HW09_Solution - EML 4507 FEA & DESGN Fall 2010 HW09...

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EML 4507 FEA & DESGN Fall 2010 HW09 Solution 12. Model the beam shown in the figure using one two–node beam finite element. a) Using the beam element stiffness matrix, set up the equation for this beam ( [ ]{ } { } K Q F ) b) Compute the angle of rotation at node 1, and write the equation of deformed shape of the beam using the shape functions. c) Explain why the answer obtained above is not likely to be very accurate. If you want to obtain a better answer for this problem using the finite element method, what would you do? Solution: (a) Let F 1 be the reaction force at the left end, F 2 the reaction force at the right end, and M 1 the reaction moment at the right end. Then, the system of matrix equation becomes 11 1 22 12 6 12 6 1 / 2 6 4 6 2 1 / 12 0.15 12 6 12 6 1 / 2 6 2 6 4 1 / 12 vF M (b) Since v 1 = v 2 = 2 = 0, we can remove 1 st , 3 rd , and 4 th rows and columns, which yields only one scalar equation as 1 1 0.15 4 12 from which we can calculate the unknown rotation angle 1 = –0.1389 rad. Using the interpolation scheme, the deformed shape of the beam is 23 21 ( ) 0.1389 2 x x x v x N L L L L f 1 2 L = 1 m EI = 0.15 N m 2
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(c) The answer is not accurate because the exact vertical deformation is a fourth-order polynomial, whereas due to FE approximation v ( x ) is a cubic function. In order to improve the solution accuracy, we need to model the beam with more number of elements. 18. A linearly varying distributed load is applied to the beam finite element of length L . The maximum value of the load at the right side is q 0 . Calculate “work equivalent” nodal forces and couples. Solution: The distributed load is a linear function.
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This note was uploaded on 06/07/2011 for the course EML 4507 taught by Professor Kim during the Fall '11 term at University of Florida.

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HW09_Solution - EML 4507 FEA & DESGN Fall 2010 HW09...

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