A Analytical Model For Vibration Period With SSI Of R/C Structures

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1 IO Journal of Mechanical and ivil Engineering (IO-JME) e-in: ,p-IN: X, Volume 12, Issue 2 Ver. II (Mar - Apr. 2015), PP A Analytical Model or Vibration Period ith I Of / tructures urkia Haithem 1, Lahbari Noureddine² 1,2, Department of ivil Engineering, Institute of ivil Engineering, Hydraulique and Architecture, University of Batna,Algeria Abstract: he effect of the fundamental period, the soil structure interaction I, and the site on the seismic behaviour of / structures is investigated using analytical model based on the Algerian seismic regulations. Hence the aim of this study is to formulate model covering the fundamental period of vibrations based on a system with continuous columns in which the deformations of structure and soil represent the degree of freedom. hear and flexural deformation are appointed for the structure, whereas relative displacement of the foundation base and rocking are meant for the soil( isolated footings for stiff and medium soil - sites 1, 2 and mat for soft soil sites 3 or 4 ).inite element method is used to analyse the response of various / frames (low, medium and high rise),assuming fixed and flexible base,(vertical, horizontal translations stiffness, rocking and torsional stiffness),and compared with Newmark osenblueth, Deleuze and Gazetas methods. eywords: soil structure interaction, foundation, / frame, eismic response, effective periods. I. Introduction Often, seismic structural design is based on rigid base assumption, and interaction with the soilfoundation system is either ignored or carried out separately, whereas in reality these systems are coupled. Ignoring the I effects may lead to erroneous structural assessment and estimates of seismic demands. his work presents a simplified and accurate formulation using finite element method applied to the analytical model based on the Algerian seismic regulation, a rapid assessment of the fundamental period of vibrations when I effects are accounted for. urthermore, it investigates the importance of I phenomena on the response of frames as function of different parameters such as soil rigidity, foundation rigidity, foundation mass, and soil mass. It is shown that for structures founded on soft soils with high relative rigidity, the I effects amplifies the dynamic response of the system. Also, it is not necessary to take into account I effects when designing a / building on stiff soil. II. ormulation Of he oil tructure Interaction - Description Of Model I Based on the formules of the period of vibration given by HOMPON [1] neglecting I effects, the new formulation taking into account I effects will be as follows: ( 2 s ) ( b 2.1 hear Mode with I ( ) In this case the bending deformation is negligible and the shear deformation is the determining factor in calculating the period. he fundamental period of vibration in hear mode is given by: ) 2 (1) here: s: hear fundamental period without I is given by [2]: s s t mr (2) : otal mass calculated from structure, soil and foundation; t : otal mass calculated from structure; r: atio of shear stiffness equal to: /, : hear stiffness with and without I. 2 4H AG 4H AG and DOI:.9790/ Page AG 4H t 1 Hg (4) 4 th g (3)

2 A analytical model for vibration period with I of / structures A: area of the section; G: Modulus of the reinforced concrete;, : security coefficient without and with I. m: atio of unit mass of building is given by: here: : foundation mass ; : soil mass. m t t t 1 t t (5) 2.2 flexural Mode with I In this case the shear deformation is negligible and the bending is the determining factor in calculating the 2 2 b 1.79H b EI period. b: lexural fundamental period without I is given by [2]: I : otal moment of inertia of the structure with I. onsequently: 1.79H gei H (6) : atio of the moment of inertia with and without I given by: here: I : Moment of inertia of the foundation. I t : Moment of inertia calculated of the structure only without I. b b b 1 I m (8) 2 I th H 1.79H b EI t gei t I I t I I 1 (9) I I I t t t (7) 2.3 ocking mode he case of isolated footings (stiff and medium soil - site 1, 2) Using the simplified method from VELEO [3,4,5], the expression of rocking stiffness from vertical and rocking stiffnesses of the soil is: i vi yi () ith: vi and i the corresponding vertical and rocking stiffnesses respectively. Y i represents the normal distance from the centroid of the i th footing to the rocking axis of the foundation. he vertical and rocking stiffnesses of the i th footing are defined by the following relations [3]: ith r ai and r mi are given as follows [3, 6] vi 4Grai 2d 1 i 1 5rai 2 (11) 3 8Gi rmi d 1 i i 2 (12) 3(1 ) rmi r ai : adius of a circular footing that has the area of the i th footing; d i : Depth of effective embedment for the i th r ai A (13) footing. r mi 4 4I (14) DOI:.9790/ Page

