Design and testing of a composite beam

8/13/2019 Design and testing of a composite beam

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MME2204

Fundamentals of Material Science 2

Faculty of Engineering

University of Malta

Composite Design

Adrian Borg (286193M)

Adrian Camilleri (540993M)

Jean Paul Sultana (226293M)


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Contents1) The Composite .................................................................................................................................... 3

2) Establishing the Young's modulus of bamboo flooring ...................................................................... 4

3) Calculating the Young's modulus for fibreglass .................................................................................. 6

4) Calculating the estimated strength of the composite ........................................................................ 6

5) Test Description and Results............................................................................................................... 8

6) Calculating the actual strength of the solid composite ...................................................................... 8

7) The percentage error in the actual strength ...................................................................................... 9

8) A discussion including a description of the mode of failure of both specimens .............................. 11

9) Conclusion ......................................................................................................................................... 12

10) Appendix ......................................................................................................................................... 13

11) References ...................................................................................................................................... 15

12) Group Efforts Page .......................................................................................................................... 16


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1) The CompositeThe composite chosen was divided into 3 main materials, a layer of acrylic, layers of bamboo

and layers of fibreglass composite. The outer layers require good compressive or tensile propertiesdepending on the side while the middle does not need to be as strong but should still be able to take

some of the load. To produce the composite the mould was made and covered in packaging tape toprevent the resin and eventually the composite form getting stuck to the mould. The several layersof bamboo and acrylic were then roughened up to improve the adhesion of the resin to the layers.The acrylic was placed first with the smooth side on the bottom, resin was applied to the rough sideand a bamboo layer was placed on top. Resin was applied onto the bamboo and another 2 layerswere added using this process. Once the last layer of bamboo was placed, resin was added onto thewood and the glass fibres were placed into the resin. Several layers of fibre were then place to finishthe composite off

Bamboo is a natural composite with a high strength to weight ratio compared. It is a very

fast growing plant so making it easier to produce compared to other types of wood. It was used inthe centre since the average density of bamboo flooring is around 0.77 g/cm 3 which is relatively low,so it is lightweight while still having good mechanical properties [1].

Acrylic is a polymer also known as polymethyl metacrylathe, it is a tough and shatterresistant. It is easily formed and moulded by heating it, shaping it when hot and cooling back toroom temperature for it to harden. It is also impact resistant and has been used as a replacement forglass (ex. in motorcycle windshields). Moreover it has good compression characteristics having acompressive modulus of 3 GPa and a compression yield strength of 95 MPa[2].

Fibreglass is a composite consisting of glass fibre reinforcement and a polymer resin matrix.It is easy to mould and has a big strength to weight ratio which makes it a popular choice forautomobiles and boat hulls among others. E-glass fibres were used having a Youngs modulus ofbetween 72 and 85 GPa and a tensile strength of 1.95 to 2.05 GPa [3]. These values are very high andwill reinforce the fibreglass composite. The resin has an average modulus of about 3.4 GPa andtensile strength ranging from 20 to 80 MPa [4]. Together the resin and glass will produce alightweight material with good tensile properties.

Figure 0 The compositeTop: Fibreglass

Middle: BambooBottom: Acrylic


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2) Establishing the Young's modulus of bamboo flooringPicture a loaded strip of bamboo hanging over a bench as

shown in the figure below along with its free body diagram. Theweight of the beam was neglected since the deflection when the

beam is loaded was taken to be the displacement of the beam's endfrom when it was unloaded, i.e. the deflection was assumed to bezero when the beam was unloaded.

Using singularity functions the following function defining theinternal moment M(x) along the x axis was defined;

[5] Illustrates how this function could be obtained and what the

Macaulay's brackets; , mean.The relationship between the internal moment and the curvature of the beam is given by thefollowing equation, its full derivation is shown in [6];

Where E is the Young's modulus in Pa, I is the moment of inertia in m 4 and y & x are thedisplacements as shown in diagram {1}. The first derivative is assumed to be small in the derivationof (2) Deflections in the experiment would be kept as small as possible to validate this assumption.

