MODELLING OF SOIL-FOUNDATION-STRUCTURE SYSTEM

Transcription

1 MODELLING OF SOIL-FOUNDATION-STRUCTURE SYSTEM Shrabony Adhikary, Yogendra Singh and D. K. Paul Department of Earthquake Engineering, IIT Roorkee 1. INTRODUCTION The behavior of structures based on compliant soils and subjected to dynamic actions depend to a large extent on the soil and foundation properties. The deformations and stresses in the supporting soil are induced due to the base shear and moment generated in the vibrating structure since in reality the structure is not fixed base. The soil deformations further lead to the modification of the structural response. The dynamic interrelationship where the response of the soil influences the motion of the structure and the response of the structure influences the motion of the soil is called soil-structure interaction. In recent times it has gained the interest of researches and engineers in the field of structural dynamics, soil dynamics wave mechanics all over the world. For the past several years efforts have been made to develop a rational procedure to incorporate SSI in the structural design. 2. SOIL-STRUCTURE INTERACTION AND ITS COMPONENTS There are two primary issues that are related to the mechanism of the soil-foundation structure interaction and it consists of inertial and kinematic interaction. Inertial interaction results from the flexibility of the soil-foundation system. In reality the structure and foundation have mass and when there is acceleration acting on the mass inertial forces will be developed. This inertial force will try to move the soil underneath the structure and when the soil is compliant the forces transmitted to it by the foundation will produce foundation movement. This is not the case for a fixed- base structure. The inertial interaction phenomenon is shown in the Fig. 1. Fig.1 Inertial Interaction Modelling of Soil-Foundation-Structure System/1

2 In the structural analysis model the flexibility of the soil-foundation system should therefore be considered. This is known as flexible foundation effects. This is one of the primary categories for SSI effects. FEMA 356, ATC-40 addresses these effects by considering the stiffness and strength properties of the underlying soil. These effects lead to the increase of vibration period and damping ratio of structures. The flexible base model is shown in Fig. 2. Fig.2 Flexible Base Modeled with Structural and Geotechnical Components of the Foundation Modeled To consider the inertial interaction inertial loading should be applied to the structure. Inertial loading depends on the foundation input motion or the base slab motion. When the structure is assumed to be fixed base the free-field motion acts as the foundation input motion. But the presence of stiff foundation in soil will cause the foundation input motion (base slab motion) to deviate from the free-field motion. Kinematic interaction reduces foundation motions relative to the free-field due to the differences in stiffness between the foundation and surrounding soil. If we consider a rigid block subjected to horizontal component of free field motion then we find that due to its axial stiffness it cannot deform incoherently. The rigid foundations act as a low pass filter by averaging out the high frequency components of the seismic motion. The two effects that have been identified are the base-slab averaging effect and the embedment effect. Embedment effect is also associated with the reduction in ground motion. The effects are very sensitive to the short period structures. The effects arising of the wave propagation considerations (an independent of the structure) are known as kinematic interaction effects. An illustration is made in the Fig. 3. Fig.3 Kinematic Interaction Modelling of Soil-Foundation-Structure System/2

3 Foundation damping effects are also the results of inertial soil structure interaction. It results from the relative movement of the foundation and the supporting soil.it is associated with the radiation of seismic waves away from the foundation and hysteric damping within the soil itself. The result can be a reduction in shaking demand. Foundation damping is shown in Fig. 4. Fig.4 Foundation Damping 2.1 Approaches for Modeling Soil Medium (Foundation Flexibility) There are classically two approaches for modeling soil media, namely Winklerian approach and Continuum approach. (i) Winkler model In Winkler s model the soil medium is represented by a no of identical but mutually independent, closely spaced, discrete, linearly elastic springs. According to this idealization, deformation of foundation due to applied load is confined to the loaded region only. Fig. 5 shows the physical representation of the Winkler foundation. Fig.5 Winkler Foundation Modelling of Soil-Foundation-Structure System/3

