EML3041: Computational Methods The Beam Deflection · PDF fileEML3041: Computational Methods The Beam Deflection-Stress ... This project will work exclusively with ... Place the unstrained

1 | P a g e

EML3041: Computational Methods

The Beam Deflection-Stress Problem Benjamin Rigsby and Autar Kaw

1. Description The project will be a semester-long exercise broken up into “mini projects”. These mini projects will be due at intervals throughout the semester. This not only allows you, the student, to get better feedback and to have a better learning experience, but also keeps you from being overwhelmed with a single-large project due at the end of the semester.

This project will work exclusively with loading of a beam - so understanding the prerequisite knowledge early-on is essential to being successful on this project. All of the given information you will need for each mini project can be found here. You may begin working on these whenever you wish; you must turn each mini project in by its respective due date. The due dates are listed next to their respective portions.

“He who would learn to fly one day must first learn to stand and walk and run and climb and dance; one cannot fly into flying.” -Friedrich Nietzsche

This project was created to help you, the student, “learn to fly”. Don’t worry though; we don’t expect you to “fly into flying”. During lectures, you will learn the concepts; you’ll learn to “stand and walk”. The homework and quizzes will further your understanding; they will teach you to “climb and dance”. These projects will require you to take your previous knowledge and applied it on your own. You’ll be responsible for making decisions; you must now “fly on your own”. When you are asked to complete a project, all of the tools you need will have been given to you. Remember what Nietzsche said, and you’ll do well in this class!

1.1 Grading This project is part of the Special Assignment/Project grade. Your solution will be graded on the following categories:

• Merit of conceptual portion

2 | P a g e

• Merit of programming portion • Format of all submissions types (See Section 1.3)

DO NOT cheat. It’s not right, and it isn’t worth it. It is much better to see the instructor or TA with whatever your problems might be than to handle them on your own by cheating. In this class, you may not receive ANY help from an individual other than the instructor or the TA. Refer to academic dishonesty policy of University of South Florida at

http://www.ugs.usf.edu/catalogs/1314/pdf/AcademicIntegrityOfStudents.pdf

1.2 Learning Objectives • Identify and implement the correct numerical method for a given problem • Improve existing programming skills • Reinforce prerequisite knowledge of mechanics • Solve real world problems

1.3 Formatting The formatting will be the same throughout the project, varying only when the submission format (programming or handwritten) changes.

See sample for formatting: http://www.eng.usf.edu/~kaw/class/EML3041/homework/sample_experimental.html

1.4 Help If you need assistance, some resources are listed below. Be sure to check out the Project Help document as it contains a checklist for your projects.

• Project help document (see Appendix A) • Instructor and TA office hours • Piazza (24/7 resource) • How do I do that in MATLAB:

http://mathforcollege.com/nm/blog_entries.html#How_do_do_that_in_MATLAB (go to end of this webpage).

2. Background You are given a cantilever beam and asked to solve a variety of scenarios. All of the information you need for the whole project is contained here. You will have to choose what you need for each problem.

3 | P a g e

Figure 1: Cantilever beam under a load at the end

Figure 2: Cantilever beam experiment setup showing dial gauge

y

B

Strain gage A Strain gage B

L

A

b

h x

Weight

4 | P a g e

Figure 3: Cantilever beam experiment setup showing strain gages at two locations

Length of beam, L = 278 mm

Height of beam, h = 1.92 mm

Width of beam, b = 28.5 mm

2.1 Equipment and Devices Experiments were preformed to provide the data used in this project. Examples of the tools used during the experiments are given here.

• Strain Gages (to measure strain) • Dial Gauge (to measure deflection) • Pencil and paper (to record data)

2.2 What to Do in the Laboratory: 1. Place the unstrained beam apparatus on a stable and flat surface.

2. Set up the strain gauge indicator:

a. Take the strain gauge wires and connect them properly (red wire to red terminal, white wire to white terminal, green wire to yellow terminal) to the strain indicator.

b. Set BRIDGE pushbutton to up (black) for ¼ bridge. c. Depress AMP ZERO pushbutton. d. Set AMP ZERO control for a reading of +- 0000. e. Depress GAGE FACTOR pushbutton. Set GAGE FACTOR controls to 2.075.

(This setting is found in the information given with the strain gauges). f. Lock GAGE FACTOR knob. g. Depress RUN pushbutton. Set BALANCE controls for reading of +- 0000. Lock

BALANCE knob. The meter reads in microstrain ( mµ )

5 | P a g e

3. To take measurements for deflection:

a. Measure out 20 positions on the beam and take their initial deflection readings b. Add weight to the hanger. c. Wait for the beam to stabilize. d. Take deflection gauge and record various readings along the beam. Record the

amount of deflection. (Do not forget to subtract the initial deflection).

