Deflection of Engineering Structures

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Nov 15, 2013 (3 years and 8 months ago)

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Deflection of Engineering Structures

Objectives

The objectives of this laboratory are to:

1.

Experimentally determine the deflection of a cantilever beam due to applied loads.

2.

Quantify the observed deviations from the assumed linearity and superposition of
so
lutions.

3.

Discuss results in terms of relative error.

Deflection, Stiffness, and Linearity

All engineering materials deflect under load, but we can look at the problem in two distinct
ways, giving us two experiments for ME 221. In the Deformation of Materia
ls Experiment, you are
introduced to
the
distortion of material that is being stretched (loaded). In that lab you see the
effects of loading on the shape of a grid marked on a “specimen” of rubber sheeting. That lab
introduces the effect of geometry and th
e concepts of strain and stress. These concepts are used
heavily in understanding why things fail

why they break. This is almost always a first topic of
lecture in an Engineering mechanics class.

The second way to look at deflection is in terms of large sc
ale structures, that is, “real
parts” like beams, floors, engine parts

you name it. Sometimes these deflections are desired and
sometimes they are not

it depends on the application. This lab will introduce you to the topics of
linear beam
bending

theory an
d superposition. As part of that, we’ll introduce some techniques for
looking at trends in data, e.g. fitting lines to see how linear the data

is
.

At the same time, we’ll have a little fun with the problem by looking at how to design an
accurate “spring” s
cale.

Beam Bending

In Engineering
M
echanics, the lectures about the analysis of beams almost always
immediately follow the lectures about stress and strain. That’s because it is an important topic too.
First, we don’t want things to break, but after that
we want to know what will happen when the load
is applied to the structure. And one of the simplest structures is a beam. We usually think of 4x6s
and I
-
beams as typical beams, but the results are applicable to other things such as shafts, axles,
cranks, l
evers, and leaf (flat) springs.

The analysis almost invariably begins with a
nonlinear

differential equation relating the
slope of the beam (i.e. the first derivative of the equation that gives us the shape of the deflected
beam) to the applied load. Bec
ause the nonlinear equation makes doing a meaningful analysis
difficult it is
linearized

by “neglecting” certain terms that are “small”. This is a common technique in
engineering analysis and
the
result is calculus you can handle that gives simple
r

equatio
ns that are
still quite useful.

A simple example would be a
cantilever

beam: a straight beam firmly supported at one end,
with the other end free (unsupported), and having a load applied at the free end (e.g. like a diving
board). Equation 1 is the deflec
tion of the free end of this beam, which in this case moves more
than the rest of the beam, hence it is named y
max
. Equation 2 is the deflection of a second example:
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the deflection of free end of a cantilever beam if the load is applied somewhere else alon
g the
beam. As you can see, loading at x=L gives us Equation 1.



EI
3
FL
=
y
3
max

(1)




x
L
EI
Fx
y
-
3
6
2
max


(2)


where

F = applied load,

E = material modulus of elasticity,

I = cross
-
section area moment of inertia,

L = length of beam, and

x = p
osition along beam (always between 0 and L).


What do equations 1 and 2 tell us? Let’s look at the factors in each. The cross section area
moment of inertia, I, is a quantity you met in Statics that represents the effects of the cross
sectional geometry of

the beam. The modulus of elasticity is a property of the material, whether it
be steel, aluminum, or something else. These two, or their product, for that matter, give us a basis
for comparing beams. Beams made of higher modulus materials (steel instead o
f aluminum or
wood) will deflect less

they’re more stiff for the same geometry.

But increasing area moment of inertia can make the beam more stiff as well. I
-
beams
achieve a very high moment of inertia without using a lot of material (i.e. low weight). For

solid
rectangular cross section beams, the moment of inertia is proportional to the thickness (or depth of
the beam)
cubed

a beam twice as deep as another is 8 times stiffer.

Another useful insight we gain from the Equation 1 is the effect of length of th
e beam. The
stiffness of a given beam is inversely proportional to the length cubed. A beam half the length of
another is 8 times stiffer. Practically speaking, if making a beam twice a
s

deep in order to stiffen it
is too expensive, maybe we can get what w
e want by simply adding supports along the beam.

What do we do when we are dealing with a beam that is supported or loaded differently
than the cantilever beam we’ve talked about so far? You can always start from the basics (the linear
differential equatio
n) and derive the result. That’s plenty of work. (You’ll get to do some for
homework exercises when you take engineering mechanics.) Or you can look it up

deflection
equations for most of the simple problems are readily available in engineering texts and h
andbooks.
If you can’t find your specific problem in a handbook, you can turn to superposition.

Superposition

Now we get to another advantage of the linear beam bending theory: superposition. An
important result you may recall from a math course in differ
ential equations is that solutions to the
same
linear
differential equation for different “inputs” (e.g. loading points) can be added to find the
effect of those inputs acting simultaneously. What does it mean? Simply this: beam problems
involving multiple

loads are solved by analyzing each load individually, which is usually easy
because the solutions to the smaller problems can be found in a handbook, and combining the
results.

This is the principl
e

of superposition, and it saves engineers a lot of work:

if you
can’t
find
your particular problem in the tables, find its “parts” (usually just easier problems) and add the
results.

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This all depends on the validity of our linearization of the beam bending problem. When the
deflections start getting large relat
ive to the depth of the beam, things start deviating from our
idealization. How much they deviate is something you’ll see in the experiment.

