Cantilever Beam Load, Bending Moment, Stress and Strain, Slope and Deflection

Topics: Beam, Second moment of area, Bending Pages: 8 (1843 words) Published: November 2, 2010
Cantilever Beam

Table of Contents
Table of Contents2
1.Introduction3
2.Theory3
2.1 Bending Moment and Stresses3
2.2 Deflection and Slopes5
3. Equipment6
4. Procedures7
4.1 Procedure 17
4.2 Procedure 28
4.3 Procedure 38
5. Results8
5.1 Results from procedure 18
5.2 Results from procedure 210
5.3 Results from procedure 312
6. Discussion and Error Analysis14
7. Conclusion15

1. Introduction
During this lab a beam was tested in order to find the relationships between load, bending moment, stress and strain, slope and the deflection in a cantilever beam which was the main objective. The main purpose was to understand the fundamental principles that have to be taken into account before designing and manufacturing a beam or using one as part of a design.

2. Theory
The theory behind this lab can be categorized to 2 different topics, bending moment and stress being the first, the second being slope and deflection. Each one is discussed below: 2.1 Bending Moment and Stresses

Picture [ 1 ]

Bending moment is a moment produced by a load applied on a surface that causes it to bend. In the case of the lab the surface is the beam and with the applied at the end of it. In order to calculate the bending moment it is necessary to draw a Free Body Diagram indicating all the forces applied upon the beam. Picture 1 is the free body diagram of the beam. F is the load applied, R is the reaction from the clamps (the bar is fixed on the bench by two G clamps) and M is the bending moment. Since we need to have equilibrium and no forces on the x-axis there are two equations we should use the following equations: If we resolve the vertical forces for the whole beam

ΣFY=0↔F-R=0 ↔
F=R (1) By taking moments anti-clockwise about right hand side for the whole beam WL+ M=0 ↔
M=-WL (2)
Equation (2) is the bending moment of the beam (measured in Nm), while the equation (1) describes the sheer force. Picture [ 2 ]
For the bending stress in the beam it is helpful to know the second moment of area (I) first. The shape of the bar is rectangular as indicated on Figure 2

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

Iz=bd312 (3)

The Second moment of area is measured in mm4
The bending stress is σx=εxE where εx is the strain and E is the Young’s modulus. (The stress is measured in N/m2) There is a last equation that connects bending stress with bending moment. MI=σxy=ER|

2.2 Deflection and Slopes

In order to find the equations for slope and deflection it is necessary to use formula (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 always smaller than the whole length L). If it is integrated once we will get the slope equation -EIdvdx=-W2x2+C1 (4) If we integrate again (4) then we have deflection equation -EI=-W6 x3+C1x+C2 (5)

By inserting boundary conditions to determine the constants of integration the final equations for the beam slope and beam deflection 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
Hence 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 following:
1) Bench
Used to place the equipment
2) Two G-Clamps
Placed parallel to each other to hold firmly the beam in place 3) One Aluminum Cantilever Beam
The main subject of the experiment, with strain gauges attached both on top and bottom. The Young’s Modulus of the beam is 70GPa. It was marked at 50mm intervals to facilitate the procedure....
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