- Email: [email protected]
A simplified empirical method for assessing seismic soil-structure interaction effects on ordinary shear-type buildings
Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
A simplified empirical method for assessing seismic soil-structure interaction effects on ordinary shear-type buildings Stefano Renzi n, Claudia Madiai, Giovanni Vannucchi Università di Firenze, Dipartimento di Ingegneria Civile e Ambientale, 50123 Firenze, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 23 May 2012 Received in revised form 16 September 2013 Accepted 18 September 2013 Available online 8 October 2013
The beneficial or detrimental effect of seismic soil-structure interaction (SSI) is still a controversial issue. A parametric analysis of the seismic SSI effects of a large number of idealised ordinary shear-type buildings was carried out with some simplifying assumptions. Results were compared to the corresponding classical fixed-base solutions. Structures were modelled as generalised single degree of freedom systems using the principle of virtual displacements and shallow squared foundations resting on different soil types were assumed. The outcomes of the numerical analyses were used as a statistical base in order to obtain simple analytical and non-dimensional relationships for estimating seismic SSI effects in terms of modified period and damping. The proposed approximated method can be used by consultants in an immediate and simplified manner in order to obtain a preliminary evaluation of SSI effects and seismic demand without devoting resources to complex analyses. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Seismic soil-structure interaction SSI simplified procedure Modified structural damping and period Modified structural seismic demand
1. Introduction Due to evidence of historical earthquakes, the importance of achieving an acceptable level of safety for ordinary shear-type buildings, as “element at risk”, is undisputed. It is also well known that seismic Soil-Structure Interaction (SSI) can play a relevant role even if the issue is not free from misconceptions. The dynamic response of a structure supported on soft soil may differ substantially in amplitude and frequency content from the response of an identical structure founded on firm ground. The main effects of seismic SSI on buildings with shallow foundations consist of an increase in the fundamental period and damping of the soil-structure system, as evidenced by many researchers [1–8]. In addition, other effects of SSI, such as foundation uplift and sliding at the soil-foundation interface, were analysed by several authors [9–16]. The latter phenomena can occur when seismically induced loads attain a limit value, the soil-foundation system initiates significant yielding and foundation permanent displacements take place, with accumulation of large residual foundation rotations and horizontal displacements. Such effects, particularly associated with strong earthquakes, involve non linearity and will not be considered in this paper.
n
Corresponding author. Tel.: þ 39 347 1945 409. E-mail address: [email protected] (S. Renzi).
0267-7261/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.soildyn.2013.09.012
In literature, different ways for studying SSI have been adopted, i.e. theoretical, physical and numerical modelling [17–19]. Despite extensive research over the past 40 years, there is still controversy regarding the role of fundamental period lengthening and damping increase due to Soil-Structure Interaction (SSI) in the seismic performance of structures founded on soft soil. As a matter of fact, the idealised design spectra of the seismic codes along with such modifications lead invariably to reduced design base shear (Fig. 1). For this reason, neglecting SSI effects is currently being suggested in many seismic codes (e.g. ATC-3-06 [20], NEHRP-97 [21]), as a conservative simplification that supposedly leads to improved safety margins [22], at least for ordinary structures. Moreover, the computational difficulties in evaluating the effects of seismic SSI discourage consultants faced with this task and, as a consequence, these effects are often neglected in engineering practice even though they may be relevant. The effect of seismic SSI has in fact been recognised as being important by many researchers [22–38]. In general, it cannot be neglected and must be evaluated on a case-by-case basis. In the opinion of the authors it should sometimes be opportunely estimated even for ordinary structures, such those investigated in this paper, for the following reasons: (i) with reference to displacement design spectra incorporated in most seismic codes, the effect of seismic SSI may induce excessive displacements to structures (this must be taken into account for example for adjacent structures, Service Limit State analyses, etc.); (ii) the modified seismic behaviour of structures, both in terms of stresses and strains, could in some cases produce a
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
101
parameters of the soil-foundation-superstructure systems and ~ the ratios T~ =T and ξ=ξ.
Fig. 1. Reduction in design base shear due to SSI ( according to NEHRP seismic coad [21]).
detrimental effect on structures, as shown by many documented evidences around the world, e.g. during the Bucharest (1977), Mexico City (1985) and Kobe (1995) earthquakes. Hence, whenever the seismic design is performed using actual or synthetic earthquakes, representative of the local seismicity, the effect of SSI could be detrimental even in terms of pseudo-spectral acceleration. Nevertheless, as mentioned before, rigorous assessment of seismic SSI is not a simple task due to the difficulties associated with the evaluation of kinematic and inertial effects. In the literature some methods are available for the evaluation of the SSI effects. The most currently used methods [2,4,5,8, 26,35,39] provide approximate solutions or charts and require the knowledge of the impedance matrices of the soil-foundation system. An exact solution that does away with the approximations of the previous available approaches has recently been proposed [40], requiring the knowledge of the dynamic impedance matrices. The aim of the present paper is to provide a practical tool, capable of offering a satisfactory trade-off between rigour and simplicity, to obtain an approximate but reliable estimate of SSI effects in terms of modified damping and period of ordinary sheartype structures with surface squared foundations. A wide range of soil-foundation-superstructure systems were examined and simple numerical relationships from statistical analyses of the obtained results are finally proposed. Even if the preliminary determination of dynamic impedance matrices was still required in the method developed in this study, the numerical relationships obtained from the authors can be applied by final users without calculating these matrices.
2. Analyses performed Extensive numerical analyses of simplified superstructurefoundation-soil models [41,42] were performed in order to evaluate SSI effects and summarise the results in equations obtained from statistical analyses. The adopted procedure can be summarised as follows: (1) Computation of soil-foundation impedance matrices. Horizontal and rocking components were numerically evaluated using the computer programme SASSI2000. (2) Idealisation of shear-type buildings. Superstructures were modelled as equivalent SDF systems having a fundamental period T and damping ratio ξ. (3) Estimate of SSI effects. Modified period T~ and damping ratio ξ~ of the analysed soil-foundation-superstructure systems were evaluated by means of a recent exact solution. (4) Statistical analyses of the results from step 3. Analytical relationships were obtained between the main non dimensional
In order to show the effectiveness of such expressions, a systematic comparison between the values of the ratios T~ =T and ~ ξ=ξ estimated by the proposed relationships and computed by applying steps 1 to 3. The SSI effects were analysed using a substructuring approach: the problem is broken down into a series of subsystems which are solved separately then reassembled using the principle of superposition, requiring as assumption soil and structural linear behaviour. A wide range of soil-foundation-superstructure configurations were examined. Shear-type buildings (up to 20 storeys) having a flexural first mode of vibration were modelled as generalised Single Degree of Freedom (SDF) systems, with squared shallow foundations resting on homogeneous subsoil. The analyses were performed under the following simplified assumptions: Regular plan and elevation shear-type buildings were considered. Since their seismic response is not significantly affected by contribution from higher modes of vibration, they were modelled as equivalent SDF systems, whose modified fundamental period was evaluated for flexural mode. Squared raft foundations were analysed; this particular shape combined with a symmetrical superstructure shows the same behaviour in terms of lateral stiffness in both x- and ydirections. Rigid foundations were considered. Homogeneous soil deposit, with constant shear stiffness and damping, was assumed.
2.1. Computation of soil-foundation impedance matrices As suggested by many Codes (e.g. FEMA 368 [43] and FEMA 369 [44]) the compliance of the subsoil was accurately evaluated by Finite Element Method (FEM) using the computer programme SASSI2000 [45,46]. The analyses were performed to estimate the horizontal and rocking foundation impedance matrices; in such preliminary analysis the foundation mass does not play a significant role, as a matter of fact, the numerous existing formula and charts proposed in the related literature [2,4,5,8,26,35,39] always refer to massless foundation. Thus, rigid and massless foundation rafts were modelled by means of 3D brick finite elements. Surface squared raft foundations of characteristic length, b¼5, 12.5 and 25 m, resting on different subsoil configurations, were considered. The subsoil is modelled as a homogeneous viscoelastic halfspace or as a homogeneous viscoelastic horizontal deposit resting on an elastic bedrock. In the latter case the bedrock was assumed at five different depths, Hdep ¼5, 10, 20, 50 m (it should be noted that for the halfspace configuration Hdep ¼ p ). In the viscoelastic halfspace configuration a viscous boundary was assumed, in order to account for radiation damping in the halfspace through the lower boundary and to avoid wave reflections. For the numerical analyses the deposit was subdivided into different sublayers. Each of them was identified by its thickness, unit weight, shear-wave velocity (VS), compressional-wave velocity (VP) and associated damping ratio. In order to evaluate the influence of soil shear-wave velocities on SSI effects, three different types of homogeneous soils were selected for the subsequent analyses, i.e. VS ¼80, 200 and 320 m/s (corresponding to effective shear modulus values of 12, 77 and
102
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
Fig. 3. General 2D model.
