**Objective 1**

From the project requirement, we are aware that the various parameters needed to calculate the theoretical stress and displacement.

The formula being used is:

The x values used here are 0 on the left side of the beam and 0 again on the right end of the beam. This is so that the maximum displacement can be at the bottom of the beam. When the load acts on the beam, it looks like this figure 1.

**Figure ****1****: Y displacement on FEM**

The value of x was taken to be as such in table 1.

**Table ****1****: Theoretical and actual displacement in the y direction**

x |
Uy Theoretical (m) |
Uy Actual (m) |

0 | 0 | 0 |

1 | -3.88284E-07 | -1.62E-07 |

2 | -7.12504E-07 | -3.49E-07 |

3 | -9.25787E-07 | -4.95E-07 |

4 | -1.00001E-06 | -5.49E-07 |

3 | -7.99224E-07 | -4.95E-07 |

2 | -6.87504E-07 | -3.49E-07 |

1 | -3.86721E-07 | -1.62E-07 |

0 | 0 | 0.00E+00 |

On the third column are the values imported from Postview. These are actual displacements calculated by the FeBio software. The following figure 2 shows the graph of x position vs. the y displacements.

**Figure ****2****: Graph of y displacement vs x position**

As it can be seen through the graph, the software predicts a displacement slightly smaller than the actual. To calculate the stress, the following equation 2 is used.

Similar to last time, we can plug the values of y in the equation 2 to find the normal stress.

The figure 3 below shows the normal stress in the x direction.

**Figure ****3****: Normal Stress concentration**

The x axis can be drawn from last time. From the graph, the maximum normal stress is found to be ±5.83Pa. The normal stress for various y values have been tabulated in the table 2 below.

**Table ****2****: Normal stress value for theoretical and computation values**

y values (m) |
Stress theoretical (Pa) |
Stress actual (Pa) |

-1 | 20 | 5.83333 |

0 | 0 | 4.77E-07 |

1 | -20 | -5.83333 |

These values from table 2 have been plotted below in figure 4.

**Figure ****4****: Graph of Normal Stress vs y displacement**

From the graph it can be seen that the actual stress that we measure from the software is much below the theoretical value. This is probably because the mesh size is really big – 1m. If we decrease the mesh size, then we might get results closer to the theoretical true value.

**Objective 2**

The goal of this section to approximate the y displacement of a cantilever beam when a uniform load is applied through Finite Element Modelling. The mesh size used in the previous section was only 1m. In this section, we calculate the maximum y displacement in the middle of the beam (x=4, y=y, z = 0) with a change in the mesh size. The table 3 below tabulates the these two categories.

**Table ****3****: Mesh size vs max y displacement**

Mesh size |
y Displacement (m) |

2 | -4.12E-07 |

1 | -5.49E-07 |

0.5 | -6.22E-07 |

0.4 | -6.33E-07 |

0.25 | -6.45E-07 |

0.1 | -6.53E-07 |

0.05 | N/A |

0.04 | N/A |

When a mesh size of 0.05 was calculated, the computer was not able to run, the software crashed. The same happened for 0.04. Therefore, no values for those two plots are available.

**Figure ****5****: Graph of y displacement vs mesh size**

From the figure 5, it can be seen that there is a general decreasing trend, however, it will approach to be a horizontal asymptote as the mesh size become small. This can already be seen as we approach 0.1 mesh size. This graph also justifies an immense discrepancy in the data from objective 1 as the mesh size used was only 1, which is very far from the horizontal asymptote.