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

Problem 1: Part 8

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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

 


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

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

 

Let   and  . Since the blue point is midway between the two,  .

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

 

Let the velocities for nodes   and   of the element be

 

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]);