Measurement of material damping with bender elements in triaxial cell

Transcription

1 Measureent of aterial daping with bender eleents in triaxial cell. Karl & W. Haegean aboratory of Soil Mechanics, Ghent University, Belgiu. Pyl & G. Degre Departent of Civil Engineering, Structural Mechanics Division, K.U.euven, Belgiu ABSTRACT: This paper presents a ethod to easure the viscous daping ratio of soils in isotropic stress conditions using a triaxial cell equipped with piezoceraic bender eleents. The technique is based on the deterination of a frequency response spectru of the soil specien by eans of a continuous sine excitation generated by the eitter eleent. Syste identification procedures, known fro odal testing, are applied for the interpretation of the spectra. Analytical expressions are found a finite eleent odel is developed to encircle the region of the resonant frequency. The test is perfored on a reconstituted undisturbed silt saple. 1 INTRODUCTION The application of piezoceraic bender eleents for easureent of daping ratio in the frequency doain in a triaxial cell under isotropic confineent is discussed in this paper. To characterize soils dynaically piezoceraic eleents have been increasingly used in the last 3 years. In an early stage piezoceraics were ainly used to generate receive copression waves. Since little inforation about the soil structure can be obtained the P-wave velocities are highly influenced by pore fluid, the piezoceraics have been cobined in different fors to generate receive shear waves. Such cobined fors of piezoceraics are known as bender eleents. Bender eleents consists of two thin piezoceraic plates rigidly bonded to a central etallic plate. Two thin conductive layers, electrodes, are glued externally to the bender. The polarization of the ceraic aterial in each plate the electrical connections are such that when a driving voltage is applied to the eleent, one plate elongates the other shortens. The net result is a bending displaceent. On the other h, when an eleent is forced to bend an electrical signal can be easured through the wires leading to the eleent. A transitter eleent a receiver eleent are respectively placed in the botto top cap of a triaxial cell. The shear strain of a pulse generated by a bender eleent is less than 1-3 %, falling in the elastic strain range of soil. In ost of the papers presented in literature bender eleents are used to easure wave velocity in the tie doain. In these ethods a pulse is eitted by a bender eleent the travel tie is deterined when the pulse arrives at the second bender eleent, with the tip located at a known distance fro the tip of the eitter. The work of Brocanelli & Rinaldi (1998) describes a ethod to easure the daping ratio shear wave velocity, using bender eleents while working in the frequency doain. This technique is revived, new theoretical forulations for the interpretation of the test finally the results of tests on two saples of a silt aterial are given. OUTINE OF THE METHOD The idea of the ethod is to bring a short cylindrical soil saple in a shear oveent. For that reason the botto bender eleent is excited with a steady sine signal of constant voltage the aplitude is easured at the receiver eleent. To ake this value independent fro the source aplitude it is noralized by this aplitude. This process is repeated at different frequencies until the whole spectru of saple response is defined. The daping ratio is estiated at the points of the curve around the natural frequency of the shear ode. For this purpose different techniques are available such as the half-power the ore general circle-fit ethod, the latter is also using the phase coponents of the resonant curves. Preliinary tests on a clay saple showed that it could be difficult to find the correct

2 peak in the response spectru corresponding with the shear ode. So it is useful to have knowledge about the frequency range were this ode doinates. The ain tasks to apply this ethod are, beside the laboratory test itself: a) finding of a suitable way to predict the shear ode frequency b) the calculation of the daping value fro the response spectru itself. 