This Implies

In order to obtain an expression for the deflection y as a function of x The above equation needs tobe integrated twice. The constants of integration could be evaluated using the following boundaryconditions;

Resulting in the following equation for the modulus should x be set to zero, i.e. y is at its maximum;

The experimental procedure would consist of choosing various values forL and F, measuring y max and calculating the Young's modulus for eachinstant. An average of the values obtained would be used in thecalculations

Two samples of bamboo where considered one had a bigger crosssection than the other. Three lengths and weights where chosen leading to 18 combinations.

Figure 1

Figure 2


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First the dimensions shown in figure {2} where established by taking multiple readings andaveraging. Then the moment of inertia MOI about the neutral axis na was found. The results areshown in the spreadsheet in figure {3}.

The results for substituting values from experimentation in equation (4) are shown in thespreadsheet illustrated in figure {4}

Figure 4

Figure 3


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3) Calculating the Young's modulus for fibreglassFor discontinuous and randomly oriented fibre reinforced composite the rule of mixtures

was found to be as follows from ref (below1);

( )

Where E CD is the composite's modulus, K is a fibre efficiency parameter, E F is the fibre's modulus, V F is the fibre's volume fraction and E M is the matrix's modulus.

Since it was hard to obtain values for the parameters required to solve the equation above fromsome form of experimentation these where averaged out from values found in literature ref(below2)

The final calculation with entered values is shown below;

4) Calculating the estimated strength of the compositeBefore any theory based on the deflections of beams is

applied, the cross section of the composite beam could betransformed to one having homogenous material, simplifyingcalculations. The choice of the material is arbitrary however usuallyone which makes up a portion of the composite beam is chosen sincecalculations will be easier. The method was obtained from [7] its proofis also shown. The method could be summarized like this; the widthsof laminates made from a different material; B, than theuntransformed material; A are multiplied by the ratio;modulus of B divided by modulus of A, once this is donethe beam could be assumed to be made entirely ofmaterial A. The new cross section's neutral axis is thesame as the one for the composite beam. Thetransformed cross section for a square cross sectionthree ply composite beam would look something like theone in figure {5}. Here the material was transformed into

one having a young's modulus of the middle sectionthroughout. Y 1 indicates the position of the neutral axis which could be found by methods used tofind the centroid of an area. Then the moment of inertia of each individual rectangle about its ownneutral axis that makes up the cross section are found out. The parallel axis theorem is applied tofind the moment of inertia about the neutral axis. All these methods are illustrated in [7] Appendix A

The beam was loaded as shown in figure {6} support reactions will not be calculated since they aretrivial. The origin of the coordinate system was set at the head of the left-most reactive force andthe directions for the axes are shown in their positive sense.

The internal moment as a function of displacement in the x direction could be defined usingsingularity functions. Detailed equations for the general case and their derivation are shown in [5].

Figure 5

Figure 6


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Then this could be equated to a term containing the 2nd derivative of the curvature equation fromequation (2) as shown below;

Integrating both sides twice and solving for the constants of integration using the followingboundary conditions would result in equation (6) which is also known as the three point bendingformula. This formula will be used to determine the force required to produce a specified deflection.Finding out the modulus of a prismatic beam made from a single material and having the same crosssection that will undergo the same deflection as the composite beam will be done later.

All of the above calculations will be entered in a spreadsheet using Microsoft excel so parameterscould be varied easily and should mistakes be present they could be eliminated without having toworking out the problem again. First the composite beam cross section will be defined as shown infigure {7}. The composite consists of three laminates stacked as shown, their heights are enteredand the uniform width b is inputted as well.

