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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2009; 38:423–437 Published online 17 November 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.858 The effect of foundation embedment on inelastic response of structures Mojtaba Mahsuli§ and M. Ali Ghannad∗,†,‡ Department of Civil Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran SUMMARY In this research, a parametric study is carried out on the effect of soil–structure interaction on the ductility and strength demand of buildings with embedded foundation. Both kinematic interaction (KI) and inertial interaction effects are considered. The sub-structure method is used in which the structure is modeled by a simplified single degree of freedom system with idealized bilinear behavior. Besides, the soil sub-structure is considered as a homogeneous half-space and is modeled by a discrete model based on the concept of cone models. The foundation is modeled as a rigid cylinder embedded in the soil with different embedment ratios. The soil–structure system is then analyzed subjected to a suit of 24 selected accelerograms recorded on alluvium deposits. An extensive parametric study is performed for a wide range of the introduced non-dimensional key parameters, which control the problem. It is concluded that foundation embedment may increase the structural demands for slender buildings especially for the case of relatively soft soils. However, the increase in ductility demands may not be significant for shallow foundations with embedment depth to radius of foundation ratios up to one. Comparing the results with and without inclusion of KI reveals that the rocking input motion due to KI plays the main role in this phenomenon. Copyright q 2008 John Wiley & Sons, Ltd. Received 16 June 2008; Revised 24 August 2008; Accepted 3 September 2008 KEY WORDS: soil–structure interaction; foundation embedment; kinematic interaction; ductility demand; cone models 1. INTRODUCTION It is well known that the flexibility of soil beneath the structure affects the response of the structure due to soil–structure interaction (SSI). This phenomenon has two main effects. First, the difference between stiffness of the foundation and the surrounding soil causes the motion experienced by the essentially rigid foundation, i.e. the foundation input motion (FIM), to differ from the free-field ∗Correspondence to: M. Ali Ghannad, Department of Civil Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran. †E-mail: ghannad@sharif.edu ‡Associate Professor. §Now, Ph.D. student at The University of British Columbia, Vancouver, BC, Canada V6T 1Z4. Copyright q 2008 John Wiley & Sons, Ltd. 424 M. MAHSULI AND M. A. GHANNAD motion (FFM). This happens even if the foundation has no mass. In other words, the FIM is the result of geometric averaging of the seismic input motion in the free field [1]. This phenomenon is called the kinematic interaction (KI) effect. Second, the flexibility of soil affects the response of the structure subjected to FIM. In fact, the soil–structure system behaves as a new system with different dynamic properties, i.e. longer natural period and usually higher damping. This is usually called inertial interaction (II) effect. The general effects of SSI on elastic response of structures were the subject of numerous studies in 1970s [2–8]. More specifically, the pioneering works by Veletsos and his co-workers [6–8] led to the tentative provisions in ATC3-06 [9], which is in fact the basis for modern regulations on seismic design of soil–structure systems [10]. The document was prepared mainly based on the findings for surface foundations and the KI effect was ignored. The effect of foundation embedment was introduced through simplified modification factors to adjust the soil dynamic stiffness. Detailed researches on embedded foundations were also conducted during almost the same period. The dynamic stiffness of embedded foundations was studied by Beredugo and Novak [11] and Elsabee et al. [12]. Also, approximate solutions were suggested by Wolf and Somaini [13]. Morray [14] studied the KI problem of embedded circular foundations parametrically for a varied range of parameters typically found in nuclear reactor design. Luco et al. [15] pointed to the influence of rocking input motion due to KI effect on the response of structures with embedded foundation. Pais and Kausel [16] performed a detailed investigation on KI for circular and rectangular embedded foundations subjected to out-of-plane shear wave using the approximation introduced by Iguchi [17]. They also concluded that the embedment usually results in a reduction in translational responses while inducing significant rocking motion, especially for vertically propagating waves. Bielak [18] also pointed to the importance of this rocking input motion for structures with deep embedment ratios. The general effect of