3 ABORATORY APPARATUS AND TEST SET-UP The test triaxial cell in which the saple is subjected to an isotropic confining water pressure the peripheral devices used in this work are shown in Figure 1. Bender eleents are ounted at the center of the botto top cap (Dyvik & Madshus 198). The signals to drive the transitter eleent are generated by a HP Dynaic Signal Analyzer aplified in a separate device to a peak-to-peak aplitude of about V p-p. The receiver bender eleent is connected to the analyzer directly. To ake coparisons possible between the applied the received signal, the driving signal to the sender is observed by the analyzer too. If the dynaic analyzer is used in the swept sine ode a frequency-span with a step width is given, the analyzer changes the source frequency autoatically, calculates the ratio between source receiver aplitudes draws the response curve. A test takes about 1. inutes. The ass of a stard etal upper cap is several ties larger than the ass of the soil specien. This would force the saple to a bending oscillation instead of the desired shear oscillation. Therefore the etal cap was redesigned a new cap was ade of a plastic aterial with a total ass of about 1 g. During the test the top cap only rests on the saple, the connection to the loading plunger is reoved. Bender eleents Top cap Saple Botto cap Saple: d = c h 3 c Figure 1. Scheatic test set-up. Signal Analyzer Aplifier Source Ch.1 Ch. NATURA FREQUENCY OF THE SHEAR MOVEMENT.1 Analytical forulations A cylindrical soil saple placed in the triaxial cell can be considered in a siplified assuption as a fixed-free bea with the ass of the top cap attached at the free end. Brocanelli & Rinaldi (1998) have shown by eans of a finite eleent analysis of a 3 c high saple that with a relatively light ass at the top, the saple defors in perfect shear in the first ode with the increase of the ass at the top, the shape of the defored specien tends to be of flexural. For a pure shear deforation an expression for the angular natural frequency of the first shear ode (ω S ) can be derived fro the wave equation for a shear wave propagating in a rod the force equilibriu at the boundaries: k T ωs = V S ω S tan (1) VS where is the ass of the top cap, T is the ass of the saple, is the saple length, k is the shear factor V S is the shear wave velocity, latter is connected to the shear odulus G the unit ass ρ by the expression: G V S = () ρ A detailed forulation can be found for instance in Graff (1991) Brocanelli & Rinaldi (1998). For long saples or large asses of the top cap, the first resonant ode of the bea will be predoinately of the bending. For the case of a pure bending deforation Cascante et al. (1998) obtained the following equation fro the Rayleigh approxiation: 3EI ω f = (3) 3 33 h 9 h t E is the odulus of elasticity, I the area oent of inertia h the distance between the end of the saple the centroid of the ass of the top cap. The basic concept in the Rayleigh ethod is the principle of conservation of energy. To apply the Rayleigh procedure, it is necessary to assue the shape of the syste in its fundaental ode of vibration. This assuption of a shape function effectively reduces the syste to a SDOF syste. Thus the frequency of vibration can be found by equating the axiu strain energy developed during the otion to the axiu kinetic energy. Detailed inforation concerning Rayleigh's ethod can be found in Clough & Penzien (1993). Because the real behavior of the saple is always influenced by shearing bending it could be diffi-

3 cult to find criteria to decide what expression is ore suitable. Therefore the Rayleigh expression was extended by a ter for the horizontal shear displaceent. This led to the following equation: 6 EI + GAkη ω = () T γ + δ with γ = η + 7η δ = h + 8 ηk 3 K+ 16 hη h + 1η The derivation of Equation can be found in the appendix. The actual shape of the ode is supposed as the superposition of a bending a shearing displaceent. η depends on the ratio between these two parts. The unit of η is ². A is the area of the cross section. The resonant frequency ω can be calculated at those η where ω becoes a iniu. This follows fro the consideration that any shape other than the true vibration shape would require the action of additional external constraints to aintain equilibriu. These extra constrains would stiffen the syste, adding to its potential strain energy, thus cause an increase in the coputed frequency. Consequently, the true vibration shape will yield the lowest frequency obtainable by Rayleigh's ethod. The solution of the extree value proble is possible but leads to a very large expression is therefore not given here. Another approach to calculate the natural frequency of the soil saple is to use the thick bea theory, the so-called Tioshenko bea theory. The Tioshenko bea includes beside the