The Golden Gate Bridge in San Francisco has a main span of length 1.28 km, one of the longest in the world. Imagine that a steel wire with this length and a cross-sectional area of 4.00 x 10-6 m^2 is laid on the bridge deck with its ends attached to the towers of the bridge, on a summer day when the temperature of the wire is 35.0°C.

Question

(a) When winter arrives, the towers stay the same distance apart and the bridge deck keeps the same shape as its expansion joints open. When the temperature drops to −10.0°C, what is the tension in the wire? Take Young's modulus for steel to be 2.00 10^11 N/m2. (The average linear expansion coefficient for steel is 1.1 10-5°C−1.)


(b) Permanent deformation occurs if the stress in the steel exceeds its elastic limit of 3.00 x 10^8 N/m2. At what temperature would the wire reach its elastic limit?

Answer 1

delta L/L = (1.1*10^-5)(45)
= 49.5 * 10^-5 meters/meter of hypothetical stretch

that delta L/L is strain
delta L/L = sigma/E which is stress/young's

stress = 4.95*10^-4 * 2*10^11
= 9.9 *10^7 N/m^2

tension = stress * area

=9.9*10^7 * 4*10^-6 = 396 N

b)
well we got 9.9*10^7 N/m^2 for a 45 degree drop
so to get 3 * 10^8 N/m^2
we need 45*30/9.9 = 136 degree drop
that is 35 - 136 = -101 deg C
(unlikely in San Francisco)