4. To take measurements for strain:

a. Place the weight on the hanger and read the corresponding strains on the two strain gages.

b. Repeat with varying weights.

2.3 Experimental Data The following experimental data is given to you. Be sure to convert it to SI units within the MATLAB code (if necessary).

Table 1 corresponds to data of deflection measured at various points along the beam under a constant load corresponding to a mass of 387 grams (see Figure 1).

Table 1: Position vs. deflection data

Position from fixed end, x (mm) Deflection, w (in)

0 0

37 0.0295

58 0.0770

79 0.1305

103 0.2045

125 0.2850

152 0.3805

178 0.4895

202 0.5630

231 0.6600

271 0.7690

Table 2 corresponds to data from strain gage A corresponding to different weights placed at the free end of the beam. The strain gage A is located at 58 mm from the fixed end.

6 | P a g e

Table 2: Strain vs mass data from strain gage B

Mass(g) Strain Gage A (µm/m)

0 0

52 78

68 110

100 194

124 222

147 260

194 362

249 470

303 570

358 680

387 702

405 730

Table 2 corresponds to data from strain gage B corresponding to different weights placed at the free end of the beam. The strain gage B is 135 mm from the fixed end.

Table 3: Strain vs mass data from strain gage B

Mass (g) Strain Gage B (µm/m)

0 0

52 58

68 80

100 154

124 175

147 185

194 245

249 316

303 379

358 485

387 450

405 500

7 | P a g e

3. Theoretical This section is outlining the physical concepts of the experiment. See the nomenclature section if you do not know what a variable represents.

3.1 Deflection The deflection of the beam is governed by:

EIxM

dxw )(=d2

2

(1)

The normal strain measured by the strain gages is given by:

EI

hxM2

)(=ε (2)

The bending moment as a function of x (measured from fixed end) due to an end load in a cantilever beam is give

( ) ( )xLPxM −= (3)

The second moment of area for a rectangular beam cross-section is given by:

12

3bhI = (4)

3.2 Stress and Strain

The normal axial stress, (σ) at location x on the top of the beam is given by

I

hxM2

)(=σ (4)

and the normal axial strain (ε) at the location x on top of the beam is given by

Eσε = (5)

3.3 Nomenclature All the applicable variables are listed here with a description of the quantity they represent. It is recommended that you use some form of these variables or names in your program (i.e. it is better to call a variable for temperature as "temp_eq" or "theta_eq" rather than "fun_eq" or something arbitrary).

8 | P a g e

𝜀𝜀: Strain �𝑚𝑚𝑚𝑚�

𝑀𝑀(𝑥𝑥): Bending Moment as a Function of 𝑥𝑥 (𝑁𝑁 ∙ 𝑚𝑚)

𝐿𝐿: Length (𝑚𝑚)

𝐴𝐴: Distance of Strain Gage A from Fixed End (𝑚𝑚)

𝐵𝐵: Distance of Strain Gage B from Fixed End (𝑚𝑚)

𝑤𝑤(𝑥𝑥): Deflection as a function of 𝑥𝑥 (𝑚𝑚)

𝐼𝐼: Second Moment of Area (𝑚𝑚4)

𝐸𝐸: Young's Modulus of Elasticity (𝑃𝑃𝑃𝑃)

𝑃𝑃: Applied Load at Free End of Beam (𝑁𝑁)

𝑚𝑚: Mass (𝑘𝑘𝑘𝑘)

𝑏𝑏: Width of Beam (𝑚𝑚)

ℎ: Thickness of Beam (𝑚𝑚)

𝑥𝑥: Position along 𝑥𝑥 axis (𝑚𝑚)

4. Mini Projects: 4.1 Some general guidelines

• Unless otherwise noted, your solution should be completed using MATLAB. • See Section 1 if you are unsure of how the project works. • You must use SI units at all times. • Follow the sample project format including cell formatting, published html format,

commenting, etc. • You may be asked for a copy of your m file. Failure to produce one may result in a

substantive grade reduction.

4.2 Exercises Mini Project One – Due Wednesday March 2, 2016 (50 points)

1. Solve the ordinary differential equation exactly subject to the proper boundary conditions. The output should be deflection as a function of location in form of a polynomial with proper numerical values for the coefficients. Assume the Young’s modulus of aluminum is 69 GPa.

2. Plot the deflection vs location polynomial from Question 1. Also, show the individual data points from Table 1 on the same plot.

3. Using the output for deflection from Question 1, find the location where the deflection is half of the defection at the free end.

4. Find the relative true error in the deflection at the sixth data point of Table 1, while assuming the analytical value as the true value.

