portfolio lab reports

MOHAMMED BAPPA LABORATORY REPORT 12 TH FEBRUARY, 2014.

SCHOOL OF SCIENCE AND ENGINEERING

A Portfolio of Three

Laboratory Reports on Mechanics of Materials - MMD2005-N

ICA1: Engineering materials Laboratory work

BEng Mechanical Engineering 2nd year

BY: MOHAMMED BAPPA – M2275573

MODULE LEADER: MANU RAMEGOWDA

DATE OF SUBMISSION: 12TH FEBRUARY, 2014.1 | P a g e

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ABSTRACT:The practical that was conducted for the mechanics of materials module consisted of three different experiments namely; Shear force in beam, Bending moment in a beam and Bending stress in a beam. It gave students some practical experience with theoretical knowledge that they were learning in lecture. The practical used two different sets of apparatus to work through three experiments and allowed students to collect enough results and data to use for this report and to generate hand calculations which will be used to verify and validate the results generated by the practical work.

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Table of FiguresFigure 1: The STR3 Hardware (Shown without the STR1 Frame)..............................................6Figure 2: Shear Force in beam apparatus.................................................................................7Figure 3: Free Diagram of the shear force in a beam apparatus..............................................7Figure 4: The STR2 Hardware (Shown without the STR1 Frame)............................................13Figure 5: Bending moment in a beam apparatus...................................................................14Figure 6: Free Body Diagram of the bending moment in a beam apparatus..........................14Figure 7: The STR5 Hardware (Shown without the STR1 Frame)............................................20Figure 8: Bending stress in a beam apparatus........................................................................21Figure 9: Free body diagram of the bending stress in a beam apparatus...............................21Figure 10: Dimensions of beam and strain gauge positions...................................................22

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Table of ContentsSCHOOL OF SCIENCE AND ENGINEERING.............................................................................1

ABSTRACT:................................................................................................................................2

Table of Figures........................................................................................................................3

EXPERIMENT 1: SHEAR FORCE IN A BEAM..............................................................................6

Abstract:...................................................................................................................................6

Introduction:............................................................................................................................ 6

Theory behind Experiment:......................................................................................................7

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

Results:.....................................................................................................................................8

Table of Readings;................................................................................................................ 8

CALCULATIONS:....................................................................................................................9

GRAPH:............................................................................................................................... 11

Discussion of Results:.............................................................................................................11

Conclusion:.............................................................................................................................12

References:............................................................................................................................ 12

EXPERIMENT 2: BENDING MOMENT IN A BEAM..................................................................13

Abstract:.................................................................................................................................13

Introduction:.......................................................................................................................... 13

Theory behind Experiment:....................................................................................................14

Experimental Procedure:........................................................................................................15

Results:...................................................................................................................................15

Table of Readings;.............................................................................................................. 15

CALCULATIONS:..................................................................................................................16

GRAPH:............................................................................................................................... 18


Conclusion:.............................................................................................................................19

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

EXPERIMENT 3: BENDING STRESS IN A BEAM.......................................................................20

Abstract:.................................................................................................................................20

Introduction:.......................................................................................................................... 20

Theory behind Experiment:....................................................................................................21

Experimental Procedure:........................................................................................................22

Results:...................................................................................................................................23

Table of Readings;.............................................................................................................. 23

CALCULATIONS:..................................................................................................................23

GRAPH:............................................................................................................................... 26


Conclusion:.............................................................................................................................27

References:............................................................................................................................ 27

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EXPERIMENT 1: SHEAR FORCE IN A BEAM

Abstract:The rationale of this experiment is to examine how shear force varies with an increasing point load.

Introduction:As the title suggests, this experiment looked into shear force in a beam and how this increased and affected the beam as the load that was applied increased. The experiment was based on ‘STR3-shear force in a beam’ structures hardware.

