Stress Matrix Visualization Wednesday, 9/4/2002

Stress Vector

Proof

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

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

Stress Visualization Method

Stress Visualization

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

Non-symmetric Square Matrix

Eigenvector Demonstration http://www.math.ucla.edu/~baker/java/hoefer/Eigendemo.htm

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

Normal Stress Visualization Method

3D Stress State

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