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No other use or distribution of this Manual is permitted. This Manual may not be sold and may not be distributed to or used by any student or other third party. No part of this Manual may be reproduced, displayed or distributed in any form or by any means, electronic or otherwise, without the prior written permission of the McGraw-Hill. 4, carry out the following tasks: (a) Develop the weak forms of the given differential equation(s) over a typical finite element, which is a geometric subdomain located between x = xa and x = xb .

5 Solution: The assembled system of equations for the pipe network are given by ⎡ ⎤ ⎧ ⎫ 1 1 1 1 ( 2a + 6a ) − 2a 0 − 6a P1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ −1 ⎥ 1 1 1 1 1 ( 2a + 3a + 2a ) −( 3a + 2a ) 0 ⎢ ⎥ P2 2a ⎢ ⎥ 1 1 1 1 1 1 P ⎪ ⎣ ⎦⎪ 0 −( 3a + 2a ) ( 3a + 2a + 2a ) − 2a ⎪ ⎩ 3⎪ ⎭ 1 1 1 1 P4 − 6a 0 − 2a ( 2a + 6a ) ⎧ ⎪ ⎪ ⎨ ⎫ Q11 + Q51 ⎪ ⎪ 1 Q2 + Q21 + Q31 ⎬ = ⎪ Q2 + Q3 + Q4 ⎪ ⎪ ⎩ 2 4 2 5 1⎪ ⎭ Q2 + Q2 The boundary conditions are: Q11 + Q51 = Q , P4 = P , and equilibrium requires that the sums of Q’s be zero: Q12 + Q21 + Q31 = 0, Q22 + Q32 + Q41 = 0 The condensed equations are obtained by condensing variable P4 out: ⎤⎧ ⎡ ⎫ ⎧ ⎫ ⎧ ⎫ 1 4 −3 0 ⎨ P1 ⎬ ⎨ Q ⎬ ⎨ 6a ·P ⎬ 1 ⎣ ⎦ −3 8 −5 P2 = 0 + 0 · P ⎩ ⎭ ⎩ ⎭ ⎩ 1 ⎭ 6a 0 −5 8 0 P3 2a · P where P = 0.

9: (Axial deformation of a bar) The governing differential equation is of the form (E and A are constant): − ∙ ¸ d du EA = 0, dx dx 0

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