9.1 INTRODUCTION [LO1]

Stresses given by relatively simple equations in the strength of materials for structures or machine members are based on the assumed continuity of the elastic medium. However, the presence of discontinuity destroys the assumed regularity of stress distribution in a member and a sudden increase in stresses occurs in the neighborhood of the discontinuity. In developing machines, it is impossible to avoid abrupt changes in cross-sections, holes, notches, shoulders, etc. Abrupt changes in cross-section also occur at the roots of gear teeth and threads of bolts. Some examples are shown in Figure 9.1.

Any such discontinuity acts as a stress raiser. Ideally, discontinuity in materials such as non-metallic inclusions in metals, casting defects, residual stresses from welding may also act as stress raisers. In this chapter, however, we shall consider only the geometric discontinuity that arises from design considerations of structures or machine parts.

Many theoretical methods and experimental techniques have been developed to determine stress concentrations in different structural and mechanical systems. In order to understand the concept, we shall begin with a plate with a centrally located hole. The plate is subjected to uniformly distributed tensile loading at the ends, as shown in Figure 9.2.

1.1 INTRODUCTION [LO1]

Mechanics is one of the oldest physical sciences, dating back to the times of Aristotle and Archimedes. The subject deals with force, displacement, and motion. The concepts of mechanics have been used to solve many mechanical and structural engineering problems through the ages. Because of its intriguing nature, many great scientists including Sir Isaac Newton and Albert Einstein delved into it for solving intricate problems in their own fields.

Engineering mechanics and mechanics of materials developed over centuries with a few experiment-based postulates and assumptions, particularly to solve engineering problems in designing machines and structural parts. Problems are many and varied. However, in most cases, the requirement is to ensure sufficient strength, stiffness, and stability of the components, and eventually those of the whole machine or structure. In order to do this, we first analyze the forces and stresses at different points in a member, and then select materials of known strength and deformation behavior, to withstand the stress distribution with tolerable deformation and stability limits. The methodology has now developed to the extent of coding that takes into account the whole field stress, strain, deformation behaviors, and material characteristics to predict the probability of failure of a component at the weakest point. Inputs from the theory of elasticity and plasticity, mathematical and computational techniques, material science, and many other branches of science are needed to develop such sophisticated coding.

The theory of elasticity too developed but as an applied mathematics topic, and engineers took very little notice of it until recently, when critical analyses of components in high-speed machinery, vehicles, aerospace technology, and many other applications became necessary. The types of problems considered in both the elementary strength of material and the theory of elasticity are similar, but the approaches are different. The strength of the materials approach is generally simple. Here the emphasis is on finding practical solutions to a problem with simplifying assumptions.

4.1 CONSTITUTIVE EQUATIONS [LO1]

So far, we have discussed the strain and stress analysis in detail. In this chapter, we shall link the stress and strain by considering the material properties in order to completely describe the elastic, plastic, elasto-plastic, visco-elastic, or other such deformation characteristics of solids. These are known as constitutive equations, or in simpler terms the stress–strain relations. There are endless varieties of materials and loading conditions, and therefore development of a general form of constitutive equation may be challenging. Here we mainly consider linear elastic solids along with their mechanical properties and deformation behavior.

Fundamental relation between stress and strain was first given by Robert Hooke in 1676 in the most simplified manner as, “Force varies as the stretch”. This implies a load–deflection relation that was later interpreted as a stress–strain relation. Following this, we can write P = kδ, where P is the force, δ is the stretch or elongation, and k is the spring constant. This can also be written for linear elastic materials as σ = E∈, where σ is the stress, ∈ is the strain, and E is the modulus of elasticity. For nonlinear elasticity, we may write in a simplistic manner σ = E∈n, where n ≠ 1.

Hooke's Law based on this fundamental relation is given as the stress–strain relation, and in its most general form, stresses are functions of all the strain components as shown in equation (4.1.1).

7.1 INTRODUCTION [LO1]

The problems of stresses and deformations in disks rotating at high speeds are important in the design of both gas and steam turbines, generators and many such rotating machinery in industry. As discussed in earlier chapters, this is another example of axisymmetric problems in polar coordinates. Although the theoretical treatment of a flat disk is simpler, in many industrial applications, disks are tapered. They are usually thicker near the hub, and their theoretical analysis is slightly more involved. We shall first take up the analysis for flat disks.

In the case of rotating disks with centrifugal force as body force, the equation of equilibrium reduces to as in equation (6.1.3).

