Problem 6.1 Compute the extensional coupling and bending matrices of a bimetallic(copper

and aluminum) strip with each layer 2 mm thick. For aluminum,

. For copper, . Assume Poisson's ratio is

0.3 for both materials.

Solution

Here the materials are metals (homogeneous)

Also, there is no temperature change.

Give the aluminum the subscript "1" and the copper the subscript "2".

We have 2 layers each have

The metal layers will be treated as composite laminates.

For aluminum layer:

For copper layer:

Problem 6.2 Given the following displacements and relation functions,

Where A, B, C, Q, and R are constants.

a) Derive the middle surface strains and curvatures.

b) Derive the strain functions as a function of x, y, and z.

c) For

evaluate

and plot the strains as a function of the thickness coordinate z.

Solution

a) Strains and curvatures of middle plane:

b) From requirement no. (a) the expressions of are the

requirements.

c) I took the limits of z from –t/2 to +t/2 (over the cross section thickness)

Results are shown below

Appendix

MATLAB code for plotting strains vs Z of requirement (C) f problem

6.2 clear;clc;close all A=0.001; B=A; C=A; Q=2; R=Q; alfa=pi; beta=alfa; x=0.5; y=x; t=0.002; z=linspace(-t/2,t/2,100);

eps_xx=-A*alfa*sin(alfa*x)*sin(beta*y)-z*(-

Q*alfa*sin(alfa*x)*sin(beta*y)); eps_yy=-beta*B*sin(alfa*x)*sin(beta*y)-z*(-

beta*R*sin(alfa*x)*sin(beta*y)); eps_xy=0.5*(beta*A*cos(alfa*x)*cos(beta*y)+alfa*B*cos(alfa*x)*cos(bet

a*y))-

z*(beta*Q*cos(alfa*x)*cos(beta*y)+alfa*R*cos(alfa*x)*cos(beta*y));

figure plot(eps_xx,z) xlabel('\epsilon_x_x') ylabel('z')

figure plot(eps_yy,z) xlabel('\epsilon_y_y') ylabel('z')

figure plot(eps_xy,z) xlabel('\epsilon_x_y') ylabel('z')