Fig. 2. Impedance scheme.
198 MPa respectively for a soil unit weight of 19 kN/m3); soil damping, ξs, was assumed equal to 5% in all cases analysed. Bedrock shear wave velocity was supposed equal to 800 m/s. Interaction between soil and foundation occurs only at their common nodes. The impedance matrices K and C of the rigid and massless foundations were computed from the foundation compliance matrices, whose columns were obtained by applying a unit force or moment in the desired direction at the foundation level (Fig. 2), and computing the resulting real and imaginary part of displacements and rotations of that point. Finally, by inverting the compliance matrices, the corresponding impedance matrices were computed. It is worth mentioning that, since the purpose of numerical analysis is to estimate impedance matrices, no seismic response analysis was performed. Fig. 4. Lateral forces assumed to obtain the shape function from static deflection.
2.2. Idealisation of shear-type buildings A set of shear-type buildings, squared in plane, with a maximum of 20 storeys, each 3 m tall, squared columns having a minimum section of 30 30 cm2, beams having rectangular sections (20 45 cm2), forming 5 m squared frames (Fig. 3) were considered in the analyses. The presence of an elevator was assumed in all the cases examined; such horizontal reinforcement gives rise to an increment in the lateral stiffness of the structure. Different types of loads were taken into account: (i) accidental loads Qk ¼2.00 kN/m2; (ii) floor self-load Gkf ¼ 4.81 kN/m2; (iii) external masonry (thickness 40 cm) Gkm ¼3.70 kN/m2. Such structures were modelled as generalised Single Degree of Freedom (SDF) systems using the principle of virtual displacements. As evidenced in literature [47], the analysis of an equivalent SDF system provides exact results for an assemblage of rigid bodies supported so that it can deflect in only one shape. The approximate natural frequency is shown to depend on the assumed deflected shape The assumed deflection can be related to a single generalised displacement z(t) through a shape function ψ(x) that approximates the fundamental vibration mode. In the analyses the deflected shape due to static forces was adopted as an approximate shape function ψ(x). The lateral force for each floor was assumed in the horizontal direction and equal to wi ¼mii, where mi is the floor mass and g the acceleration of gravity. The fundamental frequency ω0 for a building with j storeys, was estimated from the set of forces shown in Fig. 4, which leads to the following evaluation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ug∑N j ¼ 1 mj uj ð1Þ ω0 ¼ t N ∑j ¼ 1 mj u2j This equation appears in the seismic design provisions of some building codes, e.g. in the 2005 edition of the National Building
Code of Canada (NBCC) [48] and in the 2004 edition of the Mexico Federal District Code (MFDC) [49]. Following the method previously described, each superstructure was modelled as a SDF system, having equivalent height, heq, equivalent mass, meq, and equivalent stiffness, keq, with 5% viscous damping ratio. At this stage of the procedure the foundation mass, mb, was introduced, considering a 1 m thick foundation in all the cases under investigation. As previously mentioned, a linear approach was applied, ignoring the nonlinear interaction effects of the soil-foundation system. 2.3. Estimate of SSI effects The classical approach for elasto-dynamic analysis of SoilStructure Interaction [39,40] aims at replacing the actual structure by an equivalent Single Degree of Freedom (SDF) (see Section 2.2) system supported on a set of frequency-dependent springs and dashpots accounting for the stiffness and damping of the compliant soil-foundation system (see Section 2.1). The structure is described by its stiffness k, mass m, height h, and damping coefficient c (Fig. 5a). The foundation consists of a rigid shallow squared raft of characteristic dimension b (half width) and mass mb resting on a homogeneous, linearly elastic, isotropic halfspace described by a shear modulus Gs, mass density ρs, Poisson's ratio νs, and hysteretic damping ratio ξs. Translational and rotational stiffness, Kx and Kr respectively, of the compliant soil-foundation system, are modelled by a pair of frequency-dependent springs. To ensure uniform units in all stiffness terms, Kr is represented by a translational vertical spring acting at the edge of the foundation (i.e. at a distance b from the centre of the footing). Translational and rotational damping, Cx and Cr respectively, of the compliant soil-foundation system, are modelled by a pair of frequency-dependent dashpots, attached in parallel to the springs,
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
representing energy loss due to hysteretic action and wave radiation in the soil medium. The stiffness and damping terms of both the structure and the soil-foundation system can be condensed in two terms [39,40], K~ ~ The corresponding system is represented by the replaceand C. ment oscillator shown in Fig. 5b. Superstructure-foundation-soil model parameters (Hdep, VS, heq, meq, keq, b and mb) were selected in order to be representative of a wide range of actual systems; SSI analyses were performed on 563 different models. Period lengthening and effective damping of each analysed SSI system were calculated by systematic application of the exact solution recently proposed by Maravas et al. [40] (Eqs. (2) and (3) and compared to the corresponding values of the fixed-base structure. The method contains no approximations in the derivation of ~ of the ~ and effective damping, ξ, the fundamental natural period, ω, system. Furthermore, the exact frequency-varying foundation impedances may be employed. The properties of the replacement oscillator are given by ξx ξr ξ þ þ 2 ð1 þ4ξ2 Þ 2 ð1 þ 4ξ2 Þ 2 ð1 þ 4ξ2 Þ ω ω ω r x r ξ~ ¼ x 1 1 1 þ þ ω2x ð1 þ4ξ2x Þ ω2r ð1 þ 4ξ2r Þ ω2 ð1 þ 4ξ2 Þ
ð2Þ
" ω~ 2 ¼ 1 þ 4
103
2 2 2 ξ~ ξ~ ξ~ þ1þ4 þ1þ4 2 2 2 2 2 ωx ð1 þ 4ξx Þ ωr ð1 þ 4ξr Þ ω ð1 þ4ξ2 Þ
#1
ð3Þ qffiffiffiffiffiffiffi2ffi qffiffiffiffi qffiffiffi i where ωx ¼ Kmx , ωr ¼ K r b2 , ω ¼ mk , ξi ¼ ωC 2K i , (i¼x, r) with ω being mh
the circular excitation frequency.
2.4. Statistical analyses In order to perform the statistical analyses, model parameters and numerical computation results were expressed by means of the following dimensionless quantities: x1 ¼
1 heq ; ¼ s TV S
x2 ¼
b ; H dep
x3 ¼
heq ; b
x4 ¼ μ ¼
mb ; meq
T~ y¼ ; T
z¼
ξ~ ξ
where T and ξ are the fundamental period and damping of the fixed-base structure, while T~ and ξ~ are the fundamental period and damping of the numerically analysed SSI systems. The results obtained were statistically analysed, considering only the most interesting 275 cases that met the following conditions: x2 ¼ Hbdep 4 0 T~ y ¼ 4 1:05 T ξ~ z ¼ 4 1:05 ξ
Fig. 5. (a) Superstructure-foundation-soil idealised by a stick model. (b) Reduced single degree-of-freedom model (after [39,40]).
~ less than 1.05 were not included in Cases with values of T~ =T or ξ=ξ the analysis because they were considered to be of little practical significance. The statistical analyses were performed assuming the parameters y and z as dependent variables and x1, x2, x3, x4 as independent variables. A multiple regression analysis was performed in order to find the best compromise between the simplicity of the equations (minimum number of parameters of the statistical model) and their ability to provide a good estimate of the parameters (high value of R2, symmetrical distribution having mean zero of the residuals ε). A sensitivity analysis was carried out to assess the contribution of each dimensionless parameter on the dependent variables.
Fig. 6. Comparison between estimated and computed values of the ratio (a) y ¼ TT and (b) z ¼ ξξ.
