NATURAL FREQUENCIES OF SOIL-FOUNDATION SYSTEMS AND SOIL LAYERS

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1 YU HI NGH KHOA HC VÀ CÔNG NGH LN TH HCMUT 6-8/0/0 NATURAL FREQUENCIES OF SOIL-FOUNDATION SYSTEMS AND SOIL LAYERS TRNG HOÀNG NH PHNG Research Engineer, Westminster, Caliornia USA INTRODUCTION Static and dynamic loads o structures are normally applied on top soil layers through shallow and deep oundations. Soil-structure interaction could cause resonant conditions and damages to structures i the relationship between the loads o structures and the soil properties has the orm o the dynamic spring constants [], or i the structure, soil and oundations have the same value o the natural requency. This paper presents expressions o the natural circular requencies o the oundation-soil system and soil or rock layers determined by the dynamic spring constants and dynamics soil masses [,, 3 and 4], respectively.. NATURAL CIRCULAR FREQUENCIES OF FOUNDATION-SOL/ROCK SYSTEM BY DYNAMIC SPRING CONSTANTS The damped and undamped natural circular requencies o the oundation-soil system have been ound based on the dynamic spring constants [] in the vertical direction, respectively, as ollows: d b () sm d () sm where is the natural circular requency o the soil or rock layer in the vertical direction, d ro G (3) r o = the radius o the circular ooting; = Mass Density o the medium (soil or rock); m = the mass o the ooting; b = material damping constant [5]; G = Shear Modulus o the medium (soil or rock); and

2 s ( ) ( ) (4) where µ = Poisson ratio o soil or rock layer. The damped and undamped natural circular requencies or the circular ooting with rigid-base pressure distribution in the vertical direction [,, 3, 4 and 6] can be expressed as: d sm 3 3s 0. b 5 (5) d sm 3 3s (6) The damped and undamped natural circular requency o the oundation-soil system in the horizontal direction can be expressed based on Eq. () as ollows: d b nx (7) m nx d (8) m Note that the damped natural requency (Eqs., 5 and 7) only includes the material damping. In general, oundations on a layer over bedrock are normally dierent than oundations on a homogeneous hal-space in 3 ways: i. static stiness increases ii. the dynamic stiness coeicient decreases near the natural requency o the soil layer iii. Radiation damping is reduced to zero at requencies less than the natural requency o the soil layer. 3. NATURAL CIRCULAR FREQUENCIES OF SOIL OR ROCK LAYERS BY DYNAMIC SOIL MASSES The natural circular requencies o the soil or rock layers in the vertical and horizontal directions could be expressed based on the vertical and horizontal dynamic soil masses [,, 3 and 4], respectively as ollows: V s (9) sh

3 V s nx (0) h where Vs is the shear velocity; h is the thickness o the soil or rock layer. The ratio o the vertical circular requency and the horizontal circular requency is ( ) R zx () s where nx is the natural circular requency o the soil or rock layer in the horizontal direction. The natural requencies o the soil or rock layer in the vertical and horizontal directions are Vs G () sh sh nx Vs (3) h The ratio R o the natural requency Eq. () and that o [7] (or [8] or vertical oscillation), in the vertical direction is R a (n ) (4) where n is the mode o requency, i n=, Eq.(4) becomes R a 0.64 (5) The ratio R b o the natural requency Eq. () and that o [9] (or [8] or torsional oscillation), or [0], or [], in the vertical direction is R b (n ) (6) s where n is the mode o requency, i n=, Eq.(6) becomes R b s (7) The ratio R b increases with the increase in Poisson s ratio, because the value o s decreases with the increase in Poisson s ratio (Fig.). The ratio R b o the natural requency Eq. () and that o [] in the vertical direction is 3