4 (ii) Elastic continuum model The physical representation of the infinite soil media or the elastic half space will generate an elastic continuum model. A semi-infinite, homogeneous, isotropic, linear elastic solid subjected to a concentrated force acting normal to the plane boundary will represent such model. In this case, some continuous function is assumed to represent the behaviour of soil medium. This approach provides much more information on the stresses and deformations within soil mass than Winkler model. It has also the important advantage of simplicity of the input parameters, viz., modulus of elasticity and Poisson s ratio. 3. SOIL SPRING STIFFNESS FOR SHALLOW FOUNDATION The movement of the foundation is generally considered in two perpendicular horizontal directions and in vertical direction. The rotations of the same about these three directions should also be considered as shown in Fig.6. For buildings with isolated footing, below each column, three translational springs along three directions and three rotational springs about those mutually perpendicular axes should be put together to simulate the effect of soilflexibility, as suggested in well accepted literature (Gazetas,1991). The stiffness of these springs for footings resting on homogeneous elastic half space have been computed as explained in the literature. It has been observed that the stiffness of the springs are dependent on the frequency of the forcing function though stiffness properties are frequency independent. This frequency dependence is incorporated by multiplying the equivalent spring stiffness by a frequency-dependent factor. However some studies suggest that this stiffness can be considered as frequency independent with good results. Hence, the effect of such multiplication factor is not, in general, considered in studies. Degrees of freedom Table 1 Equivalent Spring Stiffness Stiffness of equivalent spring soil GL /(1 ) ( ) with A / 4L Vertical 2 Horizontal(lateral direction) GL /(2 ) (2 2.5 ) with A / 4L Horizontal(longitudinal direction) GL/(2 ) (2 2.5 ) 0.2/(0.75 ) GL 1 ( B / L) Rocking (about longitudinal) G /(1 ) I ( L / B) ( B / L) bx Rocking(about lateral) G /(1 ) I ( L / B) Torsion by 3.5GI ( B / L) ( I / B bz bz ) b b A b is the area of foundation considered,b and l, half length and half width of the rectangular foundation respectively, I bx, I by, I bz moment of inertia of the foundation area with respect to the longitudinal, lateral and vertical axis. Modelling of Soil-Foundation-Structure System/4

5 Fig. 6 Stiffness of Equivalent Soil Spring 3.1 Soil Flexibility for Shallow Bearing Foundations as considered in FEMA-356 Method 1 For shallow bearing footings that are rigid with respect to the supporting soil, an uncoupled spring model will represent the foundation stiffness (Fig. 7). The equivalent spring constants shall be calculated as specified in the Table 2 Fig.7 (a) Idealized Elasto-Plastic Load- Deformation Behavior for Soils (b) Uncoupled Spring Model for rigid footings Modelling of Soil-Foundation-Structure System/5

6 Table 2 Elastic Solution for Rigid Footing Spring Constraints Method 2 For shallow bearing foundations that are not rigid with respect to the supporting soils, a finite element representation of linear or nonlinear foundation behavior using Winkler models shall be used. Distributed vertical stiffness properties shall be calculated by dividing the total vertical stiffness by the area. Uniformly distributed rotational stiffness properties shall be calculated by dividing the total rotational stiffness of the footing by the moment of inertia of the footing in the direction of loading. Vertical and rotational stiffnesses shall be decoupled for a Winkler model. It shall be permitted to use the procedure illustrated in Fig.8 to decouple these stiffnesses. Modelling of Soil-Foundation-Structure System/6

7 Fig.8 Vertical Stiffness Modeling for Shallow Bearing Footing Method 3 For shallow bearing foundations that are flexible relative to the supporting soil, based on approved theoretical solutions for beams or plates on elastic supports, the foundation stiffness shall be permitted to be calculated by a decoupled Winkler model using a unit subgrade spring coefficient. For flexible foundation systems, the unit subgrade spring coefficient, ksv, shall be calculated by Eq.1 1.3G k sv (1) B (1 ) where, G = Shear modulus B = Width of footing = Poisson s ratio Modelling of Soil-Foundation-Structure System/7

