# Deflection Analysis of Bracket

This example shows how to analyze a 3-D mechanical part under an applied load using finite element analysis (FEA) and determine the maximal deflection.

### Create Structural Analysis Model

The first step in solving a linear elasticity problem is to create a structural analysis model. This model is a container that holds the geometry, structural material properties, damping parameters, body loads, boundary loads, boundary constraints, superelement interfaces, initial displacement and velocity, and mesh.

model = createpde("structural","static-solid");

### Import Geometry

Import an STL file of a simple bracket model using the `importGeometry`

function. This function reconstructs the faces, edges, and vertices of the model. It can merge some faces and edges, so the numbers can differ from those of the parent CAD model.

`importGeometry(model,"BracketWithHole.stl");`

Plot the geometry, displaying face labels.

figure pdegplot(model,"FaceLabels","on") view(30,30); title("Bracket with Face Labels")

figure pdegplot(model,"FaceLabels","on") view(-134,-32) title("Bracket with Face Labels, Rear View")

### Specify Structural Properties of Material

Specify Young's modulus and Poisson's ratio of the material.

structuralProperties(model,"YoungsModulus",200e9, ... "PoissonsRatio",0.3);

### Apply Boundary Conditions and Loads

The problem has two boundary conditions: the back face (face 4) is fixed, and the front face (face 8) has an applied load. All other boundary conditions, by default, are free boundaries.

structuralBC(model,"Face",4,"Constraint","fixed");

Apply a distributed load in the negative $$z$$-direction to the front face.

structuralBoundaryLoad(model,"Face",8,"SurfaceTraction",[0;0;-1e4]);

### Generate Mesh

Generate and plot a mesh.

```
generateMesh(model);
figure
pdeplot3D(model)
title("Mesh with Quadratic Tetrahedral Elements");
```

### Calculate Solution

Use the `solve`

function to calculate the solution.

result = solve(model)

result = StaticStructuralResults with properties: Displacement: [1x1 FEStruct] Strain: [1x1 FEStruct] Stress: [1x1 FEStruct] VonMisesStress: [5993x1 double] Mesh: [1x1 FEMesh]

### Examine Solution

Find the maximal deflection of the bracket in the $$z$$-direction.

```
minUz = min(result.Displacement.uz);
fprintf("Maximal deflection in the z-direction is %g meters.", minUz)
```

Maximal deflection in the z-direction is -4.43075e-05 meters.

### Plot Results Using Visualize PDE Results Live Editor Task

Visualize the displacement components and the von Mises stress by using the **Visualize PDE Results** Live Editor task. The maximal deflections are in the $$z$$-direction. Because the bracket and the load are symmetric, the *x*-displacement and *z*-displacement are symmetric, and the *y*-displacement is antisymmetric with respect to the center line.

First, create a new live script by clicking the **New Live Script** button in the **File** section on the **Home** tab.

On the **Live Editor** tab, select **Task** > **Visualize PDE Results**. This action inserts the task into your script.

To plot the *z*-displacement, follow these steps. To plot the *x*- and *y*-displacements, follow the same steps, but set **Component** to *X* and *Y*, respectively.

In the

**Select results**section of the task, select`result`

from the drop-down list.In the

**Specify data parameters**section of the task, set**Type**to*Displacement*and**Component**to*Z*.In the

**Specify visualization parameters**section of the task, clear the**Deformation**check box.

Here, the blue color represents the lowest displacement value, and the red color represents the highest displacement value. The bracket load causes face 8 to dip down, so the maximum *z*-displacement appears blue.

To plot the von Mises stress, in the **Specify data parameters** section of the task, set **Type** to *Stress* and **Component** to *von Mises*.

### Plot Results at the Command Line

You also can plot the results, such as the displacement components and the von Mises stress, at the MATLAB® command line by using the `pdeplot3D`

function.

figure pdeplot3D(model,"ColorMapData",result.Displacement.ux) title("x-displacement") colormap("jet")

figure pdeplot3D(model,"ColorMapData",result.Displacement.uy) title("y-displacement") colormap("jet")

figure pdeplot3D(model,"ColorMapData",result.Displacement.uz) title("z-displacement") colormap("jet")

figure pdeplot3D(model,"ColorMapData",result.VonMisesStress) title("von Mises stress") colormap("jet")