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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

PROBLEM1

1.1 T W E

W LEE EdV

assume uniaxialstress only ox to

w f tr e dV

Hooke's Law ice

or Alex ey ez zuex

alex ey ez 1 zney 0 solvefor Ey ee

e aCa ey ez zuee o

extey ez

2 Ey ez Ey e

Ext ey

2 Ey 2 Ey ex Ey e Inn ex

T TE i man an e am a 2AM1M ex ncsa na u µ Ex

Young'smodulus E

w f'seeidV EAfo'f dx notneededforthispart

sit E s fi dx SHquiudx SPuh L

E Equinox Put

1 s dx fo'squa dx PsatT w e Efts Yt dx fo9Wud Pulx

ELIfizzy g s d SiouxSudx Psul

1.2 a via axt

plugged im intotheequation for elasticpotential from 1.1

Bc Aco o b o

solve 24 0a in Mathematica seescript attached

via ux Eax

via ax bx c

Bc vico o c o

odes Ita o 3 o

acid z ax't PEf9x

c via axs bx ox d

Bc Aco o d o

odes IIAo CIIIo e 371 0

acid z ax't 149x note same as quadraticapproximationalsotheexactsofa

d are ax b for oexcE

ingµ cx d forEe L b o

Bc A 107 0 it E any E solvefor d as fun of a and a

odes Etta o III o

4h34 for axeand tea

4241x 42 for texel

1.3 uselinear approximation butalsorememberthat

eachconsecutiveelement needsto havecontinuous Bcs

en mas...

1.1 T W E

W LEE EdV

assume uniaxialstress only ox to

w f tr e dV

Hooke's Law ice

or Alex ey ez zuex

alex ey ez 1 zney 0 solvefor Ey ee

e aCa ey ez zuee o

extey ez

2 Ey ez Ey e

Ext ey

2 Ey 2 Ey ex Ey e Inn ex

T TE i man an e am a 2AM1M ex ncsa na u µ Ex

Young'smodulus E

w f'seeidV EAfo'f dx notneededforthispart

sit E s fi dx SHquiudx SPuh L

E Equinox Put

1 s dx fo'squa dx PsatT w e Efts Yt dx fo9Wud Pulx

ELIfizzy g s d SiouxSudx Psul

1.2 a via axt

plugged im intotheequation for elasticpotential from 1.1

Bc Aco o b o

solve 24 0a in Mathematica seescript attached

via ux Eax

via ax bx c

Bc vico o c o

odes Ita o 3 o

acid z ax't PEf9x

c via axs bx ox d

Bc Aco o d o

odes IIAo CIIIo e 371 0

acid z ax't 149x note same as quadraticapproximationalsotheexactsofa

d are ax b for oexcE

ingµ cx d forEe L b o

Bc A 107 0 it E any E solvefor d as fun of a and a

odes Etta o III o

4h34 for axeand tea

4241x 42 for texel

1.3 uselinear approximation butalsorememberthat

eachconsecutiveelement needsto havecontinuous Bcs

en mas...

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