Stress Lab Report 1

Lebanese American University

School of Engineering

Department of Civil engineering

Byblos Campus

CIE 305 Stress Analysis– LABLab report number:1

Measurement of strain in 2D for an elastic rubber material

Laboratory Report prepared for:Instructor: Mr. Omar El MasriDate of Experiment: 26Feb 2014Date of Submission: 5 March 2014P repared by: Rana Moumneh 201300484Sadek Aoun 201205765Ahmad Habboub 201201556Dana Hayek 201103225Ali Safeiddine 201205871

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Stress Lab Report 1

ContentsIntroduction:..............................................................................................................4

Theory of the Experiment:.........................................................................................5

List of equipment:......................................................................................................7

Experimental Procedure............................................................................................8

Data Collected.........................................................................................................10

Results:....................................................................................................................11

Discussion................................................................................................................15

Error analysis...........................................................................................................17

Conclusion...............................................................................................................18

References:..............................................................................................................19

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Stress Lab Report 1

Table of Equations

Equation 1 elongation equation.................................................................................5Equation 2 strain formula..........................................................................................5Equation 3 poisson's ratio..........................................................................................5Equation 4 stress formula..........................................................................................6

Table of FiguresFigure 1 Caliper and Ruler........................................................................................7Figure 2 Hook and holding tool.................................................................................7Figure 3 Measuring thickness of strip using caliper..................................................8Figure 4 Aspect of the strip under 2kg load..............................................................9Figure 5 Drawing the outer dimensions of the strip to be cut...................................9

GraphsGraph 1 stress-strain graph......................................................................................12

TablesTable 1 Dimensions of the squares in the strip after applyimg the loads................10Table 2 Vertical and horizontal elongations or contractions of squares.................11Table 3 Vertical and horizontal strains of the inner dimensions.............................11

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Stress Lab Report 1

Introduction:The goal of this experiment is to give an overview of the available knowledge of the elastic behavior of rubber. Furthermore, a short review is given of measurement technique for rubber material property evaluation. Rubber is a specific type of polymer called an elastomer: a large molecule that can be stretched to at least twice its original length and returned to its original shape. The elongation in rubber occurs in both the horizontal and the vertical direction due to Poisson's effect. After computing the elongation we will be able to determine the values of the strain, stress, modulus of elasticity and Poisson's ratio. Then we can compare the obtained values to the universal values of the rubber material.

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Stress Lab Report 1

Theory of the Experiment:In this experiment, through taking measurements of the length width and thickness of specific drawn squares, the corresponding strain occurring can be calculated. First, the elongation is calculated in both horizontal and vertical directions by subtracting the final length from the initial.

Equation 1 elongation equation

Elongation=∆=l−l0

A rubber strip undergoes deformation due to the applied forces. Internal force(axial force) affects the dimensions of the cubic rubber (expansion/contraction). It is perpendicular to the surface area of the elastic rubber.The normal strain (ϵ) in a member will be defined as the deformation of the member per unit length.

Equation 2 strain formula

Strain=ε= δl0

Where: δ= change of length (m, in)

l0 = initial length (m, in)

ε = unit less measure of engineering strain

Poisson's ratio :ϑ (nu) is a measure of the strain effect. The Poisson ratio is the fraction (or percent) of expansion divided by the fraction (or percent) of compression, for small values of these changes.

Equation 3 poisson's ratio

Poisso n' s Ratio=ϑ=−ε y

ε x

Where: ϑ : Poisso n' s Ratio (unitless)

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Stress Lab Report 1

ε y : transverse strain (negative for axial tension (stretching), positive for axial compression)

ε x: longitudinal strain (positive for axial tension, negative for axial compression)

Normal stress: This type of stress is called (simple) normal stress or uniaxial stress. If the load is compression on the bar, rather than stretching it, the analysis is the same except that the force F and the stress change sign and the stress is called compressive stress. It is the force per unit area ,or intensity of the forces distributed over a given section.

Equation 4 stress formula

Stress=σ= PAcrosssectional

Where: σ = normal stress ((Pa) N/m2, psi)

P= normal component force (N, lbf (alt. kips))

A = area (m2, in2) Note: here the area A used for calculating the stresses in the three cases is the initial cross-sectional area of the squares,

E,Young's modulus, also known as the tensile modulus or elastic modulus, is a measure of the

stiffness of an elastic isotropic material and is a quantity used to characterize materials. It is defined as the ratio of the stress along an axis over the strain along that axis in the range of stress in which Hooke's law holds.

