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2016 | Buch

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Structures and Their Analysis

verfasst von: Maurice Bernard Fuchs

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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Über dieses Buch

Addressing structures, this book presents a classic discipline in a modern setting by combining illustrated examples with insights into the solutions. It is the fruit of the author’s many years of teaching the subject and of just as many years of research into the design of optimal structures. Although intended for an advanced level of instruction it has an undergraduate course at its core. Further, the book was written with the advantage of having massive computer power in the background, an aspect which changes the entire approach to many engineering disciplines and in particular to structures. This paradigm shift has dislodged the force (flexibility) method from its former prominence and paved the way for the displacement (stiffness) method, despite the multitude of linear equations it spawns. In this book, however, both methods are taught: the force method offers a perfect vehicle for understanding structural behavior, bearing in mind that it is the displacement method which does the heavy number crunching. As a rule the book keeps things as simple as possible, conveying the basic ideas and refraining from lengthy calculations wherever possible. Further, it endeavors to unify the approach, showing that whatever applies to simple springs is equally valid for intricate frames. In addition to various design considerations, it also addresses several topics relating to optimal structures that will be of interest to students and teachers of structures.

Inhaltsverzeichnis

Frontmatter

Preliminaries

Frontmatter

Chapter 1. Equilibrium

Abstract

This chapter deals with forces and couples and recalls the basic laws of static equilibrium. We emphasize the important concept of equivalent forces and show that a system of forces is in static equilibrium if it is statically equivalent to zero. In other words, a system of forces is in static equilibrium if it is equivalent to no forces at all. We also postulate the existence of internal forces, better known as stresses, without which the principles of equilibrium do not make sense. There are a myriad of internal forces which come in pairs; no-one has ever seen them, but these forces are essential for equilibrium.

Maurice Bernard Fuchs

Chapter 2. The Diagrams

Abstract

The structures in this book are composed of beams and bars (and cables). These are long and relatively slender line elements which have a longitudinal axis passing through the centers of area of the cross-sections. The coordinate along the axis is x. On both sides of virtual sections perpendicular to the axis we find equal and opposite internal forces: normal forces n acting along the axis, shear forces s perpendicular to the axis (along the faces of the section), and equal and opposite couples, the bending moments m. Calculating these internal forces throughout the structure is the main purpose of structural analysis. This chapter shows how to compute and present the internal forces along any line element if the external (applied) forces are all known. The internal forces are functions of x, leading to the normal force distribution n(x), the shear force distribution s(x) and the bending moment distribution m(x) or diagrams, also called nsm(x) or n(x) for short. These distributions are drawn along the elements in a traditional manner, and we will show how to compute and draw these distributions or diagrams.

Maurice Bernard Fuchs

Chapter 3. Virtual Work

Abstract

Virtual Work is a concept which will help us to develop most of the properties and methods pertaining to the theory of structures. In parallel with ‘equilibrium systems’ between external and internal loads (Chap. 1), we independently construct ‘compatible systems’ composed of deformations which result from displacements. It is shown that the virtual work of the external loads through the displacements is equal to the virtual work of the internal forces through the deformations. This is the principle of virtual work. The concept is illustrated with simple examples of discrete particles, and is further developed for a beam segment under continuous and point loads undergoing an arbitrary displacement.

Maurice Bernard Fuchs

Chapter 4. Deformations

Abstract

Quite independently of what has been done till this point, in this chapter, we develop a Theory of Deformation, based on Bernoulli’s assumption of rigid sections. We show that the deformations are exactly those suggested by virtual work: \(\epsilon (x), \gamma (x)\) and \(\kappa (x)\) as a function of the displacements of the rigid ‘discs’, u(x), v(x) and \(\theta (x)\). A physical interpretation is given: \(\epsilon (x)\) is the extensional deformation (axial stain), \(\gamma (x)\) is the shear deformation (shear strain) and \(\kappa (x)\) is the bending deformation (flexural strain).

Maurice Bernard Fuchs

Chapter 5. Elasticity

Abstract

A major player in the behavior of a structure is the material it is made off. The stone columns of Greek temples or the wrought iron elements of the Eiffel tower may perform similar tasks but they do it in a very different manner. In this chapter we introduce the linear elastic materials composing the structures described in this book.

Maurice Bernard Fuchs

Chapter 6. The Unit-Load Method

Abstract

In structural analysis, displacements are not always given the importance they deserve and are often even considered a nuisance. In truth, displacements lie at the heart of how structures sustain loads. In some sense they are the engine that drives the structural response. The unit-load method is a technique that will help us to quantify displacements and rotations of the equilibrium configuration, that is, the shape of the structure after it has managed to equilibrate the applied loads. We will introduce the method with the aid of beams, and show how the unit-load method computes a displacement or a rotation at some point along the beam.

