# A Plate with stress concentration

Example created using Siemens NX 12 and Siemens NX Nastran 12

## Problem introduction and Model specifications

A plate of finite length having a hole at center and two cross sections joined by fillets. The plate is of constant thickness and made of steel (Elastic modulus (E) = 206.94 GPa and Poisson ratio (ν) = 0.288). A uniform pressure of 2 MPa is applied at both ends of the plate.

The Model specifications are as below: The thickness is 2mm, hole radius of 1.4 mm and fillet radius of 0.7 mm.

Figure 1: Model specifications (All dimensions in mm)

## Learning outcome and Output requests

This example is an introduction to modeling FEM problems using 2D shell elements in NX Nastran. The use of model symmetry is also demonstrated which reduces the problem to quarter size.  The problem includes the effect of stress concentration on the stress near the region of hole and fillets. The output results are shown in both rectangular and cylindrical co-ordinate system.

After solving the problem in NX Nastran, the displacement of the model and normal stress (especially at location of hole and fillets) are shown. The mesh convergence is also checked to make the results independent of the mesh size.

The video below contains the description about the problem introduction, model description (geometry, material, loads and constraints) and the analytical calculations.

## Analytical results

### Displacement

If we neglect the hole in the center of the plate, we can estimate the displacement of the right edge of the plate w.r.t to the plate center.

### Stress in cylindrical co-ordinate system

#### Kirsch’s equations

Kirsch equations estimate elastic stress around hole or any discontinuity in an infinite plate in one directional tension.

Stress in r, θ and rθ direction

Stress in r, θ, rθ direction at hole surface, r=a:

Stress in r, θ, rθ direction at infinity or at large distance from ‘a’

#### Steps

2. Meshing and Material
4. Results post processing
5. Results verification and validation

The first step of simulation is to create a CAD model using NX 12.0. The CAD model for this problem is a surface model which consists of only a quarter model of the problem.

The video below describes the following operations:

• Creating a new part file
• Creating a sketch and dimensioning
• Creating a surface
• Surface division

## Meshing and Material

Next step is to create a Fem file. Fem file consists of information regarding the material properties and type of meshing to be done.

The video below describes the following operations:

• Creating fem and sim file
• Solver and solution type
• Output requests: Displacement and Stress
• Properties: material (steel) and thickness
• Meshing info: 2D Mapped and free shell meshing
• Meshing quality check

Next step is to create a sim file. Sim file consists of information about the loading and constraints to be applied.

The video below describes the following operations:

• Constraints: Symmetry constraints and preventing z translation
• Model setup check
• Solve the Fem model
• Check for any errors or warnings in Analysis job information

## Results post processing

After the solution is done, a result file is created that has all the output results requested. Nastran will always calculate displacement and stress as default.

The video below describes the following post processing operations:

• Displacement of the model
• Normal stress distribution in the model
• Different contour plot options
• Identifying maximum stress at hole and fillet locations
• Results in rectangular and cylindrical co-ordinate systems

## Results verification and validation

Since we now have both the analytical and simulation results, we can easily cross check both the results to see how close the results are given by NX Nastran in comparison to the analytical results. We will also see that the results are independent of the mesh.

The video below describes the following operations:

• Comparison of analytical Vs Fem results for: Displacement of right edge, maximum stress at hole and fillet locations
• Error calculation between analytical and Fem results
• Mesh independence of results