Variation of Deflection of a Simply Supported Beam with Load, Beam Thickness and Material

Table of Contents

Abstract

Introduction

Theory:

Stress (
σ
):

Strain (
ϵ
):

Young’s Modulus (
E
):

Second Moment of Area of a Beam (
I
):

Relation between Stiffness, Thickness and Load

Pure Bending

Apparatus:

Procedure:

Results:

For Steel and Brass beams:

For 6.4 mm Aluminium beam:

Result Analysis:

Graph 1

Graph 2

Relation between Stiffness & Thickness:

Graph 3

Comparing Theoretical & Practical Values of ‘k’

Material Properties:

Graph for Steel & Brass:

Graph 4

Conclusion

References

Bending of beams under load is a common phenomenon observed in daily life situations and is a very important factor to be considered while designing a structure or a component.

The objectives of this experiment are to prove that – stiffness and the beam thickness are proportional, the relationship between modulus of elasticity, stiffness and the beam dimensions. Finally, to show that beams with different materials and thickness have different stiffness values. There are many experimental methods for this and this experiment is done with simply supported beam arrangement.

This experiment mainly focuses on the deflections shown by beams with different materials and thicknesses. By plotting graphs for deflection (z) and load (W), it is observed that the graphs are straight lines for all the beams tested. This proves that the beams have deformed in their linear, elastic region and that ‘z’ is proportional to ‘W’.

Also, for same thickness of beams with different materials (Steel, Brass and Aluminium), it is observed that the steel beam has the highest stiffness value followed by brass and then aluminium. This confirms that the material property (Young’s Modulus) can determine the stiffness.

Bending in beams is a fundamental characteristic which must be considered while designing any structure or a component. This phenomenon is observed in daily life situations and are caused by stresses generated due to loads applied. These stresses in-turn cause distortion in the length of the beam. If the beam bends in its elastic region, there is no permanent deformation in its length i.e., the beam gets back to its original length when the load is removed. Along with stresses and strains, the material property also plays a key role in determining the stiffness of a beam.

 

Theory:

Stress (
σ
):

It is the force applied to a component over a specific area and is given by,
Stress σ=FA

Strain (
ϵ
):It is the force applied to a component over a specific area and is given by,
Strain ϵ=δll

Young’s Modulus (
E
):It is the measure of stiffness of a material. (A material with higher stiffness has the higher value of Young’s Modulus).
E=σϵ

If a graph is plotted between Stress and Strain, the gradient gives the Young’s Modulus.

Second Moment of Area of a Beam (
I
):For a rectangular cross-sectional beam, the second moment of area is given by,
I=bd312

If a graph is plotted between Stress and Strain, the gradient gives the Young’s Modulus.

Relation between Stiffness, Thickness and Load

A beam with high thickness deflects less for a given load (W) than a less stiff beam. Stiffness depends on the material and dimensions of the beam. Stiffness (S) is the ratio of applied load to the deflection (z),
S=Wz N/m

Stiffness is proportional to thickness cubed, so the ratio of the stiffness to the thickness cubed is constant,
Sd3=Constant

Pure Bending

When a beam is loaded such that it bends only in the plane of applied moment, the stress distribution and curvature of the beam are related by,
MI=σy=ER

Also, deflection of a beam subjected to a point load can often be expressed in the form of,
z/W1/E=kl3I

The main component of the beam apparatus is its steel frame which holds the beam, load cell supports, moving digital deflection indicators and the cantilever support. This whole setup sits on a level bench.

The digital deflection indicators measure the deflection of the beam at any point. The cantilever support holds the beam at one end. Load cells measure distance moved but have a calibrated support spring so that each 10N of downward force moves the indicator by 1 mm. They can act as reaction force indicators when not locked and simple beam supports when locked.

A weight hanger holds the weights to load the beam at any desired point on the beam. There is a graduated scale at the top of the apparatus which helps in applying loads at repeatable distances. Storage hooks helps in the storage of unused beams.

 

Draw two blank result tables to record the deflections of the beams for different loads and thicknesses. One table is for the steel and brass beams, the other for aluminium beam because the aluminium beam quickly bends out of the range of deflection indicator with large weights. So, smaller weight divisions must be used in a separate table.

Measure the length, thickness and width of the beam and mark it at mid-span of its length using a pencil.

Choose a suitable reading close to the centre of the graduated scale of the apparatus to match with the pencil mark on the beam.

Set up the beam and two load cells. Make sure that the two load cells are equidistant from the centre of the beam and their locking pins are fitted.

The centre mark on the beam must be directly under the scale reading chosen instep 3.

The beam will now have an overhang on both the ends.

Hang the weight hanger at the centre of the beam.

A digital indicator is now placed on the upper cross member such that its contact rests directly above the weight hanger and check whether the stem is vertical and there is enough travel downwards.

Zero the indicator and start applying loads to the weight hanger in increments of 5N for steel and brass beams and 2N for aluminium beam.

