section epub:type=”chapter”> This chapter examines three applications of plasticity. The first (material selection for a spring) requires no plasticity whatever. The second (material selection for a pressure vessel) typifies plastic design of a large structure. It is unrealistic to expect no plasticity: there will always be some, at bolt holes, loading points or changes of section. The important thing is that yielding should not spread entirely through any section of the structure—plasticity must not become general. Finally, this chapter examines an instance (the rolling of metal strip) in which yielding is deliberately induced to give large-strain plasticity. We now examine three applications of plasticity. The first (material selection for a spring) requires no plasticity whatever. The second (material selection for a pressure vessel) typifies plastic design of a large structure. It is unrealistic to expect no plasticity: there will always be some, at bolt holes, loading points, or changes of section. The important thing is that yielding should not spread entirely through any section of the structure—plasticity must not become general. Finally, we examine an instance (the rolling of metal strip) in which yielding is deliberately induced, to give large-strain plasticity. Springs come in many shapes and have many purposes. One thinks of axial springs (a rubber band, for example), leaf springs, helical springs, spiral springs, torsion bars. Regardless of their shape or use, the best material for a spring of minimum volume is that with the greatest value of The argument, at its simplest, is as follows. The primary function of a spring is that of storing elastic energy and—when required—releasing it again. The elastic energy stored per unit volume in a block of material stressed uniformly to a stress σ (see Figure 9.1) is It is this that we wish to maximize. The spring will be damaged if the stress σ exceeds the yield stress or failure stress σy; the constraint is σ ≤ σy. So the maximum energy density is Torsion bars and leaf springs are less efficient than axial springs because some of the material is not fully loaded: the material at the neutral axis, for instance, is not loaded at all. Even leaf springs can take many different forms, but all of them are basically elastic beams loaded in bending. For the loading shown in Figure 13.1, the beam bending results in Chapter 7 give The elastic energy stored in the spring, per unit volume, is Figure 13.2 shows that the stress in the beam is zero along the neutral axis at its center, and is a maximum at the surface, at the midpoint of the beam (because the bending moment is biggest there). The beam bending results in Chapter 7 show that the maximum surface stress is given by Now to be successful, a spring must not undergo a permanent set during use: it must always “spring” back. The condition for this is that the maximum stress must always be less than the yield stress: Eliminating t between this and Equation (13.2) gives So if in service a spring has to undergo a given deflection δ under a force F, the ratio of Springs for a centrifugal clutch Suppose you are asked to select a material for a spring with the following application. A spring-controlled clutch like that shown in Figure 13.3 is designed to transmit 20 horsepower at 800 rpm; the clutch is to begin to pick up load at 600 rpm. The blocks are lined with Ferodo or some other friction material. When properly adjusted, the maximum deflection of the springs is to be 6.35 mm (but the friction pads may wear, and larger deflections may occur; this is a standard problem with springs—they must often withstand extra deflections without losing their sets). The force on the spring is where m is the mass of the block, r the distance of the center of gravity of the block from the center of rotation, and ω the angular velocity. The net force each block exerts on the clutch rim at full speed is where ω2 and ω1 correspond to the angular velocities at 800 and 600 rpm (the net force must be zero for ω2 = ω1, at 600 rpm). The full power transmitted by all four blocks is given by 4μsmr (
Case Studies in Yield-Limited Design
Publisher Summary
13.1 Introduction
13.2 Case Study 1: Elastic Design—Materials for Springs
. Here E is Young’s modulus and σy the failure strength of the material of the spring: its yield strength if ductile, its fracture strength, or modulus of rupture if brittle. Some materials with high values of this quantity are listed in Table 13.1.
The leaf spring
must be high enough to avoid a permanent set. This is why we have listed values of 
in Table 13.1: the best springs are made of materials with high values of this quantity. For this reason spring materials are heavily strengthened (see Chapter 11): by solid solution strengthening plus work-hardening (cold-rolled, single-phase brass, and bronze), solid solution and precipitate strengthening (spring steel), and so on. Annealing any spring material removes the work-hardening, and may cause the precipitate to coarsen (increasing the particle spacing), reducing σy and making the material useless as a spring.
Worked Example
Mechanics
) × distance moved per second by inner rim of clutch at full speed, that is
120
200
