Stress and Strain - Structural and Stress Analysis - page 155

7.4 D IRECT S TRAIN

Since no material is completely rigid, the application of loads produces distortion. An
axial tensile load, for example, will cause a structural member to increase in length,
whereas a compressive load would cause it to shorten.

Suppose that is the change in length produced by either a tensile or compressive

axial load. We now define the direct strain , ε , in the member in non-dimensional form
as the change in length per unit length of the member. Hence

δ

L 0

ε = (7.4)

where L 0 is the length of the member in its unloaded state. Clearly ε may be either a
tensile (positive) strain or a compressive (negative) strain. Equation (7.4) is applicable
only when distortions are relatively small and can be used for values of strain up to and
around 0.001, which is adequate for most structural problems. For larger values, load-
displacement relationships become complex and are therefore left for more advanced
texts.

We shall see in Section 7.7 that it is convenient to measure distortion in this non-
dimensional form since there is a direct relationship between the stress in a member
and the accompanying strain. The strain in an axially loaded member therefore
depends solely upon the level of stress in the member and is independent of its length
or cross-sectional geometry.

7.5 S HEAR S TRAIN

In Section 7.3 we established that shear loads applied to a structural member induce a
system of shear and complementary shear stresses on any small rectangular element.
The distortion in such an element due to these shear stresses does not involve a change
in length but a change in shape as shown in Fig. 7.6. We define the shear strain , γ , in the
element as the change in angle between two originally mutually perpendicular edges.
Thus in Fig. 7.6

γ = φ radians (7.5)

τ

τ Distorted

shape

f

f

τ

τ

F IGURE 7.6 Shear strain in an element