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About this book

This second edition of Structural Mechanics is an expanded and revised successor to the highly successful first edition, which over the last ten years has become a widely adopted standard first year text. The addition of five new programmes, together with some updating of the original text, now means that this book covers most of the principles of structural mechanics taught in the first and second years of civil engineering degree courses.

· Suitable for independent study or as a compliment to a traditional lecture-based course
· Adopts a programmed learning format, with a focus on student-centred learning
· Contains many examples, carefully constructed questions and graded practical problems, allowing the reader to work at their own pace, and assess their progress whilst gaining confidence in their ability to apply the principles of Structural Mechanics
· Now covering the major part of the Structural Mechanics/Analysis syllabuses of most Civil Engineering degree courses up to second year level.

Table of Contents

Programme 1. Revision of the Fundamentals of Statics

Abstract
In this programme we will review the basic principles of Statics which will be needed in following programmes. You may have learnt these principles in your previous studies, in which case this will be useful revision. If you have not studied Statics before, do not worry; by working conscientiously through the following frames and exercises you will be able to master the subsequent programmes. You will need a scale rule and a protractor. Statics is the study of bodies at rest. A body is a general word used to describe, for example, a building, a bridge or a component of a structure such as a beam, wall or foundation. All bodies are subject to the action of forces, such as their own selfweight, which are usually expressed in units of kiloNewtons (kN). It may be difficult to visualise a kN but it may be helpful to think of a small car as weighing approximately 10 kN.
Ray Hulse, Jack Cain

Programme 2. Simple Structures (Loads and Reactions)

Abstract
In this programme we will look at a number of simple structures. A structure can be defined as an assembly of structural components so arranged that loads can be supported without failure. Consequently we must study the various loads to which a structure may be subjected and the way in which the structure reacts to those loads. We must also understand what is meant by failure. Failure may imply a complete collapse. Collapse may be due to incorrect geometrical arrangement of the components comprising the structure or to a failure of one of the components because of an overloading of that component. Excessive movement may also be considered to be a failure. You should appreciate that all structures will move or deflect to a certain extent and maximum acceptable limits for deflection are normally specified for a structure at the design stage. It may be possible, however, to load a structure to such an extent that it moves as a whole, the movement continuing until the loading is changed. Such a situation indicates a lack of stability-the structure is not in equilibrium. This last aspect of failure will be studied in this programme. In subsequent programmes the behaviour of the components of a structure and the estimation of deflections will be studied.
Ray Hulse, Jack Cain

Programme 3. Pin-Jointed Frame Structures

Abstract
In the previous programme we studied the conditions necessary for the equilibrium of structures as whole bodies under the action of external forces. We can now begin to learn how structures react to those external forces, and how to calculate the values of the internal forces that develop within a structure as a consequence of the loads upon it. This programme will introduce you to frame structures and instruct you in methods of analysing such frameworks in order to determine the forces in the members. The term frame structure is applied to a number of different forms of construction. Thus the rigid assembly of columns and beams forming the skeleton of a large building may be described as a reinforced concrete frame. A lightweight pinjointed roof truss may be described as a frame structure. Our studies will, however, be confined to a specific type of frame structure as specified below, and it is plane frames complying with the following definition that we shall be considering in this programme.
Ray Hulse, Jack Cain

Programme 4. Shearing Forces and Bending Moments

Abstract
In the previous programme we studied the effect of external loads upon the members of pin jointed frames. We saw that, as a result of the loading, such members were subjected to direct forces acting along their longitudinal axes and increased or decreased in length according to the sense of those forces. In practice however, the majority of structures are not pin jointed and often loads are applied at points other than the joints between members. You will appreciate, for example, that the selfweight of members will act along their entire length. In this programme we will investigate the way in which structural members respond to such loads. The reasoning we will adopt could be applied to any member of a structure regardless of whether it be a horizontal beam, a vertical column or a member inclined at an angle other than 90 to the horizontal. For convenience, however, we will initially confine ourselves to a consideration of horizontal beams. In Programme 3 we studied the behaviour of pin-jointed frames. Why are the members of such frames subjected only to longitudinal axial forces?
Ray Hulse, Jack Cain

