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Experimental Testing for Aerospace Structures Laboratory Activity n°1 Student: Andrea Citeroni Mat: 1381071

The report details the experimental analysis performed upon a cantilever beam loaded with a calibration weight at its free end. Strain was estimated at the strain gauges’ location with the “zero” and “deflection” methods using a strain measuring system. Once the geometrical and elastic properties of the beam were evaluated, recorded data were compared to the resulting distribution of strain from the Euler-Bernoulli theory.

1. Theoretical background

In aeronautical and spatial application field, the Euler-Bernoulli linear theory of the beam can be considered an acceptable approximation of the behaviour of a cantilever beam with a lumped load. The investigation aims to validate this assumption, comparing the experimental and the analytical results.

Figure 1: beam geometry From the hypothesis of the Euler-Bernoulli theory, the bending moment distribution is linear along the beam

= ( − ) ( ) where ξ=x/L is the dimensionless span position. Stress and strain distributions by the following equations (2) and (3).

= = ( − ) ( )

= = ( − ) ( )

2. Measuring instruments Shown below the instruments used for the measurements.

o A Vernier Scale (Nonius 1/20), to which was associated an uncertainty of , .

o A Tape Measure, to which was associated an

uncertainty of , .

o Four linear Strain Gauges (with one measurement grid), fastened to the beam in order to evaluate the strain. Two models from different manufacturers were used:

three Typ3/120LY41 models from HBM industries with a gauge factor of

= 1,99 ± 0,01

a CEA-13-250UW-120 model produced

by MM Division, characterized by a gauge factor of

= 2,11 ± 0,005

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Figure 2: HBM strain gauge data-sheet

Figure 3: MM strain gauge data-sheet

o A Tester measured the resistance of the gauges.

o A Strain Gauge Measuring System, based

on Wheatstone Bridge working principles.

Figure 4: measuring system used in the activity

o A K-type Thermocouple measured the operative temperatures.

3. Test setup layout

The beam was constrained between two plates to avoid displacements or sliding. Strain gauges were applied along his span with a two-component glue on both upper and lower surfaces. Each gauge was then connected to one of the five channels of the measuring system, according to a particular Quarter Wheatstone Bridge configuration.

Figure 5: quarter bridge sketch

Figure 6: connections layout The gauges in red (numbered 1, 2 a 3) are the HBM models; the green one (numbered 4) is the MM model instead. Has to be noticed that gauges 2 and 3 are placed symmetrically on the upper and lower surface of the beam: the measured strain has to be circa equal in absolute value, but with opposite sign in the channels. This configuration allows to double the sensitivity of the bridge when gauges are on the same bridge. The thermocouple was placed near the measuring system, preventing the cable to get in touch with the equipment.

Figure 7: view of the calibration weight The calibration weight ( = ± ) was applied at the beam free end using adhesive tape, acting (approximatively) like a lumped one.

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Figure 8: global view of the test setup

4. Measurement chain 4.1. Resistance of the strain gauges

The digital tester evaluated the resistance of each strain gauge disconnected from the measuring system.

S.G. R 1 120.8 2 121.6 3 121.2 4 121.1

Table 1: strain gauges’ resistance in [Ω]

4.2. Beam geometry The length of the beam was measured with the tape taking into account the flexibility of the instrument and the uncertainty of the scale of measurement.

= ± . Section was characterized with the Vernier scale, paying attention at the orthogonality between the instrument and the beam. Due to imperfections, damages and glue leftovers on the surface, measurements were performed at different points along the beam span.

Point a t root 14.90 ± 0.025 2.95 ± 0.025

half span 14.95 ± 0.025 2.95 ± 0.025 tip 15.05 ± 0.025 2.95 ± 0.025

Table 2: beam section in [mm]

4.3. Gauges position The span position of each gauge was measured with regard to his centre.

Channel Distance [mm] span (ξ) 4 40 ± 0.5 0.104± 0.001

2 , 3 167 ± 0.5 0.436± 0.001 1 288 ± 0.5 0.752± 0.001

Table 3: gauges position

4.4. Estimation of the axial strain Data acquisition was achieved with an iterative process:

Figure 9: calibration knobs and output display

1st Step. Choose the active channel on the upper left knob;

2nd Step. Adjust the gauge factor on the lower left knob;

3rd Step. Adjust the sensitivity (setting a likely order of magnitude for the strain) on the upper right knob;

4th Step. Balance the Wheatstone Bridge using the four lowest knobs (from left to right) until the arrow on the display is on the 0, then collect the obtained sensitivity;

5th Step. Apply the load and wait a few seconds to avoid unsteady motions;

6th Step. Collect the temperature on the thermocouple;

7th Step. Read the value given by the position of the arrow on the graduated lines;

8th Step. Use the lowest knobs to balance again the Wheatstone Bridge, then collect the obtained sensitivity;

9th Step. Remove the load.

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At every iteration, output information were: sensitivity at the free-load configuration; strain; sensitivity at the loaded configuration.

In order to reduce random errors and statistically characterize the recorded data, steps from 4th to 9th were repeated 10 times for each channel.

