## Stress-Strain Relationships

Structural steels display linearly elastic properties when the load does not exceed a certain limit. Steels also are isotropic; i.e., the elastic properties are the same in all directions. The material also may be assumed homogeneous, so the smallest element of a steel member possesses the same physical property as the member. It is because of these properties that there is a linear relationship between components of stress and strain. Established experimentally (see Art. 3.8), this relationship is known as Hooke’s law. For example, in a bar subjected to axial load, the normal strain in the axial direction is proportional to the normal stress in that direction, or where E is the modulus of elasticity, or Young’s modulus.
If a steel bar is stretched, the width of the bar will be reduced to account for the increase in length (Fig. 3.14a). Thus the normal strain in the x direction is accompanied by lateral strains of opposite sign. If x is a tensile strain, for example, the lateral strains in the y and z directions are contractions. These strains are related to the normal strain and, in turn, to the normal stress by where is a constant called Poisson’s ratio.
If an element is subjected to the action of simultaneous normal stresses ƒx, ƒy, and ƒz uniformly distributed over its sides, the corresponding strains in the three directions are Similarly, shear strain  is linearly proportional to shear stress v where the constant G is the shear modulus of elasticity, or modulus of rigidity. For an
isotropic material such as steel, G is directly proportional to E: The analysis of many structures is simplified if the stresses are parallel to one plane. In some cases, such as a thin plate subject to forces along its edges that are parallel to its plane and uniformly distributed over its thickness, the stress distribution occurs all in one plane.
In this case of plane stress, one normal stress, say ƒz, is zero, and corresponding shear stresses are zero.

In a similar manner, if all deformations or strains occur within a plane, this is a condition of plane strain.