Dynamic Analysis of Partially Embedded Structures Considering Soil-Structure Interaction in Time Domain

Analysis and design of structures subjected to arbitrary dynamic loadings especially earthquakes have been studied during past decades. In practice, the effects of soil-structure interaction on the dynamic response of structures are usually neglected. In this study, the effect of soil-structure interaction on the dynamic response of structures has been examined. The substructure method using dynamic stiffness of soil is used to analyze soil-structure system. A coupled model based on finite element method and scaled boundary finite element method is applied. Finite element method is used to analyze the structure, and scaled boundary finite element method is applied in the analysis of unbounded soil region. Due to analytical solution in the radial direction, the radiation condition is satisfied exactly. The material behavior of soil and structure is assumed to be linear. The soil region is considered as a homogeneous half-space. The analysis is performed in time domain. A computer program is prepared to analyze the soil-structure system. Comparing the results with those in literature shows the exactness and competency of the proposed method.


Introduction
In a dynamic soil-structure interaction problem, the structure is supported by an unbounded soil medium subjected to a dynamic load like an earthquake.The dynamic response of the structure is affected by the interaction between the structure, foundation, and soil.
In dynamic soil-structure interaction analysis, usually the higher modes of the structure are affected significantly by soil-structure interaction SSI effects.As the influence of higher modes on the seismic response of flexible high structures with small mass remains small, the SSI effects are negligible for these structures.On the other hand for stiff and massive structures on relatively soft ground, the effects of SSI are noticeable and lead to an increase in the natural period and a change in the damping ratio of the system 1-3 .Effects of Zhang et al. presented the analysis in time domain for layered soils 42 .Tanikulu et al. extended BEM formulation for infinite nonhomogeneous media 43 .They could model only three different layers.Coupled finite element-Infinite element models have been used in dynamic soil-structure interaction analysis 44-46 .Coupled finite element/boundary element/ scaled boundary finite element model 47 has been used to solve soil-structure interaction problems.
Jeremić et al.48 have studied the effects of nonuniformity of soils in large structures where they developed various models to simulate wave propagation through soils with elastoplastic behavior.
Ghannad and Mahsuli 49 studied the effect of foundation embedment using a simplified single degree of freedom model with idealized bilinear behavior for the structure and considered the soil as a homogeneous half-space as a discrete model based on cone model concepts.The foundation is modeled as a rigid cylinder embedded in the soil.
The scaled boundary finite element method is a boundary-element method based on finite elements.This method combines the advantages of the boundary and finite element methods.It also combines the advantages of the numerical and analytical procedures.This method can be applied in both frequency and time domains 35 .This method is a semianalytical procedure which transforms the partial differential equation to an ordinary differential equation using a virtual work statement as in finite elements.In this method, no fundamental solution is required, and no singular integrals occur.Only the boundary is discretized which results in a reduction of the spatial discretization by one.The analytical solution in the radial direction permits the boundary condition at infinity to be satisfied exactly 35 .A computer program named SIMILAR based on SBFEM is presented by Wolf and Song 50 .This program calculates the dynamic stiffness of the unbounded media in frequency and time domain.
In this study, the dynamic behavior of partially embedded structures is examined.The substructure method is used, and a coupled finite element, scaled boundary finite element model is applied.The scaled boundary finite element method is used to calculate the dynamic stiffness of the soil, and the finite element method is applied to analyze the dynamic behavior of the structure.In continuation, firstly, the equation of motion of the soil structure system in total and relative displacements is introduced.The dynamic stiffness matrix of the soil is obtained using SBFEM in the second section.In the third section, an iterative procedure is presented to calculate dynamic load using dynamic stiffness matrix of the soil.Applying Newmark method, the equation of motion of the system is solved, and the displacements of the structure are obtained.It is worth noting that although the formulation in the paper is not innovative, this is the first time a complete model of structure is studied and the dynamic response of the structure is examined.Previous studies have used simplified soil model or simplified structural model and/or both.Therefore, the present results seem to be the first ones obtained based on a complete soil-structure model.Moreover, from a practical point of view, the present results could lead to an interesting conclusion in the important topic of "choosing base shear level" which is not clearly defined in practice codes Numerical examples are presented, and the final section is devoted to concluding remarks.

