Mohr’s circle (1882)

Otto Mohr (1882) showed that the normal and shear stresses on every plane through a point, plotted against each other, fall on a single circle. Its centre is the mean normal stress, its radius the largest in-plane shear stress, and its ends on the axis are the principal stresses (mohr_circle()). Turning the element by \(\theta\) moves the point by \(2\theta\) around the circle (stress_transformation()). The same paper drew three circles for a three-dimensional state, through each pair of principal stresses (principal_stresses_3d()). The example traces the circle by rotating an element, checks that the principal stresses are the eigenvalues of the stress tensor, finds the plane of maximum shear, and shows that in a pressure vessel the largest shear stress acts out of the plane of the wall.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.structural_engineering import mohr_circle, principal_stresses_3d, stress_transformation, thin_wall_vessel_stresses
from engineeringkit.structural_engineering.visualizers import plot_mohr_circle

A shaft under bending and torsion

At the surface of a shaft, bending gives \(\sigma_x\) and torsion gives \(\tau_{xy}\); there is no stress across the shaft.

sx, sy, txy = 120e6, 0.0, 50e6
center, radius, s1, s2 = mohr_circle(sx, sy, txy)
print(f"centre {center / 1e6:.1f} MPa, radius {radius / 1e6:.1f} MPa, principal {s1 / 1e6:.1f} and {s2 / 1e6:.1f} MPa")
eigen = np.linalg.eigvalsh([[sx, txy], [txy, sy]])
assert np.allclose([s2, s1], eigen)

theta = np.linspace(0, np.pi, 181)
sxp, syp, txyp = stress_transformation(sx, sy, txy, theta)
assert np.allclose(np.hypot(sxp - center, txyp), radius)
assert np.allclose(sxp + syp, sx + sy)  # the first invariant

centre 60.0 MPa, radius 78.1 MPa, principal 138.1 and -18.1 MPa

Principal planes and the plane of maximum shear

theta_p = 0.5 * np.arctan2(2 * txy, sx - sy)
on_principal = stress_transformation(sx, sy, txy, theta_p)
on_shear = stress_transformation(sx, sy, txy, theta_p + np.pi / 4)
print(f"principal plane at {np.degrees(theta_p):.1f} deg: shear {on_principal[2]:.1e} Pa")
print(f"max-shear plane at {np.degrees(theta_p) + 45:.1f} deg: shear {on_shear[2] / 1e6:.1f} MPa")
assert np.isclose(on_principal[0], s1) and abs(on_principal[2]) < 1e-6 * radius
assert np.isclose(abs(on_shear[2]), radius) and np.isclose(on_shear[0], center)

fig, axes = plt.subplots(1, 2, figsize=(10, 4))
plot_mohr_circle(sx, sy, txy, ax=axes[0])
axes[0].plot(sxp[::15], txyp[::15], ".", color="k", label="rotated element")
axes[0].set_title("Mohr's circle")
axes[1].plot(np.degrees(theta), sxp / 1e6, label=r"$\sigma_{x'}$")
axes[1].plot(np.degrees(theta), txyp / 1e6, label=r"$\tau_{x'y'}$")
axes[1].axvline(np.degrees(theta_p), color="gray", ls="--")
axes[1].set(xlabel="element rotation [deg]", ylabel="stress [MPa]", title="Stress transformation")
axes[1].legend()
fig.tight_layout()

principal plane at 19.9 deg: shear 0.0e+00 Pa
max-shear plane at 64.9 deg: shear -78.1 MPa

Three circles: a pressurised tank

The wall of a cylindrical tank carries hoop and longitudinal stresses and no in-plane shear; the radial stress is nearly zero. Both in-plane principal stresses are tensile, so the in-plane circle is small, and the largest shear acts on planes at 45 degrees through the thickness.

hoop, longitudinal = thin_wall_vessel_stresses(2e6, 0.75, 0.01)
p1, p2, p3 = principal_stresses_3d(hoop, longitudinal, 0.0)
in_plane = mohr_circle(hoop, longitudinal, 0.0)[1]
absolute = (p1 - p3) / 2
print(f"principal {p1 / 1e6:.0f}, {p2 / 1e6:.0f}, {p3 / 1e6:.0f} MPa")
print(f"in-plane max shear {in_plane / 1e6:.1f} MPa, absolute {absolute / 1e6:.1f} MPa")
assert np.isclose(absolute, hoop / 2) and absolute > in_plane

fig, ax = plt.subplots(figsize=(6, 3.5))
phi = np.linspace(0, np.pi, 181)
for a, b in ((p1, p2), (p2, p3), (p1, p3)):
    ax.plot((a + b) / 2 + (a - b) / 2 * np.cos(phi), (a - b) / 2 * np.sin(phi), color="C0")
ax.plot([p1, p2, p3], [0, 0, 0], "s", color="C3")
ax.set(xlabel="normal stress [Pa]", ylabel="shear stress [Pa]", title="Mohr's three circles for a tank wall")
ax.set_aspect("equal")
fig.tight_layout()

plt.show()

principal 150, 75, 0 MPa
in-plane max shear 37.5 MPa, absolute 75.0 MPa