The effect of foundation embedment on net horizontal foundation input motion: The case of strip foundation with incomplete contact to nearby medium

Soil Dynamics and Earthquake Engineering 96 (2017) 35–48

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The effect of foundation embedment on net horizontal foundation input motion: The case of strip foundation with incomplete contact to nearby medium

MARK

⁎

Hossein Jahankhaha, , Pouran Fallahzadeh Farashahib a b

International Institute of Earthquake Engineering and Seismology, Iran International Institute of Earthquake Engineering and Seismology, Iran

A R T I C L E I N F O

A BS T RAC T

Keywords: Kinematic interaction Embedded foundation Incomplete contact to nearby medium Response spectra Peak acceleration

This investigation is coordinated for the case of strip embedded foundation with incomplete contact to surrounding medium to uncover some aspects of kinematic interaction. The extracted results of finite element analyses are presented in three stages. First, horizontal and rotational foundation responses besides a recently introduced net horizontal foundation input motion are studied extensively. Then, the difference between net horizontal foundation input motion and free field motion is discussed here. In addition, the affected response spectra by considering such net horizontal input motion are illustrated. The results indicate that the role of frequency content of records to take kinematic interaction effects becomes highlighted for the case of incomplete contact with respect to fully bounding state. Also, kinematic interaction has considerable role in amplifying peak accelerations of net horizontal foundation input motion and related response spectra for specific cases. Such amplification is notable for the case of slender structures rested on foundations with zero contact of side walls to surrounding soil. This intensification would sometimes move traditionally designed systems, in which incomplete contact between soil and subterranean walls is ignored, to unconservative states.

1. Introduction Seismic vulnerability assessment of structures is necessary to be considered especially in earthquake prone areas. On the other hand, the compliant soil beneath supper structure may cause prominent phenomenon known as soil-structure interaction (SSI) [1,2]. This phenomenon which has attracted the attention of researchers for several decades, consists of two main parts, i.e., inertial interaction (II) and kinematic interaction (KI) [3]. The former issue accounts for adjusting period and damping of the system and the latter changes the frequency content of free field motion (FFM) [4] and hence generates new foundation input motions (FIMs). Regarding the importance of this issue, many researchers have focused on this phenomenon in recent decades. As one of the earliest work in this field, Novak et.al [5] presented an approximate analytical approach to investigate the dynamic response of embedded footing. The mentioned approach was compared to finite element solutions and also experimental results to validate its applicability. Also, Luco et.al [6] investigated the dynamic response of rigid embedded foundation under non-vertically propagating shear waves and mentioned that generated rotational and torsional motions of foundation are outcomes of embedment depth and ⁎

non-vertical incident wave fields. Wong and Luco [7] investigated the input motion variation of rectangular surface foundation under obliquely propagating waves, and tabulated ordinates of FIM for different values of foundation aspect ratios, material damping factors, Poisson ratios, and incident wave angles. The reported results indicate that the seismic wave incidence angle has significant effect on FIM whereas the effects of material damping and Poisson ratio are negligible. It is worthy to note that the effect of foundation aspect ratio on translational motion is reported to be small. That's while torsional component takes effect considerably. In addition, Kuasel et.al [8] investigated the seismic response of embedded foundation by presenting approximate spring method and compared the results to exact direct one. Moreover, Iguchi [9] proposed an approximate approach to estimate input motion of rigid embedded foundation. He explored the effect of embedment depth and incident wave angle on input motion to cylindrical foundation. With respect to the reported graphs, growing embedment depth causes decreasing in horizontal and increasing in rotational components of input motion. Pais and Kausel [10] studied dynamic response of massless foundation widely by using Iguchi's [9] approximate approach. They reported components of FIM for different foundation shapes, i.e. cylindrical and rectangular configurations, and also various

Corresponding author. E-mail addresses: [email protected] (H. Jahankhah), [email protected] (P.F. Farashahi).

http://dx.doi.org/10.1016/j.soildyn.2017.02.015 Received 5 March 2016; Received in revised form 25 February 2017; Accepted 28 February 2017 0267-7261/ © 2017 Elsevier Ltd. All rights reserved.

Soil Dynamics and Earthquake Engineering 96 (2017) 35–48

H. Jahankhah, P.F. Farashahi

that effective period is insensitive to KI whereas system damping is modified as result of this interaction phenomenon. It should be mentioned that building standard codes [36–38] address the KI effects on input motion and dynamic response of structures by proposing spectral reduction coefficients.

