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:

[itex]\Sigma[/itex]=(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:


[itex]\Sigma[/itex]=(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.

 

[rezeew]

Hi there,

Eigenvalues and eigenvectors are important concepts in linear algebra and are commonly used in engineering and physics to solve problems involving matrices and vectors. In the context of stress analysis, eigenvalues and eigenvectors can help us determine the principal stresses and their orientations, which are critical in understanding the behavior of structures under different loading conditions.

In the problem given, we are asked to find the principal stresses and their orientations for a plane stress state with known values of σx, σy, and τxy. To solve this, we can use the formula for principal stresses and principal axes:

σ1,2 = (σx + σy)/2 ± √[(σx - σy)/2]^2 + τxy^2

θ1,2 = 1/2 tan^-1(2τxy/(σx - σy))

In order to understand this solution, it is important to first understand the concept of eigenvalues and eigenvectors. An eigenvalue is a scalar value that represents the amount of stretch or compression in a particular direction, while an eigenvector is a vector that represents the direction of this stretch or compression. In the context of stress analysis, the eigenvalues represent the principal stresses, and the eigenvectors represent the principal axes of stress.

In this problem, we are given a 2x2 stress matrix, which can be represented as a linear transformation that maps a vector in the original stress state to a vector in the principal stress state. This transformation can be represented by a characteristic polynomial, which has the form:

det(A - λI) = 0

where A is the stress matrix, λ is the eigenvalue, and I is the identity matrix. Solving this equation for λ will give us the eigenvalues, which in turn can be used to find the principal stresses.

The n1 and n2 values in the solution refer to the eigenvectors corresponding to the two eigenvalues, which represent the principal axes of stress. These can be found by solving the system of equations:

(A - λI)x = 0

where x is the eigenvector. The two eigenvectors correspond to the two principal axes, and their orientations can be determined using the formula given above.

Finally, the 2x2 matrices in the solution represent the transformation matrix from the original stress state to the principal stress state. This transformation matrix is necessary because we are given only one shear stress (τxy