Simple Models of Foundation-Soil Interactions

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1 IACSIT Internatinal Jurnal f Engineering and Technlgy, Vl. 5, N. 5, Octber 013 Simple Mdels f Fundatin-Sil Interactins Shi-Shuenn Chen and Jun-Yang Shi Abstract This study aims t develp a series f simplified mass-spring-dashpt mdel t simulate unbunded sil fr a fundatin-sil system subjected t vertical, hrizntal, rcking and trsinal mtins. A grup f equivalent mdels are established by using three equivalent criteria and the cupling f hrizntal and rcking mtins is als cnsidered. An ptimal equivalent mdel is then determined t represent the best simplified mdel. The dynamic respnses f the fundatin-sil system using the ptimal equivalent mdel are cmpared with thse btained by the half-space thery and by the lumped-parameter mdels. Since the cupling f hrizntal and rcking mtins is adequately cnsidered, the ptimal equivalent mdel is fund t have mre accurate results than mst existing mdels. Mrever, the prpsed mdel is als applied t a time-histry analysis fr a building-sil interactin system subjected t hrizntal excitatins. This prpsed methd may be effectively applicable t practical sil-structure interactin prblems. Index Terms Fundatin vibratin, lumped-parameter mdel, sil-structure interactin, seismic analysis I. INTRODUCTION The mst cmmnly-used methds fr dynamic sil-structure interactin (SSI analysis are the cmplete methd and the substructure methd. The cmplete methd features in analyzing the respnse f sil and structures simultaneusly, which usually requires special bundaries t simulate the unbunded sil. In cntrast, the substructure methd features in analyzing sil and structures separately, which generally uses dynamic impedance functins t represent the sil behavir. The dynamic impedance functin is the frce-displacement relatinship fr interactin pints between fundatin and sil. Hwever, since the dynamic impedances are frequency-dependent, they can nt be used directly in time-dmain structural analysis. Hence, simplified mdels with frequency-independent parameters are extensively develped fr effectively simulating the unbunded sil in time dmain. One type f the simplified mdels is a lumped-parameter mdel, which cmprises masses, springs, and dashpts. Thrugh a curve-fitting technique r an ptimizatin prcess, the mdeling parameters f the lumped-parameter mdel are btained by minimizing the discrepancy between the impedance functins fr the mdel and that btained frm the rigrus thery. Then a lumped-parameter mdel is develped fr a fundatin-sil system while the fundatin Manuscript received April 10, 013; revised May 9, 013. This wrk was supprted in part by the Natinal Science Cuncil f the Republic f China under Grant N. NSC E The authrs are with the Department f Cnstructin Engineering, Natinal Taiwan University f Science and Technlgy, Taiwan ( sschen@mail.ntust.edu.tw, D @mail.ntust.edu.tw. is massless. Several studies [1] [11] fllwed the cnventinal methd addressed abve t develp the simplified mdels. The cnventinal methd used an ptimizatin prcess t determine the unknwn parameters f the lumped-parameter mdels s that the accuracies f thse mdels were affected by bjective functins used in ptimizatin. Since the dynamic impedances in lw-t-medium frequencies are mre imprtant n the system respnse, Wlf [7] suggested that the bjective functins need t be adapted t make the weight fr the lw frequencies higher than that fr the high frequencies. Hwever, the relatin between the frequency and the weight was nt clearly clarified. Barrs and Luc [] suggested that an iteratin methd be adpted t determine an apprpriate weight in the bjective functin until the impedance functin f the simplified mdel appraches theretical slutins well. This apprach is relatively time-cnsuming. This paper aims t presents a new simplified mdel t simulate unbunded sil fr rigid embedded fundatins underging vertical, hrizntal, rcking and trsinal mtins. Parameters used in the new mdel are mre efficiently determined withut lengthy ptimizatin analysis. Mrever, an ptimal simplified mdel is selected thrugh minimizing an errr functin which is clearly addressed and easily evaluated mre than that used in mst f cnventinal methds. The dynamic respnses f the fundatin-sil system using the new simplified mdel are cmpared with thse btained by the half-space thery and by the existing lumped-parameter mdels t examine the accuracy f the new mdel. Mrever, a time-dmain validatin fr the prpsed mdel applied t a building-sil interactin system subjected t vertical and hrizntal excitatins is als prvided. II. THEORY OF EQUIVALENT MODELS This paper mainly investigates the dynamic respnse f a rigid embedded fundatin underging vertical, hrizntal, rcking and trsinal mtins. A series f equivalent mdels shwn in Fig. 1 is used t simulate the unbunded sil. Fr vertical and hrizntal mtins, a single degree-f-freedm (SDOF mdel is used. Fr rcking and trsinal mtins, a -DOF mdel is cnsidered. The arrangement f the equivalent mdel includes a lumped mass (r inertia, a spring, and a dashpt in each directins; and tw vertical eccentricities t cnsider cupling effects. Fr a fundatin-sil system excited by harmnic frces with different frequencies, a grup f equivalent mdels are develped by cnsidering three criteria. The first criterin is equivalent static respnse, which establishes an equivalence f static displacement between the equivalent mdel and the fundatin-sil system subject t a static frce. The secnd DOI: /IJET.013.V

