Matrix Structural Analysis

The goal of this page is to support the modeling of 3D structures made up of members that can carry axial, shear, bending, or torsion forces along their length. The stiffness of members subjected to these forces is detailed on the stiffness formulas page and elaborated in a first course in solid mechanics. 

The solution procedure for the matrix analysis of structures is always the same and we will quickly see how we can analyze structures of increasing complexity, from axial elements to beams to frames, with increasingly comprehensive stiffness relationships. 

The Solution Process

The solution of matrix structural analysis problems is very mechanistic, but it relies on an application of Hooke's Law in three reference frames:

1. The element local reference frame, which describes the mechanics of a single element with coordinate axes oriented along the element's length. 

2. The element global reference frame, in which a coordinate transformation is applied to the element stiffness relationships so that the element is oriented as it will be in the completed structure. 

3. The system global reference frame, which incorporates the stiffnesses of all elements in the structure. 

The relationships between the element local, element global, and system global formulations are depicted below along with the steps in the general solution procedure. 

1D Axial Element

An example that allows us to work through the general solution procedure with the simplest possible elements is MecMovies Example 5.5. The solution of this problem using the approach outlined here is provided here:

2D Truss Element

2D Truss Element Coordinate Transformation

2D Beam Element

2D Frame Element

2D Frame Element Coordinate Transformation

1D Torsion Element

2D Grid or Grillage Element

2D Grid Element Transformation Matrix

3D Truss Element

3D Truss Element Coordinate Transformation

3D Frame Element

3D Frame Element Coordinate Transformation

Coordinate transformation for 3D elements is challenging to visualize and to perform. It will be included here in due time. 

References

If you want to learn more, take a look at the excellent textbook Matrix Structural Analysis, 2nd Ed. by McGuire, Gallagher, and Ziemian, available free at https://www.mastan2.com/.