Figure 1: The STR3 Hardware (Shown without the STR1 Frame)

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Figure 2: Shear Force in beam

apparatus

As shown above, the apparatus used for the experiment uses a beam which is cut as shown in the free body diagram below. As shown in figure 2, the beam is suspended using two supports, each of which uses a pivot. The mechanism that measures the shear force throughout the beam also prevents the beams collapse during the experiment. A series of loadings were applied to the beam, starting from 0g-500g. When each load is applied, the software automatically saves all the relevant data on the shear forces and stresses acting on the beam and records them into a file that can be accessed by the students. This data is then analysed and compared against the theoretical work and calculations.

Figure 3 above, shows the free body diagram of the shear force experiment. The cut section shows where the load will be applied on the apparatus.

Theory behind Experiment:Shear force in a beam is the force in which is acting perpendicular to the longitudinal x-axis of the beam. When looking into the beam design and determining an appropriate beam for the desired structural loadings, shear force is deemed more important than that of axial loadings (a force in which acts parallel to the longitudinal x-axis). “The shearing force at any section of a beam represents the tendency for the portion of the beam on one side of the

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section to slide or shear laterally relative to the other portion” (Code Cogs, 2004).

Sc = W (a)l

Where:

W = weight (load) applied to the beam (N),

a = Distance from the left support to the load, not the cut (m),

I = Test length of the beam (m).

This equation is used to determine the shear force acting at the point where the load is applied to the apparatus.

Experimental Procedure:The experiment followed a strict set of instructions and the method for this experiment was therefore, followed to the latter. Below is a bullet pointed list of the procedure that was followed in order to conduct the experiment in an accurate and safe manner;

1. The relevant software was set up using the computer provided for the virtual experiment mode.

2. The ‘variable hanger load’ from the property selection box.3. A 0g weight was dragged and placed at 400mm from the end of the beam (the

location at which all the loads will be placed).4. The data for the 0g load was recorded using the appropriate function.5. A table showing the relevant data appeared and this data was saved for later use.6. The process was repeated for weights of 100g-500g, all data collected were

recorded and saved.7. The results were analysed and compared to theoretical calculations.8. Graphs were produced.

Results:

Table of Readings;Mass Load Distance

from leftExperimental Shear at cut

Theoretical shear at cut

Theoretical reaction at left support

Theoretical reaction at right support

g N mm N N N N

0 0.00 400 0.05 0.00 0.00 0.00

50 0.49 400 0.46 -0.29 0.20 0.29

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100 0.98 400 1.01 -0.58 0.40 0.58

150 1.47 400 1.55 -0.87 0.60 0.87

200 1.96 400 2.03 -1.16 0.80 1.16

250 2.45 400 2.45 -1.45 1.00 1.45

300 2.94 400 3.04 -1.74 1.20 1.74

350 3.43 400 3.50 -2.03 1.40 2.03

400 3.92 400 4.03 -2.32 1.61 2.32

Table 1: Results and data generated from the software While applying various loads to the apparatus.

CALCULATIONS:To calculate the shear force;

Sc = W (a)l

Parameters given:

W = weight (load) applied to the beam = 1.96 N

a = Distance from the left support to the load, not the cut =400 mm

I = Test length of the beam = 440 mm.

I substituted these parameters into the equation;

Sc = W (a)l = 0×400440 =

Sc = W (a)l = 0.49×400440 =

Sc = W (a)l = 0.98×400440 =

Sc = W (a)l = 1.96×400440 =

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Sc = W (a)l = 2.94×400440 =

Sc = W (a)l = 2.45×400440 =

Sc = W (a)l = 2.94×400440 =

Sc = W (a)l = 3.43×400440 =

Sc = W (a)l = 3.92×400440 =

All calculations were approximated to 4 decimal places.

To calculate percentage error to show how accurate the values are;

Percentage Error = Experimental−Theoretical

Theoretical × 100%

Percentage Error = 0.46−0.44540.4454 × 100%

Percentage Error = 0.01460.4454 × 100%


Load (N) Theoretical Shear Force, Sc (N)

Experimental Shear Force (N)

% Error

0 0 0.05 0 %

0.49 0.4454 0.46 3.28 %

0.98 0.8909 1.01 13.36 %

1.47 1.3363 1.55 15.99 %

1.96 1.7818 2.03 13.92 %

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2.45 2.2272 2.45 10.00 %

2.94 2.6727 3.04 13.74 %

3.43 3.1181 3.50 12.25 %

3.92 3.5636 4.03 13.09 %

Table 2: Experimental and Theoretical (calculated) shear forces tabularised.

