Elastic Limit

What happens if you stretch a spring (or anything else) too far? If the force is small  enough, it will spring back to its original position when released, but beyond a point called its elastic limit, it will be permanently deformed and it will not spring back to its original position. If you increase the force further, you will pass the point called its ultimate strength, and then it will break.

Some materials, such as soft metals, can deform a great deal without breaking. Other materials, like ceramic or hardened steel, will break almost immediately rather than deform. All materials, though, will exhibit some elasticity as long as the deformations are kept small enough. You could make a spring out of glass, and it would indeed be “springy”, but only over a very short distance or else it would break.

Stress and Strain

– Young’s Modulus

The amount of force that it takes to stretch something like a cable will depend on its thickness. This is taken into account by defining stress as the force per cross-sectional area:

Stress will typically have units of newtons per square meter (N/m2), or pounds per square inch.

Next we need a way to describe how much a thing has stretched. It turns out that the most useful way to do this is to define strain as the fractional change in length, or

Here DL means the amount that the length has changed, and L is the original length. Notice that strain does not have any units, because it is a length divided by a length and so whatever units you are using for length will cancel out.

As long as the material is behaving elastically (meaning we haven’t stretched or compressed it too far), it will obey Hooke’s law. In term of stress and strain, Hooke’s law becomes

The constant that we called k in Hooke’s law is called Young’s Modulus, or Y, when we are talking about stress and strain. Young’s modulus is the ratio of stress to strain,


Y is a property that only depends on the material being used. Values have been measured for all common building materials, and engineers can look them up.

Shear Modulus

Young’s modulus refers to tensile forces, which are forces that stretch or compress. Now we will look at shear forces. Shear forces can be though of as forces that act sideways and try to deform the object in the way that a rectangle deforms into a parallelogram. Bolts are frequently subject to shear forces, and so it is important for engineers to know how much shear a bolt can withstand before breaking:

The shear modulus relates the shear stress with the shear force. The shear stress is the ratio of sideways deflection Dx with the height h (see the diagram below). Thus

or