embedment depths besides wave incoherency. Although investigating KI aspects within frequency domain is more common, some researchers were conducted a branch of studies in time space. For example, a time domain method, i.e. a combination of boundary element and finite element method (BEM-FEM), was proposed by Karablis and Beskos [11] to assess the dynamic behavior of flexible surface foundation subjected to obliquely incident seismic wave. The accuracy of proposed method was illustrated by various benchmarks. In another study, Gitanaros and karabalis [12] discussed the behavior of flexible embedded foundation by BEM-FEM method in frequency domain and compared the outcomes with the rigid cases. Also, a hybrid approach was proposed by Mita and Luco [13,14] and Wong [15], to indicate the dynamic response of embedded foundation with arbitrary shapes. Validation of the mentioned approach was examined by comparing to well-known methods. Iguchi and Takeda [16], in their research stream on KI, illustrated graphically the response of circular plate under Rayleigh wave. Veletsos and Prasad [17,18] presented the simple approximate closed-form formulation to express the effect of incoherent seismic waves on the response of circular and rectangular massless foundations. Another research was conducted by Kokkicons and Spyrakos [19] in frequency domain via hybrid FEM-BEM to investigate the dynamic behavior of flexible surface strip-foundation. Several numerical examples were applied to attest the accuracy of results. Iguchi conducted shaking table tests for 19 sets of ground motions to assess FIM [20]. Observations showed good agreement with numerically calculated input motions presented previously by the same author [9]. Further, Seunghyun and Stewart [21] made recommendations to modify FFM, using 29 recorded earthquakes by focusing on the surface and shallow foundations. Also, Stewart and Tileylioglu [22] proposed a model for tall buildings by addressing KI and its impact on the horizontal and rotational input motions. Spyrakos and Chaojin [23] discussed parametrically the seismic response of massive flexible strip foundation embedded in layered soil. Extracted results are compared to result of rigid foundation to demonstrate the efficiency of method. Moreover, the effect of foundation embedment on inelastic structural response was discussed by Mahsuli and Ghannad [24]. These authors tried to clarify the significant impact of rotational component on structural behavior, especially for tall and slender buildings. Further, Mylonokis, et al. [25,26] investigated the response of bridge and footing supporting bridge piers under different type of input motion excitation with respect to KI by simple engineering model. In addition, Winkler model was introduced by Gerolymos, et al. [27,28] to assess kinematically the behavior of bridge piers based on caisson foundation. The outcomes demonstrated the reliability of the proposed model by comparing to FEM results. Two simple methods were proposed by Mori and Fukuwa [29] to predict FIM. The results were validated through comparing with those extracted by available analytical methods. Moreover, Givens and Stewart [30] presented a semi-empirical model and compared obtained empirical kinematically transfer functions (TFs) with real recorded data. However, the proposed model predicted input motions conservatively in low frequencies, while underestimations were observed for the rest. Regarding the effect of rocking input motion on structural seismic design and complicated estimation of this issue, Jahankhah, et al. [31] introduced a modified translational input motion, called as Net Horizontal Foundation Input Motion (NH-FIM), with sufficient accuracy to replace both translational and rotational components. A new three dimensional finite element model was described by Torabi and Rayhani [32] to scan the dynamic behavior of soil structure systems. The results were crosschecked using centrifuge experiments [33] as benchmarks and clarified that FIM was affected by soil plasticity, structure aspect ratio, and structure-foundation stiffness ratio. Although the available solutions mostly consider the influence of KI on seismic input motions, Aviles and Prez-Rocha [34,35], in a different approach, investigated the effect of foundation embedment depth on dynamic characteristics of soilstructure system, i.e. effective period and damping. Results elucidate

2. State of the problem Although several researches were conducted to investigate the KI phenomenon, some notable points, in spite of their citation in the literature are overlooked till now. Rocking component of FIM which increases due to the embedment depth is one of those issues which has not attracted enough attention from designers. Moreover, building standard codes address just subtractive effects of KI whereas intensifications are ignored within the desired frequency range [36–38]. Also, it should be stated that generally horizontal and rocking components of FIM are studied separately whilst considering simultaneous effects of these two is necessary to define the overall acceleration history imposed on structural mass. Different phase between horizontal and rocking components which has not been recognized in the related articles would be the cause of complicated alternations in FFM. It would be discussed later that how would ignoring this fact lead to erroneous estimation of demands. The above mentioned deficiencies in FIM estimations are numerated by looking around problems which include foundations fully contacted to nearby medium. Additionally, incomplete contact length of foundation sidewall to surrounding soil, as an important parameter, has been taken hardly into account in this area of research. This issue, which practically occurs due to different methods of foundation construction and sometimes lack of soil compaction around the foundation, has been regarded just in a few articles to estimate foundation impedance functions [39–41]. Introducing such partially contact state to the problem expands the domain of formerly mentioned shortcomings in current knowledge, respectively. This investigation on FIM was conducted in two preliminary and advance stages. In the former stage, altered peak acceleration with respect to KI was inspected. This is another significant point which has not been discussed comprehensively yet. In general practice, calculated peak ground acceleration (PGA) without any change is sometimes applied in design procedures. That's while input peak acceleration would be different from PGA as a matter of KI. Ignoring KI effects in this field is not always conservative and may lead to unsafe seismic design. In the advance stage, the KI effect on response spectra would be investigated. This latter stage can provide a general influence map through which one could realize the period range of structures, which would encounter changes in response because of KI phenomenon. NHFIM is a horizontal input that can substitute both horizontal and rocking foundation input motions together [31]. The concept of this substitutive input motion is represented in Fig. 1. In this figure, part (a) shows a single degree of freedom structure with an effective mass of m, rested on a massless embedded foundation under upcoming SV shear wave field of motion. In this part, D is the embedment depth, d is the contact length, a is the half-width of the foundation and h is the hypothetical effective height of structure. Also a control point 'O' is introduced at the base of the foundation which all FIMs are reported for throughout this text. In addition, different components of displacement are shown in subsequent parts with refer to the nonmoving point 'R'. In Part (b), the kinematically induced input motions at the level of structural mass are displayed separately. This phase includes just horizontal and rotational rigid movements to the massless structurefoundation system, i.e. u0 and ϕ0(h+D), respectively. As shown in part (c), such movements would impose a history of acceleration on structural mass with an amount of [u 0̈ + ϕ0̈ × (h + D )] which in turn produce excessive displacement and rotation in foundation, i.e. uf and ϕf, and also an inter-structure deformation of us. In the concept of NHFIM, as shown in part (d), sway and rocking motions of part(b) are 36

Soil Dynamics and Earthquake Engineering 96 (2017) 35–48

H. Jahankhah, P.F. Farashahi

R m h

a

D

d

O

(a)

SV u0+ uf (φ0 + φf)(h+D)

u0 φ0(h+D)

us R

R

R

u0+ uf+φ0 (h+D) φf (h+D) us

u0+φ0(h+D) R

φ0 + φf

(b)

O

(c)

φf

O

(d)

O

(e)

O

Fig. 1. The concept of NH-FIM: (a) Soil- structure system, (b) Horizontal and rotational components of input motion at the level of structural mass, (c) Effective load and different parts of total deformation at the level of structural mass, (d) Substitutive NH-FIM, (e) Effective load and new arrangement of total deformation components under NH-FIM at the level of structural mass.

when it is expressed as acceleration time history. To clarify the results of this part, peak horizontal and rotational input accelerations, i.e. PHIA and PRIA, are reported and discussed separately. In the third and final step, response spectra aspects of KI are investigated. It should be noted that it is a common approach to separate the KI and II features of SSI in response spectra by introducing distinct correction coefficients to adjust FFM response spectrum [36–38]. Here just KI induced alternations in response spectrum is investigated.

