Stress Matrix Visualization

Wednesday, 9/4/2002

Stress Vector

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t=σ ⋅n

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σ xx σxy

σ yx σyy

⎡

⎣ ⎢ ⎤

⎦ ⎥ =1 2

2 1⎡

⎣ ⎢ ⎤

⎦ ⎥

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n=1/ 2

1/ 2

⎡

⎣ ⎢ ⎤

⎦ ⎥

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t=1

2

1 2

2 1⎡

⎣ ⎢ ⎤

⎦ ⎥ 1

1⎡

⎣ ⎢ ⎤

⎦ ⎥ =1

2

3

3⎡

⎣ ⎢ ⎤

⎦ ⎥

Proof

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t=t1t2

⎡

⎣ ⎢ ⎤

⎦ ⎥ =f1/s

f2 /s⎡

⎣ ⎢ ⎤

⎦ ⎥

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f1 =σ11s1+σ21s2

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f2 =σ12s1+σ22s2

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t=t1t2

⎡

⎣ ⎢ ⎤

⎦ ⎥ =σ11 σ12

σ 21 σ22

⎡

⎣ ⎢ ⎤

⎦ ⎥ s1/s

s2 /s⎡

⎣ ⎢ ⎤

⎦ ⎥

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n=s1/s

s2 /s⎡

⎣ ⎢ ⎤

⎦ ⎥

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t=t1t2

⎡

⎣ ⎢ ⎤

⎦ ⎥ =σ11 σ12

σ 21 σ22

⎡

⎣ ⎢ ⎤

⎦ ⎥ n1

n2

⎡

⎣ ⎢ ⎤

⎦ ⎥

Normal Stress

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(σ ⋅n)⋅n

If n is a row vector,normal_stress = (sigma*n’)’*n’

If n is a column vector,normal_stress = (sigma*n)’*n

MathematicalExpression

MATLABExpression

Shear Stress

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*nshear_stress = t - (t’*n)*n

MathematicalExpression

MATLABExpression

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t=σ⋅n

shear_stress = t−(t⋅n)n

Stress Visualization Method

Stress Visualization

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σ xx σxy

σ yx σyy

⎡

⎣ ⎢ ⎤

⎦ ⎥ =1 2

2 1⎡

⎣ ⎢ ⎤

⎦ ⎥

Principal Direction

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σ⋅n=σ nn

[Vectors, Principal] = eig (stress)

Non-symmetric Square Matrix

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A=1 2

−2 1⎡

⎣ ⎢ ⎤

⎦ ⎥

A⋅n

Eigenvector Demonstration

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

MATLAB codeS=[1,2;2,1]n=50;

hold onaxis squareS0=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

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σ11 σ12 σ13

σ21 σ 22 σ 23

σ31 σ 32 σ 33

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥ =

1 2 3

2 2 −1

3 −1 1

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

MATLAB functions

axisxlabel, ylabel, zlabel

hold

Plot

linspacemeshgrid

mesh