104
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
3. Results In the statistical model the best estimate of the variable y ¼ T~ =T was found to be represented by means of the following equation: y ¼ ax1 m1 x2 m2 x3 m3 þ1 þ ε
ð4Þ
that is lnðy 1 εÞ ¼ m1 lnðx1 Þ þ m2 lnðx2 Þ þm3 lnðx3 Þ þ lnðaÞ
ð5Þ
where the coefficients obtained from the statistical analysis are a ¼ 0:7698;
m1 ¼ 1:663;
m2 ¼ 0:1359;
m3 ¼ 0:8443
the coefficient of determination for the regression (4) is R2 ¼0.937. The comparison between estimated and computed values for the 275 cases considered is shown in Fig. 6a. The proposed analytical expression (5) shows a good agreement between estimated and computed values. The average difference is
74% over the whole range of analysis (1.05ryr2.83), with 90% of cases presenting a difference less than 710%. In particular in the range 1.05 r yr1.40, which is the range of most practical interest for engineering applications, the estimated values differs from the computed values of 7 3% (in average), with 90% of cases presenting a difference less than 75.4%. The residual ε, that is the difference between the value obtained from numerical analysis of the SSI model (computed value) and the value estimated from the proposed relationship (4) (estimated value), is a normally distributed random variable (Fig. 7a) with zero mean and standard deviation SD ¼0.098. It is evident from analysis of the coefficients of Eq. (5) that, as expected, the non-dimensional wave parameter x1 ¼ 1=s ¼ heq =TV S (which may be looked upon as a measure of the relative stiffness of the foundation and the superstructure) plays the most important role on the ratio y ¼ T~ =T. The slenderness ratio x3 ¼ heq =b, also shows a significant influence on period lengthening, while the parameter x2 ¼ b=H dep , representing the foundation
Fig. 7. Histogram for the residuals of the statistical model for: (a) y ¼ TT and (b) z ¼ ξξ.
Fig. 8. Influence of x2 (b/Hdep) on (a)
T T
and (b) ξξ.
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
105
As an example some significant comparisons are shown in Table 1 for the case 1/s equal to 0.437. The differences in terms of T~ =T are about 6%. ~ The variable z ¼ ξ=ξ was estimated by means of a more complex equation
characteristic length compared to the bedrock depth, has little influence on this parameter. The mass ratio x4 ¼ μ ¼ mb =meq does not have any influence. In order to assess the contribution of each dimensionless parameter on the estimate of y, a sensitivity analysis was done. In Fig. 8a the influence of the parameter x2 on y is presented given heq/b¼4 and μ¼ 0.60. As mentioned before, this parameter has little influence in particular for low values of 1/s (say less than 0.2) and becomes more important for higher values. The increment of the SSI effect on the fundamental period is more pronounced for lower values of x2, i.e. for deep bedrock and/ or for small foundations. Such variation is more evident as 1/s increases: i.e. given 1/s, the decrement of y with x2 is constant and shows a linear increment as 1/s increases. The effect of x3 on y is presented in Fig. 9a, given b/Hdep ¼0.5 and μ¼0.60. The increment of the fundamental period shows an exponential increment for higher values of x3, i.e. tall and slender buildings. Since the parameter x4 does not play any role in the evaluation of period increase, the sensitivity analysis for this parameter was not performed. The validity of all the steps of the method have been checked with existing solutions. In particular the dynamic impedance matrices obtained in the present paper were compared with the solution proposed by Gazetas ([36]); some differences were found, both in static and dynamic parts.
z ¼ ðm1 x4 þ m2 x3 Þx1 2 þ ðm3 x4 þm4 x3 þ m5 x2 Þx1 þ 1 þ ε
ð6Þ
that is ðz 1Þ ¼ m1 x4 x1 2 þm2 x3 x1 2 þ m3 x4 x1 þm4 x3 x1 þ m5 x2 x1 þ ε
ð7Þ
where the coefficients obtained from the statistical analysis are m1 ¼ 11:457; m2 ¼ 1:017; m3 ¼ 1:660; m4 ¼ 0:9558; m5 ¼ 4:3048 The coefficient of determination for the regression (6) is R2 ¼ 0.948. The comparison between the values obtained from the numerical models (computed values) and the values estimated from the proposed relationship (6) (estimated values) for the 275 cases considered is shown in Fig. 6b. The proposed analytical expression (7) shows a satisfactory agreement between estimated and computed values, even if in this case data are more scattered. The average difference is 714% over the whole range of analysis (1.05 ryr3.88), with 90% of cases presenting a difference less than 725%. The residual ε is a normally distributed random variable (Fig. 7b) with zero mean and standard deviation SD ¼0.286.
Fig. 9. Influence of x3 (heq/b) on (a)
T T
and (b)
ξ ξ
Table 1 Comparison with existing solutions. SDF parameters
Soil and foundation data
Results 3
heq [m]
keq [kN/m]
meq [Mg]
ξ [%]
T [s]
b [m]
VS [m/s]
γ [kN/m ]
ν []
Hdep [m]
Kx [kN/m]
Kr [kNm]
T~ T
20
57030
472.1
5
0.572
5
80
19
0.3
10
2.11E þ 05 4.06E þ 05 2.93E þ 05 3.32E þ 05
7.37E þ 06 8.85E þ06 7.04E þ 06 8.38E þ06
2.091 (Gazetas [36]) 1.963 (this study) 2.106 (Gazetas[36]) 1.990 (this study)
20
106
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
Since the proposed analytical equation is more complex, the influence of each parameter cannot be evaluated in an immediate manner. The sensitivity analysis shows in Fig. 8b the influence of the parameter x2 on z given heq/b¼ 4 and μ¼0.60. Even in this case, the increment of the SSI effect on the damping increment is more pronounced for lower values of x2, i.e. for deep bedrock and/or for small foundations. Such variation is more evident for low values of x2 and becomes rapidly less important as x2 increases. The effect of x3 on z is presented in Fig. 9b, given b/Hdep ¼0.5 and μ¼ 0.60. The damping increment is more evident for tall/ slender structures (higher values of x3). Given 1/s, the parameter x4 plays an important role in the evaluation of damping increase (Fig. 10). The sensitivity analysis shows higher damping values with increasing x4, i.e. as the mass of
the foundation increases and tends to equal the mass of the structure. This effect is more pronounced for higher values of 1/s. Once the fundamental vibrational period T~ and the viscous damping ratio ξ~ had been calculated by means of the analytical formulations, the SDF response in terms of S~ e , and S~ De , were evaluated using the elastic acceleration spectra of EC8, Type 1 [50]. As expected, the influence of seismic SSI leads to smaller accelerations in the structure and its foundation. On the other hand the increase of natural period leads to higher relative displacements, which may cause an increase in the seismic demand associated with P–Δ effects. In Fig. 11a results are presented in terms of computed and estimated pseudo-spectral acceleration ratio ðS~ e =Se Þ, (i.e. between SSI-SDF systems and SDF with fixed-base); in Fig. 11b the comparison between computed and estimated pseudo-spectral displacement ratio ðS~ De =SDe Þ, is presented. As shown in Fig. 11a, there is a good agreement between estimated and computed values. The average difference is 77.4% over the whole range of analysis. Finally good agreement is also found in Fig. 11b. The average difference is 75.7% over the whole range of analysis.
4. Conclusions
Fig. 10. Influence of x4 (mb/meq) on ξξ.
Fig. 11. Estimated versus computed values of : (a) spectral acceleration ration
Seismic SSI includes several complex effects. Of these, period lengthening and damping increase are considered particularly significant for the structural response and they were analysed in the present paper. The present paper provides a practical tool for obtaining an approximate but reliable estimate of the SSI effects in terms of modified damping and period of ordinary shear-type buildings with surface squared foundations. A sensitivity analysis to evaluate the influence of nondimensional parameters on period lengthening and effective damping was carried out. It was shown that the wave parameter (1/s) is the parameter which mostly affects fundamental period modifications. The slenderness ratio (heq/b) also plays a significant role, while the mass ratio (mb/meq) does not influence period increase. On the other hand, the effective damping exhibits remarkable influence on all the non-dimensional parameters. Results are presented as a function of 1/s. The obtained empirical relations were finally used to estimate the ratios between the pseudo-spectral acceleration and displacement of
~ Se Se
and (b) spectral displacement ratio
~ S De SDe
of SSI and fixed-base SDF.