Equation of Motion
The dynamic behavior of the structure could be described by its static stiffness matrix K and the mass matrix M .The equation of motion of the structure in total displacements in time domain is formulated as follows 35 : Considering damping matrix of the structure, C , the above equation is written as follows: where { üt }, { ut }, and {u t } are the acceleration, velocity, and displacement vectors of the structure.Subscripts are used to denote the nodes of the discretized system.As shown in Figure 1, nodes on the foundation structure interface are denoted by b, and the remaining nodes related to the structure are denoted by s. {R t } denotes the interaction forces of the unbounded soil acting on the interface nodes of soil-structure system.The interaction forces of the soil depend upon the motion relative to the effective foundation input motion{u g b }.The interaction forcedisplacement relationship in the time domain is formulated as: where S ∞ t is called the displacement unit impulse response matrix in time domain.The interaction force-displacement relationship can alternatively be written as in which M ∞ t is the acceleration unit-impulse response matrix in time domain.
Superscript ∞ denotes the unbounded medium.For an unbounded medium initially at rest, we have Substituting 2.6 in 2.1 results the equation of motion in total displacement 35 : in which { üg } is the ground motion acceleration induced to the base of the structure during an earthquake.In this paper, the equation of motion of soil-structure system in relative displacement is used

2.7
As can be seen in 2.7 , the unit impulse response matrix should be obtained a priori.The dynamic load on the right hand side of the equation is calculated a posteriori.In the next section, the unit impulse response matrix is obtained applying scaled boundary finite element method 35 .

Obtaining Acceleration Unit-Impulse Response Matrix
The force displacement relationship in the frequency domain could be written as follows 35 : where {R ω } and {u ω } are force and displacement in frequency domain.M ∞ ω is denoted as acceleration dynamic stiffness matrix in the frequency domain.The relationship between the acceleration and displacement dynamic stiffness matrices is

Mathematical Problems in Engineering
The scaled boundary finite element equation in dynamic stiffness for the unbounded medium is formulated as follows 35 : in which E 0 , E 1 , and E 2 are coefficient matrices in the Scaled Boundary Finite Element method introduced in 35 .Dividing 3.3 by iω 4 and substituting 3.2 yields 35

3.4
Applying the inverse Fourier transformation to 3.4 results in t 0

3.5
The positive definite coefficient matrix E 0 is decomposed by Cholesky's method as follows: where U is an upper-triangular matrix.Substituting 3.6 in 3.5 and premultiplying by where And the coefficient matrices are

3.9
Once obtained m ∞ t from 3.7 , the acceleration unit-impulse response matrix is obtained as In this paper, M ∞ t is obtained using the program SIMILAR presented by Jeremić et al.48 .

Calculating Dynamic Load
The dynamic load on the right hand side of 2.7 could be written as follows: where {F s r t } represents the dynamic load due to ground motion, respectively, and affects the total nodes of the system, while {F b r t } is the dynamic load related to interaction effects and affects the nodes on the foundation-structure interface denoted by b in Figure 2. The dynamic load vector on the right hand side of 2.7 could be written as follows:

{F
where M is the total mass matrix of structure, and M ∞ t is the acceleration unit-impulse response matrix.The dynamic load F b r t could be written in discrete form as follows 51 : where F r t n is the dynamic load at nth step.ü and u are the acceleration and velocity at the corresponding time step.In this paper, an iterative method is adopted to calculate F b r t n .It is supposed that the acceleration is constant at each time step, so 4.5 could be written as follows: For the first time step, the dynamic load is calculated assuming that üb 0 2.7 is solved applying Newmark scheme, and acceleration, velocity, and displacement vectors are obtained.The dynamic load is then calculated using calculated acceleration.Equation 2.7 is solved again and the magnitudes of acceleration, velocity, and displacements are obtained.This procedure is iterated until the convergence is achieved.In this study, the tolerance between two successive iterations is taken as 0.001.
The above procedure is outlined as in Table 1.
According to the above algorithm, an FORTRAN program is prepared to examine the dynamic behavior of the structure considering interaction effects.Numerical examples are presented in the next section.

Examples
2D frames on soft ground have been analyzed applying a coupled scaled boundary finite element-Finite element models.The analysis is performed in time domain and the material behavior of soil and structure is assumed to be linear.The soil-structure system is subjected to sine excitations, El Centro, and Tabas ground motions.The displacement and base shear are calculated.Base shear is assumed to be the algebraic summation of horizontal forces induced in the structure.Results are compared with those obtained by cone model.
Example 5.1.As the first example, the frames shown in Figures 2 and 3 are used in analysis.The damping ratio of the structure is considered as five percent of the mass matrix.
Structure properties are assumed as: Frame no.1: a b 6 m, h 3 m, Frame no.2: a 6 m, h 3 m.

