Introduction
Mechanics of materials is an important topic in the study of engineering mechanics. It is normally an undergraduate subject that is taught to engineering students who have previously completed prerequisite courses in statics and dynamics. This subject is also known by other names, such as strength of materials and mechanics of deformable bodies. In simple terms, mechanics of materials is the study of how forces produce stresses in solid bodies. It is an essential topic for engineers and students who are solving problems in aerospace, aeronautical, civil, and mechanical engineering.

Static and Dynamic Loading
To determine the deformations and resulting stresses in a solid body, it is imperative that the analyst understand the applied loads. There are two types of loads that are commonly encountered in the analysis of structural members. The first load type is static which includes constant forces, pressures, moments, and dead weights. The second load type is dynamic which includes varying forces, accelerations, vibration, shock, and transient dynamic forces. Real world application of dynamic loads includes impact forces, wind loads, snow loads, and earthquake loads. In the application of any type of loading, it is essential that the analyst construct free body diagrams to determine the resolution of the applied loads. When a solid body experiences combined loading, the principle of superposition can be used to add the effect of the applied loads.

Material Properties-Stress & Strain
A critical aspect of mechanics of materials is the relationship between stress and strain. Materials that obey Hooke’s law exhibit a linear relationship between stress and strain. The resulting deformation is considered elastic when the structure retains its original shape after the applied load has been removed. When the applied load is increased, and the deformation is permanent, the material behavior is called plastic. The relationship between a material’s stress and strain determines the likelihood for a structure to survive in the real world. Stresses that occur at a point are analyzed using Mohr’s circle for two or three dimensional loading.

Types of Structural Members
Mechanics of materials provides the analyst with closed form solutions for simple structural members that experience a variety of loading conditions. Prismatic bars can be loaded axially, or in torsion. A special case of a prismatic bar is a circular shaft that is loaded in bending and torsion. Beams can be loaded axially, in pure bending, or in torsion. The beam bending case is analyzed using shear and moment diagrams. As with any axial loading, it can be either tensile or compressive. Columns are loaded axially in compression, and are normally assessed for buckling stability. Thin walled pressure vessels are typically analyzed for hoop stresses.

Preparation for Advanced Topics
Understanding the basic concepts and simplified methods in mechanics of materials prepares the student for advanced study in engineering mechanics. Analyzing real world engineering structure can be done with a background in statics, dynamics, and strength of materials. Advanced topics include machine design, vibration, advanced mechanics of materials, plates and shells, elasticity, and plasticity.

Introduction
Finite element analysis (FEA) was developed decades ago to solve engineering structural problems. In its infancy, FEA was simply a matrix method to deal with simple structures that could be analyzed with rudimentary computer systems.
However, it has evolved into a widely used method for evaluating a variety of products in many engineering disciplines. FEA has been used for decades to solve complex structural, thermal, and fluid mechanics problems. Applications are typically found in civil and mechanical engineering, but FEA has also been used extensively in electrical engineering as well. Finite element analysis enables engineers to tackle real world complex assemblies and systems that are impossible to solve with simpler methods.

History
The finite element method (FEM) originated in the 1940’s. At this time, there was a need to solve more complex problems in elasticity and structural analysis. As the method evolved, various approaches were recognized that involved dividing a continuous domain into sub domains, or elements. Mathematically, the method involved solving an array of partial differential equations. During the 1950’s, further development of FEA continued in airframe and structural analysis. Research at several universities culminated in more efficient methods for solving the stiffness matrices that were the basis of the method. This research was concentrated in civil engineering applications during the 1960’s, but applications to other engineering disciplines were also beginning to appear at this time. During the late 60’s, three of the more recognized finite element codes were introduced: ANSYS, NASTRAN, and STARDYNE. Advances in computing power and software development over the years have culminated in the efficient, multi-purpose finite element (FE) programs that are available today.

The Finite Element Method
The finite element method consists of three major tasks. The first is pre-processing, where the analyst develops a mesh which consists of nodes and elements. This mesh is the basis for the analysis, and is supplemented with appropriate material properties, element properties, and boundary conditions. The second task is the solution phase, where the elements are assembled into matrices. These matrices are then solved for basic parameters, such as displacements or temperatures. The final step is post-processing, where the analyst checks the results. Part of this process involves reviewing the magnitudes and distributions of the primary solution parameters (such as displacements and stresses).

The Finite Element Model
An FE model consists of a finite number of points (nodes) and elements. The nodes are actually points in space that are used to define the elements. The elements consist of various numbers of nodes (typically from 1 to 20). These elements are defined in space by the location and connectivity of the nodes. Finite elements are actually mini structures themselves, with displacement functions that are defined by the element types. Each element is further defined by element and material properties. The density of the finite element mesh may vary throughout the model, depending on the stress gradients within the structure.

Answers from FE Models
Each node in an FE model is characterized by a specific number of degrees of freedom (DOF). For three dimensional structural problems, the maximum number of DOF is 6 (three translations and three rotations in a global coordinate system). The minimum number of DOF is one (such as temperature for a thermal model). In simple terms, static structural FE models solve the equation F = k x, where F is the applied force, k is the stiffness, and x is the displacement. All of this is done in matrix format, where thousands of DOF are calculated in a single FE model. The most common results from FEA are displacements and stresses. Nonlinear design analyses include large deflection, elastic-plastic deformation, and contact between adjacent structures.