The plastic hinge is a key concept of the theory of frames that differentiates this theory from the remaining models for structural analysis. This chapter is exclusively dedicated to define this concept and describe the different models of plastic hinges. It also discusses the differences of implementation between plastic hinges in steel frames (Sections 6.1-6.4) and those in reinforced concrete structures (Sections 6.5-6.6). This chapter is based on the ideas presented in Chapter 5 and it allows formulating the models for elasto-plastic frames that are introduced in the next chapter.
Top6.1 Elasto-Plastic Moment Vs. Curvature Relationship
Consider a simply supported beam, as the one shown in Figure 1a, subjected to a concentrated force at its mid-section; the force is monotonically increased until total collapse of the structure. The beam has a symmetric cross-section of height h, it is homogeneous and its material follows the elasto-perfect plastic constitutive model described in section 5.2. The bending moment distribution along the beam and the stress and strain distributions in the mid-section B (the most loaded) are shown in Figures 1b, 1c and 1d.
Figure 1.
a) Simply supported beam; Flexural moment distribution along the beam, strain distribution in the mid-cross-section and stress distribution at: b) Last elastic stress, c) Elasto-plastic stress, d) Ultimate stress
According to the fundamental hypothesis of the beam theory, the strain distribution in the cross-section is always linear, even if the behavior is not elastic:
(6.1.1) where
χ is the curvature and
z is the coordinate of the fiber under consideration as shown Figure 1b (see chapter 2). The elastic relationship between moment and curvature is:
(6.1.2)This last expression does not apply if the behavior is elasto-plastic. Elastic behavior in the beam ends when the maximum stress at the mid-section is equal to the yield stress as shown in Figure 1b. The normal stress distribution is still linear at this instant, and according to the Hooke's law, it is proportional to the strain values: 
.The bending moment in the section corresponding to this load is called first plastic moment Mp:
(6.1.3)S is called elastic section modulus. For larger values of the force (Figure 1c), some of the fibers in the cross-section begin yielding at constant stress. As the force and curvature increase, more and more fibers yield. As the curvature tends to infinity the bending moment tends to the ultimate bending strength of the section; this value is called ultimate moment
Mu and can be computed assuming that all the fibers in the cross-section have plastic behavior (Figure 1d):
(6.1.4)Hf is called plastic section modulus. Examples of computation of the first plastic moment and the ultimate one for different cross-sections are presented in section 6.8.
The moment curvature relationship in the mid-section of the beam is shown in Figure 2.
Figure 2. Moment-curvature relationship at the mid-section of the beam