IACSIT Internatinal Jurnal f Engineering and Technlgy, Vl. 5, N. 5, Octber 013 criterin is equivalent magnificatin factr, which establishes an equivalence f dynamic magnificatin factr (i.e., rati f displacement amplitude t static respnse between the equivalent mdel and the fundatin-sil system subject t a harmnic frce.

2 IACSIT Internatinal Jurnal f Engineering and Technlgy, Vl. 5, N. 5, Octber 013 criterin is equivalent magnificatin factr, which establishes an equivalence f dynamic magnificatin factr (i.e., rati f displacement amplitude t static respnse between the equivalent mdel and the fundatin-sil system subject t a harmnic frce. The last ne is equivalent dissipated energy, which establishes an equivalence f dynamic dissipated energy between the equivalent mdel and the fundatin-sil system subject t the harmnic frce. Fllwing the thery f equivalent mdels, the mdeling parameters can be derived fr each directin f mtins. These parameters are summarized in the fllwing. K C M K sγ [ 0 γ + (1 γ ] 4 ρv sr ac 0 γ ke γ[ cγ + (1 kγ ] 5 ρ R (1 kγ k a c k where K sγ static fundatin stiffness f trsinal mtins; k K /(GR nrmalized static stiffness; k γ and c γ dimensinless stiffness and damping cefficients f trsinal impedances, respectively. Fr rcking mtins, the mdeling parameters with stiffness K, damping cefficient C and lumped inertia M can be similarly derived as fllws. (3 K C M K sβ k a c + (1 k 0 β β 4 ρv sr ac 0 β k a c + (1 k 0 β β 5 ρ R (1 kβ (4 Fig. 1. A series f equivalent mdels. Fr vertical mtins, the mdeling parameters with stiffness K ez, damping cefficient C ez and lumped mass M ez can be shwn as belw. kez (1 kz K K ; C k c ρv R ; M ρr (1 3 ez sz ez ez z s ez where K sz static fundatin stiffness f vertical mtins; V s shear velcity f sil; R characteristic length f the fundatin; ρ sil density; k ez K ez /(GR nrmalized static stiffness with a shear mdulus G f sil; k z and c z dimensinless stiffness and damping cefficients f vertical impedances, respectively; and a 0 ωr/v s dimensinless frequency with a frcing frequency ω. Fr hrizntal mtins, the mdeling parameters with stiffness K ex, damping cefficient C ex and lumped mass M ex can be similarly expressed as belw. kex (1 kx K K ; C k c ρv R ; M ρr ( 3 ex sx ex ex x s ex where K sx static fundatin stiffness f hrizntal mtins; k ex K ex /(GR nrmalized static stiffness; k x and c x dimensinless stiffness and damping cefficients f hrizntal impedances, respectively. Fr trsinal mtins, the mdeling parameters with stiffness K, damping cefficient C and lumped inertia M can be written as belw. where K sβ static fundatin stiffness f rcking mtins; k K /(GR nrmalized static stiffness; k β and c β dimensinless stiffness and damping cefficients f rcking impedances, respectively. Fr cupled hrizntal and rcking mtins, the mdeling parameters shwn in ( and (4 are t be further mdified by cnsidering the cupling impedances in the dynamic equatins f mtins. Mre details are illustrated in the previus wrk [1]. Nte that fr a fundatin-sil system excited by harmnic frces crrespnding t N discrete frequencies, using the prpsed methd will result in N equivalent mdels. An ptimal mdel is then determined frm the established equivalent mdels t btain the mst accurate frequency-magnificatin respnse fr the fundatin-sil system subjected t harmnic frces; i.e. the ptimal mdel has the minimum errr f dynamic magnificatin factrs. An errr functin used is defined as N M M p (5 j j 1 M j ε where M dynamic magnificatin factr f the fundatin-sil system; M dynamic magnificatin factr f the fundatin with an equivalent mdel; the weighting p j (M j crrespnding t the j-th frequency pint; N number f frequency pints and the subscript dentes the directin f mtin. This paper mainly investigates the dynamic respnse f a fundatin-sil system using the ptimal equivalent mdel t effectively simulate the unbunded sil. III. VALIDATIONS ON FOUNDATION-SOIL SYSTEMS Applicatins f the ptimal equivalent mdel are t be investigated n the prblem f fundatin vibratin in this 574

IACSIT Internatinal Jurnal f Engineering and Technlgy, Vl. 5, N. 5, Octber 013 sectin. Cnsider a rigid square fundatin embedded in a slightly damped half-space with damping ratis ζ p 005 and ζ s 01.

