Role of Subgrade Reaction Modulus in Soil-Foundation-Structure Interaction in Concrete Buildings

: One of the key issues in structural and geotechnical engineering is that most parts of buildings are usually analysed separately and then the outputs are used in foundation designs. In this process, some effects are neglected. In this study, the soil–structure interaction (SSI) in foundations of concrete buildings was evaluated using the direct ﬁnite element method (DFEM). 3D models were developed and used to analyse concrete buildings with different stories constructed on soft soil. Foundation settlement, deformation of foundation, soil pressure diagram, and weight of reinforcement in the foundation were considered as the main parameters. Deformation of the foundation was analysed using the ﬁnite element method considering the effect of combined loadings (combinations of dead load, live load, and earthquake load). It is shown that by changing values of subgrade reaction modulus ( Ks ) in foundation design, the effects of SSI on tall buildings can be considered automatically. The results also show that the soil–structure interaction can cause changes in the pattern of foundation settlement, foundation deformation, and the weight of reinforcement used in foundation design. Furthermore, dishing deformation in foundation appeared in terms of SSI effects. An equation is provided to simplify considering SSI effects in foundation design. This method is practical for civil (especially structural) engineers, and they can conveniently consider these effects in foundation design without using DFEM.


Introduction
Soil deformability is one of the key parameters affecting the behaviour of tall structure.The interaction between a building, its foundation, and the soil has important effects on the behaviour of each of these components as well as the complete system.The relative stiffness of a building structure, its foundation, and the soils that support the foundation influences the stresses and displacements of both the structure and the soil.In the design of structural systems, soil-structure interaction (SSI) effects are sometimes neglected by the use of a structural model supported on a fixed base, while there has been significant progress in understanding the soil-structure interaction (SSI) effects resulting from external forces such as winds and earthquakes [1].Given the lack of information about the effects of SSI on foundation design as well as the difficulty in using software in complex problems, some effects, which have significant impacts for assessment such as SSI, have been ignored.In structural design, soil-foundation-structure interaction (SFSI) is an important parameter that should be taken into account.However, simulating SFSI often requires complex models for the soil and the foundation with a great number of degrees-of-freedom (DOF), which needs significant computational costs.Various simplified modelling strategies have Buildings 2022, 12, 540 2 of 17 been developed to consider these effects.The SSI effect can be disregarded in structures constructed on hard subgrade (rock) [2].However, in structures that are constructed on soft foundation soil, the soil-structure interaction is very important [3][4][5][6].The structural deformations and the soil displacements are not separate from each other during earthquakes [7,8].Therefore, there is a need for improved coordination of both geotechnical and structural engineering practices.The soil's response affects the structure's movement, and the structure's movement affects the soil's response [9].Once the soil deformation is investigated, it concludes that soil-structure interaction has been considered in evaluating the structural response.Documenting past seismic events is a significant step in understanding and accurately characterizing dynamic geotechnical earthquake engineering problems [10].With the expansion of the construction industry, civil engineering software has been widely used for simulating earthquake loads on structures and foundations [11][12][13][14][15]. SSI effects are usually caused by the flexibility of the soil under the foundation as well as relative vibration between the foundation and free field [16].This implies that the response of a structure constructed on a site depends on the relationship between the structural characteristics and properties of the underlying soil layers and free field motion.The standard methods to evaluate soil structure interaction effects can be classified into two groups: direct and substructure methods [17,18].In the direct analysis method, the structure, foundation, and soil are included in the one model and analysed all together in one step as a complete system.Consequently, it is unnecessary to use the principle of superposition, and the soils can be simulated using the finite element methods (FEM) [19].In this method, the structure and main parts of soil are modelled and analysed by the finite element method, and the movement of the free field can be applied at boundaries.Besides, it is possible to consider the non-linear behaviour of soil.The elastic-perfectly plastic vertical stresses have regular distributed rather than matching elastic stress values, and the elastic-plastic vertical displacements are about 25 percent greater than the corresponding elastic values [20].The substructure or indirect analysis method is the most common method used to solve the problems of soil-structure interaction.In this method, soil-structure interaction problems are sub-divided into separate parts of simpler problems.The problems are analysed separately (using the common methods), and then the results are combined by the principle of superposition to complete the analysis.In this method, it is necessary to assume linear behaviour for the structure and the soil.Shakib and Fuladgar [21] studied the effects of dynamic soil-structure interaction on the seismic response of asymmetric buildings.They found that the SSI effects increased the lateral and asymmetrical motion of the structure.The foundation was influenced by the stiffness of the soil underneath it.Gazetas proposed an equation to calculate the spring stiffness in the x, y, and z directions on a shallow foundation above the soil surface [22] (x and y axis are in direction of foundation's width and foundation's length, respectively, and the z axis is in direction of foundation's depth).A new macro-element (very small elements) for modelling the effects of SSI in the shallow foundation can be used to numerically investigate the behaviour of a wider variety of configurations that are difficult to study experimentally [23].
It has been shown that the top-floor displacement in the model with SSI effects is 2% to 3% greater than the model without soil-structure interaction [17].The structure's response to seismic loads is influenced by the relationships between the three associated systems: the structure, foundation, and soil under the foundation [24].The collaboration between geotechnical and structural engineers is therefore vital for a successful design.The subgrade reaction modulus (Ks) can be investigated as a suitable interface between the soil and the structure [25].It has a conceptual relationship between soil pressure and the structure's deformation in foundation design.The subgrade reaction modulus is not a soil constant, but depends on many factors such as the dimensions of the foundation, soil characteristics, load level, and superstructure rigidity.Winkler Theory (1867) provides a simple theory to calculate the contact stresses using the modulus of subgrade reaction, assuming elastic behaviour for soil without considering the effect of the soil-structure interaction (Equation ( 1)) [26]: where Ks, P, and S are subgrade reaction modulus, pressure, and vertical settlement, respectively.Boussinesq considered a more realistic model than the Winkler model and showed that the distribution of displacements is continuous [27].Filonenko-Borodich [28] and Hetenyi [29,30] proposed some modifications to the Winkler model.Farouk showed that the distribution of Ks is non-uniform and concentrated at the edge elements of the foundation and the value reduced in the inner parts [25].They concluded that ignoring SSI effects always led to overestimation of the inner foundation design and underestimation of the outer foundation design.Therefore, the role of Ks and consideration of SSI in finite element method are very important for practical engineers.
A review of the literature shows that the effects of the variable subgrade reaction modulus (Ks) for the foundation have not been investigated in detail.In this research, the soil-structure interaction (SSI) effects in the foundation of concrete buildings with variable Ks values were analysed using the direct finite element method (DFEM).In the following sections, the methodology is presented and the results of the application of the method to three concrete structures are evaluated and discussed.

