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

Problem 1: Part 8

Consider the curved composite beam shown in Figure 12.

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

$\cfrac{D}{Dt} \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & (C_{11}-C_{12})/2 \end{bmatrix} \begin{bmatrix} D_{11} \\ D_{22} \\ D_{33} \\ D_{23} \\ D_{31} \\ D_{12} \end{bmatrix}~.$

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,

$\theta = \cfrac{\pi}{4}~;~~ \theta_1 = \cfrac{\pi}{3}~;~~ \theta_2 = \cfrac{\pi}{6} ~.$

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

$\mathbf{v}_1 = \begin{bmatrix} v_1^x \\ v_1^y \\ \omega_1 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} ~;~~ \mathbf{v}_2 = \begin{bmatrix} v_2^x \\ v_2^y \\ \omega_2 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}$

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]);