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Determine the straims ex,ey,exy and corresponding element stresses.

A CST element has nodal co-ordinates (10, 10), (70, 35) and (75, 25) for node 1, 2 and 3 respectively. the element is 2mm thick and is of material with properties E= 70 GPa.Poission ratio is 0.3. After applying the load to the element the hodal defermation were found u1=0.01mm u2=0.03mm u3=0.02mm v1=0.04mm v2=0.02mm v3=0.04mm


Subject: Finite Element Analysis

Topic: Two Dimensional Vector Variable Problem

Difficulty: Medium

1 Answer
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(x1,y1)=(10,10)
(x2,y2)=(70,35)
(x3,y3)=(75,25)

a) Jacobian matrix
[J]=[(x2x1)(y2y1)(x3x1)(y3y1)]

=[60256515]

b)Strain displacement relation matrix is given by,
{exeyδxy}=12A[β10β20β300γ10γ20γ3γ1β1γ2β2γ3β3]{u1v1u2v2u3v3}

β1=y2y3=10
β2=y3y1=15
β3=y1y2=25

γ1=(x2x3)=5
γ2=(x3x1)=65
γ3=(x1x2)=60

2A=|1x1y11x2y21x3y3| =|110101703517525|
2A=725

$\begin{Bmatrix}e_x\\e_y\\\delta_{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 \\ -0.05 \\0.03 \\0.02 …

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