Prog1_Solution - AE420/ME471 First programming assignment...

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AE420/ME471 – First programming assignment Solution Topic: Finite element solution of 1D axi-symmetric problems Consider the simple structural problem of an axi-symmetric membrane (under constant tension T ) on an elastic foundation (with distributed stiffness k s ). The annulus-shaped membrane has an inner radius a and an outer radius b , and is subjected to a distributed transverse load Q r ( ) . The transverse deflection w r ( ) for the membrane is described by the following equilibrium equation T d 2 w dr 2 + 1 r dw dr k s w = Q for a r b . The boundary conditions involve imposed displacement values along both inner and outer radii w r = a ( ) = w a and w r = b ( ) = w b . a) Using the GWRM, derive the finite element formulation (i.e., the global stiffness matrix and global load vector) for this problem, assuming that the 1D domain a r b is discretized by N equal-size 2-node elements with node 1 at r=a and node N+1 at r=b . (Recall that, for axi-symmetric problems, the elemental area is 2 π rdr ) .
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b) Using the results found in a), derive the expression of the local stiffness matrix and local load vector (assuming that Q is constant over the element) for a generic 2-node element located between r = r 1 and r = r 2 = r 1 + h .
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c) If the expression of the potential energy for this problem is Π = 1 2 T dw dr 2 + k s w ( ) 2 r dr a b Qwr dr a b , verify the expression of the stiffness matrix and vector found in a) and b).
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d) Modify the 1D FE code provided in the notes to solve axi-symmetric membrane problems, with the value of N entered interactively at the beginning of the code and the other parameters ( a , b , T , Q , w a , w b and k s ) hardcoded in the code. Use the partitioning method to impose the essential boundary conditions. Solve the problem with a = 0.1 m , b = 0.5 m , T = 10 N / m , Q = 3 Pa , k s = 85 Pa / m , w a = 0.01 m and w b = 0.02 m . Plot (on the same graph) your numerical solution for N =1, 2, 3 and 4. Then use your code to investigate the effect of the foundation stiffness k s on the deflection of the membrane by plotting (on a single graph) the deflection w r ( ) obtained with N =20 for k s = 0,50,500 and 5000 (the other parameters remaining unchanged). Comment on your solution. Does
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Prog1_Solution - AE420/ME471 First programming assignment...

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