Lab report 1 - Biaxial Stress in a Hydraulic Cylinder PDF

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Biaxial Stress in a Hydraulic Cylinder...

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University of Technology, Sydney 48642 STRENGTH OF ENGINEERING MATERIAL LabReport1:BiaxialStressinaHydraulicCylinder

Prepared by: Nur Liyana Binti Yaacob 12779714 | Autumn 2018

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Table of Contents

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1.

HYDRAULIC CYLINDER WITH PRESSURE & TORSION EXPERIMENT ................................................. 1.1 Objective ................................................................................................................................................................. 1.2 Apparatus ................................................................................................................................................................. 1.3 Theory ..................................................................................................................................................................... 1.4 Test Procedure ........................................................................................................................................................ 1.5 Test Results ............................................................................................................................................................. 1.6 Calculations ............................................................................................................................................................. 1.7 Analysis and Interpretation of Results ..................................................................................................................

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PRESSURISED CYLINDER WITH OPEN & CLOSED EXPERIMENT ......................................................... 2.1 Objective ................................................................................................................................................................. 2.2 Equipment ............................................................................................................................................................... 2.3 Theory ..................................................................................................................................................................... 2.4 Test Procedure ........................................................................................................................................................ 2.5 Test Results ............................................................................................................................................................. 2.6 Calculations ............................................................................................................................................................. 2.7 Analysis and Interpretation of Results ..................................................................................................................

3.

References .................................................................................................................................................................

4.

Appendices ................................................................................................................................................................

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1.

HYDRAULIC CYLINDER WITH PRESSURE & TORSION EXPERIMENT

1.1 Objective • • • • •

To verify pressure in thin walled cylinder and torsion in hollow section theories. To determine E, G and V for an extruded aluminium alloy. To verify calculation of stress from strain gauge measurements. To demonstrate the validity of superposition of stress. To apply appropriate yield criteria to predict onset of yield for a general state of stress.

1.2 Apparatus An extruded aluminium alloy cylinder, inside diameter 95mm, wall thickness 3.3 mm, is subjected to internal pressure and a torque. As show in Figure 1, the torque is developed by the application of two weights (W) to a 2m long loading bar. Two rectangular strain gauge rosettes are bonded to the cylinder as shown in Figure 2.

1.3 Theory The stresses developed by the pressure are:

𝜎𝐻 =

𝑝𝑟 𝑡

; 𝜎𝐿 =

𝑝𝑟 2𝑡

; 𝜏𝑥𝑦 = 0 ---------- (1)

Where: r = inside radius t = wall thickness p = pressure

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The stresses developed by the torque are:

𝜎𝐻 = 𝜎𝐿 = 0; 𝜏𝑥𝑦 =

𝑇(𝑟+𝑡) 𝐽

--------- (2)

Where: J = Polar 2 nd Moment of Area T = Torque Using Hooke’s Law, the strains are:

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ℇ𝐻 ℇ𝐿

1

1

-v

E

-v

1

𝜎𝐻 𝜎𝐿

; ℇ𝑥𝑦 = ϒ𝑥𝑦 = 𝜏𝑥𝑦

2

------------ (3)

2g

1.4 Test Procedure 1) Record the true gauge factors (εi, εii, εiii, εiv, εv, εvi) and the strain amplifier gauge factor setting. 2) Measure the strain on each gauge for the following load cases: I. T = 0, p = 0 -> 2MPa (in increments of 0.5 MPa) II. p = 0, T = 0 -> 300N.m (in increments of 100 N.m) III. T = 200 N.m, p = 1.5 MPa 3) Record the strains on all six gauges. Note: There are no variations in the test procedure during conducting the experiment.

1.5 Test Results

Table 1

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Table 2

Table 3 Note: • • •

Yellow-filled = Input Data = I = Test Results Blue-filled = Best fit = B = Values obtained by linear regression and include the gauge factor correction Red-filled = Calculated values from best fit results.

Figure 3

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Figure 4

1.6 Calculations •

For T = 0 & p = 2.0 MPa, calculate E and V: E = 73.64 GPa v = 0.347

•

Note: Refer Appendix D for more detailed calculations. For T = 300 N.m, p = 0, calculate 𝜏𝑥𝑦 and G: 𝜏𝑥𝑦 = 6.193 MPa G = 24.76 GPa Note: Refer Appendix D for more detailed calculations.

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Results for tests [T = 0, p = 2.0 MPa] and [T = 300N.m, p = 0] added together:

εi

εii

εiii

εiv

εv

εvi

313.3

313.2

54.7

85.1

42.2

288.1

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Calculations for point A on the pressure vessel where the strain rosettes are located:

Point A

σ1 = 29.21 MPa σ2 = 12.29 MPa τabs max = 14.61 MPa σvon mises = 18.53 MPa F.S. = 10.26 Note: Refer Appendix D for more detailed calculations.