Suppse that the rigid fundatin is harmnically subjected t a vertical frce, hrizntal frce, rcking mment, and trsin, respectively, with varied frequencies.

3 IACSIT Internatinal Jurnal f Engineering and Technlgy, Vl. 5, N. 5, Octber 013 sectin. Cnsider a rigid square fundatin embedded in a slightly damped half-space with damping ratis ζ p 005 and ζ s 01. The embedment depth f fundatin is assumed t be times f characteristic length R (i.e., ne half-side length while the fundatin mass is 5ρR 3. Suppse that the rigid fundatin is harmnically subjected t a vertical frce, hrizntal frce, rcking mment, and trsin, respectively, with varied frequencies. A series f vertical, hrizntal, rcking, and trsinal impedance functin presented by Mita and Luc [13] is used in the prpsed methd t develp the ptimal equivalent mdel t simulate the damped half-space. The dynamic magnificatin factrs fr a fundatin-sil system using the ptimal equivalent mdel are t be verified with thse btained by the half-space thery. The dynamic magnificatin factr is defined as the rati f dynamic displacement amplitude t static respnse when the fundatin subjected t an external harmnic frce r mment. In additin, similar cmparisns are als made fr existing lumped parameter mdels. The analyzed results are displayed in Fig.. Fr vertical, hrizntal, rcking, and trsinal mtins, bserve in Fig. that the frequency-magnificatin respnses btained frm the ptimal equivalent mdel cnsist well with the half-space slutins even as the dimensinless frequency increases. Fr vertical mtins, the magnificatin factrs analyzed by the ptimal equivalent mdel cnsist very well with thse btained by Barrs and Luc s mdel ( DOFs and 5 parameters, especially fr the peak respnses. Fr trsinal mtins, the results analyzed by the Wu and Chen s mdel (3 DOFs and 9 parameters underestimate the trsinal magnificatin factrs in lw-frequency range but accurately estimate the high-frequency respnses. By cntrast, the trsinal respnses btained by the ptimal equivalent mdel cnsist well with the half-space slutins in all cnsidered frequencies. Fr hrizntal mtins, Wu and Chen s mdel (1 DOF and 3 parameters verestimates the hrizntal magnificatin factrs in cmparisn with the half-space slutins. Fr rcking mtins, Wu and Chen s mdel (3 DOFs and 9 parameters underestimates the rcking magnificatin factrs especially fr respnses arund the resnance frequency. This feature can be markedly bserved fr the rigid fundatin with a deep embedment rati r a large mass rati. By cntrast, gd agreements are bserved again between the results btained by the ptimal equivalent mdel and the half-space slutins. Hence, the ptimal equivalent mdel may be mre accurate than Wu and Chen s mdel due t the cupling effects n hrizntal and rcking mtins are well cnsidered in the prpsed methd. The presented results als reveal that the rtatinal respnses f embedded fundatins subjected t a harmnic mment may be significantly underestimated withut cnsidering the cupled hrizntal and rcking mtins. The ptimal equivalent mdel develped may als effectively simulate the unifrm half-space fr the rigid embedded square fundatin underging frced vibratins. Vertical magnificatin factr, M z Hrizntal magnificatin factr, M x Trsinal magnificatin factr, M γ Rcking magnificatin factr, M β Dimensinless frequency, a 0 (b hrizntal mtins.0 Wu and Chen's mdel [9] Dimensinless frequency, a 0 (c trsinal mtins.0 Wu and Chen's mdel [9] Barrs & Luc's mdel [] Dimensinless frequency, a 0 (a vertical mtins.5.0 Wu and Chen's mdel [9] Dimensinless frequency, a 0 (d rcking mtins Fig.. Dynamic respnse f fundatin-sil systems. 575

Cnsidered an idealized 3-stry building (shwn in Fig. 3 embedded in a damped half-space with unifrm stry stiffness and mass while the assciated system parameters are listed in Table I.

