Module 9: Matrix Stiffness Method

Learning Objectives

  • Define nodal degrees of freedom and assemble global force and displacement vectors.
  • Form local stiffness matrices for basic structural elements.
  • Transform element stiffness and force quantities between local and global coordinates.
  • Assemble the global stiffness matrix by degree-of-freedom connectivity.
  • Apply displacement boundary conditions and solve the reduced system.
  • Recover support reactions and member-end forces, including span-load fixed-end effects.
  • Diagnose singularity caused by unconstrained rigid-body motion or structural mechanisms.

Direct Stiffness Method

The direct stiffness method expresses each element's force-displacement relationship, transforms and assembles those element relationships into a global system, applies boundary conditions, solves nodal displacements, and then recovers reactions and member forces.

Global Structural Equilibrium

Linear elastic matrix relation between nodal forces and nodal displacements.

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

Variables

SymbolDescriptionUnit
[K][K]Global structural stiffness matrix-
{D}\{D\}Global nodal displacement vector-
{Q}\{Q\}Global nodal force vector-

Degrees of Freedom and Connectivity

A global degree of freedom is an independent nodal translation or rotation retained by the structural idealization. Element matrices are assembled by mapping each element's local/global element degrees of freedom to the corresponding global equation numbers. Shared joint degrees of freedom receive stiffness contributions from every connected element.

Local Axial-Bar Stiffness

Two-degree local stiffness matrix for a linear elastic axial member.

[k]local=AEL[1−1−11][k]_{\text{local}}=\frac{AE}{L} \begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix}

Variables

SymbolDescriptionUnit
AACross-sectional area-
EEYoung's modulus-
LLElement length-

Local and Global Coordinates

Local element coordinates align with the member and make element stiffness relations compact. Global coordinates provide one common reference for the entire structure. A transformation matrix based on direction cosines maps element displacements and forces between the two systems before assembly.

Element Stiffness Transformation

Generic orthogonal coordinate transformation for an element stiffness matrix.

[k]global=[T]T[k]local[T][k]_{\text{global}}=[T]^T[k]_{\text{local}}[T]

Variables

SymbolDescriptionUnit
[T][T]Coordinate transformation matrix-
[k]local[k]_{\text{local}}Element stiffness matrix in local coordinates-
[k]global[k]_{\text{global}}Element stiffness matrix expressed in global coordinates-

Assembly

Global stiffness assembly is an addition process. Each element stiffness term is placed into the rows and columns associated with its global degree-of-freedom numbers. Where multiple elements share a degree of freedom, their stiffness contributions add at the same global matrix location.

Interactive Exploration

Use the matrix-assembly simulation to trace one element from local stiffness through transformation and global assembly. Follow the degree-of-freedom map carefully; most manual assembly errors are indexing errors rather than algebraic errors.

Global Stiffness Matrix Assembly

Visualize how individual element stiffness matrices (k1\mathbf{k}_1 and k2\mathbf{k}_2) combine to form the global stiffness matrix (K\mathbf{K}) for a 3-node spring system.

Node 1Node 2Node 3k₁ = 10k₂ = 20
Element 1 Stiffness ($k_1$)10 kN/m
Element 2 Stiffness ($k_2$)20 kN/m

Global Stiffness Matrix $\mathbf10-100-1030-200-2020$

[
10
-10
0
-10
30
-20
0
-20
20
]

Notice the overlap at $K_22$.

Because Node 2 is shared by Element 1 and Element 2, the global stiffness at that degree of freedom is the sum of their individual stiffnesses ($k_1 + k_2 = 30$).

Symmetry and Definiteness

For a conservative linear-elastic structure, the assembled stiffness matrix is symmetric. Before sufficient displacement restraints are applied, the global matrix typically contains rigid-body modes and is therefore singular and positive semidefinite, not positive definite. After correct boundary conditions are imposed, the reduced free-degree stiffness matrix should be positive definite for a stable structure without mechanisms.

A Singular Reduced Matrix Is a Diagnostic

If the reduced stiffness matrix remains singular after intended supports are applied, investigate unrestrained rigid-body motion, internal mechanisms, duplicate or disconnected nodes, unintended member-end releases, zero stiffness properties, or incorrect constraint equations. Do not bypass the singularity numerically without correcting the structural model.

Applying Boundary Conditions

Partition the global displacement vector into unknown free displacements {Df}\{D_f\} and known restrained or prescribed displacements {Dr}\{D_r\}. The corresponding force vector contains known applied forces at free coordinates and unknown reactions at restrained coordinates.

Partitioned Stiffness Equations

System partitioned into free and restrained degree-of-freedom sets.

[KffKfrKrfKrr]{DfDr}={QfQr}\begin{bmatrix} K_{ff} & K_{fr}\\ K_{rf} & K_{rr} \end{bmatrix} \begin{Bmatrix} D_f\\ D_r \end{Bmatrix} = \begin{Bmatrix} Q_f\\ Q_r \end{Bmatrix}

Variables

SymbolDescriptionUnit
DfD_fUnknown free nodal displacements-
DrD_rKnown restrained or prescribed displacements-
QfQ_fKnown applied nodal forces at free coordinates-
QrQ_rUnknown reactions at restrained coordinates-

Free-Displacement Solution

Reduced equation including nonzero prescribed support movement.

