Finite Element Analysis

Slice continuous structures into many small elements, then assemble them to reveal overall response

The core of FEA is not to "calculate the entire beam directly", but to first discretize the structure into many small elements, write local stiffness relationships for each element, assemble them into a global system, and finally solve for displacements, strains, and stresses.

Discrete Elements Stiffness Matrix Boundary Conditions Displacement & Stress
Enter Interactive Lab

The Most Important Intuition in FEA

Calculate "elements" first, then assemble the "whole". As long as element relationships and boundary conditions are correctly defined, complex structures can be solved through assembly.

Element Discretize the continuum into small segments, making it easier to write local mechanical relationships for each segment.
Node Adjacent elements connect through nodes, where displacements are transferred and the structure is assembled.
Assembly Each element contributes its local stiffness matrix, which is finally merged into a global system of equations for solution.
Result What we really want to read are displacements, strains, stresses, and identify where it's softest and most dangerous.
Understanding the Method

The FEA workflow: "Discretize -> Assemble -> Solve -> Interpret Results"

This page uses a 1D rod element model under distributed load for teaching demonstration, preserving the core structure of FEA while allowing you to clearly see each step.

1

Discretize Mesh

Slice the rod into multiple small elements. More elements mean finer mesh, better capturing local material variations and response changes.

2

Define Element Stiffness

Each linear rod element satisfies local stiffness relationship, typically in the form of EA/L multiplied by a small matrix.

3

Apply Boundary Conditions

Fixed end has zero displacement, distributed load converts to equivalent nodal forces. Boundary conditions determine if the equations have unique solutions.

4

Read Displacements & Stresses

After solving for nodal displacements, strains and stresses can be derived within elements to identify weak zones and peak areas.

Teaching Experiment on This Page

We discretize a left-fixed rod into multiple elements and apply uniform axial load. You can also create a "soft zone" to observe how local material changes distort the overall response.

Interactive Lab

Change mesh, material, and defect position to see how displacements and stresses redistribute

This model uses real linear rod element FEA assembly. Every parameter you adjust regenerates the element stiffness matrix and solves for global displacements.

Mesh, Deformation & Stress
Top shows original rod and mesh, bottom shows scaled deformation. Colored elements indicate stress magnitude, gray shaded area shows the artificially defined soft zone.
Original Mesh Low Stress High Stress
Nodal Displacement Curve
This curve directly comes from global equation solution. Displacement increases towards the right, and soft zones cause local steepening.
How to Read Results
FEA is not just about calculating a maximum value; it tells you "where it's softer, where it's more dangerous, and where finer mesh is needed".

Waiting for solution...

When to Use

When structure is continuous, loading is complex, and you need local response

Bridges, aircraft bodies, brackets, chip packages, strata, heat dissipation structures - FEA is essential whenever you care about local displacement, stress, or temperature distribution.

①

Structure is Continuous

If the object is not a few discrete points but a continuous material, discretizing it into elements is a natural approach.

②

You Care About Local Peaks

Maximum displacement, maximum stress, thermal concentration zones, weak zones - all require spatial distribution information, not just an overall average.

③

Materials and Boundaries are Non-uniform

Different materials, constraints, and loads in different regions - these are scenarios where FEA excels.