Materials and Methods
In this study, three concrete structures with six-, eight-, and 10-stories with the same selected plan (Figure 1) were used to investigate the effects of SSI on the foundation of the structure.The structures were designed with ETABS software.ANSYS meshing was used to discretise the structure.A grid sensitivity analysis was carried on by producing ten different meshes (from 398,871 cells to 2,951,341) for the initial design using ANSYS meshing software.Analysis of the results showed that the standard deviation of the values of the base support reactions was around 0.78 % for 1,995,321 cells.Therefore, for the rest of the FEM analysis, 1,995,321 was chosen due to the fact that the computational time of the numerical analysis increased significantly with an increase in the number of cells.ANSYS structural analysis software was used to simulate and analyse the structures.The structure, foundation, and soil were modelled simultaneously in ANSYS with the above-mentioned number of elements.Linear elements with three transitional and three rotational degrees of freedom (BEAM4) were used to discretize the beams and columns.To model the slabs and shear walls, surface elements (SHELL181) with three transitional and three rotational degrees of freedom were used.In this study, three column sizes and one beam size were considered in all buildings.The roofs were assumed to be concrete slabs with 0.35 m thicknesses (Figure 2).For the foundation, volume elements 3D SOLID45 with three transitional degrees of freedom were used.The geometry of the 10-story building is shown in Figure 3.
All the beams and columns were of a rectangular shape and each story was 3 m in height.Beams, columns, and shear walls were concrete with isotropic materials such as Ex = 2.52 × 10 10 Pa, ϑ = 0.2, density (ρ) = 2400 Kg/m 3 , where Ex and ϑ are the modulus of elasticity and Poisson's ratio, respectively.Ceilings, material properties are Ex = 2 × 10 10 Pa, ϑ = 0.2, density (ρ) = 2060 Kg/m 3 .Table 1 shows the details of the used sections, Sh-W represents the shear walls in this table.⁄ .Table 1 shows the details of the used sections, Sh-W represents the shear walls in this table.
Table 1.Section details of the six-, eight-, and 10-story buildings.For all buildings, the foundation considered was 1 m thick with the modulus of elasticity, E = 2.52×10 10 Pa, Poisson's ratio, = 0.2, and density = 2400 kg/m .Three-dimensional elements in SOLID45 were also employed to model the soil.The thickness of the soil was considered to be 20 m in two layers.The Drucker-Prager model was used to describe the behaviour of the soil.Material properties for the top soil layer are modulus of elasticity (Ex) = 4×10 7 Pa, ϑ =0.35, density = 1800 Kg/m 3 , cohesion (C) = 1×10 5 Pa, angle of shearing resistance (∅) = 25°, and dilation angle (ψ) = −7°.Modulus of elasticity (Ex) = 1×10 8 Pa, ϑ = 0.30, density = 1800 Kg/m 3 , cohesion (C) = 1×10 5 Pa, angle of shearing resistance (∅) = 25°, and dilation angle (ψ) = −7°.Interface element (CONTA173) [31] with three transitional degrees of freedom was used in order to model the contact between the soil and the foundation.This element was used in these analyses to represent contact and sliding between 3-D target surfaces and a deformable surface defined by this element.The element is applicable to 3-D structural and coupled-field contact analyses.This element is located on the surfaces of 3-D solid or shell elements.Furthermore, parameters such as FKN (normal penalty stiffness factor), FTOLN (penetration tolerance factor), and ICONT (initial contact closure) were considered as 0.15 [32,33].Two different types (type 1 and type 2) of loads were applied to analyse the behaviour of the structure.Type 1 is a combination of dead, live, and earthquake loads, and type 2 is a combination of dead and live  [31] with three transitional degrees of freedom was used in order to model the contact between the soil and the foundation.This element was used in these analyses to represent contact and sliding between 3-D target surfaces and a deformable surface defined by this element.The element is applicable to 3-D structural and coupled-field contact analyses.This element is located on the surfaces of 3-D solid or shell elements.Furthermore, parameters such as FKN (normal penalty stiffness factor), FTOLN (penetration tolerance factor), and ICONT (initial contact closure) were considered as 0.15 [32,33].Two different types (type 1 and type 2) of loads were applied to analyse the behaviour of the structure.Type 1 is a combination of dead, live, and earthquake loads, and type 2 is a combination of dead and live loads.Dead loads are gravitational loads and live loads were considered for each floor, which were assigned as a pressure load with a value of 200 Kg/m 2 for each floor and 150 Kg/m 2 for the top floor.The earthquake loads were considered as an inertia acceleration with a value = 0.981 m/s 2 in the buildings' width and length directions.The base support reactions of the three types of buildings were calculated by ANSYS.The results of the numerical analysis using ANSYS were validated against ETABS software.Figure 4 shows the reaction of the 8-story building in the x-direction (F x ) under earthquake loading applied in the x-direction with ANSYS and ETABS.The static model of the sole structure was carried out to analyse the static response of the 3D fixed-base model under both load combinations.As can be seen, ANSYS results were in very good agreement with those obtained from a static analysis performed with the ETABS software.
values of base support reactions from ANSYS software were inserted in the SAFE software.The foundation was divided into 1 × 1 m 2 elements in SAFE.In this software, the foundation length was divided into 20 sections and the width of the foundation was split into 14 sections (identified with letters A-O).The values of subgrade reaction modulus (Ks) for different elements were considered to be variable [34][35][36][37][38].A trial and error approach was used by changing the values of the subgrade reaction modulus (Ks) for each element until the deformations of the foundation in ANSYS matched those of the SAFE.

