1) A differential equation with boundary conditions is given below: d² u du X3 + X +2=0 1 < x < 4 dx² dx u(1) = 1 and u(4) = 0 (a) Derive the strong and weak formulations (b) Discretize the domain in to three linear finite elements (c) Compute element matrices and vectors for this mesh discretization using linear piecewise shape functions. (do this only for weak formulation) (d) Assemble them into the global matrix and vectors (using direct approach) (e) Apply the boundary conditions to the matrix equation. (f) Solve the unknown nodal values Homework due (May 18) 1) A differential equation with boundary conditions is given below: ” w x3 da² du + X +2 0 1<< 4 dz u(1) and (4)=0 (a) Derive the strong and weak formulations (b) Discretize the domain in to three linear finite elements (c) Compute element matrices and vectors for this mesh discretization using linear piecewise shape functions. (do this only for weak formulation) (d) Assemble them into the global matrix and vectors (using direct approach) (e) Apply the boundary conditions to the matrix equation. (f) Solve the unknown nodal values 2) Generate a close-ended thin-walled tube with the following geometrical properties. (R. and Ro are inner and outer radius respectively) ⚫ Compute the Normal stress distribution in 2 direction and Von-Mises stress distribution. Calculate the von Mises stress in theory using these formulas for thin-walled elastic tube: Circumferential stress PR " t Axial stress: PR 21 " Compare and validate your finite element calculations with theory (analytical calculations as discussed in the class. above). Are there any deviations? Why? Prove that your results are correct. (It is your responsibility to prove and justify your computations if you want to get full grade from this assignment. Note also that Ansys tutorial will be useful for this problem) Dimensions Length 200mm Thickness 1 mm Ri 20mm Ro 21mm Define a new material in the Engineering Data Toolbox and assign it to your model. Linear Elastic Perform your analysis in Static Structural Toolbox using Ansys. ⚫ Choose appropriate mesh type and size ⚫ Check mesh quality (skewness and element quality) ⚫ Boundary conditions: let intramural pressure in the tube to be 1 KRa. Solutions: • Compute total deformation in tube Material E 10^6 V 0,49