Combining this with displacement equations, we have, as in equation (6.1.5), a general equation for determining the stress distribution in axisymmetric problems. This is given as

This is a nonhomogeneous differential equation. The associated homogeneous equation (complementary equation) is

The solution of this equation is Lame's equation as discussed in Chapter 6, equation (6.2.3), and taking into consideration the particular solution, the solution to equation (7.1.2) turns out to be

We may also determine the radial displacement from equation (6.2.11), and this is given as

We may therefore write the stresses and displacement for the rotating disk under one bracket as

With these introductory basic equations, we shall now set out to discuss the stress distribution and displacement in rotating disks.

6.1 INTRODUCTION [LO1]

In earlier chapters, we have discussed axisymmetric problems in two-dimensional (2D) polar coordinate systems. Thick cylinders fall into this class of problems. Cylindrical pressure vessels, hydraulic cylinders, gun-barrels, and pipes carrying fluids at high pressure develop radial and tangential stresses (circumferential). Longitudinal stresses can also be developed if the ends are closed. Therefore, ideally, this is a triaxial stress system as shown in Figure 6.1.

• (a) Circumferential or hoop stress (σθ)

• (b) Longitudinal stress (σz)

• (c) Radial stress (σr)

If the wall thickness of a hollow cylinder is less than about 10% of its radius, it may be treated as a thin cylinder. Cylinders with higher wall thickness are considered to be thick cylinders. Before analyzing the stress in a thick cylinder, we should briefly consider the stress state in thin cylinders, where radial stress is small compared to the other stresses, and this can be neglected. Stress variation across the thin wall is also negligible. Analysis of thin-walled pressure vessels may therefore be carried out on the basis of biaxial stress system. Since the presence of shear stress at the cut section would lead to incompatible distortion, the longitudinal and circumferential stresses in this case are both principal stresses. We now take another section of the cut section as shown in Figure 6.2 (a) to consider the equilibrium of the section, and this is shown in Figure 6.2 (b).

The section is acted upon by internal pressure p and the circumferential stress developed at the cut section is σθ. Force on an infinitesimal small area subtended by angle dθ at θ inclination from the horizontal axis is pridθ.

5.1 INTRODUCTION [LO1]

In any three-dimensional (3D) elasticity problem, there are 15 unknown parameters: 6 stress components, 6 strain components, and 3 displacements. There are 15 related equations: 3 equations of equilibrium, 6 compatibility equations, and 6 constitutive equations. Solutions to a particular elasticity problem require evaluation of these 15 unknown parameters using 15 equations, satisfying all the boundary conditions. As discussed in the earlier chapters, there may be displacement or stress, or mixed boundary conditions. In many cases, solutions to 3D problems are not easy analytically. Even numerical solutions may be difficult.

There are mainly three methods of simplification of solution techniques:

• (a) If the boundary conditions are in terms of stresses, stress function approach may be made as discussed in the earlier chapter. This makes the solution simpler.

• (b) Assumptions of plane stress and plane strain reduce 3D problems to two-dimensional (2D) ones and this also makes the solution simpler.

• (c) Use of St. Venant's principle and superposition principle also makes the solution of elasticity problems simpler.

An introduction to stress function approach has been discussed in Chapter 4. We therefore start our discussion on plane stress and plane strain approaches.

5.2 PLANE STRESS AND PLANE STRAIN PROBLEMS [LO2]

The idealizations of both plane stress and plane strain states are suitable for certain classes of problems that are made to reduce the complexity of solutions. We shall consider the plane stress state first.

10.1 INTRODUCTION [LO1]

There are many applications where structures or machine parts are subjected to significant changes in temperature, for example, turbine blades, high-speed rotating machinery, and boilers in thermal power plants. Large thermal stresses may be developed in such applications, and sometimes such stresses may exceed the yield limit. It is therefore necessary to make provisions in the design of components to avoid failure due to thermal stresses. If the ends of a rod or any other machine parts are rigidly fixed such that the expansion or compression is prevented and the temperature is changed, tensile or compressive stress would be set up, and in simple terms, these stresses are called thermal stresses. In a long steam pipe, expansion joints are sometimes inserted, and in bridges, one end may be rigidly fastened to the main structure while the other end rests on rollers to avoid thermal stresses. In simple terms, this may be demonstrated considering a rod of length l and cross-sectional area A fixed at both ends (Figure 10.1) and temperature is raised by ΔT. This would produce a thermal strain ∈t in the rod such that

where a is the coefficient of thermal expansion. Since the rod is not free to expand, a compressive stress σc would be developed in the rod, and this is given by

If the rise in temperature is significant, the rod may buckle, which is of serious consideration in the design of machine parts or structures. To avoid this, we need to find the critical force Pcr for buckling. Using the basic buckling criterion, this can be given for a column pin ended at both ends, by

where I is the least moment of inertia of the constant cross-section column (rod) and A is the cross-sectional area of the column.