For each load and thickness of the beam, take the readings of the deflection in the respective tables.

The deflections recorded for different beam thicknesses and loads are tabulated below.

For Steel and Brass beams:

Load W(N)

Deflection z (mm)

Steel6.4 mm

Steel

4.8 mm

Steel

3.2 mm

Brass

6.4 mm

5

0.29

0.75

2.13

0.62

10

0.55

1.42

4.29

1.24

15

0.83

2.08

6.43

1.84

20

1.11

2.71

8.52

2.44

25

1.38

3.35

10.58

3.06

30

1.68

4.00

12.70

3.78

For 6.4 mm Aluminium beam:

Load W(N)

Deflection z (mm)

2

0.29

4

0.52

6

0.79

8

1.12

10

1.48

 

 

Result Analysis:

To find the stiffness of each beam, plot a graph taking deflection (z) on Y-axis and Load (W) on X-axis and find the gradient of each graph. This gradient is equal to the inverse of the stiffness (1/S). The gradients of all the graphs are recorded in a table and their respective stiffnesses are calculated.

Graph 1

Graph 2

Relation between Stiffness & Thickness:

Material

Thickness (mm)

1/S

Stiffness (N/mm)

Steel

6.4

0.055

18.18

Steel

4.8

0.142

7.042

Steel

3.2

0.429

2.330

Brass

6.4

0.124

8.064

Aluminium

6.4

0.148

6.757

Stiffness S=Wz N/m

This graph is plotted by taking the stiffness value at constant load of 10N for different thicknesses. (6.4 mm, 4.8 mm, 3.2 mm)

D3

Stiffness (Highest Value)

262.144

18.18

110.6

7.05

32.76

2.33

Graph 3

Comparing Theoretical & Practical Values of ‘k’

Material Properties:

Material

Young’s Modulus (E)

1/E

Mild Steel

210 x 103 MPa

4.76 x 10-6

Brass

105 x 103 MPa

9.52 x 10-6

Aluminium

69 x 103 MPa

1.45 x 10-5

Graph for Steel & Brass:

Graph 4

Values obtained by Gradient of the Graph 4:

 

(z/W)

(1/E)

(k)

Steel

0.056

4.762*10–6

0.024

Brass

0.126

9.524*10–6

0.027

We know that Second Moment of Area for a rectangular cross-sectional beam,
I=bd312=19*6.4312= 4980.74 mm4
1S=z/W1/E=kl3I

Also,
z=Wl348EI
for load acting at the centre of a simply supported beam.

Therefore, the theoretical value of ‘k’ is
148=0.021

The graphs for all five beams are straight lines which confirms that the deflection is proportional to load and that they are deformed in their linear, elastic region.

Among Steel, Brass and Aluminium, steel beam has the highest value of stiffness followed by Brass and Aluminium. This proves that Stiffness can be determined by the material and its Young’s Modulus.

The beam with highest thickness (6.4 mm) has the highest value of stiffness followed by4.8 mm beam and 3.2 mm beam. This shows that the Stiffness and thickness of the beam are related and the linearity of the graph of S Versus d3 proved the equation,

StiffnessThickness3=Constant

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Thickness of Earth to Sun

Popular claims suggested that it is impossible to fold a piece of paper in half more than seven times no matter its size or thickness.  Previous mathematicians have worked out the number of folds required to reach the moon from the Earth which was worked out to be 42 folds[1] using a m thick paper. The size of the paper however, was not found. In theory, the average folds for a normal size A4 (m) paper is seven.
This is an interesting topic because it is incredible how, by exponential growth, a miniature of m thick like a piece of paper can be folded in half to reach the distance of the planet Saturn.
Britney Gallivan[2] was able to fold a piece of toilet paper of 1200 meters a number of 12 folds. She derived two mathematical expressions based on geometrical sequences, taking into consideration the amount of paper lost in every fold. These formulas make it possible to calculate the hypothetical length L and width W of a piece of paper that would be folded n times to equate the distance from the Earth to Saturn.

Gallivan established some rules that would need to be followed when folding a sheet of paper in half:    

A single rectangular sheet of paper of any size and uniform thickness can be used.
The fold line has to be in the same direction each time.
The folding process must not tear the paper.
When folded in half, the portions of the inner layers which face one another must almost touch one another.
The average thickness or structure of material of paper must remain unaffected by the folding process.
A fold is considered complete if portions of all layers lie in one straight line.

