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*:

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

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.

`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 |

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 |

PDE Modeler | Solve partial differential equations in 2-D regions |

**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.

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.

**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.

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