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Homework%20problem%204-2%20solution

# Homework%20problem%204-2%20solution - Problem 4-2...

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Problem 4-2 1 AAE352-Problem set #4 Solution to Problem 2 The configuration shown has four rod elements connected to three node points. Node 1 is fixed (u 1 =0). Node moves only in the horizontal direction and has three of the four elements attached to it. All element s have the same Young’s modulus. Element 4 has one -half the cross-sectional area of the other elements. Node 2 has only elements 2 and 4 attached to it. a) If a 1000 pound load is placed at node 2 as shown, find the system stiffness matrix for the unrestrained system. This will be in terms of a parameter EA/L. b) Find the system stiffness matrix for the restrained system. c) Find the deflections u 2 and u 3 and the reaction at node 1. These will be in terms of a parameter L/EA. d) Find the internal forces due to the 1000 pound load. Both of the answers in (a) and (b) will have the parameters E, A and L in the answers. e) Repeat parts (a), (b) and (c) when the temperature of element 1 is increased by an amount T. The coefficient of thermal expansion is and is the same for all elements. First we identify and number the three nodal displacements and then break the structure into four component rods. We then identify the internal rod forces at each end of each rod:

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Problem 4-2 2 The nodal forces are related to the element internal forces by writing force equilibrium equations at each node point. 1 11 21 31 2 22 41 3 12 32 42 P F F F P F F P F F F In matrix form this equation set reads as follows: 1 11 21 31 2 22 41 3 12 32 42 0 0 0 0 P F F F P F F P F F F There are four 3x1 vectors on the right hand side of this equilibrium matrix because there are four rod elements involved. Each element has a stiffness matrix that relates the element forces to displacements. This is necessary in this problem because it is statically indeterminate. Notice also that we have not said anything yet about boundary conditions. We have applied nodal forces, the P’s, and have taken into account the possibility that any of the displacements, u, might be an unknown. The element stiffness matrices are: 1 1 1 11 1 3 3 3 12 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 u u u F EA EA EA u u u F L L L 21 1 1 2 22 2 2 2 1 1 1 1 1 1 1 1 F u u EA EA F u u L L 31 1 1 1 3 32 3 3 3 3 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 F u u u EA EA EA F u u u L L L
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