WORLD FOR CIVIL

Topic 3.2: Stress, Strain & Hooke's Law - II

In our first topic, Static Equilibrium, we examined structures in which we assumed the members were rigid - rigid in the sense that we assumed that the member did not deform due to the applied loads and resulting forces. In real members, of course, we have deformation. That is, the length (and other dimensions) change due to applied loads and forces. In fact, if we look at a metal rod in simple tension as shown in diagram 1, we see that there will be an elongation (or deformation) due to the tension. If we then graph the tension (force) verses the deformation we obtain a result as shown in diagram 2.

In diagram 2, we see that, if our metal rod is tested by increasing the tension in the rod, the deformation increases. In the first region the deformation increases in proportion to the force. That is, if the amount of force is doubled, the amount of deformation is doubled. This is a form of Hooke's Law and could be written this way: F = k (deformation), where k is a constant depending on the material (and is sometimes called the spring constant). After enough force has been applied the material enters the plastic region - where the force and the deformation are not proportional, but rather a small amount of increase in force produces a large amount of deformation. In this region, the rod often begins to 'neck down', that is, the diameter becomes smaller as the rod is about to fail. Finally the rod actually breaks.

The point at which the Elastic Region ends is called the elastic limit, or the proportional limit. In actuality, these two points are not quite the same. The Elastic Limit is the point at which permanent deformation occurs, that is, after the elastic limit, if the force is taken off the sample, it will not return to its original size and shape, permanent deformation has occurred. The Proportional Limit is the point at which the deformation is no longer directly proportional to the applied force (Hooke's Law no longer holds). Although these two points are slightly different, we will treat them as the same in this course.

Next, rather than examining the applied force and resulting deformation, we will instead graph the axial stress verses the axial strain (diagram 3). We have defined the axial stress earlier. The axial strain is defined as the fractional change in length or Strain = (deformation of member) divided by the (original length of member) , Strain is often represented by the Greek symbol epsilon(e), and the deformation is often represented by the Greek symbol delta(d), so we may write: Strain (where L_o is the original length of the member) Strain has no units - since its length divided by length, however it is sometimes expressed as 'in./in.' in some texts.

As we see from diagram 3, the Stress verses Strain graph has the same shape and regions as the force verses deformation graph in diagram 2. In the elastic (linear) region, since stress is directly proportional to strain, the ratio of stress/strain will be a constant (and actually equal to the slope of the linear portion of the graph). This constant is known as Young's Modulus, and is usually symbolized by an E or Y. We will use E for Young's modulus. We may now write Young's Modulus = Stress/Strain, or: . (This is another form of Hooke's Law.)

The value of Young's modulus - which is a measure of the amount of force needed to produce a unit deformation - depends on the material. Young's Modulus for Steel is 30 x 10⁶ lb/in², for Aluminum E = 10 x 10⁶ lb/in², and for Brass E = 15 x 10⁶ lb/in². For more values, select: Young's Modulus - Table.
To summarize our stress/strain/Hooke's Law relationships up to this point, we have:

The last relationship is just a combination of the first three, and says simply that the amount of deformation which occurs in a member is equal to the product of the force in the member and the length of the member (usually in inches) divided by Young's Modulus for the material, and divided by the cross sectional area of the member. To see applications of these relationships, we now will look at several examples.

Continue to:
Example 1 ; Example 2 ; Example 3