AE4132 - Finite Element Analysis Spring 2022 Homework 2: Rayleigh-Ritz Method Due Friday, February 11th 2022 Problem 1 Consider the bar depicted in Figure 1. 1. Derive the corresponding expression for...

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AE4132 - Finite Element Analysis
Spring 2022
Homework 2: Rayleigh-Ritz Method
Due Friday, Fe
uary 11th 2022
Problem 1
Consider the bar depicted in Figure 1.
1. Derive the co
esponding expression for the elastic potential.
2. Use the Rayleigh-Ritz method to find approximate solutions considering the following approximate displace-
ment fields. Comment on your results.
(a) û = ax+
(b) û = ax2 + bx+ c
(c) û = ax3 + bx2 + cx+ d
(d) û = ax+ b for 0 < x < L/2 and cx+ d for L/2 < x < L
3. The last case is what we call a piece-wise linear approximation, in this case for 2 segments. Write a generic
expression for the potential co
esponding to a piece-wise linear approximation resulting from dividing the
ar into N identical segments of length L/N . Show that in the limit for N → ∞ the approximate potential
converges to the exact one. Comment on the implications of your result.
4. For the particular case where P = 400 N, q = 100 N/m, L = 2 m, A = XXXXXXXXXXm2, E = 70 GPa, and ν
= 0.3, use your expression from part 3 to create and plot the solution for N = 50 in Python. Include the
source code.
P
L
q
Figure 1: Schematics for problem 1
AE4132: Finite Element Analysis - Spring 2022, Homework 2 2
Problem 2
Consider the bars you solved in the previous homework assignment, as shown in Figure 2. Based on your knowl-
edge of the exact solution, propose û such that, by using the Rayleigh-Ritz method, would allow you to recove
the exact solution. Discuss your choice of û and solve both cases using Rayleigh-Ritz.
For the first schematic, consider the case where q = 1 lbf/in, P = 90 lbf , L = 8 ft, E = 20 Mpsi, and A
= 5 in2. Compute N(x) using a) your solution for û and b) integrating the governing equation. Plot N vs x fo
oth a) and b) on the same graph. Include your source code.
Note: If your plots overlap each other, use different line styles in the plots. For example, use plt.(x, y,
ls='-') for the first line and plt.(x, y, ls='--') for the second to plot solid and dashed lines, respectively.
P
L
q
L
q
Figure 2: Schematics for problem 2
Problem 3
Consider the beam you solved in the previous homework assignment, as shown in Figure 3.
1. Derive the co
esponding expression for the elastic potential.
2. Find an approximate solution via the Rayleigh-Ritz method using:
(a) û = ax2 + bx+ c
(b) û = a cos(bx) + c
3. Consider the particular case where q = 40 N/m, L = 1 m, E = 120 GPa, ν = 0.4, Izz = 0.8 cm
4, and
P = βqL. Calculate the total potential energy, Π, for cases (a) and (b) and plot Π vs β, for β values from
-10 to 10. Include your source code.
4. Which solution do you think is better? Why?
P
L
q
Figure 3: Beam configuration for problem 3.
Answered 2 days AfterApr 14, 2022

Solution

Sandeep Kumar answered on Apr 16 2022
11 Votes
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