The catenary (1691)

Galileo thought a hanging chain was a parabola. In 1690 Jacob Bernoulli challenged the readers of Acta Eruditorum to find its true shape, and the next year three solutions appeared there, by his brother Johann, by Leibniz and by Huygens: the curve \(y = a(\cosh(x/a) - 1)\), with \(a = H/w\) the ratio of the horizontal tension to the weight per unit length (catenary_cable()). A cable carrying a deck uniform along the span instead hangs in a parabola (parabolic_cable()), which is why suspension bridges are designed with parabolic cables. The example compares the two shapes and shows where the parabola is a good enough approximation for a cable under its own weight.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.structural_engineering import catenary_cable, parabolic_cable

A power line between two towers

w, L, f = 15.0, 300.0, 9.0  # conductor weight [N/m], span and sag [m]
H, T, length = catenary_cable(w, L, f)
Hp, Tp, length_p = parabolic_cable(w, L, f)
a = H / w
print(f"catenary: H = {H / 1e3:.2f} kN, T_max = {T / 1e3:.2f} kN, length {length:.3f} m")
print(f"parabola: H = {Hp / 1e3:.2f} kN, T_max = {Tp / 1e3:.2f} kN, length {length_p:.3f} m")
assert np.isclose(a * (np.cosh(L / (2 * a)) - 1), f)
assert np.isclose(T, H + w * f)

catenary: H = 18.77 kN, T_max = 18.91 kN, length 300.719 m
parabola: H = 18.75 kN, T_max = 18.88 kN, length 300.718 m

Deep cables: the shapes part company

ratios = np.linspace(0.02, 1.0, 60)
difference = [catenary_cable(w, L, k * L)[0] / parabolic_cable(w, L, k * L)[0] - 1 for k in ratios]
fig, axes = plt.subplots(1, 2, figsize=(10, 4))
x = np.linspace(-L / 2, L / 2, 201)
for k in (0.1, 0.5):
    Hk = catenary_cable(w, L, k * L)[0]
    ak = Hk / w
    axes[0].plot(x, ak * (np.cosh(x / ak) - np.cosh(L / (2 * ak))), label=f"catenary, sag/span {k}")
    axes[0].plot(x, k * L * (4 * x**2 / L**2 - 1), "--", label=f"parabola, sag/span {k}")
axes[0].set(xlabel="x [m]", ylabel="y [m]", title="Same span and sag")
axes[0].set_aspect("equal")
axes[0].legend(fontsize="small")
axes[1].plot(ratios, 100 * np.array(difference))
axes[1].set(xlabel="sag / span", ylabel="catenary H vs parabola [%]")
fig.tight_layout()
print(f"at sag/span 0.1 the horizontal tensions differ by {100 * difference[np.argmin(np.abs(ratios - 0.1))]:.1f}%")
assert abs(difference[0]) < 0.01 * difference[-1]

plt.show()

at sag/span 0.1 the horizontal tensions differ by 1.4%