本研究提出互制反應模擬法，以土壤剪力波速為隨機參數，針對垂直簡諧外力作用下之彈性半空間土壤與剛性基礎互制系統，進行動態反應分析及可靠度評估。分析結果顯示，基礎位移累積機率密度函數與蠻力計算法比較，其誤差可控制於5%以下，但互制反應模擬法可大幅縮減運算時間。本研究亦發現當土壤具有較高剪力波速、剛性基礎具有較低質量及外力頻率較低時，土壤剪力波速變異性對基礎位移振幅有顯著影響。另外，若採用平均剪力波速於剛性圓盤基礎與均勻土壤之互制系統中，探討簡諧外力引致之基礎垂直位移，以此位移的三倍作為容許基礎位移振幅上限時，則剛性基礎受垂直簡諧外力之位移可靠度將趨近於100%。

This study proposes the Response Method to analyze the dynamic responses and reliability of the foundation displacements in the interaction system between an elastic half-space soil and a rigid foundation. The soil- foundation system is subjected to a vertical harmonic loading considering the uncertainty of shear-wave velocity of soil. It shows that there is 5% error of the cumulative probability distribution as compared to the Brute Force Method. However, the Response Method can reduce computational time substantially. This study also found that the variability of shear-wave velocity has significant influences on the foundation displacements for the system with higher shear-wave velocity of soil, smaller foundation mass, and lower loading frequencies. Moreover, this study conducts a reliability assessment based on an allowable displacement amplitude which equals three-times displacements that computed by considering an average of random shear-wave velocities of soil. In such cases, a level of reliability is found to approach 100% for restricting the vertical foundation displacements induced by a harmonic loading lower than the allowable amplitude.

[1] Cao, Z., Wang, Y., and Li, D., "Probabilistic Approaches for Geotechnical Site Characterization and Slope Stability Analysis," Zhejiang University, Zhejiang, pp. 1-51, 2017.
[2] 雒紅麗、韓曉雷，「岩土工程可靠性分析的特點及方法綜述」，西部探礦工程，第十八卷，第三期，第38-40頁，2009。
[3] Andrieu, C.N., Doucet, A., and Jordan, M.I., "An Introduction to MCMC for Machine Learning," Machine Learning, Vol. 50, No. 1, pp. 5-43, 2003.
[4] Richard, F.E., Woods, R.E. and Hall, J.R., Jr, "Vibration of soils and foundations," Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1970.
[5] Richart, F.E., and Whitman, R.V., "Comparison of footing vibration tests with theory," J. Soil Mech. and Found. Div., ASCE, Vol. 93, No. SM 6, pp.143-168, 1967.
[6] Reissner, E., "Stationäre, axialsymmetrische, durch eine schüttelnde Masse erregte Schwingungen eines homogenen elastischen Halbraumes," Ingenieur-Archiv, Vol. 7, No. 6, pp. 381-396, 1936.
[7] Hsieh, T.K., "Foundation vibrations," Proc. Institution of Civil Engineers, Vol. 22, No. 2, pp. 211-226, 1962.
[8] Lysmer, J., "Vertical motion of rigid footings," Dept. of Civil Eng., University of Michigan, Ph.D. Dissertation, 1965.

[9] Luco, J.E., and Westmann, R.A., "Dynamic response of circular footings," J. of Eng. Mech., ASCE, Vol. 97, No. EM 5, pp. 1381-1395, 1971.
[10] Veletsos, A.S., and Wei, T.Y., "Lateral and rocking vibration of footings," Earthquake Eng. and Structural Dyn., Vol.97, No. SM 9, pp.1227-1248, 1971.
[11] Novák, M., and Beredugo, Y.O., "Vertical vibration of embedded footings," J. Soil Mech. and Found. Div., Vol. 98, No. SM 12, 1972.
[12] Lysmer, J., Tabatabaie-Raissi, M., Tajirian, F., Vahdani, S., and Ostadan, F., "SASSI: A system for analysis of soil-structure interaction," University of California, Berkeley, 1981.
[13] Lysmer, J., "Analytical procedures in soil dynamics," Geotechnical. Eng. Div. Specialty Conf. on Earthquake Eng. and Soil Dyn., ASCE, Pasadena, California, Vol. 80, pp. 12243, 1978.
[14] Phoon, K.K., "Towards reliability-based design for geotechnical engineering," Special Lecture for Korean Geotechnical Society, Seoul, Vol. 9, pp. 1-23, 2004.
[15] Lutes, L.D., Sarkani, S., and Jin, S., "Response variability of an SSI system with uncertain structural and soil properties," Eng. Structures, Vol. 22, No. 6, pp. 605-620, 2000.
[16] Wolf, J.P., "Spring‐dashpot‐mass models for foundation vibrations," Earthquake Eng. and Structural Dyn., Vol. 26, No. 9, pp. 931-949, 1997.

[17] Chaudhuri, S.R., and Gupta, V.K., "Variability in seismic response of secondary systems due to uncertain soil properties," Eng. Structures, Vol. 24, No. 12, pp. 1601-1613, 2002.
[18] Moghaddasi, M., Cubrinovski, M., Chase, J.G., Pampanin, S., and Carr, A., "Probabilistic evaluation of soil–foundation–structure interaction effects on seismic structural response," Earthquake Eng. and Structural Dyn., Vol. 40, No. 2, pp. 135-154, 2011.
[19] Chen, S.S., and Shi, J.Y., "Simplified model for vertical vibrations of surface foundations," J. of geotechnical and geoenvironment Eng., Vol. 132, No. 5, pp. 651-655, 2006.
[20] Ang A.H.S., and Tang, W.H., "Probability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering," Wiley, 2007.
[21] Building Seismic Safety Council (BSSC), "The 2003 NEHRP recommended provisions for new buildings and other structures," Part I: (Provisions) and Part II (Commentary), FEMA 450, 2003.
[22] 陳煌銘，振動基礎分析與設計，科技圖書股份有限公司，文笙書局，台北，第169-186頁，1997。