Matrix Analysis of Structures

Introduction to the Direct Stiffness Method, the foundation of modern structural analysis software.

Introduction

Definition

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.

Key Advantages

  • 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 is:
{P}=[K]{D} \{P\} = [K] \{D\}
Where:
  • {P}\{P\} is the global nodal force vector (applied loads).
  • [K][K] is the global stiffness matrix of the structure.
  • {D}\{D\} is the global nodal displacement vector (unknowns).

Procedure for Matrix Analysis

Procedure

  1. Define the global coordinate system (X,Y,ZX, Y, Z).
  2. 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).
  3. Formulate the local stiffness matrix [k′][k'] for each member based on its properties (A,E,I,LA, E, I, L).
  4. Transform the local stiffness matrix [k′][k'] into the global coordinate system [k]=[T]T[k′][T][k] = [T]^T [k'] [T], where [T][T] is the transformation matrix containing direction cosines of the member.
  5. Assemble the global stiffness matrix [K][K] by adding the contributions of each member's global stiffness matrix [k][k] to the corresponding DOFs in [K][K].
  6. Assemble the global force vector {P}\{P\} from applied nodal loads and equivalent nodal loads (from members with distributed loads).
  7. Apply boundary conditions to the global system of equations [K]{D}={P}[K]\{D\} = \{P\}. Partition the matrices to separate the unknown displacements {Du}\{D_u\} from the known displacements {Dk}\{D_k\}.
  8. Solve for the unknown displacements: {Du}=[K11]−1({Pk}−[K12]{Dk})\{D_u\} = [K_{11}]^{-1} (\{P_k\} - [K_{12}]\{D_k\}).
  9. Calculate the support reactions: {Pu}=[K21]{Du}+[K22]{Dk}\{P_u\} = [K_{21}]\{D_u\} + [K_{22}]\{D_k\}.
  10. Extract the member end displacements in local coordinates {d′}=[T]{d}\{d'\} = [T]\{d\} and use the member stiffness equation {p′}=[k′]{d′}+{FEM}\{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

Because member stiffnesses are derived in local coordinates [k′][k'], they must be transformed into the global coordinate system [k][k] before they can be assembled into the structure's global stiffness matrix [K][K].
[k]=[T]T[k′][T] [k] = [T]^T [k'] [T]
The transformation matrix [T][T] is composed of the direction cosines (the cosines of the angles between the local axes and the global axes) that geometrically map the local vectors to the global vectors.

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)

A planar truss member only carries axial force. It has 2 degrees of freedom at each node (local x' and y' translations). The 4×44 \times 4 local stiffness matrix [k′][k'] is:
[k′]=AEL[10−100000−10100000] [k'] = \frac{AE}{L} \begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}

Beam Analysis (2D)

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). The 4×44 \times 4 local stiffness matrix [k′][k'] is:
[k′]=EIL3[126L−126L6L4L2−6L2L2−12−6L12−6L6L2L2−6L4L2] [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}

Frame Analysis (2D)

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 6×66 \times 6 local stiffness matrix [k′][k'] combines the truss and beam matrices:
[k′]=[AEL00−AEL00012EIL36EIL20−12EIL36EIL206EIL24EIL0−6EIL22EIL−AEL00AEL000−12EIL3−6EIL2012EIL3−6EIL206EIL22EIL0−6EIL24EIL] [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}

Interactive Tool: Matrix Assembly Simulator

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

Direct Stiffness Assembly Simulation

Walk through the core concept of matrix structural analysis: assembling local stiffness matrices into a global matrix.

1. Define the Structure

Consider a 1D axial structure with two spring members (Member 1 and Member 2) connected in series at three nodes (Node 1, Node 2, Node 3).

Node 1u₁Node 2u₂Node 3u₃Member 1 (k₁)Member 2 (k₂)

Boundary Conditions and Solving

Imposing Support Conditions

Once the global stiffness matrix [K][K] is assembled, the system equation [F]=[K][D][F] = [K][D] cannot be solved immediately because [K][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][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][D], these are transformed back into local coordinates for each member. The element stiffness equation [f]=[k][d][f] = [k][d] 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.
  • It involves assembling a global stiffness matrix from local member matrices.
  • The fundamental equation [K]{D}={P}[K]\{D\} = \{P\} relates applied forces, structural stiffness, and resulting displacements.
  • The process systematically determines unknown nodal displacements and then calculates internal member forces and support reactions.
  • Understanding kinematic indeterminacy and the difference between local and global coordinates is essential for forming the transformation matrix.
  • Matrix analysis formulates the Displacement Method using matrix algebra, making it ideal for computer algorithms.
  • The Direct Stiffness Method operates by assembling individual element stiffness matrices into a global structure stiffness matrix.
  • Coordinate transformations are necessary to align local member axes with global structural axes before assembly.
  • 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.
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