Macaulay’s singularity functions (1919)

Integrating \(EIy'' = M\) for a beam with several loads used to mean a separate moment equation, and two constants of integration, for every stretch between loads, all joined by continuity conditions. William Herrick Macaulay (1919), following Clebsch (1862), wrote the moment once for the whole beam with brackets \(\langle x - a\rangle^n\) that vanish to the left of \(a\), so only two constants remain. Beam adds the unknown reactions to those two constants and solves equilibrium and the support conditions together, so statically indeterminate beams need no special treatment. The example solves a propped cantilever under a point load, a couple and a triangular load, and checks it against the older method of superposition (superpose_beam_results()).

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.structural_engineering import Beam, superpose_beam_results

A propped cantilever under three kinds of load

L, E, I = 8.0, 200e9, 1.2e-4
loads = {"point_loads": [(3.0, 40e3)], "moments": [(6.0, 30e3)], "distributed_loads": [(4.0, 8.0, 0.0, 15e3)]}
propped = Beam(L, E, I, ((0.0, "fixed"), (L, "roller")))
result = propped.solve(**loads, n=801)
(RA, RB), (MA,) = result.reactions, result.support_moments
print(f"reactions {RA / 1e3:.2f} kN and {RB / 1e3:.2f} kN, wall moment {MA / 1e3:.2f} kN m")
print(f"largest deflection {result.deflection.min() * 1e3:.2f} mm at x = {result.x[np.argmin(result.deflection)]:.2f} m")
assert np.isclose(RA + RB, 40e3 + 15e3 * 4 / 2)
assert abs(result.deflection[0]) < 1e-15 and abs(result.slope[0]) < 1e-15 and abs(result.deflection[-1]) < 1e-15

reactions 34.66 kN and 35.34 kN, wall moment -67.25 kN m
largest deflection -7.39 mm at x = 4.39 m

The same beam by superposition

Remove the roller: the cantilever under the loads deflects at its tip, and the redundant reaction is the tip force that pushes it back.

cantilever = Beam(L, E, I, ((0.0, "fixed"),))
free = cantilever.solve(**loads, n=801)
unit = cantilever.solve(point_loads=[(L, -1.0)], n=801)  # an upward unit force at the tip
R = -free.deflection[-1] / unit.deflection[-1]
combined = superpose_beam_results(free, cantilever.solve(point_loads=[(L, -R)], n=801))
print(f"redundant by superposition: {R / 1e3:.2f} kN")
assert np.isclose(R, RB)
assert np.allclose(combined.deflection, result.deflection, atol=1e-12)

fig, axes = result.plot()
axes[0].set_ylabel("shear [N]")
axes[1].set_ylabel("moment [N m]")
axes[2].set_ylabel("deflection [m]")
axes[0].set_title("Propped cantilever by singularity functions")
fig.tight_layout()

plt.show()

redundant by superposition: 35.34 kN