Programme 5. Stress Analysis (Direct Stress)

Abstract
In Programme 3 we saw how external loads on a pin-jointed frame structure induced axial forces in the individual members of the frame. In Programme 4 we saw how external loads induced shearing forces and bending moments in beams. We now begin to look more closely at the way in which the material of which the structure is made reacts to the effects of loading. We start in this programme by considering in detail the behaviour of structural members subjected only to axial tensile or compressive forces. The effect of shearing forces and bending moments will be considered in subsequent programmes. We have already noted in previous programmes that members subjected to tensile forces increase in length and members subjected to compressive forces decrease in length. It is true to say that whenever a force is applied to a piece of material, that material will change shape. The change may only be microscopic but in all cases a loading is accompanied by deformation. Thus all structures when loaded will deform, the amount of deformation depending on factors such as the magnitude of the load and the type of material used in the structure. In this programme we will learn how to calculate the effect of axial loading on some simple structures and structural elements.
Ray Hulse, Jack Cain

Programme 6. Bending Stress

Abstract
In Programme 4 you learnt how to determine the shear force and bending moment at any section of a loaded beam. In future work in structural design you will learn how to design beams capable of withstanding the effects of shear force and bending moment. As an introduction to beam design we will, in this programme, develop and study the equations that can be used to analyse the bending stresses that result from the application of bending moment. Although the theory that we will develop can be applied to other types of structural elements, we will restrict our study to a consideration of beams only.
Ray Hulse, Jack Cain

Programme 7. Combined Bending and Direct Stress

Abstract
In Programmes 5 and 6 we learnt how to calculate uniform stress due to direct axial loads and bending stresses resulting from bending moments. In this programme we will see how to determine the stresses at any point in a structural member which is subjected to a combination of both direct axial loading and bending moments. There are many practical situations where both axial loads and bending moments act in combination on a structural member. Can you think of any? The list of situations is almost endless but you might have thought about columns with load applied eccentrically to the centroidal axis of the section; walls subjected to their own self-weight plus lateral loading from wind or retained earth, etc. We will look at some of these examples in detail so if you cannot immediately appreciate that these are circumstances where combined stresses aris
Ray Hulse, Jack Cain

Programme 8. Shear Stress

Abstract
So far we have shown how to calculate direct stresses arising from direct axial loading acting on a structural member, and bending stresses within structural elements which are subjected to bending moments. In Programme 7 we also saw how to calculate stresses arising from combinations of axial loading and bending moments. However there are other stresses which can arise within a structural element and in this programme we are going to examine a further type of stress. To be precise we will study the development of shear stress and see how shear stresses can arise and how to calculate the magnitude and distribution of such stresses for a range of common problems. The development of shear stress is related to the existence of shearing forces and the magnitude of such stress depends upon the magnitude and distribution of the shearing forces. Hence it is important that, in the case of a beam, you can sketch the shape of the shearing force diagram. If you are not confident about this, revise Programme 4 before proceeding. Now, lets see how and where shear stresses can arise.
Ray Hulse, Jack Cain

Programme 9. Torsional Stress

Abstract
Our study of stress would not be complete without examining the stresses that can occur within a structural element due to torsional or twisting effects. In this programme we are going to determine the governing equations that will define the behaviour and response of an element which is subjected to torsion, and see how these equations can be applied to the solution of a number of common problems. A complete treatment of torsional stress for all shapes of structural sections can be quite complex and is beyond the scope of this text. We will however look at the analysis of circular sections whereby the fundamental analytical expressions can be developed and the basic principles established. Hollow circular or solid circular sections subject to torsional or twisting effects are quite common in everyday life. Can you think of an example? The drive shaft of a car connecting the engine to the rear axle is a very common example. The rotation of the shaft will cause twisting, resulting in the development of torsional stresses. Other examples could include the propeller shaft of a ship or aircraft. On a more simple level the application of a spanner to tighten up a steel bolt would result in torsional stresses which, if excessive, could lead to fracture of the bolt, as anybody who has tried to release a rusty bolt well knows.
Ray Hulse, Jack Cain