12.1 INTRODUCTION [LO1]

There are in general two approaches to solving equilibrium problems in solid mechanics: Eulerian and Lagrangian. The first approach deals with vectors such as force and moments, and considers the static equilibrium and compatibility equations to solve the problems. In the second approach, scalars such as work and energy are used, and here solutions to problems are based on the principle of conservation of energy. There are many situations where the second approach is more advantageous, and here some powerful methods, such as the method of virtual work, based on this approach, are used.

Eulerian and Lagrangian approaches to solving solid mechanics problems are much more involved. However, here we have chosen to describe these in a simplified manner, which is suitable as a prologue to the present discussion on energy methods.

In mechanics, energy is defined as the capacity to do work, and this may exist in different forms. We are concerned here with elastic strain energy, which is a form of potential energy stored in a body on which some work is done by externally applied forces. Here it is assumed that the material remains elastic when work has been done so that all the energy is recoverable and no permanent deformation occurs. This means that strain energy U = work done. If the load is applied gradually in straining, the material load–extension graph is as shown in Figure 12.1, and we may write U = ½ Pδ.

The hatched portion of the load–extension graph represents the strain energy and the unhatched portion ABD represents the complementary energy that is utilized in some advanced energy methods of solution.

8.1 INTRODUCTION [LO1]

In simple words, the application of a torque on a prismatic member causes twisting or torsion. This causes shear stress if a torque alone is applied. However, this is rarely true in practical cases. A circular bar, used to transmit torque between a prime mover and a machine, is a typical example of a torsion member. However, in many applications, a torque along with a bending moment and axial loading are applied, and there we need to combine these effects and find the principal stresses. A typical example of such combined stresses is a propeller shaft. Torsional problems are important in many applications both in industry and in our daily life. Therefore, we consider torsion alone in this chapter in some detail.

Torsional problems for circular members are generally solved assuming that plane sections normal to the axis of the bar remain plane even after twisting. This assumption was first made by Coulomb intuitively in 1784, and he came up with a correct usable equation for members with circular sections. However, this assumption does not apply to bars with a noncircular cross-section. Navier attempted to solve torsional problems with noncircular sections using Coulomb's assumption and came up with an erroneous solution. The correct solution was provided by St. Venant in 1853 using a warping function. Much later, in 1903, Prandtl came up with a membrane analogy method that could solve problems with any complicated cross-section. First, we shall consider torsional problems with circular cross-sections.

8.2 TORSION OF MEMBERS WITH CIRCULAR CROSS-SECTION [LO2]

The torsion analysis of members with a circular cross-section starts with simplified assumptions made by Coulomb. In order to establish a relation between the applied torque and shear stress developed and the angle of twist in such cases, the following assumptions are made:

• 1. Material is homogeneous and isotropic.

• 2. Plane sections perpendicular to the axis of a circular member remain plane after twisting. No warping or distortion of the parallel planes occurs.

11.1 INTRODUCTION [LO1]

Stresses developed at the contact between two loaded elastic bodies are generally localized and most machine parts or structures are designed based on the stresses in the main body. However, there are many important machine members where the localized stresses developed at the contact between curved surfaces with initially limited contact area play an important role in their design. Ball or roller bearings, gears, cams, and valve tappets of internal combustion engines are some of the examples of machine parts where contact stresses must be taken into account in order to predict their failure probability.

The localized contact stresses that develop between two curved bodies as they are loaded with small deformations are often referred to as Hertzian stresses, following the work of H. Hertz (1881), who first solved these contact problems elegantly more than a century ago. Since then the topic has received a good deal of attention by the researchers due to its importance in engineering practice and science. Much work has been done on the stress distribution at the Hertzian contact surfaces and sub-surfaces. Ball bearings and gear teeth often fail by pitting. Hertzian stress analysis can precisely locate the depth at which maximum shear stress occurs where cracks may initiate and propagate leading to failure. Thus, a remedy to such failures may be prescribed in terms of limiting stresses. In many rolling contact problems, failure occurs with the initiation of a tiny crack that eventually grows due to repeated contacts. Analysis of crack initiation and growth is often based on Hertzian stress analysis. In this chapter, we shall consider the basics and application of contact stress analysis, beginning with some basic elasticity theory necessary for such analyses.