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1.7 Analysis and Interpretation of Results •

Comparison for the value v, G and E from test results with published value:

Modulus of Elasticity, E (GPa)

Shear Modulus, G (GPa)

Poisson’s Ratio, v

Published Value (PV)

73.10

27.00

0.35

Calculated Value (CV)

74.64

24.76

0.347

Percentage Difference ((PV – CV)/CV x 100) (%)

-2.06

9.05

0.86

Table 5 From Table 5, there are minor percentage differences in these values that has been detected. The differences might have been cause by some factors. One of the factors is the placement of the strain gauge rosettes on the surface of the hydraulic cylinder may been slightly difference to the ideal angles used for the published value. Another factor is the hot glue material that were being used to attach the strain gauge rosettes on the hydraulic cylinder might affect the strain detected by the system. However, the percentage differences are relatively small and is below 20% which is not a bad result for the experiment. Note: Published value is referred from Appendix E. •

Relationship of v, G and E (G = E/2(1+v): From the calculated values, v, G and E are related by the equation; 𝐺=

𝐸

From the above equation in conjunction with the calculated values for E and v results in G = 27.66 GPa. This agrees with the calculated G = 24.76 GPa as the results only differ approximately 11.71%. The value of 𝜏𝑥𝑦 and other values used to calculate the G’s value also need to be taken into consideration of the inconsistencies in the value of G. The rounding off and number of significant figures also could make the results differ.

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Comparison of values from 1.6.3 with test values for [T = 300 N.m, p = 2.0 MPa]:

Values, p(MPa), T(N.m)

εi

εii

εiii

εiv

εv

εvi

p=2.0, T=300 (best fit results)

313.00

310.00

55.00

88.00

44.00

289.00

P=2.0, T=0 + p=0, T=300 313.30

313.20

54.70

85.10

42.20

288.10

Percentage Difference (%)

-1.02

0.55

3.41

4.27

0.31

-0.09

Table 6

Table 6 above shows the comparison results between values obtained by using superposition method and accumulation using torque only and pressure only. The overall percentage variance is relatively small with the highest being ± 4.27% which is indicates the error was minimize. However, the small difference could have been due to the measuring device or the strain gauge rosettes itself. The expanding of rosette when pressure or torque being exerted could affect the initial condition which leads to the percentage difference that we could observed from Table 6 above. Method of superposition is undeniable since the errors were small. As a conclusion, the test results support the principle of superposition. The error from the combination of both values can be more accurate by increasing the measuring device’s sensitivity. Another way of reducing errors would be conducting the experiment in longer interval for the strain gauge rosette to completely retract back to its initial condition.

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Principal stresses, principal orientation, absolute maximum shear stress & von Mises stress calculations using theoretical stress analysis methods:

Percentage Difference, (%) Theoretical Values

Strain gauge A

Comparison

Column (T)

(T)-(A)

(A) σ1 (MPa)

28.79

27.82

0.97

σ2 (MPa)

14.39

13.68

0.71

15.54

14.61

0.93

14.40

18.53

-4.13

13.19

10.26

2.93

τabs max (MPa) σvon mises (MPa) F.S.

Table 6

Using theoretical values of E, G and v along with the calculated values for the hoop, longitudinal and shear stresses, the factor of safety can be determined by assuming if the yield stress 190 MPa. The theoretical values are compared with measured values in the above Table 7. As we can observe from Table 7, the average percentage difference is very low and below 5% variance. The highest difference was only up to 2.93%. Hence, we can conclude that using the experimental analysis is a very effective method to confirm the experimental results. Note: Refer Appendix D for more detailed calculations.

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2.

PRESSURISED CYLINDER WITH OPEN & CLOSED EXPERIMENT 2.1 Objective • To verify the theory of transformation of strain. • To verify thin walled cylindrical pressure vessel theory. • To apply yield criteria to predict onset of yield.

2.2 Equipment The thin walled cylinder apparatus is a mild steel hydraulically pressurised cylinder. Open and closed end conditions are controlled by trust knob on the front of the cylinder. Figure 5 shows when knob is ‘wound in’ against the piston the pressure force on the piston is reacted by external frame and there is no axial load on the cylinder walls. So that it is known as open end conditions. When the thrust knob is ‘wound out’, the pressure force on the piston is reacted by the cylinder and an axial stress is developed in the cylinder walls. This is known as closed end condition. Foil type strain gauges are bounded to the cylinder as shown in figure 6. For all gauges, the gauge factor is 2.00 and the gauge length is 6mm. The cylinder dimensions are O/D=86.43mm, I/D= 79.98 mm.

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2.3 Theory “Thin-walled” refers to a vessel having an inner radius to wall thickness ratio of 10 or more (r/t≥10). As wall thickness / internal radius...