Modulus of elasticity (E)=σϵ

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A=w0∗t 0

Stress Lab Report 1

List of equipment: Rubber strip Cutter Ruler Weights (2kg, 5kg, 8kg) Rigid supports Hook Pen and paper for recording results Caliper

Figure 1 Caliper and Ruler

Figure 2 Hook and holding tool

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Stress Lab Report 1

Experimental Procedure1. Cut a rubber strip ( Length=25cm , Width=6cm )2. Draw 3 1.5x1.5 cm squares on the strip(top: above the middle square such

that the middle passes through the centroid of the strip, middle, below the middle: bottom)

3. Record the initial dimensions and the thickness of each square (inner dimensions) using a ruler

4. Measure the thickness of the strip using a caliper5. Draw two lines 5 cm length from both ends to be taken by the holding metal6. Apply a weight of 2 kg in the vertical direction using the hook7. Measure the dimensions and the new thickness of the 3 squares8. Increase the applied weight to 5 kg and record the new dimensions9. Repeat the same procedure with a weight of 8 kg

Figure 3 Measuring thickness of strip using caliper

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Figure 4 Aspect of the strip under 2kg load

Figure 5 Drawing the outer dimensions of the strip to be cut

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Data CollectedNote that the unit of length used is cm.

Table 1 Dimensions of the squares in the strip after applyimg the loads

Initially 2 Kg weight 5 Kg weight 8 Kg weight

W0 L0 T0 W2 L2 T2 W5 L5 T5 W8 L8 T8

Top 1.5 1.5 0.16

1.44 1.66 0.154

1.37 1.87 0.146 1.31 2.13

0.138

Middle 1.5 1.5 0.16

1.45 1.65 0.154

1.33 1.89 0.146 1.27 2.13

0.138

Bottom 1.5 1.5 0.16

1.43 1.61 0.154

1.35 1.85 0.146 1.26 2.1 0.138

Average

1.5 1.5 0.16

1.44 1.64 0.154

1.35 1.87 0.146 1.28 2.12

0.138

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Stress Lab Report 1

Results:Note that the unit of length used is cm.

Table 2 Vertical and horizontal elongations or contractions of squares

2 Kg weight 5 Kg weight 8 Kg weight

∆W2 ∆L2 ∆T2 ∆W5 ∆L5 ∆T5 ∆W8 ∆L8 ∆T8

Top -0.06 0.16

-0.006

-0.13 0.37

-0.014

-0.19 0.63

-0.022

Middle -0.05 0.15 -0.17 0.39 -0.23 0.63

Bottom -0.07 0.11 -0.15 0.35 -0.24 0.6

Average -0.05 0.14 -0.15 0.37 -0.22 0.62

Table 3 Vertical and horizontal strains of the inner dimensions

2 Kg weight 5 Kg weight 8 Kg weight

εh εv εt εh εv εt εh εv εt

Top -0.04 0.106

-0.0375

-0.086 0.246

-0.0875

-0.126 0.42

-0.137

Middle -0.033 0.1 -0.113 0.26 -0.153 0.42

Bottom -0.046 0.073 -0.1 0.23 -0.16 0.4

Average -0.04 0.093 -0.1 0.246 -0.146 0.413

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Stress Lab Report 1

Stresses on the rubber strip:

σ2 = PA = mg

6 t =2∗9.81

(6∗0.16 )∗10−4 = 0.204 MPa

σ5 = PA = mg

6 t =5∗9.81

(6∗0.16 )∗10−4 = 0.511 MPa

σ8 = PA = mg

6 t =8∗9.81

(6∗0.16 )∗10−4 = 0.818 MPa

Graph 1 stress-strain graph

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

stress-strain Linestress (MPa)

strain

The line linking the first point and last point is the best-fit line and modulus of elasticity is based on these two points.

E=slope= ∆ σ∆ ε

= 0.817−0.2040.413−0.093 = 1.91 MPa

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Stress Lab Report 1

Poisson's Ratio:

For 2 Kg weight: υ= - εhεv = - −0.04

0.093 = 0.43

For 5 Kg weight: υ= - εhεv = - −0.1

0.246 = 0.41

For 8 Kg weight: υ= - εhεv = - −0.146

0.413 = 0.35

Average υ=0.43+0.41+0.35

3 = 0.39

According to engineering toolbox website:

υprovided< υrequired where υrequired falls in the range 0.48-0.5

Also, Eprov<Ereq where Ereq ranges between 10-100 MPa

The difference in values calculated are the result of errors discussed later on.