2.1 MATHEMATICAL PRELIMINARIES [LO1]

In any scientific or engineering field of study, knowledge of some mathematical techniques and methods are essential. Solid mechanics is no exception. To develop proper formulation methods and solution techniques for elasticity problems, it is necessary to have an appropriate mathematical background. In this chapter, we shall discuss Cartesian tensors, which have a special significance in the discussion of stress, strain, and displacement fields, and their manipulation. Other mathematical details will be discussed as and when they are required in solving different problems.

Tensors may be defined in a number of ways. One simple definition is that a tensor is a physical quantity that is governed by certain transformation laws when the coordinate system is changed. A tensor is invariant under any change of coordinate system, but its components along the coordinate axes change with the changed coordinate system. Tensors of order zero are called scalars. Common examples of scalars are temperature, density, Young's modulus, or Poisson's ratio. They have a single magnitude at each point in space, and they are invariant with coordinate transformations. A typical example of scalars is often taken as temperature T at a point in space with coordinates (x, y, z) represented as T(x, y, z). Temperature at the same point does not change if we choose a different coordinate system (x′, y′, z′) represented as T′(x′, y′, z′) and we may say

T=T′. (2.1.1)

Tensors of first order are vectors, and we know that a vector has a magnitude and a direction. A typical example of a vector is a velocity vector V. It is sometimes taken as a convention to represent a vector by a bold letter. Consider the velocity vector V in (x, y, z) coordinate system.

Solid mechanics, compared to mechanics of materials or strength of materials, is generally considered to be a higher level course. It is usually offered in higher semester to senior students. There are many textbooks available on solid mechanics, but they generally include a large part of theory of elasticity with in depth mathematical formulations. The usual prerequisites are one or two semester course on elementary strength of materials and a thorough mathematical background, including scalar, vector, and tensor field theory and cartesian and curvilinear index notation. The difference in levels between these books and elementary texts on strength of materials is generally formidable. However, in our experience of teaching this course for many years at premier institutes like IIT Kharagpur and Jadavpur University, despite its complexity, senior students generally cope well with the course using the readily available textbooks.

However, there is a vast student population pursuing mechanical, civil, or allied engineering disciplines across the country in colleges where AICTE curriculum is followed. Through several years of interaction with this group of students, we have found that there is no suitable textbook that suits their requirements. The book is primarily aimed at this group of students, attempting to bridge the gap between complex formulations in the theory of elasticity and elementary strength of materials in a simplified manner for better understanding. Index notations have been avoided, and the mathematical derivations are restricted to second-order differential equations, their solution methodologies, and only a few special functions, such as stress function and Laplacian operators.

The text follows more or less the AICTE guidelines and consists of twelve chapters. The first five chapters introduce the engineering aspects of solid mechanics and establish the basic theorems of elasticity, governing equations, and their solution methodologies. The next four chapters discuss thick cylinders, rotating disks, torsion of members with both circular and noncircular cross-sections, and stress concentration in some depth using the elasticity approaches. Thermoelasticity is an important issue in the design of high-speed machinery and many other engineering applications. This is dealt with in some detail in the tenth chapter. Problems on contact between curved bodies in two-dimensional and three-dimensional situations can be challenging, and they have wide applications in mechanical engineering such as in bearing and gear technology.

3.1 STATE OF STRESS AT A POINT [LO1]

When a body is subjected to external forces, its behavior depends on the magnitude and distribution of forces and properties of the body material. Depending on these factors, the body may deform elastically or plastically, or it may fracture. The body may also fail by fatigue when subjected to repetitive loading. Here we are primarily interested in elastic deformation of materials.

In order to establish the concept of stress and stress at a point, let us consider a straight bar of uniform cross-section of area A and subjected to uniaxial force F as shown in Figure 3.1. Stress at a typical section A - A′ is normally given as σ = F/A. This is true only if the force is uniformly distributed over the area A, but this is rarely true. Therefore, definition of stress must be considered by progressively reducing the area until it is small enough such that the force may be considered to be uniformly distributed.

To understand this, consider a body subjected to external forces P1, P2, P3, and P4 as shown in Figure 3.2. If we now cut the body in two pieces,

Internal forces f1, f2, f3, etc. are developed to keep the pieces in equilibrium. Now consider an infinitesimal element of area ΔA Dat the cut section and let the resultant force on the element be Δf.