bending effects, shear rotary inertia effects. A suitable solution of the syste of differential equations of a cantilever bea with a ass elastically ounted at the free end can be found in Rossit & aura (1). Siplified to the case of a rigid ounted ass, this solution is given by the following deterinant expression: δ + α = α sinα δ cosα α δ Ω ηλ r cosα δ sinα α ε β β sinh β ε cosh β β ε Ω ηλ () cosh β ε sinh β β r with = r F = Ω T Ω ε = Ω ηλ + α = EI 3 ω Ω = T T I η = η λ 1 H + β H F H = (1 A + λ) Ω η δ = Ω ηλ α β = H + (1 + ν ) λ = k H F The natural circular frequency ω T can be extracted fro this expression. T, A, E, I, ν, k Figure. Siplified syste with paraeters of Equation.. Finite eleent odel The given analytical expressions describe only the first flexural ode. Even though the kind of excitation suggests that this ode is doinating, a finite eleent odel was used to get an idea what other odes appear especially to check how they interact with the flexural ode of interest. Furtherore the validity of the Rayleigh the Tioshenko forulation can be evaluated. The syste of the cylindrical saple the top cap was odeled in the three diensional space with the FEM-progra ABAQUS. All nodes in the botto face of the saple were fixed. The discretization of saple cap was done by nearly cubic brick eleents of about length. inear elasticity was chosen for the behavior of the soil saple the cap. The latter was odeled with the actual properties of the plastic aterial, i.e. a ass of 1 g, an elasticity odulus of 3 MPa, a height of 19 a diaeter of. The bender eleents itself the excitation were not included since the odeling focuses on the extraction of the natural frequencies shapes of the odes. The first five odes together with a verbal description are given in Table 1. This table is valid for the reconstituted silt saple S1 at 1 kpa confining pressure for the coefficient C A =. Fro the FEM-calculation it is seen that the lowest ode is indeed a flexural ode doinated by shear deforation. The frequency of the second ode sees to be sufficient higher so not interacting with the first ode.

4 Table 1. Natural frequencies found by finite eleent calculation for saple S1, 1 kpa C A =. Mode 1 Mode Mode 3 Mode Mode The estiation of the natural frequencies in the experiental part of this work is always done with the FEM-value k = Hz 131 Hz 171 Hz 3 Hz 1 Hz flexural torsional longi- flexural torsional, tudinal nd order Poisson's ratio The shear wave velocity is deterined by tie arrival easureents on the real saple so the shear odulus is known. However the analytical expressions as well as the finite eleent odel need the elasticity odulus too. An experiental easureent using pulses of copression waves was not done. Therefore, the elasticity odulus was calculated assuing a Poisson's ratio of ν =.9. The finite eleent analysis is perfored with different ν values to evaluate the influence of ν on the natural frequencies. The differences between the natural frequencies for the lowest ode, doinated by shear deforation, were not crucial. Therefore an assuption of ν =.9 sees to be acceptable is used in all calculations. Shear coefficient k A shear coefficient k is needed in the Equations. The coefficient is a diensionless quantity, dependent on the shape of the cross section is introduced to account for the fact that the shear stress shear strain are not uniforly distributed over the cross section. According to the definition, k is the ratio of the average shear strain on a section to the shear strain at the centroid. Beside the siple, Poisson's ratio independent assuption k =.9, Cowper (1966) gives the following expression for a circular cross section: ( + ν ) 6 1 k = (6) 7 + 6ν This gives for a Poisson's ratio of ν =.9, k =.886. Because in a FEM calculation the values of shear stress shear strain are known at each node the k coefficient can be calculated by these data too. The shear stress at each node in a cross section was taken averaged. This value was divided by the shear stress in the iddle of the section provided a k factor. The k values for the different sections were again averaged. The final k factor calculated at the frequency of the 1 st ode was k =.839, which is less than Cowper's shear coefficient. This sees reasonable since Cowper (1966) points out that his values of k are ost satisfactory for static low-frequency deforations of beas not for high-frequency vibrations as in our case..3 Influence of the confining water pressure on the natural frequency of the