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Dynamic Response of Embedded Foundations in Layered Halfspace: A Cone Model Approach

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Abstract

This paper presents the evaluation of the dynamic stiffness coefficient (the stiffness and damping coefficient) of a rigid and massless foundation embedded in a layered elastic half-space. Linear hysteric material damping is introduced in the model using the correspondence principle. Based on the strength of the material approach with one-dimensional wave propagation in cones (cone model), horizontal and vertical dynamic stiffness coefficients were evaluated. To check the accuracy of the model, the dynamic stiffness coefficient of foundations resting on and/or embedded in layered half-space was evaluated using a cone model and validated with published results based on rigorous analysis. A parametric study is also carried out to investigate the influence of shear wave velocity, depth of embedment, and thickness of the top layer on the dynamic response of the foundation embedded in layered half-space. The results of the cone model analyses are presented in terms of stiffness coefficient K(ao) and damping coefficient C(ao) varying with dimensionless frequency (ao) for both horizontal and vertical modes of vibration. The results of the cone model provide physical insight with sufficient generality, making it convenient to use for various foundation vibration problems.

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All data, models, and code generated or used during the study appear in the submitted article.

References

  1. Hsieh TK (1962) Foundation vibrations. Proc Inst Civ Eng 22(2):211–226

    Google Scholar 

  2. Lysmer J (1965) Vertical motions of rigid footings. Department of Civil Engineering, University of Michigan Report to WES Contract Report No. 3–115 under Contract No. DA-22-079-eng-340

  3. Richart FE, Whitman RV (1967) Comparison of footing vibration tests with theory. J Soil Mech Found Div 93(SM6):143–168

    Article  Google Scholar 

  4. Chehab AG, El Naggar MH (2004) Response of block foundations to impact loads. J Sound Vib 276(1–2):293–310. https://doi.org/10.1016/j.jsv.2003.07.028

    Article  Google Scholar 

  5. Chen SS, Shi JY (2006) Simplified model for vertical vibrations of surface foundations. J Geotech Geoenv Eng 132(5):651–655. https://doi.org/10.1061/(ASCE)1090-0241(2006)132:5(651)

    Article  Google Scholar 

  6. Chen SS, Shi JY (2007) Simplified model for torsional foundation vibrations. Soil Dyn Earthq Eng 27(3):250–258. https://doi.org/10.1016/j.soildyn.2006.06.007

    Article  MathSciNet  Google Scholar 

  7. Quinlan P (1954) The elastic theory of soil dynamics. In: Symposium on dynamic testing of soils. ASTM International, pp 3–34

  8. Thomson WT, Kobori T (1963) Dynamical compliance of rectangular foundations on an elastic half-space. J Appl Mech 30(4):579–584. https://doi.org/10.1115/1.3636622

    Article  Google Scholar 

  9. Karasudhi P, Keer LM, Lee SL (1968) Vibratory motion of a body on an elastic half plane. Appl Mech 35(4):697–705. https://doi.org/10.1115/1.3601294

    Article  Google Scholar 

  10. Veletosos A, Nair VV (1974) Response of torsionally excited foundations. J Geotech Eng Div 100(4):476–482

    Article  Google Scholar 

  11. Luco JE, Westmann RA (1971) Dynamic response of circular footings. J Eng Mech 97:1381–1395

    Google Scholar 

  12. Awojobi AO, Tabiowo PH (1976) Vertical vibration of rigid bodies with rectangular bases on elastic media. Earthq Eng Struct Dyn 4(5):439–454. https://doi.org/10.1002/eqe.4290040503

    Article  Google Scholar 

  13. Ang AH, Harper GN (1964) Analysis of contained plastic flow in plane solids. J Eng Mech 90(5):397–420

    Google Scholar 

  14. Agabein ME, Parmelee RA, Lee SL (1968) A model for the study of soil-structure interaction. In: Proceedings of 8th conference international association bridge & structure engineering, New York

  15. Krizek RJ, Gupta DC, Parmelee RA (1972) Coupled sliding and rocking of embedded foundations. ASCE J Soil Mech Found Div 98(SM12):1347–1135

    Article  Google Scholar 

  16. Lysmer J, Kuhlemeyer RL (1969) Finite dynamic model for infinite media. J Eng Mech 95(4):859–878

    Google Scholar 

  17. Das R, Manna B, Banerjee A (2023) Spectral element formulation for rock-socketed mono-pile under horizontal dynamic loads. Soil Dyn Earthq Eng 169:107863. https://doi.org/10.1016/j.soildyn.2023.107863

