136561 Mechanics of Materials Beam Deflection Test

8/12/2019 136561 Mechanics of Materials Beam Deflection Test

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Mechanics of Materials Laboratory

Beam Deflection Test

David Clark Group C:

David Clark

Jacob Parton

Zachary Tyler

Andrew Smith

10 !0 !00"


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#bstract

If a beam is supported at two points, and a load is applied anywhere on the beam,

the resulting deformation can be mathematically estimated. Due to improperexperimental setup, the actual results experienced varied substantially when compared

against the theoretical values. The following procedure explains how the theoretical and

actual values were determined, as well as suggestions for improving upon the experiment.

The percent error remained relatively small, around 10%, for locations close to supports.

s much as !0% error was experienced when analy"ing positions closer to the center of

the beam.

#


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Table of Contents

1. Introduction & ack!round ............................................................ "

1.1. #eneral ack!round .............................................................. "1.$. Determination o% Curvature .................................................... "

1. . Central 'oadin! ..................................................................... "

1.". (verhan!in! 'oad) ................................................................ *

$. +,ui-ment and Procedure ............................................................

$.1. +,ui-ment ..............................................................................

$.$. +/-eriment Setu- ...................................................................

$. . Central 'oadin! ...................................................................... 0

$.". (verhan!in! 'oad) ................................................................ 0

. Data Analy)i) & Calculation) ....................................................... 2

.1. Central 'oadin! ...................................................................... 2

.$. (verhan!in! 'oad) .............................................................. 11

". 3e)ult) ........................................................................................ 1$

4. Conclu)ion) ................................................................................. 1

*. 3e%erence) .................................................................................. 1"

. 3aw 5ote) ................................................................................... 1"

!


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1$ %ntroduction & Back'round

1$1$General Back'round

If a beam is supported at two points, and a load is applied anywhere on the beam,

deformation will occur. $hen these loads are applied either longitudinally outside or

inside of the supports, this elastic bending can be mathematically predicted based on

material properties and geometry.

1$!$Determination of Curvature

urvature at any point on the beam is calculated from the moment of loading &'(,

the stiffness of the material &)(, and the first moment of inertia &I.( The followingexpression defines the curvature in these parameters as 1*+, where + is the radius of

curvature.

I E M =

1

Equation 1

) uation 1 does not account for shearing stresses.

urvature can also be found using calculus. Defining y as the deflection and x as

the position along the longitudinal axis, the expression becomes

#!

#

#

#

1

1

+

=

dxdy

dx yd

Equation 2

1$($ Central Loadin'

entral loading on a beam can be thought of as a simple beam with two supports

as shown below.

-


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Figure 1

pplying e uilibrium to the free body e uivalent of igure 1, several expressions

can be derived to mathematically explain central loading.

#0

##0

0

P R R P R F

P R L R

L P M

R F

aycay y

C C A

ax x

=+==+

=+====+

Equation 3, 4, and 5

igure # and ! act as free body diagrams for the section between / and /

respectively.

Figure 2

Figure 3

olving the reactions between / and / , e uation 1 can be expressed as


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L x L L P x P

dx yd

I E

L x

x P dx

yd I E

+=

=

###

#0

#

#

#

#

#

Equation 6, 7

Integrating twice, ) uation 2 becomes

L x L

C xC x L P x P

y I E

L xC xC

x P y I E

+++=

++=

#-1#

#0

1#

-!

#!

#1

!

Equation 8, 9

To determine the constants, conditions at certain positions on the beam can be

applied. 3nowing the deflection at each of the supports, as well as the slope at the top of

the curve is "ero, the constants can be derived to

-412!

012

!

-#

!#

#

1 L P

C L P C C L P

C ====

Equation 10, 11, 12, and 13

ombining ) uations 4 and 5 with 10 through 1!, the expressions for deflection

can be expressed as

L x L L P x L P x L P x P

y I E

L x

x L P x P y I E

++=

=

#-412!

-1#

#0

121#!##!

#!

Equation 14, 15

1$)$*verhan'in' Loads6verhanging loading on a beam is similar to that of central loading. In

overhanging loading, a simple beam is supported with two supports and two loads as

shown below.

2


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Figure 4

7sing similar methods used previously for central loading, the e uation for

determination of deflection as a function of position, load, length, stiffness, and geometry

can be derived as

( ) ( ) L x xba L P xa P ba L x P

y I E ++= 0#2#2

#!

Equation 16

!$ +,uipment and -rocedure

!$1$+,uipment

1$ .rame /ith Movable nife +d'e upports

#. Metal beam 8 In this experiment, #0#-9T2 aluminum was tested. The beam

should be fairly rectangular, thin, and long. pecific dimensions aredependant to the si"e of the test frame and available weights.

!. Calipers2 Dial Ga'es2 and a Tape Measure 8 alipers should be used to

measure the width and thic:ness of the beam. Dial gages will be used to

measure deflection along the length of the beam. The tape measure is used

to measure the length of the test region.

-. 3an'ers and 4ei'hts 8

!$!$+5periment etup

et the :nife supports at determined positions along the frame and mount the

beam to be tested. The material, width, thic:ness, and length between supports should be

measured and recorded for later use.