saple The water in the triaxial cell used to apply an isotropic stress to the saple, is considered as an incopressible inviscid fluid. An open water surface does not exist in the closed cell. In this case the influence of the fluid to the natural frequencies of a rigid cylinder, surrounded by the fluid, can be taken into account by introducing an additional ass to the syste. This ass results fro soe of the fluids particles being peranently displaced by the intruding body can be quantified following Wilson (198): a d = C Aρ Fπ (7) a denotes the added ass per unit length, C A is a nondiensional added ass coefficient, ρ F is the fluid density d the diaeter of the cylinder. It was observed that, as the cylinder length becoes uch larger than its diaeter, the value of C A approaches a theoretical liit of unity. For shorter cylinders different authors give coefficients depending on the length to diaeter ratio l/d. For instance Hafner (1977): 1 C A = (8) 1+ ( d ) l Equation 8 is given for the case of a oving cylinder with two free ends. Even though the considered syste of saple top cap is fixed at one side it can be assued that this equation gives a reasonable approxiation. The length l is assued with the total length of saple top cap. So resonance frequency calculations, including the effects of the confining water, can be easily perfored using a virtual ass coposed of the actual body ass the above given added ass. VISCOUS MATERIA DAMPING The basis for the analysis of the frequency response of the soil saple is the identification of different odes of vibration at resonance. The daping ratio D is calculated at these points of the response spectru in the neighbourhood of a resonance peak..1 Aplitude of the response spectru The ost coon ethod of easuring daping uses the relative width of the response spectru. Us-

5 ing the quantities indicated on the curve in Figure 3, the logarithic decreent δ the daping ratio D, can be calculated fro (Richart et al. 197): πd π f f1 A 1 D δ = = (9) 1 D f A A 1 D When D is sall, the last ter (1-D ). can be taken as equal to 1.. Further siplification is possible, if A is chosen equal to A /. : f f D = (1) 1 f The application of latter expression is usually called the half-power ethod. Knowing this point akes it possible to deterine the necessary angles α. ω 1 I c α 1 α ω at Resonance ω Re Figure. Nyquist plot used in the circle-fit-ethod. 6 TESTING ON SIT SAMPES Figure 3. Resonant curve with variables for half-power ethod.. Circle-Fit ethod The circle-fit ethod, described in Ewins (1988) is able to calculate the daping ratio with very few points around the resonance peak the aplitude of the peak has only little influence on the result. This is an advantage in cases were different odes have frequencies close to each other. The Nyquist plot of the response spectru of a single degree of freedo syste leads to a circle as shown in Figure. Even though the saple is not such a syste it behaves for selected frequency sections in the sae way. The aterial daping can be calculated fro points close to the axiu aplitude using the following expression: ω ω1 D = (11) α α1 ω ω tan + ω1 tan with: ω, angular frequency corresponding to the axiu sweep angular velocity; ω 1, ω, angular frequencies; α 1, α, angles at both sides of ω. A circle is fitted to the points of the response curve close to the resonant frequency to find the center. The bender eleent test was perfored on two silt saples fro the sae site. One saple was reconstituted (S1) the other undisturbed (S). Both were subjected to different isotropic stress levels by an external water pressure. The backpressure inside the saple was set to 1 kpa. The external isotropic pressure was therefore chosen always 1 kpa above the value of the target effective stress. The absorption of water by the saple during consolidation was easured taken into account for the calculation of the saple ass. The shear wave velocity was deterined by the analysis of the wave arrival at each stress level. The paraeters of the saple before the installation in the cell are given in Table. The properties of the top cap are repeated in Table 3. Table. Paraeters of the tested silt saples. Saple S1 S reconstituted undisturbed ength, [] Diaeter, d [] Distance bender eleents, tip to tip []. Mass, T [g] Unit ass, ρ [kg/³] Table 3. Properties of the top cap. Height [] 19 Diaeter, d [] Mass, [g].7 Figures 6 present the easured response spectra, Tables -7 Figure 7 the predicted easured natural frequencies.