GRAPH:A scale of 1 to 2 was used on both the Load vs. Shear Force axes respectively.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Load vs Theoretical and experimental shear force

Theoretical shear force (N)Experimental Shear Force (N)

shear force (N)

Load

(N)

Graph 1: Applied load versus experimental and theoretical shear stress.

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Graph 2: Load against Theoretical shear at cut.

Discussion of Results:From the table above, it can be deduced that the Load is directly proportional to the shear force. This means that, as the load increases the shear force increases as well. For instance; a load of 0.98N results in a shear force of 0.8909 and at load of 3.92N the shear force is 3.5636 N.

Comparing the theoretical and experimental values from the table, It can be noted that there is negligible difference between their values. For example, the difference between the experimental shear force value of 1.55 N and theoretical shear force value 1.33 N of a load of 1.47N was 0.22 N which is negligible. This shows that this experiment was done carefully with little errors. In addition, the theoretical values are less than the experimental values.

The shape of the graph is linear. This means that the relationship between load and shear force is linear. More so, as the load increase the shear force increases. From the graph, it can be noted that the values started from the origin and increases as the load and shear force increases. The percentage errors for all values were less than 16%. The percentage error is low and this shows that this experiment is reliable and was done carefully. Equation 2 that was used to measure the shear force (Sc) at the cut accurately predicted the behaviour of the beam.

Conclusion:All in all, from the results obtained during the experiment it can be concluded that load is directly proportional to the shear force which means that as the load increases the shear force increases as well. The percentage errors were less than 16% which proves that this experiment was done carefully and the values can be reliable. The graph shows that the experimental values are more than the theoretical values.

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References:Books:

1. E.P Popov. (1999). Engineering Mechanics of Solids. 2nd ed. Chapter 20, Prentice Hall (Singapore).

Websites:

1. NTU.edu.sg. (n.d). Ultimate Load of a Beam under Pure Bending. [Online]. Available: http://www.ntu.edu.sg/home/masundi/projects/Lab-Projects/M312-Report1.pdf. Last Accessed: 9th March, 2014.

2. Momade, H. (2011). Shear Force, Bending Moment, Deflection Beams, and Strut Apparatus Test. Available: http://www.academia.edu/3671106/Shear_Force_Bending_Moment_Deflection_Beams_Strut_Apparatus_Test. Last Accessed: 9th March, 2014.

EXPERIMENT 2: BENDING MOMENT IN A BEAM

Abstract:The rationale of this experiment is to examine how bending moment varies with at the point of loading.

Introduction:

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http://www.academia.edu/3671106/Shear_Force_Bending_Moment_Deflection_Beams_Strut_Apparatus_Test

http://www.academia.edu/3671106/Shear_Force_Bending_Moment_Deflection_Beams_Strut_Apparatus_Test

http://www.ntu.edu.sg/home/masundi/projects/Lab-Projects/M312-Report1.pdf


This experiment looked into bending moments and how they are affected by the increasing larger loads applied to the beam. This experiment was based on the ‘STR2-Bending moment in a beam’ structures hardware.

As shown above, the apparatus used for the experiment uses a beam which is ‘cut’ as shown in the free body diagram below. As shown in figure 5, the beam is suspended using two supports, each of which uses a pivot. The mechanism that measures the shear force throughout force throughout the beam also prevents the beams collapse during the experiment. A series of loadings were applied to the beam, starting from 0g-500g. When each load is applied, the software automatically saves all the relevant data on the shear forces and stresses acting on the beam and records them into a file that can be accessed by the students. This data is then analysed and compared against the theoretical work and calculations.

