1. Stress Matrix Visualization
Wednesday, 9/4/2002

2. Stress Vector

3. Proof

4. Normal Stress normal_stress = (sigma*n’)’*n’
Mathematical
Expression
If n is a row vector,
normal_stress = (sigma*n’)’*n’
If n is a column vector,
normal_stress = (sigma*n)’*n
MATLAB
Expression

5. Shear Stress t = (sigma*n’)’ shear_stress = t - (t*n’)*n t = sigma*n
Mathematical
Expression
If all vectors expressed as row vectors,
t = (sigma*n’)’
shear_stress = t - (t*n’)*n
If all vectors expressed as column vectors,
t = sigma*n
shear_stress = t - (t’*n)*n
MATLAB
Expression

6. Stress Visualization Method

7. Stress Visualization

8. Principal Direction
[Vectors, Principal] = eig (stress)

9. Non-symmetric Square Matrix

10. Eigenvector Demonstration

11. MATLAB code S=[1,2;2,1] n=50; hold on axis square
S0=sqrt(sum(sum(S.^2)));
axis([-S0*2,S0*2,-S0*2,S0*2])
theta = linspace( 0, pi*2, n );
plot(S0*cos(theta),S0*sin(theta),'r-');
for i=1:n
t = theta(i);
x0 = S0*cos(t);
y0 = S0*sin(t);
x1 = x0 + S(1,1)*cos(t) + S(1,2)*sin(t);
y1 = y0 + S(2,1)*cos(t) + S(2,2)*sin(t);
plot([x0,x1],[y0,y1])
end

12. Normal Stress Visualization Method

13. 3D Stress State

14. MATLAB functions axis xlabel, ylabel, zlabel hold Plot linspace
meshgrid
mesh