Engineering 101
Linking Experiments to Models through the
Bridge Design Exercise
Prof. Subramaniam (“Subby”) D. Rajan, Prof. Narayanan Neithalath and Amie
Baisley
Graduate Students: Kirk Vance, Matt Aguayo, Tejas Ashani, Joseph Harrington and
Canio Hoffarth
electrical, computer and energy engineering
20-Jan-2010
What are Experiments?


Tests to determine the relationship between (input) variables
and (output) responses
Example 1: What is the effect of dowel diameter on the weight
of the bridge?
– Model: The entire bridge system
– Input Variable: Dowel diameter
– Output Response: Weight of the bridge

Example 2: What is the effect of dowel diameter on the
maximum deflection of the bridge deck?
– Model: The entire bridge system
– Input Variable: Dowel diameter
– Output Response: Deflection of the bridge deck at various locations

2
What are Models?

Relationship between (input) variables and (output) responses
– Simple equation
– Model described by one or more complex equation(s) – differential
equation(s), integral equation(s), …

Example 1: What is the effect of dowel diameter on the weight
of the bridge?
n
d2 
WBRIDGE  WOTHER  WDOWELS  WOTHER   g i Li  i 
i 1
 4 

Example 2: What is the effect of dowel diameter on the
maximum deflection of the bridge deck?
– Needs a model whose solution can be described by several linear,
algebraic equations
3
What is a System?

Dictionary definitions
– a set of connected things or parts forming a complex whole, in particular
– a set of principles or procedures according to which something is done; an
organized scheme or method

Traits of a system
– has structure, its parts or components are directly or indirectly interact
with each other
– has behavior (where input and output are linked)
4
Questions


Q1: Draw a diagram that shows the components of the bridge
system, establishes the boundary and identifies the
surroundings.
Q2: Describe the bridge system with particular attention to (a) its
functionalities, (b) how the different components interact with
each other and (c) how the bridge system behaves.
5
Engineering Process or Product Design
Analysis
Experiments
Analysis
Model
Engineering Process
or Product
Design
Optimization
Toolbox
Design
Model
6
Verification and Validation


Models need to be validated and verified before they can be
used with any confidence
Verification: Are you building it right?
– Is the theory/principle embodied in the model implemented correctly?
F  ma
g = 9.81 m/s2
7
Verification and Validation

Validation: Are you building the right thing?
– Do the results from the model correlate well with experimental results?
Trial
M
m
(kg) (kg)
Exp.
a
(m/s2)
Model
a
(m/s2)
% error
1
2
3
8
Questions





Q3: Describe what a bridge model could be, by identifying the
input variables and output responses.
Q4: Identify the characteristics of each input variable. Describe
how you would obtain the values of these variables.
Q5: Identify the characteristics of each output response. What is
the purpose of each output response?
Q6: Give examples of engineering processes and products?
Q7: Describe the linkages between experiments and modeling.
9
Case Study
10
Case Study

Develop a model to predict the tip deflection (displacement) of a
cantilever beam due to a tip load. Use experiments to validate
the model.
y, v
P
A
B 
x
B
L
11
Case Study: Basic Steps



Use a sound scientific or engineering principle to develop the
model. What parameters will be a part of this model – input and
output variables?
Design experiment(s) to verify the model.
Design experiment(s) to validate the model.
12
Case Study: Principle/Theory

Euler-Bernoulli Beam Theory (w/o derivation)
Differential
Equation
d 2v( x)
M ( x)

2
dx
E ( x) I ( x)
v(x): vertical displacement
M(x): Bending moment
E(x): Young’s modulus
I(x): Moment of inertia
L: length of the beam
y, v
M
A
M
v
B
x
dv

dx
u
L
Boundary
Conditions
v( x  0)  0
v( x  L)  0
13
Case Study: Cantilever Beam
y, v
P
A
B
x
L
Boundary
Conditions
v( x  0)  0
dv
( x  0)  0
dx
Integrating
twice and
using the BCs
2
Px
v( x) 
 x  3L 
6 EI
14
Case Study: The Model
2
Px
v( x) 
 x  3L 
6 EI
Para. Remarks
P
The applied load at the tip of the beam
E
Material property that needs to be
found
I
Cross-sectional property that needs to
be computed
L
Length of the beam that needs to be
measured
x
Location where the displacement is
computed
15
Case Study: Modulus of Elasticity