[Kff]{Df}={Qf}−[Kfr]{Dr}[K_{ff}]\{D_f\}=\{Q_f\}-[K_{fr}]\{D_r\}

Variables

SymbolDescriptionUnit
[Kff][K_{ff}]Reduced stiffness matrix for free degrees of freedom-
[Kfr][K_{fr}]Coupling stiffness between free and restrained coordinates-

Equivalent Nodal Loads and Fixed-End Effects

Loads applied along an element must be represented consistently in the global nodal equations. A common frame-element procedure calculates fixed-end actions for the restrained member, uses the corresponding equivalent nodal load vector in assembly with the selected sign convention, then adds the appropriate fixed-end component back during final member-force recovery.

Member-Force Recovery

After solving global nodal displacements, extract the displacement components belonging to each element, transform them to the element's local coordinates, and evaluate the local element force relation. Include fixed-end or initial-strain effects when present.

Direct Stiffness Method Workflow

Treat the matrix method as a deterministic pipeline. Each step creates data required by the next, and every stage has a physical check: element stiffness, coordinate transformation, assembly, constraints, displacement solution, reaction recovery, and member equilibrium.

Direct Stiffness Method WorkflowStructural geometry, elements, properties, supports, and loads → Number nodes, elements, and global degrees of freedom; Number nodes, elements, and global degrees of freedom → Form each local element stiffness matrix; Form each local element stiffness matrix → Transform element stiffness and load quantities to global coordinates; Transform element stiffness and load quantities to global coordinates → Assemble global stiffness matrix and nodal load vector; Assemble global stiffness matrix and nodal load vector → Apply restrained and prescribed displacement boundary conditions; Apply restrained and prescribed displacement boundary conditions → Reduced free-DOF stiffness nonsingular and stable?; Reduced free-DOF stiffness nonsingular and stable? — Yes → Solve free nodal displacements; Reduced free-DOF stiffness nonsingular and stable? — No → Diagnose rigid-body mode, mechanism, release, connectivity, or property error; Diagnose rigid-body mode, mechanism, release, connectivity, or property error → Number nodes, elements, and global degrees of freedom; Solve free nodal displacements → Recover support reactions; Recover support reactions → Recover local member-end forces and fixed-end effects; Recover local member-end forces and fixed-end effects → Global equilibrium, member equilibrium, and expected deformation satisfied?; Global equilibrium, member equilibrium, and expected deformation satisfied? — Yes → Verified structural response; Global equilibrium, member equilibrium, and expected deformation satisfied? — No → Review DOF mapping, transformations, units, loads, and constraints; Review DOF mapping, transformations, units, loads, and constraints → Number nodes, elements, and global degrees of freedom

Structural geometry, elements, properties, supports, and loads → Number nodes, elements, and global degrees of freedom; Number nodes, elements, and global degrees of freedom → Form each local element stiffness matrix; Form each local element stiffness matrix → Transform element stiffness and load quantities to global coordinates; Transform element stiffness and load quantities to global coordinates → Assemble global stiffness matrix and nodal load vector; Assemble global stiffness matrix and nodal load vector → Apply restrained and prescribed displacement boundary conditions; Apply restrained and prescribed displacement boundary conditions → Reduced free-DOF stiffness nonsingular and stable?; Reduced free-DOF stiffness nonsingular and stable? — Yes → Solve free nodal displacements; Reduced free-DOF stiffness nonsingular and stable? — No → Diagnose rigid-body mode, mechanism, release, connectivity, or property error; Diagnose rigid-body mode, mechanism, release, connectivity, or property error → Number nodes, elements, and global degrees of freedom; Solve free nodal displacements → Recover support reactions; Recover support reactions → Recover local member-end forces and fixed-end effects; Recover local member-end forces and fixed-end effects → Global equilibrium, member equilibrium, and expected deformation satisfied?; Global equilibrium, member equilibrium, and expected deformation satisfied? — Yes → Verified structural response; Global equilibrium, member equilibrium, and expected deformation satisfied? — No → Review DOF mapping, transformations, units, loads, and constraints; Review DOF mapping, transformations, units, loads, and constraints → Number nodes, elements, and global degrees of freedom

  • Structural geometry, elements, properties, supports, and loads: terminator
  • Number nodes, elements, and global degrees of freedom: process
  • Form each local element stiffness matrix: process
  • Transform element stiffness and load quantities to global coordinates: process
  • Assemble global stiffness matrix and nodal load vector: process
  • Apply restrained and prescribed displacement boundary conditions: process
  • Reduced free-DOF stiffness nonsingular and stable?: decision
  • Diagnose rigid-body mode, mechanism, release, connectivity, or property error: process
  • Solve free nodal displacements: process
  • Recover support reactions: process
  • Recover local member-end forces and fixed-end effects: process
  • Global equilibrium, member equilibrium, and expected deformation satisfied?: decision
  • Review DOF mapping, transformations, units, loads, and constraints: process
  • Verified structural response: terminator

Software Does Not Remove Modeling Responsibility

Matrix stiffness is the foundation of many structural-analysis programs, but an exact matrix solution to the wrong connectivity, support condition, stiffness, load vector, or release model is still the wrong structural answer. Use equilibrium, deformation shape, symmetry, and simplified hand estimates as independent checks.

Key Takeaways
  • The direct stiffness method assembles element force-displacement relationships into one global equilibrium system.
  • Local element matrices must be transformed and mapped to the correct global degrees of freedom before assembly.
  • The unconstrained global matrix is normally singular because rigid-body modes are present; the properly restrained reduced matrix is positive definite only for a stable mechanism-free model.
  • Prescribed support movements enter the reduced force equation through the coupling matrix.
  • Member span loads require consistent equivalent-nodal-load and fixed-end-force treatment.
  • Final validation includes global equilibrium, reactions, member equilibrium, deformation shape, and independent engineering checks.

References