A CST element has nodal coordinates (10,10) , (70, 35) for nodes 1,2 and 3 respectively. The element is 2 mm thick and is of material with properties E=70 GPA. Poisson's ratio os 0.3. After applying the load to the element the nodal deformation were found to be $u_1=0.01mm, \ \ v_1=-0.04mm \ \ u_2=0.03mm, v_2=0.02mm, \ \ u_3=-0.02m, \ \ u_1=-0.04mm$

Determine the strains $e_x,e_y,e_z$ and corresponding element stresses.

Mumbai University > Mechanical Engineering > Sem 6 > Finite Element Analysis

Marks: 10M

Year: Dac 2016

$\hspace{1cm} x_1= 10 mm\hspace{1cm} y_1= 10 mm\\ \hspace{1cm} x_2= 70 mm\hspace{1cm} y_2= 35 mm\\ \hspace{1cm} x_3= 75 mm\hspace{1cm} y_1= 25 mm$

$\beta_1=y_2-y_3 = 10\hspace{2.5cm} r_1= x_3-x_2=5\\ \beta_2=y_3-y_1 = 15\hspace{2.5cm} r_2= x_1-x_3=65\\ \beta_1=y_1-y_2 = 25\hspace{2.5cm} r_3= x_2-x_1=60$

$2A=\begin{vmatrix} \ 1 & 10 & 10 \\ \ 1 & 70 & 35 \\ \ 1 & 75 & 25 \\ \end{vmatrix}=-725 mm^2$

$B=\frac{1}{2A}\begin{bmatrix} \ \beta_1 & 0 & \beta_2 & 0 & \beta_3 & 0 \\ \ 0_1 & r_1 & 0 & r_2 & 0 & r_3 \\ \ r_1 & \beta_1 & r_2 & \beta_2 & r_3 & \beta_3 \\ \end{bmatrix}$

$=\frac{-1}{725}$

$\begin{bmatrix} \ 10 & 0 & 15 & 0 & -25 & 0 \\ \ 0 & 5 & 0 & -65 & 0 & 60 \\s \ 5 & 10 & -65 & 15 & 60 & -25 \\ \end{bmatrix}$

[e]=[B][u]

$\begin {Bmatrix} \ ex \\ \ ey \\ \ xy \\ \end{Bmatrix}=\frac{-1}{725}$

$\begin{bmatrix} \ 10 & 0 & 15 & 0 & -25 & 0 \\ \ 0 & 5 & 0 & -65 & 0 & 60 \\ \ 5 & 10 & -65 & 15 & 60 & -25 \\ \end{bmatrix}$

$\begin {Bmatrix} \ 0.01 \\ \ -0.04 \\ \ 0.03 \\ \ 0.02 \\ \ -0.02 \\ \ -0.04 \\ \end{Bmatrix}$

$\hspace{1.6cm}=\begin {bmatrix} \ -0.00144 \\ \ 0.00538 \\ \ 0.00304 \\ \end{bmatrix}$

$D=\frac{E}{1-v^2}\begin{bmatrix} \ 1 & v & 0 \\ \ v & 1 & 0 \\ \ 0 & 0 & \frac{1-v}{2} \\ \end{bmatrix}$

$\hspace{0.6cm} =76.923\times10^3\begin{bmatrix} \ 1 & 0.3 & 0 \\ \ 0.3 & 1 & 0 \\ \ 0 & 0 & 0.35 \\ \end{bmatrix}$

$[\sigma]$=[D][e]

$\hspace{0.6cm} =76.923\times10^3\begin{bmatrix} \ 1 & 0.3 & 0 \\ \ 0.3 & 1 & 0 \\ \ 0 & 0 & 0.35 \\ \end{bmatrix}$

$\begin{Bmatrix} \ -1.44 \\ \ -5.38 \\ \ 3.03 \\ \end{Bmatrix}\times10^{-3}$

$\begin{Bmatrix} \ \sigma_x \\ \ \sigma_y \\ \ z_{xy} \\ \end{Bmatrix}=$

$\begin{bmatrix} \ 12.732 \\ \ 380.371 \\ \ 81.698 \\ \end{bmatrix}N/mm^2$