9 | P a g e

5. Only using data in Table 1, use any scientific numerical method learned in class to find the second derivative of the deflection with respect to location at sixth data point. Use this and other constant values then to estimate the bending moment at this point while assuming the Young’s modulus of aluminum is 69 GPa. Find the absolute true error in the bending moment, while assuming the analytical value as exact.

Mini Project Two – Due Wednesday March 30, 2016 (50 points) 1. The deflection as a function of location data in Table 1 can be regressed to

33

2210 xaxaxaaw +++= . The equations needed to find the constants of the polynomial

regression model are given below in the matrix form.

=

∑

∑

∑

∑

∑∑

∑∑

∑

∑

∑

∑

∑∑∑∑

∑∑∑

=

=

=

=

==

==

=

=

=

=

====

===

n

iii

n

iii

n

iii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

wx

wx

wx

w

aaaa

xx

xx

x

x

x

x

xxxx

xxxn

1

3

1

2

1

1

3

2

1

0

1

6

1

5

1

5

1

4

1

4

1

3

1

3

1

2

1

4

1

3

1

2

1

1

3

1

2

1.

Solve the above equations to find the constants of regression model. Check if you get the same constants by using the polyfit command.

2. a. Only using the data in Table 1, use cubic spline interpolation to find the second

derivative of the deflection with respect to location at the sixth data point. b. Use this found value in part (a) and value of other constants then to estimate the

bending moment at the sixth data point assuming the Young’s modulus of aluminum is 69 GPa.

c. Find the bending moment at the sixth data point using the analytical model from Equation (3)

d. Find the absolute percentage true error in the bending moment, while assuming the analytical value as exact.

3. You can relate how experimental data can be used to find material properties. By hand, on a separate sheet of paper, derive the regression formula in symbols for Young’s modulus by regressing strain in gage A to applied load (assume applied load is known more accurately than the strain in gage A). Note that there is also only one constant of regression in the model.

4. Use the regression model you derived in #3 and data from Table 2 to estimate the Young’s modulus of the material. Plot the strain in gage A vs. load data that shows individual data points and the plot of the regression curve.

10 | P a g e

Appendix A: Project Help This handout is meant to help students approach engineering problems effectively and efficiently. Without the proper approach, engineering problems can be very confusing. The following guidelines are written with common correct and incorrect approaches in mind. Remembering and implementing these approaches can not only help you find a solution faster, but it can increase your understanding of the problem and its conceptual basis. Most of these guidelines are not relegated to this class; you can use them in any engineering class!

How to approach solving problems on paper • Start with what you know. If you don't know where to start, start with what you know. It's

a little bit like connecting the dots. You can’t connect the dots until you have written some down.

o Look at the information you're given.

o Look at the applicable equations.

What are the restrictions on these equations?

o Be methodical in your approach.

Often students will say, “I don’t know anything about this!” Typically, this is because they don’t know what they know and what they don’t know. Start with what you know!

• Use dimensional analysis as a hint.

o If you can't find a mistake in your work, check the unit continuity in the problem.

o If you don't know how to solve a problem, determine the units of the solution and then look to see what units you're missing in the solution.

• Don't cut corners! This WILL hurt you sooner or later.

How to approach programming • Start with what you know.

o If you're having trouble programming a problem, start by working through the problem on paper.

o Don’t try to think up the whole program in your head and then type it out!

• When translating the problem solution into a program, display each part of the code. Fix one piece at a time.

• Avoid using “;” at end of statements while debugging the program. You can add the “;” later when the program is finalized.

• Look at the ‘How do I do that in MATLAB series’.

• Use the MATLAB help site (http://www.mathworks.com/help/matlab/) to look up error codes, syntax, etc.

11 | P a g e

o If you're looking for syntax examples, click the "example" links on the right side of MathWorks sections for a sample program.

Common mistakes • Hard coding • Incorrect format • Misunderstanding the conceptual (paper) solution • Inefficient program debugging • Cutoff errors • Unit errors/no units

How to check your submission • Did you follow the general format as given in the sample project? • Did you follow the cell format as given in the sample project? • Did you publish in MATLAB in HTML format? • Are you writing proper and reasonable comments? • Are you CLEARLY identifying your method for each problem? • Be sure to put these comments on their own lines as seen in the sample project mfile (not

at the end of the code). • All statements should be suppressed. So, you have to show input and output variables

using fprintf and disp statements. • The input data also needs to be displayed. • See this blog to see how to display vectors. • Did you check for cut off errors in the printed published file? • Did you use proper units where appropriate? • Are you using variables properly so as not to hard code? (i.e. the program should still

work if ANY of the input data is changed in ANY way)