I A mechanical part of square section of dimension a and of thickness e is subjected...

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I A mechanical part of square section of dimension a and of thickness e is subjected to the loading described in the following figure: 3 2 X The material constituting the part is a linear elastic of Young's modulus E and poisson coefficient V. The effect of gravity will be neglected. Part I: Variational Formulation: 1. Knowing that the thickness e of the part is very small compared to its side a (e << a), what hypothesis can be adopted to reduce the problem to a two-dimensional one? Justify it. 2. Using this hypothesis, write the equilibrium, the boundary conditions and the behavior law. 3. Write the integral form of Galerkin type associated with this problem 4. Give, from this integral form, the weak integral form and simplify it by introducing the boundary conditions Part II: Finite Element modelization : The plate is discretized with a single quadrangular element at 4 nodes 1. Discretize the weak integral form 2. Deduce (without developing the calculations) the stiffness matrix and the vector of external forces. 3. What is the size of the final system to be solved after the introduction of boundary conditions? 2 99 N nt g: I A mechanical part of square section of dimension a and of thickness e is subjected to the loading described in the following figure: 3 2 X The material constituting the part is a linear elastic of Young's modulus E and poisson coefficient V. The effect of gravity will be neglected. Part I: Variational Formulation: 1. Knowing that the thickness e of the part is very small compared to its side a (e << a), what hypothesis can be adopted to reduce the problem to a two-dimensional one? Justify it. 2. Using this hypothesis, write the equilibrium, the boundary conditions and the behavior law. 3. Write the integral form of Galerkin type associated with this problem 4. Give, from this integral form, the weak integral form and simplify it by introducing the boundary conditions Part II: Finite Element modelization : The plate is discretized with a single quadrangular element at 4 nodes 1. Discretize the weak integral form 2. Deduce (without developing the calculations) the stiffness matrix and the vector of external forces. 3. What is the size of the final system to be solved after the introduction of boundary conditions? 2 99 N nt g:

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