Properties of the transformed section are found out using the principles defined before, shown infigure {8}. The calculated values correspond to those shown in figure {5}

In order to calculate the force expected to produce aspecified deflection using equation (6) and the

modulus of a homogenous beam having the same

Figure 8

Figure 7

Figure 9


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Figure 11

original cross section, the properties of the prismatic beam need to be defined as shown in figure{9}. Finding out the modulus of a prismatic beam made from a single material and having the samecross section as the composite beam, and that will undergo the same deflection could simply bedone by equating the modulus and moment of inertia of this beam and the beam with thetransformed cross section, and solving for the modulus of the new beam. This was done andincluded in the results, shown in figure {10}.

5) Test Description and ResultsThe test carried out on the sample is known as a three point

flexural test in the industry. The machinery used is displayed in figure {11},the holders below the specimen move upward, whilst the point in themiddle stays stationery. The results of the machine could be outputted in aMicrosoft excel spreadsheet as was the case in our test. This enabled us toeasily interpret data using graphs and calculations. We were mainlyinterested in the force and corresponding deflections. A graph of flexuralload vs. extension was plotted for both samples; this can be seen in theappendix

6) Calculating the actual strength of the solid compositeOnce again a form of equation (6) (E was made subject) was used and the beam was

considered to be made of a single material in order to find out a single value for its modulus.Microsoft excel was used to calculate the modulus since it was very convenient. The table in figure{12} shows the result using only the final extension and force values, whilst figure {13} shows thevalue of the calculated modulus as found out by taking the average of the modulii which werecalculated for each corresponding force deflection value given in the lab excel sheet for deflectionsgreater than 5 mm, since the modulii for deflections less than 5 varied widely.

Figure 13Figure 12

Figure 10


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7) The percentage error in the actual strengthThe method for the calculation of the percentage error in our calculations starts off tfrom

the equation for the Youngs Modulus with deflection of the beam;

We can note that to achieve the error in the Youngs Modulus, E, we must investigate with respectthe uncertainties in the four containing variables of the equations;

Therefore to obtain the uncertainty for the variable I, its general eqaution for a rectangular cross-section is used;

Incorporating the equation for uncertainty[10];

[ ]

To obtain the uncertainty value for the Force variable, we must do this analytically from the graphobtained from the test-machine. The plot obtained is a scattered plot, whereby using the features ofMicrosoft Excel, a trend line is produced along the plot and its equation found.

The equation of the trend line was found to be;


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Where the y-values indicate force applied and the x-values the deflection of the beam. Thedifferences for numerous points along the graph between the estimated, using the trend line, andthe actual obtained results was found and tabulated below.

DeflectionEstimated/Actual

ForceDifference in Forces

3.14mmYe = 103.123

8.734NYa = 94.389

4.42mmYe = 162.442

15.499NYa = 146.943

4.72mmYe = 176.345

10.698NYa = 165.547

5.45mmYe = 210.175

13.277NYa = 165.647

6.07mmYe = 238.908

11.516NYa = 227.392

6.51mmYe = 259.299

5.753NYa = 253.546

7.08mmYe = 285.714

11.491NYa = 274.223

7.67mmYe = 313.057

10.051NYa = 303.006

8.48mmYe = 350.595

8.183NYa = 358.778

9.56mmYe = 400.645

6.110NYa = 394.535

10.54mm Ye = 446.061 6.779NYa = 452.840

11.45mmYe = 488.233

3.255NYa = 491.488

12.31mmYe = 528.088

4.605NYa = 532.693

13.55mmYe = 585.554

7.031NYa = 578.523

14.28mmYe = 619.384

0.993NYa = 620.377

Thus from the differences in forces a standard deviation was found and this was taken to be theuncertainty of the force variable.

Once these uncertainties for Force and Moment Inertia were found, we can return to the YoungsModulus Equation and find its derivatives with respect to the containing variables;


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The Uncertainty equation can be incorporated at this stage,

[ ]

Once this uncerta inty is obtained we can carry on to find the percentage error of the actual YoungsModulus found.

