The Mechanics of Engineering Structures

The following sections are included:

• Contents
• Preface

The following sections are included:

• Introduction
• Units and Conventions
• Elasticity
• Structures
• Yielding
• Concluding Remarks
• References

In this chapter the basic properties of areas that underpin many of the topics that appear throughout this book are introduced. It will be seen later how the present topic is applied to the cross-sections of beams in bending, shafts in torsion and long struts under compressive loading. Examples are selected to show what is meant by the terms ‘centroid’, the ‘first and second moments’ of their plane section areas. Two theorems are derived that enable the transfer of second moments of area between parallel and perpendicular axes within a given cross-section. Both analytical and graphical techniques are available for dealing with co-ordinate rotations. Exercises are given to enable the reader to gain familiarity with these terms and techniques.

The following analyses are restricted to systems of two-dimensional forces all lying in the same plane. The distinction is made between forces which pass through a common point and those that do not. The conditions given refer, firstly, to when each co-planar force system is in equilibrium. Their nonequilibrium conditions follow in which a resultant force and a moment is identified. Applications to free-body diagrams include frames, plates and beams.

There are a limited number of forces which, when applied to structural materials, cause their fibres to suffer stress and become strained. These ‘forces’ refer to tension, compression, shear, bending and torsion. This chapter's title refers to the basic stress and strain arising from the simplest applications of an axial tension (or compression) to a bar and of a tangential shear force to a block. These two contrasting loadings will be examined here in terms of the manner in which both their stress and strain are defined and to the relationship that exists between stress and strain up to the yield point and beyond. The method of ‘designing against’ these forces through the use of safety factors is given within typical practical applications including a similar loading of composite bars. The parallel analyses of bending and torsion appear later in Chapters 5 and 7 as these are more complicated involving stress and strain gradients. In practice, these four sources of stress and strain often act in combination. Various accounts when dealing with their more common combinations are considered here and in Chapters 11 and 12, where both the analytical and graphical methods are employed. An additional source arising from constrained thermal expansion is also given here.

A stress state may exist in one, two or three dimensions, which is more commonly referred to as being, respectively, a uniaxial, biaxial or triaxial state. The elastic constants provide constitutive relations, which connect each stress state to strain. Strain is usually three-dimensional for each stress state. For example, under uniaxial tension a rectangular bar suffers contractions to its width and thickness as it extends. A plate thins as it is stretched biaxially in perpendicular directions within its plane. In contrast, a pure shear distortion (see Fig. 3.4) would not produce thinning unless it is combined with an axial loading. The engineering strains, both normal and shear, are defined as the ratio between each distortion and the original length, using the formulae in Eqs 3.2 and 3.7b. At its most general, a constitutive relation can provide the three-dimensional strains for a stress state arising from any combination of loading: tension, shear, bending and torsion. The relation requires that the normal and shear stress components be identified either within a Cartesian or polar co-ordinate system that is more suited to the geometry of the solid subjected to these loads. Finally, it is in the meaning of the term linear elasticity that all distortion is recoverable when the solid is relieved of its loading. Here the loading must lie within the elastic limit within which the analyses of stress and strain are simplified with the solid responding in a linearly elastic manner to its loading and unloading. While most metals and their alloys behave in this way, nonmetals may not. We shall examine here and later in Chapter 12 how ductile metals behave when loaded beyond the elastic limit and where brittle, elastic failure applies to solids and fibrous composites incapable of sustaining large extensions.

The two influences upon the design of a loaded beam lie in the distributions of shear force and bending moment along its length. These two distributions were shown within their corresponding diagrams constructed in Chapter 2 (see § 2.5, p. 47). The force and moment are the external ‘loads’ upon the beam each to be reacted by its cross section. The section does this by becoming stressed under each loading mode, distorting away from its original shape and position due to the strain within it. Here we shall examine the relationships that exist between stress and strain for elastic distortions arising from beam curvature and transverse shearing. The stress expressions provided by this analysis are crucial for designing load-bearing beams with the required degree of safety.

Various methods are available for calculating the deflected shape of a transversely loaded beam. The flexure equation uses the beam's curvature to express its slope and deflection in the length. The moment area method interprets an integration of the said equation within the area of its bending moment diagram. Integrating by Macaulay's step function method accounts for discontinuities in the bending moment diagram. When Hooke's law applies so too do the superposition principle and Maxwell's reciprocal law. Each is used to connect many applied loads to the elastic displacements produced as a sum of their separated effects.

The shear mode of deformation in a bar under axial torsion is distinct, theoretically, from the shear mode in a bar or beam under transverse forces. The latter was considered in Chapter 5 where an examination of the distribution in shear stress was made for solid sections and thin-walled sections in open and closed forms. Here we shall examine the nature of the shear stress when similar sections bear axial torsion. At an introductory level it is sufficient to restrict the analysis to solid and hollow circular sections, with particular applications to stepped, tapered and composite bars. Thereafter, torsion of thin-walled tubes of any shape and open sections shaped from thin strip extent the complexity of the theory.

There are many forms of structural buckling. The most common form considered here is the flexural lateral buckling which occurs in slender struts under axial compression. The Euler mathematical theory of elastic buckling provides the buckling load but is idealised in that it does not limit the material's stress. Strictly, this theory can only be applied reliably to long thin members that are prone to buckling under low elastic stress levels. The stress increases for shorter struts to a theoretical cut off at the yield stress where direct compression influences the flexural buckling load. Here a number of empirical formulae are available to estimate buckling loads for so-called imperfect struts, the bases for these being a stress summation from these two causes. The Euler theory may be also be modified in a similar manner to provide useful accounts of initial imperfections including eccentric loading, lack of initial straightness and the presence of lateral loading in addition to axial compression. Various standards provide formulae restricted to shorter struts or for applications extending over a wider range from short to medium-length struts. A degree of safety is usually admitted in which Euler's theory is recognised as providing a reliable upper limit to the buckling load for long struts.

