Abstract. 1 Introduction

Transcription

1 Preliminary investigation of using transformed damping coefficients in time domain analysis D.-W. Chang Department of Civil Engineering, Tamkang University, Tamsui, Taiwan 251, Republic of China Abstract In this paper, applications of the frequency dependent damping to time domain analysis are presented with the use of Fourier transform. The proposed transformed damping models are compared with conventional models simulating vertical excitations of a SDOF lump-mass system. The result implies that the transformed models is similar to the conventional models under steady-state loads. Modeling of the damping ratio is very important for the model simulating structural response under an impact. Pile response due to axial loads can be qualitatively achieved by solving the wave equation with these models and the finite difference method. 1 Introduction Damping mechanism which dissipates the energy as the velocity of motion and strain varied is important to dynamic analyses in solid mechanics. Two types of damping (viscous and hysterestic) are commonly considered according to the dependence of strain-rate or frequency. It is generally believed that the internal dissipation of energy due to particle friction in soils is of a hysteretic nature, but the loss of energy due to propagation of waves away from the region of interest, known as radiation damping, depends on the frequency of waves. Discussions of the soil damping can be found in Christian et d.,* and Schmidt.^ Presuming that strains are small, linear viscosity and linear hysteresis are denoted to describe the soil behavior. These models is easily converted using the modal

2 624 Soil Dynamics and Earthquake Engineering analysis with critical damping ratios (%) at various modes. With numerical discrete models on hand, the viscous damping in dynamic equilibrium equations can be evaluated using the model of Rayleigh damping^ and its alternatives. These models evolved from modal analysis compute the viscosity using the orthogonality of modal shapes with modal damping ratios. By assuming a hysteretic type of material damping ratio for the frequencies of interest, the viscous terms are simply evaluated with the proportionalities of mass and stiffness at system resonance. To reproduce radiation damping of the system, a number of damping models simulating the boundary conditions have been suggested. Discussion of these models can be found in Roesset.4 If a simplified lump mass system with sets of spring and dashpot is used to reproduce the vibrations of foundation under harmonic loads, the amount of radiation damping can be represented by an equivalent damping ratio at each load. A state-of-art summary of the regarding research are recently provided by Gazetas.^ 2 Conventional Damping Models As mentioned previously, Rayleigh damping model is often used in modeling the dynamic structural response. Mathematic expression of this model is C = am+p# (1) where C, M and K respectively represents viscous, mass and stiffness term in the dynamic equilibrium equation; a and p are the damping parameters which depend on the circular frequency (co) of the harmonic load and the critical damping ratio ( ), where a = co and p = /co. By equating the motion of a linear viscous system to that of a linear hysteretic sytsem, the critical damping ratio ( ) can be related to hysteretic damping ratio (D) such that = DCOQ/CO, where CDQ is fundamental natural circular frequency of system. The associated equivalent viscous coefficient ceq is obtained as 2DA7o). Neglecting differences between D and, and considering resonance of the system. Parameters a and p can be rewritten as follows, a = Dcoo ; p-d/oo (2) Equation 2 can be achieved by obtaining a significant frequency from response spectra of the system at a constant material damping. Alternatives of the Rayleigh damping are the stiffness-proportional and the mass-proportional damping which simulates the viscosity using independent terms. Simplification of the damping variations and predetermination of the modal vibrations are disadvantages for these models. Other available damping models are suggested by Roesset et al. ^ as the weighted modal damping and complex damping. It is necessary to point out that most of the dynamic soil testing in lab calculate the material damping ratio by two methods; 1. Compute the logarithmic decrement at