8 4. SOIL MODELLING FOR PILE FOUNDATION 4.1 Winkler Model The analytical subgrade reaction model, typically known as the Winkler model, can be used for the analysis of piles under lateral loading (Fleming et al. 1985). The pile group is investigated in terms of: (1) horizontal stiffness of the soil at the level of the pile cap and surrounding the piles, and (2) soil to pile vertical stiffness interaction. The finite element model used in the pile group analysis is presented in Fig. 9. The pile can be modeled as beam elements and the surrounding soil modeled as an array of uncoupled spring elements using truss elements according to the discrete finite element as shown in Figure 10. It can be shown that the accuracy in the finite element model concerning the lateral response of piles is directly dependent on the pile discretization. For the given study (Silva) six beam elements with a length of D/6 were used at the pile head, and using a quadratic regression, the element sizes were increased to D/2 at the pile tip elevation, as shown in Fig.10. Fig.9 Pile Group Finite Element Model Horizontal Soil Model Surrounding Piles Effects of the soil surrounding the piles in the horizontal direction were modeled in terms of elements with axial stiffness only. These elements were placed only on one side of the pile with equal axial stiffness in compression and tension. A bilinear relation between the horizontal soil pressure and lateral displacement, as shown in Figure 11, was used to idealize the soil strength. The analyses described in this section were performed using the soil types (Coefficient of Subgrade Reaction, k S for Soft Soil: 3000 kn/m 3 ; Stiff Soil: 30,000 kn/m 3 ). The bilinear horizontal soil model expressed in terms of the soil pressure was given by (Pender 1978; Poulos 1971) ; (2) Modelling of Soil-Foundation-Structure System/8

9 (a) (b) Fig.10 Finite Element Model for Single Piles Fig.11 Bilinear Soil Model The soil spring stiffness at any depth was obtained according to the relation (3) where Ks is the equivalent spring stiffness, k S is the coefficient of subgrade reaction given in kn/m 3, and Δ Z is the spacing between the springs at a depth Z. Assuming the coefficient of subgrade reaction was normalized in terms of a nominal pile diameter of 1.80m, then k S may be expressed in terms of the nominal pile diameter D* and the pile section diameter D. Thus, the soil normalized spring stiffness was given by (4) In Eq. (4) the limiting soil pressure, p ult, was obtained according to the relation (Fleming 1985) 5 (5) (6) Modelling of Soil-Foundation-Structure System/9

10 where K p, is the passive earth pressure coefficient, ϕ s is the soil friction angle, which for sands is usually taken as 35 o. Based on this soil friction angle, K p then has a value of In Eq. (5) σ v is the vertical effective stress at a depth Z expressed in terms of the soil unit weight, γ, by the relation (7) Change of the soil properties during cyclic response may lead to permanent deformations in the soil layer, and the soil ultimate strength and stiffness decay with each cycle Horizontal Soil Model Surrounding Pile Cap The seismic response of the pile group was also characterized in terms of the passive pressure that develops in front of the pile cap. This was conducted by positioning a spring in front of the pile cap as illustrated in Fig.9. The pile cap spring stiffness was computed according to the expression (8) where H cap and W cap are the height and width of the pile cap, respectively Vertical End Bearing Stiffness The soil structure interaction in the vertical direction of the piles was modeled using vertical elements with axial stiffness. Modeling of the soil-structure interaction in the axial direction is described next. The vertical end bearing stiffness, K vb, (only for piles in compression) was given by (9) This expression is similar to Eq. (4) but the distance between the springs is equal to half the pile diameter. This indicates that the vertical end bearing resistance was modeled equally to the horizontal spring stiffness, which is located at the bottom of the piles Vertical Skin Friction Resistance Stiffness The vertical skin friction resistance stiffness, K vfi, was given by (Pender 1978; Poulos 1971) 1.8. (10) Where η is the pile ratio, and λ is the pile-soil stiffness ratio given by (11) (12) Modelling of Soil-Foundation-Structure System/10

11 where E s_tip is the soil young modulus at the pile tip. In Eq. (10) α is an expression that was used in order to distribute the effects of the soil vertical stiffness along the length of the piles. As described in this section these elements were positioned along the piles according to these different conditions: (1) a single vertical spring was positioned at the bottom of the piles (see Fig. 12a), and (2) springs were distributed vertically along the length of the piles (see Fig.12b). These conditions are described next Single Vertical Spring at Bottom of Piles For the condition where a single vertical spring was positioned at bottom of the piles α=1.00. In this case the vertical soil structure interaction (i.e. skin friction and end bearing) formulation was defined as depicted in Fig. 12a Distributed Vertical Springs Along Length of Piles For distributed vertical springs positioned along the full length the total skin friction resistance given by the piles α is given by (13) In Eq. (13) α is significantly smaller than 1 and summation of all α s along the length of the pile is nearly 1.00, which indicates that the total stiffness given by these two approaches are similar and can be compared directly. In addition, the vertical stiffness increases along the length of the piles in terms of the relation Z/L, which is to take into account the change in the soil Youngs s modulus along the length of the piles. Fig.12 Model for vertical soil springs Modelling of Soil-Foundation-Structure System/11

