Figure 1: Cantilever beam loaded by a uniformly-distributed load. The grey box on the left represents the solid wall on which the beam is mounted. The blue arrows represent the load, w.
A cantilever beam is loaded by a uniformly-distributed load (UDL) (Fig. 1). Due to the applied force, the beam deforms (is no longer perfectly straight), but the deformation is con-sidered to be so small relative to the length of the beam that there is no horizontal movement (although there is rotation). The de ection of the beam (the vertical distance between the original location of the centroid and the deformed location of the centroid) at any point along the beam can be calculated using:
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v =
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wx2
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24EI x2 + 6L2 4Lx :
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(1)
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Here v is the de ection (up is de ned to be positive), w is the uniformly-distributed load (force per unit length) (down is de ned to be positive), x is the distance from the wall, E is the Young`s modulus for the beam material, I is the second-moment of area for the beam shape and L is the beam`s total length. For this problem, the values of the parameters are:
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w
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=
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3 kN=m
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E
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=
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200 GPa
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I
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=
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986:0 106 mm4
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L
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=
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10:5 m
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For this assessment item, you must:
- Solve Eq. (1) using hand calculations to determine the values of v at x = 0, x = L=2 and x = L.
- Solve Eq. (1) using MATLAB to determine the values of v at x = 0, x = L=2 and x = L. This can be done using the Command Window with screenshots of the work, or using a code written in the MATLAB editor that outputs the results to the Command Window. You must also produce MATLAB code which:
- Calculates the value of v from one end of the beam to the other end at 1 mm intervals.
- Determines the values of v at x = 0, x = L=2 and x = L from the calculation of Re-quirement 3 and reports the results to the Command Window. The calculations of v from Requirement 2 must be completely independent of these calculations (i.e. should not use any of the same functions).
- Compares the two di erent MATLAB methods (Requirements 2 and 4) of calculating v to the hand calculations (Requirement 1) to determine the accuracy of the MATLAB calculations (verify the MATLAB calculations).
- Reports to the Command Window the outcome of the comparison.
- Plots the undeformed shape of the beam along with the deformed shape of the beam on the same image. Also plots the deformations at x = 0, x = L=2 and x = L on the same image.
- Outputs to an ASCII le the distance along the beam and the corresponding de ection as two columns. The perfect le would contain su cient information for someone to know exactly what the numbers represent.
- Has appropriate comments throughout.
Your submission will be assessed using the following scheme. Note that you are marked based on how well you perform for each category, so the correct answer determined in a basic way will receive half marks and the correct answer determined using an excellent method/code will receive full marks.
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Quality of hand calculations
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5 marks
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Quality of basic MATLAB calculations (Requirement 2)
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5 marks
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Quality of header(s) and comments
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5 marks
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Quality of v calculation at many locations
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10 marks
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Quality of completing Requirement 4
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5 marks
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Quality of veri cation based on hand calculations
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5 marks
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Quality of plot
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10 marks
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Quality of le output
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10 marks
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Quality of the code
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5 marks
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Figure 2: Cantilever beam loaded by a point load. The grey box on the left represents the solid wall on which the beam is mounted. The blue arrow represents the load, P .
Repeat Question 1, but the cantilever beam is loaded by a point load at some distance along the beam. The de ection of the beam at any point along the beam can be calculated using:
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v = (
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Px2
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(3a x) ;
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0 x a
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6EI
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Pa2
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(3x
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a) ;
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a < x
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L
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(2)
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6EI
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where v is the de ection (up is de ned to be positive), P is the point load (force) (down is de ned to be positive), x is the distance from the wall, E is the Young`s modulus for the beam material, I is the second-moment of area for the beam shape, a is the distance from the wall to the point load and L is the beam`s total length. For this problem, the values of the parameters are:
E = 180 GPa
I = 554:0 106 mm4
a = 13:4 m
L = 16:5 m
For this assessment item, you must:
- Solve Eq. (2) using hand calculations to determine the values of v at x = 0, x = L=2 and x = L.
- Solve Eq. (2) using MATLAB to determine the values of v at x = 0, x = L=2 and x = L. This can be done using the Command Window with screenshots of the work, or using a code written in the MATLAB editor that outputs the results to the Command Window.