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Figure 6 above, shows the free body diagram of the bending moment experiment. The cut section shows where the load will be applied on the apparatus.

Theory behind Experiment:Bending moments occur when moments are applied to a structural element. The moments induce a ‘bend’ in the structural element. The analysis of these bending moments is of a particular importance within the engineering industry, as any structural element that has a moment introduced to it must be capable of withstanding such a moment or risk structural collapse. Analysing bending moments in a beam will allow the engineer to determine where the greatest amount of bending will occur (centrally in a beam with a simple UDL loading).

BM = Wa(l−a)l

Where:

W = Weight/ load applied to the beam (N),

A = Distance from the left hand support (m)’

L= test length of the beam (m).

This equation used to determine the bending moment within the beam as the load applied to the beam is increased.

Experimental Procedure:The experiment followed a strict set of instructions and the method for this experiment was therefore, followed to the latter. Below is a bullet pointed list of the procedure that was followed in order to conduct the experiment in an accurate and safe manner;

1. Using the computer provided, the software was set up to ‘virtual experiment mode’.2. ‘The variable hanger load’ option was selected from the property selection box.

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3. The 0g load was removed from the cut section and replaced with a 100g load located in the tool box area. A force diagram and resultant force graph appeared to confirm the change in loadings.

4. The data collected was recorded automatically by the software for the 100g load and saved consequently.

5. This process was repeated for the loads of 200g – 500g. All data collected was recorded and saved.

6. The results were analysed and compared to the theoretical calculations. Graphs were included.

Results:

Table of Readings;Mass Load Distance

from left

Experimental Force

Theoretical Force at cut

Experimental bending moment at cut

Theoretical bending moment at cut

Theoretical reaction at left support

g

N mm N N Nm Nm N

0 0.00 400 0.01 0.00 0.00 0.00 0.00

50 0.49 400 0.44 0.32 0.06 0.04 0.20

100 0.98 400 0.96 0.65 0.12 0.08 0.40

150 1.47 400 1.47 0.97 0.18 0.12 0.60

200 1.96 400 1.98 1.30 0.25 0.16 0.80

250 2.45 400 2.50 1.62 0.31 0.20 1.00

300 2.94 400 2.99 1.95 0.37 0.24 1.20

350 3.43 400 3.50 2.27 0.44 0.28 1.40

400 3.92 400 4.01 2.60 0.50 0.32 1.61

Table 3: Results and data generated from the software while applying the various loads to the apparatus.

CALCULATIONS:To calculate Bending moment;

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BM = Wa (l−a)l

Parameters given;

W = Weight/ load applied to the beam (N) = 0.49 N

A = Distance from the left hand support (m) = 400mm = 0.400m

L= test length of the beam (m) = 580mm = 0.580m

BM = Wa(l−a)l

= 0.00×0.400(0.580−0.400)

0.580 =

BM = Wa(l−a)l

= = 0.49×0.400(0.580−0.400)

0.580 =

BM = Wa (l−a)l

= 0.98×0.400 (0.580−0.400)0.580

=

BM = Wa (l−a)l

= 1.47×0.400 (0.580−0.400)0.580

=

BM = Wa(l−a)l

= 1.96×0.400(0.580−0.400)

0.580 =

BM = Wa(l−a)l

= 2.45×0.400(0.580−0.400)

0.580 =

BM = Wa (l−a)l

= 2.94×0.400 (0.580−0.400)0.580

=

BM = Wa(l−a)l

= 3.43×0.400 (0.580−0.400)0.580

=

BM = Wa(l−a)l

= 3.92×0.400(0.580−0.400)

0.580 =


To calculate percentage error to show how accurate the values are;

Percentage Error = Experimental−Theoretical

Theoretical × 100%

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Percentage Error = 0.44−0.4260.426 × 100%

Percentage Error = 0.0140.4454 × 100%

Then, I followed the same pattern to calculate for other consecutive values.