Results
The analysis was repeated for three tall (six, eight, and 10-story) buildings founded on the selected soil.For an analysis of the results, the deformation of the buildings was elaborated in this section.The six, eight, and 10-story buildings were analysed based on type 1 and 2 loadings.The behaviour of the structure under type 1 and type 2 loadings is presented and compared in Figures 5 and 6, respectively.All units in Figures 5 and 6 are in meters.Figure 5a shows the deformations of the eight-story structure with type 1 loading.It can be seen that the total deformation was in the horizontal direction, which is the direction of the applied earthquake load.Figure 5b shows the foundation of the selected model and its deformation based on applied type 1 loading.The middle of the foundation experienced a significant deformation (15% more than corners of the foundation) in the zdirection.Figure 5c and d  SAFE software was used to analyse and design different foundations.The obtained values of base support reactions from ANSYS software were inserted in the SAFE software.The foundation was divided into 1 × 1 m 2 elements in SAFE.In this software, the foundation length was divided into 20 sections and the width of the foundation was split into 14 sections (identified with letters A-O).The values of subgrade reaction modulus (Ks) for different elements were considered to be variable [34][35][36][37][38].A trial and error approach was used by changing the values of the subgrade reaction modulus (Ks) for each element until the deformations of the foundation in ANSYS matched those of the SAFE.

