Vectors and Bridge Design
Grade Level
& Duration:
11/12; 2 50minute class
periods
(separated by
several days
to allow
completion of
bridge design
and analysis)
Subject:
Honors Pre-calculus
Prepared By:
Carol Clinton
Analyze Learners
Overview & Purpose (STEMcinnati theme)
In this two-part lesson, students use vector
mathematics to learn key concepts of structural
design. Then, they apply the knowledge to design
and analyze models of bridges. Computer models
are used to test the models, and compare
designs. Extension: This lesson could link with
an extracurricular bridge construction competition
where participating students construct and test
actual wood bridges.
Applications to the real world : Existing local
bridge trusses are shown and used for the vector
analysis demonstration. Then students design
and analyze a bridge truss.
Societal impact: The numerous local bridges
across the Ohio River are vital to everyday life of
local residents, and others who depend on goods
transported across the river on interstate
highways. As our infrastructure ages, new
bridges will need to be designed and built.
Career connections:
The tools and methods used in the lesson are
commonly used by engineers in the actual design
of bridges
Education Standards Addressed
Number Operations
MA-HS-1.3.1
Students will solve real-world and mathematical problems to specified accuracy levels by
simplifying expressions with real numbers involving addition, subtraction, multiplication,
division, absolute value, integer exponents, roots (square, cube) and factorials.
Measuring Physical Attributes
MA-HS-2.1.2
Students will describe how a change in one or more dimensions of a geometric figure
affects the perimeter, area and volume of the figure.
MA-HS-2.1.3
Students will apply definitions and properties of right triangle relationships (right triangle
trigonometry and the Pythagorean theorem) to determine length and angle measures to
solve real-world and mathematical problems.
Shapes and Relationships
MA-HS-3.1.3
Students will analyze and apply angle relationships (e.g., linear pairs, vertical,
complementary, supplementary, corresponding and alternate interior angles) in real-world
and mathematical problems.
MA-HS-3.1.7
Students will solve real-world and mathematical problems by applying properties of triangles
(e.g., Triangle Sum theorem and Isosceles Triangle theorems).
Patterns, Relations and Functions
MA-HS-5.1.1
Students will identify multiple representations (tables, graphs, equations) of functions
(linear, quadratic, absolute value, exponential) in real-world or mathematical problems.
Teacher Guide
Select Goals and
Student Guide
Objectives
Goals and
Objectives
(Specify
skills/information that
will be learned.)
Select
Instructional
Strategies –
Information
(Give and/or
demonstrate necessary
information)
Students will:
- use the Pythagorean formula to calculate lengths of
members of a tress that they design
- use trig relationships and vector properties to solve
for reactions to load applied to their truss, and for the
resulting internal member forces
- represent their truss in a scale drawing
- use computer models to simulate their truss, and
conduct a structural analysis using the computer
model
Materials Needed
 Paper
 Pencil
 Pre and post tests and worksheets (1
per student)
 Interwrite tablets
 Classroom computer with Internet
connection
Day 1
Catch: video of bridge failures (see list at bottom of lesson plan)
Inform of objectives:
Review:
• Pythagorean theory,
• Trig relationships (particularly sine and cosine),
• Vector properties (solving for component vectors, then combining vectors).
Introduce concepts (see http://www.eecs.usma.edu/bridgecontest/pdfs/la3.pdf for details)
• Equilibrium: Σ Fx = 0, Σ Fy = 0
• Structural models: represent the structure
° definitions (truss – structure made of joints connected by members, joint – point where members
connect, member – load carrying part of truss)
• Steps in constructing and analyzing models:
° draw truss members (to scale) and indicate joints with circles
° show loads and reactions ( Σ moments – forces times distances -- to get reactions)
• Calculate reactions and internal forces in members (using free body diagrams as necessary – these
consider one joint or truss section at a time, using information you know to solve for unknown pieces)
Demonstrate with truss sections from local bridges.