replaced with a single translational component with an amplitude of u0+ϕ0(h+D). By this substitution it is expected that the input acceleration history, experienced by structural mass, holds unchanged with respect to part(c). Hence alike inertial force and subsequently same total response may be recorded in the system. The expected sub-parts of system total deformation are shown in part (e). In fact, this latter modified motion would provide more reasonable overall judgments on the role of KI in system input motion. It should be noted that the single degree of freedom system in Fig. 1, could be assumed as a replacement for the first mode of every multi-degree of freedom super structure. Hence NH-FIM is suitable to use in the case of systems for which the first mode of vibration can sufficiently reflect the overall response ordinates of the system. Another point is that, since the state of having rigid embedded foundation rested on linear elastic medium and bounding the response in the range of small deformations holds valid, NH-FIM can properly express the imposed acceleration on structural mass. This fact maintains sound regardless of superstructure behavior [42]. Thus soil-foundation linearity and also ignoring second order effects are two other pre assumptions behind the above concept. In this article, KI effects on FIMs of the strip embedded foundation are investigated. The foundation is considered as a rigid inclusion with permission to have incomplete contact between sidewall and surrounding soil. The field of motion includes shear waves with upward propagation direction. The results are examined in three steps. At first kinematically induced FIMs are presented in frequency domain. The TFs for horizontal, rocking, and NH-FIM are discussed separately. Secondly, peak net horizontal input acceleration (PNHIA) is reported as well, which would be compared with the well-known parameter, PGA. This latter parameter is considered as peak value of NH-FIM

3. Method of analysis This section describes the method used in this article to investigate the KI effects on FIM, PNHIA, and structural response spectra. 1. An instant pulse is applied at the base of the model. Then horizontal and rocking time histories of foundation response, under aforesaid excitation, are extracted. This step is conducted in the absence and presence of foundation. 2. Second step deals with calculating TFs. As mentioned above, three components of motions will be presented in this study. Two of them correspond to translational and rocking motions at similar degrees of freedom attributed to point 'O' of Fig. 1a). The third is the NHFIM that the two first can be replaced by it. For this purpose and as the initial phase, fourier transforms of translational FIMs, calculated in step 1 at point 'O', is divided by similar ordinates from FFMs. Hence, horizontal TFs regarding KI effects are derived. The same procedure is then followed for rotational TFs with additionally multiplying results by (h+D). This latter scaling would interpret the results as the translation which will be imposed to center of mass due to rocking input motion. Finally, the NH-FIM TFs are obtained

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Soil Dynamics and Earthquake Engineering 96 (2017) 35–48

H. Jahankhah, P.F. Farashahi

by combination of these two, through algebraic summation. 3. A set of twenty FFM real records is selected. 4. The time history of records is converted to frequency domain. 5. By product of the NH-FIM TFs from step 2 and each Fourier transform of records from step 4, the frequency domain representation of KI induced FIMs, from ensemble ground motion, is achieved. 6. The frequency domain FIMs of step 5 are converted to time domain. 7. PNHIAs, from step 6, are scaled by PGA of respective FFM. 8. Response spectra of the NH-FIM records, derived in step 6, are extracted and normalized to response spectra of the FFM records to obtain spectral magnification coefficients as a result of KI effects. 4. Modelling and validation Finite element method [43] was utilized to analyse the two dimensional model numerically. Applying appropriate absorbing boundary conditions is a prerequisite to conduct numerical analysis. So, here, the free field soil columns are utilized at sides to simulate 1D propagation of shear wave. Those columns are joined the main middle part of the model by viscous elements [44]. Furthermore, viscoelastic boundaries are added at the base of the model to create an integrated transmitting boundary condition in the perimeter. Although the absorbing boundary elements at the base and sides of the model are common in current practice, their well-known performance deficiencies may unfavourably reduce results accuracy [43]. One of those deficiencies is the limited capability to transmit outgoing waves; especially surface waves. To overcome the problem, the main model size is arranged so that before fading foundation response, the unwanted reflected waves from boundaries do not appear at the foundation position. The total effective response time is set to 25 s. This allows a proper frequency resolution in target TFs as frequency increment size in TFs just depends on total time of the record. Fig. 2 illustrates a sample deformed shape at an arbitrary time where the model is excited with an input pulse at the base of the model. It should be noted that results are reported in non-dimensional format in this study, thus they would be applicable for every set of soil characteristics with ʋ=0.3 and different states on the premise that the dimensionless parameters fall within the ranges reported in the following sections. In this research, four values of 0.5, 1, 1.5, and 2 are considered for embedment depth ratio, D/a. The values 0.5 and 2 could be considered as illustrative of shallowly and deeply embedded foundations, correspondingly. In addition, side-wall contact length ratio to soil, d/D, is presented as another dimensionless parameter. This parameter varies between 0, as a representative of sidewall - nearby soil contactless condition, to 1 which shows the complete contact situation. Two values of 0.25 and 0.5 are also included within the mentioned range as well. Further, h/a introduced as the hypothetical aspect ratio of structure is considered here. 1 and 4 are two values assumed for this dimensionless parameter that are representatives of squat and slender structures respectively.

Fig. 3. Comparing results from this study to that of Benchmarks [45]: (a) Horizontal dynamic stiffness and damping, (b) Rocking dynamic stiffness and damping, (c) Coupled dynamic stiffness and damping.

Before extracting mentioned TFs, it would be necessary to verify the reliability of the above introduced model. For this aim, the impedance functions along with the kinematically induced FIM components for the case of full contact state are compared to the previous findings which are defined as the benchmarks in this field. Specific state with D/ a=1, and d/D=1 was considered for model validation in Fig. 3. The model dimensions, mesh size and soil characteristics are reported in Table 1. However, as stated before, because of dimensionless representation of results, the graphs are not restricted to the set of parameters shown in Table 1. Parts (a), (b) and (c) of Fig. 3 depict the calculated impedance functions of this research against those reported by Wang and Rajapakse [45] using boundary element method for horizontal, rocking, and coupled terms, respectively. The vertical axis of the mentioned figures shows dynamic stiffness and damping which are normalized to soil shear modulus, G, and half-width of the foundation, a. In this figure Kh, KR and KRh are horizontal, rotational and coupled terms in frequency dependent stiffness matrix. Also Ch, CR and CRh are similar terms in damping matrix. Moreover, horizontal axis shows nondimensional frequency of excitation, a0=ω×a/Vs, where ω is frequency of excitation and Vs is shear wave velocity of the medium. As can be seen in the whole frequency range, in spite of using different methods, a reasonable match is achieved for both stiffness and damping

Fig. 2. Deformed shape of the model at an instant arbitrary time under impulse loading.