4.5. Recorded data The choice of the proper graduated scale on the system output display, depended on the set sensitivity of the channel: for the 1st one with = , it was used the lower

scale where each interval was worth μ ; for the 2nd and the 3rd ones with = , it was

used the upper scale where each interval was worth μ ;

for the 4th one with = , it was used the lower scale where each interval was worth μ .

The uncertainty associated to the measure was considered to be a quarter of the interval.

4.5.1. First channel Set: = , = = ±

Measu

re T [°C]

Deflection Method Zero Method Sensitivity

S [V] Strain [µs]

Sensitivity [V]

Strain [µs]

1st 23.4 49541 152 49685 144 2nd 23.2 49540 152 49684 144 3rd 23.2 49539 152 49684 145 4th 23.3 49539 152 49684 145 5th 23.3 49539 152 49684 145 6th 23.2 49540 152 49685 145 7th 23.3 49540 152 49684 144 8th 23.4 49539 152 49684 145 9th 23.3 49540 152 49684 144

10th 23.4 49539 152 49684 145 Table 4a: data acquired from the strain gauge n°4

4.5.2. Second channel

Set: = , = = ±

Measu

re T [°C]

Deflection Method Zero Method Sensitivity

S [V] Strain [µs]

Sensitivity S [V]

Strain [µs]

1st 23.4 49172 +320 49493 321 2nd 23.4 49170 +320 49494 324 3rd 23.4 49171 +320 49491 320 4th 23.4 49169 +320 49490 321 5th 23.4 49167 +320 49490 323 6th 23.4 49168 +320 49490 322 7th 23.4 49168 +320 49491 323 8th 23.4 49167 +320 49493 326 9th 23.3 49167 +320 49492 325

10th 23.3 49166 +320 49492 326 Table 4b: data acquired from the strain gauge n°2

4.5.3. Third channel Set: = , = = ±

Measu

re T [°C]

Deflection Method Zero Method Sensitivity

S [V] Strain [µs]

Sensitivity S [V]

Strain [µs]

1st 23.8 50760 -320 50431 -329 2nd 23.6 50759 -320 50436 -323 3rd 23.4 50758 -320 50435 -323 4th 23.4 50759 -320 50434 -325 5th 23.4 50759 -320 50436 -323 6th 23.5 50758 -320 50431 -327 7th 23.4 50755 -320 50353 -320 8th 23.4 50754 -320 50432 -322 9th 23.4 50753 -320 50431 -322

10th 23.3 50754 -320 50433 -321 Table 4c: data acquired from the strain gauge n°3

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4.5.4. Fourth channel

Set: = , = = ±

Measu

re T [°C]

Deflection Method Zero Method Sensitivity

S [V] Strain [µs]

Sensitivity S [V]

Strain [µs]

1st 23.6 48947 -560 48418 -529 2nd 23.6 48960 -560 48417 -543 3rd 23.6 48941 -520 48421 -520 4th 23.3 48938 -520 48419 -519 5th 23.4 48946 -560 48419 -527 6th 23.3 48942 -520 48420 -522 7th 23.4 48941 -520 48353 -588 8th 23.5 48944 -560 48423 -521 9th 23.6 48947 -560 48417 -530

10th 23.6 48947 -560 48414 -533 Table 4d: data acquired from the strain gauge n°1

In the Zero method, the strain was evaluated as

= −

5. Thermal Compensation The deformation of the gauge due to thermal effects induced by the currents was evaluated as

= −

Figure 10: thermal compensation for the HBM gauge

Figure 11: thermal compensation for the MM gauge For HBM gauge ( ≅ °) was used the relation

= − , + − , ∗ + ,∗ ± , /°

and for MM gauge ( ≅ ° )

= −3,32 ∗ 10 + 2,72 − 6,38 ∗ 10 + 3,55 ∗ 10 − 3,99 ∗ 10 ± 0,27 /°

For example, when ≅ , °

= − , ± , for the 2nd channel; = − , ± , for the 1st channel.

Since the were smaller than the uncertainties associated to the graduated scales of the measuring system, thermal compensation was neglected.

6. Results

6.1. Introduction to the statistical approach

The mean value for each set of measures was evaluated as

= ∑ = ( ) where was the value of the i-th measurement. Then the deviation from the mean value of each was evaluated as

= − ( ) In order to estimate the uncertainty associated to , the Standard Deviation was taken into account.

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= ∑ 2=1− 1 ( ) The Sum of Squared Residuals approach was also considered in the evaluation on uncertainties: for a defined variable = ( , … , ), function of direct measures with their uncertainties , the total uncertainty can be measured as

= ( )

6.2. Strain

Eq. (4), (5) and (6) were used. The 7th measure of Zero method in Table 4d, much higher than the others, was not considered.

