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Tutorial 4 Force Method II

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Civil Engineering
ENCI 451

ENCI 551 Tutorial 4 Solution October 13, 2012 Question 1. For the beam shown, determine the bending moments at A, E, B, F, and C for: A) the loads shown, B) a settlement at B of b/100. Solution: First, let us find the degree of static indeterminacy, i. Because i is very small (i.e. typically i < 4), we can apply the force method. Part A) Loads shown above Step 1: Find i, and make the structure determinate Two extra unknowns/redundant/releases are needed to be eliminated so the structure becomes determinate. In this case, we will convert A into a pin support and hinge B: Step 2: Apply the given external loads and find the displacements at the releases, D1and D 2 ENCI 551 Tutorial 4 Solution October 13, 2012 From equation B.3 in Appendix B of the textbook: Step 3: Find the flexibility coefficients by applying a unit load/moment at the releases F1= 1: Using equations B.9 and B.10 in Appendix B of the textbook: Similarly with F2= 1: ENCI 551 Tutorial 4 Solution October 13, 2012 So the flexibility matrix is: Step 4: Solve for the unknown release forces using displacement compatibility Because this is a continuous beam with three or more supports, the trick here to solve for F 1 and F using displacement compatibility is using the three moment equation in its general 2 form (Equation 4.15, pg. 118, Ghali, Neville and Brown, 2009): Where i represents each of the i force releases from Step 2, hence in this problem, i = 1, 2. For i = 1, hence F1= M :A Note: f01nd F D0 NOT EXIST! So, Let us call this equation A For i = 2, hence F2= M :B 2
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