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H. Jahankhah, P.F. Farashahi

For all presented TFs in Fig. 6, a general attenuating trend can be observed by increasing dimensionless frequency a0, from starting point zero. After that, fluctuations occur and also in a few states TFs may reach an amount of unity or more. In addition, these plots illustrate that horizontal FIM is sensitive to variation of embedment depth in a way that make the lower and upper bounds of the results diminish and rise respectively. Also, as shown in these curves, contact length of foundation to soil has significant impact on TFs and cause different trends for full and partial contact conditions in the studied frequency range. For example, in low frequencies and before a distinctive threshold for each contact length, partial contact decreases the amplitude of TFs. However, opposite trend is observed beyond this limit where horizontal foundation input motion approaches to FFM again. This tendency is amplified for the case of zero contact length ratios. Further, the mentioned threshold diminishes as the contact length descends and embedment ratio increases. As an example, the mentioned verges are shown for the case of d/D=0.25 by star labels. Although the movement of the star along the abscissa seems to be short, its impact on the effective frequency content of the record is notable. This is due to the fact that the ordinates of the above curves for values of a0 less than 3 are more influential on FIM. To make this point clear suppose a system with an amount of Vs/a more than 30, which is a practical assumption. Such state along with a0 < 3 would cover frequencies of excitation less than 15 Hz. This frequency range enfolds natural frequencies of almost all common soil-structure systems. Another important point is that for specific frequency ranges the ordinates take values beyond unity. This takes place especially for the case of zero contact length. However, such trend has not been reported in the literature of KI for the case of fully bounded foundations to surrounding soil. This phenomenon can be interpreted through findings in the field of racking ratio of open cavities under upcoming shear waves [46,47]. This coefficient is introduced as the ratio of shear deformation of an open cavity to that of the same region in free field. It is known that this ratio takes values more than unity [48]. On the other hand, as the contact length of side walls tends to zero, the rigidity constraint weakens and foundation-soil system approaches to an open cavity. Hence, stronger horizontal FIM with respect to FFM could be justified as well. Fig. 7 depicts the TF's of kinematically induced rotational component of the foundation scaled to (h+D) and normalized by FFM. Again the results are sketched for various contact lengths of foundation to soil against a0. The figure also consists of results belonging to different values of D/a and d/D. In addition, two rows of graphs are depicted in Fig. 7, e.g. (a), (b), corresponding to two values of slenderness ratio, h/ a=1, 4. The reported TFs in Fig. 7 initiate from zero, increase with frequency to the peak, then get to a valley, and after that rise again. This trend is observed for all rocking components except for the case of D/a=0.5 and d/D=1. When embedment ratio grows, the peak values of the whole TFs increase and appear in lower frequencies. Contact length ratio dictates different trends to rocking input motion. Except for the case of D/a=0.5, for low values of a0, partial contact of side wall, i.e. d/ D=0.5, leads to lower rotational ordinates of input motion in comparison with full and non-contact states. In addition, the lower the contact length, the lower the rotational input motion can be seen for high frequencies in specific embedment depth ratios. Also, the slenderness effects of superstructure on rocking motion component can be followed in separate rows as (a) and (b) in Fig. 7. Variation of rotational FIM for the case of squatty structure, h/a=1, and shallow foundation, D/a=0.5, is not as prominent as that of the slender structure, h/a=4, and deep foundation, D/a=2. The last point of frequency domain representation of results belongs to NH-FIM. According to original description [31], this synthetic component can replace both horizontal and rotational components of FIM with sufficient accuracy for systems in which first mode of super structure dominates the total response. Fig. 8 presents the NH-FIM component for aforesaid ranges of non-dimensional

Table 1 Model and material characteristics. Model mesh size

Foundation depth

Foundation width

Soil density

Soil shear modulus

0.25 m

D= 3 m

2a= 6 m

ρ=1.65 gr/ cm3

G= 1520N/ cm2

Fig. 4. Comparing results from this study to that of Benchmark [10]: (a) Translational foundation input motion, (b) Rotational foundation input motion. (D/a=1, d/D=1).

functions. In addition, Fig. 4(a) and (b) present other validations for the case of horizontal and rocking components of FIM. In this figure the resulting kinematically induced horizontal TFs in current study are examined against corresponding results reported by Pais and Kausel [10]. In part (a), this figure illustrates the normalized horizontal FIM to FFM, i.e. u0/uff, in frequency domain. Also part (b) shows the rotational component of FIM normalized to FFM, i.e. ϕ0*a/uff, versus a0. Comparing the graphs in Fig. 4 illustrates that the model simulation would be suitable enough to reach the scope of this research. The small discrepancy is attributed to limitations of approximate method used in the benchmark reference [10]. This validation was done for the case of strip foundation with ʋ=0.3. To inspect how material damping would affect target TFs, the results from two distinct models, i.e. one with zero and the other with 3% Rayleigh damping, are compared in Fig. 5. The figures show normalized horizontal and rocking FIM's against a0. As can be seen the effect of material damping on KI is minor. Similar remark was previously reported by other researchers [35]. In the following sections, TFs are depicted for 3% Rayleigh damping. 5. Result and discussion Analyses were accomplished parametrically and extracted graphs are reported in three parts. First part covers frequency domain where major aspects of the results are depicted in TF format. Second and third parts review time domain key characteristics of FIMs by inspecting peak input acceleration and response spectra changes, respectively. 5.1. Frequency domain aspects of KI Fig. 6 illustrates the normalized translational component of FIM imposed on rigid embedded strip foundation. This figure expresses the results for various contact lengths of foundation to surrounding soil against a0. The figure also consists of results belonging to different embedment ratios, D/a.

