In finite element analysis, the loads are often applied to the elements rather than to the grid points. Examples of such loadings include the edge load on a CQUAD4 element or a pressure load on one of the faces of a solid element.
When you convert these element loads to grid point loads, a common tendency is to spread them equally at the connected grid points. Let us look at an example of a typical plate element with uniform in-plane loading at one of the edges as shown in Figure 9-27. The total load (P) is to be uniformly distributed along the edge of the element with an intensity of P/l, where l is the length of the edge. This same load is to be applied to six different elements-CQUAD4, CTRIA3, CQUAD8, CTRIA6, CQUADR, and CTRIAR. If you lump these loads equally to the grid points, the load at each node is P/2 for the CQUAD4, CTRIA3, CQUADR and CTRIAR, and P/3 for the CQUAD8 and CTRIA6. Since the geometry and boundary condition are identical for the six models, and the loads are “seemingly equivalent,” you will expect the results (displacements, stresses, etc.) to be identical. However, in general, this is not the case. The results for the cases using CQUAD8, CTRIA6, CQUADR, and CTRIAR are different from the results using CQUAD4 and CTRIA3. The reason for this is because these “seemingly equivalent” loads are actually not quite equivalent. They are considered as lumped loads. In order to obtain the correct load distribution, these loads need to be converted to consistent loads.
The equivalent loads at the grid points computed from the element loads are known as consistent loads, and they are calculated by applying the principle of virtual work. The same shape function that is used in deriving the element stiffness is used for arriving at this load-and hence the word consistent load. They are a function of the element types and the applied loads. Depending on the element, these lumped loads in general are not equal to the consistent loads. The consistent loads for each of these elements for a uniform in-plane edge load of P/l are shown next to each element. In this case, the lumped loads and consistent loads are the same for the CQUAD4 and CTRIA3. They are not the same for the CQUAD8, CTRIA6, CQUADR and CTRIAR. As you can see, they are substantially different from the lumped loads that were discussed in the previous paragraph. It is interesting to note that the loads are distributed as 1/6, 4/6, and 1/6 along the edge for the CQUAD8 and CTRIA6, which is quite different from the lumped load approach. For the CQUADR and CTRIAR, an additional moment Pl/12 is needed to arrive at the consistent loads. When the corresponding consistent loads shown in Figure 9-27 are applied to each of the respective elements, the force distributions are then equivalent for all the elements.
|Figure 9-27 Consistent Loads Due to Uniform In-Plane Element Edge Load|
Consider another example with solid elements. A load P is to be applied evenly as an outward pressure load (P/A) to a surface of each of the four solid elements shown in Figure 9-28. Again, if you lump these loads equally to the grid points, the lumped load at each node is P/4 for the CHEXA (with eight nodes) and CPENTA (with six nodes), and P/8 for the CHEXA (with 20 nodes) and CPENTA (with 15 nodes). The consistent loads are shown in Figure 9-28. The consistent and lumped loads are the same for the CHEXA (with eight nodes) and the CPENTA (with six nodes) for this case. The consistent loads for the CHEXA (with 20 nodes) and CPENTA (with 15 nodes) are shown in Figure 9-28(c) and Figure 9-28(d), respectively. The load distribution along each edge is -.0833P, .3333P and -.08333P, or a -1/4/-1 ratio, which is quite counter intuitive, especially for the sign change. Fortunately, you do not have to calculate this consistent load for the pressure load. This consistent load is generated automatically inside MSC Nastran by using the PLOAD4 entry.
|Figure 9-28 Consistent Loads Due to Uniform Loads on a Solid Element Face|
As you can see, consistent loads are functions of the element types and applied loads.
All of the content in this blog post has been directly extracted from Chapter 9 of the MSC Nastran 2012 Linear Static Analysis User's Guide.