Talk with class about why bridges might be designed the way they are (site conditions, length of span,
need for clearance for ship traffic or other factors) also how bridge materials and their maintenance
requirements make a difference in design and life-time cost. Encourage them to look at bridges
around them and bring observations to discuss on Day 2.
Distribute and review truss design and analysis project.
Day 2:
Review results. Students whose bridge truss designs worked (resulting forces were within the
allowables) use the interwrite tablets and/or smartboard to demonstrate their design to the class using
the computer truss test software. If time is available, can use the computer software to show effect of
load placement (top chord vs. bottom). Discuss how site limitations and other factors can drive bridge
design (including loading).
Utilize Technology
•Classroom computer with projector
Optional: SmartBoard (interactive whiteboard
operates with classroom computer)
• Computer structural design and test program
(either Johns Hopkins truss or West Point Bridge –
this lesson lends better to Johns Hopkins, but could
be modified to use West Point Bridge)
• Internet bridge test videos and pictures of local
bridges
Interwrite tablets
(interactive with the
smartboard)
Other Resources
(e.g. Web, books, etc.)
Cincinnati bridges:
http://www.cincinnati-transit.net/
http://www.roadfan.com/cinbri.html
Design and test software programs:
Johns Hopkins truss program (free) http://www.jhu.edu/~virtlab/bridge/truss.htm
West Point Bridge Designer 2007 (free)
http://bridgecontest.usma.edu/download.htm
Other helpful sites:
http://www.bridgesite.com/funand.htm - links
to many sites, including bridge design tips at
http://abcdpittsburgh.org/kids/kids.htm
ModelSmart software to design and
simulate testing of structures (bridges,
towers) http://www.preengineering.com/modelsmart/ms.html
Require Learner
Participation
Project outlined in “Vectors
and Bridge Design Packet.”
Activity
(Describe the
independent activity to
reinforce this lesson)
Students will design a truss to meet specified criteria
(dimensions, loading). They will
- draw their truss structure (labeling load, reactions,
members, joints);
- solve for reactions;
- solve for member lengths and internal forces, using
vector analysis and trig properties;
- compare internal forces to “allowable” forces (what
the material of construction can withstand); and
- compute strength to weight ratio.
Students whose truss had only “allowable” forces in
all members will use the interwrite tablets to show
their structures to the class.
Evaluate
(Assessment)
(Steps to check for
student understanding)
Essential questions:
- how do you calculate
reactions? ( Σ moments –
forces times distances -- to
get reactions)
- if you know angle and one
side’s magnitude (either length
or force vector) how do you
calculate other sides? (trig to
define vector components and
side lengths);
- how do you solve for
unknown vector forces and
directions (create free body
diagrams, use Σ F = 0)
Pre-and post assessments
Truss Design Packets
Bridge Failures:
Tacoma Narrows
http://www.youtube.com/watch?v=j-zczJXSxnw
Additional Notes
I-35 W Minneapolis – August 2007
http://www.eng.uc.edu/bridge/videos/ - news clip about the UC competition, also some testing
http://www.teachertube.com/view_video.php?viewkey=4a8c7bbbf9015a755dc7 – toothpick bridge testing (failures)
Reflection after teaching: Student feedback (and scores on the post-test) indicated that I should have spent more time explaining internal forces
and free body diagrams. I took time in Day 2 to use the computer program to show the effect of top chord vs. bottom chord loading. We used
one of the designs that a student had created , and experimented with putting the load all on the top point, then all on the bottom point. Before
running the model I asked the students to predict what would happen – how the load would be distributed under each scenario. There was a large
difference, which was interesting to the students.
This lesson was intended to link with a planned Physics class and after school bridge competition project that did not end up occurring. I think the
lesson would have been much better if they had built actual bridges and tested them. This lesson could easily be the introduction and class-room
analysis of designs, giving clues about design to build.
Honestly, I think this lesson would have been better for undergraduate engineers, or a subgroup such as a bridge competition team rather than a
wide-spectrum math class. The students who liked it (ones who are interested in becoming engineers) really liked it, but others did not. Maybe
that’s a good conclusion -- for them to understand that structural design may not be the best choice for them.