Fig. 5. comparing the horizontal and rotational components with and without considering Rayleigh damping for the case of D/a=1 and d/D=1.

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Soil Dynamics and Earthquake Engineering 96 (2017) 35–48

H. Jahankhah, P.F. Farashahi

Fig. 6. The ordinate ratios of horizontal FIM to FFM in frequency domain arranged by different embedment ratios for various side wall contact lengths to nearby soil.

Fig. 7. The ordinate ratios of rotational FIM to FFM in frequency domain arranged by different embedment ratios for various side wall contact lengths to nearby soil; a) h/a=1, b) h/ a=4.

Fig. 8. The ordinate ratios of NH-FIM to FFM in frequency domain arranged by different embedment ratios for various side wall contact lengths to nearby soil; a) h /a=1, b) h/a=4.

5.2. Time domain characteristics of KI

parameters, D/a, d/D, and h/a against a0. The presented trends in Fig. 8 were expected regarding the observed tendencies of TFs in Figs. 6 and 7 and also different phase of these two. Amplification can be seen for most of foundation depths and contact lengths, in specific frequency ranges which intensify by increasing slenderness ratio as well. This intensification sometimes makes NH-FIM become several times stronger than it's free field counter -part. The worst case can be followed for D/a=2, h/a=4, and d/ D=0 for specific values of a0. In that case, the NH-FIM at structural mass center is about four times higher than free field excitation. Also, de-amplification is seen for shallow to semi-deep foundations, i.e. D/ a=0.5 and 1, especially for the case of full contact length and squatty structure. In addition, it is noteworthy to state that NH-FIM reach zero value for specific cases at particular frequencies, e.g. D/a=1, h/a=4, and d/D=0.

This part will discuss the effect of KI on FFM time histories and related PNHIAs. For this purpose, the KI effects on two selected sample records are investigated. Fig. 9(a) shows the FFM respective time histories, the descriptions of which will later be presented in Table 2. The records, nominated R1 and R2, possess peak accelerations of about 0.31g and 0.25g respectively. The Normalized Fourier transforms of the records are presented in Fig. 9(b). According to this figure, R1 is much richer in high frequencies rather than R2. These two records are used to derive PNHIA for different sets of nondimensional frequencies. For this purpose, in Fig. 9(c) and (d), NH-FIMs resulted from KI are depicted. First row of charts, i.e. Fig. 9(c), relates to D/a=1 and h/a=1 whereas the time histories for systems with D/a=2 and h/a=4 are presented in second row, i.e. Fig. 9(d). In addition, four above

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Soil Dynamics and Earthquake Engineering 96 (2017) 35–48

H. Jahankhah, P.F. Farashahi

(b)

(a)

(c)

D/a=1 h/a=1

(d)

D/a=2 h/a=4

Fig. 9. Effect of KI on two sample FFM real records, Vs/a=10: (a, b) FFM records and respective Fourier transforms, (c) NH-FIMs for the case D/a=1, h/a=1, (d) NH-FIMs for the case D/a=2, h/a=4, and different d/D labeled on the graphs. Table 2 Descriptions of real selected records. Station

Geology

Earthquake Date

Magnitude

Epicentral Distance (km)

Component

PGA (g)

El Centro-Irrigation Distinct Taft _ Lincoln School Tunnel Figueroa _ 445 Figueroa St. Ave. of the stars _ 1901 Ave. of the Stars Meloland_ Interstate 8 Overpass

Alluvium Alluvium Alluvium Silt and Sand Layers Alluvium

Imperial Valley, May 18, 1940 Kern County, July 21, 1952 San Fernando, February 9, 1971 San Fernando, February 9, 1971

6.3(ML) 7.7(MS) 6.5(ML) 6.5(ML)

8 56 41 38

S90W, S00E 308, 218 N52E, S38W N46W, S44W

0.21 0.31(R1) 0.15, 0.18 0.14, 0.12 0.14, 0.15

6.6(ML)

21

360, 270

0.31, 0.30

Bond Corner _ Heighways 98 and 115

Alluvium

6.6(ML)

3

140, 230

0.51, 0.78

Alhambra _ Freemont School

Alluvium

6.1(ML)

7

270, 180

0.41, 0.30

Altadena _ Eaton Canyon Park

Alluvium

6.1(ML)

13

90, 360

0.15, 0.30

Burbank_ California Fedral Saving Building Holister _ South and Pine

Alluvium

Imperial Valley, October 15, 1979 Imperial Valley, October 15, 1979 Whitter_ Narrows, October 1, 1987 Whitter_ Narrows, October 1, 1987 Whitter_ Narrows, October 1, 1987 Loma Prieta, October 17, 1989

6.1(ML)

26

250, 340

0.23, 0.19

7.1(MS

48

90, 180

0.25 (R2)0.21

Alluvium

including the previously introduced records, R1 and R2, is investigated in this research. The descriptions of mentioned records are listed in Table 2. The extracted results are normalized to PGA of respective records. The obtained results are presented in Fig. 10. This figure illustrates the graphs in three separate groups. The first group, i.e. Fig. 10(a), presents PNHIA. The second and third groups, i.e. Fig. 10(b) and (c), enclose PHIA and PRIA, accordingly. In each group a matrix with two rows and four columns of graphs is rendered against different ratios of Vs/a. This latter parameter varies from 10 to 100. In current practice, meeting less values than the Lower band and larger values than the upper bound of this range, would occur in rare occasions. Graphs are reported for various embedment depths and contact length ratios in rows and columns, respectively. Moreover, just aspect ratios of 1 and 4

mentioned values of d/D are discussed for both conditions. As can be seen, R1 is much more affected by KI than R2. This point can be followed by concentration on the variation of PNHIA with respect to PGA. For R1 the PNHIA varies from 0.23 to 0.87. That's while for R2 this range is bounded to values of 0.21 and 0.51. In these cases, the crucial states of 0.87 g and 0.51 g belong to zero side contact, deep foundation and slender structure, i.e. d/D=0, D/a=2 and h/a=4. Hence modification percentage for deep foundation and tall structure, D/a=2 and h/a=4, is considerable particularly for R1. Also, it should be emphasized that unexpectedly in this latter case, the value of PNHIA for partial contact situation, d/D=0.5, is smaller than both full contact and no contact conditions. In order to gain better understanding of PNHIA variations under kinematic interaction, a set of 20 ground motion time histories,