4 IACSIT Internatinal Jurnal f Engineering and Technlgy, Vl. 5, N. 5, Octber 013 IV. VALIDATIONS ON BUILDING-SOIL SYSTEMS Applicatins f the ptimal equivalent mdel are t be investigated n the prblem f seismic excitatin in this sectin. Cnsidered an idealized 3-stry building (shwn in Fig. 3 embedded in a damped half-space with unifrm stry stiffness and mass while the assciated system parameters are listed in Table I. In the analytical mdels, the hrizntal grund acceleratin histry recrded frm Statin TCU045 during 1999 Taiwan Chi-Chi earthquake is t be cnsidered. As shwn in Fig. 4, the respnses f a simplified building-sil system using the ptimal equivalent mdel are t be btained by a mdal analysis methd in the time dmain, which are accmplished by a cmmercial prgram, SAP000 [14]. Mrever, the respnses f the real building-sil system are further analyzed by using a numerical prgram, SASSI [15]. This prgram SASSI features in using a special transmitting bundary t simulate the sil layer and a flexible vlume methd t slve dynamic sil-structure interactin prblems. Fr the 3-strey structures, the hrizntal acceleratin histry at the tp flr is displayed in Fig. 5 while the assciated flr respnse spectrum is shwn in Fig. 6. Observe that the histry respnses f the simplified system mstly agree well with that f the real system. N marked differences n the acceleratins r phases can be fund. The spectral curves f the simplified system als cnsist well with that f the real system. In summary, fr the building structure underging hrizntal excitatins, the seismic respnse f the simplified system agrees well with that f the real system. TABLE I: SYSTEM PARAMETERS OF NUMERICAL EXAMPLES Zne Parameter Pissn s rati: Shear-wave velcity: 500 m/s Unifrm Material density: Mg/m3 Half-space Damping rati fr P-wave: 005 Damping rati fr S-wave: 01 Side length: 0 m Embedment depth: 5 m Fundatin mass: 5000 Mg Square Fundatin inertia: Mg-m Fundatin Static stiffness: kn/m Static stiffness: kn-m Strey mass: 000 Mg Strey inertia: Mg-m Upper structure Strey stiffness: kn/m Mdel damping rati: 5% Fig. 4. Case study f an idealized three-stry building: Simplified System Fig. 5. Hrizntal acceleratin histry f the tp flr. 30 Simplified system (by SAP000 Real system (by SASSI Abslute spectral acceleratin (m/sec Frequency (Hz Fig. 6. Hrizntal respnse spectrum f the tp flr. V. CONCLUSIONS A simplified mdel is presented t simulate the unbunded sil fr rigid embedded fundatins. The mdeling parameters are determined by the thery f equivalent mdel Fig. 3. Case study f an idealized three-stry building: Real System 576

IACSIT Internatinal Jurnal f Engineering and Technlgy, Vl. 5, N. 5, Octber 013 withut using lengthy ptimizatin.

This new methd may be effectively applied t practical prblems invlving sil-structure interactin, such as machine fundatin vibratin and seismic structural analysis. [11] W. H.

Shi, Simplified Mdels fr Dynamic Sil-Structure Interactin Systems, Ph. D Dissertatin, Dept. Cnstructin En

