Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 8

Problem 1: Part 8

Consider the curved composite beam shown in Figure 12.

File:CompositeShell.png Figure 12. Composite Beam.

Assume that the beam has been shaped into an arc of a circle. The material of the beam is a transversely isotropic fiber composite material with the fibers running along the length of the beam. The rate constitutive relation of the material is given by

$\frac{D}{Dt}\left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{23}\\ {\sigma }_{31}\\ {\sigma }_{12}\end{array}\right]=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{12}& C_{11}& C_{13}& 0& 0& 0\\ C_{13}& C_{13}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{44}& 0\\ 0& 0& 0& 0& 0& (C_{11}-C_{12})/2\end{array}\right]\left[\begin{array}{c}{D}_{11}\\ D_{22}\\ D_{33}\\ D_{23}\\ D_{31}\\ D_{12}\end{array}\right].$

The problem becomes easier to solve if we consider numerical values of the parameters. Let the local nodes numbers of element $5$ be $1$ for node $5$, and $2$ for node $6$.

Let us assume that the beam is divided into six equal sectors. Then,

$\theta =\frac{\pi }{4};{\theta }_1=\frac{\pi }{3};{\theta }_2=\frac{\pi }{6}.$

Let $r_1=1$ and $r_2=1.2$. Since the blue point is midway between the two, $r=1.1$.

Also, let the components of the stiffness matrix of the composite be

$C_{11}=10;C_{33}=100;C_{12}=6;C_{13}=60;C_{44}=30.$

Let the velocities for nodes $1$ and $2$ of the element be

${𝐯}_1=\left[\begin{array}{c}{v}_1^x\\ v_1^y\\ {\omega }_1\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right];{𝐯}_2=\left[\begin{array}{c}{v}_2^x\\ v_2^y\\ {\omega }_2\end{array}\right]=\left[\begin{array}{c}2\\ 1\\ 1\end{array}\right]$

The Maple code for these initializations is shown below

> with(linalg):
> #
> # Input geometry
> #
> theta:= evalf(Pi/4):
> theta1 := evalf(theta + Pi/12):
> theta2 := evalf(theta - Pi/12):
> r1:= 1; r2 := 1.2; r:= (r1+r2)/2;
> #
> # Input material properties
> #
> C11 := 10; C33:= 100; C12:= 6; C13 := 60; C44 := 30;
> CC := (C11-C12)/2;
> #
> # Input velocities
> #
> vx1 := 1; vy1 := 2; w1 := 1;
> vx2 := 2; vy2 := 1; w2 := 1;
> v1master := linalg[matrix](3,1,[vx1,vy1,w1]);
> v2master := linalg[matrix](3,1,[vx2,vy2,w2]);