 Hooke's law is the fundamental equation in this course relating deformation to loading. However, in order to obtain a deformation from it, we need to perform an integration of the strain state over the domain of a structure. And we all know how much you guys like to integrate. One way to avoid unnecessary integration is to recognize instances of repetition. Rather than integrating the strain distribution over the domain of a structure, every time we solve a problem, we can recognize repeating structural elements and perform the integration once. In this way, we can produce a shortcut relating force and displacement. The first structural element we will produce a force-displacement relation for is an axial-loaded member. Take this generic rod with the uniform cross-section and length L. If we apply an axial load P to this rod, it will deform. It will elongate by a given displacement which we will denote as a lower case, delta. Before we can apply Hooke's law, we need to establish a local coordinate system for our rod. We will do so by aligning the x-axis with the axis of the rod. The directions of the y and z-axis do not matter as much due to the axisymmetric nature of the problem, so we will just draw them as shown here. We can now look at Hooke's law in relation to our problem. All loading in our problem is along the axis of the rod, and there are no constraints on deformation due to the Poisson effect. As a result, there will be no stresses in either the y or the z directions. We are also not considering a temperature change at this time, so delta T also reduces to zero. Hooke's law for this problem can then be simplified as follows. We also know that axial loading produces a state of uniform stress and strain, as long as we are sufficiently far away from the boundary conditions according to Saint-Venant's principle. Integration of the stress and strain state become trivial when they are both uniform, resulting in the following two equations. If we substitute the results from integrating our stress and strain distributions into our simplified Hooke's law, we get the following relationship. Rearranging this in terms of deformation, we obtain that delta is equal to P times L divided by E times A. We now have our first force displacement relationship.