2. 1 HOW MANY ELEMENTS? How to choose element size? –Critically important in obtaining good results –Mesh refinement improves solution accuracy. –How small is good enough? No. of elements Stress or displacement Exact value Acceptable mesh size Need mesh refinement

3. 2 HOW MANY ELEMENTS cont. Convergence rate Richardson extrapolation –Calculate the function of interest at three different meshes –Let h 1, h 2, and h 3 be the sizes of elements, ordered by h 1 > h 2 > h 3 –Usually h 3 =ph 2 =p 2 h 1 –Then –Convergence rate  : indicates how fast the solution will converge

4. 3 EXAMPLE Calculating the maximum stress in an FE model, we used a 2x2, 4x4, 8x8, and 16x16 elements. The results are given in the table. Estimate the true maximum stress. For convenience you can assume that the largest h value is 1. Using first three results get Using the last three get To compare g’s need to multiply second one by 2 1.593 =-7.762 Mesh sizeMax stress 3x271.53 6x475.63 12x877.35 24x1677.92

5. 4 BOUNDARY CONDITIONS. Applying displacement boundary conditions –FE model should be properly restrained so that it is not free to move in any direction even if there are no applied forces in that direction –Errors in BC will not disappear no matter how much you refine the model –Any unexplained high stress may be due to a wrong boundary condition Fix center node Rigid-bar elements Plate Fix all nodes Plate Not allowed to translate/rotateNot allowed to translate

6. 5 BOUNDARY CONDITIONS cont. Example of error in BC 1 2 3 2 L x y 1,000 N L L L 11 2 3 3 1 2 3 (a) Improper case(b) Proper case

7. 6 APPLIED LOADS Applying external forces –Forces are applied through a complex mechanism –It is often simplified when the interest region is far from the load application location –FE results near the load application location are not accurate due to approximation involved in the force Applying a concentrated force –Theoretically infinite stress (zero area) –Practically, all forces are distributed in a region –Concentrated force in FE is an idealization of distributed forces in a small region

8. 7 APPLIED LOADS cont. Note that the distributed forces are converted to the equivalent nodal forces. All applied forces must be converted to the equivalent nodal forces because the RHS of finite element matrix equations is the vector of nodal forces. (a) Concentrated force(b) Distributed forces

9. 8 APPLIED LOADS cont. St. Venant’s principle –If the region of interest is relatively far from the force location, the stress distribution tends to be independent of the actual mode of application of the force  min = 0.973  ave  min = 0.668  ave  min = 0.198  ave  max = 1.027  ave  max = 1.387  ave  max = 2.575  ave 0.25b 0.5b b b

10. 9 APPLIED LOADS cont. Applying a couple to a plane solid Applying a force through shaft C (a) Beam element F F d (b) Plane solid elements Force Bar elements Plate p max Plate Hole

11. 10 APPLIED LOADS cont. Plate with a hole example –All nodes on the left edge are fixed in x-direction –node at the center of the left edge is fixed both in x- and y-direction –uniform pressure 600 psi, which is equivalent to the 300 lb, is applied on the right edge