Flat plates supporting a one-dimensional axial compression can buckle laterally in a similar manner to struts considered in the previous chapter. The buckling mode for plates becomes more complex when all four sides are supported with an in-plane loading that is two-dimensional. The latter includes special cases of shear buckling arising in the thin, vertical walls of a beam section under transverse shear or within the individual plates of both open and closed thin-walled sections placed under axial torsion. Torsional buckling of thin-walled circular tubes is treated separately with its sensitivity to a number of geometrical factors including the ratios between tube length, radius and thickness. Local buckling of the cross section can also occur alongside the global, flexural buckling of a plate strut. For example, the limb tips of an equal-angle section rotate as the strut bows laterally from end to end when supporting an axial compression. The following buckling modes are described from their relevance to engineering design.

In this chapter it is shown how great use is made of the balance between work done by external forces and energy stored within a structure that bears load. This energy balance leads to the two theorems of Castigliano and the principle of virtual work that provide for the deflection of a structure. Theory is presented here between the various approaches following the particular to the general. Thus, from the direct application of the conservation of energy the deflection beneath a single load point is found. The two theorems mentioned apply to many loads. They allow the deflection to be found at each point where a load is applied and vice versa. Virtual work principles are wider ranging, allowing the deflections to be found at points beneath or between loads. These techniques are applied to close- and opencoiled helical springs, leaf springs, curved beams, rings and pin-jointed frames. The potential energy component of energy conservation appears where loads are impacted upon their respective structures. One notable advantage of adopting an energy approach is that being a scalar the energy stored is the sum of its contributions from all sources. The latter refers to all the modes of deformation that apply in mechanics: tension, compression, shear, torsion and bending. The chapter begins with the derivation of the strain energy expressions under each mode applied separately and then for various combined modes.

In this chapter we consider methods for transforming stress and strain states from one orientation to another lying in the same plane. It will be seen that there are a number of methods available to make the required transformation. Among them are analytical, graphical and matrix methods all of which will be given here with illustrative examples. The choice between which of the alternative techniques to adopt for this type of analysis is left to the reader given that within themselves they serve to check one another. The topic becomes important to ensuring a safe engineering design based upon the greatest (principal) stress induced by any manner of combined loadings. The latter might include tension, bending, torsion and shear applied to components operating under service conditions. These analyses enable the principal stress to be factored safely to a lower level than the limiting stress imposed by the material under load.

Despite the advances in developing new composite materials for loadbearing applications, ductile metals and alloys have retained their usefulness for a similar role unceasingly. This is largely due to the capacity of metals to survive overloads without resulting in failure. The same cannot be said of man-made composites that would either fail or become unserviceable with the permanent damage caused through overloading. While simple stress states can be factored for safety, there remains the risk of overloading a material when it has to bear a combination of loads, arising from, say, tension, bending, shear and torsion in action simultaneously. Across these two classes of material, metals and composites, a safe design would avoid yielding and fracture. Hence this chapter's title refers to yield criteria for metals and strength criteria for brittle non-metals. They are placed together here because the respective criteria for their limiting elasticity often take similar forms. For example, the von Mises criterion may refer to both the yielding and fracture of an isotropic solid. However, real materials are often anisotropic through their processing, as with the rolling of metals, or by design, as with the enhanced strength aligned with the fibres in a composite. Certain one- and twodimensional anisotropic criteria are material specific but general three-dimensional criteria limit both directional yielding and strength in a common formulation.

This introduction to finite elements will show how to derive the appropriate stiffness matrix, connecting ‘force’ to ‘displacement’, for: (i) a bar under uniaxial stressing, (ii) a beam in bending, (iii) a shaft under torsion and (iv) plates under plane stress and plane strain, using the theory of previous chapters. The approach used in the finite element method (FEM) refers to part of the uniform bar, beam, shaft or plate as an element and to its ends/corners as the nodes. When using the stiffness or displacement method of analysis the displacements at the element's nodes are the unknowns. These nodal-point displacements can be found from their relation to the nodal forces. The stiffness matrix, which connects nodal forces to displacements, has a unique form depending upon the number of degrees of freedom for the element in question. Once the displacements are known, the strains follow from the strain–displacement relations. Finally, knowing the material's elastic constants the stresses corresponding to these strains are found from applying the appropriate form of Hooke's law.

It will be shown how to develop the stiffness matrices for simple bar, beam and shaft elements for (i)-(iii) above using a routine procedure. The FEM requires the sub-division of a body into many smaller elements interconnected throughout at the nodal points. Examples will show how to assemble an overall stiffness matrix for two or more bar or beam elements connected in series. The initial unstrained shape of the body is thus discretised within an assemblage of elements. When the assembly is elastically strained, the behaviour of the entire body may be computed from the known elastic behaviour of its elements. We shall see that the stiffness method assumes a displacement function for the loaded body. This is to provide for equilibrium and compatibility of its internal stress and strain distributions and to match external boundary conditions for force and displacement. Having assembled the overall stiffness matrix, the displacements at the nodes will follow from finding the inverse of the stiffness matrix.

The following section is included:

• INDEX