## Distributed Force On Nodes Finite Element

In many "thin" structures modeled as continuous bodies the appearance of "skinny" elements is inevitable on account of computational economy reasons. An example is provided by the three-dimensional modeling of layered composites in aerospace and mechanical engineering problems.

§8.2.3. Physical Interfaces

A physical interface, resulting from example from a change in material, should also be an interelement boundary. That is, elements must not cross interfaces. See Figure 8.3.

No OK

No OK

§8.2.4. Preferred Shapes

In two-dimensional FE modeling, if you have a choice between triangles and quadrilaterals with similar nodal arrangement, prefer quadrilaterals. Triangles are quite convenient for mesh generation, mesh transitions, rounding up corners, and the like. But sometimes triangles can be avoided altogether with some thought. One of the homework exercises is oriented along these lines.

In three dimensional FE modeling, prefer strongly bricks over wedges, and wedges over tetrahedra. The latter should be used only if there is no viable alternative. The main problem with tetrahedra and wedges is that they can produce wrong stress results even if the displacement solution looks reasonable.

§8.3. DIRECT LUMPING OF DISTRIBUTED LOADS

In practical structural problems, distributed loads are more common than concentrated (point) loads.2 Distributed loads may be of surface or volume type.

Distributed surface loads (called surface tractions in continuum mechanics) are associated with actions such as wind or water pressure, lift in airplanes, live loads on bridges, and the like. They are measured in force per unit area.

Volume loads (called body forces in continuum mechanics) are associated with own weight (gravity), inertial, centrifugal, thermal, prestress or electromagnetic effects. They are measured in force per unit volume.

2 In fact, one of the objectives of a good design is to avoid or alleviate stress concentrations produced by concentrated forces.

Nodal force f at 3 is set to P, the magnitude of the crosshatched area under the load curve. This area goes halfway over adjacent element sides

Figure 8.4. NbN direct lumping of distributed load, illustrated for a 2D problem.

A derived type: line loads, result from the integration of surface loads along one transverse direction, or of volume loads along two transverse directions. Line loads are measured in force per unit length.

Whatever their nature or source, distributed loads must be converted to consistent nodal forces for FEM analysis. These forces eventually end up in the right-hand side of the master stiffness equations.

The meaning of "consistent" can be made precise through variational arguments, by requiring that the distributed loads and the nodal forces produce the same external work. Since this requires the introduction of external work functionals, the topic is deferred to Part II. However, a simpler approach called direct load lumping, or simply load lumping, is often used by structural engineers in lieu of the more mathematically impeccable but complicated variational approach. Two variants of this technique are described below for distributed surface loads.

The node by node (NbN) lumping method is graphically explained in Figure 8.4. This example shows a distributed surface loading acting on the straight boundary of a two-dimensional FE mesh. (The load is assumed to have been integrated through the thickness normal to the figure, so it is actually a line load measured as force per unit length.)

The procedure is also called tributary region or contributing region method. For the example of Figure 8.4, each boundary node is assigned a tributary region around it that extends halfway to the adjacent nodes. The force contribution P of the cross-hatched area is directly assigned to node 3.

This method has the advantage of not requiring the computation of centroids, as required in the EbE technique discussed in the next subsection. For this reason it is often preferred in hand computations. It can be extended to three-dimensional meshes as well as volume loads.3 It should

3 The computation of tributary areas and volumes can be done through the so-called Voronoi diagrams.

Force P has magnitude of I liofnrMitnn l/"-\«""»/-l

Force P has magnitude of I liofnrMitnn l/"-\«""»/-l

Figure 8.5. EbE direct lumping of distributed load, illustrated for a 2D problem.

be avoided, however, when the applied forces vary rapidly (within element length scales) or act only over portions of the tributary regions.

§8.3.2. Element by Element (EbE) Lumping

In this variant the distributed loads are divided over element domains. The resultant load is assigned to the centroid of the load diagram, and apportioned to the element nodes by statics. A node force is obtained by adding the contributions from all elements meeting at that node. The procedure is illustrated in Figure 8.5, which shows details of the computation over segment 2-3. The total force at node 3, for instance, would that contributed by segments 2-3 and 3-4.

If applicable, the EbE procedure is more accurate than NbN lumping. In fact it agrees with the consistent node lumping for simple elements that possess only corner nodes. In those cases it is not affected by the sharpness of the load variation and can be used for point loads that are not applied at the nodes.

The procedure is not applicable if the centroidal resultant load cannot be apportioned by statics. This happens if the element has midside faces or internal nodes in addition to corner nodes, or if it has rotational degrees of freedom. For those elements the variational approach is preferable.

§8.4. BOUNDARY CONDITIONS

The key distinction between essential and natural boundary conditions (BC) was introduced in the previous Chapter. The distinction is explained in Part II from a variational standpoint. In this Chapter we discuss next the simplest essential boundary conditions in structural mechanics from a physical standpoint. This makes them relevant to problems with which a structural engineer is familiar. Because of the informal setting, the ensuing discussion relies heavily on examples.