Mass Load Experimental bending moment at cut

Calculated Theoretical

Bending Moment% Error

g N Nm Nm

0 0.00 0.00 0.000

50 0.49 0.06 0.061

100 0.98 0.12 0.122

150 1.47 0.18 0.182

200 1.96 0.25 0.243

250 2.45 0.31 0.304

300 2.94 0.37 0.365

350 3.43 0.44 0.426

400 3.92 0.50 0.486

Table 4: Experimental and Theoretically calculated bending moments.

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GRAPH:A scale of 0.1 to 1 on the bending moment Axis (N/m) and Load (N)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

0.4

0.5

0.6

Experimental bending moment at cut NmCalculated Bending Moment (N/m) N/m

Load (N)

Bend

ing

Mom

ent (

N/m

)

Graph 3: Bending moments versus loads applied.

Graph 4: Experimental Force against Theoretical Bending moment at cut.

Discussion of Results:From the table above, it can be deduced that the load is directly proportional to the

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bending moment of the beam. This simply implies that an increase in load results in a corresponding increase in the bending moment (both experimental and theoretical). For Example; A load of 1.96N results in a theoretical value of 0.243 N and a load of 3.43 N results in theoretical value of 0.426 N.

The percentage errors range from 0% to 2.88% which shows that the experiment was done carefully and the values are reliable. There is negligible difference between the experimental values and the calculated theoretical values. For instance; for a load of 0.49N, the experimental value was 0.18N and the theoretical value was 0.182. The difference is 0.002 which is negligible. This shows that that the experiment was done with little or no errors which proves accuracy and reliability of the values.

From the graph, it can be noted that from the origin the experimental and theoretical values are together but as it gets higher they deviate slightly away from each other. Furthermore, the graph shows that the experimental bending moment is slightly higher than the values of the theoretical bending moment. Equation 1 which was used to measure the bending moment (BM) at the cut accurately predicted the behaviour of the beam.

Conclusion:All in all, from the results obtained during the experiment it can be concluded that the load is directly proportional to the bending moment of the beam. This entails that as the load increases, the bending moment increases as well. The percentage errors which ranges from 0% to 2.88% proves the reliably and accuracy of this experiment. There is negligible difference between the experimental bending moment and the theoretical bending moment.

References: Books

1. Hibbeler, R. C. (2005). Mechanics of Material, Sixth Edition, Prentice Hall.

Websites

1. Board, J. (2010). Bending Beam. [Online]. Available: http://dcomm.cxc.lsu.edu/portfolios/09fall/jboard2/Bending_lab_report_Final.pdf. Last Accessed: 12th March, 2014.

2. Fang, S. et Redon, A et VanHorn, A. D. (2010). Estimation of Bending Beam in box-beam bridges using cross-sectional deflections. [Online]. Available: http://digital.lib.lehigh.edu/fritz/pdf/322_2.pdf. Last Accessed: 12th March, 2014.

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http://digital.lib.lehigh.edu/fritz/pdf/322_2.pdf

http://dcomm.cxc.lsu.edu/portfolios/09fall/jboard2/Bending_lab_report_Final.pdf


EXPERIMENT 3: BENDING STRESS IN A BEAM

Abstract:The rationale of this experiment was to examine the relationship between bending moment and strain.

Introduction:This experiment takes a look into how increasing loads affect the bending stress within a beam that has such loads applied. This experiment is based on the ‘STR5-Bending Stress in a beam’ structures hardware.

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The beam that was used for this experiment consisted of an inverted T-section beam. This beam is supported by resting on two simple supports. With this arrangement of supports, the load applied by the frame is applied in two separate locations as shown in the free body diagram below and between the two loading points are multiple strain gauges that measure the strain at each area. The arrangement of the beam also allows for a constant bending moment in the area surrounding the strain gauges. As part of the experiment, a series of loadings were applied to the beam section, and the software would collect all relevant data for the experiment for later analysis. This data would then be compared to theoretical work done by the student.

Figure 9 above, shows the free body diagram of the bending stress experiment. The diagram shows where the loads will be placed via the frame work, with strain gauges in-between.