3 Soil Dynamics and Earthquake Engineering 625 free vibration after the resonant excitation, 2. Compute the energy ratio from cyclic stress-strain hysteresis loop. The possible variation of these material damping with strain rate or loading frequency must be taken into accounted for these damping models. 3 Transformed damping model The dynamic deformational characteristics of soils are well known to be affected by many factors such as the state of stress, loading history, number of loading cycles, the aging and the degree of saturation etc.,&c. Although the strain rate was reported to be relatively unimportant for damping ratio of sands (Hardin and Drnevich^), research work conducted by Czajkowski and Vinson;^ Isenhower and Stokoe;^ and Sousa and Monismith^ showed that the strain-rate dependence is indeed apparent for certain soils in a range of frequency. By modeling damping ratio with frequency, D(co); a transformed damping function C(t) representing the equivalent viscous coefficient at time domain is herein proposed. To obtain C(t), mathematic functions of damping ratio are suggested as follows, D((o) = 0-*co ; D(co) = a + Wco ; D(w) = a /of (3) Parameters a and b in above equations can be determined from the experimental data with curve fitting procedudure. Disscussions of damping variation with frequency for various viscous damper can be found in Makris and Constantinou.i * Substituting D(co) to equation c = 2DA7co where an equivalent viscous coefficient of the hysterestic system is represented; the frequency related viscous coefficient c(co) is established. Applying c(co) with the Fourier transform and the Residue theorem, the correspondent transformed damping coefficients C(t) are obtained as, C(t) = i af ; C(t) = -6ft + i af ; C(t) = - i aff / 2 (4) For simplification, above equations are respectively denoted as transformed model J%, # and C in following study. To use the complex equations in timedomain analysis, magnitudes of transformed damping coefficients ought to be calculated. Details of the derivation and the discussion of these equatins can be found in Chang and Lu. ^ 4 Response of SDOF lump mass system Comparison study is conducted simulating the vertical response of a SDOF lump mass system, in which dimensionless mass and spring constant are kept as 2 and 10 under arbitrary loading for both types of models. Implicit formulations are used to integrate the dynamic equilibrium equation. Table 1 summarizes the damping characteristics selected in the study. 5% damping ratio is assigned to

4 626 Soil Dynamics and Earthquake Engineering conventioanl models for reasonable comparison. In each vibration mode, both transient and steady-state response are studied varying the duration (Td) or the period (T) of loads with respect to fundamental natural period (Tn) of the system. Fig. 1 depicts the dimensionless displacement history at an impulse with Ray lei gh damping. Effects of the load duration are revealed in each figure. The response curves under one- and ten-cycle periodic loads are shown in Figs. 2 and 3 for Rayleigh damping. Fig. 4 to Fig. 6 exert the displacements at an impulse with transformed models #, # and C. Studies are also conducted for transformed models and for mass- and stiffness-proportional dampings under various loading. When load duration of the impulsive load is equal to Tn, the displacement decaying after load duration is found particularly small for transformed models ^t and #. Excessive amount of damping in models # and C will yield small steady-state vibrations under steady-state loading. Transient response under impulsive load and one-cycle sinusoidal load are observed for all models in the cases where Td > Tn. Constant amplitudes are found for conventional models and transformed model & in the cases of ten-cycle loading where T > Tn. When Td (or T) < Tn, these models produce similar response history. In such cases, very small vibrations are resulted by steady-state excitation. 5 Response of axially loaded pile The response of an axially loaded pile is investigated thereafter. Based on wave equations simulating the dynamic equilibrium of the pile segments, the displacements along the pile can be calculated using finite difference approximation with a set of soil springs and dashpots along the pile. Table 2 lists characters of the pile and the surrounding soils. For simplicity, previous damping parameters a and b are adopted. Nevertheless, modification of the damping parameters is necessary for pratical applications since the soil damper herein is correspondent to the radiation damping. Four types of loads are considered; l.monotonic load, 2. impulsive load, 3 one-cycle sinusoidal load, and 4. five-cycle sinusoidal load. The displacement histories at top of the pile under these loads with transformed models C are shown in Fig. 7. Oscillations along the curves are noticeable. Experimental and theoretic results obtained by Lee etal.^ had shown similar oscillations for monotonically loading condition. Force-displacement hysteresis at the pile top under harmonic load from these models are then plotted in Fig. 8. With constant load, a nonlinear hardening hysteretic load-displacement can be obtained from model C.

