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Consider the square finite element with nodes as shown. The proposed two dimensional shape functions will be based on the one

 

I need a solution for this homework questions please!

Consider the square finite element with nodes as shown. The proposed two dimensional shape functions will be based on the one 6) Determine if this element type is edge compatible with elements of dimensional shape functions: the same type. \[ \begin{array}{l} N_{1}(\xi)=1-3 \xi+2 \xi^{2} \\ N_{2}(\xi)=4 \xi-4 \xi^{2} \\ N_{3}(\xi)=-\xi+2 \xi^{2} \end{array} \] which have \[ \begin{array}{lll} N_{1}(0)=1 & N_{1}(0.5)=0 & N_{1}(1)=0 \\ N_{2}(0)=0 & N_{2}(0.5)=1 & N_{2}(1)=0 \\ N_{3}(0)=0 & N_{3}(0.5)=0 & N_{3}(1)=1 \end{array} \] 7) Determine if this element type is edge compatible with constant strain triangular elements. The two dimensional shape functions are given by \( \quad N_{i j}=N_{y}(\xi, \eta)=N_{i}(\xi) \cdot N_{j}(\eta) \) and the displacement functions are given by \[ \begin{array}{l} u=u(\xi, \eta)=N_{11} u_{1}+N_{12} u_{2}+N_{13} u_{3}+N_{21} u_{4}+N_{22} u_{5}+N_{23} u_{6}+N_{31} u_{7}+N_{32} u_{8}+N_{33} u_{9} \\ u=u(\xi, \eta)=u_{00}+u_{01} \eta+u_{02} \eta^{2}+u_{10} \xi+u_{11} \xi \eta+u_{12} \xi \eta^{2}+u_{20} \xi^{2}+u_{21} \xi^{2} \eta+u_{22} \xi^{2} \eta^{2} \\ v=v(\xi, \eta)=N_{11} v_{1}+N_{12} v_{2}+N_{13} v_{3}+N_{21} v_{4}+N_{22} v_{5}+N_{23} v_{6}+N_{31} v_{7}+N_{32} v_{8}+N_{33} v_{9} \\ v=v(\xi, \eta)=v_{00}+v_{01} \eta+v_{02} \eta^{2}+v_{10} \xi+v_{11} \xi \eta+v_{12} \xi \eta^{2}+v_{20} \xi^{2}+v_{21} \xi^{2} \eta+v_{22} \xi^{2} \eta^{2} \end{array} \] 8) Determine the equivalent nodal forces for a uniformly distributed surface force along one edge. 1) Determine if these displacement functions can model rigid body motion. 2) Determine if these displacement functions can model constant strain. 3) Determine the most complex forms (highest order polynomials) for stress that these functions can model. 4) Determine the size of the element stiffness matrix. 5) Without actually integrating anything, show how to determine the stiffness matrix of an element that is \( \mathrm{b} \) by \( \mathrm{h} \) instead of square.


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1) These displacement functions cannot model rigid body motion as they do not account for rotations. 2) These displacement functions can model constan
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