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AEE/CEE/EGM/MEE 546 Finite Element Analysis I Assignment #4 Gaussian Quadrature Develop and integrate the stiffness matrix of the Physical Element shown below

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AEE/CEE/EGM/MEE 546 Finite Element Analysis I
Assignment #4 Gaussian Quadrature

Develop and integrate the stiffness matrix of the Physical Element shown below. This element is a four-node plane stress quadrilateral element with two degrees of freedom per node. The stiffness matrix is given by:
[?]=∫∫[?]?[?][?]det[?]??????1−11−1
in which
det [J] is the determinate of the Jacobian Matrix:
[?]=[ ????????????????]
[B] is the strain-displacement matrix with {?}=[?]{?}
and
[D] is the linear isotropic stiffness matrix:
[?]=?1−?2[1?0?10001−?2]
Use the following material properties: E = 70 GPa, ????0.33.
Assume the element is 2.5 mm thick.

The nodal coordinate (x,y) of the physical element are:
Node 1 (-0.25, -1.0) Node 2 (1.0, -0.50) Node 3 (0.75, 0.25) Node 4 (-1.0, 0.5)
Recall the shape functions for the element are:
?1=14(1−?)(1−?) ?2=14(1+?)(1−?) ?3=14(1+?)(1+?) ?4=14(1−?)(1+?)
Use 2x2 Gaussian Quadrature to evaluate the stiffness matrix. You’ll want to do this in software; you may choose the program/language to use (Matlab, Mathematica, Python, EXCEL, Julia, VBA, etc.). The evaluation should be completed as in a FEA code: evaluate the sub-matrices that make up [K] at a given integration point, then multiply these matrices of constants to obtain [K] at the integration point. Your final solution should be the sum of the [K] matrices at the integration points. It must be clear from the output how the integration was performed.
Upload the following to your DropBox on Isidore by Monday, Nov 7, at the beginning of class:
1. Your code, EXCEL file, etc, with the file-name format LastName – Assignment 4.xxx, in which xxx is the suitable extension for your file type. For example, Matlab code the instructor wrote would be “whitney-Assignment 4.m”
2. A text, WORD, or pdf file showing any intermediate output from your code and your final element stiffness matrix. Use the file-name format LastName – Assignment 4 Output.xxx,” again with xxx being the appropriate extension.
Hints
1. Include comments in your code or spreadsheet as necessary for clarity.
2. Create output at intermediate steps in the process. This will help diagnose errors.
3. Make painfully obvious the quantities in the output. It’s fine to print out the value of a variable or matrix, but make sure it’s obvious what that variable or matrix is – don’t just print, for example, “q = 6.7” and expect a reader to know what q is.