Cantilever Beam Essay

1. Introduction

During this lab a beam was tested in order to happen the relationships between burden. flexing minute. emphasis and strain. incline and the warp in a cantilever beam which was the chief aim. The chief intent was to understand the cardinal rules that have to be taken into history before planing and fabricating a beam or utilizing one as portion of a design.

2. Theory
The theory behind this lab can be categorized to 2 different subjects. flexing minute and emphasis being the first. the 2nd being incline and warp. Each one is discussed below:

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2. 1 Bending Moment and Stresses

Bending minute is a minute produced by a burden applied on a surface that causes it to flex. In the instance of the lab the surface is the beam and with the applied at the terminal of it. In order to cipher the bending minute it is necessary to pull a Free Body Diagram bespeaking all the forces applied upon the beam. Picture 1 is the free organic structure diagram of the beam. F is the burden applied. R is the reaction from the clinchs ( the saloon is fixed on the bench by two G clinchs ) and M is the bending minute. Since we need to hold equilibrium and no forces on the x-axis there are two equations we should utilize the undermentioned equations: If we resolve the perpendicular forces for the whole beam

?FY=0-F-R=0 –
F=R ( 1 ) By taking minutes anti-clockwise about right manus side for the whole beam WL+ M=0 –
M=-WL ( 2 )
Equation ( 2 ) is the flexing minute of the beam ( measured in Nm ) . while the equation ( 1 ) describes the sheer force. Picture [ 2 ]
For the bending emphasis in the beam it is helpful to cognize the 2nd minute of country ( I ) foremost. The form of the saloon is rectangular as indicated on Figure



Iz=y2dA=-d2d2y2 bdy=by33-d2d2=bd324+bd324-

Iz=bd312 ( 3 )

The Second minute of country is measured in mm4
The bending emphasis is ?x=?xE where ?x is the strain and E is the Young’s modulus. ( The emphasis is measured in N/m2 ) There is a last equation that connects flexing emphasis with flexing minute. MI=?xy=ER|

2. 2 Deflection and Slopes

In order to happen the equations for incline and warp it is necessary to utilize expression ( 1 ) and ( 2 ) . It is known that M =-EId2vdx2. Thus the bending equation can be rewritten as -EId2vdx2=-Wx ( x is a random length of the beam ever smaller than the whole length L ) . If it is integrated once we will acquire the incline equation
-EIdvdx=-W2x2+C1 ( 4 ) If we integrate once more ( 4 ) so we have deflection equation -EI=-W6 x3+C1x+C2 ( 5 )

By infixing boundary conditions to find the invariables of integrating the concluding equations for the beam incline and beam warp can be determined. At x=L: dv/dx=0 and v=0
4-0=-W2 L2+C1-
C1=WL22

5-0=-W6 L3+WL22 L+C2-
C2=-WL33
Therefore 4 and 5 can be re-written as
Beam Slope: dvdx=W2EI ( x2-L2 ) [ m ]
Beam Deflection: v=W6EI L-x2 ( 2L+x ) [ m ]



3. Equipment
The equipment used during the lab session is the followers:
1 ) Bench
Used to put the equipment


2 ) Two G-Clamps
Placed analogues to each other to keep steadfastly the beam in topographic point
3 ) One Aluminum Cantilever Beam
The chief topic of the experiment. with strain gages attached both on top and underside. The Young’s Modulus of the beam is 70GPa. It was marked at 50mm intervals to ease the process.
Picture 3



4 ) Strain gage span amplifier with digital read-out
The strain amplifier provides the readings of the difference in strain between the top and bottom surfaces of the beam ( measured in micro strains ) .
5 ) One hanger with weights ( 10 blocks of 100 gms each ) . It is used as the burden that is applied at the terminal of the cantilever saloon. The hanger helps in using the burden ( there is a hook at the left terminal of the beam ) .
6 ) Dial Gauge
The dial gage was used to mensurate the warp caused by the burden. It measured in 0. 01mm



4. Procedures
Unlike other labs during this one three different processs had to be followed. Each one of them will be described individually in this subdivision.

4. 1 Procedure 1

During the first process the cantilever had to be clamped at its terminal with the border of the bench so that the length of the beam was 500mm. Then the dial gauged must be placed underneath the beam and be calibrated so that the measuring recorded is every bit accurate as possible. Finally the hanger must be placed at the hook on the terminal of the beam and weight it with at least 8 different tonss. For each burden the measuring of the strain and the warp due to the burden should be recorded.

4. 2 Procedure 2

The 2nd process is really similar to the first 1. The difference is that the cantilever is clamped farther along so that the entire length is 400mm and the strain gages should non be within 50mm of the clamped subdivision of the beam. Again here at the terminal of the process the measurings of strain and warp have to be recorded.

4. 3 Procedure 3

During the 3rd process. ab initio the beam must be clamped so that it is 500mm long. Then use two different tonss at the terminal ( i. e. 500g and 1000g ) and mensurate the warp every bit good as the warp for zero tonss. After that the positioned of the dial gage must be changed by 50mm and so re-measured the value of warp for each burden as described above. This procedure must be repeated at least 5 times. ( Note that the strain during this process is non necessary ) .

5. Consequences

Because the processs followed were three and from each one the measurings were different the subdivision of the consequences will be divided into three different 1s.

x

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