formulate the problem of a vibratory mass on an elastic half-space. It was .... The vertical vibration of an actual foundation embedded in a soil described.
Downloaded from ascelibrary.org by King Abdullah University of Science and Technology Library on 11/10/16. Copyright ASCE. For personal use only; all rights reserved.
VERTICAL VIBRATION OF M A C H I N E
FOUNDATIONS
By Ali H. Nayfeh1 and Samir J. Serhan 2 ABSTRACT: A perturbation technique, the method of multiple scales, is used to obtain an approximate analytical solution of the nonlinear dynamic response of foundations on soils. The analysis takes into account the nonlinearity of the soil material, radiation, hysteretic and viscous damping, and the effect of embedment. Closed-form expressions are derived for the response, which show clearly the effects of the various parameters involved in soil dynamics. A closed-form expression is presented for the constant settlement of the nonlinear soil structure caused by the vibration. In addition to large responses accompanying primary or main resonances (the excitation frequency Cl is near the natural frequency of the foundation 0 and sgn (u) = — 1 when u < 0 (2) Here, u is the vertical displacement from the original position (not the static equilibrium one), T is the time, the overdot indicates the derivative with Q(T>
FIG. 3. Actual Foundation of Mass m Embedded in Soil Classified by Its Parameters G, p, and v 58
J. Geotech. Engrg., 1989, 115(1): 56-74
Downloaded from ascelibrary.org by King Abdullah University of Science and Technology Library on 11/10/16. Copyright ASCE. For personal use only; all rights reserved.
Q(T)
F/2
F/2
•^7™777
*
•
yy?
w*
FIG. 4. Equivalent Foundation System respect to T, and m is the mass of the foundation, the machinery, and the soil vibrating in phase with the foundation (effective mass of the soil). The problem of whether to include the effective mass of the soil or not received considerable attention. Hsieh (1962) developed values for the effective mass of the soil for different modes of vibration. Richart and Whitman (1967) emphasized the unimportance of the effective mass of the soil, because soil particles under vibration are moving in different directions with different accelerations. However, for high Poisson's ratios, the effective mass of the soil may become important. For a detailed discussion of the magnitude of the friction force due to embedment, the reader is referred to Chae (1970), Den Hartog (1931), Richart and Whitman (1967), Sridharan et al. (1981), and Stokoe (1972). Testing of soil samples in triaxial machines shows that the variation of the deviatoric stress o^ — a3 with the principal strain e, is not linear. For numerical calculations, one can either smooth the experimental data and use an interpolation scheme or use a theoretical model such as the hyperbolic model (3)
CTi — 0 (superharmonic resonance of order two) or Cl = 2o>0 (subharmonic resonance of order one-half). This means that by arbitrarily designing the foundation system to have a natural frequency larger or smaller than the excitation frequency, we are not on the safe side. 2. A closed-form expression is presented for the constant settlement of the nonlinear soil structure caused by the vibration. The results show that the frequency and amplitude are not independent, a characteristic of nonlinear behavior, as observed at the Waterways Experiment Station. 3. The results of the perturbation solution are in good agreement with the results of numerical simulation. 4. The embedment decreases the amplitude of vertical vibration of foundations in agreement with the results of Chae (1970). The present study needs to be extended to more than one degree of freedom. Coupling sliding and rocking vibrations of foundations is important in the design of nuclear power plants and machine foundations. Coupled modes of vibration are needed to handle the case of nonalignment of machinery on foundations. The present analysis represents the vibratory motion of soils by a finite lumped model characterized by a mass, a linear damper, and a spring. A better solution could be obtained by extending the analysis to the case of propagation of nonlinear waves in an elastic half-space. APPENDIX I.
