INFLUENCE OF GROUND INHOMOGENEITY ON WIND INDUCED GROUND VIBRATIONS. Abstract

Transcription

1 INFLUENCE OF GROUND INHOMOGENEITY ON WIND INDUCED GROUND VIBRATIONS Mohammad Mohammadi, National Cente fo Physical Acoustics, Univesity of Mississippi, MS Caig J. Hicey, National Cente fo Physical Acoustics, Univesity of Mississippi, MS Richad Raspet, National Cente fo Physical Acoustics, Univesity of Mississippi, MS Vahid Nadeyan, National Cente fo Physical Acoustics, Univesity of Mississippi, MS Abstact The mechanical behavio of the nea suface of the gound (vadose one, citical one) is equied fo many geotechnical applications such as oadway constuction, tafficability, foundation design, and agicultual poduction. The mechanical behavio may also be indiectly elated to tanspot acoss this citical inteface such as; wate infiltation and gas flux which ae impotant to hydological and climate studies. Mechanical souces fo investigating nea suface of the gound included high fequency electomechanical souces and loudspeaes. We attempt to exploit the pessue fluctuations poduced by wind as a souce fo investigating nea suface soil popeties. In a pevious study, the gound was modeled as an elastic homogenous half space and a pediction model was developed fo the wind-induced gound motion. Howeve, gound is a layeed o inhomogeneous medium and will have a diffeent esponse due to the suface pessue. In this pesentation, we discuss diffeent inds of theoetical inhomogeneous gounds and thei available analytical esponses due to a suface load. Futhemoe, we use Comsol-Multiphysics, a commecial finite element pacage, to simulate a moe ealistic gound consisting of a laye ove a half-space. A new esponse function is deived by intepolating the Comsol output to substitute in the wind-gound coupling equations. A compaison between the homogeneous and non-homogeneous gound pedictions ae used to evaluate the impotance of layeing on wind-induced gound vibations. Intoduction Nea-suface geophysics focuses on geophysical methods to study popeties and layes of the nea-suface egion of the eath. Seismic field measuement is one of the most common suface-based methods which equie a souce to geneate seismic vibations. In this pape, we conside wind as a souce and investigate gound motion to study nea suface gound popeties. Nadeyan et al. (2015) utilied the developed theoy by Yu et al. (2011) to pedict wind pessue at the gound suface based on the wind velocity and then the fequency-dependent coelation of wind noise founded by Shields (2005) to estimate the distibution of wind pessue on the suface. The theoy of elasticity fo a homogeneous elastic half-space unde a quasi-static point load was applied to the pedicted wind noise distibution to develop a quasi-static model fo wind-induced gound displacements. A vast amount of liteatue exists on the calculation of displacements and stesses in a half-space with diffeent types of theoetical inhomogeneity. In this pape, two types of idealiations ae consideed fo inhomogeneity in the gound and the pedictions fo the suface displacements ae used in the wind-

2 gound model. Kassi (1970), Booe et al. (1985) and One (1990) consideed an idealied elastic halfspace in which the shea modulus o Young s modulus incease with a powe of depth and Poisson s atio emains constant. The analytic defomation esponse of this model due to a vetical suface load is used in the oiginal wind-gound equation and the esults ae compaed with the pevious study. Yue et al. (1988), Enian (1989) and Pan et al. (2007) idealied the gound as a layeed half-space made up of elastic layes laying ove an elastic isotopic half-space. In ode to simplify the modeling pocess, we conside a single elastic laye ove an elastic homogeneous half-space as ou layeed gound. Since thee is no closed fom analytic solution fo the defomations of ou model, a simila finite element model is built in Comsol-Multiphysics. We apply an intepolation function of the computed data points by Comsol to the wind-gound model. Finally, we compae new and pevious esults and discuss the capability of the windgound measuements in studying nea-suface gound. In this study, we eplace the homogeneous half space model with theoetical non-homogeneous models in ode to dive the pedictions close to eality. Compaison between the homogeneous and nonhomogeneous pedictions povides insight into the use of wind induced seismic to study nea-suface gound popeties. Wind-Gound Coupling Theoy Nadeyan et al. developed a wind-gound coupling theoy by combining following theoies: the gound (half-space) defomations due to suface loads (Landau and Lifshit, 1986), the wind pessue spectum at the gound suface fom the measued wind velocity pofile (Kaichnan, 1956, Raspet et al., 2008, Yu et al., 2011), and the distibution of souces associated with wind tubulence ove the gound suface applied by the wave numbe-dependent coelation function of the wind noise in the downwind and cosswind diections (Shields, 2005). Combining these theoies leads to following equations fo the powe specta of the vetical and hoiontal components of the wind-induced gound suface displacements due to the vetical pessue (Nadeyan et al., 2015). U 2 u u 2 u u p 2 ( ) exp( x x )cos( x x )exp( y y ) dxdx dydy U p ( ) (1b) exp( x x )cos( x x )exp( y y ) dxdx dydy whee U and U ae hoiontal and vetical displacements at the obsevation point of the pessue, is wind wave numbe, and E ae Poisson s atio and Young s modulus, espectively, ( x, y) and ( x, y ) ae coodinates of two andom points on the gound suface with espect to the obsevation point of the pessue, and ae x y and x y espectively, p is the Powe Spectal Density (PSD) of wind pessue on the gound, and ae two constant numbes based on the wind velocity. Homogeneous Gound In the study by Nadeyan et al. (2015), the expessions fo the gound defomation, is given by Eq. 2. The gound is assumed as a homogeneous linealy elastic medium bounded by an infinite plane on one (1a)