Results
The analysis was repeated for three tall (six, eight, and 10-story) buildings founded on the selected soil.For an analysis of the results, the deformation of the buildings was elaborated in this section.The six, eight, and 10-story buildings were analysed based on type 1 and 2 loadings.The behaviour of the structure under type 1 and type 2 loadings is presented and compared in Figures 5 and 6, respectively.All units in Figures 5 and 6 are in meters.Figure 5a shows the deformations of the eight-story structure with type 1 loading.It can be seen that the total deformation was in the horizontal direction, which is the direction of the applied earthquake load.Figure 5b shows the foundation of the selected model and its deformation based on applied type 1 loading.The middle of the foundation experienced a significant deformation (15% more than corners of the foundation) in the z-direction.Figure 5c,d show the deformation of the foundation in the z-direction with constant Ks and with variable Ks, respectively.It can be seen that the maximum deformation of the model with constant Ks occurred in the corner of the foundation; however, in the model with variable K S , it was observed in the middle of the foundation.Deformation of the eight-story structure with type 2 loading is shown in Figure 6a.It can be seen that the maximum deformation occurred at the centre of the foundation.According to Figure 6b, the minimum deformation was at the corner of the foundation.Finally, the deformation of the foundation with constant Ks (Figure 6c) was compared with the foundation with variable Ks (Figure 6d).The difference between the maximum and minimum deformations in the constant Ks model was 14 mm, while for the variable Ks model, it was 8 mm.This clearly shows that the consideration of variable Ks can lead to different deformations in the foundations.Moreover, the deformation of the foundation was in good agreement with the proposed pattern of Ks provided in Chapter 4 of ATC FEMA 273 [39].Furthermore, depression (or dishing) effects were observed in the foundation, which results from total deformation of the foundation considering the SSI effects.
to Figure 6b, the minimum deformation was at the corner of the foundation.Finally, the deformation of the foundation with constant Ks (Figure 6c) was compared with the foundation with variable Ks (Figure 6d).The difference between the maximum and minimum deformations in the constant Ks model was 14 mm, while for the variable Ks model, it was 8 mm.This clearly shows that the consideration of variable Ks can lead to different deformations in the foundations.Moreover, the deformation of the foundation was in good agreement with the proposed pattern of Ks provided in Chapter 4 of ATC FEMA 273 [39].Furthermore, depression (or dishing) effects were observed in the foundation, which results from total deformation of the foundation considering the SSI effects.Then, the model with variable Ks was analysed and the weight of the used reinforcements was calculated.The results are presented in Table 2.In another case, the foundation was modelled with a constant subgrade reaction modulus of Ks = 1 × 10 4 kN/m 3 and the weight of the used reinforcements was obtained (Table 2).It can be seen the value of the same Ks in the first row and the domain of variable values of Ks in the second row related to six-, eight-, and 10-story buildings in this table.The last column of this table shows the used weight of reinforcement for buildings with different numbers of stories, with the same Ks and with variable Ks.It can be seen that by using variable Ks, the weight of the required reinforcement of the foundation was reduced by 32%, 37%, and 38% for six-, eight-, and 10-story buildings respectively and it is economical for engineers [40].Nowadays reliability analysis is considered for uncertainty in exact values of soil parameters [41][42][43] and it will be considered in next study in combination with soil-structure interaction effects.Then, the model with variable Ks was analysed and the weight of the used reinforcements was calculated.The results are presented in Table 2.In another case, the foundation was modelled with a constant subgrade reaction modulus of Ks = 1×10 4 kN/m 3 and the weight of the used reinforcements was obtained (Table 2).It can be seen the value of the same Ks in the first row and the domain of variable values of Ks in the second row related to six-, eight-, and 10-story buildings in this table.The last column of this table shows the used weight of reinforcement for buildings with different numbers of stories, with the same Ks and with variable Ks.It can be seen that by using variable Ks, the weight of the required reinforcement of the foundation was reduced by 32%, 37%, and 38% for six-, eight-, and 10-story buildings respectively and it is economical for engineers [40].Nowadays reliability analysis is considered for uncertainty in exact values of soil parameters  There is a correlation between soil elasticity modulus and modulus of subgrade reaction in foundation design [44], which can be modified in different zones of a foundation.An equation is presented from variable values of subgrade reaction modulus to consider SSI effects in foundation design.Regions and variable values of subgrade reaction modulus are obtained from Equation (2).
The dimensionless parameters a and b are respectively the coefficient of stories and the coefficient of different zones of foundation on soft soil.Coefficient of story values (a) are obtained from the following equation.a = 0.25(3n + 3) by a = 0.30(3n + 3) ( where n is the number of building stories. Coefficient values of different zones of foundation (b) were derived from Figure 7 and Table 3 together.The minimum values for a and b coefficients could keep the design on the safe side.There is a correlation between soil elasticity modulus and modulus of subgrade reaction in foundation design [44], which can be modified in different zones of a foundation.An equation is presented from variable values of subgrade reaction modulus to consider SSI effects in foundation design.Regions and variable values of subgrade reaction modulus are obtained from Equation (2).
where is the number of building stories.
Coefficient values of different zones of foundation (b) were derived from Figure 7 and Table 3 together.The minimum values for a and b coefficients could keep the design on the safe side.

