Design

Stress Strain Relationship

Stress Strain Relationship

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A member is under axial loading when a force acts along its axis. The internal force is normal to the plane of the section and the corresponding stress the rod is experiencing is described as a normal stress. The stress is obtained by dividing the magnitude of the resultant of the internal forces distributed over the cross section by the area of the cross section, thus it represents the average value of the stress over the cross section.

Average Normal Stress:
 

Another important aspect of design relates to the deformations caused by the loads applied to a member. The ratio of deformation (Dl) over the length of the rod (l) is called the strain and is denoted by e. In the case of a member of constant cross sectional area with length l, the strain remains constant along the entire member and may be obtained by dividing the total deformation by its length.

Nominal or Engineering Strain:
 


True Strain: 

where: 



Generalized Hooke’s Law
 

The generalized Hooke’s Law can be used to predict the deformations caused in a given material by an arbitrary combination of stresses.

The linear relationship between stress and strain applies for 

Original Specimen

where:

E is the Young’s Modulus
n is the Poisson Ratio

The generalized Hooke’s Law also reveals that strain can exist without stress. For example, if the member is experiencing a load in the y-direction (which in turn causes a stress in the y-direction), the Hooke’s Law shows that strain in the x-direction does not equal to zero. This is because as material is being pulled outward by the y-plane, the material in the x-plane moves inward to fill in the space once occupied, just like an elastic band becomes thinner as you try to pull it apart. In this situation, the x-plane does not have any external force acting on them but they experience a change in length. Therefore, it is valid to say that strain exist without stress in the x-plane.