Question 18.P.13: Figure P18.13 shows a three-node triangular element ABC unde...

Dynamics of Structures

Figure P18.13 shows a three-node triangular element ABC under plane stress. The coordinates of the nodes are given in centimeters and the thickness of the element is $2 \mathrm{~cm}$ . The modulus of elasticity is $200,000 \mathrm{MPa}$ and the Poisson’s ratio is $0.3$ . Obtain the stiffness and mass matrices for the element. The distributed traction force acting perpendicular to side $\mathrm{BC}$ varies from $100 \mathrm{MPa}$ at $\mathrm{B}$ to $200 \mathrm{MPa}$ at $\mathrm{C}$ . Find the equivalent nodal loads.

p18.13

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\begin{aligned}&\mathbf{K}=10^{7}\left[\begin{array}{rrrrrrr}1.8393 & 0.8571 & -1.1181 & -0.1978 & -0.7212 & -0.6593 \\0.8571 & 1.7286 & -0.0879 & 0.3319 & -0.7692 & -2.0604 \\-1.1181 & -0.0879 & 1.5989 & -0.5714 & -0.4808 & 0.6593 \\-0.1978 & 0.3319 & -0.5714 & 1.0418 & 0.7692 & -1.3736 \\-0.7212 & -0.7692 & -0.4808 & 0.7692 & 1.2019 & 0 \\-0.6593 & -2.0604 & 0.6593 & -1.3736 & 0 & 3.4341\end{array}\right] \\&\mathbf{M}=0.26\left[\begin{array}{rrrrrr}2 & 0 & 1 & 0 & 1 & 0 \\0 & 2 & 0 & 1 & 0 & 1 \\1 & 0 & 2 & 0 & 1 & 0 \\0 & 1 & 0 & 2 & 0 & 1 \\1 & 0 & 1 & 0 & 2 & 0 \\0 & 1 & 0 & 1 & 0 & 2\end{array}\right]\end{aligned}

The nodal loads are given by $p_{1}=2.924 \times 10^{5}, \quad p_{2}=3.655 \times 10^{5} \mathrm{~N}$

These loads act in a direction perpendicular to side $\mathrm{BC}$ .

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