Analyse the bar using the galerkin finite element method

Part 1 – Civil Engineering Numerical Analysis
H(m) = 0.3(m).

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1. Treating the bar illustrated above as being a one dimensional axial bar where plane sections remain plane and parallel, analyse the bar using the Galerkin finite element method. The coding solution for linear h-refinement and its manual solution for a limited number of linear elements areon Blackboard for your reference.

Consider h-refinement using three-nodedquadratic elements, produce the manual solution for when the domain includes of 2 equal spacing quadratic elements. Generate the outcomes to produce:

a. Nodal displacements

b. Displacement at the left quarter of each element.

c. Normal stress at the left quarter of each element.

2. Treating the bar illustrated above as being a two dimensional plane stress domain, analyse using the Galerkin finite element method. The distributed axial force is spread uniformly across the width of the bar and on the bar edge. Due to symmetry, the analysis is performed for the top half of the bar. The other half is the mirror image about the indicated axis of symmetry.

Use 4-noded quadrilateral elements,produce the manual solution for when the domain which describes the top half of the bar, includes of 2 equal spacing quadrilateral elements. Generate the outcomes to produce:

a. Nodal displacements.
b. Nodal stresses.

3. Extend the two-noded linearMatlab program provided to produce the corresponding solution for three-noded quadratic elements. You need to amend all sub function m-files, but up to calculation of nodal displacement vector only for the main m-file. You do not need to work on stress recovery, convergencerate and error estimation in the main m-file.

Hints:

1. Time management: Questions 1 and 2 focus on detailed calculation for FEM understanding. Question 3 practises you lightly for programming FEM using Matlab. With multiple members in group, you can consider to start on all questions at once and do them concurrently. All 3 questions are independent from each other. Assign the work share efficiently among your group to take good use of the number of members.

2. Approach and result confirmation: Question 1 is an upgrade of the solution that we fully detailed for linear elements in class,you are advised to follow closely solution steps for linear elements to generate the solution for quadratic elements. For Question 2, the solution derivation in Matlab coding is already on Blackboard. Have a good reference to this coding solution to generate the pairing manual calculation. For assurance of solution accuracy, you should verify your manual calculation with the corresponding program outputs.

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