8) A discussion including a description of the mode of failure of bothspecimens

There are numerous mechanism thorugh which failure of a structure can occur[11], indesigning a composit structure the design stage is crucial in taking into account all the possiblefrailures that can occur. Such common modes of failure range from Brittle Failure, where the loadexceeds the Ultimate Tensile Strength of the structure and cracks or fractures propagate throughtthe sample. There is also Ductile Failure where the sample id displaced greatly in reaction to theloads applied, and can deform both elastically as well as plastically if its Yield Limit is exceeded.Other modes can also be present in the non-cyclic load application as our sample has undergone,such as Buckling, where the sample bends,bulges and bows under load; Ductile Fracture where thesample fractures during plastic deformation and finally Elastic Distortion where the sample returnsto its original shape after it fails to perform as designed.

With respect to our uniform beam of 0.6m, we clearly saw the sample deform elastically as expecteddue to the Bamboo core, which has a great tensile capability, together with the Fibre-Glass. TheSample was seen the continuously deform elastically with no signs of failure, brittle or plasticdeformation of any kind in either of thedifferent material laminas.


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10) Appendix

Images of the composite beams:


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y = 46.343x - 42.934

y = 13.686x + 15.956

-100

0

100

200

300

400

500

600

700

-2.00 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00

Flexural Load N Vs Extension mm

Load (no coupling)

Load (coupling)

Linear (Load (no coupling))

Linear (Load (coupling))


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11) References[1] bamboo floors. (2003). Bamboo Brochure. [online] Available:www.bamboofloors.com.au/Bamboo%20Brochure.pdf. Last accessed 22nd May 2013.

[2] Compressive Strength Testing of Plastics. [online] Available:www.matweb.com/reference/compressivestrength.aspx. Last accessed 22nd May 2013.

[3] E-Glass Fibre. [online] Available:www.azom.com/properties.aspx?ArticleID=764. Last accessed 22nd May 2013.

[4] Tim Pepper, Ashland Chemical Company. Polyester Resins. [online] Available:polycomp.mse.iastate.edu/files/2012/01/5-Polyester-Resins.pdf. Last accessed 22nd May 2013.

[5]Ferdinand P. Beer, E. Russel Johnston, J.R., John T. Dewolf & David F. Mazurek (2009). Mechanics

of Materials : Deflection of Beams (Fifth Edition). USA: Mc Graw Hill.

[6]Victor Dias Da Silva (2006). Mechanics and Strength of Materials : Bending Deflections . Portugal:Springer.

[7]R.C. Hibbeler(2009). Mechanics of Materials: Bending (eight edition). USA: Prentice Hall.

[8] William D. Callister & David G. Rethwisch. (2011). Materials Science and Engineering . (eightedition). Asia: John Wiley & Sons.

[9] Winona State University. Hand Lay-up. [online] Available:

www.google.com.mt/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDMQFjAA&url=http%3A%2F%2Fcourse1.winona.edu%2Fkdennehy%2FENGR390%2FTopics%2Flayup.pptx&ei=7PqbUcS8GIn-PLLkgbgK&usg=AFQjCNH-CTiT. Last accessed 22nd May 2013.

[10] Scheid, F.Schaum's outline of theory and problems of numerical analysis . In-text: (Scheid, 1968).Bibliography: Scheid, F. (1968) Schaum's outline of theory and problems of numerical analysis. NewYork: McGraw-Hill.

[11] Craig, B. Material Failure Modes, Part IBibliography: Craig, B. (2005) Material EASE, Material Failure Modes part 1. Rome, New York

Published (online): http://ammtiac.alionscience.com/pdf/2005MaterialEASE29.pdf
http://ammtiac.alionscience.com/pdf/2005MaterialEASE29.pdfhttp://ammtiac.alionscience.com/pdf/2005MaterialEASE29.pdf


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12) Group Efforts Page

Adrian Borg286193M

33%

Adrian Camilleri540993M

33%

Jean Paul Sultana226293M

33%

Design and testing of a composite beam

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