12 4.2 p-y Method The simplicity and versatility of the Winkler soil model makes it a popular method of modeling piles today, particularly in seismic engineering. A pseudostatic version of the Winkler model, known as the p-y method is commonly recommended in design codes for seismic engineering. In this method the soil characteristics in the soil-pile interaction has been described by three types of nonlinear resistance-displacement curves that is, (a) p-y curves, which describe the relationship between the lateral soil pressure (horizontal force per unit length of pile) and the corresponding pile displacement; (b) t-z curve, which describe the relationship between skin friction (vertical force per unit length of pile) and the relative vertical displacement between the pile and the soil; and (c) q-z curves, which describe the relationship between the bearing stress (vertical force on effective pile tip area) at the pile tip and the pile tip settlement. All three types of curves assume the soil behavior to be nonlinear. Again the Winker model assumes that these springs are uncoupled, that is the stiffness of one spring does not affect another. Analytical solution for the soil-pile system accounting for nonlinear soil behavior is quite complicated and they need advanced computer program for analysis. The nonlinear pile and nonlinear soil spring behavior can be solved numerically using nonlinear finite element method. Details of the curves for different soil types are given in API-RP2A For more details on the method Reese and Impe (2011) can be referred. Fig.13Modeling of soil-pile foundation using p-y method 5. Effect of Foundation Soil Flexibility on Response of Structures Flexibility reduces the overall stiffness of the structure and increases the natural period of the system. Considerable change in spectral acceleration with natural period is observed from the Modelling of Soil-Foundation-Structure System/12

13 shape of the response spectrum curve. Thus the change in natural period may alter the seismic response of any structure considerably. In addition to this, soil medium imparts damping due to its inherent characteristics. Moreover, the relationship between the periods of vibration of structure and that of supporting soil is important regarding the seismic response of the structure. The displacements increase with the increase in soil-flexibility. These show that the soil structure interaction should be accounted for in the analysis of dynamic behavior of structures, in practice. Hence, soil structure interaction under dynamic loads is an important aspect to predict the overall structural response. References 1. American Petroleum Institute (API RP2A-WSD: 2000). Recommended practice for planning, designing and construction fixed offshore platforms-working stress design. 2000, Washington, D.C. 2. FEMA 356.(2000), Prestandard and Commentry for the Seismic Rehabilitation of Buildings, Federal Emergency Management Agency, Building Seismic Safety Council, Washington, D.C. 3. Fleming, W. G. K., Weltman, A. J., Randolph, M. F., Elson, W. K. (1985), Piling Engineering, John Wiley & Sons, Inc., New York, Gazetas, G. (1991), Formulas and Charts for Impedances of Surface and Embedded Foundations, Journal of Geotechnical Engineering., 117, 9: Pender, M. J., (1978) Aseismic Pile Foundation Design Analysis, Bulletin of the New Zealand National Society for Earthquake Engineering, 11, 2: Poulos, H. G. (1971), Behavior of Laterally Loaded Piles: I - Single Piles, Proceedings of the American Society of Civil Engineers, 97, No. SM5, May 1971: Reese, L.C., and Van Impe, W.F Single Piles and Pile Groups under Lateral Loading, Balkema. 8. Silva, Pedro F. Seismic Evaluation of Full-Moment Connection CISS Piles/Foundation Systems Silva, P. F., and Manzari, M. T., Nonlinear Pushover Analysis of Bridge Columns Supported on Full-Moment Connection CISS Piles on Clays, Earthquake Spectra, 24: Acknowledgement The authors would like to acknowledge that the information give in this lecture notes has been collected and edited from literature and would not be used for any commercial purpose but for educating masses on the subject. Most of the material has been collected from FEMA 356 and Silva and Manzari Modelling of Soil-Foundation-Structure System/13