Two-Bender Receivers Frequency Domain Analysis in Resonant-Column Tests

Javier Camacho-Tauta1, Jaime A. Santos2 and António Viana da Fonseca3

1Associate Professor, Nueva Granada Military University, KR 11 # 101-80, Bogotá, Colombia; [email protected] 2Associate Professor, University of Lisbon, Av. Rovisco Pais, 1049-001, Lisboa, Portugal; [email protected] 3Associate Professor, University of Porto, Rua Dr. Roberto Frias, s/n 4200-465, Porto, Portugal; [email protected]

ABSTRACT: The installation of bender elements (BE) in different geotechnical testing devices has been increasing in recent years. One of the most widespread combination in this matter, is the use of BE in resonant-column (RC) equipment. Taking into account that the piezoceramic elements can work alternatively as a transmitter or as a receiver, a complementary testing procedure is proposed by using the same RC-BE device. In this case, the perturbation signal is produced by the RC oscillator, while the response of the system is measured by the pair of BE located at each end of the specimen. The analysis of the received data by use of frequency domain techniques allows to obtain the dynamic properties of the soil specimen. A RC apparatus capable of producing and transmitting a signal with a specific frequency band (i.e. sine sweep or random noise) was equipped with BE and a software designed to perform this type of combined tests (TRFD). A laboratory test programme of RC and TRFD combined tests was conducted in order to study the feasibility of this hybrid method. Results indicate that the method could be useful as complementary tool to evaluate the small-strain stiffness of soil when performing RC and BE tests.

INTRODUCTION

Bender elements (BE) systems are installed in a variety of devices: oedometers, direct shear devices, triaxial cells, cyclic triaxial apparatuses, stress-path cells, among others (Ferreira 2008). The use of BE in resonant-column (RC) devices was an early application because the RC method is the reference laboratory test to evaluate the dynamic properties of soils (ASTM International 2002). Some published works show acceptable agreement between both tests (Brignoli et al. 1996; Dyvik and Madshus 1985; Ferreira et al. 2006; Nakagawa et al. 1996; Nazarian and Baig 1995; Souto et al. 1996; Yamashita et al. 2009). However, the BE test remains without a standard because there is not a unique method (Viana da Fonseca et al. 2009). One of the most important issues in BE testing is the lack of knowledge of the actual behavior of the BE elements inside the soil specimen (Lee and Santamarina 2005). Rio (2006) measured the BE response to excitations using a laser Doppler Velocimeter (LDV).

1 This work concluded that the actual movement of the transmitter depends on its own resonant frequency and the stiffening of the medium, rather than the vibration frequency of the input signal. Measurements by miniature accelerometers along the travel path between the BE transmitter and BE receiver (Camacho-Tauta et al. 2011), show that the waveform is substantially different from the originated by the function generator even for locations near the transmitter. Pallara et al.(2008) compared the input signal used to excite the bender transmitter and its actual movement measured by a laser diode. These authors suggest to use other source types and limit the use of BE only as receivers. Besides the difficulties associated to measure the soil stiffness, few authors have propose a method to compute the damping ratio of the soil based on BE techniques (Brocanelli and Rinaldi 1998; Karl 2005). Piezoceramic elements can work alternatively as transmitter or receiver, depending on the wiring and polarity combinations (Lings and Greening 2001). When serial-connected, a BE transmitter can works as receiver. Despite this arrangement is designed to measure P-waves, the receiver has enough sensibility to detect bending movements (Camacho-Tauta 2011).

Taking into account the common arrange of RC-BE systems, the aforementioned difficulties in the interpretation of BE tests, and the feasibility of use the two BE as receivers, this work proposes a complementary test in which the vibration produced by the RC-oscillator is detected by the BE-receivers located at the two ends of the cylindrical specimen. This test can be performed simultaneously with the RC test. Under these conditions, the system can be analyzed as a single-input multiple-output (SIMO) dynamic system. Thus, two transfer functions can be computed, one for each end of the specimen. The system resonant frequency and damping ratio can be identified simultaneously by a curve-fitting process or separately by the peak-picking method (Peeters and De Roeck 2001) and the half-power method (Clough and Penzien 1995), respectively.

THEORETICAL BACKGROUND

Resonant-column test by frequency domain methods (RCFD)

The RC test is the most common laboratory testing method used for measuring the small-strain dynamic properties of soils. A cylindrical soil specimen with height h and rotational mass inertia J, is subjected to torsional harmonic loading. The amplitude T0 and frequency ω of the torque are controlled while a motion transducer measures the resulting rotation level θ0, which is ϕTθ radians out of phase regarding the torque. The shear-wave velocity Vs and the damping ratio ξ of the soil are obtained by means of the transfer function H(ω) of the RC system (Santos 1999):

Where JA, ξA and ωA are the calibration factors of the active end: rotational mass

2 inertia, damping ratio and angular resonant frequency, respectively.