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!$($Central Loadin'


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tre) "*)e +oad ($#! 'age 1 (in! 'age 2 (in! 'age 3 (in! 'age 4 (in!2,500 6,500 13,500 17,500

6 Actual 6.666 6.666 6.666 6.666 6.6666 Theoretical 6.666 6.666 6.666 6.666 6.666

6 +rror 6.668 6.668 6.668 6.6681 Actual 6.116 96.66& 96.660 96.660 96.66"1 Theoretical 6.116 96.66& 96.66* 96.66* 96.66&1 +rror 16.2$8 &1.&18 &1.&18 "2. 28$ Actual 6.*16 96.61 96.6"1 96.6"1 96.612$ Theoretical 6.*16 96.614 96.6&" 96.6&" 96.614$ +rror 1&.&48 $1.&*8 $1.&*8 $0.&68& Actual 1.116 96.6&6 96.6 * 96.6 4 96.6&"& Theoretical 1.116 96.6$ 96.6*1 96.6*1 96.6$& +rror 2.2$8 $&.*$8 $$.668 $*.1 8" Actual 1.*16 96.6"4 96.116 96.116 96.646

" Theoretical 1.*16 96.6"6 96.602 96.602 96.6&2" +rror 1&.*08 $&.&*8 $&.&*8 $ .2$84 Actual $.116 96.642 96.1"4 96.1"4 96.6**4 Theoretical $.116 96.64$ 96.11 96.11 96.6414 +rror 1&. &8 $".608 $".608 $0.0"8

De&$ection Data &or -entra$ +oading

"a#$e 3

De&$ection .esu$ting on a -entra$$* +oaded Beam

96.1*6

96.1"6

96.1$6

96.166

96.606

96.6*6

96.6"6

96.6$6

6.666

6.666 $.666 ".666 *.666 0.666 16.666 1$.666 1".666 1*.666 10.666 $6.666

%osition (inc es!

D e

& $ e c

t i o n

( i n c

e s

!

Ste- 6Ste- 1Ste- $Ste- &Ste- "Ste- 4Ste- 6 TheoreticalSte- 1 TheoreticalSte- $ TheoreticalSte- & TheoreticalSte- " TheoreticalSte- 4 Theoretical

Figure 7

10


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($!$*verhan'in' Loads

$6.6661.6*6

6.1"61&.6661&.666Di)tance %rom ri!ht )u--ort to ed!e

Beam DimensionsTe)t 'en!th

7idth

Thickne))Di)tance %rom le%t )u--ort to ed!e

"a#$e 4

/1 $.4/$ 16/& 1 .4

%osition o& gages

"a#$e 5

tre) "*)e +oad ($#! 'age 1 (in! 'age 2 (in! 'age 3 (in!2,500 10,000 17,500

6 Actual 6.666 6.666 6.666 6.6666 Theoretical 6.666 6.666 6.666 6.6666 +rror 6.668 6.668 6.6681 Actual 6.116 6.61" 6.6&4 6.6141 Theoretical 6.116 6.61$ 6.6$0 6.61$1 +rror 1$.0$8 $&."68 $6.008

$ Actual 6.&&6 6.6"& 6.16 6.6"0$ Theoretical 6.&&6 6.6& 6.604 6.6&$ +rror 14.418 $4. 48 $0.2"8& Actual 6.446 6.6 6 6.1 $ 6.6 *& Theoretical 6.446 6.6*$ 6.1"$ 6.6*$& +rror 1$.0$8 $1.$08 $$."28" Actual 6. 6 6.161 6.$4$ 6.11$" Theoretical 6. 6 6.60 6.122 6.60" +rror 1*.$ 8 $*.2$8 $0.2"84 Actual 6.226 6.1&1 6.&$6 6.1"$4 Theoretical 6.226 6.11$ 6.$44 6.11$

4 +rror 1 .&68 $4.&*8 $ .148

De&$ection Data &or / er anging +oads

"a#$e 6

11


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De&$ection .esu$ting &rom / er anging +oads

6.666

6.646

6.166

6.146

6.$66

6.$46

6.&66

6.&46

$ " * 0 16 1$ 1" 1* 10

%osition (inc es!

D e

& $ e c

t i o n

( i n c

e s

!

Ste- 6Ste- 1Ste- $Ste- &Ste- "Ste- 4

Ste- 6 TheoreticalSte- 1 TheoreticalSte- $ TheoreticalSte- & TheoreticalSte- " TheoreticalSte- 4 Theoretical

Figure 8

)$ 6esults

The theoretical results were not as expected or experienced. There was significant

error between the actual results and theoretical value, especially as the distance studied

approached the midpoint of the beam. Though the difference in inches was small, the

percent error could be as high as !0%.

The main source of error within this experiment occurs due to the improper

testing procedure. s seen in igure 5, the theory used within this exercise is based upon

a beam with one fixed support allowing one degree of freedom, a second support

allowing two degrees of freedom, and a central load.

1#


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Figure 9

This produces dramatically different results when compared against the actual

setup. $hen using two :nife supports, the setup contains two supports allowing two

degrees of freedom and a central load. This is pictured in igure 10.

Figure 10

ince both ends are under9constrained, the analysis for the experiment with the above

theory is not accurate.

nother cause of error in the theoretical is the effect of gravity on the beam. $ith

no applied load, the e uations above would return a "ero result. This is inaccurate for

beams that are not specifically supported such that gravitational factors are overcome.

7$ Conclusions

$hen an load is applied to a beam, either centrally over at another point, the

deflection can be mathematically estimated. Due to the error that occurred in this

exercise, it is clear that margins in safety factors, as well as thorough testing, is needed

when utili"ing beam design. It is also important to ensure the scope of the testing closely

models real9world practicality.

1!


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"$ 6eferences

>ilbert, ?. and . @. armen. A hapter 11 B /eam Deflection Test.A ' )* ) !;0 B

'echanics of 'aterials @aboratory 'anual. ?une #000.

8$ 6a/ 9otes

Figure 11

1-


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Figure 12

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136561 Mechanics of Materials Beam Deflection Test

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