3 side (half-space). The vetical and adial suface defomations of the model due to a nomal suface load, as a function of adius fom a delta function foce souce ae (Landau and Lifshit, 1986): (1 )(1 ) 1 u ( ) F (2a) E (1 )(1 ) 1 u ( ) F (2b) E whee is the adial distance fom the load s cente, and E ae Poisson s atio and Young s modulus, espectively, and F is the vetical foce. Eq. 1 can be completed with these expessions and numeically calculated ove tuncated anges. Inhomogeneous Gound Continuously Inceasing Rigidity One (1990) consideed an idealied elastic half-space in which the shea modulus inceases with an abitay powe of depth and Poisson s atio is constant. The hoiontal suface displacement of this model due to a vetical point foce equals to eo because of eo shea modulus at the suface. The vetical displacement is as follow: n G G0 (3a) u F 4 G 1 0 ( n 1) ( n 1) whee G is shea modulus, is the depth with the positive diection towad the half-space, u is the vetical displacement, F is the vetical point foce, is the adial distance fom the load s cente, and the following estictions apply; 1/3 < n < 2/3 and n ( 1/ ) 2. Fig.1a shows compaison of gound suface 7 defomation pedicted using Eq. 3 and Eq. 2 with G 0 10 due to equal vetical loads ( F 100N ). The vetical suface displacements fo the homogeneous gound and thee diffeent inhomogeneous gounds in which shea modulus inceases at diffeent ates as a function of depth ae shown in Fig. 1a. a) b) (3b) Figue 1: (a) Vetical suface displacements and (b) thei powe spectal density fo homogeneous and inhomogeneous half-space models.

4 In the neafield, inhomogeneous models displacements ae geate than homogeneous model. It is consistent with the eo value of shea modulus at the suface used in the inhomogeneous model. The suface displacement deceases faste with inceasing distance fom the souce as the powe of shea modulus function inceases. The defomation of the gound suface ae equal fo the homogenaous and inhomogeneous models at a distance of 1m coesponding to the depth at which the shea modulus is equal fo all of them. The pedicted wind diven suface defomation fo the homogeneous and inhomogeneous gounds ae shown in Fig. 1b. The slope of the lines deceases as the igidity inceases faste with depth. It shows that the influence of gadually changing igidity with depth is manifested in the slope of the PSD wind coupling esults. Layeed Gound To ou nowledge, thee is no closed fom solution fo the defomation esponse of a layeed halfspace subjected to a nomal static suface load. Theefoe, a model simulating an elastic laye ove a homogeneous half-space subjected to a vetical static suface load is calculated in Comsol. The soil and souce popeties ae listed in Table 1. Fig. 2a and 3a compae the suface defomations fo diffeent top laye thicness. Two homogeneous half-space models ae consideed; the Soft-Homo model having the lowe Young s modulus and Had-Homo model having the highe Young s modulus. In Eq. 2, both vetical and hoiontal displacements of the homogeneous half spaces (had and soft) ae popotional to 1 /. Hence, they ae paallel with fixed diffeence (blue and puple lines in Fig. 2a and 3a). Fo the layeed models, the displacements coespond to those of Soft-Homo in the neafield and to the displacements of Had-Homo in the fa field. The tansition occus at distances close to the souce as the laye thicness deceases. Table 1: Simulation Paametes Vetical Foce Young s Modulus (Pa) Load Poisson s Density (N ) Bottom 3 Radius (m) Top Laye Ratio ( g m ) Half-space a) b) Figue 2: (a) Hoiontal suface displacements and (b) thei powe spectal density fo homogeneous and layeed half-space models. The wind-diven defomation of the gound ae shown in Fig. 2b and 3b. An intepolation function based on finite numbe of points of the lines in Fig. 2a and Fig. 3a ae used in ode to geneate a numeical esponse function of adial distance fom the souce. The esponse functions ae substituted in the windgound equations (Eq.1) and numeically calculated. At low fequencies, the 2-layeed models behavio

