Susan Faraji

Method of initial functions for thick shells

Abstract In the present work for circular cylindrical shells, three-dimensional elasticity equations are solved by assuming Taylor series expansions, in the radial direction, for the stresses and displacements. Depending upon the number of terms retained in the expansion, different order shell theories are derived., classical theories (referred to as eighth-order), the shear deformation-transverse normal stress theories (referred to as tenth-order), and higher order theories (referred to as twelfth-order). In each case, by carrying out the symbolic algebra using the digital computer, partial differential equations are derived. The procedure was carried out in detail for the case of a circular cylindrical shell with no loading on the interior surface and a given pressure distribution on the exterior surface. Then, numerical comparisons are made between the current theories and various shell theories, as well as the exact (three dimensional) theory. Thus, using this method with its associated computer programs, one can realize a spectrum of approximate shell theories ranging from the classical thin shell, through all current thick shell theories, right up to the three-dimensional elastic theories.

Inelastic Response of Structures

The conventional design approach works with factored loads and reduced material capacities such as strength. When subjected to service loads, the structure is detailed such that it behaves elastically. For extreme loadings, the structure is allowed to experience a limited amount of deformation beyond the elastic limit. This deformation is called “inelastic” since in contrast to elastic deformation, when the loading is removed, the structure does not return to its original position. Up to this point in the text, we have assumed the behavior to be elastic. In Chap. 10, we included geometric nonlinear effects but still assumed elastic behavior. Here, we introduce an additional effect, inelastic behavior. We start with an in-depth discussion of the stress–strain behavior of structural steels and concrete, apply these ideas to beams subjected to inelastic bending, and then develop an analysis procedure to determine the inelastic response of frame-type structures. This approach allows one to estimate the “maximum” loading that a structure can support, i.e., the “limit load.” Examples illustrating the influence of inelastic behavior on the ultimate capacity are included.

Statically Determinate Truss Structures

We begin this chapter by reviewing the historical development of truss structures. Trusses have played a key role in the expansion of the highway and railroad systems during the past two centuries. From a mechanics perspective, they are ideal structures for introducing the concepts of equilibrium and displacement. We deal first with the issues of stability and static determinacy, and then move on to describe manual and computer-based techniques for determining the internal forces generated by external loads. A computational scheme for determining the displacements of truss structures is presented next. Given a structure, one needs information concerning how the internal forces vary as the external live load is repositioned on the structure for the design phase. This type of information is provided by an influence line. We introduce influence lines in the last section of this chapter and illustrate how they are constructed for typical trusses. This book focuses on linear elastic structural analysis. Although nonlinear structural analysis is playing an increasingly more important rule in structural design, we believe an understanding of linear analysis is essential before discussing the topic of nonlinear analysis.

Vertical Retaining Wall Structures

Vertical wall type structures function as barriers whose purpose is to prevent a material from entering a certain space. Typical applications are embankment walls, bridge abutments, and as underground basement walls. Structural Engineers are responsible for the design of these structures. The loading acting on a retaining wall is generally due to the soil that is confined behind the wall. Various theories have been proposed in the literature, and it appears that all the theories predict similar loading results. In this chapter, we describe the Rankine theory that is fairly simple to apply. We present the governing equations for various design scenarios, and illustrate their application to typical retaining structures. The most critical concerns for retaining walls are ensuring stability with respect to sliding and overturning, and identifying the regions of positive and negative moment in the wall segments. Some of the material developed in Chap. 7 is also applicable for retaining wall structures.

The Force Method

Up to this point, we have focused on the analysis of statically determinate structures because the analysis process is fairly straightforward; only the force equilibrium equations are required to determine the member forces. However, there is another category of structures, called statically indeterminate structures, which are also employed in practice. Indeterminate structures require another set of equations, in addition to the force equilibrium equations, in order to solve for the member forces. There are two general methods for analyzing indeterminate structures, the force (flexibility) method and the displacement (stiffness) method. The force method is more suited to hand computation whereas the displacement method is more procedural and easily automated using a digital computer.

The Displacement Method

The previous chapter dealt with the force method, one of two procedures for analyzing statically indeterminate structures. In this chapter, we describe the second procedure, referred to as the displacement method. This method works with equilibrium equations expressed in terms of variables that correspond to displacement measures that define the position of a structure, such as translations and rotations of certain points on the structure. We start by briefly introducing the method specialized for frame-type structures and then apply it to truss, beam, and frame structures. Our focus in this chapter is on deriving analytical solutions and using these solutions to explain structural behavior trends. We also include a discussion of the effect of geometrically nonlinear behavior on the stiffness. Later in Chap. 12, we describe how the method can be transformed to a computer-based analysis procedure.

Vertical Loads on Multistory Buildings

The previous chapter dealt with issues related to the lateral loadings on building systems. In that chapter, we described how one can represent the global lateral loading as loads acting on the individual frames contained in the building system. We focus in this chapter on how one treats vertical loads such as gravity loads. Gravity loads applied to a floor slab are converted to distributed loads acting on the beams which support the slab. Since the floor slab loads involve both dead and live loads, one needs to investigate various floor slab loading patterns in order to establish the maximum values of the design parameters such as bending moment. We apply Muller-Breslau principle for this task. The last section of the chapter contains a case study which illustrates the process of combining lateral and vertical loading, and demonstrates the sensitivity of the structural design to the type of structural system.

Statically Determinate Plane Frames

Plane frame structures are composed of structural members which lie in a single plane. When loaded in this plane, they are subjected to both bending and axial action. Of particular interest are the shear and moment distributions for the members due to gravity and lateral loadings. We describe in this chapter analysis strategies for typical statically determinate single story frames. Numerous examples illustrating the response are presented to provide the reader with insight as to the behavior o\f these structural types. We also describe how the Method of Virtual Forces can be applied to compute displacements of frames. The theory for frame structures is based on the theory of beams presented in Chap. 3. Later in Chaps. 9, 10, and 15, we extend the discussion to deal with statically indeterminate frames and space frames.

Statically Determinate Beams

Our focus in this chapter is on describing how beams behave under transverse loading, i.e., when the loading acts normal to the longitudinal axes. This problem is called the “beam bending” problem. The first step in the analysis of a statically determinate beam is the determination of the reactions. Given the reactions, one can establish the internal forces using equilibrium-based procedures. These forces generate deformations that cause the beam to displace. We discuss in detail the relationship between the internal forces and the corresponding displacements and describe two quantitative analysis procedures for establishing the displacements due to a particular loading. The last section of the chapter presents some basic analysis strategies employed in the design of beams such as influence lines and global envelopes.

Approximate Methods for Estimating Forces in Statically Indeterminate Structures

In this chapter, we describe some approximate methods for estimating the forces in indeterminate structures. We start with multi-span beams subjected to gravity loading. Next, we treat rigid frame structures under gravity loading. Then, we consider rigid frame structures under lateral loading. For this case, we distinguish between short and tall buildings. For short buildings, we first describe the portal method, an empirical procedure, for estimating the shear forces in the columns, and then present an approximate stiffness approach which is more exact but less convenient to apply. For tall buildings, we model them as beams and use beam theory to estimate the forces in the columns. With all the approximate methods, our goal is to use simple hand calculation-based methods to estimate the forces which are needed for preliminary design and also for checking computer-based analysis methods.