5 IACSIT Internatinal Jurnal f Engineering and Technlgy, Vl. 5, N. 5, Octber 013 withut using lengthy ptimizatin. Fr a fundatin-sil system subjected t frced vibratins, the dynamic magnificatin respnses analyzed by using the ptimal equivalent mdel cnsist well with thse btained frm the half-space thery and existing lumped-parameter mdels. Fr an idealized building structures subjected t hrizntal excitatins, the flr respnse f a simplified building-sil system using the prpsed mdel may als agree with thse btained frm the numerical prgram. This new methd may be effectively applied t practical prblems invlving sil-structure interactin, such as machine fundatin vibratin and seismic structural analysis. [11] W. H. Wu and W. H. Lee, Nested lumped-parameter mdels fr fundatin vibratins, Earthquake Engineering & Structural Dynamics, vl. 33, n. 9, pp , July 004. [1] J. Y. Shi, Simplified Mdels fr Dynamic Sil-Structure Interactin Systems, Ph. D Dissertatin, Dept. Cnstructin Eng., Natinal Taiwan University f Science and Technlgy, Taiwan, 006. [13] A. Mita and J. E. Luc, Impedance functins and input mtins fr embedded square fundatins, Jurnal f Getechnical Engineering, ASCE, vl. 115, n.4, pp , April [14] E. L. Wilsn and A. Habibullah, SAP000 Nnlinear V.8 Integrated Sftware fr Structural Analysis and Design (CD-ROM, Cmputer and Structures, Inc., Berkeley, Calif., 00. [15] J. Lysmer, M. Tabatabaie-Raissi, F. Tajirian, S. Vahdani, and F. Ostadan, SASSI: a system fr analysis f sil-structure interactin prblems, Reprt UCB/GT81-0, University f Califrnia, Berkeley, REFERENCES [1] J. P. Wlf and D. R. Smaini, Apprximate dynamic mdel f embedded fundatin in time dmain, Earthquake Engineering & Structural Dynamics, vl. 14, n. 5, pp , Sep [] F. C. P. de Barrs and J. E. Luc, Discrete mdels fr vertical vibratins f surface and embedded fundatins, Earthquake Engineering & Structural Dynamics, vl. 19, n., pp , Feb [3] W. Y. Jean, T. W. Lin, and J. Penzien, System parameters f sil fundatin fr time dmain dynamic analysis, Earthquake Engineering & Structural Dynamics, vl. 19, n. 4, pp , May [4] J. P. Wlf, Cnsistent lumped-parameter mdels fr unbunded sil: physical representatin, Earthquake Engineering & Structural Dynamics, vl. 0, n. 1, pp. 11-3, Jan [5] J. P. Wlf, Cnsistent lumped-parameter mdels fr unbunded sil: Frequency-independent stiffness, damping and mass matrices, Earthquake Engineering & Structural Dynamics, vl. 0, n. 1, pp , Jan [6] J. P. Wlf and A. Parness, Lumped-parameter mdel fr a rigid cylindrical fundatin embedded in a sil layer n rigid rck, Earthquake Engineering & Structural Dynamics, vl. 1, n.1, pp , Dec [7] J. P. Wlf, Fundatin Vibratin Analysis Using Simple Physical Mdels, Englewd Cliffs, New Jersey: Prentice-Hall, 1994, ch.. [8] J. P. Wlf, Spring-dashpt-mass mdels fr fundatin vibratins, Earthquake Engineering & Structural Dynamics, vl. 6, n. 9, pp , Sep [9] W. H. Wu and C. Y. Chen, Simple lumped-parameter mdels f fundatin using mass-spring-dashpt scillatrs, Jurnal f Chinese Institute f Engineers, vl. 4, n. 6, pp , Nv [10] W. H. Wu and W. H. Lee, Systematic lumped-parameter mdels fr fundatins based n plynmial-fractin apprximatin, Earthquake Engineering & Structural Dynamics, vl. 31, n. 7, pp , July 00. Shi-Shuenn Chen received his dctrate in civil engineering frm the University f Califrnia, Berkeley, is frmer President f the Natinal Taiwan University f Science and Technlgy (Taiwan Tech. He established the Taiwan Building Technlgy Center in Taiwan while pursuing a First-Class University t fster the develpment f building technlgies at the cutting edge f research. Dr. Chen serves currently as the president f the Chinese Institute f Civil and Hydraulic Engineering, the president f the Assciatin f Natinal Universities f Science and Technlgy, as well as the president f the Chinese Assciatin fr Crss-Strait Exchanges and Cperatin in Higher and Vcatinal Educatin. He is als the editr-in-chief f the Jurnal f the Chinese Institute f Engineers. Dr. Chen als served previusly as a member f the Public Cnstructin Cmmissin, under the Executive Yuan, and f the Supervisry Bard f Science Educatin, under the Ministry f Educatin f the ROC. Dr. Chen s previus psitins als include thse f deputy directr general in the Natinal Expressway Engineering Bureau, president f the Chinese Assciatin f Structure Engineers. Hnrs received by Dr. Chen include the Outstanding Engineering Prfessr Award frm the Chinese Institute f Engineers and the Excellent Research Paper Award frm the Chinese Institute f Civil and Hydraulic Engineering. Jun-Yang Shi received his dctrate in cnstructin engineering frm the Natinal Taiwan University f Science and Technlgy (Taiwan Tech. Currently, he is a pst-dctral researcher in Taiwan Building Technlgy Center in Taiwan Tech. Dr. Shi has studied the fllwing areas: earthquake engineering and structural dynamics. 577

Exercise 3 Identification of parameters of the vibrating system with one degree of freedom

Exercise 3 Identification of parameters of the vibrating system with one degree of freedom Exercise 3 Identificatin f parameters f the vibrating system with ne degree f freedm Gal T determine the value f the damping cefficient, the stiffness cefficient and the amplitude f the vibratin excitatin