Simmons and Ruge’s bonded strain gauge (1938)

Stress cannot be measured, only strain. In 1936 Edward Simmons at Caltech, and in April 1938 Arthur Ruge at MIT, independently cemented fine resistance wire to a specimen, so that its resistance changed with the strain of the surface beneath; their names became the “SR-4” gauge. One gauge measures strain in one direction; a rosette of three at known angles gives the whole plane strain state (strain_rosette()), and Hooke’s law turns it into stress (plane_stress_from_strains()). The example simulates a rosette on a shaft under bending and torsion and recovers the torque from the three readings.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.structural_engineering import mohr_circle, plane_stress_from_strains, strain_rosette
from engineeringkit.structural_engineering.visualizers import plot_mohr_circle

Readings of a 0-45-90 rosette on a shaft surface

E, nu, G = 200e9, 0.3, 200e9 / 2.6
d, M, T = 0.05, 1.2e3, 2.0e3
sx, txy = 32 * M / (np.pi * d**3), 16 * T / (np.pi * d**3)
ex, ey, gxy = sx / E, -nu * sx / E, txy / G
angles = np.radians([0.0, 45.0, 90.0])
readings = ex * np.cos(angles) ** 2 + ey * np.sin(angles) ** 2 + gxy * np.sin(angles) * np.cos(angles)
rng = np.random.default_rng(1938)
noisy = readings + rng.normal(0, 2e-6, 3)  # a few microstrain of noise
print("gauge readings [microstrain]:", np.round(noisy * 1e6, 1))

gauge readings [microstrain]: [ 490.   702.8 -146.5]

Back to stress and torque

strains = strain_rosette(noisy, angles)
stress = plane_stress_from_strains(*strains, E, nu)
torque = stress[2] * np.pi * d**3 / 16
print(f"recovered: sigma_x {stress[0] / 1e6:.1f} MPa (true {sx / 1e6:.1f}), tau_xy {stress[2] / 1e6:.1f} MPa (true {txy / 1e6:.1f})")
print(f"torque {torque:.0f} N m (applied {T:.0f} N m)")
exact = plane_stress_from_strains(*strain_rosette(readings, angles), E, nu)
assert np.allclose(exact, (sx, 0.0, txy), atol=1e-3 * sx)
assert abs(torque - T) < 0.05 * T

_, radius, s1, s2 = mohr_circle(*stress)
print(f"principal stresses {s1 / 1e6:.1f} and {s2 / 1e6:.1f} MPa")
ax = plot_mohr_circle(*stress)
ax.set_title("Stress from a strain-gauge rosette")

plt.show()

recovered: sigma_x 98.0 MPa (true 97.8), tau_xy 81.7 MPa (true 81.5)
torque 2005 N m (applied 2000 N m)
principal stresses 144.3 and -46.2 MPa