4 R b A (8) s where A is the ampliication actor []. I the ampliication actor A is equal to.0, the Poisson s ratio is 0.7 or cases o rock medium, then the values o Eqs. (), (6) and (8) are also equal to (Fig.). The ampliication actors [3] due to the lithology unit or hard rock, sot rock, hard soil, medium soil, sot soil, and very sot soil are,.,.7,,., and.8, respectively Ratio Ra and Rb Poisson's Ratio s Ra Rb Figure. Variation o Ra, Rb and s with Poisson s ratio. The relationships between thickness o the soil or rock layer and the natural requency have been investigated by [4] and [5]. r b h (9) a where a = 57.67, and b = [4]; and a=58.9 and b=-0.95 [5]. Variation o natural requencies with soil thickness with two dierent shear wave velocities (80 m/s and 360 m/s) and Poisson s Ratios (0.3 and 0.4) by various authors [4, 0, Truong, 0 (Eq.()); and 5] are shown in Fig.. Note that: (i) the results o using Eq.() with shear modulus o 80m/s and Poisson s Ratio o 0.3 (Truong-) are very close with those by [5] as shown in Fig., and (ii) In Fig., Truong-, Truong-3 and Truong-4 are or cases using Eq. () with shear wave velocity o 80m/s and 4

5 Poisson s ratio o 0.4, shear wave velocity o 360 m/s with Poisson s ratios o 0.3 and 0.4, respectively Natural Frequency, Soil Thickness, h Dinesh Dobry- Dobry- Truong- Truong- Truong-3 Truong-4 Iba-Von Figure. Variation o Frequency, with Soil Thickness with Dierent Soil Shear Velocities and Poisson s Ratios by Various Authors. For a slope, the natural circular requency o the soil or rock layer becomes: V s 90 x s (0) sh 80 where x is the angle o the slope orming with the vertical line in degrees. The natural requency o the soil or rock layer is Vs 90 x s () sh 80 The ratio o the natural requency o Eq.() to that o [6] and [7] is R zs.6 90 x () s 80 The ratio Rzs increases with the increases in slope angles and Poisson s Ratio (Fig.3). 5

6 Ratio Rzs Angle (Deg.) Mu Figure 3. Variation o Rzs with Slope Angle and Poisson s ratio. 4. EFFECT OF TWO-LAYERED SYSTEM In general, or the two-layered system, the harder the second soil or rock layer, the higher the natural requency o the irst soil or rock layer is. The stiness o the irst soil or rock layer increases with the increase in ratio o the stiness o the second layer to that o the irst layer based on the work by [8]. Thereore, the requency o the irst soil or rock layer could be expressed as: Vs Vs Vs h G Vs Vs h sh 0V h h sh 0V h h s s (3) where subscripts and denote the properties o the irst and second layers, respectively, e.g. h and h are the thicknesses o the irst and second layers, respectively. 5. CONCLUSIONS 6

