A rectangular element used in plane stress analysis has corners whose coordinates in metres referred to an Oxy axes system are 1(−2, −1), 2(2, −1), 3(2, 1), 4(−2, 1). The displacements of the corners (in metres) are

$\begin{aligned}&u_1=0.001 \quad u_2=0.003 \quad u_3=-0.003 \quad u_4=0 \\&v_1=-0.004 \quad v_2=-0.002 \quad v_3=0.001 \quad v_4=0.001\end{aligned}$

If Young’s modulus is 200 000 N/mm² and Poisson’s ratio is 0.3 calculate the strains at the centre of the element.

Question

A rectangular element used in plane stress analysis has corners whose coordinates in metres referred to an Oxy axes system are 1(−2, −1), 2(2, −1), 3(2, 1), 4(−2, 1). The displacements of the corners (in metres) are

[latex]\begin{aligned}&u_1=0.001 \quad u_2=0.003 \quad u_3=-0.003 \quad u_4=0 \&v_1=-0.004 \quad v_2=-0.002 \quad v_3=0.001 \quad v_4=0.001\end{aligned}[/latex]

If Young’s modulus is 200 000 N/mm² and Poisson’s ratio is 0.3 calculate the strains at the centre of the element.

Accepted Answer

Suitable displacement functions are:

[latex]\begin{aligned}&u=\alpha_1+\alpha_2 x+\alpha_3 y+\alpha_4 x y \&v=\alpha_5+\alpha_6 x+\alpha_7 y+\alpha_8 x y\end{aligned}[/latex]

Then

[latex]\begin{aligned}&u_1=\alpha_1-2 \alpha_2-\alpha_3+2 \alpha_4=0.001 (i) \&u_2=\alpha_1+2 \alpha_2-\alpha_3-2 \alpha_4=0.003 (ii)\&u_3=\alpha_1+2 \alpha_2+\alpha_3+2 \alpha_4=-0.003 (iii) \&u_4=\alpha_1-2 \alpha_2+\alpha_3-2 \alpha_4=0 (iv)\end{aligned}[/latex]

Subtracting Eq. (ii) from Eq. (i)
[latex]\alpha_2-\alpha_4[/latex] = 0.0005 (v)
Subtracting Eq. (iv) from Eq. (iii)
[latex]\alpha_2+\alpha_4[/latex] = −0.00075 (vi)
Subtracting Eq. (vi) from Eq. (v)
[latex]\alpha_4[/latex]= −0.000625
Then, from either of Eqs (v) or (vi)
[latex]\alpha_2[/latex] = −0.000125
Adding Eqs (i) and (ii)
[latex]\alpha_1-\alpha_3[/latex]= 0.002 (vii)
Adding Eqs (iii) and (iv)
[latex]\alpha_1+\alpha_3[/latex] = −0.0015 (viii)
Adding Eqs (vii) and (viii)
[latex]\alpha_1 [/latex]= 0.00025

Then from either of Eqs (vii) or (viii)
[latex]\alpha_3 [/latex] = −0.00175
Similarly

[latex]\begin{aligned}&\alpha_5=-0.001 \&\alpha_6=0.00025 \&\alpha_7=0.002 \&\alpha_8=-0.00025\end{aligned}[/latex]

Then

[latex]\begin{aligned}&u_i=0.00025-0.00125 x-0.00175 y-0.000625 x y \&v_i=-0.001+0.00025 x+0.002 y-0.00025 x y\end{aligned}[/latex]

From Eqs (1.18) and (1.20)

[latex]\left.\begin{array}{rl}\varepsilon_x & =\frac{\partial u}{\partial x} \\varepsilon_y & =\frac{\partial v}{\partial y} \\varepsilon_z & =\frac{\partial w}{\partial z}\end{array} ight\}[/latex] (1.18)

[latex]\left.\begin{array}{l}\gamma_{x z}=\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z} \\gamma_{x y}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y} \\gamma_{y z}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\end{array} ight\}[/latex] (1.20)

[latex]\begin{aligned}\varepsilon_x &=\frac{\partial u}{\partial x}=-0.000125-0.000625 y \\varepsilon_y &=\frac{\partial v}{\partial y}=0.002-0.00025 x \\gamma_{x y} &=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=-0.0015-0.000625 x-0.00025 y\end{aligned}[/latex]

At the centre of the element where x = y = 0

[latex]\varepsilon_x=-0.000125 \quad \varepsilon_y=0.002 \quad \gamma_{x y}=-0.0015[/latex]

Question p.6.12: A rectangular element used in plane stress analysis has corn...

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