Channel 1st 2nd 3rd 4th Deflection

Method 152.00 320.00 -320.00 -544.00 Zero

Method 144.60 323.10 -323.50 -527.11 Table 5: mean value of strain in [µs]

1st channel 2nd channel 3rd channel 4th channel

Deflection Method

\ \ \

-16.00 -16.00 24.00 24.00 -16.00 24.00 24.00 -16.00 -16.00 -16.00

Zero Method -0.60 -2.10 -5.50 -1.89 -0.60 0.90 0.50 -15.89 0.40 -3.10 0.50 7.11 0.40 -2.10 -1.50 8.11 0.40 -0.10 0.50 0.11 0.40 -1.10 -3.50 5.11

-0.60 -0.10 3.50 \ 0.40 2.90 1.50 6.11

-0.60 1.90 1.50 -2.89 0.40 2.90 2.50 -5.89

Table 6: deviations

σ 1st channel

2nd channel

3rd channel

4th channel

Deflection Method \ \ \ 20.66

Zero Method 0.52 2.13 2.76 7.74 Table 7: standard deviation associated to the strain in [µs]

6.3. Evaluation of the Young modulus

From Table 2 it was obtained = , ± . . The Young modulus of the beam was then evaluated with eq. (2) at each gauge’s position using the mean strain value of the relative channel.

Deflection Method Zero Method 1st channel 56.47 59.36 2nd channel 61.00 60.41 3rd channel 61.00 60.34 4th channel 57.00 58.83

Table 8: Young modulus in [GPa] Deflection Method Zero Method

58.87 59.73 ± 2.47 ± 0.77

Table 9: standard deviation of the Young Modulus in [GPa] The SSR formula (eq. (7)) was applied to

= ( − ) given the partial differential expressions:

= ( − )

= ( − )

= −

= − ( − )

= − ( − )

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= − ( − )

Deflection Method Zero Method 1st channel ± 3.00 ± 3.33 2nd channel ± 1.90 ± 3.78 3rd channel ± 1.90 ± 4.11 4th channel ± 10.90 ± 3.36

Table 10: RSS uncertainties

7. Discussion of the results Table 7 shows that for the Deflection method was obtained a constant measure in each of the first three channels. For the Zero method instead, the strain was the difference of two measures (i.e. the sensitivities of the system), each one affected by a certain error, inducing fluctuations of data. As concerned the 4th channel, a too high value for the deviation was obtained because the set of measures wasn’t a normal distribution. Nevertheless, being approximatively equals to a quarter of interval on the graduated scale it was considered an acceptable result. The Zero Method instead, gave smaller deviations because the strain depended on sensitivities and not on the uncertainty associated to the scale. From Table 9, the constitutive material of the beam was supposed to be Aluminium. Since typical values are from up to , the strain measurements led to a quite large error in the Young modulus esteem. Deflection Method Zero Method

= 8.3 % 6.6 % = 17,6 % 15.9 %

Table 11: error on Young Modulus esteem Probable reasons could have been: The quality of the fastening between gauges and

structure;

The span position of the gauges, which were considered non-dimensional;

The presence of eventual pre- or residual stresses on the beam which could have altered the measured strain;

The parallax error in the measures acquisition; The hysteresis of the measuring system, because

of the recurring balances; Bad electrical connections; Fluctuations of the tension in the Wheatestone

Bridges; Environmental disturbances, such as the thermal

one measured by the thermocouple or the movement of the system during the acquisition of data;

The fact that was calculated with the

approximated formula given by the Euler-Bernoulli theory (in which, for example, was approximated too);

The propagation of errors associated to the

measures, being function of six parameters with their uncertainties.

The error associated to the use of materials different from the calibration one was neglected.

= −−

For HBM gauges (calibrated with steel) = , % and, since ≅ , the error was about

. For MM gauges instead, the calibration material was exactly an aluminium alloy. The gravity on the beam wasn’t considered because the balance of the Wheatestone Bridge at each measurement was performed with the structure constrained. To consider its effects, the structure should have been weighed separately from the measuring system.

8. Comparison with analytical results For the Young modules given by Table 9, the strain distribution was evaluated with eq. (3) and plotted on a − ( ) plane.

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Figure 12a: strain distribution (Deflection method)

Figure 12b: strain distribution (Zero method)

Deflection Method Euler-Bernoulli 1st channel 152.00 145.8 2nd channel 320.00 333.92 3rd channel 4th channel 544.00 526.74

Table 11: comparison of strain values Zero Method Euler-Bernoulli

1st channel 144.60 143.70 2nd channel 323.1 326.80 3rd channel 323.5 4th channel 527.11 519.20

Table 12: comparison of strain values The error of approximation committed in the Euler-Bernoulli theory, with regard to the displacement method, was about:

, % in the 1st channel; , % in the 2nd and 3rd channels;

, % in the 4th channel; And with regard to the Zero method was about:

, % in the 1st channel; , % in the 2nd channel; % in the 3rd channels; , % in the 4th channel.

Since the errors were quite small, the Euler-Bernoulli theory can be considered a good approximation for the behaviour of the cantilever beam analysed in the activity.

9. Concluding Considerations It was noticed that in the definition of the global uncertainty , the only quantity which wasn’t directly measured was the calibration weight. in the laboratory, an electronic scale with the uncertainty of ± gave

= ± that is an increase of about the 10% compared to the nominal one of . Since the Young modulus relation is proportional to the weight itself, the same increase of was expected. In fact, for the deflection method the new Young Modulus was

= . ± , which is quite similar to an Aluminium alloy’s one.