5 6 Conclusions Soil Dynamics and Earthquake Engineering 627 The transformed damping coefficient shall increase rapidly with time as the damping ratio decreases nonlinearly with frequency. For vertical vibrations of a SDOF lump mass system, the transformed model would allow rational solutions for transient and steady-state response under periodic loads. For impulsive loading conditions where the load duration equals to the fundamental natural period of the system, use of the transformed model with mathematic fuction to describe damping variation needs extreme precaution. In the simulation of axially loaded piling, qualitative results can be obtained from these models. The rapid increase of damping coefficient would produce a nonlinear hysteretic loaddisplacemnt relation. Applications of the model with versatile mathematic equations to interpretate different physics are suggested for further research. References 1. Christian, J.T., Roesset, J.M. and Desai, C.S., Two- and threedimensional dynamic analyses, Chapter 20, Numerical Methods in GeotechnicalEngineering, ed C.S. Desai & J.T. Christian, McGraw-Hill, Inc., pp , Schmidt, L.C., Vibration theory, Chapter 2, Analysis and Design of Foundations for Vibrations, ed P.J. Moore, pp , A.A.Balkema, Rotterdam, Netherlands, pp , Lord Rayleigh, Theory of Sound, Vol. 1, Dover Publications, New York, USA, Roesset J.M., A Rreview of Soil-Structure Interaction, Seismic Safety Margins Research Program Project III - Soil-Structure Interaction, Research Report No. UCRL-15262, Lawrence Livermore Laboratory, 125 pp, Gazetas, G., Foundation vibrations, Chapter 15, Foundation Engineer ing Handbook, ed Fang, H.Y., Van Nostrand Reihold Pub., Roesset, J.M., Whitman, R.V. and Dobry R., Modal analysis for structures with foundation interaction, J. Struct. Div., ASCE, Vol. 99, No. ST3, pp , Hardin, B.O. and Drnevich, V.P., Shear modulus and damping in soils: measurement and parameter effects, Journal of Soil Mechanics and Foundations Division, Vol. 98, No. SM6, pp , Czajkowski, R.L. and Vinson, T.S., Dynamic properties of frozen silt under cyclic loading, Journal of Geotechnical Engineering, ASCE, Vol. 106, No. GT9, pp , Isenhower, W.M. and Stokoe, K.H., Strain-rate dependent shear modulus of San Francisco Bay mud, Proc. Int. Conf. Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, Vol. 2, pp , 1981.

6 628 Soil Dynamics and Earthquake Engineering 10. Sousa J.B. and Monismith C.L., Dynamic response of paving materials, Transportation Research Record, Vol. 1136, pp , Makris, N. and Constantinou, M.C., Application of fractional calculus in modeling viscous damper, Earthquake, Blast and Impact - Measurement and Effects of Vibration, ed SECED, Elsevier Applied Science, pp , Chang, D.W. and Lu, Y.L., Application of Transformed Damping in Modeling Axially Loaded Pile, Research Report, Dept. of Civil Eng., Tamkang University, Taiwan, ROC, 154 pp., (in Chinese) 13. Lee, S.L., Chow, Y.K., Karunaratne, G.P. and Wong, K.Y., Rational wave equation model for pile-driving analysis, Journal of Geotechnical Engineering, ASCE, Vol. 114, No.3, pp , Table 1 Parameters used in transformed damping model Model \ Parameter A # C a b N/A Table 2 Characters of pile and soil in piling study Character Value Character Value Pile Length, m 30.5 Cross Section Area of Pile, cm^ Pile Modulus, kg/cm^ 2.12* 10^ Soil Spring / ) Constant, Kg/cm" Pile Density, Mg/m^ 2.45 Damping Parameters* as suggested in Table 1 * Parameters used herein are not proposed to match any of the actual damping values.

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10 632 Soil Dynamics and Earthquake Engineering (a) Models Dynamic Loading Time (sec) 1.80 A (b) Impulsive I 000-L, A r _ n^» V ff Dynamic Loading Time (sec) of Pile (cm) (b) Model (B rt of Pile (cm) , (c) One cycle l\ IA, 0. )0 VpoJ Dynamic r Loading Time (cm) -1 8E+ 5 J of Pile (cm) (c) Model c (d) Five cycle Dynamic Loading Time (sec) of Pile (cm) Figure 7 Response fistory at pile top under various loads from transformed model c Figure 8 Cyclic force-displacement relationships at pile top