🏗️ MATLAB Truss Analysis — 2D & 3D Finite Element Method (FEM)

Linear static analysis of planar (2D) and space (3D) truss structures using the Finite Element Method and the Direct Stiffness Method — implemented from scratch in pure MATLAB.

Features · Theory · Inputs · Results · Getting Started · Assumptions

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📌 Overview

This repository provides two standalone MATLAB scripts for truss structural analysis:

Solver	Structure	DOFs per Node	Element Type
truss2D.m	Planar truss	2 (UX, UY)	2-node axial bar
truss3D.m	Space truss	3 (UX, UY, UZ)	2-node axial bar

Both solvers implement the complete linear static finite element workflow — from model definition (nodes, elements, loads, boundary conditions) through global stiffness matrix assembly, system solution, and post-processing of axial forces, stresses, strains, and yield-stress utilization — without relying on any external structural-analysis solver or toolbox.

The project is intended for educational, computational mechanics, structural analysis, and engineering programming applications.

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✨ Features

Feature	2D	3D
Direct stiffness method	✅	✅
Element stiffness matrix formulation	✅	✅
Global stiffness matrix assembly	✅	✅
Nodal load definition	✅ (magnitude + angle)	✅ (components Fx Fy Fz)
Flexible boundary-condition definition	✅	✅ (per-DOF flags)
Static nodal displacement solution	✅	✅
Element elongation (µm)	✅	✅
Axial force (N)	✅	✅
Normal stress (MPa)	✅	✅
Strain (microstrain, µε)	✅	✅
Yield-stress utilization (%)	✅	✅
Members > 100% utilization highlighted in red	✅	✅
Undeformed + deformed geometry plot	✅	✅
Node & element labeling	✅	✅
Command-window result reporting	✅	✅

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📐 Analysis Method

The solver follows the standard linear static finite element procedure for truss structures.

The global governing system is:

$$ [K]{u} = {F} $$

where:

Symbol	Meaning
[K]	Global stiffness matrix
{u}	Global nodal displacement vector
{F}	Global nodal force vector

2D Truss Element

With c = cos(θ) and s = sin(θ) derived from the element orientation:

$$ k_e = \frac{AE}{L} \begin{bmatrix} c^2 & cs & -c^2 & -cs \\ cs & s^2 & -cs & -s^2 \\ -c^2 & -cs & c^2 & cs \\ -cs & -s^2 & cs & s^2 \end{bmatrix} $$

3D Truss Element

The 3D formulation uses the direction cosines of the element axis:

$$ c_x = \frac{\Delta x}{L}, \qquad c_y = \frac{\Delta y}{L}, \qquad c_z = \frac{\Delta z}{L} $$

The element stiffness matrix is written as:

$$ k_e = \frac{AE}{L} \begin{bmatrix} \lambda & -\lambda \\ -\lambda & \lambda \end{bmatrix} $$

with:

$$ \lambda = \begin{bmatrix} c_x^2 & c_xc_y & c_xc_z \\ c_xc_y & c_y^2 & c_yc_z \\ c_xc_z & c_yc_z & c_z^2 \end{bmatrix} $$

Solution Strategy

1. Assemble the full global stiffness matrix [K] by direct stiffness summation.
2. Apply boundary conditions by eliminating the rows and columns of restrained DOFs.
3. Solve the reduced system for free-DOF displacements.
4. Recover the complete displacement vector (zero at restrained DOFs).

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🧾 Input Definition

All model data are defined directly at the top of each script.

Common Units

Quantity	Unit
Coordinates	m
Cross-sectional area A	m²
Young's modulus E	Pa
Yield stress	Pa
Forces	N
Force angle (2D only)	degrees

2D Inputs (truss2D.m)

Nodes — [x y] per row:

data{1,1}.nodes = [0 0
                   3.8 9.12/7
                   ...];

Elements — [node1 node2 A E yieldStress]:

data{1,1}.elements = [1 2 0.0005 200e9 20e7
                      ...];

Forces — [nodeNumber F angle], with components computed as Fx = F·cos(θ), Fy = F·sin(θ) (θ in degrees):

data{1,1}.forces = [1 5700 -90
                    ...];

Supports — [nodeNumber type orien]:

type	orien	Restrained DOFs
1	1	UX only
1	2	UY only
2	any	UX + UY (pin)

data{1,1}.supports = [1 2 0      % pin at node 1
                      8 1 2];    % UY restraint at node 8

3D Inputs (truss3D.m)

Nodes — [x y z] per row.

Elements — [node1 node2 A E yieldStress] (same format as 2D).