Maurice Bernard Fuchs

The Structures

Frontmatter

Chapter 7. Types of Structures

Abstract

Structures are composed of prismatic shaped elements, such as bars, beams, and columns, rigidly or partially connected to one another or to the ground. We will classify these structures by the type of internal loads found inside the structure when external loads are applied to them or when they are subjected to temperature fluctuations. The most general type of skeletal structures are frames, a linear subset of which are beams. Trusses, which have only normal internal forces, constitute a class apart. This chapter also discusses the type of connections that can exist between elements and between the structure and ground.

Maurice Bernard Fuchs

Chapter 8. Structural Analysis

Abstract

Analyzing a structure is synonymous to computing its internal forces. There are several approaches to the analysis of a structure but they can all be reduced to two: the force method and the displacement method. Whilst the former was historically paramount and is still important for simplified models of structures, the displacement method is nowadays the main analysis tool. In this chapter we will analyze a simple structure by both methods to highlight their main features.

Maurice Bernard Fuchs

Chapter 9. Qualitative Analysis and Design

Abstract

Before embarking on a full analysis of a structure, it is often useful to have an idea of what the final result should look like. Qualitative analysis can give you the general form of the nsm-diagrams without actually performing an analysis, that is, without going through the calculations that provide the values of the internal forces or reactions. For complex structures under complicated loading this may be easier said than done. But our aim is also didactic. It will assist us in performing and especially understanding the results of a real analysis. We will be mainly concerned with redundant beams and with some elementary frame examples. The method is to a large extent visual, and it uses an holistic approach in the sense that we employ everything we know about structural analysis, especially equilibrium, in order to improve the approximate result we are working on. The key to performing a qualitative analysis is Hooke’s law for a beam element in bending \(m=(EI)\;\kappa \), where EI is considered constant.

Maurice Bernard Fuchs

Flexibility

Frontmatter

Chapter 10. Flexibility Coefficients

Abstract

In this chapter we will apply virtual work using equilibrium systems and compatible systems taken from a same real structure. For every deformed shape of the structure we can derive an equilibrium system and a compatible system. Taking the equilibrium system from one deformed shape and the compatible system from another deformed shape, and vice versa, will eventually lead to Betti’s reciprocal theorem and to Maxwell’s flexibility coefficients.

Maurice Bernard Fuchs

Chapter 11. Redundancy

Abstract

From an analysis aspect, determining the degree of static redundancy of a structure is a prerequisite of the force method. As such, this chapter can be considered as a first step in the force method. For the displacement method we do not need to know the degree of redundancy. Redundancy also tells us how far the structure is removed from being statically determinate, a condition which is notoriously unsafe. As such, the redundancy is also important from a design viewpoint irrespective of the method of analysis. It tells us something regarding the behavior of the structure. Recognizing the redundancy, that is, safety, stiffness, prestressing issues and the like, or designing for determinacy in order to prevent locked stresses, will be discussed at the end of the chapter.

Maurice Bernard Fuchs

Chapter 12. The Force Method

Abstract

The Force method is the first of the two main analysis methods presented in this book. It is intended for relatively simple structures of modest redundancy. Although the method is quite general it leads to heavy computations for complex structures. Historically it was practically the only method available until the advent of computational techniques. This paved the way to the systematic and relatively simple Displacement method which is intended for computerized solutions. The Force method has retained its position in structural analysis because it is intuitive and it lends itself to a better understanding of the behaviour of a structure and its response to applied loads and applied deformations such as heating.

Maurice Bernard Fuchs

Chapter 13. Applied Strains and Initial Stresses

Abstract

In the present chapter we show how a redundant structure under temperature variation and similar events is analyzed. The force method is used in its standard form, but instead of applying loads we apply deformations. As we will show, this leads to some simplifications but also to intriguing final deformation distributions.

Maurice Bernard Fuchs

Stiffness

Frontmatter

Chapter 14. Introducing the Stiffness Method

Abstract

The stiffness or displacement method for analyzing structures is exemplified by means of trusses. This type of structure has been relatively neglected until now because, in the view of the author, truss analysis by the force method is numerically too cumbersome. The stiffness method, on the other hand, is totally independent of the type of structure and applies with equal ease to trusses, beams and frames. We, of course, bear in mind that the computations are all computerized.

Maurice Bernard Fuchs

Chapter 15. Element Stiffness Matrices

Abstract

In the stiffness or displacement method for analyzing a structure, we start by figuratively dissecting the structure into simple segments which are called elements. If we know the stiffness matrices of these elements the problem is by and large solved (usually by a computer). In this chapter we will mainly show how we can compute the stiffness matrices of the ubiquitous elements of skeletal structures: beams, rods and springs.