You must also produce MATLAB code which:
- Veri es the compatibility of the two formulae in Eq. (2) by con rming that v is continuous at a [i.e. both formulae yield the same value of v(x = a)].
- Calculates the value of v from one end of the beam to the other end at 1 mm intervals.
- Determines the values of v at x = 0, x = L=2 and x = L from the calculation of Re-quirement 4 and reports the results to the Command Window. The calculations of v from Requirement 2 must be completely independent of these calculations (i.e. should not use any of the same functions).
- Compares the two di erent MATLAB methods (Requirements 2 and 5) of calculating v to the hand calculations (Requirement 1) to determine the accuracy of the MATLAB calculations (verify the MATLAB calculations).
- Reports to the Command Window the outcome of the comparison.
- Plots the undeformed shape of the beam along with the deformed shape of the beam on the same image. Also plots the deformations at x = 0, x = L=2 and x = L on the same image.
- Has appropriate comments throughout.
Your submission will be assessed using the following scheme. Note that you are marked based on how well you perform for each category, so the correct answer determined in a basic way will receive half marks and the correct answer determined using an excellent method/code will receive full marks.
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Quality of hand calculations
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5 marks
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Quality of header(s) and comments
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5 marks
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Quality of MATLAB calculations
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10 marks
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Quality of veri cations
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5 marks
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Quality of the code
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10 marks
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You are to write a brief report (about 100 words, excluding any code), which includes:
- A description of an instance during the writing of your code for this assignment where there was a problem (e.g. a bug, an error, an unexpected result) or the most challenging aspect to overcome.
- What steps you took to overcome the problem or challenge (including any code you wrote to test the problem/challenge).
- The code before you resolved the problem or challenge, highlighting the line(s) of code where the problem or challenge occurred.
- The code after you resolved the problem/overcame the challenge.
- Code from the MATLAB editor is to be copied into Word; screenshots should be taken of the Command window.
The problem report will be assessed using the following scheme:
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Description
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Marks
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Excellent description of a di cult problem and e ective solution
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5
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Good description of a moderate problem and useful solution
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4
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Reasonable description of a genuine problem with a pragmatic solution
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3
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Poor description or Problem is not very challenging or Solution is not e ec-
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2
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tive/e cient
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Problem is trivial or Solution is poor
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1
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No report or description does not include a genuine problem
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0
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Submission
Submit your code, with the output le that is produced in Question 1, by the due date to the StudyDesk. Submit your hand calculations as a pdf le (this le does NOT require selectable text: it merely needs to be able to be opened). Submit your problem report as a pdf le that contains selectable text, i.e. you can use a mouse cursor to highlight text which you can copy and paste (your assignment will not be marked if the problem report pdf le does not satisfy this requirement). Note that:
You do not need to rename your les when uploading: the system automatically segregates di erent students` submissions.
You can submit all your MATLAB les in a *.zip le or individually (do not archive in another le format such as rar or 7z). Any zipped le MUST have the extension *.zip. You must submit your pdf le(s) separately.
You will NOT receive any con rmation of receipt. If you can see that the les have uploaded, then you have successfully submitted your assignment. There is no need to click a send for marking" button, but you will have to click a button con rming that the submission is your own work.
After your les are successfully submitted, the system will automatically attempt to submit each le to Turnitin. The system can ONLY submit your le to Turnitin if it has been SUCCESSFULLY SUBMITTED for marking. You will almost certainly receive a warning or error message from Turnitin:
{ *.zip les WILL ALWAYS return an error message (Turnitin cannot extract the les to process them)
{ *.m les SOMETIMES return an error message (although they contain plain text, *.m is not a standard le extension which Turnitin recognises)
{ *.mat les WILL ALWAYS return an error message (it is a binary format which Turnitin doesn`t recognise)
{ *.pdf les without selectable text MAY return an error/warning message (there is no content that Turnitin can process).
If you receive an error message from Turnitin, there is NO DOUBT: your le has DEFINITELY BEEN SUCCESSFULLY SUBMITTED for marking by the markers.