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Fig. 10. (a) PNHIAs normalized to PGAs (b) PHIAs normalized to PGAs (c) PRIAs normalized to PGAs for selected set of records; Results for individual records in grey style, average results in black and average plus/minus standard deviation in red.

accelerations of NH-FIMs are notably sensitive to contact length ratio, especially for the case of D/a=2. So reducing d/D increases PNHIA and causes higher values of input acceleration being attributed to the case with d/D=0. The highest acceleration ratios are observed for the system with D/a=2 and d/D=0. For this case, through almost whole variation range of Vs/a, intensification is notable. There among the ensemble, specific records may reach values four times higher than that of their original states in the free field. This point is surprising as such magnification is far beyond common safety factors. To decode the reasons of such phenomenon, the roles of PHIA and PRIA are reviewed

are considered as illustrative samples for squat and slender structures. In every chart of Fig. 10, 20 normalized graphs, associated with respective records, are depicted in grey style together, firstly. Then the average (μ), and just in Fig. 10(a), average plus and average minus the standard deviation (μ+Ϭ, μ- Ϭ) curves are specified over basic ensemble results by thick black and red graphs respectively. Four values of contact length and also embedment depth ratio pursuant to previous sections are considered in this area. Graphs in Fig. 10(a) reveal that generally the effects of KI on PNHIA amplify as D/a increases. Moreover, results illustrate that peak 42

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h/a=1

h/a=4

Fig. 11. Average values of normalized PNHIA arranged by different embedment ratios for various side wall contact lengths to nearby soil.

h/a=1

h/a=4

Fig. 12. COV of PNHIAs for different embedment ratios and various side-wall contact lengths to nearby soil.

When PHIA and PRIA are combined, the out of phase characteristic of these two leads in the inequality equation of |PNHIA| ≤| PHIA| +|PRIA|. In addition, the following points can be stated about the combination of these two: – As for the case of D/a=1 and h/a=1, the average values of PRIA are less than that of PHIA, in their combination, the latter one plays a more highlighted role. – In contrary, for the cases with D/a=2, h/a=4 and specially d/D=0, 0.25, the ensemble average of PRIA takes values higher than that of PHIA. Hence, in such cases the role of PRIA is more significant than PHIA.

separately. Fig. 10(b) shows the normalized peak horizontal acceleration by peak ground acceleration, i.e. PHIA/PGA, versus Vs/a. The results for different records are depicted in gray, while the average is sketched in black. The following points can be witnessed: – Through an overall view, it can be seen that the ensemble average takes values about unity in the whole range of Vs/a. – For all cases there are some records for which the normalized PHIA exceeds unity. – For cases with d/D=0 and low values of Vs/a, even the average ratios tend to values beyond 1. – The dispersion of results doesn’t differ so much as Vs/a varies. Fig. 10c) show the normalized peak rotational acceleration by peak ground acceleration, i.e. PRIA/PGA, versus Vs/a. Again the results for various records are depicted in gray and the average is sketched in black. The major concluding statements about this figure are listed as bellow: – At the first glance, it can be seen that the dispersion of results is much more than that of Fig. 10b). – For cases with D/a=1, h/a=1, d/D=0, 0.25 and for a number of records, one could find regions where the normalized PRIA experience values beyond 1. Also for cases with D/a=2, h/a=4, all values of d/D and for a number of records, the PRIA reaches levels 3 times of PGA.

Fig. 11 summarizes briefly the averages of normalized PNHIA which are highlighted by black lines in Fig. 10(a). The impact of different embedment depth ratios is presented in this figure where normalized PNHIA by PGA is drawn against Vs/a. Every chart consists of four curves related to four different values of contact length ratio. It should be considered that obtained results are related to both slender and squatty structures, i.e. h/a=4 and h/a=1, respectively. It can be seen that, for each D/a there is a threshold in the abscissa, Vs/a, after which PNHIAs approach to same values. The threshold values increase by growing D/a ratio. The magnification percentage with respect to PGA increases when embedment depth ratio rises and length of sidewall-soil contact reduces.

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(a)

(b)

(c)

D/a=1 h/a=1

(d )

D/a=2 h/a=4

Fig. 13. Effect of KI on response spectra of two sample FFM real records. Results are reported for Vs/a=10 and different values of d/D: (a, b) FFM records and respective response spectra, (c) Normalized acceleration response spectra for the case D/a=1, h/a=1, (d) Normalized acceleration response spectra for the case D/a=2, h/a=4.

Fig. 14. Normalized NH-FIM response spectra for selected set of records; Results for individual records in grey style, average results in black and average plus/minus standard deviation in red. (Vs/a=10).

tioned before, KI would be a source of notable changes in foundation input motions. Hence, considering the KI impacts on response spectra is unavoidable. In this regard, the answer to the question that,”How KI effects would be reflected on response spectra?” will be discussed in this section. First, the two previously introduced sample records, R1 and R2, were utilized to uncover the changed response spectra by considering NH-FIM. Fig. 13 consists of three rows of charts to demonstrate the KI effects on structural response spectra. FFM records and their related response spectra are reported in the first row of Fig. 13 entitled (a) and (b), respectively. As can be seen record R1 possess higher PGA than that of R2. However response spectra of record R2 take higher values for systems with periods longer than one second. In another words, records R1 and R2 are intentionally selected to be rich in short and long periods, correspondingly. Following rows show normalized response spectra of NH-FIM records to FFM, under the influence of KI. The set of system characteristics are labeled on the

Charts of Fig. 12 are expressed to provide deeper understanding of results dispersion. Hence, a well-known statistical parameter, i.e. coefficient of variation (COV), is plotted versus Vs/a. This parameter is defined as the ratio of standard deviation to average value. It should be noted that lower values for COV may be interpreted as lower recorddependency of the results. It means that for lower values of this statistical parameter, average values of results can better be representatives of general trends. As it is shown in Fig. 12, dispersion increases due to the reduction of contact length ratio. The more the embedment depth ratio, the more the COV is observed in this figure as well. Also, it can be seen that the dispersion and hence record dependency of results increases for deeper foundation and higher values of Vs/a. As it is known, seismic response spectra of structures are the most common tools used in safe seismic designs. The derivation of such spectra requires feeding input motions. On the other hand, as men44

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Fig. 15. Average values of normalized NH-FIM response spectra arranged by different foundation embedments and structure aspect ratios for various side wall contact lengths to nearby soil. (Vs/a=10).