Programme 10. Stress Transformations and Mohr’s Circle of Stress

Abstract
In Programmes 5 to 9 we learnt how to calculate the stress within structural members when subjected to direct axial loading, bending, shear and torsion. The theories that we have developed are applicable to a wide range of problems and the formulae that we have used are widely utilised in the design of many types of structural component using different materials of construction. In Programme 7 we made use of the principle of superposition to calculate the stresses due to a combination of direct and bending stresses and saw that in such situations we could calculate the components of stress separately and simply add the components together to give the final set of stresses. However, many structural components are subject to much more complex stress situations than those we have already considered. For example, we have already seen that the cranked cantilever beam in Frame 2 of Programme 9 is stressed from a combination of bending, shear and torsion, and indeed there are many practical situations where such complex stress combinations exist.
Ray Hulse, Jack Cain

Programme 11. Composite Sections

Abstract
In all of the programmes so far that have been concerned with stress analysis we have looked at problems with structural elements fabricated from a single type of material. There are, however, a number of structural situations where a component may be fabricated from more than one type of material, each of which will have its own set of physical properties, such as Youngs Modulus of Elasticity, and hence the response of a composite structure will be governed by the composite action of the materials out of which the structure is fabricated. Typical composite structures are shown in the figures below and in this programme we will look at some aspects of stress analysis in such structures. Figure (a) shows a very common example of composite construction: the cross-section of a reinforced concrete beam where the flexural strength is derived from the composite action of reinforcing steel which has high tensile strength together with concrete which has high compressive strength.
Ray Hulse, Jack Cain

Programme 12. Beam Deflections and Rotations

Abstract
Most of the previous seven programmes have been concerned with the analysis of different fonns of stress and the application of the theories which we have developed to the analysis and design of different fanTIs of structure and structural components. In the design of structures the primary requirement is to ensure that the structure or structural component can adequately resist the loading to which it is being subjected. Usually this means that under normal loading conditions the stresses set up within the structure must be less than some permissible value, that value being the failure stress of the material divided by some appropriate factor of safety. This aspect of design is concerned with designing for strength. However, there are other aspects of design that are important, and in the case of the design of beams another consideration is the value of the vertical deflections that will occur when such a beam is loaded. This programme is intended as an introduction to the analytical techniques used for calculating deflections in beams and also for calculating the rotations at critical locations along the length of a beam. Can you think why it is necessary to calculate the vertical deflections that take place when a beam is loaded?
Ray Hulse, Jack Cain

Programme 13. Strain Energy

Abstract
In all forms of structure the application of an external loading system will give rise to an internal stress system and internal deformations. The work done by the loading system in deforming the structure will be conserved as energy within the structure. This energy, known as strain energy, is normally recoverable when the structure is unloaded provided that the elements of the structure have been stressed only within the elastic range. You can illustrate this for yourself by stretching an elastic band and then letting go of the ends. In stretching the elastic the release of energy will be quite apparent particularly if you only let go of one end! A different form of strain energy due to bending can be illustrated by taking a plastic ruler and rotating both ends. If you then release the ends suddenly, the ruler will straighten as the stored energy is released.
Ray Hulse, Jack Cain

Programme 14. Virtual Work

Abstract
This programme is intended as an introduction to a method of structural analysis which is arguably the most powerful analytical technique for solving a very wide range of structural problems. We will examine the basis of the method and its application in the calculation of deflections of framed structures and beams. We have already, in Programme 12, looked at one method of determining the deflections of beams. The method that we will now investigate is an alternative and more versatile technique known as the Principle of Virtual Work. The general principle of Virtual Work can be seen by considering a set of concurrent forces acting on a small particle at a point P as shown below. The forces will have a resultant R, also shown on the diagram.
Ray Hulse, Jack Cain