the foundation embedment on the structural response through simplified methods was also studied by Bielak [19], Kausel et al. [20] and more recently by Aviles and Perez-Rocha [21] and Takewaki et al. [22] among others. Almost all aforementioned researches were conducted on linear soil–structure systems. The SSI effect on the response of nonlinear structures, however, was not studied in detail. Examples of early works are those made by Veletsos and Verbic [23], Bielak [18] and Muller and Keintzel [24]. In recent years, by development of performance-based design philosophy, the SSI effect on inelastic response of structures has attracted much more attention [25–30]. However, most of the researches are focused on surface foundations, which incorporate no KI effect. Moreover, it is generally believed by researchers that total KI effect is somehow beneficial for the structure [31, 32]. The NEHRP provisions [10] on SSI also ignore the KI effect. Besides, FEMA 440 [33] supports this idea implicitly by just paying attention to the reducing effect of KI due to base slab averaging and foundation embedment. However, it seems that despite the reducing effect of KI on the translational component of the FIM, the resulting rocking component may increase the structural demands especially for soil–structure systems with deep embedded foundations [34]. Here, the subject is studied parametrically for a set of non-dimensional parameters, which well define the whole considered soil–structure system. 2. SOIL–STRUCTURE MODEL Figure 1(a) illustrates the soil–structure system considered in this study in which the soil beneath the structure is considered as a homogeneous half-space. The super-structure is modeled as a single degree of freedom (SDOF) system with height h, mass m and mass moment of inertia I , Copyright q 2008 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:423–437 DOI: 10.1002/eqe EFFECT OF FOUNDATION EMBEDMENT 425 Figure 1. (a) The soil–structure system; (b) the basic soil–structure model; and (c) soil–structure model with soil material damping. which may be considered to be the effective values for the first mode of vibration of a real multi degree of freedom system. This is, in fact, the basic model used by current design codes and seismic performance evaluation regulations [10, 33]. The foundation is considered to be rigid with embedment depth e and mass and mass moment of inertia m f and I f , respectively. The basic model for such a soil–structure system is shown in Figure 1(b) in which the structure is replaced by an elasto-plastic spring along with a dashpot with coefficients k and c, respectively. The soil is considered to be in full contact with the foundation and is modeled as a discrete model based on the concept of cone models for embedded foundations [35]. Two degrees of freedom (DOF) are introduced in this model for the foundation, namely, sway, u f , and rocking, �. Consequently, by considering an additional internal DOF for the soil model, �1, a 4-DOF model is formed for the whole soil–structure system as shown in Figure 1(b). The internal DOF allows the frequency dependency of the soil stiffness also to be taken into account while all the coefficients in the model are frequency independent. These coefficients are defined as follows: k0h = 8�v 2 s r 2−� ( 1+ e r ) , c0h = r vs �0hk0h (1) k0r = 8�v 2 s r 3 3(1−�) [ 1+2.3e r +0.58 (e r )3] , c0r = r vs �0r kr (2) c1r = r vs �1r kr , I1r = ( r vs )2 �1r kr (3) where �, �, vs and r are the specific mass, Poisson’s ratio, shear wave velocity of the soil and the radius of the cylindrical foundation, respectively. Besides, �0h,�0r , �1r and �1r are non-dimensional coefficients of the discrete model in terms of e/r and are calculated using the following formulae: �0h = 0.68+0.57 √ e r (4a) �0r = 0.15631 e r −0.08906 (e r )2−0.00874(e r )3 (4b) Copyright q 2008 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:423–437 DOI: 10.1002/eqe 426 M. MAHSULI AND M. A. GHANNAD �1r = 0.4+0.03 (e r )2 (5a) �1r = 0.33+0.1 (e r )2 (5b) Sway springs and dashpots are connected to the super-structure model with the following eccen- tricities in order to account for the coupling terms between the sway and rocking DOFs in the stiffness matrix of the embedded foundation: fk = 0.25e (6a) fc = 0.32e+0.03e (e r )2 (6b) Using the basic soil–structure model in Figure 1(b), the more complex model of Figure 1(c) can be generated to consider the soil material damping as well. In this model, every spring and dashpot in the basic model is augmented with a dashpot and a mass, respectively [35]. Elasto-plastic behavior is considered for the structure while all the soil representative springs behave elastically. The effect of soil nonlinearity, however, may be approximately introduced into the model through equivalent linear approach in which a degraded shear wave velocity, consistent with the estimated strain level in soil, is used for the soil medium [36]. This is the same approach used in NEHRP 2003 [10] and FEMA 440 [33] where the strain level in soil is implicitly related to the peak ground acceleration. The model of Figure 1(c) is used directly in a time domain analysis subjected to sway and rocking components of FIM, ug and �g , as shown in the figure. Details on the computation of these components are introduced in the following sections. 