6 Table. Saple S1 (reconstituted), easured predicted natural frequencies, easured daping ratios. Effective Stress, p eff [kpa] 1 3 Resonant frequencies: C A = Rayleigh Equation, f Ray. [Hz] Tioshenko Equation, f TBT [Hz] FEM, f FEM [Hz] Figure. Frequency response spectra for saple S1 (reconstituted). C A =.8 Rayleigh Equation, f Ray. [Hz] Tioshenko Equation, f TBT [Hz] FEM, f FEM [Hz] Experiental result, f eas. [Hz] Daping ratio: Half-power ethod, D HPM [%] Circle-fit ethod, D CFM [%] Table 6. Saple S (undisturbed), saple conditions. Effective Stress, p eff [kpa] 1 3 Water absorption [g] Effective saple ass, T,eff [g] Shear wave velocity, V S [/s] Figure 6. Frequency response spectra for saple S (undisturbed). Table 7. Saple S (undisturbed), easured predicted natural frequencies, easured daping ratios. Effective Stress, p eff [kpa] 1 3 Resonant frequencies: C A = Rayleigh Equation, f Ray. [Hz] Tioshenko Equation, f TBT [Hz] FEM, f FEM [Hz] C A =.8 Rayleigh Equation, f Ray. [Hz] Tioshenko Equation, f TBT [Hz] FEM, f FEM [Hz] Experiental result, f eas. [Hz] Daping ratio: Half-power ethod, D HPM [%] Circle-fit ethod, D CFM [%] Figure 7. Calculated easured resonant frequencies for the first flexural ode of saple S (undisturbed), added ass effects are included (C A =.8). Table. Saple S1 (reconstituted), saple conditions. Effective Stress, p eff [kpa] 1 3 Water absorption [g] Effective saple ass, T,eff [g] Shear wave velocity, V S [/s] The theoretical estiation of the natural frequencies of the first ode was done with an added ass coefficient of C A =.8, which corresponds to a length to diaeter ratio of.96, as well with no added ass (C A = ) to allow coparisons. As expected, the frequencies including the added ass coponent are lower than the values without the additional ass. For the sae C A, the results fro the Rayleigh expression show a good agreeent with the FEMvalues. The frequencies calculated by the Tioshenko equation are always higher. The test results show that the natural frequencies the shear wave velocities increase with an in-

7 creasing cell pressure. The aplitude ratios of the observed peaks decrease with increasing stress level. The predicted frequencies are in all cases larger than the easured. The closest agreeent to the experiental results is achieved at C A =.8 by the Rayleigh equation the FEM. The daping ratios are in the range of 7. to 9.9 % for S1 between % for S. The deviation between the values fro the half-power ethod the circle-fit ethod are in nearly all cases less than 1 %. A dependence of the daping ratio on the confining stress could be not noticed. The daping values are high in coparison with results of field laboratory tests at low shear strains presented in literature. Therefore a resonant colun test is perfored on an undisturbed silt saple fro the sae boring as saple S. A shear strain of about 1-3 % was chosen easureents were taken after isotropic consolidation at, 1, kpa. This test gave for all cases a viscous daping ratio in the range of 1. to 1. %. This eans the gap between the daping values out of both ethods is quite significant still under investigation. 7 CONCUSIONS AND REMARKS The application to easure the daping ratio in the frequency doain by eans of bender eleents installed in the triaxial cell is discussed in this paper. Two basic probles of the test results should be noticed here. The first is the deviation of the calculated the easured resonant frequencies. The assuption of a bea with top-ass at its one end, even if shear effects are taken into account, ight be a relatively crude way to describe the very short, about 3 c long, saple the top cap. The finite eleent odel can partly overcoe this geoetrical proble, but also in this case, influences of the rubber ebrane the not fully rigid ounting of the botto cap are not taken into account. Furtherore the consideration of the water in the cell by an additional ass ight not be sufficiently correct. If the viscosity of the water is taken into account, a frictional (viscous) drag force, which is proportional to the velocity of the oving cylinder, has to be introduced (Wilson 198). The used coefficient C A for the added ass depends, besides the cylinder geoetry, also on the Reynolds nuber the cylinder roughness. A