REFERENCES
Anandakrishnan, M., and Krishnaswamy, N. R. (1973). "Response of embedded footings to vertical vibrations." J. Soil Mech. Found., ASCE, 99(SM10), 863881. Barkan, D. D. (1962). Dynamics of bases and foundations. McGraw-Hill Book Co., Inc., New York, N.Y. Bycroft, G. N. (1956). "Forced vibrations of a rigid circular plate on a semi-infinite elastic space or an elastic stratum." Trans. Royal Soc. London, Series A, 248, 327-368. Chae, Y. S. (1970). "Dynamic behaviour of embedded foundation-soil systems." Presented at 49th Ann. Mtg. of U.S. Highway Res. Board. Chandra, S. (1979). "Analysis of beams and plates on nonlinear subgrades." thesis presented to I. I. T. Kanpur, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Chandra, S., Madhav, M. R., and Iyengar, N. G. R. (1984). "Trapezoidal footings on nonlinear subgrades." Int. J. Num. Anal. Meth. in Geomech., 8, 519-529. Das, B. M. (1983). Fundamentals of soil dynamics. Elsevier Science Publishing Co., New York, N.Y. Den Hartog, J. P. (1931). "Forced vibrations with combined coulomb and viscous friction." Trans., ASME, APM-53-9, 107-115. Dobry, R., and Gazetas, G. (1986). "Dynamic response of arbitrarily shaped foundations." J. Geotech. Engrg., ASCE, 112(2), 109-135. 71
J. Geotech. Engrg., 1989, 115(1): 56-74
Downloaded from ascelibrary.org by King Abdullah University of Science and Technology Library on 11/10/16. Copyright ASCE. For personal use only; all rights reserved.
Dominguez, J., and Roesset, J. M. (1978). "Dynamic stiffness of rectangular foundations." Res. Rept. R78-20, Dept. of Civ. Engrg., Massachusetts Inst. Tech., Cambridge, Mass. Eastwood, W. (1953). "Vibration in foundations." Struct. Engrg., 31, 82-98. Funston, N. E., and Hall, W. J. (1967). "Footing vibration with nonlinear subgrade support." J. Soil Mech. Found., ASCE, (SM5), 191-211. Gazetas, G., and Roesset, J. M. (1976). "Forced vibrations of strip footings on layered soils." Proc. ASCE Specialty Conf. Methods of Structural Analysis, Madison, Wis., 115-131. Gazetas, G. (1983). "Analysis of machine foundation vibrations: State of the art." J. Soil Dyn. Earthquake Engrg., 2(1), 2-42. Gazetas, G., Dobry, R., and Tassoulas, J. L. (1985). "Vertical response of arbitrarily shaped embedded foundations." J. Geotech. Engrg., ASCE, 111(6), 750-771. Hardin, B. O. (1965). "The nature of damping in sands." J. Soil Mech. F.ound., ASCE, 91(SM1), 63-88. Hsieh, T. K. (1962). "Foundation vibrations." Proc. Inst. Civ. Engr. 22, 211-226. Jennings, P. C. (1974). "Periodic response of a general yielding structure." J. Engrg. Mech., ASCE, 90(EM2), 131-166. Karasudhi, P., Keer, L. M., and Lee, S. L. (1968). "Vibratory motion of a body on an elastic half plane." J. Appl. Mech., ASME, 35E, 697-705. Kausel, E., and Roesset, J. M. (1975). "Dynamic stiffness of circular foundations." J. Engrg. Mech., ASCE, 101(EM6), 771-785. Luco, J. E., and Westmann, R. A. (1972). "Dynamic response of a rigid footing bonded to an elastic half space." J. Appl. Mech., ASME, 527-534. Lutes, L. D. (1970). "Approximate technique for testing random vibration of hysteretic systems." J. Acoust. Soc. Am., 48(1), Part 2, 299-306. Lysmer, J. (1980). "Foundation vibrations with soil damping." Proc. 2nd ASCE Conf. Civ. Engrg. and Nuclear Power, Knoxville, Tenn., Vol. II, 1—18. Lysmer, J., and Kuhlemeyer, L. (1969). "Finite dynamic model for infinite media." J. Engrg. Mech., ASCE, 95(EM4), 859-877. Massalas, C. (1977). "Fundamental frequency of vibration of a beam on nonlinear elastic foundation." J. Sound Vibration, 54, 613-615. Nayfeh, A. H. (1973). Perturbation methods. John Wiley and Sons, Inc., New York, N.Y. Nayfeh, A. H. (1981). Introduction to perturbation techniques. John Wiley and Sons, Inc., New York, N.Y. Nayfeh, A. H. (1983). "The response of single degree of freedom systems with quadratic and cubic non-linearities to a subharmonic excitation." J. Sound Vibration 89(4), 457-470. Nayfeh, A. H. (1984). "Combination