Deflection of Beams

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Nov 15, 2013 (3 years and 9 months ago)

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Name:






Date:







#
-




M
EC

-

251


Strength of Materials



Deflection of Beams



The College of New Jersey









I. Introduction


The purpose of this experiment is to measure the deflec
tion of simply supported
and cantilever beams, as seen in Figures 1 and 2. The deflection will be recorded upon
application of several different loading conditions on beams with different material and
section. The experimental results are to be compared
with analytical results in order to
confirm the theories of beam deflection. This is accomplished by computation of the
percent difference (errors).


The main objective of the experiment is to conclude that deflection of beams is a
function of the follo
wing parameters:



1. Loading conditions



2. Type of structure (supports)



3. Material properties



4. Section properties (area moment of inertia)



5. Size (length)


Simply Supported Beam

w(x) = Deflection in Y Direction


浡m

=⁄ 晬fc瑩潮⁡琠te湴e爠潦 Bea洠⡦潲⁳y浭e瑲tc潡摩湧






Figure 1



Cantilever Beam

w(x) = Deflection in Y Direction


浡m

=⁄ 晬fc瑩潮⁡琠t牥e⁅湤 ⁂ea洠m




Figure 2



II. Equipment and Supplies

It is impo
rtant that lab members familiarize themselves with the following
equipment prior to starting the experiment.



-
Test frame and simple supports for beam analysis


-
Extensometer/dial indicators


-
Aluminum beam, (0.25 x 1 x 36 in)


-
Steel beam #1 (thin), (0.1
875 x 1 x 36 in)


-
Steel beam #2 (thick), (0.25 x 1 x 36 in)


-
Laboratory loading apparatus (support system and cradles)


III. Procedure


NOTE
: In the interest of time, two groups will be working on this experiment
simultaneously. Your group might be st
arting with either part A, simply supported beam
analysis, or part B consisting of experimentation on the cantilever specimens. For both
parts of this experiment, it is important to follow the steps shown to minimize the time of
experimentation.


WARNIN
G
: For all deflection scenarios, an excessive load or lengthy experiment
duration may permanently deform the specimen.


Part A
-

Simply Supported Beam Analysis (Refer to Figure 3 below)

1)

Set up the thin steel beam in a 34” simply supported arrangement so
th
at one end rests on the roller support while the other end rests on the
knife edge. (Although the beam is 36" long, we save 1" on each side
of the supports as overhang.)

2)

Install a dial indicator 10" from the left support and another at the
center of the b
eam (17" from the left support).

3)

Record the initial readings of the dial indicators.

4)

Install one of the five simply supported load distribution fixtures
(displayed in Appendix A). This is accomplished by carefully placing
the cradles on the supports.

5)

Recor
d the new values displayed on the dial indicators.

6)

Repeat steps 1 through 5 for both the thick steel and aluminum beams.

7)

Repeat steps 1 through 6 until all of the simply supported load
distribution fixtures have been tested on all three specimens.

8)

Once all

data has been collected, calculate the theoretical deflection
values at the points where the dial indicators had been placed.

9)

Using the theoretical and experimental deflection values, compute the
percent errors.

10)

Fill in the tables found at the end of this

lab guide




Figure 3. Simple Support Setup




Part B
-

Cantilever Beam Analysis (Refer to Figure 4 on the following page)

1)

Measure and record the weight of the thick steel specimen.

2)

Rotate the left support so that both supports are oriented in the same
direction.

3)

Place a 1/4" spacer in the right support

4)

Set up the thick steel beam in the left support bracket so that it
produces a 27” cantilever, but allow its free end to rest on the right
support for the time being.


5)

Install two dial indicators, the fir
st at the center of the cantilever and
the second at 26".


6)

Record the initial readings of the dial indicators while the beam is
resting on the right support.

7)

Install one of the three cantilever load distribution fixtures (displayed
in Appendix B). This is

accomplished by carefully placing the cradles
on the supports.

8)

Slowly slide the right support over and out of the way so that the
specimen acts as a cantilever. Make sure not to displace the support to
the point where the cradle is no longer braced by it
.

9)

Record the deflection on both dial indicators.

10)

It is critical that the support, loading and measurement setup remain as
unaltered as possible in order to eliminate minor disturbances for
comparison purposes. In order to do so, leave the loading bracket

on
the supports, and rearrange the point loads in order to create the
second load distribution.

11)

Record deflection values.

12)

Repeat steps 10 and 11 for the third and final load distribution.

13)

Once all data has been collected, calculate the theoretical deflect
ion
values at the points where the dial indicators had been placed. It is
important to note that for these cantilever scenarios, the weight of the
beam is acting as a uniform load across its span. This must be
addressed in the deflection calculations

14)

Us
ing the theoretical and experimental deflection values, compute the
percent errors.

15)

Fill in the tables found at the end of this lab guide.

Figure 4. Cantilever Support Setup




IV. Results and Discussion


Using equations from the back of the strength o
f materials text, calculate the
deflection for the simply supported beams subjected to uniform load,
linearly varying
triangle load, and sinusoidal load. In addition, determine theoretical deflection for the
three cantilever load scenarios. For the unpro
vided cases, such as the cosin
e
load
ing
,
develop the equation of deflection using integration method. The equation for the
linearly varying reverse triangle load is provided at the end of this hand out. Remember
to use superposition to calculate total de
flection when necessary. Compare theoretical
deflection data to the actual deflection obtained in this experiment in order to calculate
percent error.


