Matrix Analysis of Structures - Theory & Concepts

Matrix Analysis of Structures

Learning Objectives

Understand the principles of the Direct Stiffness Method for analyzing complex structures.
Formulate the fundamental stiffness equations relating forces, stiffness matrices, and displacements.
Distinguish between global and local coordinate systems and apply transformation matrices.
Define the degrees of freedom (kinematic indeterminacy) for trusses, beams, and frames.
Understand the assembly of global stiffness matrices and the application of boundary conditions.

Matrix Analysis, specifically the Direct Stiffness Method, is a systematic, computer-oriented approach to solving statically indeterminate structures. It relies on assembling the global stiffness matrix of a structure from the individual stiffness matrices of its members.

Introduction

Definition

Key Advantages

Advantages of Matrix Analysis

Highly systematic and easily programmed for computers.
Handles very large, complex structures with thousands of degrees of freedom.
Automatically accounts for member properties (area, inertia, length, modulus) and joint coordinates.
Directly yields joint displacements and member forces.

The Direct Stiffness Method

The fundamental equation of the stiffness method forms the backbone of matrix structural analysis, linking applied external forces to structural stiffness and resulting displacements.

Fundamental Equation of the Stiffness Method

Relates the global nodal force vector, global stiffness matrix, and global nodal displacement vector.

\{P\} = [K] \{D\}

Variables

Symbol	Description	Unit
$\{P\}$	Global nodal force vector (applied loads)	-
$[K]$	Global stiffness matrix of the structure	-
$\{D\}$	Global nodal displacement vector (unknowns)	-

Procedure for Matrix Analysis

Direct Stiffness Method Analysis Steps

STEP-BY-STEP

Define the global coordinate system ( $X, Y, Z$ ).
Number the nodes and elements. Identify the Degrees of Freedom (DOFs) at each node (translations and rotations). Unrestrained DOFs are unknowns; restrained DOFs (supports) have known displacements (usually zero).
Formulate the local stiffness matrix $[k']$ for each member based on its properties ( $A, E, I, L$ ).
Transform the local stiffness matrix $[k']$ into the global coordinate system $[k] = [T]^T [k'] [T]$ , where $[T]$ is the transformation matrix containing direction cosines of the member.
Assemble the global stiffness matrix $[K]$ by adding the contributions of each member's global stiffness matrix $[k]$ to the corresponding DOFs in $[K]$ .
Assemble the global force vector $\{P\}$ from applied nodal loads and equivalent nodal loads (from members with distributed loads).
Apply boundary conditions to the global system of equations $[K]\{D\} = \{P\}$ . Partition the matrices to separate the unknown displacements $\{D_u\}$ from the known displacements $\{D_k\}$ .
Solve for the unknown displacements: $\{D_u\} = [K_{11}]^{-1} (\{P_k\} - [K_{12}]\{D_k\})$ .
Calculate the support reactions: $\{P_u\} = [K_{21}]\{D_u\} + [K_{22}]\{D_k\}$ .
Extract the member end displacements in local coordinates $\{d'\} = [T]\{d\}$ and use the member stiffness equation $\{p'\} = [k']\{d'\} + \{FEM\}$ to find internal member forces.

Global Coordinates vs Local Coordinates

The distinction between the coordinate system of the entire structure and the coordinate systems of individual members.

Coordinate Systems

Matrix analysis requires tracking forces and displacements in two different frameworks.

Global Coordinate System (X, Y, Z): A single, fixed Cartesian coordinate system used for the entire structure. All nodal coordinates, applied joint loads, and final joint displacements are defined relative to this system. It ensures that when members are connected at a joint, their equilibrium and compatibility are evaluated in the same reference frame.
Local Coordinate System (x', y', z'): A unique coordinate system attached to each individual member. Typically, the local x'-axis is aligned along the longitudinal axis of the member (from the "near" node to the "far" node). The local y' and z' axes correspond to the principal axes of the cross-section. Internal forces (axial, shear, bending moment) are always calculated and interpreted in this local system.

Transformation Matrix Equation

Because member stiffnesses are derived in local coordinates, they must be transformed into the global coordinate system before they can be assembled into the structure's global stiffness matrix.

[k] = [T]^T [k'] [T]

Variables

Symbol	Description	Unit
$[k]$	Element stiffness matrix in global coordinates	-
$[T]$	Transformation matrix composed of direction cosines	-
$[T]^T$	Transpose of the transformation matrix	-
$[k']$	Element stiffness matrix in local coordinates	-

Kinematic Indeterminacy (Degrees of Freedom)

Degrees of Freedom (DOF)

The number of independent coordinates required to completely specify the displaced shape of a structure is called the number of degrees of freedom (DOF) or kinematic indeterminacy.

In a 2D truss, each node has 2 DOFs (translation in X and Y).
In a 2D beam, each node has 2 DOFs (translation in Y and rotation about Z).
In a 2D frame, each node has 3 DOFs (translation in X, translation in Y, and rotation about Z).

Element Stiffness Matrices

The fundamental building blocks of matrix analysis, defined in the local coordinate system.

