Read Segel Sections 4.3 and 5.1.
-1. (10pts) Perform the final steps of the computation of the "pure bending" solution (that we omitted in class). Be sure to number equations and state at each step what facts you are making use of.
0. (4pts) What's the simplest &epsilonij-field you can come up with that does not satisfy the compatibility conditions?
1. (a) (4pts) (What is the radius of a brass bar of circular cross-section that
has the same flexural rigidity as a brass bar of rectangular cross-section
with dimensions 2.5cm (x1 direction) by 0.5cm (x2 direction)?
Note this is the "tall" way - the way in which it is more rigid.
The bending torque is about the x2 axis.
(b) (2pts) Is more or less brass needed in the circular case? By what factor?
(c) (2pts) Given a certain amount of material, and a required length of a beam,
what cross-sectional shape maximizes the flexural rigidity? Justify your
answer, and say if the optimal shape you describe is practical.
2. (4pts) Make qualitative sketches of the deflection of our clamped brass bar and its 1st, 2nd, and 3rd derivatives over the entire length of the bar, including the part on the table. Note any jumps and/or spikes in the derivatives. What can you conclude about the distribution of force of the bar on the table?
3. (a) (5pts for matrix, 4pts for first root of determinant.) Compute the frequency of the lowest transverse vibrational mode
of our clamped brass bar. You may need to use numerical methods in the end.
(b) (4pts) Compare your answer with the frequency we measured in class
for the actual brass bar.
(Data on blackboards of
4/18 and
4/20.)
Comment on any discrepancy.