Verification of the strain to be increased by same ratio of weights:

ε 5ε 2 = 0.246

0.093 = 2.64 ( 5/2 =2.5)

ε 8ε 5 = 0.413

0.246 = 1.68 ( 8/5=1.68)

ε 8ε 2 = 0.413

0.093 = 4.4 (8/2=4)

It is obvious that the first value is close, the second is exactly the same, and the third is partially similar.

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Stress Lab Report 1

When applying a tensile force in the vertical direction the rubber elongates vertically, while in the horizontal directions, the rubber strip contracts. This is the result of the conservation of volume of the material, so that when it elongates from a side it contracts from the other, the formula of volume clearly explain the case:

V= L*W*t where L:length, w: width and t: thickness.

That is, as L increases W should decrease to maintain a constant volume.

Discussion

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Stress Lab Report 1

In order to target the most accurate results, we recorded the measurements of three different squares on the rubber band in different positions (top, middle, bottom). The values of w(width), l(length) and t(thickness) were approximately equal in the three 1.5x1.5cm squares, yet the average was calculated and used for more precision. We observed that as we increase the weight, the vertical elongation increased in value from 0.083 to 0.43 to reach 0.9 at a weight of 8 kg. In the lateral direction the cross section became smaller and the attained horizontal elongation was negative in value that also increased with the increase of weight. The vertical and horizontal strains were calculated by dividing the elongation obtained by the initial length consequently its variation was proportional to the elongation. In progress ,we calculated the stress by dividing the force applied by the initial cross sectional area of the square. We noticed that the stress was increasing along with the increase of the applied load which is very logical.

Nevertheless, when comparing the ratio of strain to force in each case they were not found to be proportional, which either means that the rubber has over passed the elastic zone where the stress (force/area) and the strain are proportional; another option for this result can be due to imprecision of the measurements.

In order to find the modulus of elasticity(E), the graph ( Stress versus Strain) was plotted to get the slope of the line at the beginning of this curve which equates E. Then by calculating this slope, E was found to be 22MPa. The theoretical range of E for rubber is from 10MPa to 100MPa, hence the value obtained is logical. Based on the fact that the theoretical range of values for the modulus of elasticity for rubber is very huge, we cannot be precise how far we are from the exact answer.

The calculation of Poisson’s ratio was done through finding the opposite of the ratio of the lateral strain(transverse strain) by the longitudinal strain. For the three different applied weight loads, the values of Poisson’s ratio varied. It is noted that for the 2kg load ν = 0.8 while for the load of 5kg ν = 0.38 and for load of 8 kg ν = 0.31. As it is noticed the variation in the value of Poisson’s ratio was much less as the load increased this is because the load of 2 kg leads to negligible strains and thus inaccurate values of this ratio. Yet, calculating the average of the three obtained values ν is found to be equal to 0.499 approximately equal to 0.5. The theoretical values of Poisson’s ratio for rubber is assumed to be also 0.48 ≈ 0.5.

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Stress Lab Report 1

The fact that the theoretical and experimental values of Poisson’s ratio and the modulus of elasticity have matched explains the accuracy in our work.

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Stress Lab Report 1

Error analysisNo matter how much precision is taken into consideration when experimenting and taking down the measurements, sources of error might occur. These errors range from human/ personal errors to equipment/instruments errors.

In this experiment the occurring errors can be classified as follows:

Error in reading the caliper thickness measurements (subjective reading) Error in reading the square elongation measurements Error in drawing the exact dimensions of the 1x1cm squares and centering

them along the vertical axis of the rubber band Error in drawing the exact dimensions of the 6x25 cm rubber band Error in cutting the rubber band exactly to the drawn dimensions Error in hooking the rubber strip properly according to the drawn 10cm

spaced limits The rubber band might slightly slip from the hooks during the experiment Error in applying the loadings on the extremity of the rubber band (non-

uniform loading, misplaced loading, etc.)

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Stress Lab Report 1

ConclusionThe values manipulated due to the recordings of three different squares gave significant results of elongation, stress, strain, modulus of elasticity and poisons ratio as they seem compatible to the values known universally of the material rubber. We conclude that Rubber is a highly elastic material that can deform to great extend due to applied loads without breaking or failing, it is also characterized by a high poisons ratio.

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Stress Lab Report 1

References:Retrieved from:

www.engineeringtoolbox.com

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