1.  Introduction

Structural failure in plates forming the exterior of structures is usually initiated at the most highly stressed points of the structure, typically near a sharp corner or a hole. For example, an aircraft window, as seen in Figure 1, is one of the most highly stressed areas in an aircraft. Other examples include windows in ships or submarines.

Figure 1. Aircraft fuselage window (left), and a ship window (right).

For this reason, estimating the stress concentration introduced by geometrical discontinuities, such as holes, is of practical importance in the field of structural integrity monitoring. In this assignment, the finite element method is used to examine the stress field in the vicinity of the hole that is present in a plate. The results are compared with the predictions from the Theory of Elasticity; Topic 1.

2.  Assignment Aim

The aims of this assignment are the following:

•   To show how the finite element method can be used to solve stress analysis problems.

•   To obtain practical experience in using the commercial finite element package ABAQUS

•   To demonstrate critical analysis of the output obtained from the finite element analysis calculations, commenting  on  mesh  convergence,  element  behaviour,  and justifying technical arguments with experimental and theoretical results from literature.

3.  Stress Solution in the Vicinity of a Hole in an Infinite Elastic Plate

The stress field around the hole in an infinite elastic plate under uniaxial tension has been derived in lectures. In the (r, θ) coordinate system, these components are [1]:

This solution indicates that the maximum stress concentration factor (SCF) S, in this plate is in the vicinity of the hole:

The plate, of course, does not have to be loaded under uniaxial tension. Cases of biaxial loading (Figure 2) and shear are common – e.g. pure shear is obtained by imposing σ2   = −σ1. For these cases, the solution of (1) and superposition can be used to obtain exact solutions.

Figure 2. Plate with a hole, subjected to abiaxial load.

As we’ll see in class, the stress field can be found by superimposing the solutions due to σ 1  and σ2.

Figure 3. Solutions due to σ 1  (left) and σ2 (right).

The contributions due to σ 1  are:

The contributions due to σ2  are:

Making use of the below relationships, the contributions due to σ2  can be expressed in terms of θ 1  and σ 1 :

Therefore, the stress fields are:

4.  Coursework Tasks and Report

For your FE model setup, assume that the plate is thin and is made of an aluminium alloy, with material properties E = 70 GPa, and ν = 0.33. The stress biaxiality ratio is given in the table below, according to the first letter of your first name.

Table 1. Stress biaxiality ratios according to the first letter of your first name.

Your report should include the following sections:

1.   Description of the finite element model setup:

•   Geometry of the plate

•   Boundary conditions

•   Element type and justification of choice

•   Mesh configuration used and mesh convergence, including the physical quantity used for monitoring convergence, and the convergence criterion and convergence threshold used. NB: Typically, the quantity used for judging convergence is either the quantity of interest,i.e. something that you need to investigate, or a quantity that is most sensitive to mesh density. The convergence criterion is a numerical measurement that shows you how fast you are approaching mesh-independence of results. The convergence threshold is a target to reach mesh independence of results.

You can assume that the person reading your report is familiar with Abaqus. For the above items, include diagrams to support your descriptions.

2.   Post-Processing and Examination of Results

•   Distribution (field output) of the three components of the stress field in the plate from the FE analysis.

•   Plots of: from  the  FE  analysis  and  theory (superimposed), where ax  is the maximum applied stress (this will be either σ1  or σ2  depending on your stress biaxiality ratio) andr is the radial distance from the centre of the hole.

3.   A discussion on:

•   The location and magnitude of the maximum stress concentration factor (SCF), defined as where is the maximum stress component in the vicinity of the hole. Which stress component (σrr or σθθ or τrθ ) is largest, and where? How do your FE analysis calculations compare with the theoretical predictions for the location and magnitude of S? What are the discrepancies, what are the reasons?

•   The effect of the plate dimensions on the stress concentration factor (SCF) for the biaxiality ratio assigned to you. How do you expect the finite dimensions of the plate to affect the stress concentration observed at the hole, and for what reason?

For both discussion items, provide quantitative arguments, and support them with references and your own calculations/analyses.