Stress and Strain Relations

Stress results in a fractional change in length or volume, defined as strain. Hooke's law states that stress is directly proportional to strain, but ONLY up to a limit called the proportionality constant. If the material in question exhibits linear-elastic behaviour, and Hooke's law applies, then a linear variation of normal strain may be assumed to occur during bending. Within the elastic limit of the beam material, the deflection, D, at midspan is

where I is the second moment of area and E is the modulus of elasticity.


The modulus of elasticity, or Young's modulus, is the ratio of change in stress to change in strain during elastic deformation and thus is the slope of the stress-strain graph until the material reaches its proportional load. Using the relation above and the results from the load-deflection plot of a linear-elastic material, the modulus of elasticity can be determined.

If the load only slightly exceeds the proportional load, then the material may still respond elastically, but the response to the load is no longer linearly related to the applied load. When the applied load permanently changes the properties of the material, the specimen is said to be exhibiting plastic deformation. Eventually, the load to which the material is subjected will be at the material's ultimate load. At this point, the material will either fail (if it is a brittle material) or it will continue to deflect (if it is ductile) until it finally ruptures.

The maximum load that a material can sustain before it no longer behaves linear-elastically is the proportional load (Ppl)

The bending stress, or the amount of stress to which the beam is subjected while it is behaving in a linear-elastic manner, is given by

where y is the distance of the point of consideration from the neutral axis.

When we have a beam (with depth, d) that is symmetric about its x-x (strong) axis, the neutral axis is located at the middle of the beam.
In this case, y=d/2 since the maximum bending stress occurs at the top and bottom fibers of the beam.

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