Complex Geometry in Structural Analysis

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An example of a finite element analysis software showing stress analysis results.

Finite element analysis (FEA), or the finite element method (FEM), is a numerical technique for analyzing the stresses and reactions in structures. Its computational power enables it to be used to solve structures with complex geometries.  Traditional handwritten methods require simple structures in order to make vital assumptions and idealizations such as localizing the mass at its center of gravity or disregarding negligible geometry.  For example, a simple cantilever beam loaded at its end may be analyzed with introductory statics techniques: summing moments about an end to solve for support reactions, cutting the member to solve for an internal force, etc.  Handwritten approaches would reduce the shape of the beam to a line and consider all the mass to be located in the middle.  Though these assumptions may help in solving a beginner level statics problem, they prove to be inadequate in the analysis of structures with more complex design.  For example, solving for the specific stress and deformation of a point on a gear tooth in a rack and pinion system can only be attempted with a more detailed and computationally cumbersome method.  Calculating the stresses of this system would prove to be arduous with handwritten methods.  Through the modern power of computers however, the process of analyzing tiny sections within the object not only becomes a viable option, but can now be performed with a great amount of accuracy.

The basis of FEA starts with dividing the structure into smaller shapes of simpler form.  For example, the profile of a bridge may be divided into a puzzle of triangles fitted together.  This mesh of finite elements is much easier to analyze.  By studying the behavior of the smaller bits of the whole, the total behavior of the bridge may be determined.  Furthermore, each element is defined by points or “nodes” which have an explicitly calculated and known position.  Thus any other location within the element may be estimated from the values of the enclosing nodes.  Generally these elements come in triangular shapes providing only three nodes of information.  However, other models may use rectangular elements with as many as eight nodes.  Each node has the ability to move within the coordinate system of the structure.  However, in order to facilitate a solvable system, certain nodes along the structure’s exterior edge may be classified as “fixed,” making their position immovable.  The process of FEA determines mathematical equations to model the movement or “displacement” of each node as a function of x, y, and z coordinates.  All of this spatial information is stored in a node-specific matrix, known as the “stiffness matrix.”  Each element contains a standard number of nodes, each defined by its stiffness matrix, which contains mathematical formulas that model its movement.  From each node’s position, the displacement of the element may also be implicitly determined.  Different degrees of accuracy can be used within the same object by using varying numbers of elements.  The more elements defined in an area, the more accurate the analysis will be.  As in the gear example, the teeth and areas near this edge will be composed of several tiny elements, while the area near the center of the gear would not need to be very accurate and are therefore composed of fewer and larger elements.  From this basic information, higher level knowledge regarding the element’s stress, strain, and energy may be calculated.  Once all this data is determined for the elements, the analysis can then determine what is happening in the larger model.

With the abilities of FEA, the physical reaction of a body may be calculated at any point in the structure.  One of the many powers of this method of analysis is the ability to change an object’s material properties.  This allows the user to model element behavior as well as geometry; this effects the element stiffness matrices to adjust to a material profile.  While it remains vital to retain in-depth engineering knowledge, as will be discussed later, engineers may now design buildings without reference to tables, allowing the computer to do the majority of the mechanics of materials.  

Basic Structural Analysis (8:56)

This video discusses the Finite Element Method, using Autodesk Robot Structural Analysis Professional. It demonstrates the creation of a 2D frame and shows how to set loads: dead, live and wind (2:52). In addition, one can view the deformed shape of structure to see if the load is causing the beam and columns to bend in the right directions (5:37). And finally, a quick rule of thumb for calculating moment is shown (6:26). The dataset for this example is available for download below.