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# 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:

`$\left(\lambda +\mu \right)\nabla \left(\nabla \cdot u\right)+\mu {\nabla }^{2}u+f=\rho \frac{{\partial }^{2}u}{\partial {t}^{2}}$`

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:

`$\mu =\frac{E}{2\left(1+\nu \right)},\text{ }\lambda =\frac{\nu E}{\left(1+\nu \right)\left(1-2\nu \right)},\text{ }f=\left(\begin{array}{l}{f}_{x}\\ {f}_{y}\end{array}\right)$`

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.

## Functions

expand all

 `createpde` Create model `structuralProperties` Assign structural properties of material for structural model `structuralDamping` Specify damping parameters for transient structural model `structuralBodyLoad` Specify body load for structural model `structuralBoundaryLoad` Specify boundary loads for structural model `structuralBC` Specify boundary conditions for structural model `structuralIC` Set initial conditions for a transient structural model `solve` Solve heat transfer or structural analysis problem `assembleFEMatrices` Assemble finite element matrices
 `evaluateStress` Evaluate stress for dynamic structural analysis problem `evaluateStrain` Evaluate strain for dynamic structural analysis problem `evaluateVonMisesStress` Evaluate von Mises stress for dynamic structural analysis problem `evaluateReaction` Evaluate reaction forces on boundary `evaluatePrincipalStress` Evaluate principal stress at nodal locations `evaluatePrincipalStrain` Evaluate principal strain at nodal locations
 `interpolateDisplacement` Interpolate displacement at arbitrary spatial locations `interpolateVelocity` Interpolate velocity at arbitrary spatial locations for all time steps for transient structural model `interpolateAcceleration` Interpolate acceleration at arbitrary spatial locations for all time steps for transient structural model `interpolateStress` Interpolate stress at arbitrary spatial locations `interpolateStrain` Interpolate strain at arbitrary spatial locations `interpolateVonMisesStress` Interpolate von Mises stress at arbitrary spatial locations
 `pdeplot` Plot solution or mesh for 2-D geometry `pdeplot3D` Plot solution or surface mesh for 3-D geometry
 `findStructuralProperties` Find structural material properties assigned to geometric region `findStructuralDamping` Find damping model assigned to structural dynamics model `findStructuralBC` Find structural boundary conditions and boundary loads assigned to geometric region `findStructuralIC` Find initial displacement and velocity assigned to geometric region `findBodyLoad` Find body load assigned to geometric region

## Objects

 `StructuralModel` Structural model object `StaticStructuralResults` Static structural solution and its derived quantities `TransientStructuralResults` Transient structural solution and its derived quantities `ModalStructuralResults` Structural modal analysis solution

## Properties

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

## Apps

 PDE Modeler Solve partial differential equations in 2-D regions

## Topics

### 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.

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