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Analysis of a circular plate simply supported on a knife edge and subjected to a load F distributed around a circle. Abstract

NomenclatureSymbol C1, C2, C3 D E F Lf Lo Mr M Q R2 R1 Z r t y er e d dr r Description Constants Flexural Stiffness Youngs Modulus of Elasticity Load Length Original length Radial Moment Hoop Moment Reaction Force Maximum Radius Radius of loading circle Distance to fibre of interest Radius Thickness Central deflection Radial Strain Hoop Strain Pi Slope Radial Stress Hoop Stress Poissons Ratio Deflection Units -m N/m2 N m m N/m N/m N m m m m m m ----N/m2 N/m2 -m

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ContentsSection No. 1 2 3 4 4.1 4.2 4.2.1 4.2.2 5 6 7 8 9 Section Title Introduction Objectives Theoretical Analysis Experimental Work Description of apparatus Experimental Procedure Plate Deflection Experiment Strain Gauge Experiment Results Discussion Conclusion References Appendices Page No. 1 1 2 5 5 5 5 5 6 8 8 8 8

1. IntroductionThe purpose of this report is to study the effectiveness of theoretical analysis against experimental testing of a circular plate. The steel plate was simply supported on a knife edge and subjected to a load within its elastic limit. In doing so, the deflections, stresses and strains were obtained from two experiments. The longitudinal stresses were considered zero because deflections were restricted to no greater than half the plate thickness (Hearn, 1997). The experimental and theoretical analysis is described in the following sections.

2. ObjectivesThe key elements of this report are as follows: Measure the deflection of the plate along a radial line. Measure the load-deflection at a particular point. Compare these findings with theoretical values. Establish the principle strains and stresses. Plot the principal stresses across the diameter of the plate and superimpose the experimental values.

3. Theoretical analysisAssumptions: 1. 2. 3. 4. Normals to middle surface stay normal after loading. No stretching of the neutral plane. Plane stresses are assumed. No deflection due to shear. 2

Fundamental deflection equation:

(

(

))

(1)

Substituting in Q above: Integrate with respect to r: Divide by r: Integrate again: Divide by r and integrate again:

( ( ( )

( )

))

(2)

(

) (3) (4)

C1, C2 and C3 can be found by apply boundary conditions: r =R1 @ Mr =0 r =R2 @ Mr =0 r =0 @ =0 C3 =0

Figure 1: Annular ring with total load F distributed around inner radius. (load acting at centre)

In order to find C1 & C2, Mr is derived as follows: Applying the B.C for both radii yields: [ And [ ] ( ]

(

)

(5)

Subtracting to eliminate C1 gives:

(

))

And hence: Subbing C1 and C2 into eqn. (4):( (

(

(

))

))

( 3

(

))

Fundamental theoretical Hoop & Radial stress: ( )

Experimental Hoop and Radial stress from engineering tables page 108: ( ) ( )

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4. Experimental work4.1 Experimental apparatusThe apparatus used is shown in figs. 2 and 3. The equipment consists of a ridged table (4) and a load cell on a moving stage (2). A cylindrical knife edge (3) supports the steel plate. A plunger (5) with a steel ball (6) on the moving stage applies force to the hole in the steel plate. Forces are measured off an analogue gauge (1) on the side of the machine.

1 2

3 4

4.2

Experimental procedure4.2.1 The steel plate (7) was loaded onto the knife edge and a steel ball was placed into the inner hole of the plate. The moving stage was brought near the plate and the plunger was placed between the ball and the stage. The stage was moved until a 20N force was applied to the plate. A Linear Voltage Displacement Transducer (8) was used for the load-deflection test. Slip gauges were used to change the distance of the L.V.D.T out from the plate and the voltages given off were recorded and converted into Figure 2 plunger and plate apparatus displacement readings. The slip gauges moved the L.V.D.T out from the inner circle in increments of 5mm. An equipment check was carried out at a radius of 18.6mm and the plate was loaded from 0-700N in increments of 100N, taking voltage readings at each interval. 4.2.2 For the load-strain test the plate was loaded into the machine the same way as the loaddisplacement test but the steel plate had two rosettes of three strain gauges attached to it as shown in fig. 4. The strain gauges were hooked up to a computer monitoring the strains produced. The plate was loaded in increments of 100N from 0700N and readings were recorded at each interval. A printout of the strains was obtained and Mohrs stress circles were drawn accordingly.Figure 4: Steel plate with strain gauges arrangement. Figure 3 experimental apparatus used

5 6 7 8

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5. ResultsThe data in fig. 5 shows the relationship between the actual and theoretical deflection as the plate was loaded in intervals of 100N from 0-700N.

Calibration Graph0.4 0.35 Deflection (mm) 0.3 0.25 0.2 0.15 0.1 0.05 0 0 200 400 Force (N) 600 800 Experimental theroretical

Figure 5

The data in fig. 6 shows the results of the theoretical vs. the actual deflection of the plate under a load of 700N.

1 0.9 0.8 0.7 Deflection 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30

y = 0.0123x + 0.4798

Experimental Theoretical Linear (Experimental) y = 0.0143x + 0.0836 Linear (Theoretical)

40

50

Distance from centre

Figure 6

The data in fig. 7 shows the stain gauge values as the plate was loaded from 0-700N in increments of 100N. With the strains plotted it was possible to calculate the and r.

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800 y = -9.5258x - 67.945 700 y = 3.4115x - 12.042 600 500 Force (N) 400 300 200 100 0 -100 -100 0 -200 100 Strain () 200 300 y = 3.2042x + 5.5498 Series 2 Series 1 Series 3 Linear (Series 2) Linear (Series 1) Linear (Series 3)

Figure 7

The data in fig. 8 shows the hoop stress vs. the radial stress over the range of the plate.

200000000 180000000 160000000 140000000 120000000 Stress 100000000 80000000 60000000 40000000 20000000 0 -60 -40 -20 0 Radius 20 40 60 Sigma r Simgs Theta Series3 Series4 Series5 Series6

Figure 8

7

8

9

6. DiscussionThe calibration graph proves that the plate was loaded linear elastically, up until that point it was an assumption. The theoretical results deviate from the experimental results. Possible reasons for this are as follows. The deflection curves are similar and linear but they did not match up, the percentage error of ...... reasons for not matching up. The steel plate used had multiple holes around the circumference outside the knife edge. These holes could have potentially changed the properties of the simply supported steel plate with a central hole. The plate could not have been centred perfectly before the load was applied. The thickness was assumed to be 2mm throughout the plate, in reality there may be a slight deviation in its thickness.

7. Conclusion Experimentally it would have been of benefit if more points had been analysised. I would have preferred if a plate was used with just the central hole and the radial ones left out. Theoretically the work of Hearn and Benham and Crawford was confirmed. The preload should be taken away from the max load to get the actual deflection calculated. The objectives of this study were accomplished.

8. ReferencesBenham, P.P., Crawford, R.J., and Armstrong, C.G. (1996) Mechanics of Engineering Materials, 2nd ed., Harlow, England: Pearson/Prentice Hall. An introduction to the mechanics of elastic and plastic deformation of solids and structural materials, 3nd edition, E.J. Hearn, 1997, Butterworth Heineman, University of Warwick, UK

9. Appendices

10

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e0

e45

-77Figure 9

502

e90

e0

e45

-52.4Figure 10

802.4

11

12

Mohrs Strain Circle. [ [ * * , [ * { [ [ ] [ ] ] + + ] * +] ] [ ]} +

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