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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 Description Units

C1, C2, C3 Constants --D Flexural Stiffness mE Young’s Modulus of Elasticity N/m2

F Load NLf Length mLo Original length mMr Radial Moment N/mMθ Hoop Moment N/mQ Reaction Force NR2 Maximum Radius mR1 Radius of loading circle mZ Distance to fibre of interest mr Radius mt Thickness my Central deflection mer Radial Strain --eθ Hoop Strain --π Pi --dωdr Slope --

σr Radial Stress N/m2

σθ Hoop Stress N/m2

ν Poisson’s Ratio --ω Deflection m

ContentsSection No. Section Title Page No.

1 Introduction 12 Objectives 13 Theoretical Analysis 24 Experimental Work 5

4.1 Description of apparatus 54.2 Experimental Procedure 5

4.2.1 Plate Deflection Experiment 54.2.2 Strain Gauge Experiment 5

5 Results 66 Discussion 87 Conclusion 88 References 89 Appendices 8

1. Introduction The 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. Objectives The 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 analysis Assumptions:

1. Normals to middle surface stay normal after loading.2. No stretching of the neutral plane.3. Plane stresses are assumed.

4. No deflection due to shear.

Fundamental deflection equation: ddr (1

r.ddr (r . dwdr ))=−Q

D(1)

Q=D F2π

D= E t 3

12(1−ν2)

Substituting in Q above:ddr (1

rrdr (r d ωdr ))= −F

2πD(2)

Integrate with respect to r:1rddr (r d ωdr )= −F

2πDLnr+C1

Divide by r: r ( d ωdr )= −F2πD ( r

2Lnr− r

4 )+C1 r2

2+C2

Integrate again: d ωdr

=−Fr8πD

(2Lnr−1 )+C1r

2+C2

r (3)

Divide by r and integrate again: ω=−Fr2

8 πD(Lnr−1 )+

C1r2

4+C2 lnr+C3 (4)

C1, C2 and C3 can be found by apply boundary conditions:

r =R1 @ Mr =0 r =R2 @ Mr =0 r =0 @ ω =0 ∴ C3 =0

In order to find C1 & C2, Mr is derived as follows: D( d2wd r2 + ν

rdwdr )=M r (5)

Applying the B.C for both radii yields:

−F8πD

[2(1+ν ) logeR+ (1−ν ) ]+C1

2(1+ν )−

R2 (1−ν )=0

And

−F8πD

[2(1+ν ) logeR1+(1−ν ) ]+C1

2(1+ν )−

R12 (1−ν )=0

R2R1

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

Subtracting to eliminate C1 gives: C2=F

2D ( (1+ν )(1−ν ) ( R2

2R12

R22−R1

2 lnR1

R2 ))And hence: C1=

F2D

Subbing C1 and C2 into eqn. (4):

ω=−Fr2

8 πD(Lnr−1 )+ F

2D¿¿

Fundamental theoretical Hoop & Radial stress:

Ɛr=L f−LoL0

=(R+Z ) (θ )−Rθ

Rθ=ZR

= ddr ( dwdr ) ∴Ɛr=

Z d2wd r2

Ɛθ=2π (r+Zθ )−2πr

2 πr=Zθr

θ=tanθ=dwdr

∴Ɛθ=Zrdwdr

σ r=(Ɛr+ν Ɛθ )E

(1−ν2 ) σ θ=¿

(Ɛθ+ν Ɛr ) E(1−ν2 )

Experimental Hoop and Radial stress from engineering tables page 108:

σ r=−Et

2 (1−ν2) ( d2wdr2 + ν

rdwdr ) σ θ=

−Et2 (1−ν2 ) ( dwdr 1

r+ν d

2wdr 2 )

4. Experimental work

4.1 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.

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 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 load-displacement 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 0-700N and readings were recorded at each interval. A printout of the strains was obtained and Mohr’s stress circles were drawn accordingly.

5. Results The 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.

1 2 3

4 5 618.5mm

29.0mm

Figure 2 experimental apparatus used

Figure 3 plunger and plate apparatus

Figure 4: Steel plate with strain gauges arrangement.

0 100 200 300 400 500 600 700 8000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Calibration Graph

Experimentaltheroretical

Force (N)

Defle

ction

(mm

)

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.

5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

f(x) = 0.0143315017595561 x + 0.0836059566366584

f(x) = NaN x + NaN

ExperimentalLinear (Experimental)TheoreticalLinear (Theoretical)

Distance from centre

Defle

ction

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 σθ and σr.

-100 -50 0 50 100 150 200 2500

100

200

300

400

500

600

700

800

f(x) = 3.41146945193639 x − 12.0421955867499f(x) = 3.20418745719597 x + 5.5498483514333

f(x) = − 9.52580654824312 x − 67.9447623041668

Series 2Linear (Series 2)Series 1Linear (Series 1)Series 3Linear (Series 3)

Strain (µɛ)

Forc

e (N

)

Figure 7

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

-50 -40 -30 -20 -10 0 10 20 30 40 500

20000000

40000000

60000000

80000000

100000000

120000000

140000000

160000000

180000000

200000000

Sigma rSimgs ThetaSeries6Series8Series10Series12

Radius

Stre

Figure 8

6. Discussion The 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. References Benham, 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