Unilateral Damage in Reinforced Concrete Frames

One of the main applications of the lumped damage mechanics or the damage mechanics of dual systems is the earthquake vulnerability assessment of structures. This means not only the consideration of the inertia forces but, mainly, the adequate description of crack propagation under general cyclic loading. Chapter 9 described the concept of unilateral damage (i.e. the appearance of distinct and independent sets of cracks after loading reversals). This phenomenon can also be observed in RC structures, and the models presented in Chapters 10 and 11 do not describe it; thus, they should be used only in the cases of mono sign loadings. The first goal of this chapter is the generalization of the damage models, including unilateral effects; the next one consists of the development of lumped damage models for tridimensional analysis of RC frames. Finally, some guidelines for the use of the damage models in industrial applications are presented.

The Plastic Hinge

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.

Fundamental Concepts of Strength of Materials

This chapter presents the concepts of strength of materials that are relevant to the analysis of frames. These are the modified Timoshenko theory of elastic beams (Sections 2.1-2.3) and the Euler-Bernoulli one (Section 2.4). These concepts are not presented as in the conventional textbooks of strength of materials. Instead, the formulations are described using the scheme that is customary in the theory of elasticity and that was described in Chapter 1 (Section 1.1.1) (i.e. in terms of kinematics, statics, and constitutive equations). Kinematics is the branch of mechanics that studies the movement of solids and structures without considering its causes. Statics studies the equilibrium of forces; the basic tool for this analysis is the principle of virtual work. The constitutive model that describes a one-to-one relationship between stresses and deformations completes the formulation of the elastic beam problem. Finally, in Section 2.5, some concepts of the elementary theory of torsion needed for the formulation of tridimensional frames are recalled.

Industrial Applications

The application of structural analysis techniques to solve real engineering problems is an entirely independent discipline by itself that cannot be properly presented in a book of structural mechanics. However, it is important to give an overview of how mathematical models can help make engineering decisions. This is the subject of the current chapter. The context of the presentation is that of earthquake safety assessment. Of course, this is not the only industrial application of the fracture and damage mechanics of frames, but it is a very representative one and a good example of it. The chapter is organized as follows. First, the problem is presented and a protocol to solve it is described in Section 13.1. Then, an academic software that can be accessed via Internet is described in Section 13.2. This program is used to solve some examples of real structures in the last section of the chapter.

Damage Mechanics of Dual Systems

This chapter begins with the presentation of some experimental results on RC specimens using a special technique called “digital image correlation.” Then, it describes a damage model for RC walls. Next, the model is generalized to include elements with any aspect ratio; finally, the analysis of dual system is described and some numerical simulations are presented. Notice that Section 3.4 described the elastic behavior of dual systems; in section 7.3 that model was extended to include plastic deformations. The goal of this chapter is to generalize that model, including cracking propagation described by the Griffith criterion or its modified version.

Elasto-Plastic Frames

The goal of this chapter is to describe how the concepts presented in Chapters 5 and 6 can be included in the mathematical models for the elastic plastic analysis of frame structures. The numerical implementation of such an analysis is described in Chapter 8. The models presented in this chapter cover applications for reinforced concrete frames, shear walls, wide beams, and dual systems, as well as steel structures. Both cases, planar and tridimensional analyses, are considered. However, this chapter does not yet describe the numerical and computational analysis of elasto-plastic structures; this is the subject of the next chapter.

Analysis of Elastic Frames

The formulation of a mathematical model that describes some physical phenomenon is just a first step; a second one, equally important, is the development of numerical procedures that transform this model into a potentially predictive tool with practical engineering applications; no computer software can be developed without robust numerical algorithms. This chapter describes some of these procedures in the case of elastic frames. First, it considers the direct stiffness method that permits the analysis of linear elastic and quasi static structures (Section 4.1); then, the procedure is extended to the more complex cases of nonlinear structures (Section 4.2) and dynamic loading (Section 4.3).

Elastic Frames

This chapter describes the mathematical models for the analysis of elastic frames. Again, these theories are not presented as in the conventional textbooks of structural analysis but using the scheme of continuum mechanics and the theory of elasticity that was described in Chapter 1 (Section 1.1.1). The reason is that the conventional presentation is not suitable for generalization of the case of inelastic structures, specifically for fracture and damage mechanics of frames. Several classes of elastic frames are described in this chapter: planar (Sections 3.1-3.4), tridimensional (Section 3.5), linear, nonlinear, based on the Euler-Bernoulli beam theory (Section 3.3), based on the formulation of Timoshenko (Section 3.4), and under quasi-static or dynamic forces.

Lumped Damage Mechanics

This chapter, the last of the book, describes how to use the concepts of damage mechanics for the description of the behavior of tubular frame structures. In the first section of this chapter, the concept of damage of a plastic hinge is used to describe local buckling evolution. It also shows that the technique of the variation of the elastic stiffness, described in Chapter 10, can be utilized to measure the degree of local buckling in the metallic elements. This first section is restricted to the analysis of frames subjected to mono-sign loadings. The second one deals with the behavior of the structures under general loadings in the plane. It shows that in the case of cyclic loadings with reversal of sign a new phenomenon appears: “counter-buckling”; in metaphoric terms, counter-buckling can be described as “ironing the wrinkles.” In this section, this effect is characterized and modeled introducing the concept of “local buckling driving rotation.” Finally, in the third section of the chapter, the analysis of tridimensional frames is addressed.

Fundamental Concepts of Fracture and Continuum Damage Mechanics

Some fundamental concepts of fracture mechanics, those needed for the description of concrete cracking in framed structures, are presented in a simplified way in the first section of this chapter. The second section introduces the fundamentals of continuum damage mechanics. The third section describes a physical phenomenon called localization; this is a very important effect during the process of structural collapse. In this section, it is shown that damage mechanics can lead to ill-posed mathematical problems. Finally, the relationship between localization and ill-posedness is discussed. In this chapter, fracture and damage mechanics or localization concepts are not yet applied to the analysis of framed structures; that is the subject of the following chapters.