7 The natural circular requency based on the dynamic spring constant is inversely proportional to the total mass o the ooting, and not to the square root o the total mass o the ooting as in the current literature. The natural requency based on the vertical dynamic soil mass is greater about than that o the commonly used ormula o the natural requency and depends on the Poisson s ratio, shear velocity and thickness o the soil or rock layer. REFERENCES. Truong, H.V.P. - Dynamic Spring Constants and Eect o Frequency on Footing Vibration, Proc. 4 th Asian Regional Conerence on Soil Mechanics and Geotechnical Engineering, 3-7 May 0, Hong Kong, Paper No Truong, H.V.P. - Added Soil Mass and Ratio o Damping For Vertical and Horizontal Vibration, Proc. th Conerence on Science and Technology, University o Technology in Saigon, Vietnam National University, Vietnam, (009) Truong, H.V.P. - Eects o Damping and Dynamic Soil Mass on Footing Vibration, Proc. GeoShanghai International Conerence in Shanghai in June 00 (ASCE -GSPs), Truong, H.V.P. - Dynamic Soil Mass and Resonant Frequency o Pile Foundations, Proc. 4 th Asian Regional Conerence on Soil Mechanics and Geotechnical Engineering, 3-7 May 0, Hong Kong, Paper No Dobry, R. and Gazetas, G.. - Dynamics Stiness and Damping o Foundations by Simple Methods - Vibration Problems in Geotechnical Engineering o Symposium Sponsored by the Geotechnical Engineering Division, in Conjunction with the ASCE Convention, Nagendra, M.V. and Sridaran, A. - Response o Circular Footings to Vertical Vibrations - Journal o Geotechnical Engineering Division, ASCE, Vol. 07, 98, pp Richart, F.E.; Woods,R.D., and Hall, J.R. - Vibrations o Soils and Foundations - Prentice- Hall, 970, 44p. 8. Bycrot, G.N. - Forced Vibrations o a Rigid Circular Plate on a Semi-Ininite Elastic Space on an Elastic Stratum - Philosophical Transaction, Royal London, Series A, Vol.48, 956, pp Madabhushi, S.P.G. - Frequency o a Horizontal Soil Layer, Part II: Saturated Sand Bed - CUED/IbSOILS/TR73, 994, 88 p. 0. Dobry, R., Oweis, I., and Urzura, A. - Simpliied Procedures or Estimating the Fundamental Period o a Soil Proile. Bulletin o the Seismological Society o America, Vol. 66, No.4, Aug. 976, pp Kramer, S.L. - Geotechnical Earthquake Engineering - Prentice Hall, Saita, J.; Bautista, M.L.P. and Nakamura, Y. - On Relationship between the Estimated Strong Motion Characteristics o Surace Layer and the Earthquake Damage. Case Study at Intramuros, Metro Manila - Proceedings o 3 th World Conerence on Earthquake Engineering, Vancouver, B.C. Canada, 004, pp Garcia-Rodriguez, M.J.; Havenith, H.B. and Benito, B.- Evaluation o Earthquake- Triggered Landslides in El Salvador Using A GIS-Based Newmark Model - Proc. 4 th World Conerence on Earthquake Engineering, October, -7, 008, Beijing, China, 8p. 7

8 4. Dinesh, B.V., Nair, G.I, Prasad, A.G.V., Nakkeran, P.V. and Radhakrishna, M.C. - Relation between Sedimentary Layer Thickness and Fundamental Frequency o the H/V Spectra or Bangalore City. - Current Science, Vol. 97, No.7, 0 Oct. 009: Iba-Von Seht, M. and Wohlenberg, J. - Microtremor Measurements Used to Map Thickness o Sot Sediments - Bulletin o the Seismology Society o America, 999, pp Makdisi, F.I. and Seed, H.B. - Simpliied Procedure or Evaluating Embankment Response - Journal o the Geotechnical Engineering Division, ASCE, Vol. 05, No.GT, 979, pp Okamoto, S. - Introduction to Earthquake Engineering.- University o Tokyo Press, 984, pp Kagawa, T., Krat, L.M. - Machine Foundations on Layered Soil Deposits - Proceeding o 0 th International Conerence on Soil Mechanics and Foundation Engineering, Stockholm, 98, pp SUMMARY NATURAL FREQUENCIES OF SOIL-FOUNDATION SYSTEMS AND SOIL LAYERS Natural requency o the soil-oundation system, simply and directly based on the dynamic spring constant, increases with the increase in area o the ooting; shear modulus, soil density and Poisson s ratio, and decreases with the increase in the applied load. Resonant condition is taking place when the structure, the soil-oundation system and the soil layer all have the same natural requency. The ratio between the natural requencies o the soil layers in the vertical and horizontal directions based on the dynamic soil mass increases with the increase in Poisson s ratio. Comparison between the new expressions o the natural requencies o the soil layers based on the dynamic soil masses and the current expressions o the natural requencies in the vertical and horizontal directions are also carried out. New expression o the vertical natural requency o the soil layer by including the eect o the two-layered system is also proposed. Keywords: Natural Frequency, spring constant, shear modulus, Poisson s ratio, soil dynamics. 8

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