3. KEY PARAMETERS Basically, the response of the soil–structure system depends on the size of the structure, its dynamic characteristics and the soil profile as well as the applied excitation. It is shown that the effect of these factors can be best described by the following non-dimensional parameters [8, 37]: 1. A non-dimensional frequency as an index for the structure to soil stiffness ratio defined as: a0 = �h vS (7) where � is the natural circular frequency of the fixed-base structure. It can be shown that the practical range of a0 for ordinary building-type structures is from zero for the fixed-base structure to about two for cases with predominant SSI effect [28, 29]. 2. Aspect ratio of the building defined as h/r . 3. Embedment ratio of the foundation defined as e/r . 4. Ductility demand of the structure defined as: �= um uy (8) where um and uy are the maximum displacement caused by a specific base excitation and the yield displacement of the structural stiffness, respectively. Copyright q 2008 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:423–437 DOI: 10.1002/eqe EFFECT OF FOUNDATION EMBEDMENT 427 5. Structure to soil mass ratio index defined as: m¯ = m �r2h (9) 6. The ratio of the mass of the foundation to that of the structure defined as m f /m. 7. Poisson’s ratio of the soil indicated by �. 8. Material damping ratios of the soil and the structure indicated by �0 and �s , respectively. The first three items are the key parameters that define the principal SSI effect including both KI and II. The fourth one controls the level of nonlinearity in the structure. The other parameters, however, are of less importance and may be set to some typical values for ordinary buildings [37]. Here, the following values are assigned to these parameters: m¯ =0.5, m f /m =0.1, �=0.25, �0 =0.05, �s =0.05 (10) 4. KINEMATIC INTERACTION Generally, as a result of KI, two different FIM components are produced: horizontal FIM, ug , and rocking FIM, �g . Horizontal FIM component generally decreases in comparison with FFM especially for more embedment depths; however, rocking FIM amplitude has a crescendo as the depth of embedment increases. These components are in fact the input motion to the soil–structure model introduced in Figure 1(c). Here, the method proposed by Meek and Wolf [38] is used to evaluate FIM components. The method is based on the concept of double-cone models. Double cones are used to represent a disk embedded in a full space. An embedded foundation is then replaced by a stack of N disks commencing from the lowermost point of the foundation, e, and continuing to the ground surface. In order to provide stress-free conditions on the ground surface, another stack of N disks, which are the mirror images of the former disks, are considered on the other side of the ground surface as demonstrated in Figure 2. These mirror image disks are excited by the same excitations as the original disks; therefore, stress-free conditions on the ground surface will be guaranteed. By using the green functions at the level of each disk and its mirror image, the N × N flexibility matrix of the free field is evaluated. The inverse of this flexibility matrix is the dynamic stiffness matrix of the free field, S f . Now, by extracting the excavated part of the soil from the model and inserting the rigid foundation, the dynamic stiffness of the embedded foundation can be evaluated. By virtue of the rigidity of the inserted foundation, the dimension of the stiffness matrix is reduced from N to 2 for the introduced sway-rocking foundation model. This can be accomplished by utilizing an N ×2 kinematic conditions matrix, which is calculated based on the foundation geometry. Thus, the dynamic stiffness matrix of the rigid foundation, Sg , is calculated as follows: Sg =ATS f A+�2M (11) in which A and M are the above-mentioned kinematic conditions matrix and the mass matrix of the excavated part of the soil, respectively. Subsequently, the FIM vector is evaluated using the following equation: ug =Sg−1ATS f u f (12) Copyright q 2008 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:423–437 DOI: 10.1002/eqe 428 M. MAHSULI AND M. A. GHANNAD Figure 2. Model of the embedded foundation comprising stack of N disks and their mirror image. where u f is the N ×1 vector of the FFM evaluated at the level of the disks and ug is the 2×1 vector of the FIM comprising the two components of the sway and rocking motions: ug = [ ug �g ] (13) In deriving Equation (12), use is made of the following relationship between the dynamic stiffness and motion of the free-field state as well as those of the foundation: ATS f u f =Sgug (14) 5. METHOD OF ANALYSIS The introduced soil–structure model has the capability of being used directly in a time domain analysis. Here, inelastic response of soil–structure systems subjected to 24 free-field strong motions recorded on alluvium deposits are studied using the step by step integration by the Newmark method. Details of the selected ground motions are listed in Table I. For each ground motion, a group of 2500 different soil–structure systems covering 100 different fixed-base periods and a wide range of defined key parameters are considered. This includes systems with three values of non-dimensional frequency (a0 =0,1,2), three values of aspect ratio (h/r =1,3,5) and four values of embedment ratio (e/r =0,0.5,1,2). It should be mentioned that though buildings with aspect ratio h/r =5 or embedment ratio e/r =2 are rare in practice, they are used here as extreme cases to show the trends and possible effects. The response of each system is investigated both with and without inclusion of KI effect. In each analysis, the structure is analyzed to reach the ductility levels of 2 and 6 and an analysis for the elastic case is done as well. For any given case, the yield force of the structure, Fy , is calculated by iteration in order to reach the target ductility in the structure, as a part of the soil–structure system, within 1% of accuracy. A certain procedure is followed to investigate the SSI effect on the ductility demand of the structure. For each soil–structure model, first the yield strength demand of the super-structure in the fixed-base state, i.e. ignoring the soil effect, is calculated to reach a specific ductility level when Copyright q 2008 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:423–437 DOI: 10.1002/eqe EFFECT OF FOUNDATION EMBEDMENT 429 Table I. Selected ground motions recorded at alluvium sites. Station Geology Earthquake date Magnitude Epicentral distance (km) Component PGA (g) El Centro— irrigation distinct Alluvium Imperial Valley, 18 May 1940 6.3 (ML ) 8 S90W, S00E 0.21, 0.31 Taft—Lincoln school tunnel Alluvium Kern County, 21 July 1952 7.7 (MS) 56 308, 218 0.15, 0.18 Figueroa—445 Figueroa St. Alluvium San Fernando, 9 February 1971 6.5 (ML ) 41 N52E, S38W 0.14, 0.12 Ave. of the Stars—1901 Ave. of the Stars Silt and sand layers San Fernando, 9 February 1971 6.5 (ML ) 38 N46W, S44W 0.14, 0.15 Meloland— Interstate 8 overpass Alluvium Imperial Valley, 15 October 1979 6.6 (ML ) 21 36, 270 0.31, 0.30 Bond Corner— Highways 98 and 115 Alluvium Imperial Valley, 15 October 1979 6.6 (ML ) 3 14, 230 0.51, 0.78 Alhambra— Freemont school Alluvium Whitter-Narrows, 1 October 1987 6.1 (ML ) 7 27, 180 0.41, 0.30 Altadena—Eaton Canyon park Alluvium Whitter-Narrows, 1 October 1987 6.1 (ML ) 13 90, 360 0.15, 0.30 Burbank— California Federal Saving Building Alluvium Whitter-Narrows, 1 October 1987 6.1 (ML ) 26 250, 340 0.23, 0.19 Los Angeles— Baldwin Hills Alluvium over shale Whitter-Narrows, 1 October 1987 6.1 (ML ) 27 90, 360 0.06, 0.13 Capitola—Fire Station Alluvium Loma Prieta, 17 October 1989 7.1 (MS) 9 90, 360 0.44, 0.53 Hollister—South and Pine Alluvium Loma Prieta, 17 October 1989 7.1 (MS) 48 90, 180 0.25, 0.21 subjected to a specified free-field ground motion. Then, the ductility demand of the structure, as a part of the soil–structure system, is calculated for soil–structure systems with different values of a0, h/r and e/r , providing the same yield strength for the structure as calculated in the fixed-base state. As the input motion to the soil–structure system, both the FFM and the resulted FIM are used in order to investigate the SSI effect with and without KI. In each case, the difference between the ductility demand of the fixed-base model and that of the structure as a part of the soil–structure system reflects the problem that does exists in conventional design methodology, i.e. the difference between our expectation of structural behavior as a fixed-base model and the way that structures behave in reality when located on flexible soil. The procedure described above is summarized as follows: 1. Select a free-field ground motion. 2. Consider a soil–structure system with a specific set of non-dimensional frequency, a0, aspect ratio, h/r and embedment ratio, e/r . Copyright q 2008 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:423–437 DOI: 10.1002/eqe 430 M. MAHSULI AND M. A. GHANNAD 3. Select a target ductility demand for the fixed-base structure. 4. Select the natural period of the fixed-base structure. 5. Analyze the fixed-base structure subjected to the selected ground motion and calculate the resulting yield strength demand, Fy , to reach the target ductility demand with 1% tolerance. 6. Analyze the soil–structure system with the same yield strength