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
the SSI systems and the corresponding values of the fixed-base systems (S~ e =Se and S~ De =SDe respectively). Comparisons with the analogous values computed by applying steps 1 to 3 of the above described procedure show good agreement. The analytical expressions proposed in this paper represent the best compromise between simplicity and ability to estimate the fundamental parameters governing seismic SSI. The outcomes of the present research can be used by consultants in order to obtain a preliminary estimate of seismic SSI effects without devoting resources to complex analyses. References [1] Veletsos AS, Wei YT. Lateral and rocking vibration of footings. Journal of the Soil Mechanics and Foundations Division 1971;97(SM9):1227–48. [2] Jennings PC, Bielak J. Dynamics of building-soil interaction. Bulletin of the Seismological Society of America 1973;63(1):9–48. [3] Luco JE. Impedance functions of a rigid foundation on a layered medium. Nuclear Engineering and Design 1974;31:204–17. [4] Bielak J. Dynamic behavior of structures with embedded foundations. Earthquake Engineering and Structural Dynamics 1975;3:259–74. [5] Veletsos AS, Nair VV. Seismic interaction of structures on hysteretic foundations. Journal of Structural Engineering, 101101; 109–29. [6] A.S. Veletsos, A.M. Prasad, Y. Tang. Design approaches for soil-structure interaction, National center for earthquake engineering research. Technical report NCEER-88-0031; 1988. [7] Gazetas G. Formulas and charts for impedances of surface and embedded foundations. Journal of Geotechnical Engineering 1991;117(9):1363–81. [8] Wolf JP. Foundation vibration analysis using simple physical models. Englewood Cliffs: Prentice-Hall; 1985. [9] Paolucci R, Shirato M, Yilmaz T. Seismic behaviour of shallow foundations: Shaking table experiments vs numerical modelling. Earthquake Engineering and Structural Dynamics 2008;37:577–95. [10] Paolucci R. Simplified evaluation of earthquake-induced permanent displacements of shallow foundations. Journal of Earthquake Engineering 1997;13:563–79. [11] Nova R, Montrasio L. Settlement of shallow foundations on sand. Geotechnique 1991;41(2):243–56. [12] Cremer C, Pecker A, Davenne L. Modeling of nonlinear dynamic behaviour of a shallow strip foundation with macro-element. Journal of Earthquake Engineering 2002;6(2):175–212. [13] Gazetas G, Apostolou M, Anastasopoulos IX. Seiemic uplifting of foundations on soft soil, with examples from Adapazari. In: Foundations: innovations, observations, design, and practice, british geotechnical association. Thomas Telford; 37–50. [14] Psycharis I. Dynamic behaviour of rocking structures allowed to uplift. Journal of Earthquake Engineering and Structural Dynamics 1983;11. [15] Chopra A, Yim S. Simplified earthquake analysis of structures with foundation uplift. Journal of Structural Engineering 1985:906–30. [16] Maugeri M, Musumeci G, Novità D, Taylor CA. Shaking table test of failure of a shallow foundation subjected to an eccentric load. Soil Dynamics and Earthquake Engineering 2000;20:435–44. [17] Abate G, Massimino MR, Maugeri M, Muir Wood D. Numerical modelling of a shaking table test for soil-foundation-superstructure interaction by means of a soil constitutive model implemented in a fem code. Geotechnical and Geological Engineering Journal 2010;28(1):37–59. [18] Saouma V, Miura F, Lebon G, Yagome Y. A simplified 3D model for soilstructure interaction with radiation damping and free field input. Bulletin of Earthquake Engineering 2011;9(5):1387–402. [19] Rayhani MT, El Naggar MH. Physical and numerical modeling of seismic soilstructure interaction in layered soils. Geotechnical and Geological Engineering Journal 2012;30(2):331–42. [20] ATC-3-06. Amended tentative provisions for the development of seismic regulations for buildings. ATC Publication ATC 3-06, NBS Special Publication 510, NSF Publication 78-8, Applied Technology Council. US Government Printing Office: Washington, DC; 1978. [21] Building seismic safety council, NEHRP. Recommended provisions for the development of seismic regulations for new buildings (and other structures). National earthquake hazards reduction programme: Washington, DC; 1985, 1988, 1991, 1994 (1997, 2000). [22] Mylonakis G, Gazetas G. Seismic soil-structure interaction: beneficial or detrimental? Journal of Earthquake Engineering 2000;4:277–301.
107
[23] Wu WH, Smith HA. Efficient modal analysis for structures with soil-structure interaction. Earthquake Engineering and Structural Dynamics 1995;24:283–99. [24] Wu WH. Equivalent fixed-base models for soil-structure interaction systems. Soil Dynamics and Earthquake Engineering 1997;16:323–36. [25] Wolf JP. Spring-dashpot-mass models for foundation vibrations. Earthquake Engineering and Structural Dynamics 1997;26:931–49. [26] Aviles J, Perez-Rocha LE. Effect of foundation embedment during building-soil structure interaction. Earthquake Engineering and Structural Dynamics 1998;27:1523–40. [27] Stewart JP, Seed RB, Fenves GL. Seismic soil-structure interaction in buildings. II: empirical findings. Journal of Geotechnical and Geoenvironmental Engineering 1999;125(1):38–48. [28] Ambrosini RD, Riera JD, Danesi RF. On the influence of foundation flexibility on the seismic response of structures. Computers and Geotechnics 2000;27:179–97. [29] Aviles J, Suarez M. Effective periods and dampings of building-foundation systems including seismic wave effects. Engineering Structures 2002;24:553–62. [30] Takewaki I, Takeda N, Uetani K. Fast practical evaluation of soil structure interaction of embedded structures. Soil Dynamics and Earthquake Engineering 2003;23:195–202. [31] Barcena A, Esteva L. Influence of dynamic soil-structure interaction on the nonlinear response and seismic reliability of multistory systems. Earthquake Engineering and Structural Dynamics 2007;36:327–46. [32] Chatterjee P, Basu B. Some analytical results on lateral dynamic stiffness for footings supported on hysteretic soil medium. Soil Dynamics and Earthquake Engineering 2008;28(1):36–43. [33] Pitilakis D, Dietz M, Wood DM, Clouteau D, Modaressi A. Numerical simulation of dynamic soil-structure interaction in shaking table testing. Soil Dynamics and Earthquake Engineering 2008;28(6):453–67. [34] Livaoglu R. Investigation of seismic behavior of fluid-rectangular tank-soil/ foundation systems in frequency domain. Soil Dynamics and Earthquake Engineering 2008;28:132–46. [35] Veletsos AS, Meek JW. Dynamic behaviour of building-foundation systems. Earthquake Engineering and Structural Dynamics 1974;3:121–38. [36] Gazetas G. Analysis of machine foundation vibrations: state of the art. International Journal of Soil Dynamics and Earthquake Engineering 1983;2:2–42. [37] Avilés J, Pérez-Rocha LE. Evaluation of interaction effects on the system period and the system damping due to foundation embedment and layer depth. Soil Dynamics and Earthquake Engineering 1996;15:11–27. [38] Mylonakis G, Nikolau S, Gazetas G. Footing under seismic loading: Analysis and design issues with emphasis on bridge foundations. Soil Dynamics and Earthquake Engineering 2006;26:824–53. [39] Veletsos AS. Dynamics of Structure-Foundation Systems. Hall, W.J.: Prentice Hall; 1977 (Structural and Geotechnical Mechanics). [40] G. Maravas, G. Mylonakis, D.L. Karabalis. Dynamic characteristics of simple structures on piles and footings. In: Proceedings of the forth international conference on earthquake geotechnical engineering, Paper no. 1672; 2007. [41] S. Renzi. Influence of dynamic soil-structure interaction analyses on shear buildings [Doctoral Thesis]. Department of Architecture. Germany: Civil Engineering and Environmental Sciences of the Technische Universität Carolo-Wilhelmina zu Braunschweig. Italy: Faculty of Engineering Department of Civil Engineering of the University of Florence ; 2010. [42] Renzi S, Mylonakis G, Madiai C, Vannucchi G. Influence of soil-structure interaction on seismic response of shear buildings. In: Proceedings of the fifth international conference on recent advances in geotechnical earthquake engineering and soil dynamics and symposium in honor of professor IM Idriss, San Diego, CA; 2010. [43] FEMA 368. The 2000 NEHRP Recommended Provisions For New Buildings And Other Structures Part 1:Provisions. NEHRP; 2000. [44] FEMA 369. The 2000 NEHRP Recommended Provisions For New Buildings And Other Structures Part 2:Commentary. NEHRP; 2000. [45] Lysmer J, Ostadan F, Chin C. SASSI2000 Theoretical manual – Revision 1. Geotechnical Engineering Division. Civil Engineering Department – University of California: Berkeley; 1999. [46] Lysmer J, Tabatabaie-Raissi M, Tajirian F, Vahdani S, Ostadan F. SASSI-A System for Analysis of Soil-Structure Interaction. Report N. UCB/GT/81-02, Geotechnical Engineering. University of California: Berkeley; 1981. [47] Chopra A. Dynamics of structures – theory and applications to earthquake engineering. third ed.. New Jersey: Prentice Hall, Englewood Cliffs; 305–36. [48] Canadian commission on building and fires code. The National Building Code of Canada. Ottawa: National Research Council; 2005. [49] Government of the federal district. Complementary technical norms for seismic design. Mexico: Official Gazette of the Federal District; 2004. [50] Eurocode 8: Design of structures for earthquake resistance – Part 5: Foundations, retaining structures and geotechnical aspects.