What is modulus of elasticity or Young’s modulus (E)?
– In a one-dimensional state of stress it is constant of proportionality
between the normal stress and the normal strain and has the units of
stress.
Stress-strain curve
(ductile material)

4
1
2
3
5
E

16
Case Study: Moment of Inertia

What is moment of inertia, I?
– The second moment of area (or, moment of inertia) is a measure of a
beam’s cross-sectional shape’s resistance to bending.
y
Y
Yc
I x   y 2 dA
A
dx
x
x
C
y
dy
Xc
I y   x 2 dA
A
h
x
w
y
O
X
wh3
I x   y dA 
12
A
2
w3 h
I y   x dA 
12
A
2
17
Experiment
Measure the width, w, and thickness, t, of a
steel plate
y
t
w
z
18
Raw Measurement Data
Caliper 1
Width (W)
(in)
1.114
1.1135
1.1145
1.1145
1.114
1.113
1.115
1.114
1.113
1.113
1.113
Thickness (T)
(in)
0.03
0.03
0.0305
0.03
0.0305
0.0305
0.0305
0.03
0.03
0.0305
0.0305
Caliper 2
Width (W)
(in)
1.115
1.115
1.115
1.115
1.115
1.115
1.114
1.114
1.113
1.113
1.113
Thickness (T)
(in)
0.031
0.034
0.029
0.03
0.034
0.033
0.032
0.031
0.031
0.031
0.031
Measurements taken at 11 different locations
19
Raw Measurement Data
Histogram Plot
20
Statistical Analysis of Data
Caliper 1
Width (in)
Caliper 2
Thickness (in)
Width (in)
Thickness (in)
# of readings
(n)
11
11
11
11
Mean
1.1138
0.0303
1.1143
0.0315
Median
1.114
0.0305
1.115
0.031
Standard
Deviation
0.0007198
0.0002611
0.000905
0.00157
 : mean
 : standard deviation
1 n
   xi
n i 1
1 n 2
n
2

x



i
n  1 i 1
n 1
21
Questions





Q8: What is sample size?
Q9: What is mean? What is another name for mean?
Q10: What is median?
Q11: What is standard deviation?
Q12: Write a few sentences on the quality of the thickness and
width data for the steel plate.
22
Normal Distribution
Probability Density
Function*
1
f X  x,  ,   
e
 2
  x   2 


2
 2 


Function whose graph is a
continuous curve over a range of
values that x can take. It has the
units of probability rate (not
probability). x is called random
variable.
Area under curve between x1 and
x2 gives the probability that x lies
in the interval x1 and x2.
6
68-95-99.7 rule: 1, 2, 3 standard deviations from mean
*Excel terminology: Probability Mass Function
23
Cumulative Distribution Function
x
FX ( x) 


f X (z) dz
What is the probability that a
random width value is between
1.113 in and 1.114 in?
Pr[1.113  x  1.114]
 FX (1.114)  FX (1.113)
 0.6  0.15  0.45
24
Questions


Q13: Normal distribution is often called bell curve. Are there
other types of distribution?
Q14: Identify and rank the effect of the random variables in the
equation for tip deflection.
Px 2
v( x) 
 x  3L 
6 EI
25
Experiment 2
Measure the tip displacement of an
aluminum cantilever beam
26
Raw Experimental Data
27
Case Study: Model Verification
28
Case Study: Model Validation
Published Elastic Modulus of Aluminum (6016-T6) = 1.01(107) psi
E Published  E Computed
Diff 
E Computed
29
Forensic Engineering
30
One-Parameter Regression Analysis


Objective: Use the model and experimental data to determine
the Young’s modulus of aluminum.
Find E
to min f ( E )    
n
i 1
exp
i

FEA
i

2
E L  E  EU
31
References





Do an internet search using these keywords – system, model, experiment,
verification, validation, statistical quantities.
Engineering Statistics: http://www.itl.nist.gov/div898/handbook/
http://www.mathsisfun.com/links/curriculum-high-school-statistics.html
http://www.stevespanglerscience.com/lab/experiments
http://en.wikipedia.org/wiki/Verification_and_validation
32