Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May., 2016
Instructor: G. K. Ananthasuresh , Room 106, ME Building, suresh at mecheng.iisc.ernet.in
Lectures: Tu, Th: 08:30 AM - 10:00 AM; Venue: ME Lecture Hall

Homework #6
Assigned: Feb. 25th, 2016
Due: Mar. 3rd, 2016
Points: 50
Additional points for work that is beyond instructor's expectation!
Look up Homework #1 of 2010 offering of this course and the solution posted there to know what is meant by "beyond instructor's expectation".

1. 20 points
   Consider a chain of length 1.3 m hung between points A (0,0.5) and B (0.5,0.3) under gravity. If the chain has 1 kg/m mass density, compute the equilibrium shape of the chain if there is rigid floor along the x-axis. Solve the problem analytically first and then compare with the discretized numerical solution given by fmincon in Matlab. What do the Lagrange multiplers imply in this problem?
2. 20 points
   Consider a beam of length L, Young's modulus E, second moment of area I. It has built-in support at x = 0 and pinned at x = L. The beam is made of two halves pinned at the middle. There is a downward transverse force at x = L/4 and an upward transverse force at x = 3L/4. Solve for the deflection of the beam using the minimum potential energy principle. Write all equations and unknowns clearly before you solve for the unknowns.
3. 10 points
   Write the Euler-Lagrangew equations for .
   It is not easy to solve the resulting equation. So, try the substitutions: t = ln x and q = y, and solve the resulting Euler-Lagrange equation.