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Theory behind Experiment:Bending stress is a form of normal stress in structural mechanics. Bending stress is caused when a load is applied to a body or structural member and caused it to bend. It is at this point that result in bending stress or the normal stress at the point of bending and such, bending stress can be viewed as a more specific type of normal stress. It can also be determined that the value of the bending stress will vary linearly from the neutral axis as the distance from the axis changes.

E = σε

MI ×σy

Where:

σ = Stress (Nm-1)

ε = Strain

E = Young’s Modulus (Nm-1)

M = Bending Moment (Nm)

I = Second Moment of Area of the beam.

Y = Distance from the neutral axis.

These equations can be used to determine various factors such as the stress and strain acting within the beam as a result of the loadings and bending moment as a result of each of the loading.

Experimental Procedure:

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The experiment followed a strict set of instructions and the method for this experiment was therefore, followed to the latter. Below is a bullet pointed list of the procedure that was followed in order to conduct the experiment in an accurate and safe manner;

1. To begin with, the software was set up for ‘virtual experiment mode’ on the computer provided for this experiment.

2. The section dimensions option was used within the tool box.3. The correct dimension for the beam was entered.4. The second moment of Area for the beam and neutral axis values were noted and

automatically shown on the software.5. The initial load to 0N was set and the data were recorded/ saved. A result table

appeared (and added within this report).6. The loads of 100N-500N were repeated.7. The results of the experiment were analysed and compared to the theoretical results

calculated on paper.

Results:

Table of Readings;

Average Strain Values

Gauge numbers

Vertical position Average strains where bending moment is:

(nominal)

(actual)

mm mm 0.1 Nm

0.1 Nm

0.1 Nm

0.0 Nm

0.0 Nm

0.0 Nm

8.7 Nm

0.0 Nm

8.7 Nm

17.4 Nm

26.3 Nm

35.0 Nm

43.7 Nm

52.5 Nm

61.3 Nm

70.1 Nm

1 0.0 ??? -0.1 -0.2 -0.3 0.4 0.0 0.3 -0.2 -0.1 -52.3

-114.5

-156.9

-161.4

-162.6

-161.8

-161.8

-161.8

2,3 8.0 ??? -0.1 0.1 0.1 0.0 -0.2 0.0 0.1 0.1 -35.7 -78.9 -

118.0-

156.2-

185.9-

208.6-

227.0-

246.24,5 23.0 ??? 0.2 -0.1 0.1 0.0 0.0 0.2 -0.2 0.2 -6.5 -14.5 -22.4 -29.7 -36.6 -44.4 -51.6 -58.96,7 31.7 ??? 0.1 0.0 -0.1 0.2 -0.2 0.0 0.0 0.2 11.1 23.3 34.6 46.0 56.3 69.4 80.7 91.98,9 38.1 ??? 0.1 0.1 0.1 -0.2 -0.3 0.0 -0.1 0.0 24.0 51.9 77.7 102.7 127.0 155.6 180.3 205.1

Table 5: Results generated by the software.

Load

Bending Moment

Gauge 1 Strain

Gauge 2 Strain

Gauge 3 Strain

Gauge 4 Strain

Gauge 5 Strain

Gauge 6 Strain

Gauge 7 Strain

Gauge 8 Strain

Gauge 9 Strain

Gauge 10 Strain

N Nm 0.1 0.0 0.3 0.2 -0.1 0.3 0.1 0.4 -0.3 -0.1 0.0 --49.6 8.7 -52.3 -35.3 -36.2 -5.3 -7.6 12.4 9.7 25.8 22.2 --99.6 17.4 -114.5 -77.9 -79.8 -13.0 -16.0 24.7 21.9 53.6 50.2 --150. 26.3 -156.9 -117.8 -118.2 -21.2 -23.6 33.8 35.4 76.9 78.5 --

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4200.3 35.0 -161.4 -155.3 -157.2 -27.7 -31.8 46.8 45.2 103.6 101.7 --

249.6 43.7 -162.6 -177.6 -194.1 -34.0 -39.3 57.0 55.6 128.2 125.7 --

299.8 52.5 -161.8 -181.6 -235.6 -42.2 -46.7 67.3 71.5 154.0 157.2 --

350.2 61.3 -161.8 -183.0 -271.1 -50.3 -52.9 75.0 86.5 175.3 185.3 --

400.4 70.1 -161.8 -183.4 -309.0 -57.2 -60.7 86.1 97.7 200.2 210.0 --

Table 6: Results generated by software.

