Calculating the Forces on the Structural Members of a Bridge Design

(due date October 21)

Consider the structure shown below. Assume that the entire weight of the structure is resting at the two end points where the opposing forces have been labelled f1 and f2. Assume also that the rigid beams of the structure are made from a very light material so that their weight can be ignored. The angles indicated are all measured in degrees. Using the laws of statics, calculate the stress in each beam of the structure.

If you are not familiar with the laws of statics, here is all you need to know:

  1. The structure is obviously not moving, and neither are any of its joints or members; hence the term static. So the forces on each junction must add up to zero.
  2. Using trigonometry it is possible to decompose any force acting on any junction into the sum of forces in the x and y directions.
  3. If a rigid beam is being compressed, then it exerts a force equal to the compression outward along the direction of the beam.
  4. If a rigid beam is being pulled, then it exerts a force equal to the compression inward along the direction of the beam.
  5. Considering the forces acting on all the junctions yields a linear system of equations which can be solved simultaneously.

Be sure to note that you seem to have more equations than variables; this embarrassment of riches can cause problems.

Which of the beams experiences the greatest compression? How does shortening or lengthening this beam by a small amount change the stress which it undergoes, given the same design and weight load? Be careful here! Changing the length of one beam effect the lengths of some neighbouring beams as well. It is up to you exactly how to modify the design when changing the length of the beam.


Instructor

Mike Zabrocki
Email address: zabrocki@yorku.ca
Department of Mathematics and Statistics
York University
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