The transfer function of the input towards the output isolates the inherent dynamic properties VS and ξ, of a mechanical structure (Richardson 1999). The transfer function data can be obtained by means of experimental modal analysis techniques (Ewins 1984; Ramsey 1975). This alternative is suitable for small-strains, in which the system behaves linearly and consists in applying a sine sweep or random noise with frequency content in a bandwidth around resonance. The transfer function between torque T(t) and rotation θ(t) is computed from a number of time series by:

Where is the Fourier transform operator and * is the complex conjugate function. In order to cancel external noise, the transfer function is an average a multiple tests. The transfer function is obtained from experimental data using Eq. 2 and a curve fitting of the complex data provides the unknown soil parameters of the right side of Eq. 1: the shear-wave velocity VS and the damping ratio ξ.

Two-Receiver test by frequency domain methods (TRFD)

A resonant-column apparatus equipped with a bender element system represents an advantage because it is possible to compare results from both methods. Furthermore, this hybrid apparatus is able to perform a wider variety of tests procedures. The excitation source is the oscillator of the RC apparatus and the two bender element (located at the upper and lower ends of the specimen) receivers measure the response of the system. This procedure in which parts of the resonant-column system are used in combination with parts of the bender element system and the results are analyzed in frequency domain, is designated here as two-receiver frequency domain (TRFD) test. The transfer function HU(ω) between torque T(t) and response of the upper BE U(t) is computed by Eq. 3. Similarly, the transfer function HB(ω) between torque T(t) and response of the bottom BE B(t) is computed by Eq. 4.

These transfer functions of the torsional vibration system can be approximately represented by a single-degree of freedom (SDOF) system (Cascante 1996; Khan et al. 2008) in which and ξS is the damping ratio of the system and KU and KB are the voltage-to-bending ratio of the upper and bottom BE receivers, respectively:

(6) Due to the traditional location of the bender element in the extreme caps (near the axis of rotation as can be seen in Figure 1a), the deformation of the bender element is

b R max L w r

a) b) limited and the shape of the deformation is not pure bending (Figure 1b). Nevertheless, this deformation is sufficient to produce an electrical response of the BE.

FIG. 1. Deformation of BE due to the torsion of the specimen: a) top view: rotation of the top cap and BE; b) approximate shape of deformation

By using a curve-fitting method (Meza-Fajardo and Lai 2007; Rix and Meng 2005), it is possible to obtain the resonant frequency, the total damping ratio and the voltage- to-bending ratio of each of the two transfer functions. Then, the shear-wave velocity and the damping of the soil are obtained with a very good agreement by Eq. 7 and Eq. 8, respectively (Camacho-Tauta 2011):

EQUIPMENT

A conventional RC apparatus was improved by including an arbitrary function generator, a data acquisition card, a pair of bender elements and an electronic filter/amplifier. In addition, an application in LabView was developed for control, data acquisition and signal processing. The system controls the equipment, acquires data and processes the information automatically for a variety of tests including resonant-column and two-receiver methods. Figure 2 illustrates a schematic view of the RC-BE system. The signal parameters specified by the program are sent to the

4 function generator, which in turn produces the input signal. The input signal is amplified and conducted to the coils of the resonant-column apparatus. The excitation produces the vibration of the soil specimen and the movements are recorded by the two bender elements located at the ends of the specimen and by a motion transducer located at the active end of the RC apparatus. The actual current in coils is measured by a voltage through a resistance in the current circuit. The electrical signals produced by the motion transducer, bender elements and current in coils are filtered, amplified and then acquired according to the sampling parameters defined by the user in the software. The function generator produces the trigger to synchronize the readings with the starting of the input signal. Finally, the acquired raw data is processed by frequency domain techniques and the dynamic soil properties estimated by curve-fitting of the transfer functions.

FIG. 2. Schematic view of the combined RC-BE system.

TESTING METHODS

RCFD and TRFD tests can be performed simultaneously because both methods use the same source of vibration and signal excitation. The RCFD test records the rotation of the top cap by a motion transducer, while the TRFD test records the response by the BE located at each end of the specimen.

The frequency domain analysis requires a continuous signal to mobilize the coils of the resonant-column system. Two types of signals are available in the program: sine sweep and random noise. The parameters of the input signals for sine sweep and random noise are: amplitude, start frequency, stop frequency and number of tests. In the RCFD test, the applied torque and resulting rotation signals are processed by means of Eq. 2 and the resulting transfer function is compared against Eq. 1. The dynamic soil properties are obtained by curve-fitting of Eq. 1. In the TRFD test, the

5 applied torque and the vibration at each end of the specimen are measured and processed by use of Eq. 3 and Eq. 4. By using the transfer functions (Eq. 5 and Eq. 6), the resonant frequency and the total damping ratio of the system are computed. The shear-wave velocity and the damping ratio are then obtained by use of Eq. 7 and Eq. 8, respectively.