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
A simplified empirical method for assessing seismic soil-structure interaction effects on ordinary shear-type buildings Stefano Renzi n, Claudia Madiai, Giovanni Vannucchi Università di Firenze, Dipartimento di Ingegneria Civile e Ambientale, 50123 Firenze, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 23 May 2012 Received in revised form 16 September 2013 Accepted 18 September 2013 Available online 8 October 2013
The beneficial or detrimental effect of seismic soil-structure interaction (SSI) is still a controversial issue. A parametric analysis of the seismic SSI effects of a large number of idealised ordinary shear-type buildings was carried out with some simplifying assumptions. Results were compared to the corresponding classical fixed-base solutions. Structures were modelled as generalised single degree of freedom systems using the principle of virtual displacements and shallow squared foundations resting on different soil types were assumed. The outcomes of the numerical analyses were used as a statistical base in order to obtain simple analytical and non-dimensional relationships for estimating seismic SSI effects in terms of modified period and damping. The proposed approximated method can be used by consultants in an immediate and simplified manner in order to obtain a preliminary evaluation of SSI effects and seismic demand without devoting resources to complex analyses. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Seismic soil-structure interaction SSI simplified procedure Modified structural damping and period Modified structural seismic demand
1. Introduction Due to evidence of historical earthquakes, the importance of achieving an acceptable level of safety for ordinary shear-type buildings, as “element at risk”, is undisputed. It is also well known that seismic Soil-Structure Interaction (SSI) can play a relevant role even if the issue is not free from misconceptions. The dynamic response of a structure supported on soft soil may differ substantially in amplitude and frequency content from the response of an identical structure founded on firm ground. The main effects of seismic SSI on buildings with shallow foundations consist of an increase in the fundamental period and damping of the soil-structure system, as evidenced by many researchers [1–8]. In addition, other effects of SSI, such as foundation uplift and sliding at the soil-foundation interface, were analysed by several authors [9–16]. The latter phenomena can occur when seismically induced loads attain a limit value, the soil-foundation system initiates significant yielding and foundation permanent displacements take place, with accumulation of large residual foundation rotations and horizontal displacements. Such effects, particularly associated with strong earthquakes, involve non linearity and will not be considered in this paper.
n
Corresponding author. Tel.: þ 39 347 1945 409. E-mail address: [email protected] (S. Renzi).
0267-7261/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.soildyn.2013.09.012
In literature, different ways for studying SSI have been adopted, i.e. theoretical, physical and numerical modelling [17–19]. Despite extensive research over the past 40 years, there is still controversy regarding the role of fundamental period lengthening and damping increase due to Soil-Structure Interaction (SSI) in the seismic performance of structures founded on soft soil. As a matter of fact, the idealised design spectra of the seismic codes along with such modifications lead invariably to reduced design base shear (Fig. 1). For this reason, neglecting SSI effects is currently being suggested in many seismic codes (e.g. ATC-3-06 [20], NEHRP-97 [21]), as a conservative simplification that supposedly leads to improved safety margins [22], at least for ordinary structures. Moreover, the computational difficulties in evaluating the effects of seismic SSI discourage consultants faced with this task and, as a consequence, these effects are often neglected in engineering practice even though they may be relevant. The effect of seismic SSI has in fact been recognised as being important by many researchers [22–38]. In general, it cannot be neglected and must be evaluated on a case-by-case basis. In the opinion of the authors it should sometimes be opportunely estimated even for ordinary structures, such those investigated in this paper, for the following reasons: (i) with reference to displacement design spectra incorporated in most seismic codes, the effect of seismic SSI may induce excessive displacements to structures (this must be taken into account for example for adjacent structures, Service Limit State analyses, etc.); (ii) the modified seismic behaviour of structures, both in terms of stresses and strains, could in some cases produce a
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
101
parameters of the soil-foundation-superstructure systems and ~ the ratios T~ =T and ξ=ξ.
Fig. 1. Reduction in design base shear due to SSI ( according to NEHRP seismic coad [21]).
detrimental effect on structures, as shown by many documented evidences around the world, e.g. during the Bucharest (1977), Mexico City (1985) and Kobe (1995) earthquakes. Hence, whenever the seismic design is performed using actual or synthetic earthquakes, representative of the local seismicity, the effect of SSI could be detrimental even in terms of pseudo-spectral acceleration. Nevertheless, as mentioned before, rigorous assessment of seismic SSI is not a simple task due to the difficulties associated with the evaluation of kinematic and inertial effects. In the literature some methods are available for the evaluation of the SSI effects. The most currently used methods [2,4,5,8, 26,35,39] provide approximate solutions or charts and require the knowledge of the impedance matrices of the soil-foundation system. An exact solution that does away with the approximations of the previous available approaches has recently been proposed [40], requiring the knowledge of the dynamic impedance matrices. The aim of the present paper is to provide a practical tool, capable of offering a satisfactory trade-off between rigour and simplicity, to obtain an approximate but reliable estimate of SSI effects in terms of modified damping and period of ordinary sheartype structures with surface squared foundations. A wide range of soil-foundation-superstructure systems were examined and simple numerical relationships from statistical analyses of the obtained results are finally proposed. Even if the preliminary determination of dynamic impedance matrices was still required in the method developed in this study, the numerical relationships obtained from the authors can be applied by final users without calculating these matrices.
2. Analyses performed Extensive numerical analyses of simplified superstructurefoundation-soil models [41,42] were performed in order to evaluate SSI effects and summarise the results in equations obtained from statistical analyses. The adopted procedure can be summarised as follows: (1) Computation of soil-foundation impedance matrices. Horizontal and rocking components were numerically evaluated using the computer programme SASSI2000. (2) Idealisation of shear-type buildings. Superstructures were modelled as equivalent SDF systems having a fundamental period T and damping ratio ξ. (3) Estimate of SSI effects. Modified period T~ and damping ratio ξ~ of the analysed soil-foundation-superstructure systems were evaluated by means of a recent exact solution. (4) Statistical analyses of the results from step 3. Analytical relationships were obtained between the main non dimensional
In order to show the effectiveness of such expressions, a systematic comparison between the values of the ratios T~ =T and ~ ξ=ξ estimated by the proposed relationships and computed by applying steps 1 to 3. The SSI effects were analysed using a substructuring approach: the problem is broken down into a series of subsystems which are solved separately then reassembled using the principle of superposition, requiring as assumption soil and structural linear behaviour. A wide range of soil-foundation-superstructure configurations were examined. Shear-type buildings (up to 20 storeys) having a flexural first mode of vibration were modelled as generalised Single Degree of Freedom (SDF) systems, with squared shallow foundations resting on homogeneous subsoil. The analyses were performed under the following simplified assumptions: Regular plan and elevation shear-type buildings were considered. Since their seismic response is not significantly affected by contribution from higher modes of vibration, they were modelled as equivalent SDF systems, whose modified fundamental period was evaluated for flexural mode. Squared raft foundations were analysed; this particular shape combined with a symmetrical superstructure shows the same behaviour in terms of lateral stiffness in both x- and ydirections. Rigid foundations were considered. Homogeneous soil deposit, with constant shear stiffness and damping, was assumed.
2.1. Computation of soil-foundation impedance matrices As suggested by many Codes (e.g. FEMA 368 [43] and FEMA 369 [44]) the compliance of the subsoil was accurately evaluated by Finite Element Method (FEM) using the computer programme SASSI2000 [45,46]. The analyses were performed to estimate the horizontal and rocking foundation impedance matrices; in such preliminary analysis the foundation mass does not play a significant role, as a matter of fact, the numerous existing formula and charts proposed in the related literature [2,4,5,8,26,35,39] always refer to massless foundation. Thus, rigid and massless foundation rafts were modelled by means of 3D brick finite elements. Surface squared raft foundations of characteristic length, b¼5, 12.5 and 25 m, resting on different subsoil configurations, were considered. The subsoil is modelled as a homogeneous viscoelastic halfspace or as a homogeneous viscoelastic horizontal deposit resting on an elastic bedrock. In the latter case the bedrock was assumed at five different depths, Hdep ¼5, 10, 20, 50 m (it should be noted that for the halfspace configuration Hdep ¼ p ). In the viscoelastic halfspace configuration a viscous boundary was assumed, in order to account for radiation damping in the halfspace through the lower boundary and to avoid wave reflections. For the numerical analyses the deposit was subdivided into different sublayers. Each of them was identified by its thickness, unit weight, shear-wave velocity (VS), compressional-wave velocity (VP) and associated damping ratio. In order to evaluate the influence of soil shear-wave velocities on SSI effects, three different types of homogeneous soils were selected for the subsequent analyses, i.e. VS ¼80, 200 and 320 m/s (corresponding to effective shear modulus values of 12, 77 and
102
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
Fig. 3. General 2D model.
