 In developing the force-displacement relationship for an axial loaded member, such as the tensile rod shown here, we established that the deformation and the resulting stress distribution due to the resultant tensile load P are both uniform. But in doing so, we actually cheated a little. We only looked at the deformation far away from any point of load application. Why does this matter? We'll take a closer look at the point of load application in our axial rod. We simply have the end of the structure drawn with the resultant force P indicated. This is easy to draw but is in no way an accurate representation of the boundary conditions of a real axial loaded member. There are numerous ways of introducing load into real structures. In the case of our rod, one of the ends could be built into a larger, relatively rigid structure while the other end may have load introduced via pin bearing through a pin connection to another structure. More complex boundary conditions involving two or more fasteners could also be conceived. Looking at each of these boundary conditions, it should be obvious that they will not all behave the same locally at the connection. If we take a closer look at the built-in end of the rod, it is easy to see that this type of boundary condition locally constrains the lateral deformation of the rod. This restriction and deformation will result in shear stresses locally that eventually redistribute to a state of uniform normal stress as you move away from the boundary condition. Similarly, for the pin connection end of the rod, load introduction occurs through the pin. Thus stresses in the rod are initially concentrated around that pin and have to flow around the pin hole before they can redistribute into a more uniform stress state further along the rod. First observed by the French scientist, Barry de Saint-Venant. This observation is referred to as Saint-Venant's principle. It states that at a distance sufficiently removed from a boundary condition, the stress in a structure will be the same for any boundary condition that produces an equivalent resultant load. Looking at a typical engineering problem then, we are faced with having a simplified force-displacement relationship that we can apply to the structure, but also the knowledge that there is a region within the structure where that relationship is not really correct. We can of course ignore the influence of the boundary conditions and calculate a deflection using this force-displacement relationship, but we should have some way to assess if neglecting this influence is acceptable. In the problem shown here, we can highlight a length of the beam influenced by each of the two boundary conditions and denote the length of these regions as k. The question becomes, how big is k? It turns out that the answer to that question is not entirely straightforward and can depend a lot on the type of constraint provided by the particular boundary condition. However, a general rule of thumb is that the length k is equal to the largest cross-sectional dimension. So if the beam shown here had a rectangular cross-section of width a and height b, then k would be approximately equal to b. Add a bit of engineering judgment with this rule of thumb and you may just be able to decide if your assumption to neglect the influence of the boundary condition is valid.