Finite element analysis (FEA) might bring to mind either colorful contours or matrix math, and while those are both important parts of FEA, the purpose of this page is to give an engineer’s perspective on finite element analysis to bridge the gap between the equations and the colors.

{F} = [K] {x}

This page focuses on structural finite element analysis, but many other types of systems can be simulated with FEA, including acoustics, heat transfer, electromagnetics, fluid mechanics, and various combinations of these physics and others.

Reducing structural FEA down to one sentence might look something like this:

Solid mechanics equations are solved on a grid of interconnected elements representing the geometry of interest, subject to choices of material properties, loads and constraints, and solution type, providing the engineer with information on motion and/or stress.

The following sections will break down this single sentence to explain, piece-by-piece, how the engineer interacts with finite element analysis.

## Solid mechanics equations are solved…

The basic form of the equations in FEA will be familiar to most engineers, including Hooke’s law, which is the relation between the force from a spring, F, the spring constant, k, and the displacement of that spring, x.

{F} = [K] {x}

Though the math gets more complicated as implemented, for most practical engineering applications, the numerical solution of solid mechanics equations is handled by commercial software such as Abaqus, Algor, Ansys, COMSOL, Nastran, and others. Access to and expertise with these programs is not nearly as common as other engineering tools such as Excel, MATLAB, or AutoCAD.

Though commercial solvers can efficiently solve the necessary complex mathematics, the answer returned is only as good as the inputs chosen by the user. The analyst must determine the geometric representation, the material properties, the loads and constraints, and the solution type.

## ….on a grid of interconnected elements representing the geometry of interest…

As the name “finite element analysis” implies, the geometry is broken down into small elements over which the solution is analyzed, in what is collectively referred to as a “mesh” or a “grid.” The software solves equations across multiple elements, approximating the spatial variation of the solution, and ensuring that what each part of the domain thinks is happening agrees with what its neighbors think is happening.

Though most FEA software environments will include tools that will discretize a geometry into a mesh, the analyst must make important decisions to ensure that models are both accurate and efficient. Increasing the number of elements in a model can improve resolution of stresses and strains, but more elements require a greater computational overhead to solve the equations at the additional points.

#### Elements vary in dimensionality, with a given 3-D model often including a combination of 1-D, 2-D, and 3-D elements:

• 1-D elements are typically used to model slender entities including
• Beams
• Trusses
• Bolts
• Pipes
• 2-D elements are typically used to model thin entities such as:
•  Sheet metal
•  Thin-walled pressure vessels
• Pipe walls
• Composite layups
• 3-D elements directly model something in space. Uses may include:
• Forged or cast components
• Thick-walled vessels and nozzles
• Components with nonuniform thickness
• Detailed welds
• Anything that cannot be accurately or efficiently simplified to 1-D or 2-D elements

#### A successful mesh will:

• Accurately resolve the geometric features of interest
• Use the correct element types for the application
• Consist of elements that are not excessively warped or mis-shapen
• Use a high enough resolution (small enough elements) to adequately resolve stresses and strains without introducing unnecessary computational expense.

## Equations are solved… subject to choices of material properties…

Making our geometry behave as it should requires that we pick the right material properties to model the physics of interest

• Elastic modulus, E, and Poisson’s ratio, ν, tells the solver how springy the elements are
• Density gives the elements mass, which is critical when solving for:
• Harmonic motion
• The effect of gravity forces
• The time history of motion
• A yield stress and a stress-strain curve in the plastic region allows realistic prediction of residual stresses and deformation after yield.
• A Thermal expansion coefficient allows consideration of thermal displacement and thermally-induced stresses

Material properties can be set to vary with temperature or other solution variables, and solvers support advanced material behaviors such as creep, swelling, complex plasticity models, and damage modeling.

## Equations are solved… subject to choices of… loads and constraints

Loads and constraints define how the model interacts with the rest of the world.

• Concentrated forces or moments applied at a small set of points
• Body forces such as gravity

#### Constraints include:

• Fixed, pinned, sliding, or symmetry boundary conditions
• Spring constraints – for a non-rigid connection to the world
• Contact boundaries – to restrict motion in one direction but not another

## Equations are solved… subject to choices of… solution type, providing the engineer with information on motion and/or stress.

Different solution types provide different information to the user:

• Static analysis solves for the steady-state displaced shape and stress state of a system when subject to a combination of loads
• Transient analysis solves for a time history of system motion and stress when subject to loads and constraints, with consideration of inertia and damping effects
• Modal analysis solves for a system’s normal modes, providing information on mode shapes at the resonant frequencies of the system
• Steady state dynamics solves for a steady-state system response when subject to oscillating loads at multiple frequencies
• Linear buckling analysis is used to determine the likelihood that something will experience a buckling failure when subject to prescribed loads and constraints

Calculated stresses can be used to calculate design suitability or predict failure locations through calculation of fatigue life or comparison with allowable stresses. Such evaluations are often conducted in the context of codes such as ASME BPVC, API, Eurocode, and AISC.

Calculated displacements can be used to ensure that a system will perform the service for which it is designed.

Consideration of natural frequencies and mode shapes of a system can be important to ensure, for example, that sufficient frequency separation is built into a design to avoid resonance with a known excitation source.