MATERIALS

To evaluate the performance of the proposed method, a specimen of an industrially- processed kaolinite was reconstituted in laboratory by vertical onedimensional consolidation. The initial water content was determined previously as well as the liquid and plastic limits. The soil was mixed with water in proportions to obtain a water content equivalent to 1.5 times the liquid limit in order to eliminate any previous history stress. Then, the mixture was poured into a cylindrical mold, which rests on a bed of fine gravel covered by filter paper. The set was placed in a water tank and the soil was gradually loaded until reach a vertical pressure of 50 kPa. After the consolidation process, the specimen was extruded and cut. A small and controlled vertical load was applied in order to insert bender elements and blades of the top cap into the soil. Once full contact between top cap and specimen was achieved, the membrane was fixed to the top cap with two o-rings. Finally, the triaxial chamber was placed, filled with water and pressurized up to 20 kPa. Vertical load was adjusted accordingly to impose an initial isotropic stress state.

The specimen was subjected to consolidation stages under isotropic confinement of 50, 100, 200 and 400 kPa. During the consolidation process, the pore pressure, vertical deformation and volumetric change were measured until to identify the end of the primary consolidation. Table 1 shows the physical characteristics and initial properties of the soil. RCFD and TRFD test were performed at the end of the consolidation process for four different isotropic confinements.

TABLE 1. Initial physical properties of the soil specimen

Description Unit Value Diameter mm 72.0 Length mm 105.5 Total mass g 737.3 Water content % 31.1 Specific gravity of - 2.62 solids Plasticity index - 17

RESULTS AND DISCUSSION

The excitation level was adjusted to obtain small shear strain (γ) levels. The shear- wave velocity and the damping ratio were estimated by means of the three transfer functions. Figure 3 shows the transfer function of the RCFD test. The model of Eq. 1

6 was adjusted to experimental data in order to obtain the soil parameters indicated in Table 2. Figure 4 shows the transfer function between torque and vibration in the upper bender element. This data was used to obtain the soil parameters by adjusting the model of Eq. 5 and solving Eq. 7 and Eq. 8. Finally, in Figure 5 is presented the transfer function between torque and vibration in the bottom bender element together with the adjusted model. Again, the soil parameters were estimated by adjusting the model of Eq. 6 and solving Eq. 7 and Eq. 8. From the Table 2, it can be observed that the agreement between RCFD and TRFD methods is satisfactory for practical purposes. Even though that the TRFD method uses an approximate solution based on a SDOF model and the scatter near 950 Rad/s (due to noise caused by harmonics of 50 HZ from the electrical network of the laboratory), the model adjusted well to the experimental data as can be seen in Figures 4 and 5. Almost all velocities were slightly lower than the values obtained by the RC method. The relative error did not exceed -0.3%, which is negligible. There is no evidence of increases of relative differences neither as function of confinement or as function of shear-wave velocity.

0.6 Measured Curve-fitting 0.5

0.4 ] m N /

d 0.3 a R [

0 900 950 1000 1050 1100  [Rad/s]

FIG. 3. Transfer function of RCFD test, p’=400 kPa

Regarding the damping ratio, the estimates obtained from the upper-end BE were higher than the same values obtained by the RC test. The maximum difference was 0.36%. There was no clear trend of this difference with the variation of the confinement pressure. The values of damping ratio estimated from the bottom-end BE were in the same order of difference. Differences between those damping ratios values could be due the damping eigenvalues of each BE.

TABLE 2. Shear-wave velocity and damping ratio by three transfer functions

p’ (kPa) γ Vs (m/s) ξ (%)

H HU HB H HU HB

7 200 Measured Curve-fitting

150 ] m N / 100 V [

0 900 950 1000 1050 1100  [Rad/s] 50 7.4x10-6 157.41 157.15 157.43 2.36 2.72 2.08 100 7.2x10-6 192.97 192.71 192.81 1.69 1.76 1.60 200 6.0x10-6 237.93 237.23 237.44 1.38 1.50 1.48 400 5.1x10-6 290.71 289.91 290.26 1.13 1.33 1.30

FIG. 4. Transfer function between Upper BE and Torque, p’=400 kPa

800 Measured Curve-fitting

600 ] m N / 400 V [

0 900 950 1000 1050 1100  [Rad/s]

FIG. 5. Transfer function between Bottom BE and Torque, p’=400 kPa

CONCLUSIONS

8 A new system was developed including the basic data acquisition and signal processing for laboratory dynamic testing of soils. The system controls the equipment, acquires data and processes the information automatically for resonant- column and the combined method proposed in this paper.

A combined RC-BE test was proposed in this research. The new test takes advantage of the resonant-column apparatus equipped by BE transducers. RCFD and TRFD tests can be performed simultaneously. By using a single-degree of freedom (SDOF) model, a very close approximation of the dynamic properties of the soil can be found.

Comparison between RCFD and TRFD results show that this combined test could be used as a complementary procedure, which might assist when dynamic properties by the RC test are difficult to determine.

ACKNOWLEDGMENTS

This paper is a result of the financial support due to the research project POCI/ECM/55589/2004 – “Dynamic properties in residual soils from granite: strain- rate, frequency & time effects” and developed under the research activities of ICIST of IST and CEC from FEUP, supported by pluri-annual funding from FCT (Portuguese Science and Technology Foundation). First author acknowledges the financial support provided by the Nueva Granada Military University (Project INP- ING-1575).

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