This web page provides guidance for the MENG 421 homework assignment to find the displacement and stress in a truncated cone.

Click for graphic of Cone.Using Matlab, you will calculate the displacement and stress in a truncated cone that is constrained at one end and loaded at the other. This is the problem that was introduced in class. There are four parts:

- Describe the problem on the cover sheet.
- Solve the problem exactly using calculus (d5 = -5.269e-3 in).
- Solve using one element (d = -4.68e-3 in)
- Don't solve the cone using two elements, but report the results given in class
- Solve with Matlab using four elements, but set up the matrix on your cover sheet
- In a table, compare the error in displacement by using one, two, and four finite elements.
## Data

- Diameters: Left = 2 in, right = 1 in
- Length: 12 in
- Material: steel, E = 29e6 psi
- Load: 20 kip compression
- Constraint: left end
## Details

- On the cover sheets, state the problem with Given and Find.
- Use a linear shape function to find the diameter as a function of distance along the cone.
D = N_{1}D_{1}+ N_{2}D_{2}- Find the area as (pi/4)* diameter^2.
- Find the exact displacement by calculus.
- Find the displacement using one finite element.
- Copy one of your previous axial programs and name it cone4.m
- Change cone4.m to give the four-element solution.
- Using this shape function, define in Matlab, the diameter,
dia, as a function of position,pos, along the cone. Then use the diameter as usual to calculate the area. Then define an array k on a single line. For example:len = 3 * ones(1,4) % element length pos = (1:4)*3 - 1.5 % position of each element center dia = 2 - pos / 12; % diameter area = (pi * dia .^2)/4 ex = 29e6 % elastic modulus k = area * ex ./ len % make k a vectork is now a vector with the appropriate value for each of the four elements.- Try each of the above Matlab expressions by typing them into Matlab in interactive mode. Before you use them in your program, be sure you understand what each does.
- Now, for the stiffness matrix, define the diagonal elements with the k vector.
k11 = k(1) k22 = k(1)+k(2)However, use the k vector itself off the diagonal.stiff = [ k11 -k(1) 0 0 0 % stiffness matrix -k(1) k22 -k(2) 0 0- On the cover sheet compare the displacements (giving percent error) for the one-element, two-element, and four-element models to the exact solution.

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Last revised: February 27, 2004 -- Copyright © 1997-2004 ARMiller