ore detailed investigation could help to find a ore exact theoretical solution. On the other h the prediction of the natural frequency is only used to find the right resonant peak aong several, so the calculations are already sufficient for a successful selection. Brocanelli & Rinaldi (1998) used air to apply the confining pressure to the saple, which has indeed no affect on the saple oveent. The second proble concerns the difference between the results of the resonant colun device the bender eleent test. Attention should be given also here to the influence of the confining water pressure on the resulting daping ratio, which is not taken into account yet. ACKNOWEDGEMENTS The results presented in this paper have been obtained within the frae of the STWW-project IWT1 'Traffic induced vibrations in buildings'. The financial support of the Ministry of the Fleish Counity is gratefully acknowledged. REFERENCES Brocanelli, D. & Rinaldi, V Measureent of low-strain aterial daping wave velocity with bender eleents in the frequency doain. Canadian Geotechnical Journal 3: Cascante, G., Santaarina, C. & Yassir, N Flexural excitation in a stard torsional-resonant colun device. Canadian Geotechnical Journal 3: Clough, R.W. & Penzien, J Dynaics of structures ( nd edition). New York: McGraw-Hill. Cowper, G.R The Shear Coefficient in Tioshenko's Bea Theory. Journal of Applied Mechanics June 1966: Dyvik, R. & Madshus, Ch ab Measureents of G ax using Bender Eleents. Advances in the Art of Testing Soils Under Cyclic Conditions; Proc. ASCE, Detroit, October 198: New York: ASCE. Ewins, D.J Modal Testing: Theory Practice. Taunton: Research Studies Press. Graff, K. F Wave Motion in elastic Solids. New York: Dover Publications. Hafner, E Untersuchung der hydrodynaischen Kräfte auf Baukörper i Tiefwasserbereich des Meeres. Stuttgart: Institut für Wasserbau der Universität Stuttgart. Richart, F.E., Hall, J.R. & Woods, R.D Vibration of soils foundations. Englewood Cliffs, N.J.: Prentice- Hall. Rossit, C.A. & aura, P.A.A. 1. Transverse, noral odes of vibration of a cantilever Tioshenko bea with a ass elastically ounted at the free end. Journal of the Acoustical Society of Aerica 11(6): Wilson, J.F Dynaics of offshore structures. New York: John Wiley & Sons. APPENDIX Resonant frequency: Rayleigh s ethod The ode shape for the horizontal bending displaceent y B of the specien at elevation x is assued to be a third-order polynoial: y B + x 3 = a + a1x + a x a3 (A1) At the lower plate (corresponding to x = ) the displaceent the tangent are y() = y'() =, respectively. Hence, a = a 1 =. Neglecting

8 the oent at the top end x =, EI y''() =. Then, Equation A1 becoes y B [ x] = α x 3 for x (A) where α is a constant given by α = a /3 = -a 3. The horizontal displaceent of a rigid ass placed above the specien is estiated fro the horizontal displaceent y() the tangent y'() at the top of the specien: y B [ + 3( x ) ] = α for x > (A3) The ode shape for the shearing displaceent y S is assued to be linear. This is described by: = βx for x (A) y S = β for x > (A) y S The superposition of bending shearing follows then with: [ x] βx y( x) = α x 3 + for x [ + 3( x ] β y( x) = α ) + for x > (A6) (A7) The axiu internal potential energy E pot is coputed by taking into consideration the internal energy of the bending shearing coponent of the defored specien: E pot = 1 1 EI ( y'') dx + GAk Introducing A6 in A8 gives: E pot 3 = 6EIα K 1 K+ GAk α ( y') dx α β + β (A8) (A9) The axiu kinetic energy E kin is coputed by presuing haronic oscillation with frequency ω. The coponent for the specien is E kin, 1 6 = ω (α + 1α h + αβ K 3 K+ 6αβ h + 9α h + β ) (A13) Finally, the circular resonant frequency for the flexural ode is coputed by equating the axiu internal potential energy E pot the axiu kinetic energy E kin, T + E kin, : 6 EI + GAkη ω = (A1) T γ + δ with γ = η + 7η δ = h + 8 ηk 3 K+ 16 hη h + 1η To the sae tie α β are replaced by η=β/α. Top Cap Saple E,G= T G= a) h T x E= E,G= Figure A1. Assued ode shapes a) bending b) shearing. b) E kin, T 1 = ρω A y dx (A1) or using A6: Ekin, = Tω α + αβ + β (A11) T 7 6 The kinetic energy for a concentrated ass added at a distance h above the specien can be evaluated using Equation A7 at x = h + : 1 = (A1) E kin, yh+ ω