How did experimental deflections compare to calculated values? For a given
loading, were all of the
percent errors for the three specimens either positive or negative?
If not, where might the discrepancy lie? While both the thin steel and aluminum
specimens were analyzed as simply supported beams, why was only the thick steel beam
tested as a cantileve
r? Comment on any sources of error that might be present in the
experiment.





Appendix A


Simply Supported Beam

With Uniform Loading

Beam

Loading

Deflection at

Center (in.)

Deflection 10" from left

support (in.)


Steel (Thin)

E=30x10
6

Measured (in.)

-
0.280

-
0.211

Calculated (in.)

-
0.276

-
0.222

Percent Error

-
1.45%

+4.95%


Steel (Thick)

E=30x10
6

Measured (in.)

-
0.116

-
0.0940

Calculated (in.)

-
0.120

-
0.0982

Percent Error

+3.33%

+4.28%


Aluminum

E=10x10
6

Measured (in.)

-
0.329

-
0.264

Calculated

(in.)

-
0.334

-
0.268

Percent Error

+1.50%

+1.49%









Simply Supported Beam

with Linearly Varying Triangle Load

Beam

Loading

Deflection at

Center (in.)

Deflection 10" from left

support (in.)


Steel (Thin)

E=30x10
6

Measured (in.)

-
0.598

-
0.462

C
alculated (in.)

-
0.630

-
0.502

Percent Error

+5.08%

+7.97%


Steel (Thick)

E=30x10
6

Measured (in.)

-
0.256

-
0.214

Calculated (in.)

-
0.274

-
0.243

Percent Error

+6.57%

+11.93%


Aluminum

E=10x10
6

Measured (in.)

-
0.757

-
0.564

Calculated (in.)

-
0.762

-
0.
609

Percent Error

+0.66%

+7.39%








Simply Supported Beam

with Linearly Varying Reverse Triangle Load

Beam

Loading

Deflection at

center (in.)

Deflection 10" from left

support (in.)


Steel (Thin)

E=30x10
6

Measured (in.)

-
0.465

-
0.368

Calculated
(in.)

-
0.475

-
0.384

Percent Error

+2.11%

+4.17%


Steel (Thick)

E=30x10
6

Measured (in.)

-
0.197

-
0.154

Calculated (in.)

-
0.207

-
0.164

Percent Error

+4.83%

+6.10%


Aluminum

E=10x10
6

Measured (in.)

-
0.576

-
0.462

Calculated (in.)

-
0.575

-
0.464

Perce
nt Error

-
0.17%

+0.43%





Simply Supported Beam

with Sinusoidal Loading

Beam

Loading

Deflection at

center (in.)

Deflection 10" from left

support (in.)


Steel (Thin)

E=30x10
6

Measured (in.)

-
0.642

-
0.513

Calculated (in.)

-
0.712

-
0.569

Percent Err
or

+9.83%

+9.84%


Steel (Thick)

E=30x10
6

Measured (in.)

-
0.270

-
0.216

Calculated (in.)

-
0.287

-
0.230

Percent Error

+5.92%

+6.08%


Aluminum

E=10x10
6

Measured (in.)

-
0.783

-
0.631

Calculated (in.)

-
0.861

-
0.688

Percent Error

+9.06%

+8.28%











Simply Supported Beam

with Cosin
e
Loading

Beam

Loading

Deflection at

center (in.)

Deflection 10" from left

support (in.)


Steel (Thin)

E=30x10
6

Measured (in.)

-
0.453

-
0.363

Calculated (in.)

-
0.711

-
0.417

Percent Error

+36.3%

+12.9%


Steel (Thick)

E=30x10
6

Measured (in.)

-
0.188

-
0.152

Calculated (in.)

-
0.310

-
0.183

Percent Error

+39.4%

+16.9%


Aluminum

E=10x10
6

Measured (in.)

-
0.556

-
0.451

Calculated (in.)

-
0.861

-
0.502

Percent Error

+35.4%

+10.2%








Appendix B


Cantilever Beam

With
Uniform and End Loading

Beam

Loading

Deflection at

center (in.)

Deflection at

end (in.)


Steel (Thick)

E=30x10
6

Measured (in.)

-
0.198

-
0.581

Calculated (in.)

-
0.213

-
0.588

Percent Error

+7.04%

1.19%













Cantilever Beam

With Partial Uniform

Load and Two Point Loads (1)

Beam

Loading

Deflection at

center (in.)

Deflection at

end (in.)


Steel (Thick)

E=30x10
6

Measured (in.)

0.182

-
0.549

Calculated (in.)

0.190

-
0.556

Percent Error

+4.21%

+1.26%












Cantilever Beam

With Partial Un
iform Load and Two Point Loads (2)

Beam

Loading

Deflection at

center (in.)

Deflection at

end (in.)


Steel (Thick)

E=30x10
6

Measured (in.)

-
0.104

-
0.301

Calculated (in.)

-
0.113

-
0.317

Percent Error

+7.96%

+5.05%