Name: Petar Galabov

Student ID: 8041205

Coursework submitted as part of the MACE30051Unit

School of Mechanical, Aerospace and Civil Engineering

University of Manchester

Problem description

For the purposes of this experiment a thin flat plate carrying a tensile stress has been examined. Due to the fact that the plate was symmetric only one quadrant of it was modelled. Two different cases were analyzed – with a circular hole and with a crack in the middle of the plate. The mesh was suitably refined in the region adjacent to the surface of the hole where largest stress gradients were expected. The plate is 56mm long and 20mm wide with thickness equal to 1mm. The Young’s Modulus was equal to 7.0 × 104 MPa and the Poisson ratio was 0.25. The applied uniform tensile stress was equal to 25 MPa.

Results

Case 1

Initially the effect on applying a uniform tensile stress on the plate was examined by using a coarse mesh. Contour plots of the stresses in both X and Y directions are shown on the figure below. Figure 1: Stress distributions in X-direction (left) and Y-direction (right)

As it can be expected, the stress in both directions is highest at the edge of the hole along the plane of symmetry. The Stress Concentration Factor (SCF) can be calculated by taking the ratio of the tensile stress at the edge of the hole to the applied uniform stress. The maximum stress is equal to 98.8855 MPa, which gives Kg = 3.96. The ratio of the diameter of the hole (d) to the width of the plate (H) is equal to 0.5 which gives Kn = Kg(1 – d/H) = 1.98. Using the theoretical equations for the Stress Concentration Factors given, they are calculated to be 4.314 and 2.157, which is not quite consistent with the numerical solution.

A graph showing the variation of the stresses in the X and Y direction along the horizontal line of symmetry can be seen below. It is clear again that the stress is highest at the edge of the hole and is gradually decreasing towards the end of the plate. Figure 2: Variation of X and Y stresses with distance

In order to obtain a more accurate solution the mesh was refined. The refinement has the effect of increasing the number of elements in the mesh which means that the number of calculations done by the processor is also increased. Plots of the X and Y stresses are shown below.

Figure 3: Stress distributions in X-direction (left) and Y-direction (right)

As it can be seen, the maximum stress this time was calculated to be 106.731 MPa which gives a percentage difference of around 7.5% with the solutions obtained using the coarse mesh. This leads to SCF values of Kg = 4.27 and Kn = Kg(1 – d/H) = 2.13. Those values obtained for the Stress Concentration Factors are in much closer agreement with the theoretical ones.

The variation of stresses in both directions along the symmetry line is shown on the graph below. Again, the stress is highest at the edge of the hole and decreases towards the end of the plate.

Figure 4: Variation of X and Y stresses with distance

Case 2

This part of the experiment involved the modelling of a thin flat plate, geometrically similar to the one