Maurice Bernard Fuchs

Chapter 16. Change of Coordinates

Abstract

Element stiffness matrices are developed in local coordinates which are often not the same as the system coordinates of the structure. A typical example is when the elements of the structure have different inclinations. This requires a change of coordinates. We show in this chapter how congruent transformations accommodate such cases and several relevant ones.

Maurice Bernard Fuchs

Chapter 17. Assembling the System Stiffness Matrix

Abstract

Having decomposed the structures into nodes and elements, it is clear that knowing the nodal displacements will give us the internal loads in the elements. The displacements at the closed degrees of freedom are known, but we must still compute the displacements at the open degrees of freedom. This is done by writing and solving the equilibrium equations at the open degrees of freedom. We show, in this chapter, that instead of writing the equilibrium equations from first principles, we can instead assemble the system stiffness matrix by a relatively simple algorithmic procedure.

Maurice Bernard Fuchs

Chapter 18. Loads on Elements

Abstract

At this juncture we have the major ingredients of the stiffness method. There is, however, one important aspect that has still not been addressed. You will remember that, all that has been done relates to structures with point forces and point couples applied to the open degrees of freedom. But, more often than not loads are applied to elements, be they concentrated loads or distributed ones. In this chapter we will see how such element loads can be replaced by equivalent loads applied at the nodes.

Maurice Bernard Fuchs

Four Additional Topics

Frontmatter

Chapter 19. A Strain Energy Theorem—Moving Supports

Abstract

Consider an elastic beam with a set of given supports, subjected to external loads, such as the cantilever beam in Fig. 19.1a which is subjected to distributed and point forces. We now want to move the simple support to maximize the ‘stiffness’ of the beam with respect to that loading. As is customary in this sort of problem, we will seek to position the support such as to minimize the ‘strain energy’ U of the loaded beam, which is a measure of its ‘flexibility’. We have not discussed the notion of strain energy in a structure until now so we will start with defining that quantity.

Maurice Bernard Fuchs

Chapter 20. Fully-Stressed Trusses

Abstract

The notion of fully-stressed structures is very appealing in structural design. The basic definition of a fully-stressed structure is a structure where every part is stressed to the maximum permissible stress of the material it is made of. An intuitive premise of most designers is that such a structure, even if not optimal, cannot be that bad. However, when we try to design such a structure, we note that unless we relax the definition of fully-stressed design (f.s.d.) it is not that easy. We will address the complication in the case of trusses. Although redundant trusses can in general not be fully stressed we present an intriguing exception: the circle-chord truss. That structure under some restrictions is both redundant and fully stressed.

Maurice Bernard Fuchs

Chapter 21. Frames Viewed as Generalized Trusses

Abstract

A truss is often considered as a structure relatively different from a frame. The truss is composed of unimodal normal elements and when properly triangulated it opposes the external forces applied at the nodes by normal internal forces, constant in every element. It is shown that a frame can likewise be considered as being composed of unimodal normal, shear and moment elements and when subjected to nodal loads it behaves in many respects as a generalized truss.

Maurice Bernard Fuchs

Chapter 22. Explicit Analysis

Abstract

Given a structure, we have to solve equations in order to calculate the distribution of the internal forces in that structure, be it the compatibility equations if one uses the force method, or the equilibrium equations of the displacement method. One way or the other, we solve a system of linear equations. This is what makes structural analysis an implicit affair. In the realm of design we often need to redesign and hence reanalyze the structure several times before reaching a satisfactory solution. This may prove cumbersome. There are, however, relations which allow us to write the internal forces of a truss explicitly in terms of the stiffnesses thus bypassing any need for reanalysis. The explicit equations have a major shortcoming but they are theoretically very interesting. We will present here the explicit expressions of the internal forces as a function of the stiffnesses of the bars of the truss and indicate a way to prove the relations.

Maurice Bernard Fuchs

Epilogue

Frontmatter

Chapter 23. Plus ça Change

Abstract

As a young student of structures I was often mystified by the different number of unknowns of a structural problem: R=M-N if solved by the force method or N if solved by the displacement method. Well, in truth, an analysis of a structure solves the structure equations in 2M+N unknowns. The symmetric coefficient matrix of these equations comprises the three pillars of structures: equilibrium, deformations and elasticity. The force and displacement methods are subsets of the structure equations. This is exemplified by a simple assemblage of linear springs.

Maurice Bernard Fuchs

Titel: Structures and Their Analysis
verfasst von: Maurice Bernard Fuchs
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-31081-7
Print ISBN: 978-3-319-31079-4
DOI: https://doi.org/10.1007/978-3-319-31081-7

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