Matrix Structural Analysis (Introduction) - Theory & Concepts

Matrix Structural Analysis (Introduction) - Theory & Concepts

Matrix Structural Analysis is the foundation of all modern structural analysis software. It provides a systematic method for analyzing complex structures (trusses, beams, frames, grids) by formulating the equilibrium equations in matrix form.

Stiffness Method

Also known as the Displacement Method. It is preferred for computer implementation because the procedure is the same regardless of the degree of indeterminacy.

Global Stiffness Equation

The fundamental equation of the stiffness method.

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

Where:

$[K]$ : Global Stiffness Matrix (System Property).
$\{D\}$ : Global Displacement Vector (Unknowns).
$\{F\}$ : Global Force Vector (Applied Loads).

Procedure

The process of Matrix Structural Analysis involves building the mathematical model of the structure piece by piece, then solving it simultaneously.

Procedure

STEP-BY-STEP

Discretize the Structure: Identify nodes (joints) and elements (members). Number them systematically.
Define Degrees of Freedom (DOF): Identify unknown displacements (translations and rotations) at each node.
Form Element Stiffness Matrices ( $k$ ): Calculate the stiffness matrix for each member in its local coordinate system.
Transformation: Transform element stiffness matrices from local to global coordinates using rotation matrices ( $T$ ).

[k]_G = [T]^T [k]_L [T]

Procedure

STEP-BY-STEP

Assemble Global Stiffness Matrix ( $K$ ): Add terms from element stiffness matrices into the corresponding positions in the global matrix based on node connectivity.
Apply Boundary Conditions: Reduce the matrix by removing rows/columns corresponding to zero displacements (supports).
Solve for Displacements ( $D$ ): Invert $[K]$ and multiply by $\{F\}$ .

\{D\} = [K]^{-1} \{F\}

Procedure

STEP-BY-STEP

Calculate Member Forces (Local Force Recovery): Once the global nodal displacements ( $\{D\}$ ) are found, the internal end forces ( $\{f\}$ ) for each individual member (like shear, moment, and axial force) must be calculated in its own local coordinate system. This is done by extracting the relevant global displacements for that specific member ( $\{d\}$ ), multiplying by the local stiffness matrix ( $[k']$ ), and adding any fixed-end forces ( $\{FEM\}$ ) caused by loads applied directly along the member span.

\{f\} = [k'] \{d\} + \{FEM\}

Stiffness Matrices for Truss and Beam Elements

Truss Element (2D)

A truss member has 2 degrees of freedom (DOF) at each node (horizontal

u

, vertical

v

). Total 4 DOF per element.

[k] = \frac{EA}{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}

Where

c = \cos \theta

s = \sin \theta

Beam Element (2D)

A beam member has 2 DOF at each node (vertical

v

, rotation

\theta

). Total 4 DOF per element (neglecting axial deformation).

[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}

Global vs. Local Coordinate Systems

In matrix analysis, members are oriented in various directions. Therefore, equations written for a single member must be transformed into a common, structure-wide coordinate system before they can be assembled.

Coordinate Transformation

Local Coordinates ( $x', y'$ ): Aligned with the axis of each individual member. The local $x'$ axis runs along the length of the member, and $y'$ is perpendicular to it. The local stiffness matrix $[k']$ relates forces and displacements purely within this local frame.
Global Coordinates ( $X, Y$ ): The overarching fixed coordinate system applied to the entire structure.
Transformation Matrix ( $[T]$ ): A matrix containing the direction cosines (cosine and sine of the angle $\theta$ between the local and global axes) of the member. It is used to convert the local stiffness matrix $[k']$ into the global stiffness matrix $[k]$ for that member.

The relationship for transformation is:

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

This ensures that when member matrices are added together at a common node, they are summing forces acting in the exact same global directions.

Assembly of Global Stiffness Matrix

The assembly process is crucial. Each element connects two nodes (e.g., node

i

and node

j

). The terms in the element stiffness matrix corresponding to node

i

are added to the global matrix locations for node

i

, and similarly for node

j

and the cross-terms (

i-j

Node Numbering and Matrix Bandwidth

How you choose to number the nodes in your structure drastically affects the computational efficiency of solving the matrix equations.

Bandwidth: The non-zero elements in a global stiffness matrix $[K]$ tend to cluster around the main diagonal. The "bandwidth" is the maximum distance from the main diagonal to a non-zero element.
Efficiency: Solving large systems of equations is faster and requires less computer memory if the bandwidth is small.
Optimal Numbering Rule: To minimize bandwidth, nodes should be numbered such that the maximum numerical difference between any two connected nodes in the entire structure is kept as small as possible. For example, numbering nodes across the short dimension of a rectangular grid is much better than numbering them along the long dimension.

Application of Boundary Conditions

Before the global equilibrium equations (

[Q] = [K][D]

) can be solved, boundary conditions (known displacements at supports) must be applied. Otherwise, the global stiffness matrix

[K]

is singular (non-invertible) because the structure is free to move as a rigid body.

Methods for Boundary Conditions

Partitioning Method: The global matrices are partitioned into known and unknown displacements and forces. The equations corresponding to known zero displacements are mathematically eliminated or separated, resulting in a reduced, invertible stiffness matrix.
Penalty Method: A very large stiffness value (e.g., $10^{20}$ ) is added to the diagonal element of the global stiffness matrix corresponding to the restrained degree of freedom, effectively enforcing a zero displacement without altering the size of the matrix.

Key Takeaways

Stiffness Method solves $[K]\{D\} = \{F\}$ for unknown displacements.
Degrees of Freedom (DOF): Independent displacements (translations/rotations) that define the deformed shape.
Matrix Analysis is systematic and scalable, ideal for computer software.
Assembly: Combining element stiffness matrices into a global matrix based on connectivity.
Transformation: Rotating local element coordinates to align with the global system.
Node Numbering: Proper node numbering minimizes matrix bandwidth, leading to faster computational solution times and lower memory usage.

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