3 A analytical model for vibration period with I of / structures he case of mat (soft soil - site 3, 4) rom the simplified method of VELEO [4,5] described by OULUMIA [7] and based on the text of A-3 [8]; he expression of rocking stiffness of rectangular footing can be expressed by [3]: 3 8Gr m (15) 3(1 ) here: : ocking stiffness of rectangular footing; G: hear modulus of soil beneath the i th footing; : he Poisson's ratio of soil. r m : adius of the circle of the equivalent foundation calculated as follows: ith: A : Area of the section of the foundation. hen: r m A I (16) 8G AI 3 (17) 3(1 ) he final formulation for the fundamental period of vibration, taking into account I effect will be: 2 2 m b III. ormulation Of I o einforced oncrete rames In the following, an approximate formula for the lateral drift of the frame is determined by considering the interaction of soil structure. he assumptions of the method of analysis of rigid frame were adopted [9]. igure 1 shows the frame after deflection under lateral forces. he total lateral displacement of a level U is equal to the sum of the displacement without I and the displacement due to the interaction U r [2, ]. U = U+ U (19) here: U : he total lateral displacement with I; U: he displacement at the n th floor of the built structure without I. U = U + U g (20) U, U g : he displacement of the columns due to the bending mode and the displacement of the beams due to the shear mode. U : he displacement due to the translation and rocking of the foundation [11, 12, 13]. he lateral displacement U without I is calculated by [2]: mr (18) Vh²[( N U 12EN he linear rigidities of columns and beams are: 1) ( N g N g 1) ] (21) I I g and g (22) h L V: shear force at the base of the structure without I. I c, I g : moment of inertia of the columns and beams respectively N : Number of columns. E: Modulus of elasticity of the concrete. h: story height. L: length of bay. he total displacement taking into account the I is calculated by simplified method from VELEO [3] as: U V V here: V : he reduced shear force corresponding to the soil structure interaction (with I). M 0 : he moment due to the lateral forces without I is: (2HV)/3 M 0H U DOI:.9790/ Page (23)

4 : he rocking stiffness of the foundation with I. H: total height of the structure. hen the shear deformation due to the lateral displacement will be: A analytical model for vibration period with I of / structures U V MO H V here: : he shear deformation without I calculated by: = U / h (24) Hence the shear stiffness with I is: he hear stiffness without I [2] is: And consequently the stiffness ratio r is: V V 3 M O 3 2H 12EN ( Nc 1) s h[( N 1) N g g ] (26) (25) r Is the stiffness corrector ratio with effects I 2H 3 3 2H (27) IV. Numerical Application 4.1 haracteristics of dimensionless parameters he characteristic parameters of the interaction model are defined as well as the intervals of typical values for building structures as follows [14]: - atio of the foundation mass to the structure mass: 0 / t atio of the moment of inertia of the foundation to the mass moment of inertia of the structure: 0 I / I tr Damping ratio for the fixed-base structure and the soil = 0.07, which is a conventional value adopted for the most buildings and soils (I effects are not sensitive to the. ixed base structural damping ratio [23]) - Poisson's ratio for the soil: µ =0.20, 0.25 and 0.4 which are representative values for stiff, medium and soft soils, respectively. - atio of the shear stiffness: 1 r elative mass density between the structure and the soil: 2 / t 5. - lenderness ratio of the structure: H / = 2 to Assumptions In the case of structures without I, the assumption of fixed base is used to estimate the fundamental period of vibrations. his is assessed according to the PA code [15] for different categories of sites. In the case of structures with I, the soil is modelled by springs: horizontal, vertical and rocking. o determine the stiffness, the methods of NEMA - OENBLUEH, DELEUZE and GAZEA [16, 17] are applied. he shear modulus of the soil G is given three values, the density of soil is set at 2t/m 3 and the coefficient of critical damping is taken as = 7%; able. 1 summarizes the different values. V. esults - able 2 presents values of fundamental natural periods calculated by different methods: exact method, PA code [15], ADELI model without soil-structure interaction. - It shows a good correlation between exact solution and ADELI model [1] without I: a difference of 2.5% is observed in all sites for the ratio exact / ADELI with a deviation of 0.09, ig It can be observed from the results that the interaction effects are negligible (1/ < 0.) in stiff soil and outstanding in medium and soft oil (1/ > 0.), able 3 (ig.3a).hese results are in good correlation with those obtained by MAUMI, ABAABAIEA [18] and MIHAEL JAME GIVEN [11]. - he incorporation of I and number of stories tends to increase the fundamental period by 26.3% in medium soil 2 and 27.9% in soft soil 3 4 as showed in [19] DOI:.9790/ Page