Fig. 2. Impedance scheme.
198 MPa respectively for a soil unit weight of 19 kN/m3); soil damping, ξs, was assumed equal to 5% in all cases analysed. Bedrock shear wave velocity was supposed equal to 800 m/s. Interaction between soil and foundation occurs only at their common nodes. The impedance matrices K and C of the rigid and massless foundations were computed from the foundation compliance matrices, whose columns were obtained by applying a unit force or moment in the desired direction at the foundation level (Fig. 2), and computing the resulting real and imaginary part of displacements and rotations of that point. Finally, by inverting the compliance matrices, the corresponding impedance matrices were computed. It is worth mentioning that, since the purpose of numerical analysis is to estimate impedance matrices, no seismic response analysis was performed. Fig. 4. Lateral forces assumed to obtain the shape function from static deflection.
2.2. Idealisation of shear-type buildings A set of shear-type buildings, squared in plane, with a maximum of 20 storeys, each 3 m tall, squared columns having a minimum section of 30 30 cm2, beams having rectangular sections (20 45 cm2), forming 5 m squared frames (Fig. 3) were considered in the analyses. The presence of an elevator was assumed in all the cases examined; such horizontal reinforcement gives rise to an increment in the lateral stiffness of the structure. Different types of loads were taken into account: (i) accidental loads Qk ¼2.00 kN/m2; (ii) floor self-load Gkf ¼ 4.81 kN/m2; (iii) external masonry (thickness 40 cm) Gkm ¼3.70 kN/m2. Such structures were modelled as generalised Single Degree of Freedom (SDF) systems using the principle of virtual displacements. As evidenced in literature [47], the analysis of an equivalent SDF system provides exact results for an assemblage of rigid bodies supported so that it can deflect in only one shape. The approximate natural frequency is shown to depend on the assumed deflected shape The assumed deflection can be related to a single generalised displacement z(t) through a shape function ψ(x) that approximates the fundamental vibration mode. In the analyses the deflected shape due to static forces was adopted as an approximate shape function ψ(x). The lateral force for each floor was assumed in the horizontal direction and equal to wi ¼mii, where mi is the floor mass and g the acceleration of gravity. The fundamental frequency ω0 for a building with j storeys, was estimated from the set of forces shown in Fig. 4, which leads to the following evaluation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ug∑N j ¼ 1 mj uj ð1Þ ω0 ¼ t N ∑j ¼ 1 mj u2j This equation appears in the seismic design provisions of some building codes, e.g. in the 2005 edition of the National Building
Code of Canada (NBCC) [48] and in the 2004 edition of the Mexico Federal District Code (MFDC) [49]. Following the method previously described, each superstructure was modelled as a SDF system, having equivalent height, heq, equivalent mass, meq, and equivalent stiffness, keq, with 5% viscous damping ratio. At this stage of the procedure the foundation mass, mb, was introduced, considering a 1 m thick foundation in all the cases under investigation. As previously mentioned, a linear approach was applied, ignoring the nonlinear interaction effects of the soil-foundation system. 2.3. Estimate of SSI effects The classical approach for elasto-dynamic analysis of SoilStructure Interaction [39,40] aims at replacing the actual structure by an equivalent Single Degree of Freedom (SDF) (see Section 2.2) system supported on a set of frequency-dependent springs and dashpots accounting for the stiffness and damping of the compliant soil-foundation system (see Section 2.1). The structure is described by its stiffness k, mass m, height h, and damping coefficient c (Fig. 5a). The foundation consists of a rigid shallow squared raft of characteristic dimension b (half width) and mass mb resting on a homogeneous, linearly elastic, isotropic halfspace described by a shear modulus Gs, mass density ρs, Poisson's ratio νs, and hysteretic damping ratio ξs. Translational and rotational stiffness, Kx and Kr respectively, of the compliant soil-foundation system, are modelled by a pair of frequency-dependent springs. To ensure uniform units in all stiffness terms, Kr is represented by a translational vertical spring acting at the edge of the foundation (i.e. at a distance b from the centre of the footing). Translational and rotational damping, Cx and Cr respectively, of the compliant soil-foundation system, are modelled by a pair of frequency-dependent dashpots, attached in parallel to the springs,
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
representing energy loss due to hysteretic action and wave radiation in the soil medium. The stiffness and damping terms of both the structure and the soil-foundation system can be condensed in two terms [39,40], K~ ~ The corresponding system is represented by the replaceand C. ment oscillator shown in Fig. 5b. Superstructure-foundation-soil model parameters (Hdep, VS, heq, meq, keq, b and mb) were selected in order to be representative of a wide range of actual systems; SSI analyses were performed on 563 different models. Period lengthening and effective damping of each analysed SSI system were calculated by systematic application of the exact solution recently proposed by Maravas et al. [40] (Eqs. (2) and (3) and compared to the corresponding values of the fixed-base structure. The method contains no approximations in the derivation of ~ of the ~ and effective damping, ξ, the fundamental natural period, ω, system. Furthermore, the exact frequency-varying foundation impedances may be employed. The properties of the replacement oscillator are given by ξx ξr ξ þ þ 2 ð1 þ4ξ2 Þ 2 ð1 þ 4ξ2 Þ 2 ð1 þ 4ξ2 Þ ω ω ω r x r ξ~ ¼ x 1 1 1 þ þ ω2x ð1 þ4ξ2x Þ ω2r ð1 þ 4ξ2r Þ ω2 ð1 þ 4ξ2 Þ
ð2Þ
" ω~ 2 ¼ 1 þ 4
103
2 2 2 ξ~ ξ~ ξ~ þ1þ4 þ1þ4 2 2 2 2 2 ωx ð1 þ 4ξx Þ ωr ð1 þ 4ξr Þ ω ð1 þ4ξ2 Þ
#1
ð3Þ qffiffiffiffiffiffiffi2ffi qffiffiffiffi qffiffiffi i where ωx ¼ Kmx , ωr ¼ K r b2 , ω ¼ mk , ξi ¼ ωC 2K i , (i¼x, r) with ω being mh
the circular excitation frequency.
2.4. Statistical analyses In order to perform the statistical analyses, model parameters and numerical computation results were expressed by means of the following dimensionless quantities: x1 ¼
1 heq ; ¼ s TV S
x2 ¼
b ; H dep
x3 ¼
heq ; b
x4 ¼ μ ¼
mb ; meq
T~ y¼ ; T
z¼
ξ~ ξ
where T and ξ are the fundamental period and damping of the fixed-base structure, while T~ and ξ~ are the fundamental period and damping of the numerically analysed SSI systems. The results obtained were statistically analysed, considering only the most interesting 275 cases that met the following conditions: x2 ¼ Hbdep 4 0 T~ y ¼ 4 1:05 T ξ~ z ¼ 4 1:05 ξ
Fig. 5. (a) Superstructure-foundation-soil idealised by a stick model. (b) Reduced single degree-of-freedom model (after [39,40]).
~ less than 1.05 were not included in Cases with values of T~ =T or ξ=ξ the analysis because they were considered to be of little practical significance. The statistical analyses were performed assuming the parameters y and z as dependent variables and x1, x2, x3, x4 as independent variables. A multiple regression analysis was performed in order to find the best compromise between the simplicity of the equations (minimum number of parameters of the statistical model) and their ability to provide a good estimate of the parameters (high value of R2, symmetrical distribution having mean zero of the residuals ε). A sensitivity analysis was carried out to assess the contribution of each dimensionless parameter on the dependent variables.
Fig. 6. Comparison between estimated and computed values of the ratio (a) y ¼ TT and (b) z ¼ ξξ.
