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Structural Mechanics

Solve static linear elasticity problems for 3-D solids, as well as 2-D plane stress and plane strain 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.

(λ+μ)(u)+μ2u+f=0

Here, vector u is the displacement, 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.

μ=E2(1+ν),λ=νE(1+ν)(12ν),f=(fxfy)

A typical programmatic workflow for solving a linear elasticity problem includes the following 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 body loads.

  • Specify boundary loads and constraints.

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

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.

Functions

createpdeCreate a model
structuralPropertiesAssign structural properties of a material for a structural model
structuralBodyLoadSpecify body load for a structural model
structuralBoundaryLoadSpecify boundary loads for a structural model
structuralBC Specify boundary conditions for a structural model
solveSolve heat transfer or structural analysis problem
interpolateDisplacementInterpolate displacement at arbitrary spatial locations
interpolateStressInterpolate stress at arbitrary spatial locations
interpolateStrainInterpolate strain at arbitrary spatial locations
interpolateVonMisesStressInterpolate von Mises stress at arbitrary spatial locations
evaluateReactionEvaluate reaction forces on boundary
evaluatePrincipalStressEvaluate principal stress at nodal locations
evaluatePrincipalStrainEvaluate principal strain at nodal 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
findStructuralBCFind structural boundary conditions and boundary loads assigned to geometric region
findBodyLoadFind body load assigned to geometric region

Using Objects

StructuralModelStructural model object
StaticStructuralResultsStatic structural solution and its derived quantities

Properties

StructuralMaterialAssignment PropertiesStructural material property assignments
BodyLoadAssignment PropertiesBody load assignments
StructuralBC Properties Boundary condition or boundary load for structural analysis model

Apps

PDE ModelerSolve partial differential equations in 2-D regions

Topics

Programmatic Workflow

Deflection Analysis of a 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.

Deflection of a Piezoelectric Actuator

Solve a coupled elasticity-electrostatics problem.

Clamped, Square Isotropic Plate With a Uniform Pressure Load

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

Dynamics of a Damped Cantilever Beam

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

Dynamic Analysis of a Clamped Beam

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

Vibration of a Square Plate

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

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.

Elliptic Equations

Mathematical definition and discussion of the elliptic equation

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