# Structural FEA - understanding the fundamentals

• blue24
In summary: PDE's that can be solvedIn structural analysis, the fundamental equation I am solving is the Equation of Motion:F=m\ddot{x}+c\dot{x}+kxThis is an ordinary differential equation, not a PDE. Are PDE’s involved in structural analysis?This is correct if the system you are studying has a single degree of freedom. For a lumped-parameter system with multiple degrees of freedom, this expands into the more general matrix version quoted by @FEAnalyst:##M \ddot{u} + C \dot{u} +Ku=F##.In the case
blue24
TL;DR Summary
Are partial differential equations involved in transient, structural FEA?
I am a mechanical an engineer with a few years of experience. Most of the work I do is transient, structural finite element analysis. I have gotten reasonably competent at building models and pumping out results, but I regularly come across gaps in my fundamental knowledge. I have been doing some reading on the basics of finite element analysis because I want to understand more of its mathematical foundations.
My understanding is that the finite element method is often (primarily?) used to solve partial differential equations, which get pretty hairy. In structural analysis, the fundamental equation I am solving is the Equation of Motion:

$$F=m\ddot{x}+c\dot{x}+kx$$

This is an ordinary differential equation, not a PDE. Are PDE’s involved in structural analysis?

Actually FEA programs don't solve PDEs directly which may be counterintuitive. Instead they calculate so called stiffness matrix using the formula derived from weak form of PDE (in solid mechanics Navier's equation of motion). That's why when people write their own simple FEA programs (not only for 1D elements but also plane stress/strain) they don't have to implement PDEs. Dynamic analysis solves the following matrix formula: ##M \ddot{u} + C \dot{u} +Ku=F##. In static analysis this reduces to ##Ku=F##. To find out how FEA works it's best to understand some simple examples of hand calculations for beam and plane stress elements.

SCP, blue24 and berkeman
blue24 said:
Summary: Are partial differential equations involved in transient, structural FEA?

This is an ordinary differential equation, not a PDE. Are PDE’s involved in structural analysis
you are correct. This is an ordinary differential equation and in structural analysis PDEs are involved (partial derivatives with respect to x, y, z coordinates and, sometimes, time)

blue24
Thanks both of you. I know the strain displacement equations are PDE's. Is that where the PDE's come in? So in general, for a dynamic structural analysis,

Step 1 - Solve an ODE for displacements, based on applied boundary conditions and applied loading

Step 2 - Solve PDE's for strain from stress

I'm trying to put this in my own "layman's" terms, to make sure I understand. Thanks for the help!

Ok, so this question is still lingering in my mind - where do the PDE's come in for structural analysis? FEAnalyst says that FEA programs don't solve PDE's directly. I think I understand that part. But what is the PDE that is being "indirectly" solved? Is it the strain displacement equations?

blue24 said:
In structural analysis, the fundamental equation I am solving is the Equation of Motion:

$$F=m\ddot{x}+c\dot{x}+kx$$
This is an ordinary differential equation, not a PDE. Are PDE’s involved in structural analysis?
This is only correct if the system you are studying has a single degree of freedom. For a lumped-parameter system with multiple degrees of freedom, this expands into the more general matrix version quoted by @FEAnalyst:
FEAnalyst said:
##M \ddot{u} + C \dot{u} +Ku=F##.
In the case of the more general distributed-parameter system, you get a PDE. However, computers don't like differential equations, and the numerical solution requires discretization. Discretization is what you are doing when you are meshing your system. The meshed (i.e. - discretized) version of the PDE returns us to the matrix equation above, and this is what your FEA program ultimately solves.

SCP said:
This is only correct if the system you are studying has a single degree of freedom. For a lumped-parameter system with multiple degrees of freedom, this expands into the more general matrix version quoted by @FEAnalyst:

In the case of the more general distributed-parameter system, you get a PDE. However, computers don't like differential equations, and the numerical solution requires discretization. Discretization is what you are doing when you are meshing your system. The meshed (i.e. - discretized) version of the PDE returns us to the matrix equation above, and this is what your FEA program ultimately solves.

blue24 said:
Ok, so this question is still lingering in my mind - where do the PDE's come in for structural analysis? FEAnalyst says that FEA programs don't solve PDE's directly. I think I understand that part. But what is the PDE that is being "indirectly" solved? Is it the strain displacement equations?
The underlying PDE is the Cauchy momentum equation.

It is just Newton II expressed for a continuum. Note that the ##\mathbf f## in it contains body forces only, like gravity. The more typical types of external forces come into play as boundary conditions for stress.

Edit: clarifications

Last edited:
SCP

## What is Structural FEA?

Structural Finite Element Analysis (FEA) is a computerized method for predicting how a product reacts to real-world forces, vibration, heat, fluid flow, and other physical effects. It uses numerical techniques to solve complex equations and simulate the behavior of a product or structure under various loading conditions.

## What are the fundamentals of Structural FEA?

The fundamentals of Structural FEA include understanding the basic principles of mechanics, such as stress and strain, as well as the properties of materials. It also involves knowledge of the finite element method, which is used to discretize the structure into smaller elements for analysis. Additionally, understanding the boundary conditions and loading conditions is crucial in FEA.

## What are the benefits of using Structural FEA?

Structural FEA allows engineers to analyze and predict the behavior of a product or structure before it is physically built. This can save time and money in the design process by identifying potential issues and allowing for modifications to be made. It also allows for optimization of designs to improve performance and reduce weight.

## What are the limitations of Structural FEA?

One of the main limitations of Structural FEA is the accuracy of the results. The analysis is based on assumptions and simplifications, so the results may not always match the real-world behavior of the structure. Additionally, FEA requires a significant amount of computational power and can be time-consuming to set up and run.

## What are some common applications of Structural FEA?

Structural FEA is commonly used in industries such as aerospace, automotive, and civil engineering to analyze and design structures such as buildings, bridges, and aircraft. It is also used in the design and optimization of mechanical components, such as engine parts and machine components.

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