## Stressstrain relationships

Before commencing a study of the material in this chapter it is best to become familiar with the main terms used to describe the stress-strain relationships of a material. It is useful to begin by examining a typical stress-strain plot obtained for a metal (Fig. 4.1).

Results such as those indicated in the figure would normally be obtained by subjecting a specimen of the metal to a tensile test and plotting the values of tensile strain against the nominal values of tensile stress, as the stress-strain relationship obtained is equally applicable in tension or compression in the case of a metal.

Note Nominal stress = actual load/original cross-sectional area of specimen, i.e. no allowance is made for reduction in area, due to necking, as the load is increased.

From the plot it is seen that in the early stages of loading, up to point B, the stress is proportional to the strain. Unloading tests can also demonstrate that, up to the point A, the metal is elastic in that it will return to its original dimensions if the load is removed. The limiting stress at which elasticity effects are not quite complete is known as the elastic limit, represented by

Fig. 4.1 Stress-strain relationships.

Plastic

Fig. 4.1 Stress-strain relationships.

point A. The limiting stress at which linearity between stress and strain ceases is known as the limit of proportionality, point B.

In most metals points A and B occur so close together that they are generally assumed to coincide, i.e. elastic limit is assumed equal to the limit of proportionality.

Point C in Fig. 4.1a represents the yield point, i.e. the stress value at which there is a sudden drop of load, as illustrated, or the stress value at which there is a continuing extension with no further significant increase in load.

Figure 4.1a can be approximated to Fig. 4.1b which represents the ideal elastic-plastic material. In this diagram, point 1 represents the limit of elasticity and proportionality and the point at which plastic behaviour occurs. The form of the compressive stress-strain relationships typical for all types of soil up to their peak values is as shown in Fig. 4.1c.

It is seen that the stress-strain relationship of a soil is never linear and, in order to obtain solutions, the designer is forced either to assume the idealised conditions of Fig. 4.1b or to solve a particular problem directly from the results of tests that subject samples of the soil to conditions that closely resemble those that are expected to apply in situ.

In most soil problems the induced stresses are either low enough to be well below the yield stress of the soil and it can be assumed that the soil will behave elastically (e.g. immediate settlement problems), or they are high enough for the soil to fail by plastic yield (bearing capacity and earth pressure problems), where it can be assumed that the soil will behave as a plastic material.

With soils, even further assumptions must be made if one is to obtain a solution. Generally it is assumed that the soil is both homogeneous and isotropic. As with the assumption of perfect elasticity these theoretical relationships do not apply in practice but can lead to realistic results when sensibly applied.

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