Suspension Bridge

A flexible massless cable in a suspension bridge is subject to uniform loading along the x-axis. The weight of the load per unit length of the cable is ω, and the tension in the cable at the center of the bridge (at x = 0) is T₀ (see Figure 1.1).

Figure 1.1

a) Find the shape of the cable at equilibrium.

b) What is the tension T(x) in the cable at position x at equilibrium?

SOLUTION

a) We use an elementary method to solve this problem. The conditions for a static equilibrium are

(1)

(2)

(see Figure 1.2). (1) and (2) can be rewritten in the form

(3)

(4)

Figure 1.2

Integrating (3) and (4), we obtain

(5)

(6)

At x = 0, θ = 0, so C₀ = 0, C₁ = T₀ and dividing (5) by (6), we have

(7)

From (7) we find the shape of the suspension bridge, which is parabolic

(8)

b) To find the tension T(x) at x ≠ 0 ( θ ≠ 0) multiply (5) by (6).

(9)

(10)

So

(11)