Structural Mechanics

Solve linear static, dynamic, and modal analysis problems

Navier partial differential equations describe the displacement field as a function of body forces and structural properties of the material. Knowing the displacement field, you can calculate the strain and stress fields:


Here, vector u is the displacement, ρ is the mass density, μ is the shear modulus, λ is the Lame modulus of the material, and f is a vector of volume forces. The shear modulus and Lame modulus can be expressed via the Young's (elastic) modulus E and the Poisson's ratio ν:


A typical programmatic workflow for solving a linear elasticity problem includes these steps:

  • Create a special structural analysis container for a solid (3-D), plane stress, or plane strain model.

  • Define 2-D or 3-D geometry and mesh it.

  • Assign structural properties of the material, such as Young's modulus, Poisson's ratio, and mass density.

  • Specify a damping model and its values for a dynamic problem.

  • Specify gravitational acceleration as a body load.

  • Specify boundary loads and constraints.

  • Specify initial displacement and velocity for a dynamic problem.

  • Solve the problem and plot results, such as displacement, velocity, acceleration, stress, strain, von Mises stress, principal stress and strain.

For modal analysis problems, use the same steps for creating a model and specifying materials and boundary constraints. In this case, the solver finds natural frequencies and mode shapes of a structure.

For plane stress and plane strain problems, you also can use the PDE Modeler app. The app includes geometry creation and preset modes for applications.


expand all

createpdeCreate model
structuralPropertiesAssign structural properties of material for structural model
structuralDampingSpecify damping parameters for transient structural model
structuralBodyLoadSpecify body load for structural model
structuralBoundaryLoadSpecify boundary loads for structural model
structuralBC Specify boundary conditions for structural model
structuralICSet initial conditions for a transient structural model
solveSolve heat transfer or structural analysis problem
assembleFEMatricesAssemble finite element matrices
evaluateStressEvaluate stress for dynamic structural analysis problem
evaluateStrainEvaluate strain for dynamic structural analysis problem
evaluateVonMisesStressEvaluate von Mises stress for dynamic structural analysis problem
evaluateReactionEvaluate reaction forces on boundary
evaluatePrincipalStressEvaluate principal stress at nodal locations
evaluatePrincipalStrainEvaluate principal strain at nodal locations
interpolateDisplacementInterpolate displacement at arbitrary spatial locations
interpolateVelocityInterpolate velocity at arbitrary spatial locations for all time steps for transient structural model
interpolateAccelerationInterpolate acceleration at arbitrary spatial locations for all time steps for transient structural model
interpolateStressInterpolate stress at arbitrary spatial locations
interpolateStrainInterpolate strain at arbitrary spatial locations
interpolateVonMisesStressInterpolate von Mises stress at arbitrary spatial locations
pdeplotPlot solution or mesh for 2-D geometry
pdeplot3DPlot solution or surface mesh for 3-D geometry
findStructuralPropertiesFind structural material properties assigned to geometric region
findStructuralDampingFind damping model assigned to structural dynamics model
findStructuralBCFind structural boundary conditions and boundary loads assigned to geometric region
findStructuralICFind initial displacement and velocity assigned to geometric region
findBodyLoadFind body load assigned to geometric region


StructuralModelStructural model object
StaticStructuralResultsStatic structural solution and its derived quantities
TransientStructuralResultsTransient structural solution and its derived quantities
ModalStructuralResultsStructural modal analysis solution


StructuralMaterialAssignment PropertiesStructural material property assignments
StructuralDampingAssignment PropertiesDamping assignment for a structural analysis model
BodyLoadAssignment PropertiesBody load assignments
StructuralBC Properties Boundary condition or boundary load for structural analysis model
GeometricStructuralICs PropertiesInitial displacement and velocity over a region
NodalStructuralICs Properties Initial displacement and velocity at mesh nodes


PDE ModelerSolve partial differential equations in 2-D regions


Programmatic Workflow

Deflection Analysis of Bracket

Analyze a 3-D mechanical part under an applied load and determine the maximal deflection.

Stress Concentration in Plate with Circular Hole

Perform a 2-D plane-stress elasticity analysis.

Structural Dynamics of Tuning Fork

Perform modal and transient analysis of a tuning fork.

Modal Superposition Method for Structural Dynamics Problem

Use modal analysis results to compute the transient response of a thin 3-D plate under a harmonic load at the center.

Thermal Deflection of Bimetallic Beam

Solve a coupled thermo-elasticity problem.

Deflection of Piezoelectric Actuator

Solve a coupled elasticity-electrostatics problem.

Clamped, Square Isotropic Plate with Uniform Pressure Load

Calculate the deflection of a structural plate acted on by a pressure loading.

Dynamics of Damped Cantilever Beam

Include damping in the transient analysis of a simple cantilever beam.

Dynamic Analysis of Clamped Beam

Analyze the dynamic behavior of a beam clamped at both ends and loaded with a uniform pressure load.

Vibration of Square Plate

Calculate the vibration modes and frequencies of a 3-D simply supported, square, elastic plate.

Finite Element Analysis of Electrostatically Actuated MEMS Device

Perform coupled electro-mechanical finite element analysis of an electrostatically actuated micro-electro-mechanical (MEMS) device.

PDE Modeler App Workflow

von Mises Effective Stress and Displacements

Use the PDE Modeler app to compute the von Mises effective stress and displacements for a steel plate clamped along an inset at one corner and pulled along a rounded cut at the opposite corner.

Algorithms and Other Theory

Linear Elasticity Equations

Linear elasticity equations for plane stress, plane strain, and 3-D problems.