HW7 - 3. For the two-dimensional case, Hooke's law for an...

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MCE 561 Computational Methods in Solid Mechanics Homework Assignment 7 Due April 4, 2011 1. For the linear triangular element shown, calculate the stiffness matrix for an orthotropic plane stress elasticity model. It is not necessary to formally integrate the expressions, so simply use the results shown on page 427 along with equation (11.4.6) in the text. Ans. ( 29 ( 29 . . . , 2 1 , 4 1 , 2 2 4 1 , 4 4 1 66 12 12 11 11 12 66 12 12 11 66 11 11 11 C K C K C C K C C K - = - = + = + = 2. Since rigid body motion (RBM) leads to zero strains thus implying zero stresses, the finite element formulation should produce zero nodal forces under such deformation. Using the standard element equation } ]{ [ } { u K F = with [ K ] from the previous problem, explicitly verify that this is indeed true. Note that RBM would correspond to a two-dimensional displacement field of the form ϖ - = y u u o and ϖ + = x v v o , where o o v u and are the translational components and ϖ is the rotational term (all constants). In order to simplify the problem, drop the rotation term ( 0 = ϖ ) and only consider the translational terms in your proof.
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Unformatted text preview: 3. For the two-dimensional case, Hooke's law for an orthotropic material including thermoelastic effects reads + = 2 66 22 12 12 11 T T e e e C C C C C y x xy y x xy y x where C ij are the usual elastic moduli, T is the temperature change and x and y are the material thermal expansion coefficients . Using the weak form/Ritz-Galerkin scheme, develop the finite element equation for this case. Your final result should look similar to what was developed in class (isothermal case), except that temperature terms will now appear in the boundary integral involving the traction components and in a new domain integral. (1,1) (3,1) (1,2) 1 2 3 x y...
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This note was uploaded on 10/03/2011 for the course MCE 561 taught by Professor Sadd during the Spring '11 term at Rhode Island.

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