CE 3101
Fall 2011
Homework 06 – Optimal Design of a Nonprismatic Beam
(Solving a 4thorder ordinary differential equation using the “shooting method”)
Due Tuesday, 18 October 2011, 11:55 P.M.
This is a GROUP assignment.
Abstract
In selfselected groups of up to three members, you will write an Excel spreadsheet program to assist in the
optimal design of a
nonprismatic beam
with mixed end conditions and a nonuniform load. A brief letter
report is required in addition to the working code. A full written report is
NOT
requested or required
1
.
Group Evaluation and the Time Card
After completing this assignment you are each required to INDIVIDUALLY complete the
Homework 06
Group Evaluation
, found on the course
Moodle
web site. Your evaluation activity must be completed by 6:00
P.M., Friday, 21 October 2011, or you risk the imposition of an egregious, 5point, penalty
2
.
1
Background
In this group assignment, you will determine the optimal
3
design for a
nonprismatic beam
carrying a non
uniform load. A nonprismatic beam is a beam with dimensions, shape, or materials, that change along its
length. You will focus on changing dimensions – specifically the height of the beam. Oh, and by the way,
to complete this problem you must solve a 4thorder ordinary differential equation with mixed boundary
conditions using the shooting method
This problem is similar to a problem given in past semesters.
However, your class has demonstrated
exceptional abilities, so we made this a more challenging
4
, and we hope more interesting, version of the
problem
5
. To compensate for the potential extra work, the requirements for the written report have been
substantially reduced – i.e. almost eliminated.
2
The Setup
Consider a 40 meter long beam
6
carrying a nonuniform distributed load (see Figure 1). Define a coordinate
system where the left end of the beam is at location
x
=
0
[
m
]
, and the right end of the beam is at
x
=
40
[
m
]
.
The positive yaxis points upward
7
, as is usually the case. The beam is simply supported
8
at the left end
(at
x
=
0
[
m
]
), and cantilevered
9
on the right end (at
x
=
40
[
m
]
).
1
Did you see that a full written report is NOT required for this assignment?
2
Did you see that there will be an individual 5point penalty assessed for not completing the group evaluation on time?
3
For this problem, we define
optimal
as the minimum mass (weight) beam while meeting a set of engineering constraints.
Using a cost per kilogram of fabricated steel offers a simple, and remarkably good, firstorder approximation for the constructed
cost. Of course, the unit cost of fabricated steel is higher than the unit cost of raw steel sitting on a shop floor.
4
If you ignore hints (5) and (17), you deserve the pain and frustration that you will suffer.
In some societies this fate is
called
justice
, in some childrearing books it is called
natural consequences
.
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 Spring '11
 Barnes
 Boundary conditions, Dr. Theodore Galambos

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