Principal stress calculator

Plane stress

The stress tensor is written as a 2 by 2 matrix with units: S = [sigma_x, tau_xy; tau_xy, sigma_y]. The function eigenvals(S) returns the principal stresses in ascending order, so the major one is the second. eigenvecs(S) returns unit vectors along the principal directions, one per row in the same order. For sigma_x = 80 MPa, sigma_y = -40 MPa and tau_xy = 25 MPa, the engine gives:

The Mohr's circle check

The same results by the classical construction: the circle's centre is the mean normal stress (sigma_x + sigma_y) / 2, its radius is R = sqrt(((sigma_x - sigma_y) / 2)^2 + tau_xy^2), the principal stresses are the centre plus and minus the radius, and the angle is atan(2 tau_xy / (sigma_x - sigma_y)) / 2.

The eigenvalue and the circle agree to the decimals shown, and the maximum shear stress is the radius of the circle.

A 3D stress state

For a general stress state the tensor is 3 by 3, and the same two calls work. With T = [60, 20, -10; 20, -30, 15; -10, 15, 40] MPa the three principal stresses are 66.15 MPa, 41.72 MPa and -37.87 MPa. Mohr's circle has no simple 3D form, so the check here is an invariant: the trace of the tensor, 70 MPa, equals the sum of the principal stresses, 70 MPa, as it must for any rotation of the axes.

Why eigenvalues give principal stresses

Rotating the axes changes the components of the stress tensor but not the state of stress. In the principal axes the tensor is diagonal: no shear, only normal stresses. Finding those axes is the eigenvalue problem of the matrix. Because the stress tensor is symmetric, its eigenvalues are real and its eigenvectors are at right angles to each other, which is why the principal planes are always perpendicular.

The sheet source

The sample as it ships (this page adds three lines that pick out the 3D principal stresses):

# Principal stresses

The stress state at a point is a symmetric tensor. Its eigenvalues are the principal
stresses and its eigenvectors are the principal directions, so `eigenvals` and `eigenvecs`
give both in one call each. Mohr's circle checks the result.

## Plane stress

sigma_x: 80 MPa
sigma_y: -40 MPa
tau_xy: 25 MPa

S: [sigma_x, tau_xy; tau_xy, sigma_y]

The eigenvalues come in ascending order, so the major principal stress is the last one.

sigma_p: eigenvals(S)
sigma_1: sigma_p.2
sigma_2: sigma_p.1
tau_max: (sigma_1 - sigma_2)/2

Each row of `eigenvecs` is a unit vector along one principal direction, in the same order.
The angle of the major direction from the x axis (a direction, so the sign of the vector does not matter):

V: eigenvecs(S)
theta_p: atan(V.(2, 2)/V.(2, 1))

### Check with Mohr's circle

sigma_avg: (sigma_x + sigma_y)/2
R: sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2) -> MPa
sigma_1c: sigma_avg + R
sigma_2c: sigma_avg - R
theta_pc: atan(2*tau_xy/(sigma_x - sigma_y))/2

## A general 3D stress state

T: [60 MPa, 20 MPa, -10 MPa; 20 MPa, -30 MPa, 15 MPa; -10 MPa, 15 MPa, 40 MPa]
sigma_3D: eigenvals(T)
N: eigenvecs(T)

The sum of the principal stresses equals the first invariant, the trace of the tensor:

I_1: trace(T)
I_1c: sum(sigma_3D)

s_a: sigma_3D.1 | smallest
s_b: sigma_3D.2 | middle
s_c: sigma_3D.3 | largest