5 A analytical model for vibration period with I of / structures - he effect of soil-structure interaction is to be considered when the following criterion is satisfied: H/ (Vs.) > he values of factor 1/ for the seismic behaviour of / structures, according to the Algerian code PA2013 considering the I effect, are given only for soft soil site 3, (ig.3b). - able 4 presents values of the natural periods with I obtained by the methods of NEMA - OENBLUEH, DELEUZE and GAZEA and by the proposed model. - he fundamental periods of vibration obtained by the proposed model gives good results compared to GAZEA, DELEUZE and NEMA- OENBLUEH methods: 5.57% for site 2 and 6.01% for site 3, (ig.4). VI. Discussions - Influence of the ratio / t : the period of vibrations increase with the increase of the ratio / t,about 7.4% in soft and medium soil. - Influence of the ratio I / I t : no notification < 1% as showed by[35]. - Influence of the ratio / t : increase of the period with the increase of the mass soil about 27% (ig.5),[] presents an increase of 20% - Influence of the ratio D/: no notification: > 2%. - he variation of lateral natural period due to incorporation I increases with the reduction in stiffness of soil. It is minimum in case of stiff soil ( 1 ) and maximum in soft soil ( 3 and 4 ) about 75%. A maximum increase of more than about 78% is noted in [21] and 70% in [22] VII. onclusion - hen considering I effects, the soil flexibility and number of stories have an influence on the naturel period. - Natural period of / system including I effects increases when the ground is softer. - It is not necessary to consider the effect of soil structure interaction for seismic design of reinforced concrete frame buildings founded on stiff soil. Hence it is possible to include the soil-structure interaction effects in the analysis of multi-story building response by other means such as incorporating a few modifications to the fixed base condition. hese modifications include mass of soil, inertia of foundation, ratio of shear stiffness and slenderness. - As 1/ increases, the significance of I effects increases. - inally, it is essential to consider the effect of soil-structure interactions for seismic design of reinforced concrete frame for: 1/ > 0. eferences [1]. hompson., theory of vibration with application, 2nd edition, Prentice Hall,Englewood cliffs, New jersey,(1981). [2]. Adeli H, Approximate formulae for period of vibrations of building systems, civil engineering for practising and design engineers, Vol4 Nb1 pp93-128, junnary [3]. Victor.D, la construction en zone sismique, Paris, Editions le Moniteur [4]. Veletsos, A.., and Meek, J., Dynamic behavior of building-foundation systems, J.Earthquake Engrg. truct. Dyn., 3(2), , [5]. Veletsos, A.., and Nair, V. V, eismic interaction of structures on hysteretic foundations,j. truct. Engrg., AE, 1(1), 9 129, [6]. halil.l, adek.marwan, hahrouri.isam, Influence de l interaction sol structure I sur la fréquence fondamentale des bâtiments, XXIV Emes encontres universitaires de génie civil, Montpellier, 1-2 juin (2006). [7]..ouloumiac, Interaction sol-structure, méthode simplifiée, ocotec, [8]. A-3, entative Provisions for the Development of seismic regulations for buildings, ch.6, [9]. Norris..H, ilbur.j.b, Utku., Elementary tructural Analysis, Mc Graw Hill, []. Ayman Ismail, Effect of oil lexibility on eismic Performance of 3-D rames, IO Journal of Mechanical and ivil Engineering (IO-JME) e-in: ,p-IN: X, Volume 11, Issue 4 Ver. II, PP , Jul- Aug [11]. Michael James Givens, Dynamic oil-tructure Interaction of Instrumented Buildings and est tructures, doctoral dissertation, university of alifornia - Los Angeles, 2013 [12]. Mylonakis, George and Gazetas, George, eismic soil structure interaction: Beneficial or Detrimental?, Journal of Earthquake Engineering, 4: 3, , 2000 [13]. E.N.ovithis,.D. Pitilakis, G.E. Mylonakis, eismic analysis of coupled soil pile structure systems leading to the definition of a pseudo natural I frequency, oil dynmacis and earthquake engineering 05-15, 2009 [14]. Javier Avilés, Luis E. Pérez-ocha, Evaluation of interaction effects on the system period and the system damping due to foundation embedment and layer depth, oil Dynamics and Earthquake Engineering ,Elsevier cience Limited, [15]. ègles Parasismiques Algériennes 99-03, OPU, Algérie, [16]. Gazetas, G. and Mylonakis, G, eismic oil-tructure Interaction: New Evidence and Emerging Issues, Emerging Issues Paper, Geotechnical pecial Publication No 75, AE, Vol III., pp , [17]. Gazetas, G. and Mylonakis, G., eismic soil-structure interaction: beneficial or detrimental, Journal Earthquake, vol. 4, No 3, , DOI:.9790/ Page

