strain distribution as shown in Figure 3.1(b) satisfies Bernoulli's compatibility
hypothesis, which assumes a plane-cross section to remain plane after the bending deformation.
Third, Hooke's linear law is applicable to both concrete and reinforcing bar as shown in Figure
3.1(e) and (f). As a result, rigorous solutions can be obtained not only up to the yield strength
of a flexural member, but also the flexural deformations.
Hooke's constitutive law is generally assumed to be applicable to both concrete and rebars
up to the yielding of steel and, therefore, up to the service load stage . Bending theory based on
Hooke's law is called the linear bending theory and is studied in this section. The constitutive
laws of concrete and reinforcement are definitely nonlinear at the ultimate load stage. Bending
theory based on a nonlinear stress-strain relationship of concrete will be called the nonlinear
bending theory and will be studied in Section 3.2.
A reinforced concrete member subjected to bending is expected to crack before the service
load stage. The stresses and strains in the rebars at the cracked sections should be greater than