Fig. 16. COV of normalized response spectra arranged by different foundation embedments and structure aspect ratios for various side wall contact lengths to nearby soil.

figure. This normalized parameter is assigned as "RS Ratio" on vertical axis of following graphs. In addition, it should be noticed that Vs/a=10 is considered for presented results in Figs. 13 through 16. By considering the trends of modified response spectra it can be found that both records are affected by KI whereas these changes are more obvious for R1. Also it is observed that KI induces amplifying effects on response spectra in specific period ranges except for the case with D/a=1, h/a=1, and contact length ratios of 0.5 and 1. Such magnifications sometimes reach extraordinary levels especially for short contact length of side-wall to neighbor medium. It should be mentioned that this substantive amplification is diminished for higher values of Vs/a as will be discussed in succeeding parts. As expected, for all investigated situations in this part, normalized response spectra converge to unity at long periods. In order to generalize the tendencies of response spectra under KI, the normalized affected response spectra of NH-FIM for mentioned group of 20 records are depicted in Fig. 14. This figure consists of two rows and four columns. The effect of contact length of sidewall-soil is investigated within columns and the rows deal with embedment depths and structure aspect ratios. The results for each record of mentioned group is drawn in grey color whereas average and average plus/ minus

the standard deviation are highlighted by black and red colors respectively. The significant impact of KI on response spectra could be figured out from presented charts in Fig. 14. The trend in altered response spectra for non-contact sidewall state is the most critical among other states. That's while the amplification factor approaches to values about 2 and 4 for squatty and slender structures respectively. The subtractive effect of KI on response spectra can be observed for the case with foundation full contact to soil, in low periods especially for the case of D/a=1 and h/a=1. To comprehend better, average of the results are replotted in Fig. 15. Again, different nondimensional values of contact length, embedment depth, and also structure height are taken into account. As can be seen, shallow foundation with D/a=0.5, involves less effects and approaches to unity in lower periods in comparison with other states. Additionally, zero contact length for this case unexpectedly behaves similar to full contact within short period ranges, i.e. lower than 0.1 s. It should be emphasized that partial contact with d/ D=0.5 results in lower response spectral ratios with respect to complete contact state at periods longer than 0.5 s. The other point is that with larger embedment ratios, a vaster region of spectrum is affected by KI.

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Fig. 17. (a): Effect of KI on of two sample FFM real records and respective response spectra, Vs/a=30: (1) Normalized acceleration response spectra for the case D/a=1, h/a=1, and different values of d/D labeled on the graphs, (2) Normalized acceleration response spectra for the case D/a=2 h/a=4, and different values of d/D labeled on the graphs. (b): Effect of KI on of two sample FFM real records and respective response spectra, Vs/a=50: (1) Normalized acceleration response spectra for the case D/a=1, h/a=1, and different values of d/D labeled on the graphs, (2) Normalized acceleration response spectra for the case D/a=2, h/a=4, and different values of d/D labeled on the graphs.

required to watch the graphs of Fig. 17(a) and (b) with those presented in Fig. 13 together. It can be observed that the fluctuations of normalized spectra at long periods reduce notably when the Vs/a value increases. This trend is shown obviously for the case of zero contact length state with Vs/a=50. In addition, R2 which is introduced as low frequency content record is hardly affected by KI at Vs/a=50 and it behaves like FFM. It is interesting to note that period bands in which the response spectra would be affected by KI become smaller having larger values of Vs/a. However, for the state of D/a=2, h/a=4, and d/ D=0, PNHIA of the system with Vs/a=50 takes larger value in comparison to the cases with Vs/a=30 and 10. This witness points to a fact that the results in short period systems for the noncontact state of side-wall and soil may be substantially higher than that of FFM for larger values of Vs/a. The ensemble average representation of the results would help to draw overall conclusion for higher values of Vs/a. In this regards, Fig. 18(a) and (b) summarize changes in response spectra averages versus period for different depths, contact lengths, and structure slenderness ratios. Presented charts in Fig. 18(a) unfold the spectral trends for Vs/a=30 whereas the results for Vs/a=50 are indicated in Fig. 18(b). By inspecting Fig. 18(a) and (b) it can be found that shallow

This progressive trend, in the worst case, resulted in magnification value of about 3 for period of vibration about 1 s in systems with D/a=2 and h/a=4. In order to show the dispersion of response spectra within the studied period ranges statistical COV, as described before, is plotted against period in Fig. 16. The graphs of this figure are presented for different values of non-dimensional parameters similar to Fig. 15. According to presented graphs, independence of results from record variations can be seen clearly at high periods for the case of full contact. Also, the lower the contact length ratio, the more the fluctuations can be observed in most cases. In other words, when contact length diminishes record-dependency of result increases. It is noteworthy that for low periods the above trend would face some exceptions. In order to expand findings from KI effects on spectral ordinates, two other values of Vs/a are also investigated here. Figs. 17 and 18 deal with the altered response spectra, affected by soil-foundation interaction for the illustrative sets of non-dimensional parameters. The response spectral ratios, based on previously introduced records, R1 and R2, are appeared in Fig. 17(a) and (b) for Vs/a=30 and 50 respectively. In each figure two rows and four columns are presented which cover systems with various nondimensional parameters. To gain better insight into the Vs/a effects on response spectra, it is