104
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
3. Results In the statistical model the best estimate of the variable y ¼ T~ =T was found to be represented by means of the following equation: y ¼ ax1 m1 x2 m2 x3 m3 þ1 þ ε
ð4Þ
that is lnðy 1 εÞ ¼ m1 lnðx1 Þ þ m2 lnðx2 Þ þm3 lnðx3 Þ þ lnðaÞ
ð5Þ
where the coefficients obtained from the statistical analysis are a ¼ 0:7698;
m1 ¼ 1:663;
m2 ¼ 0:1359;
m3 ¼ 0:8443
the coefficient of determination for the regression (4) is R2 ¼0.937. The comparison between estimated and computed values for the 275 cases considered is shown in Fig. 6a. The proposed analytical expression (5) shows a good agreement between estimated and computed values. The average difference is
74% over the whole range of analysis (1.05ryr2.83), with 90% of cases presenting a difference less than 710%. In particular in the range 1.05 r yr1.40, which is the range of most practical interest for engineering applications, the estimated values differs from the computed values of 7 3% (in average), with 90% of cases presenting a difference less than 75.4%. The residual ε, that is the difference between the value obtained from numerical analysis of the SSI model (computed value) and the value estimated from the proposed relationship (4) (estimated value), is a normally distributed random variable (Fig. 7a) with zero mean and standard deviation SD ¼0.098. It is evident from analysis of the coefficients of Eq. (5) that, as expected, the non-dimensional wave parameter x1 ¼ 1=s ¼ heq =TV S (which may be looked upon as a measure of the relative stiffness of the foundation and the superstructure) plays the most important role on the ratio y ¼ T~ =T. The slenderness ratio x3 ¼ heq =b, also shows a significant influence on period lengthening, while the parameter x2 ¼ b=H dep , representing the foundation
Fig. 7. Histogram for the residuals of the statistical model for: (a) y ¼ TT and (b) z ¼ ξξ.
Fig. 8. Influence of x2 (b/Hdep) on (a)
T T
and (b) ξξ.
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
105
As an example some significant comparisons are shown in Table 1 for the case 1/s equal to 0.437. The differences in terms of T~ =T are about 6%. ~ The variable z ¼ ξ=ξ was estimated by means of a more complex equation
characteristic length compared to the bedrock depth, has little influence on this parameter. The mass ratio x4 ¼ μ ¼ mb =meq does not have any influence. In order to assess the contribution of each dimensionless parameter on the estimate of y, a sensitivity analysis was done. In Fig. 8a the influence of the parameter x2 on y is presented given heq/b¼4 and μ¼ 0.60. As mentioned before, this parameter has little influence in particular for low values of 1/s (say less than 0.2) and becomes more important for higher values. The increment of the SSI effect on the fundamental period is more pronounced for lower values of x2, i.e. for deep bedrock and/ or for small foundations. Such variation is more evident as 1/s increases: i.e. given 1/s, the decrement of y with x2 is constant and shows a linear increment as 1/s increases. The effect of x3 on y is presented in Fig. 9a, given b/Hdep ¼0.5 and μ¼0.60. The increment of the fundamental period shows an exponential increment for higher values of x3, i.e. tall and slender buildings. Since the parameter x4 does not play any role in the evaluation of period increase, the sensitivity analysis for this parameter was not performed. The validity of all the steps of the method have been checked with existing solutions. In particular the dynamic impedance matrices obtained in the present paper were compared with the solution proposed by Gazetas ([36]); some differences were found, both in static and dynamic parts.
z ¼ ðm1 x4 þ m2 x3 Þx1 2 þ ðm3 x4 þm4 x3 þ m5 x2 Þx1 þ 1 þ ε
ð6Þ
that is ðz 1Þ ¼ m1 x4 x1 2 þm2 x3 x1 2 þ m3 x4 x1 þm4 x3 x1 þ m5 x2 x1 þ ε
ð7Þ
where the coefficients obtained from the statistical analysis are m1 ¼ 11:457; m2 ¼ 1:017; m3 ¼ 1:660; m4 ¼ 0:9558; m5 ¼ 4:3048 The coefficient of determination for the regression (6) is R2 ¼ 0.948. The comparison between the values obtained from the numerical models (computed values) and the values estimated from the proposed relationship (6) (estimated values) for the 275 cases considered is shown in Fig. 6b. The proposed analytical expression (7) shows a satisfactory agreement between estimated and computed values, even if in this case data are more scattered. The average difference is 714% over the whole range of analysis (1.05 ryr3.88), with 90% of cases presenting a difference less than 725%. The residual ε is a normally distributed random variable (Fig. 7b) with zero mean and standard deviation SD ¼0.286.
Fig. 9. Influence of x3 (heq/b) on (a)
T T
and (b)
ξ ξ
Table 1 Comparison with existing solutions. SDF parameters
Soil and foundation data
Results 3
heq [m]
keq [kN/m]
meq [Mg]
ξ [%]
T [s]
b [m]
VS [m/s]
γ [kN/m ]
ν []
Hdep [m]
Kx [kN/m]
Kr [kNm]
T~ T
20
57030
472.1
5
0.572
5
80
19
0.3
10
2.11E þ 05 4.06E þ 05 2.93E þ 05 3.32E þ 05
7.37E þ 06 8.85E þ06 7.04E þ 06 8.38E þ06
2.091 (Gazetas [36]) 1.963 (this study) 2.106 (Gazetas[36]) 1.990 (this study)
20
106
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
Since the proposed analytical equation is more complex, the influence of each parameter cannot be evaluated in an immediate manner. The sensitivity analysis shows in Fig. 8b the influence of the parameter x2 on z given heq/b¼ 4 and μ¼0.60. Even in this case, the increment of the SSI effect on the damping increment is more pronounced for lower values of x2, i.e. for deep bedrock and/or for small foundations. Such variation is more evident for low values of x2 and becomes rapidly less important as x2 increases. The effect of x3 on z is presented in Fig. 9b, given b/Hdep ¼0.5 and μ¼ 0.60. The damping increment is more evident for tall/ slender structures (higher values of x3). Given 1/s, the parameter x4 plays an important role in the evaluation of damping increase (Fig. 10). The sensitivity analysis shows higher damping values with increasing x4, i.e. as the mass of
the foundation increases and tends to equal the mass of the structure. This effect is more pronounced for higher values of 1/s. Once the fundamental vibrational period T~ and the viscous damping ratio ξ~ had been calculated by means of the analytical formulations, the SDF response in terms of S~ e , and S~ De , were evaluated using the elastic acceleration spectra of EC8, Type 1 [50]. As expected, the influence of seismic SSI leads to smaller accelerations in the structure and its foundation. On the other hand the increase of natural period leads to higher relative displacements, which may cause an increase in the seismic demand associated with P–Δ effects. In Fig. 11a results are presented in terms of computed and estimated pseudo-spectral acceleration ratio ðS~ e =Se Þ, (i.e. between SSI-SDF systems and SDF with fixed-base); in Fig. 11b the comparison between computed and estimated pseudo-spectral displacement ratio ðS~ De =SDe Þ, is presented. As shown in Fig. 11a, there is a good agreement between estimated and computed values. The average difference is 77.4% over the whole range of analysis. Finally good agreement is also found in Fig. 11b. The average difference is 75.7% over the whole range of analysis.
4. Conclusions
Fig. 10. Influence of x4 (mb/meq) on ξξ.
Fig. 11. Estimated versus computed values of : (a) spectral acceleration ration
Seismic SSI includes several complex effects. Of these, period lengthening and damping increase are considered particularly significant for the structural response and they were analysed in the present paper. The present paper provides a practical tool for obtaining an approximate but reliable estimate of the SSI effects in terms of modified damping and period of ordinary shear-type buildings with surface squared foundations. A sensitivity analysis to evaluate the influence of nondimensional parameters on period lengthening and effective damping was carried out. It was shown that the wave parameter (1/s) is the parameter which mostly affects fundamental period modifications. The slenderness ratio (heq/b) also plays a significant role, while the mass ratio (mb/meq) does not influence period increase. On the other hand, the effective damping exhibits remarkable influence on all the non-dimensional parameters. Results are presented as a function of 1/s. The obtained empirical relations were finally used to estimate the ratios between the pseudo-spectral acceleration and displacement of
~ Se Se
and (b) spectral displacement ratio
~ S De SDe
of SSI and fixed-base SDF.
