COMP 208: Computers in Engineering
Fall, 2017
Assignment 6: De Castigliano
Due Date
Assignment 6 is due on December 7, 2017 at 23:59. The cutoff is automated and is exactly at this
time. Assignments within the following hour will be marked late but accepted without penalty.
The assignment is to be done individually. You can collaborate on understanding the problem but
you must write the solutions independently. Submissions might be subject to being checked by
plagiarism detection software.
Introduction:
You are an engineer in charge of selecting the material of an airplane wing with an engine
centered along it. Before going into time-consuming detailed calculations and costly
prototyping, you would like to obtain an order of magnitude estimation (a value that makes
sense) for the elastic modulus (also known as Young’s Modulus) of a material.
The simplest way to go about this without relying on Finite Element Analysis software is to use
De Castigliano’s strain energy method applied to a linear-elastic model.
You choose to model your wing as a cantilever beam under uniform. lift force, Y, with a force
representing the engine’s weight, W, which acts in the middle of the wing. Note: this is an
oversimplified model but should give a result within an order of magnitude.
Using De Castigliano’s strain energy method, you find the follow relation:
𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 = 𝛿 = 1𝐸𝐼[∫ [𝑊 ∗𝑥 ∗ (𝑥 − 𝐿2)− 𝑌 ∗ 𝑥
To save time in your calculations, you decide that you are to write a C program to help you determine
the optimal elastic modulus thus allowing you to select a material with greater ease.
For the entire assignment, assume the following:
I = 0.0225m4
W = 50000N
Y = 12000N/m
L=25m
Part A - Numerical Integration (50 Marks):
For this part use:
E = 72*109 Pa
You must write a program that given E, I, L, Y and W, will output y. Your output should look like the
following. (It is negative due to Castigliano’s method).
The wing deflection is -1.155398m
You must do this part using a numerical integration method for (n = integration panels = 1000). Use the
Trapezoidal method. Do not make it specific to this function but use a general form. that will integrate an
arbitrary function.
Guidelines:
1. Your program will not receive user input or write to a file. You can simply hardcode the values
given into your code to save time in testing. You may make use of global variables or the
#define to set the values.
2. Your program must contain at least 3 functions.
a. 2 functions that will compute the double values:
b. A function that returns the integrand of any function between a specified range using
the trapezoidal method. Your function should make use of a function pointer, as to pass
the functions described in a). Here is a sample function prototype using the type
definition DfD as covered in the class notes:
double integration (DfD f, double x0, double x1, int n)
Part B - Root-Finding (50 Marks):
Now that your code in Part A) can compute wing deflections, your next step is to write code that will
find the optimal elastic modulus (Young’s Modulus) through a root-finding method (it is possible to do
this without root-finding but for this assignment, you are required to use a root finding algorithm). In
this particular case, you can use the bisection rule.
Assuming that δ=Function (E), you can apply bisection method to find the optimal modulus E, given y.
You will be looking for the elastic modulus that corresponds to a deflection of:
𝛿 = 0.5𝑚
Your output must look like:
The optimal elastic modulus is 166.377 GPa
Guidelines:
1. Re-use your code in part A appropriately. Hint: If you convert your main function from Part A
into its own function (i.e. double deflection) this will save you time.
2. Write a function called bisection that will return the elastic modulus.
a. Use the range, 70GPa < E < 250GPa.
b. The tolerance for this part will be 0.001
3. Remember to modify δ=Function (E) so that a root (f(E)=0) exists.
Requirements
The programs must be written in C
Define and use the functions as described above.
Use meaningful variable names
Comment and indent your code. It is your responsibility to make it readable to the grader
Submit only the source files (.c) and name your files A6a_123456789.c and A6b_123456789.c
where 1234567898 is replaced by your student ID number.