Truss Analysis (2D)

Truss Member Stiffness

A planar truss member only carries axial force. It has 2 degrees of freedom at each node (local x' and y' translations).

2D Truss Local Stiffness Matrix

The 4x4 local stiffness matrix for a 2D truss element.

[k'] = \frac{AE}{L} \begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}

Variables

Symbol	Description	Unit
$[k']$	Local stiffness matrix	-
$A$	Cross-sectional area of the member	-
$E$	Modulus of elasticity of the material	-
$L$	Length of the member	-

Beam Analysis (2D)

Beam Member Stiffness

A planar beam member resists shear and bending but is typically assumed to be axially rigid in basic beam formulations. It has 2 degrees of freedom at each node (local y' translation and rotation).

2D Beam Local Stiffness Matrix

The 4x4 local stiffness matrix for a 2D beam element.

[k'] = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix}

Variables

Symbol	Description	Unit
$[k']$	Local stiffness matrix	-
$E$	Modulus of elasticity of the material	-
$I$	Moment of inertia of the cross-section	-
$L$	Length of the member	-

Frame Analysis (2D)

Frame Member Stiffness

A planar frame member resists axial force, shear, and bending. It has 3 degrees of freedom at each node (local x' translation, y' translation, and rotation). The local stiffness matrix combines the truss and beam matrices.

2D Frame Local Stiffness Matrix

The 6x6 local stiffness matrix for a 2D frame element.

[k'] = \begin{bmatrix} \frac{AE}{L} & 0 & 0 & -\frac{AE}{L} & 0 & 0 \\ 0 & \frac{12EI}{L^3} & \frac{6EI}{L^2} & 0 & -\frac{12EI}{L^3} & \frac{6EI}{L^2} \\ 0 & \frac{6EI}{L^2} & \frac{4EI}{L} & 0 & -\frac{6EI}{L^2} & \frac{2EI}{L} \\ -\frac{AE}{L} & 0 & 0 & \frac{AE}{L} & 0 & 0 \\ 0 & -\frac{12EI}{L^3} & -\frac{6EI}{L^2} & 0 & \frac{12EI}{L^3} & -\frac{6EI}{L^2} \\ 0 & \frac{6EI}{L^2} & \frac{2EI}{L} & 0 & -\frac{6EI}{L^2} & \frac{4EI}{L} \end{bmatrix}

Variables

Symbol	Description	Unit
$[k']$	Local stiffness matrix	-
$A$	Cross-sectional area of the member	-
$E$	Modulus of elasticity	-
$I$	Moment of inertia	-
$L$	Length of the member	-

Interactive Tool: Matrix Assembly Simulator

Interactive Simulation

Observe how individual member stiffness matrices assemble into the global stiffness matrix for a simple structure using the simulator below.

Boundary Conditions and Solving

Imposing Support Conditions

Once the global stiffness matrix $[K]$ is assembled, the system equation $\{P\} = [K]\{D\}$ cannot be solved immediately because $[K]$ is singular (its determinant is zero), reflecting that the structure is unconstrained and capable of rigid body motion.

Boundary conditions (supports) are imposed by setting the corresponding displacements in $\{D\}$ to zero. The rows and columns corresponding to these zero displacements are partitioned out, resulting in a reduced, non-singular stiffness matrix that can be inverted to solve for the unknown nodal displacements.

Member Forces

Post-Processing

After solving for the global nodal displacements $\{D\}$ , these are transformed back into local coordinates for each member. The element stiffness equation $\{p'\} = [k']\{d'\} + \{FEM\}$ is then applied to calculate the final internal member forces (axial, shear, and moment at the ends of the element).

Comparison of Matrix Methods

Matrix structural analysis primarily utilizes two dual methods: the Flexibility Method and the Stiffness Method.

Flexibility vs. Stiffness Method

Flexibility (Force) Method: The primary unknowns are redundant forces. It requires determining static determinacy, selecting redundant forces, and establishing compatibility equations. The flexibility matrix must be inverted. It is difficult to automate because the choice of redundant forces is not unique.
Stiffness (Displacement) Method: The primary unknowns are joint displacements. It requires determining kinematic determinacy (degrees of freedom) and establishing equilibrium equations at the nodes. The global stiffness matrix is assembled and inverted. It is highly systematic, unique for any structure, and therefore the universal basis for modern computer structural analysis software (FEM).

Key Takeaways

Matrix analysis uses the Direct Stiffness Method to solve complex structures via computer.
Matrix analysis formulates the Displacement Method using matrix algebra, making it ideal for computer algorithms.
It involves assembling a global stiffness matrix from local member matrices.
Coordinate transformations are necessary to align local member axes with global structural axes before assembly.
The fundamental equation $\{P\} = [K]\{D\}$ relates applied forces, structural stiffness, and resulting displacements.
Boundary conditions must be applied to the singular global matrix to solve for unknown nodal displacements, followed by back-substitution to find internal member forces.
Understanding kinematic indeterminacy and the difference between local and global coordinates is essential for forming the transformation matrix.

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