Finite elements for 2‐d problems
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This document discusses finite element analysis for 2D problems. It describes common 2D elements like linear triangles, quadratic triangles, linear quadrilaterals, and quadratic quadrilaterals. It explains how shape functions are used to derive element stiffness matrices and calculate displacements, strains, and stresses within each element from the nodal values. Key elements covered include the constant strain triangle (CST), linear strain triangle (LST), linear quadrilateral (Q4), and quadratic quadrilateral (Q8). The document also discusses calculating von Mises stresses and using contour plots to visualize stress results.
Finite elements for 2‐d problems
- 1. Finite Elements for 2‐D Problems General Formula for the Stiffness Matrix f ff Displacements (u, v) in a plane element are interpolated from nodal displacements (ui, vi) using shape functions Ni as follows, where N is the shape function matrix, u the displacement vector and d the nodal displacement vector. Here we have assumed that u depends on the nodal values of u only, and v on nodal values of v only. Most commonly employed 2‐D elements are linear or quadratic triangles and quadrilaterals.
- 2. Constant Strain Triangle (CST or T3) This is the simplest 2 D element, which is also called linear triangular element. This is the simplest 2‐D element which is also called linear triangular element For this element, we have three nodes at the vertices of the triangle, which are numbered around the element in the counterclockwise direction. Each node has two degrees of freedom (can move in the x and y directions). The displacements u and v are assumed to be linear functions within the element, that is, where bi (i = 1, 2, ..., 6) are constants. From these, the strains are found to be, which are constant throughout the element.
- 4. The shape functions (linear functions in x and y) are and d is the area of the triangle. is the area of the triangle.
- 5. The strain‐displacement relations are written as where xij = xi ‐ xj and yij = yi ‐ yj (i, j = 1, 2, 3). Again, we see constant strains within the element. From stress‐strain relation, we see that stresses obtained using the CST element are also constant. The element stiffness matrix for the CST element, in which t is the thickness of the element. Notice that k for CST is a 6 by 6 symmetric matrix. i i
- 7. The plot for shape function N1 is shown in the following figure. N2 and N3 have similar features. We have two coordinate systems for the element: the global coordinates (x, y) and the natural coordinates . The relation between the two is given by where xij = xi ‐ xj and yij = yi ‐ yj (i, j = 1, 2, 3) as defined earlier.
- 8. Displacement u or v on the element can be viewed as functions of (x, y) or . Using the chain rule for derivatives, we have, g where J is called the Jacobian matrix of the transformation, and is expressed as h Ji ll d h J bi i f h f i di d where and A is the area of the triangular element.
- 15. In the natural coordinate system the eight shape functions are, Again, we have at any point inside the element. Again we have at any point inside the element The displacement field is given by which are quadratic functions over the element. Strains and stresses over a quadratic quadrilateral element are quadratic functions, which are better representations.
- 17. Stress Calculation The stress in an element is determined by the following relation, y g where B is the strain‐nodal displacement matrix and d is the nodal displacement vector which is known for each element once the global FE equation has been solved. Stresses can be evaluated at any point inside the element (such as the center) or at the nodes. Contour plots are usually used in FEA software packages (during post‐ p y p g ( gp process) for users to visually inspect the stress results. The von Mises Stress: The von Mises stress is the effective or equivalent stress for 2‐D and 3‐D stress The von Mises stress is the effective or equivalent stress for 2 D and 3 D stress analysis. in which and are the three principle stresses at the considered point in a structure.