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Lecture 13(: Constrained Optimization & Project Discussion
In order to define some terms involved in constrained optimization, let us consider the following example:
Consider a simply supported beam with rectangular section subjected to a uniform load. Let say we want to find:
The required dimensions (b and h) of the beam of section (1)
Such that:
The weight of the beam is minimized, (2)
and the following conditions are satisfied:
The maximum normal stress d" all (3-a)
The maximum shear stress d" all (3-b)
The maximum deflection d" all (3-c)
Where:
L, E, all, all, all are given. (4)
Statement in (1) lists the design variables . In our example they are the dimensions of the beam cross-section. Statement in (2) defines the objective function which is the weight of the beam. If we assume that the beam is made of a homogeneous material, then the volume of the beam can be considered as the objective function, i.e. b*h*L. Statements in (3) lists the design constraints and statement in (4) gives the design parameters which are known values.
We will not discuss the solution of constrained optimization problems in the class. We will however use Mathematica and Excel to solve this kind of problems in the lab.
Design Project Discussion :Optimum Design of a Walkway Truss
Objective:
To design a minimum cost walkway truss (shape and dimensions) with the following constraints:
The stress in each member does not exceed the allowable stress. The force in each member does not exceed the buckling force
2m d" h e" 4m where h is the height of the truss
1mm d" t e" 5mm where t is the thickness of the member cross-section (see assumption #5 below)
You can not use more than three different cross-section sizes
Assumptions:
The length of the walkway, L = 20m
The total load is uniform = 2*live load of pedestrians (assume that the walkway is fully crowded)
The above load is represented by concentrated loads over the upper (or lower) joints of the truss
The truss members are made of structural steel (ASTM-A36). Assume that all = y/f.s., where y is the yield stress and f.s. is the factor of safety = 3.
Use a factor of safety = 3 for the
The cross-section is tubular with diameter, d and thickness, t = 0.1 d
The cost of the truss construction = 50 SR * nj +300 SR * nm, where nj = number of joints and nm = number of members
Your truss comes here
h
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Project Reporting
I- Progress Report: To be submitted not later than 11 May 2003. It contains:
Sections 1and 4 of the final report (see below).
A table of the member forces in an arbitrary truss (not the optimum one).
II- Final Report: To be submitted during the last lab. It contains:
Title Page: It should contain the title of the project, the names of team members in addition to the other essential information such as the course name, department name, date, etc.
Summary: It gives a very brief overview and description of the major finding(s) of the report. Presumably, a busy engineer could read the summary and understand the major points of the report without actually reading through the entire technical document.
Table of Contents: It lists the major sections in the report and the page numbers which they begin on. The Table of Contents also lists the References and Appendices.
Problem Description: This section describes the design problem and the objective of the solution. A complete listing of the design variables, design constraints, design parameters, assumptions and all necessary drawings and design details.
Design Procedure: This section includes a description of the model and/or design methodology used to arrive at the optimum design. Include a list of all assumptions and equations. The methodology may include iteration schemes, flowcharts, or other step-by-step procedures.
Analysis: A sample calculation should be included with appropriate units to verify the correctness of calculations performed by a computer model. Any computer codes or output should be placed in this section.
Results: The results in the form of tables and figures are given in this section. The minimum requirement is to include 3 tables containing the following design results for 3 different truss configurations (each student is assigned one configuration):
force in each member
cross-section of each member (d & t)
normal stress in each member
height of the truss
cost of the truss
Conclusions & Recommendations: Restate the key points from the previous sections of the report. Emphasize the major points you feel are important. Discuss recommendations for improving the final design.
References: A list of sources used or referred to in the report. They could be: books, periodicals, internet sites, class handouts, etc.
Appendices: It contains the details that are useful to the reader but which you do not want to include in the main part of the report).
III- Presentation: During the last lab, each group will give a 15-min presentation (5 min/student) + 5 min for discussion.
Project Grading:
Progress report: 10%
Final report 50%
Teamwork 10%
Presentation 10%
Design economy 20% for the most economical, 15% for 2nd and 10% for the last. In case the design does not satisfy all constraints, 0% will be given for this component.
( Important: This handout is only a summery of the lecture. The student should take detailed notes during the class and refer to the textbook for more examples and discussion.
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