Linear Fits and Residual Error

Since we’re talking about linear responses, we’ll take the opportunity to discuss ho
w you
analyze supposedly linear data. Our immediate application is the bending of beams, but it is also
needed in the calibration of sensors and measuring devices. When looking at data that is linear, we
are usually interested in the slope of the line thro
ugh the data. This is our
sensitivity
, or in the case
of beam bending, the reciprocal of the stiffness. This slope is found by doing a
regression
, which
can be done with a hand calculator or computer software like Matlab and Excel
1
.

The next problem after
getting the sensitivity is to quantify the quality of the fit. One method
is simply to plot the
residual error
, the difference between our actual data and the line we fit to it. In
a spreadsheet we simply set up
four
columns for the inputs (e.g. applied lo
ads), the outputs
(measured deflection), a predicted output (using the results from the regression) and the difference
of the two “outputs” (the residual error). Usually the residual error means more if we convert it to a
percent relative error
: divide the

residual error by the range of the output (i.e. the difference
between the maximum output value and the minimum value) and multiply by 100.

The plot of the either the residual error or the percent relative error can tell us a few things.
First, if we get
a straight horizontal line, we have an exact fit with no error. That’s unlikely: in the
Metrology lab you were introduced to precision error and here we are certainly likely to see some
scatter due to our resolution, uncertainty in the masses, and mistakes
. Second, if the plot of the
error shows a distinct curve (i.e. looks like you could fit a curve to the data), then we know our data
is probably not linear. It may also still show precision error.

Finally, the plot of the residual error gives us a useful s
tatistic the nicely sums up the quality
of the data. Simply look at the chart, pick off the maximum excursion from zero error (it might be
negative

take the absolute value) and we have a measure of the accuracy of device. For example,
we may observe a 0.5%

maximum relative error. That means we can expect to be able to predict
the output, for a given input (e.g. an applied load), to within +/
-
0.5%.

The “percent error” part of this puts some perspective on the “quality” of devices in general.
For example, if

I wish to purchase an electronic scale, I’ll quickly find out that scales accurate to
0.2%, regardless of whether that’s for 10 gram or 10 kilogram capacity, are common and cheap,
around $100. Scales to 0.01% are more expensive, around $500. Scales to 0.0
01%, cost even
more. And so on.

We’ll see today how well we do with a “homemade” spring scale based on a cantilever
beam.

Procedure

You will be testing the accuracy of a cantilever beam as a spring scale. After some
demonstration of the concepts of stiffn
ess, linearity and superposition, you’ll take some data to
calibrate the spring scale and look at the question posed in the assignment. You will apply a



1

You can always plot data and estimate the slope
by hand, but we’re talking about numerical
analysis here.

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number of loads at the end of the beam, and deflection will be measured by two different methods.
How d
ifferent can they be, you wonder? You’ll see

one method is almost 10 times more accurate
than the other, yet both use the same measurement device (a simple plastic ruler). Also, as a sort
of a contest, you will work out the calibration so you can predict t
he weight of two unknown objects.

Assignment

You’ll be able to do most of the analysis in the lab under the guidance of your TA. Be sure
to bring a floppy disk
or thumb drive
so you can take the results with you. At least one member in
each group should br
ing a laptop computer for analyzing the data during the lab period.

Next week we will discuss the writing assignment in lab. In the meantime, prepare two
figures on one page with the lab title in the header and your name, section and date as for all
assig
nments. For the first figure, plot the residual errors in predicted mass in grams that you
calculated for the measurements with the pointer and without. That’s two data series in the same
graph. In the second figure, plot the residual errors in predicted d
isplacement in millimeters for
measurements with the pointer and without.

The x axis data for both figures will the applied mass in
grams.

For each figure,

use the user
-
defined style we created for the Sample Analysis. Then
format each series as
circle
mar
kers with no line segments
, sized 7 point and colored black
. For the
data with the pointer
,

use filled (solid) circles. For the data without the pointer, use empty circles
(white
back
ground
, black foreground
)
. For each series, add a
2
nd

order trend line
wi
thout the
equation. Right click on each trend line and format them to use the next
-
to
-
thinnest
line
weight
.

In the legend
of each figure
(add it if you don’t have it already), we want the series titles to
be “With Pointer” and “Without Pointer”,
or
“Fixed

Scale” and “Moveable Scale”, or something
similarly descriptive and short. If you selected the
column headings with the data when you created
the graphs, you can edit these headings and they’ll be updated in the legend. Otherwise,
right click
in the figur
e and select “Source Data” and change the series names.
Now
left click once on the
legend, then right click on
a
legend entry

for one of the trendlines and select “clear”. This deletes
the trendline from the legend. Do the same for the other trendline.


F
inally, format all legend entries and axis numbers as 12 point font. Set the axis titles to 14
point font
. (W
e’ll let you try
writing these

titles.
)

When you paste to Word, the recommended
method is “Paste Special” as an enhance metafile picture.
Select “F
ormat Picture” (not “Edit
Picture”) and set them to 5.5”
wide
(or 5” if necessary), with locked aspect ratio. W
rite
descriptive
figure titles in Word, but be sure they mention that the figure shows 2
nd

order trendlines with the
data.
No other text is neede
d at this time.

Turn this in with your Deflection and Superposition Worksheet.

References

Shigley, Joseph E., and Larry D. Mitchell,
Mechanical Engineering Design,

4
th

Edition,
McGraw
-
Hill, New York. 1983.