tones in the response of single-degree-of-freedom syterns with quadratic and cubic nonlinearities." J. Sound Vibration 92(3), 379-386. Nayfeh, A. H., and Mook, D. T. (1979). Nonlinear oscillations. John Wiley and Sons, Inc., New York, N.Y. Novak, M., and Beredugo, Y. O. (1972). "Vertical vibration of embedded footings." J. Soil Mech. Found., ASCE, 98(SM12), 1291-1310. Novak, M., and Sachs, K. (1973). "Torsional and coupled vibrations of embedded footings." Int. J. Earthquake Struct. Dyn., 2(1), 11-33. Ramberg, W., and Osgood, W. T. (1943). "Description of stress-strain curves by three parameters." Tech. Note 902, Nat. Advisory Comm. Aeronaut. Reissner, E. (1963). "Stationare axialsymmetrische durch eine Schuttelnde Masseeregte Schwingungen eines homogenen elastischen Halbraumes." Ingenieu-Archiv, 7, Part 6, 381-396. Richart, F. E. (1962). "Foundation vibrations." Trans. ASCE, 111, 863-898. Richart, F. E., and Whitman, R. V. (1967). "Comparison of footing vibration tests with theory." J. Soil Mech. Foundat., ASCE, 93(SM6), paper 5568. Richart, F. E., Woods, R. D., and Hall, J. R. (1970). Vibrations of soils and foundations. Prentice-Hall, Inc., Englewood Cliffs, N.J. 72
J. Geotech. Engrg., 1989, 115(1): 56-74
Downloaded from ascelibrary.org by King Abdullah University of Science and Technology Library on 11/10/16. Copyright ASCE. For personal use only; all rights reserved.
Riicker, W. (1982). "Dynamic behavior of rigid foundations of arbitrary shape on a half space." Earthquake Engrg. Struct. Dyn., 10, 675-690. Sridharan, A., Nagendra, M. V., and Chinnaswamy, C. (1981). "Embedded foundations under vertical vibration." J. Geotech. Engrg., ASCE, 107(GT10), 14291434. Stokoe, K. H. (1972). "Dynamic response of embedded foundations," thesis presented to the University of Michigan, Ann Arbor, Mich., in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Sung, T. Y. (1953). "Vibrations in semi-infinite solids due to periodic surface loading." Symp. Dynamic Testing of Soils, Special Tech. Publications No. 1561, ASTM, 35. Veletsos, A. J., and Verbic, B. (1974). "Basic response functions for elastic foundations." J. Engrg. Mech... ASCE, 100(EM2), 189-201. Wen, Y.-K. (1975). "Approximate method for nonlinear random vibration." J. Engrg. Mech., ASCE, 101(EM4), 389-401. Wen, Y.-K. (1976). "Method for random vibration of hysteretic systems." J. Engrg. Mech., ASCE, 102(EM2), 249-263. Whitman, R. V., and Richart, F. E., Jr. (1967). "Design procedures for dynamically loaded footings." J. Soil Mech. Found., ASCE, 93(SM6), 169-193. Wong, H. L., and Luco, J. E. (1976). "Dynamic response of rigid foundations of arbitrary shape." Earthquake Engrg. Struct. Dyn., 4, 579-587. Woods, R. D., Barnett, N. E., and Sagesser, R. (1974). "Holography—A new tool for soil dynamics." J. Geotech. Engrg., ASCE, 100(GT11), 1231-1247. Wu, T. H. (1971). Soil dynamics. Allyn and Bacon, Boston, Mass. APPENDIX II.
NOTATION
The following a c e F F / F0 G g k m m0 Q q t u uc us u v wQ x
= = = = = = = = = = = = = = = = = = = = = =
x
=
symbols are used in this paper:
amplitude of response; damping coefficient; eccentricity; Coulomb damping force; nondimensional amplitude of excitation; scaled, nondimensional amplitude of excitation; amplitude of excitation; shear modulus; gravitational acceleration; linear spring constant; mass of foundation system; eccentric mass; forcing function; nonlinear restoring force; nondimensional time; vertical displacement; characteristic displacement; static settlement; velocity; displacement from static equilibrium position; natural frequency; scaled, nondimensional vertical displacement from static equilibrium position; nondimensional vertical displacement from static equilibrium position; 73
J. Geotech. Engrg., 1989, 115(1): 56-74
Downloaded from ascelibrary.org by King Abdullah University of Science and Technology Library on 11/10/16. Copyright ASCE. For personal use only; all rights reserved.
z
= =