S. Renzi et al. / Soil Dynamics and Earthquake Engineering 55 (2013) 100–107
the SSI systems and the corresponding values of the fixed-base systems (S~ e =Se and S~ De =SDe respectively). Comparisons with the analogous values computed by applying steps 1 to 3 of the above described procedure show good agreement. The analytical expressions proposed in this paper represent the best compromise between simplicity and ability to estimate the fundamental parameters governing seismic SSI. The outcomes of the present research can be used by consultants in order to obtain a preliminary estimate of seismic SSI effects without devoting resources to complex analyses. References [1] Veletsos AS, Wei YT. Lateral and rocking vibration of footings. Journal of the Soil Mechanics and Foundations Division 1971;97(SM9):1227–48. [2] Jennings PC, Bielak J. Dynamics of building-soil interaction. Bulletin of the Seismological Society of America 1973;63(1):9–48. [3] Luco JE. Impedance functions of a rigid foundation on a layered medium. Nuclear Engineering and Design 1974;31:204–17. [4] Bielak J. Dynamic behavior of structures with embedded foundations. Earthquake Engineering and Structural Dynamics 1975;3:259–74. [5] Veletsos AS, Nair VV. Seismic interaction of structures on hysteretic foundations. Journal of Structural Engineering, 101101; 109–29. [6] A.S. Veletsos, A.M. Prasad, Y. Tang. Design approaches for soil-structure interaction, National center for earthquake engineering research. Technical report NCEER-88-0031; 1988. [7] Gazetas G. Formulas and charts for impedances of surface and embedded foundations. Journal of Geotechnical Engineering 1991;117(9):1363–81. [8] Wolf JP. Foundation vibration analysis using simple physical models. Englewood Cliffs: Prentice-Hall; 1985. [9] Paolucci R, Shirato M, Yilmaz T. Seismic behaviour of shallow foundations: Shaking table experiments vs numerical modelling. Earthquake Engineering and Structural Dynamics 2008;37:577–95. [10] Paolucci R. Simplified evaluation of earthquake-induced permanent displacements of shallow foundations. Journal of Earthquake Engineering 1997;13:563–79. [11] Nova R, Montrasio L. Settlement of shallow foundations on sand. Geotechnique 1991;41(2):243–56. [12] Cremer C, Pecker A, Davenne L. Modeling of nonlinear dynamic behaviour of a shallow strip foundation with macro-element. Journal of Earthquake Engineering 2002;6(2):175–212. [13] Gazetas G, Apostolou M, Anastasopoulos IX. Seiemic uplifting of foundations on soft soil, with examples from Adapazari. In: Foundations: innovations, observations, design, and practice, british geotechnical association. Thomas Telford; 37–50. [14] Psycharis I. Dynamic behaviour of rocking structures allowed to uplift. Journal of Earthquake Engineering and Structural Dynamics 1983;11. [15] Chopra A, Yim S. Simplified earthquake analysis of structures with foundation uplift. Journal of Structural Engineering 1985:906–30. [16] Maugeri M, Musumeci G, Novità D, Taylor CA. Shaking table test of failure of a shallow foundation subjected to an eccentric load. Soil Dynamics and Earthquake Engineering 2000;20:435–44. [17] Abate G, Massimino MR, Maugeri M, Muir Wood D. Numerical modelling of a shaking table test for soil-foundation-superstructure interaction by means of a soil constitutive model implemented in a fem code. Geotechnical and Geological Engineering Journal 2010;28(1):37–59. [18] Saouma V, Miura F, Lebon G, Yagome Y. A simplified 3D model for soilstructure interaction with radiation damping and free field input. Bulletin of Earthquake Engineering 2011;9(5):1387–402. [19] Rayhani MT, El Naggar MH. Physical and numerical modeling of seismic soilstructure interaction in layered soils. Geotechnical and Geological Engineering Journal 2012;30(2):331–42. [20] ATC-3-06. Amended tentative provisions for the development of seismic regulations for buildings. ATC Publication ATC 3-06, NBS Special Publication 510, NSF Publication 78-8, Applied Technology Council. US Government Printing Office: Washington, DC; 1978. [21] Building seismic safety council, NEHRP. Recommended provisions for the development of seismic regulations for new buildings (and other structures). National earthquake hazards reduction programme: Washington, DC; 1985, 1988, 1991, 1994 (1997, 2000). [22] Mylonakis G, Gazetas G. Seismic soil-structure interaction: beneficial or detrimental? Journal of Earthquake Engineering 2000;4:277–301.
107
[23] Wu WH, Smith HA. Efficient modal analysis for structures with soil-structure interaction. Earthquake Engineering and Structural Dynamics 1995;24:283–99. [24] Wu WH. Equivalent fixed-base models for soil-structure interaction systems. Soil Dynamics and Earthquake Engineering 1997;16:323–36. [25] Wolf JP. Spring-dashpot-mass models for foundation vibrations. Earthquake Engineering and Structural Dynamics 1997;26:931–49. [26] Aviles J, Perez-Rocha LE. Effect of foundation embedment during building-soil structure interaction. Earthquake Engineering and Structural Dynamics 1998;27:1523–40. [27] Stewart JP, Seed RB, Fenves GL. Seismic soil-structure interaction in buildings. II: empirical findings. Journal of Geotechnical and Geoenvironmental Engineering 1999;125(1):38–48. [28] Ambrosini RD, Riera JD, Danesi RF. On the influence of foundation flexibility on the seismic response of structures. Computers and Geotechnics 2000;27:179–97. [29] Aviles J, Suarez M. Effective periods and dampings of building-foundation systems including seismic wave effects. Engineering Structures 2002;24:553–62. [30] Takewaki I, Takeda N, Uetani K. Fast practical evaluation of soil structure interaction of embedded structures. Soil Dynamics and Earthquake Engineering 2003;23:195–202. [31] Barcena A, Esteva L. Influence of dynamic soil-structure interaction on the nonlinear response and seismic reliability of multistory systems. Earthquake Engineering and Structural Dynamics 2007;36:327–46. [32] Chatterjee P, Basu B. Some analytical results on lateral dynamic stiffness for footings supported on hysteretic soil medium. Soil Dynamics and Earthquake Engineering 2008;28(1):36–43. [33] Pitilakis D, Dietz M, Wood DM, Clouteau D, Modaressi A. Numerical simulation of dynamic soil-structure interaction in shaking table testing. Soil Dynamics and Earthquake Engineering 2008;28(6):453–67. [34] Livaoglu R. Investigation of seismic behavior of fluid-rectangular tank-soil/ foundation systems in frequency domain. Soil Dynamics and Earthquake Engineering 2008;28:132–46. [35] Veletsos AS, Meek JW. Dynamic behaviour of building-foundation systems. Earthquake Engineering and Structural Dynamics 1974;3:121–38. [36] Gazetas G. Analysis of machine foundation vibrations: state of the art. International Journal of Soil Dynamics and Earthquake Engineering 1983;2:2–42. [37] Avilés J, Pérez-Rocha LE. Evaluation of interaction effects on the system period and the system damping due to foundation embedment and layer depth. Soil Dynamics and Earthquake Engineering 1996;15:11–27. [38] Mylonakis G, Nikolau S, Gazetas G. Footing under seismic loading: Analysis and design issues with emphasis on bridge foundations. Soil Dynamics and Earthquake Engineering 2006;26:824–53. [39] Veletsos AS. Dynamics of Structure-Foundation Systems. Hall, W.J.: Prentice Hall; 1977 (Structural and Geotechnical Mechanics). [40] G. Maravas, G. Mylonakis, D.L. Karabalis. Dynamic characteristics of simple structures on piles and footings. In: Proceedings of the forth international conference on earthquake geotechnical engineering, Paper no. 1672; 2007. [41] S. Renzi. Influence of dynamic soil-structure interaction analyses on shear buildings [Doctoral Thesis]. Department of Architecture. Germany: Civil Engineering and Environmental Sciences of the Technische Universität Carolo-Wilhelmina zu Braunschweig. Italy: Faculty of Engineering Department of Civil Engineering of the University of Florence ; 2010. [42] Renzi S, Mylonakis G, Madiai C, Vannucchi G. Influence of soil-structure interaction on seismic response of shear buildings. In: Proceedings of the fifth international conference on recent advances in geotechnical earthquake engineering and soil dynamics and symposium in honor of professor IM Idriss, San Diego, CA; 2010. [43] FEMA 368. The 2000 NEHRP Recommended Provisions For New Buildings And Other Structures Part 1:Provisions. NEHRP; 2000. [44] FEMA 369. The 2000 NEHRP Recommended Provisions For New Buildings And Other Structures Part 2:Commentary. NEHRP; 2000. [45] Lysmer J, Ostadan F, Chin C. SASSI2000 Theoretical manual – Revision 1. Geotechnical Engineering Division. Civil Engineering Department – University of California: Berkeley; 1999. [46] Lysmer J, Tabatabaie-Raissi M, Tajirian F, Vahdani S, Ostadan F. SASSI-A System for Analysis of Soil-Structure Interaction. Report N. UCB/GT/81-02, Geotechnical Engineering. University of California: Berkeley; 1981. [47] Chopra A. Dynamics of structures – theory and applications to earthquake engineering. third ed.. New Jersey: Prentice Hall, Englewood Cliffs; 305–36. [48] Canadian commission on building and fires code. The National Building Code of Canada. Ottawa: National Research Council; 2005. [49] Government of the federal district. Complementary technical norms for seismic design. Mexico: Official Gazette of the Federal District; 2004. [50] Eurocode 8: Design of structures for earthquake resistance – Part 5: Foundations, retaining structures and geotechnical aspects.