# Simple Stress and Strain

The application of loads to structural members induce deformations and internal resisting forces within the materials. The intensity of these forces is known as the stress in the material and is measured as the force per unit area of the cross-sections which is normally given the symbol a when it acts perpendicular to the surface of a cross-section and t when it acts parallel to the surface. Different types of force cause different types and distributions of stress for example: axial stress, bending stress, shear stress, torsional stress and combined stress.

Consider the case of simple stress due to an axial load P which is supported by a column of cross-sectional area A and original length L as shown in Figure 2.1. The applied force induces an internal stress c such that:

P=(<rx A) and hence <7 = Pi A (Le, load/unit area) The deformation induced by the stress is quantified by relating the change in length to the original length and is known as the strain in the material normally given the symbol e where:

S=(change in length/original length)=(S/L)

Note: the strain is dimensionless since the units of 5 and L are the same.

The relationship between stress and strain was first established by Robert Hook in 1676 who determined that in an elastic material the strain is proportional to the stress. The general form of a stress/strain graph is as shown in Figure 2.2. Slrjín (Í: )

Figure 2.2

The point at which this graph ceases to obey Hook's Law and becomes non-linear is the 'elastic limit' or 'proportional limit'.

A typical stress-strain curve for concrete is shown in Figure 2.3(a). This is a non-linear curve in which the peak stress is developed at a compressive strain of approximately 0.002 (depending upon the strength of the concrete) with an ultimate strain of approximately 0.0035. There is no clearly defined elastic range over which the stress varies linearly with the strain. Such stress/strain curves are typical of brittle materials.

A typical stress-strain curve for hot-rolled mild steel is shown in Figure 2.3(b). When a test specimen of mild steel reinforcing bar is subjected to an axial tension in a testing machine, the stress/strain relationship is linearly elastic until the value of stress reaches a yield value, e.g. 250 N/mm2.

At this point an appreciable increase in the stretching of the sample occurs at constant load: this is known as yielding. During the process of yielding a molecular change takes place in the material which has the effect of hardening the steel. After approximately 5% strain has occurred sufficient strain-hardening will have developed to enable the steel to carry a further increase in load until a maximum load is reached.

The stress-strain curve falls after this point due to a local reduction in the diameter of the sample (known as necking) with a consequent smaller cross-sectional area and load carrying capacity. Eventually the sample fractures at approximately 35% strain.

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IoaJ filling {riidiw mui«lKIKMirt*Wtt» IflmwVingoflhc SirtiMi(ff) Slnln(f)

SirtiMi(ff) Slnln(f)

Figure 2.3

The characteristics of the stress/strain curves are fundamental to the development and use of structural analysis techniques. A number of frequently used material properties relating to these characteristics are defined in Sections 2.1.2 to 2.1.6.