CALCULATIONS:Experimental values, Bending of Beam with variable load. Beam section-Inverted T section, Material: 69.0GPa, 2nd Moment of Area: 58.01×109 m4 (these values have been given by the software itself, theoretical 2nd moment of Area value will be calculated below).

y̅ × Total Area = Area 1 × its distance from X –X axis + Area 2 × its distance from X – X axis.

Area 1 = 31.7 × 6.4 = 202.88 mm2

Area 2 = 38.1 ×6.4 = 243.84 mm2

AT = 202.88 +243.84 = 446.72 mm2

y̅ × 446.72 = 202.88 × 22.25 + 243.84 × 3.2

y̅ = 202.88×22.25+243.84×3.2

446.72 =

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Figure 11: Beam T-section

The software calculated this value to be 26.25mm. This is due to the fact that it took it from the top of T-section (at C from 11) while I have taken it from the bottom (at D from figure 11).

Total height of the beam= 26.25 + 11.85 = 38.1 mm

IXX= IXX base + Ah2 = bd3

12 + (Area) × (y̅ -y2)2

IXX = 38.1×6.43

12 + (6.4 ×38.1) × (11.85 – 3.2)2 + 6.4×31.73

12 + (6.4 × 31.7) × (11.85 – 22.25)2

As shown, this value for the second moment of area is identical to that calculated by the software.

Due to the symmetrical nature of the beam in question, the reaction forces at each end of the beam are equal to half the total load applied to the beam (i.e. if the total load is 99.6N then the reaction force at each end is equal to 49.6N).

E = σε

MI = σy

σ =M . yI

Where:

σ = Stress (Nm-1)

ε = Strain

E = Young’s Modulus (Nm-1)

M = Bending Moment (Nm)

I = Second Moment of Area of the beam.

Y = Distance from the neutral axis.

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Experimental bending Stress

σ =8.7×26.25×10−3

58.009×10−9 = 3.937×106 Nm-1

Theoretical Bending Stress

σ =8.7×11.85×10−3

58.009×10−9 = 1.777×106 Nm-1

Then, I followed the same pattern to calculate for other consecutive values.


Load Bending moment Experimental bending Stress

Theoretical Bending Stress

N Nm Nm Nm

0.1 0 0.0453 ×106 0.0204×106

49.6 8.7 3.937×106 1.777×106

99.6 17.4 7.874×106 3.554×106

150.4 26.3 11.901×106 5.373×106

200.3 35.0 15.838×106 7.150×106

249.6 43.7 19.775×106 8.927×106

299.8 52.5 23.757×106 10.725×106

350.2 61.3 27.739×106 12.522×106

400.4 70.1 31.721×106 14.320×106

GRAPH:A scale of 5 to 100 on the bending moment Axis (N/m) and Load (N) respectively.

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0 50 100 150 200 250 300 350 400 4500

5

10

15

20

25

30

35

Experimental bending Stress NmLinear (Experimental bending Stress Nm)Theoretical Bending Stress NmLinear (Theoretical Bending Stress Nm)

Load (N)

Bend

ing

Stre

ss (

Nm)

Graph 6: Strain against Vertical position (nominal).

Discussion of Results:From the table above, it can be deduced that the experimental bending stress values are higher than the theoretical bending stress values. More so, the difference between the

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http://academic.uprm.edu/pcaceres/Courses/INME4011/MD-3A.pdf

http://www.doitpoms.ac.uk/tlplib/beam_bending/bend_moments.php

http://dcomm.cxc.lsu.edu/portfolios/09fall/jboard2/Bending_lab_report_Final.pdf

portfolio lab reports

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