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.
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
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
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
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!
Let us call this equation A
For i = 2, hence F2= M :B