6 Medium oil - 2 tiff oil - 1 oil ype Number of Bay Number of stories tory Height (m) tory idth (m) A analytical model for vibration period with I of / structures [18]. Hamid reza abatabaiefar, Ali Massumi, A simplified method to determine seismic responses of reinforced concrete moment resisting building frames under influence of soil-structure interaction. oil dynamics and erathquake engineering, ,20. [19]. H. hinmayi, B. Jayalekshmi, oil-structure interaction analysis of frame shear wall buildings over raft foundations under seismic loading, International Journal of cientific & Engineering esearch Volume 4, Issue 5, May-2013 [20]. Jayalekshmi B., hinmayi H., Effect of soil flexibility on lateral natural period in framed buildings with shear wall, International Journal of Innovative esearch in cience, Engineering and echnology Vol. 2, Issue 6, June [21]. I. raus & D. Džakić, oil-structure interaction effects on seismic behaviour of reinforced concrete frames, 50 E EEE University of Osijek, aculty of ivil Engineering Osijek,roatia [22]. Jian Zhang, Yuchuan ang, Dimensional analysis of structures with translating and rocking foundations under near fault ground motions, oil dynamics and earthquake engineering , aptions to tables oil type able. 1: Geotechnical specification of the utilized soils in research. Elastic Module E (n/m²) hear Module G (n/m²) Poisson atio Mass Density (n.²/m 4 ) ol (Bars) tiff - ite Medium - ite oft - ite 3 and Dimensional specification of the studied frames able. 2: Variation of fundamental lateral natural period without I undamental natural periods without soil-structure interaction I (s) omparison hear ave Vs(m/s) Exact PA 2003 ADELI Model exact / PA2003 ADELI / PA2003 exact / ADELI and 4 6b 6b 6b 6b 2s 2s 3s 3s 4s 4s 5s 5s 6s 7s Mean = Devation = Mean = Devation= Mean = Devation=0.09 able. 3: actor of the relative stiffness between structure and soil. 1/ = H / Vs oil ype rame ype PA 2003 Proposed Model Mean = Mean = Mean = Mean = DOI:.9790/ Page

7 oil ype rame ype b: bay and s:storey Veletsos Deleuze - Newmark Gazetas oft oil 3 and 4 A analytical model for vibration period with I of / structures Mean = Mean = able. 4: Variation of fundamental lateral natural period with I. (4a) = 2 t Natural Periods with oil-tructure Interaction I Proposed Model with Isolated footings = 2 t I = 0 I = 0.05 I t I = 0.1 I t = 0 t =0.25 t =0.5 t = 0 t =0.25 t =0.5 t = 0 t =0.25 t =0.5 t oft oil 3 and 4 r = 1. Medium oil - 2 r = DOI:.9790/ Page

8 Medium oil- 2 r = 1.05 oil ype rame ype b: bay and s:storey Veletsos Deleuze - Newmark Gazetas A analytical model for vibration period with I of / structures (4b) = 5 t Natural Periods with oil-tructure Interaction I Proposed Model with Isolated footings = 5 t I = 0 I = 0.05 I t I = 0.1 I t = 0 t =0.25 oft oil 3 and 4 r = 1. t =0.5 t = 0 t =0.25 t =0.5 t = 0 t =0.25 t =0.5 t aptions to figures ig. 1: chematic illustration of I Model. DOI:.9790/ Page

9 A analytical model for vibration period with I of / structures ig. 2: Variation of change in period of vibration without I. (3a) ig. 3: Variation of the relative stiffness between structure and soil. DOI:.9790/ Page

10 A analytical model for vibration period with I of / structures (3b) ig. 4: Variation of change in period of vibration with I considering s = 2 t, I f = 0.05 I t and f = 0.25 t. (4a) (4b) ig. 5: Periods of soil-structure systems for various soil mass considering = 0.25 t, I f = 0.05 I t. (5a) DOI:.9790/ Page

11 A analytical model for vibration period with I of / structures (5b) DOI:.9790/ Page

INFLUENCE OF THE SOIL-STRUCTURE INTERACTION ON THE SEISMIC BEHAVIOR OF BUILDINGS ON SHALLOW FOUNDATIONS

Geotech., Const. Mat. and Env., ISSN:2186-2982(P), 2186-2990(O), Japan INFLUENCE OF THE SOIL-STRUCTURE INTERACTION ON THE SEISMIC BEHAVIOR OF BUILDINGS ON SHALLOW FOUNDATIONS Z. Benadla,. Hamdaoui, S.