5.1
The soil properties are ρ 1600 kg/m 3 , V s 150 m/s, υ 0.3.Firstly, a dynamic analysis is performed ignoring SSI effects .The natural frequencies and periods of the structures are calculated and presented in Tables 3 and 4. Then the soil-structure system is subjected to sine excitations with unit amplitude.The loading frequency is con-  sidered to be variable and selected so that it would be close to natural frequencies of the structure.The harmonic load used in the analysis could be expressed as follows:

Mathematical Problems in Engineering
where T is the period of sine function.The SSI effect on dynamic response of the structure is examined.Figures 4,5,6,7,8,9,10,11,12,13,14,15,and 16 show the results obtained in analysis.Figures 17,18,19,and 20 show the variation of displacement and base shear versus period of dynamic load.The maximums peaks represent the obtained magnitudes with loading period close to the first and the second natural periods of the structure.As can be seen considering SSI effects, the maximum displacement and base shear are decreased.In Tables 5  and 6 the percentage of the relative reduction of displacement and base shear due to first and second modes is presented.
It is observed that considering SSI effect leads to reduction in displacement and base shear.The reduction in displacement and base shear is more significant when the loading frequency is close to natural frequencies of the structure.As shown in Tables 5 and 6, the   percentage of relative reduction in displacement and base shear is more significant for the second mode than the first one.It could be concluded that SSI effect is more pronounced for higher modes of the structure.Comparing the results presented in Tables 5 and 6 shows that the relative reduction is more significant for farme no. 2. It could be concluded that SSI effects are more significant for stiff structures.
Example 5.2.In the second example, the fames are subjected to El Centro ground motion.It is worth noting that the predominant period of El Centro ground motion is 0.56 sec which is close to the first natural period of frame no. 2. The results are given in Figures 21,22,23,and 24.As can be observed in Tables 7 and 8, the relative reduction in displacement and shear base is more significant for frame no. 2. It can be concluded that when the predominant period of the earthquake is close to natural period of the structure, considering SSI effects leads to more significant reduction, and the dynamic response of the structure is more affected.Example 5.3.In the third example, the frames are subjected to Tabas ground motion.The predominant period of Tabas ground motion is 0.2 sec which is close to the second natural period of frame no. 2. The results are given in Figures 25,26,27,and 28.As it is observed, considering SSI effect has a pronounced effect on results Tables 9 and 10 .

Conclusion
Analysis and design of structures subjected to arbitrary dynamic loadings especially earthquakes have been studied during past decades.In practice, the effects of soil-structure interaction on the dynamic response of structures are usually neglected.In this paper, a coupled scaled boundary finite element-finite element model is presented to examine the dynamic response of the structure considering soil-structure interaction.The substructure method is used to analyze the soil-structure interaction problem.The analysis is performed in time domain.
The material behavior of soil and structure is assumed to be linear.The scaled boundary finite element method is used to calculate the dynamic stiffness of the soil, and the finite element method is applied to analyze the dynamic behavior of the structure.2D frames have been analyzed using the proposed model.The results are compared with those obtained by cone model.Considering SSI effect leads to reduction in displacement and base shear.When the system is subjected to sine excitation, the reduction in displacement and base shear is more significant when the loading frequency is close to natural frequencies of the structure.The reduction in displacement and base shear is more significant for the second mode than the first one, thus considering SSI in dynamic analysis of the structure affects the higher modes more significantly.It is observed that when the soil-structure system is subjected to an earthquake whose predominant period is close to natural period of the structure, considering SSI effects leads to more significant reduction, and the dynamic response of the structure is more affected.It is obvious that considering SSI effects results in more effective design without decreasing safety margin.

Soil-structure interface s b Figure 1 :
Figure 1: Soil-structure system.

Figure 4 :
Figure 4: Comparison of displacement at the top of frame no. 1 subjected to sine excitation T 1.2 s .

Figure 5 :Figure 6 :Figure 7 :
Figure 5: Comparison of displacement at the top of frame no. 1 subjected to sine excitation T 1.1 s .

Figure 19 :
Figure 19: Comparison of displacement at top of frame no. 2.

Table 1 :
The Newmark algorithm and iteration procedure used in the analysis.

Table 2 :
Properties of structural elements.

Table 3 :
Natural frequencies and periods of first five modes of frame no. 1.

Table 4 :
Natural frequencies and periods of first five modes of frame no. 2.

Table 5 :
Relative reduction of displacement and base shear considering SSI effects for frame no. 1 subjected to sine excitation.

Table 6 :
Relative reduction of displacement and base shear considering SSI effects for frame no. 2 subjected to sine excitation.

Table 7 :
Relative reduction of displacement and base shear considering SSI effects for frame no. 1 subjected to El Centro ground motion.

Table 8 :
Relative reduction of displacement and base shear considering SSI effects for frame no. 2 subjected to El Centro ground motion.

Table 9 :
Relative reduction of displacement and base shear considering SSI effects for frame no. 1 subjected to Tabas ground motion.

Table 10 :
Relative reduction of displacement and base shear considering SSI effects for frame no. 2 subjected to Tabas ground motion.