Need help understanding eigenvalues and eigenvectors

[Cisneros778]

Homework Statement

Find the principal stresses and the orientation for the principal axis of stress for the following cases of plane stress.

σ_x = 4,000 psi
σ_y = 0 psi
τ_xy = 8,000 psi

Homework Equations

See picture.

The Attempt at a Solution

https://mail.google.com/mail/u/0/?ui=2&ik=bc68d58ae7&view=att&th=139dbef260c42514&attid=0.1&disp=inline&realattid=f_h79p3pz70&safe=1&zw

I solved this problem using Mohr's Circle. However, the solution to the problem is different and I would like to understand it.
I do not know what the steps mean. Why does the determinate of that function must equal zero?
And what is the n1 and n2 about?
Finally, the 2x2 matrices adds two shear stresses where I was only given one τ_xy where does this other value come from?

 

[Cisneros778]

Any help please?

 

[cmmcnamara]

I'm not sure if anyone else is having trouble here, but your picture doesn't seem to be loading?

 

[Chestermiller]

Homework Statement

Find the principal stresses and the orientation for the principal axis of stress for the following cases of plane stress.

σ_x = 4,000 psi
σ_y = 0 psi
τ_xy = 8,000 psi

Homework Equations

See picture.

The Attempt at a Solution

https://mail.google.com/mail/u/0/?ui=2&ik=bc68d58ae7&view=att&th=139dbef260c42514&attid=0.1&disp=inline&realattid=f_h79p3pz70&safe=1&zw

I solved this problem using Mohr's Circle. However, the solution to the problem is different and I would like to understand it.
I do not know what the steps mean. Why does the determinate of that function must equal zero?
And what is the n1 and n2 about?
Finally, the 2x2 matrices adds two shear stresses where I was only given one τ_xy where does this other value come from?

In terms of components and unit vectors, the stress tensor can be written as a sum of terms (similar to a vector) as follows:

σ = σ_xx i_xi_x + τ_xy i_xi_y+ τ_yx i_yi_x+ σ_yy i_yi_y

Since the stress tensor is symmetric, τ_yx = τ_xy

Therefore, the stress tensor is given by:

σ = σ_xx i_xi_x + τ_xy i_xi_y+ τ_xy i_yi_x+ σ_yy i_yi_y

This is where the other τ_xy you were asking about comes from.


If n is a unit vector oriented in some arbitrary horizontal direction within your material, then, in terms of its components in the x and y directions, n can be written as the following sum of terms:

n = n_x i_x + n_y i_y

According to the so-called Cauchy stress relationship, the traction (force per unit area) acting on a plane perpendicular to the unit normal n is obtained by dotting the stress tensor σ with the unit normal n:

\Sigma=(σ_xx n_x + τ_xy n_y) i_x + (τ_xy n_x + σ_yy n_y) i_y

If n corresponds to one of the principal direction of stress, then the traction on the plane normal to n is perpendicular to the plane, and parallel to n:


\Sigma=(σ_xx n_x + τ_xy n_y) i_x + (τ_xy n_x + σ_yy n_y) i_y = λ (n_x i_x + n_y i_y)

where λ is the principal stress.

From the above equation, we get:

σ_xx n_x + τ_xy n_y = λ n_x

τ_xy n_x + σ_yy n_y) = λ n_y

This defines your eigenvalue problem. The components of the normal define the eigenvector, and the principal stress defines the eigenvalue.