Conclusions
In this research, the effects of SSI on the foundation of concrete buildings were investigated by using DFEM.The main conclusions drawn from this research are as follows: (1) Assigning variable values of subgrade reaction modulus in different parts of a foundation based on Equation ( 2) is a convenient method for considering soil-structure interaction in foundation design without using the direct finite element method in modelling.(2) By using variable Ks, the weight of the required reinforcement of the foundation can be reduced considerably.(3) The results of load combinations illustrate that the total deformations were in the direction of the applied earthquake load.In particular, consideration of the full interaction led to an overall increase in deformability of the soil-foundation-structure system compared to the fixed-base structure.In addition, the middle of the foundation experienced a significant deformation in the z-direction (in the direction of the foundation depth).
(4) By considering the soil-structure interaction, a dishing effect was observed.The deformation of the foundation was in good agreement with the pattern of Ks provided in Chapter 4 of ATC FEMA273.(5) The results presented in this paper improved the current understanding of the SSI in the foundations of concrete buildings.
This suggested method can be used for concrete buildings but for steel buildings, it should be considered in future studies.In addition, all of our analyses were deterministic, while the role of uncertainty in soil parameters could be considered by probabilistic analyses.

Acknowledgments:
The authors would like to acknowledge the support from the Department of Civil and Environment Engineering of Shiraz University of Technology as well as the Department of Engineering of the University of Exeter for this project.

Conflicts of Interest:
The authors declare no conflict of interest.

Buildings 2022 , 17 Figure 1 .
Figure 1.The plan of the buildings with the locations of the shear walls.

Figure 2 .
Figure 2. Details of the floor slabs.

Figure 1 . 17 Figure 1 .
Figure 1.The plan of the buildings with the locations of the shear walls.

Figure 2 .
Figure 2. Details of the floor slabs.Figure 2. Details of the floor slabs.