Hence, the length L of the paper influences the number of times it can be folded in half.
Hypothesis – The distance from the Earth to Saturn will be obtained by folding a piece of paper ≥50 times with a hypothetically large enough paper.
This exploration used exponential growth and logarithms in order to find out the number of folds required to reach Saturn. Before any calculation was done, it was indispensable to collect all the data required for the investigation. All the values were used were in the international system of units (metres) and standard form in order to keep the exploration standardized.
Taking into consideration the elliptical orbits of the planets, sometimes they are closer to Earth and sometimes they are further away. Therefore, the value used during this mathematical exploration was the mean value of when they are the furthest apart and the closest together. Astronomical units (AU) are the standard unit measure used when dealing with distances within the Solar System. 1 AU is equal to the distance from the Earth to the Sun which is equivalent tom.
Distance from the Earth to Saturn when closer together – 8.00 AU[3]
Distance from the Earth to Saturn when further apart – 11.0 AU3
Mean distance from the Earth to Saturn –  AU
        m
The thickness of a normal A4 paper (0.210 297m) was calculated by taking a measurement of an office pack of 500 pages (80gsm) which was found to be about 0.05m. The thickness of each individual page is calculated by dividing the total thickness (0.05m) by the amount of sheets (500); giving a result of m.
Whenever a paper is folded in half, the number of layers is doubled so the thickness increases by two. When there is only one layer of paper (not folded), its thickness is m. Once it is folded in half for the first time, its thickness will be multiplied by 2 hence,
m
Folding it one more time means multiplying it by two again,
m
Thus, an expression can be established, showing the exponential growth;

The expression can be represented in a graph to illustrate graphically the exponential growth because of folding.
Graph 1:

From Graph 1, it is possible to visualise how, something that seems unrealistic like folding a sheet of paper to reach Saturn becomes possible. The graph also illustrates how rapidly exponential growth occurs.
Since the expression needs to be equal to the distance from the Earth to Saturn to work out (the number of folds), an equation to find can be solved:

Folds
To find, the rules of logarithms were put in place due to the exponential nature of the equation. The answer has been rounded up to 54 because it is not possible to have a half fold.
Gallivan derived the following formula for the minimum length of a piece of paper of thickness t to be folded n times in a single direction

To prove this formula, it is neccesary to understand that after each fold, some part of the paper is lost and becomes a rounded edge. I folded an A4 sheet of paper seven times in order to illustrate this:
As you can see from the picture, there is a rounded edge on the side which is paper being lost and is not contributing to the real thickness but just joining the layers. The curved portion becomes bigger in correlation with the number of folds and begins to take a greater area of the volume of the paper.
At the first fold, a semicircle of radius t (thickness) is formed, which has a perimeter . Thus, units of the paper are being used in the fold.  A paper smaller than this cannot be folded since there is not enough paper to form the fold. After the fold, there is a two-layer sheet of paper with a thickness of 2t. Another fold results in folding the second layer over the first layer. The second layer has a radius of , so it uses units of paper. The total amount of paper used by the second fold, for both layers, is resulting in a four-layer piece of paper.
The ith fold begings with layers, and folding the jth layer uses units of paper. Hence, the total length of paper used for the ith fold is given by

Therefore, to obtain the total length of paper required for n number of folds, sum this over i from 1 to n, which gives Gallivan’s formula:

The thickness can be substituted into t and the number of folds can be substituted into n

which gives L to be equal to 1.70-1028m.
The other expression proposed by Gallivan can be used to calculate the width of the paper.

If the length lost in the radii of earlier folds is not considered, the length lost must be considered in the last fold. At the final fold n, the side of the square must be at least equal to the length lost in the final fold which is (amount of length lost in each fold. Taking into consideration that the total area of the sheet (area = nb of sheets in penultimate step area of square in penultimate step) is preserved, Gallivan’s equation can be derived:

Again, the thickness and number of folds can be substituted and an answer for W can be found

giving W to be 2.69-1020m.
In conclusion, the initial hypothesis was right since the number of folds was 54 which is, indeed, greater than 50. The hypothetical paper that could, in theory, be folded 54 times so that its thickness equates the distance from the Earth to Saturn of m would be long and wide (taking into consideration that its thickness would bem). The dimensions of this paper would be bigger than the actual distance from the Earth to Saturn so, unfortunately, we do not have a paper that big that would allow us to reach to Saturn just by folding it in half.
This mathematical exploration used logarithms to find out the number of folds needed to reach Saturn with a m thick paper. However, the dimensions of this sheet of paper would be too big and hence, impossible to find in the Earth’s surface. Nevertheless, the exploration could have looked at using a thinner piece of paper to see if its dimensions would have been smaller and perhaps, we would have been able to find it in the surface of the Earth and we would have been able to reach Saturn.
References    
Astronomy, S. (2012). How Far away is Saturn? [Online] Space.com. Available at: http://www.space.com/18477-how-far-away-is-saturn.html [Accessed 16 Jan. 2017].
IFLScience. (2016). Fold A Piece of Paper in Half 103 Times and It Will Be As Thick As the UNIVERSE. [Online] Available at: http://www.iflscience.com/space/fold-piece-paper-half-103-times-and-it-will-be-thick-universe/ [Accessed 16 Jan. 2017].
Pomonahistorical.org. (2002). Folding Paper in Half Twelve Times. [Online] Available at: http://pomonahistorical.org/12times.htm [Accessed 16 Jan. 2017].