Forces — direct Cartesian components [nodeNumber Fx Fy Fz]:

data{1,1}.forces = [1 0 0 -25000
                    ...];

Supports — [nodeNumber Dx Dy Dz], where each translational DOF is defined independently:

Flag	Meaning
0	Free
1	Restrained

For example [3 1 1 1] fully fixes node 3 in X, Y, and Z.

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⚙️ Analysis Workflow

1. Define nodal coordinates.
2. Define element connectivity, area, elastic modulus, and yield stress.
3. Compute element lengths and orientation parameters (angle / direction cosines).
4. Build element stiffness matrices.
5. Assemble the global stiffness matrix.
6. Assemble the global load vector.
7. Apply boundary conditions.
8. Solve the reduced linear system.
9. Recover the full displacement vector.
10. Compute element elongation and axial force.
11. Compute element stress and strain.
12. Compute yield-stress utilization.
13. Visualize undeformed and deformed configurations.

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📊 Results

Per-element results are reported in the MATLAB command window:

Quantity	Symbol	Unit	Sign Convention
Elongation	ΔL	µm	positive = elongation
Axial force	N	N	positive = tension
Normal stress	σ = N/A	MPa	positive = tension
Strain	ε	µε (microstrain)	positive = elongation
Utilization	—	%	always positive

Yield-stress utilization is computed as:

$$ \text{Utilization} = \left| \frac{\sigma}{\sigma_y} \right| \times 100 $$

Members whose utilization exceeds 100% are drawn in red in the structure plot.

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🎨 Visualization

Each script produces a labeled plot of the structure:

Item	Representation
Undeformed members	Blue (red if utilization > 100%)
Deformed configuration	Magenta overlay (indicative visual aid)
Node numbers	Labeled at node positions
Element numbers	Labeled at member midpoints
Supports	Marked at restrained nodes

3D support marker colors indicate the restrained DOF combination:

Dx Dy Dz	Color
1 1 1	⚫ Black
1 1 0	⚪ White
1 0 0	🟢 Green
0 1 0	🔵 Cyan
0 0 1	🟣 Magenta
0 1 1	🫒 Olive
1 0 1	🟠 Orange
0 0 0	🔵 Blue (free)

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📁 Project Structure

MATLAB-Truss-Analysis/
├── 2D/
│   └── truss2D.m
├── 3D/
│   └── truss3D.m
├── LICENSE
└── README.md

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📸 Results & Visualization

2D Planar Truss

Finite Element Analysis (FEA) of a 2D planar truss, showing the deformed configuration.

3D Space Truss

Three-dimensional space truss analysis using the Finite Element Method (FEM), including structural deformation and visualization of failed elements.

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🚀 Getting Started

Requirements

• MATLAB (base installation — no toolboxes required)

Installation

git clone https://github.com/amirhh-04/MATLAB-Truss-Analysis.git
cd MATLAB-Truss-Analysis

Usage

% 2D planar truss
run('2D/truss2D.m')

% 3D space truss
run('3D/truss3D.m')

To model your own structure, edit the Inputs Section at the top of the corresponding script (nodes, elements, forces, supports) and re-run it.

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📖 Assumptions & Scope

The implementation performs linear static analysis of ideal truss structures.

Model assumptions

• Members carry axial force only (pin-jointed idealization)
• Linear elastic material behavior
• Constant cross-sectional properties per element
• Small displacements
• Static loads
• Translational DOFs only

Not included

• Geometric nonlinearity
• Material plasticity
• Buckling analysis
• Dynamic / transient / modal analysis
• Fatigue analysis
• Joint slip modeling
• Design-code compliance checking

Notes:

• Reported strain is expressed in microstrain (µε).
• The deformed-geometry overlay is provided as a qualitative visual aid alongside the exact numerical element results.
• The nodal force vector [K]{u} is computed internally; explicit support reaction reporting is planned for a future release.

The project should be considered an educational and computational-mechanics implementation, not a certified structural design package.

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🗺️ Roadmap

• Support reaction reporting
• Deformation scale factor control
• Improved input validation
• Verification & benchmark examples
• Integration of the GUI version
• Expanded material and section libraries
• Structural optimization features

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🤝 Contributing

Issues and pull requests are welcome — feel free to open an issue for bug reports, benchmark models, or feature suggestions.

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📄 License

This project is licensed under the MIT License — see the LICENSE file for details.

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👤 Author

Amirhh

🐙 GitHub	amirhh-04
🌐 Website	amirhh.ir
💼 LinkedIn	amirhh

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🔍 Keywords

MATLAB · Finite Element Method · FEM · FEA · Finite Element Analysis · Truss Analysis · Truss Solver · 2D Truss · 3D Truss · Space Truss · Structural Analysis · Structural Engineering · Direct Stiffness Method · Stiffness Matrix · Computational Mechanics · Mechanical Engineering