The roof covering a walkway in a northern city is slanted to allow melting snow to drain off. If the roof deflects too much under a heavy snow load, the water will not be able to run off, thus causing maintenance problems. We are tasked with determining the minimum moment of inertia for the steel support beam (E = 29,000 ksi) in order to prevent water from ponding on the roof. Because this is a serviceability issue rather than a strength issue, we use the maximum likely unfactored snow load. Since the slope of the beam is very small, —0.05, we can reasonably approximate the beam as horizontal for all of our calculations. Note that our answers will be in terms of the moment of inertia, I, since we have not yet chosen the beam cross-section.
a. Guess the displaced shape under uniform snow load.
b. Estimate the peak slope by considering only the span between the supports, AB. Will this provide an upper bound, a lower bound, or just an approximation for the actual peak vertical displacement? Justify your answer.
c. Derive the expressions for the shear and moment for the full beam under a uniform load along its entire length.
d. Derive the expressions for the slope and displacement for the full beam under a uniform load along its entire length.
e. Plot the expressions for the displacement to show the displaced shape.
f. Identify at least three features of the displaced shape that suggest you have a reasonable answer.
g. Determine the peak positive slope and its location.
h. Evaluate the slope prediction using your results from parts (b) and (f).
i. As long as the peak positive slope does not exceed the initial negative slope of the roof, —0.05, the roof will always have a downward slope and the water will drain. Determine the minimum moment of inertia required to ensure this.
J. Comment on why your guess in part (a) was or was not close to the solution in part (e).