5 is close to a homogeneous half space with popeties of the bottom laye. As the fequency inceases, the layeed model become close to a homogeneous half space with popeties of the top laye. It is losing sensitivity to the hade bottom half space. This tansition taes place at lowe fequencies fo the thice top layes. This obsevation agees with the geneal behavio of Rayleigh waves whee, the longe wavelengths penetate deepe and ae sensitive to deepe popeties of the gound wheeas, shote wavelengths epesent moe shallow suface popeties. a) b) Figue 3: (a) Vetical suface displacements and (b) thei powe spectal density fo homogeneous and layeed half-space models. Conclusion The wind-gound theoy popose a theoetical model in ode to tansfe the diving pessue petubations on the gound suface to the gound vibations. In the pevious study, the gound was modeled as an elastic homogeneous half-space. This pape investigate the same theoy fo inhomogeneous gounds modeled by two common types of idealiation fo the gound inhomogeneity. Fist, we conside a gound with gadually inceasing shea modulus. A theoetical pediction fo this type of inhomogeneity by One is used instead of the elastic esponse of the homogeneous half-space in oiginal wind gound equations. The influence of this change can be seen on the ate of change of PSDs with fequency. The PSD of the displacements deceases slowe with fequency as the igidity inceases faste with depth. Next type of theoetical inhomogeneity is modeled in Comsol by a soft elastic laye ove a hade half space in tems of Young s modulus. The displacements of the simulation due to a vetical suface load at a seies of points on a adial line ae used to find an appopiate esponse function because a closed fom function is needed in the wind-gound equations. The simulation esults show a tansient behavio fom the esponse of the half space with top laye popeties in the neafield to the esponse of the half space with bottom laye popeties in the fa field. This tansition occus at close distance to the souce fo thinne top layes. The wind-induced layeed gound motion pedictions behave simila to a half space built of bottom laye mateial at low fequencies and convege to the behavio of a half space built of top laye mateial at high fequencies. The ate of this tansition o the slope of the lines elative to fequency can be used as an implication of the top laye thicness. Refeences Booe, J.R., Balaam, N.P. and Davis, E.H., The Behaviou of an Elastic Non homogeneous Halfspace. Pat I line and point loads. Intenational Jounal fo Numeical and Analytical Methods in Geomechanics, 9(4), pp

6 Enian, P., Static Response of a Tansvesely Isotopic and Layeed Half-space to Geneal Suface Loads. Physics of the Eath and Planetay Inteios, 54(3-4), pp Kassi, M.K., The Reissne-Sagoci Poblem fo a Non-homogeneous Solid. Intenational Jounal of Engineeing Science, 8(10), pp Kaichnan, R. H. (1956), Pessue Fluctuations in Tubulent Flow ove a Flat Plate, J. Acoust. Soc. Am., 28(3), Landau, L. D., and E. M. Lifshit (1986), Theoy of Elasticity, Couse of Theo. Phys., vol. 7, 3 ed., Buttewoth-Heinemann, Oxfod. Nadeyan, V., Hicey, C.J. and Raspet, R., Wind induced Gound Motion. Jounal of Geophysical Reseach: Solid Eath, 121(2), pp One, M., Vetical and Hoiontal Defomation of an Inhomogeneous Elastic Half space. Intenational Jounal fo Numeical and Analytical Methods in Geomechanics, 14(9), pp Pan, E., Bevis, M., Han, F., Zhou, H. and Zhu, R., Suface Defomation due to Loading of a Layeed Elastic Half-space: A Rapid Numeical Kenel Based on a Cicula Loading Element. Geophysical Jounal Intenational, 171(1), pp Raspet, R., J. Yu, and J. Webste (2008), Low Fequency Wind Noise Contibutions in Measuement Micophones, J. Acoust. Soc. Am., 123(3), , doi: / Shields, F. D. (2005), Low-fequency Wind Noise Coelation in Micophone Aays, J. Acoust. Soc. Am., 117(6), , doi: / Sung, T., 1954, Januay. Vibations in Semi-infinite Solids due to Peiodic Suface Loading. In Symposium on dynamic testing of soils. ASTM Intenational. Yu, J., R. Raspet, J. Webste, and J. Abbott (2011), Wind Noise Measued at the Gound Suface, J. Acoust. Soc. Am., 129(2), , doi: / Yue, Z.Q. and Wang, R., Static Solution fo Tansvesely Isotopic Elastic N-layeed Systems. Acta Scientiaum Natualium, Univesitatis Peinensis, 24, pp Acnowledgment This wo was suppoted by the U.S. Depatment of Agicultue unde Non-Assistance Coopeative Ageement