    Article  Google Scholar 

  18. Hadjian AH, Luco JE, Tsai NC (1974) Soil-structure interaction: continuum or finite element? Nucl Eng Design 31(2):151–167. https://doi.org/10.1016/0029-5493(75)90138-7

    Article  Google Scholar 

  19. Jakub M (1977) Dynamic stiffness of foundations: 2-Dyi. 3D Solutions

    Google Scholar 

  20. Barkan DD (1965) Basic problems of the dynamics of bases and foundations (review of the conference on the dynamics of bases and foundations). Soil Mech Found Eng 2(6):376–378. https://doi.org/10.1007/BF01704871

    Article  Google Scholar 

  21. Crouse CB, Hushmand B, Luco JE, Wong HL (1990) Foundation dynamic stiffness coefficient: theory versus experiment. J Geotech Eng 116(3):432–449. https://doi.org/10.1061/(ASCE)0733-9410(1990)116:3(432)

    Article  Google Scholar 

  22. Dobry R, Gazetas G, Stokoe KH (1986) Dynamic response of arbitrarily shaped foundations: experimental verification. J Geotech Eng 112(2):136–154. https://doi.org/10.1061/(ASCE)0733-9410(1986)112:2(136)

    Article  Google Scholar 

  23. Gazetas G, Stokoe KH (1991) Free vibration of embedded foundations: theory versus experiment. J Geotech Eng 117(9):1382–1401. https://doi.org/10.1061/(ASCE)0733-9410(1991)117:9(1382)

    Article  Google Scholar 

  24. Nii Y (1987) Experimental half-space dynamic stiffness. J Geotech Eng 113(11):1359–1373. https://doi.org/10.1061/(ASCE)0733-9410(1987)113:11(1359)

    Article  Google Scholar 

  25. Baidya DK, Muralikrishna G (2000) Dynamic response of foundation on finite stratum—an experimental investigation. Indian Geotech J 30(4):327–350

    Google Scholar 

  26. Baidya DK, Krishna GM (2001) Investigation of resonant frequency and amplitude of vibrating footing resting on a layered soil system. Geotech Test J 24(4):409–417. https://doi.org/10.1520/GTJ11138J

    Article  Google Scholar 

  27. Baidya DK, Sridharan A (1994) Stiffnesses of the foundations embedded into elastic stratum. Indian Geotech J 24(4):353–367

    Google Scholar 

  28. Sridharan A, Nagendra MV, Chinnaswamy C (1981) Embedded foundations under vertical vibration. J. Geotech. Geoenv. Eng. 107:1429–34

    Google Scholar 

  29. Halabian AM, Ghasemi S, Mohasseb S (2022) Rocking response of shallow foundations in time domain using cone model theory. J Earthq Eng 26(6):2822–2845

    Article  Google Scholar 

  30. Ehlers G (1942) The effect of soil flexibility on vibrating systems. Beton und Eisen 41(21/22):197–203

    Google Scholar 

  31. Meek J, Veletsos AS (1974) Simple models for foundations in lateral and rocking motion. In: Proceedings of the 5th World Conference on Earthquake Engineering, pp 2610–2631

  32. Veletsos AS, Verbič B (1973) Vibration of viscoelastic foundations. Earthq Eng Struct Dyn 2(1):87–102. https://doi.org/10.1002/eqe.4290020108

    Article  Google Scholar 

  33. Fry ZB (1963) Development and evaluation of soil bearing capacity. Found. of Struct. WES

    Google Scholar 

  34. Veletsos AS, Wei YT (1971) Lateral and rocking vibration of footings. J Soil Mech Found Div 97(SM9):1227–1248

    Article  Google Scholar 

  35. Wolf JP, Somaini DR (1986) Approximate dynamic model of embedded foundation in time domain. Earthq Eng Struc Dyn 14(5):683–703. https://doi.org/10.1002/eqe.4290140502

    Article  Google Scholar 

  36. Meek JW, Wolf JP (1992) Cone models for soil layer on rigid rock. II. J Geotech Eng 118(5):686–703. https://doi.org/10.1061/(ASCE)0733-9410(1992)118:5(686)

    Article  Google Scholar 

  37. Meek JW, Wolf JP (1993) Cone models for nearly incompressible soil. Earthq Eng Struct Dyn 22(8):649–663. https://doi.org/10.1002/eqe.4290220802

    Article  Google Scholar 

  38. Meek JW, Wolf JP (1994) Cone models for embedded foundation. J Geotech Eng 120(1):60–80. https://doi.org/10.1061/(ASCE)0733-9410(1994)120:1(60)

    Article  Google Scholar 

  39. Wolf JP, Meek JW (1994) Dynamic stiffness of foundation on layered soil halfspace using cone frustum. Earthq Eng Struct Dyn 23:1079–1095. https://doi.org/10.1002/eqe.4290231004