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Fig. 18. (a): Average values of normalized NH-FIM response spectra arranged by different foundation embedment depths and structure aspect ratios for various side wall contact lengths to nearby soil. (Vs/a=30), (b): Average values of normalized NH-FIM response spectra arranged by different foundation embedment depths and structure aspect ratios for various side wall contact lengths to nearby soil. (Vs/a=50).

horizontal and rotational degrees of freedom and also a recently introduced Net Horizontal Foundation Input Motion (NH-FIM) with respect to free field motion were estimated. According to original definition, this latter motion can substitute both translational and rocking motions with single excitation. At the next step, a group of real records under the influence of KI was converted to NH-FIMs. The peak accelerations of new excitation histories were explored under the title of peak net horizontal input acceleration (PNHIA) and compared with peak ground acceleration (PGA). Finally, response spectra of altered records were drawn out versus period to show the modified trends when KI is taken into account. Pursuant to presented results, the following main conclusions can be deduced:

foundations with D/a=0.5 and 1 have taken less influence from KI as the Vs/a value increases. Normalized NH-FIM response spectra would approach to unity as the value of Vs/a rises. This point can be concluded with respect to Figs. 16, 18(a) and (b). Although, still amplifications of response spectra can be seen for some cases, deamplification occurs for both squatty and slender structures at short periods with d/D=0.5, 1, D/a=1.5, 2 and both Vs/a=30 and 50. Also, it can be seen that detachment of side-wall from surroundings, brings about the critical state for deep foundations where intensification takes place for both values of Vs/a at short periods. 6. Conclusion

•

The effect of kinematic interaction on foundation input motions, both in frequency and time domain was investigated here. In this regard, a 2D numerical analysis was conducted parametrically for the strip embedded foundation excited by vertical propagating shear wave. This analysis was done by considering variable contact length of sidewall to nearby soil and different embedment depth ratios. It should be noted that the slenderness ratio of hypothetical superstructure was regarded to investigate the input motion changes as well. In order to extract the desired results, three major steps were carried out in this article. First, transfer functions (TFs) of foundation input motions in

• •

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TFs show amplification trends in specific regions of investigated frequency ranges. This can be followed for most of studied cases and a0 less than 5. These amplifications are generally intensified by increasing foundation embedment depth, structural slenderness ratio and also by reducing contact length of foundation to surrounding medium. It is concluded that, in time domain, the PNHIA would substantially be higher than PGA. This would occur when contact length of sidewall to nearby soil tends to zero and also where embedment depth and slenderness ratio increase. The order of difference is sensitive to

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• • •

•

frequency content of free filed records. It is found that response spectra are affected by KI considerably regarding incomplete contact of side-walls to surroundings. Maximum changes on response spectra occur at short periods in the case of zero contact between side-walls and surroundings. For rare cases, e.g. D/a=2, D/d=0.5, h/a=4 and Vs/a=30, possessing partial contact lengths to nearby medium results in lower response spectral ratios with respect to full contact states. However, states with no contact between sidewall and soil, in most cases, make spectral ordinates to reach values several times higher than that of FFM. As another point, it should be stated that the Vs/a values manage the response spectra variations under the influence of KI. This governing role is in such a way that response spectra take less effects from KI phenomenon as Vs/a increases.

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Consequently, KI effects on foundation input motion and response spectra, when the contact of foundation to environs is incomplete, are distinct and ignoring this phenomenon may lead to unsafe seismic design. At last it should be emphasized that the validity of results reported here is restricted by several assumptions among which the system linearity would have the crucial importance. References [1] Housner GW. Interaction of building and ground during an earthquake. Bull Seism Soc Am 1957;47:179–86. [2] Kausel AS. Early history of soil–structure interaction. Soil Dyn Earthq Eng 2010;30:822–32. [3] Wolf JP. Dynamic soil structure interaction. Englewood Cliffs, NJ: Prentice-Hall; 1985. [4] Bielak J. Dynamic behavior of structures with embedded foundations. Earthq Eng Struct Dyn 1974;3:259–74. [5] Novak MD, Beredugo YO. Vertical vibration of embedded footings. J Soil Mech Found Div 1972;98(12):1291–310. [6] Luco JE. Torsional response of structures for SH waves. The case of hemispherical foundations. Bull Seismol Soc Am 1976;66(1):109–23. [7] Wong HL, Luco JE. Dynamic response of rectangular foundations to obliquely incident seismic waves. Earthq Eng Struct Dyn 1978;6:3–16. [8] Kausel E, Whitman RV, Morray JP, Elsabb F. Effects of horizontally travelling waves in soil-structure interaction. Nucl Eng Des 1978;48:377–92. [9] Iguchi M. an approximate analysis of input motion for rigid embedded foundation. Trans Archit Inst Jpn 1982;315:61–75. [10] Pais A, Kausel E. Stochastic response of foundations, Report No. R8506. Cambridge, MA: Massachusetts Institute of Technology; 1985. [11] Karabalis DL, Beskos DE. Dynamic response of 3-D flexible foundations by time domain BEM and FEM. Soil Dyn Earthq Eng 1985;4(2):91–101. [12] Gaitanaros A, Karabalis DL. Dynamic response of 3-D flexible foundations by frequency domain BEM and FEM. Earthq Eng Struct Dyn 1988;16:653–74. [13] Mita A, Luco J. Dynamic response of a square foundation embedded in an elastic half-space. Soil Dyn Earthq Eng 1989;8(2):54–67. [14] Mita A, Luco J. Impedance function and input motion for embedded structures. J Geotech Eng (ASCE) 1989;115(4):491–503. [15] Luco JE, Wong HL. Seismic response of foundations embedded in a layered halfspace. Earthq Eng Struct Dyn 1987;15:233–47. [16] Iguchi M, Takeda M. Seismic response of flexible plate on viscoelastic medium to Rayleigh wave excitation. Proceeding of the Ninth World Conference on Earthquake Engineering. Tokyo, Japan; 1988. [17] Veletsos AS, Prasad AM. Seismic interaction of structures and soils. J Struct Eng (ASCE) 1989;115(4):935–56. [18] Veletsos AS, Prasad AM, Wu WH. Transfer functions for rigid rectangular

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