Programme 15. Moment Distribution and Indeterminate Structures

Abstract
So far we have looked at how to solve problems of a structural nature where the structures have been statically determinate: that is, they can be solved using the fundamental principles of statics and the equations of statics. Many structures are not as simple, in fann or construction, as those that we have considered and it is not possible to apply the techniques that we have considered to their solution. However, to design these structures it is still necessary to establish the magnitude and direction of the supporting reactive forces and the magnitude and nature of the internal shear forces and bending moments which develop to resist the external loading system. This programme will therefore introduce one of the most common techniques for the solution of structures which are not statically determinate. The particular method that we will develop is known as the method of moment distribution and is a manual solution method. In a later programme we will develop another solution technique which is the basis of modern computer methods of analysis. Can you recall and write down the necessary condition for a structure to be considered to be externally statically determinate?
Ray Hulse, Jack Cain

Programme 16. The Slope-Deflection Method

Abstract
In the previous programme we looked at the method of moment distribution and showed how we could use the method to analyse beam and frame type structures which are statically indeterminate. The method gave us a means of calculating, by a process of successive reduction, the end moments in the members of those structures and hence, using normal principles of statics, the support reactions and bending moments and shearing forces within the members. In this programme we will introduce an alternative technique of solution known as the Slope Deflection Method. This alternative method will enable us to solve the same type of problems but has the added advantage that we will be able to calculate the deformations of (deflections and rotations) as well as the internal forces within the structure. This particular technique involves the solution of sets of simultaneous equations: the more sizeable and complex the structure the more equations will have to be solved. Hence it is only possible to manually solve structures of limited size using this method. However, the method can be readily adapted such that the equations can be solved by computer and, adapted in this way, is in fact the basis of modern computer techniques of structural analysis. Before we start can you recall the expression that defines the stiffness ofa member in a rigidly jointed structure?
Ray Hulse, Jack Cain

Programme 17. Influence Lines

Abstract
In all the previous programmes we have studied structures, or structural components, where the loading system is considered to be fIxed in position on the structure. There are, however, some types of engineering problems where the load system could be located at one of a number of possible positions and, indeed, may actually move across the structure. Can you think of a situation where a structure may be subject to a moving load? I A train crossing a rail bridge I You may equally have thought about a car or a number of cars simultaneously crossing a road bridge. If you were designing such a bridge constructed as,Alternatively, what about easing housing shortages by promoting housebuilding on land that is currently occupied by run-down industrial premises or using agricultural land just outside the boundaries of each urban area?
Ray Hulse, Jack Cain

Programme 18. Elastic Buckling of Axially Loaded Compression Members

Abstract
In Programme 5 we examined the behaviour of axially loaded structural members including those subject to compressive loads; such as columns within buildings and struts within pin-jointed framed structures. We saw how any externally applied axial load would result in the development of an internal elastic stress system. Such structural members would continue to resist any increase in load by developing proportionately higher internal stresses providing that the material is not strained beyond the elastic range. If the loading is increased beyond this limit then some form of material failure will eventually occur such as crushing of the concrete in a concrete column or yielding of the steel in a steelwork stanchion.
Ray Hulse, Jack Cain

Programme 19. Plastic Analysis

Abstract
In this programme we will develop the principles of plastic theory which are widely used in the analysis and design of steel structures including portal frames, multi-bay frames and continuous beams. All the techniques of analysis and design that we have developed so far have been based on the principles of linear elastic theory, i.e. that stress is directly proportional to strain and that stress and strain are directly related by Youngs Modulus ofElasticity. However, if we consider the behaviour of steel as a structural material we know that once the material is strained beyond the limit of proportionality its stress strain behaviour changes and stress is no longer related to strain by Youngs Modulus of Elasticity. If a sample ofmild steel is strained beyond the limit ofproportionality what term do we use to describe the next stage of its stress strain behaviour? I The plastic stage I The figure below shows the idealised stress strain behaviour of mild steel. We can see that initially it behaves in a linear elastic manner but once the strain exceeds the limit of proportionality the material yields and it continues to strain without any corresponding increase in stress. We assume that the stress strain behaviour is identical in both tension and compression.
Ray Hulse, Jack Cain
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