    Article  Google Scholar 

  40. Jaya KP, Prasad AM (2002) Embedded foundation in layered soil under dynamic excitations. Soil Dyn Earthq Eng 22(6):485–498. https://doi.org/10.1016/S0267-7261(02)00032-5

    Article  Google Scholar 

  41. Wolf JP (1994) Foundation vibration analysis using simple physical models. Pearson Education

    Google Scholar 

  42. Pradhan PK, Baidya DK, Ghosh DP (2003) Dynamic stiffness coefficient of circular foundation resting on layered soil using cone model. Electron J Geotech Eng 8:1

    Google Scholar 

  43. Pradhan PK, Mandal A, Baidya DK, Ghosh DP (2008) Dynamic response of machine foundation on layered soil: cone model versus experiments. Geotech Geol Eng 26:453–468. https://doi.org/10.1007/s10706-008-9181-8

    Article  Google Scholar 

  44. Wolf JP, Deeks AJ (2004) Foundation vibration analysis: a strength of materials approach. Elsevier

    Google Scholar 

  45. Pal AS, Baidya DK (2019) Effect of soil layering on coupled dynamic response and active length of piles embedded in layered soil using cone model. Indian Geotech J 49(1):50–57. https://doi.org/10.1007/s40098-017-0292-0

    Article  Google Scholar 

  46. Khakpour M, Hajialilue Bonab M (2020) Soil-structure-interaction using cone model in the time domain for horizontal and vertical motions in layered half-space. J Earthq Eng 24(4):529–554. https://doi.org/10.1080/13632469.2018.1441766

    Article  Google Scholar 

  47. Hajari F, Rahnema H (2023) Prediction of horizontal and torsional vibrations of foundations resting on saturated porous media using cone model. Indian Geotech J 53(1):154–169. https://doi.org/10.1007/s40098-022-00665-5

    Article  Google Scholar 

  48. Ahmad S, Bharadwaj A (1991) Horizontal impedance of embedded strip foundations in layered soil. J Geotech Eng 117(7):1021–1041. https://doi.org/10.1061/(ASCE)0733-9410(1991)117:7(1021)

    Article  Google Scholar 

  49. Apsel RJ, Luco JE (1987) Dynamic stiffness coefficient for foundations embedded in a layered medium: an integral equation approach. Earthq Eng Struct Dyn 15(2):213–231. https://doi.org/10.1002/eqe.4290150205

    Article  Google Scholar 

  50. Ahmad S, Rupani AK (1999) Horizontal impedance of square foundation in layered soil. Soil Dyn Earthq Eng 18(1):59–69. https://doi.org/10.1016/S0267-7261(98)00028-1

    Article  Google Scholar 

  51. Beredugo YO, Novak M (1972) Coupled horizontal and rocking vibration of embedded footings. Can Geotech J 9(4):477–497. https://doi.org/10.1139/t72-046

    Article  Google Scholar 

  52. Dominguez J, Roesset JM (1978) Dynamic stiffness of rectangular foundations. Report R78–20. Department of Civil Engineering

  53. Han Z, Lin G, Li J (2015) Dynamic stiffness coefficient for arbitrary-shaped rigid foundation embedded in anisotropic multilayered soil. J Eng Mech 141(11):04015045. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000915

    Article  Google Scholar 

  54. Novák M, Beredugo YO (1972) Vertical vibration of embedded footings. J Soil Mech Found Div 98:1291–1331

    Article  Google Scholar 

  55. Sasmal SK, Pradhan PK (2021) A critical review of the cone model for analysis of machine foundations under translational and rotational motion. Aust J Multi-Discipl Eng 17(1):97–106. https://doi.org/10.1080/14488388.2021.1923431

    Article  Google Scholar 

  56. Kausel E, Roesset JM (1975) Dynamic stiffness of circular foundations. J Eng Mech 101(6):771–785

    Google Scholar 

  57. Lin G, Han Z, Li J (2013) An efficient approach for the dynamic impedance of surface footing on layered half-space. Soil Dyn Earthq Eng 49:39–51. https://doi.org/10.1016/j.soildyn.2013.01.008

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    Susmita Panda & Bappaditya Manna

  2. Department of Civil Engineering, Veer Surendra Sai University of Technology, Burla, Odisha, 768018, India

    Pradip Kumar Pradhan

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  1. Susmita Panda
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Panda, S., Pradhan, P.K. & Manna, B. Dynamic Response of Embedded Foundations in Layered Halfspace: A Cone Model Approach. Indian Geotech J (2023). https://doi.org/10.1007/s40098-023-00825-1

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