Figure 2 .
Figure 2. Details of the floor slabs.Figure 2. Details of the floor slabs.

Figure 3 .
Figure 3.The geometry of the 10-story building.

Figure 3 .
Figure 3.The geometry of the 10-story building.For all buildings, the foundation considered was 1 m thick with the modulus of elasticity, E = 2.52 × 10 10 Pa, Poisson's ratio, v = 0.2, and density ρ = 2400 kg/m 3 .Threedimensional elements in SOLID45 were also employed to model the soil.The thickness of the soil was considered to be 20 m in two layers.The Drucker-Prager model was used to describe the behaviour of the soil.Material properties for the top soil layer are modulus of elasticity (Ex) = 4 × 10 7 Pa, ϑ = 0.35, density = 1800 Kg/m 3 , cohesion (C) = 1 × 10 5 Pa, angle of shearing resistance (∅) = 25 • , and dilation angle (ψ) = −7 • .Modulus of elasticity (Ex) = 1 × 10 8 Pa, ϑ = 0.30, density = 1800 Kg/m 3 , cohesion (C) = 1 × 10 5 Pa, angle of shearing resistance (∅) = 25 • , and dilation angle (ψ) = −7 • .Interface element (CONTA173)[31] with three transitional degrees of freedom was used in order to model the contact between the soil and the foundation.This element was used in these analyses to represent contact and sliding between 3-D target surfaces and a deformable surface defined by this element.The element is applicable to 3-D structural and coupled-field contact analyses.This element is located on the surfaces of 3-D solid or shell elements.Furthermore, parameters such as FKN (normal penalty stiffness factor), FTOLN (penetration tolerance factor), and ICONT (initial contact closure) were considered as 0.15[32,33].Two different types (type 1 and type 2) of loads were applied to analyse the behaviour of the structure.Type 1 is a combination of dead, live, and earthquake loads, and type 2 is a combination of dead and live loads.Dead loads are gravitational loads and live loads were considered for each floor, which were assigned as a pressure load with a value of 200 Kg/m 2 for each floor and 150 Kg/m 2 for the top floor.The earthquake loads were considered as an inertia acceleration with a value = 0.981 m/s 2 in the buildings' width and length directions.The base support reactions of the three types of buildings were calculated by ANSYS.The results of the numerical analysis using ANSYS were validated against ETABS software.Figure4shows the reaction of the 8-story building in the x-direction (F x ) under earthquake loading applied in the x-direction with ANSYS and ETABS.The static model of the sole structure was carried out to analyse the static response of the 3D fixed-base model under both load combinations.As can be seen, ANSYS results were in very good agreement with those obtained from a static analysis performed with the ETABS software.

Figure 4 .
Figure 4. Comparison of the predicted Fx reactions for the 8-story building under earthquake loading in ANSYS and ETABS.

Figure 4 .
Figure 4. Comparison of the predicted F x reactions for the 8-story building under earthquake loading in ANSYS and ETABS.

Figure 5 .
Figure 5. Deformation of the eight-story building with type 1 loading (a) structure, (b) foundation in ANSYS, (c) foundation (with the constant Ks) in SAFE, (d) foundation (with variable Ks) in SAFE.

Figure 6 .
Figure 6.Deformation of the eight-story building with type 2 load (a) structure, (b) foundation in ANSYS, (c) foundation (with the constant Ks) in SAFE, (d) foundation (with variable Ks) in SAFE.

Figure 6 .
Figure 6.Deformation of the eight-story building with type 2 load (a) structure, (b) foundation in ANSYS, (c) foundation (with the constant Ks) in SAFE, (d) foundation (with variable Ks) in SAFE.
a and b are respectively the coefficient of stories and the coefficient of different zones of foundation on soft soil.Coefficient of story values (a) are obtained from the following equation.a = 0.25(3 + 3) a = 0.30(3 + 3)

Figure 7 .Table 3 .Figure 7 .Table 3 .
Figure 7. Different zones of sub grade reaction modulus values to calculate the b coefficient.

Table 1 .
Section details of the six-, eight-, and 10-story buildings.

Table 2 .
The used weight of reinforcement (Wr) with constant and variable K S values.