Homework #7
Assigned: Mar. 8, 2009
Due: Mar. 17, 2009
Points: 60

1. 60 points
   Write a size-optimization program for a straight slender beam for general boundary conditions and mesh size. You need to design the beam for maximum stiffness for a given volume of material. Use either the width or depth of the rectangular cross-section for the beam as the design variable. Use the optimality criteria method. Run your program for a number of meshes to see the effect of the size of the mesh. Try different loads and boundary conditions and upper and lower limits on the area of cross-section to get a 'feel' for optimality in this problem.
   Use the following data.
   • Length of the beam 2 m
   • Bounds on the width and depth: 10 cm and 1 cm.
   • Volume of material = 2 x 0.05 x 0.05 m³.
   • Assume a loading that is not too high or too low. A good guidline is to ensure that the maximum deflection does not exceed the depth or width of the beam nor is it less than one fourth of the larger of width or depth.
   The following should be submitted.
   • Problem statement and the details of derivation that leads to the program implementation.
   • The program with all its m-files and inpur data files zipped into a single file.
   • A paper copy of the result of the optimal beam for fixed-fixed boundary condition and uniform loading. Include the results of different mesh sizes for this problem. Clearly mark the axes and indicate units. Write down the strain energy value under each plot of optimal beam profile.
   • Extra credit if you submit a paper copy solution for additional boundary conditions and loading.
   The beam finite element Matlab files are here .