9.1 Introduction to Classification

We rely on machine learning (ML) to make critical decisions or predictions in the modern world. It is very important to understand how computers by using ML make these predictions. Usually, the predictions made by ML models are classified into two types, i.e., classification and regression. The ML models use various techniques to predict the outcome of an event by analyzing already available data. As machines learn from data, the type of training or input data plays a crucial role in deciding the machine's capability to make accurate decisions and predictions. Usually, this data is available in two forms, i.e., labeled and unlabeled. In label data, we know the value of the output attribute for the sample input attributes, while in unlabeled data, we do not have the output attribute value.

For analyzing labeled data, supervised learning is used. Classification and regression are the two types of supervised learning techniques used to predict the outcome of an unknown instance by analyzing the available labeled input instances. Classification is applied when the outcome is finite or discrete, while the regression model is applied when the outcome is infinite or continuous. For example, a classification model is used to predict whether a customer will buy a product or not. Here the outcome is finite, i.e., buying the product or not buying. In this case, the regression model predicts the number of products that the customer may buy. Here the outcome is infinite, i.e., all possible numbers, since the term quantity refers to a set of continuous numbers.

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.

Second-wave feminism arrived late to economics. It initially permitted criticism of how Gary Becker positioned gendered inequalities in families as rational choices, not as injustices. Methodologies were heterogenous. ‘Equity’ approaches, like Barbara Bergmann’s, engaged statistical analysis and extended Becker-style rational choice theory to reposition gendered inequalities as effects of unfair decision constraints. ‘Critical’ approaches, like Nancy Folbre’s, focused on deficits in the valuation of women’s care, quantifying the full economic worth of care-work, with policies for provisioning in response to needs. Quickly, feminist economists recentred poverty, focusing on ‘lone motherhood’ in the US in its connection with race, and on empowering global South development consistent with justice for women and girls in poor families. Methods developed by Esther Duflo and the ‘poor economists’ included institutional descriptions of poverty traps, with randomised controlled trials studying how incentives affect family agency. However, local knowledge could not easily apply to larger regions. As for the US, just as Becker mobilised controversial 1970s sociobiology against women’s liberation to rationalise women’s specialisation in household labour as an effect of biological comparative advantage in bearing children, categories of binary gender and binary biological difference initially prevented feminist economists from studying injustices experienced by queer families.

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).

Limited research has been devoted to investigating assumptions about competition dynamics established through a neoliberal lens. Advocates argue that competition fosters innovation and benefits consumers by incentivizing private enterprises to develop better products or services at competitive prices compared to their rivals. Critics argue that competition exacerbates inequality by disproportionately rewarding high achievers. Rewarding high achievers reflects the meritocratic aspect of competition, which has been widely assumed to be rooted in the individualistic culture of Western countries. Contrary to this assumption, the ideology of meritocratic competition thrived in ancient collectivist Asian countries. Moreover, the assumed linear relationship between individualism, competition, and inequality is contradicted by economic literature, which suggests more individualistic nations display lower income inequality. Despite extensive economic and cultural examination of competition, competition’s political dimensions remain understudied. This interdisciplinary book challenges conventional assumptions about competition, synthesizing evidence across economics, culture, and politics.

Compared with Britain, industrial transformation occurred more slowly in nineteenth-century France and Italy, forcing two early marginalists from the Lausanne school to pay continued attention to family poverty among the agrarian masses. Although Léon Walras and Vilfredo Pareto wanted to explain and resolve family impoverishment by securing a market whose outcomes correlated with what families deserved, the two economists diverged on the causes of family impoverishment, and on the best ways to respond. Walras’s ‘social economics’ rejected a popular view that family ‘immorality’ was the cause of family impoverishment, instead identifying badly designed government policy as the key factor. Pareto’s studies of population suggested the opposite position. Assuming government corruption and protective policies had been dismantled, Pareto assigned primary responsibility for poverty to egoistical parents, who should have anticipated cyclical economic decline before having children. Describing Malthus’s rejection of contraception as ‘not very scientific’, Pareto studied ‘people as they are’, finding families to be already limiting fertility through delayed sexual union or contraceptive knowledge. This suggested to Pareto that poverty would disappear spontaneously. Neither Walras nor Pareto explained how to manage existing family destitution or unanticipated economic crisis, and they did not problematise the many structural impediments to escaping one’s class.

In Victorian times, the family’s problems were viewed by one influential British economist, Alfred Marshall, through the lens of public consternation with urban family impoverishment. Marshall provided a critical extension of the population studies of late eighteenth-century clergyman, T. Robert Malthus, for whom poverty occurred when family reproduction exceeded the capacity of agricultural production. On the one hand, Marshall argued that Malthus had not recognised how larger populations could be sustained by the productivity gains of industrialisation. On the other hand, Marshall extended Malthus’s criticisms of the Old Poor Laws to the New laws too, which were rejected for encouraging poor families to have more children than they could adequately sustain. Marshall also followed Malthus by rejecting calls by Annie Besant, Charles Bradlaugh, and others for working-class access to contraceptive knowledge and birth-control techniques. Describing and evaluating class-based behaviour in factory families, artisanal families, and families of the higher class, Marshall identified the effects on labour productivity and living standards of patterns of family formation, fertility, mortality, household–market labour division, educational investment, and aged care provision. However, his policies supported gendered divides, overlooking how male breadwinning did not convert into an adequate family income, and rejecting activist demands for women’s rights.

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.

In this chapter, I explore competition among students and parents in a North Korean elementary school. Despite the perception that competition is discouraged in socialist countries like North Korea, it is prevalent as a means to motivate citizens to increase productivity. During my childhood in North Korea, competing with friends was commonplace. Teachers encouraged competition as a method to motivate students to study hard. While capitalist societies openly embrace competition, in North Korea, it exists in a visible but unspoken form. People are encouraged to compete to "praise the Great leaders" rather than for personal goals. The norms and meaning of competition tend to vary depending on the context, as illustrated by my childhood experiences. I highlight competition in three areas: (a) competition in classes through publicized performance scores, (b) competition among students to meet material quotas (e.g., papers, apricot stones, copper, etc.) through "mini assignments," and (c) competition for student leadership positions among parents through bribery.