Citation: Khosravifardshirazi, A.;

Johari, A.; Javadi, A.A.; Khanjanpour,

M.H.; Khosravifardshirazi, B.;

Akrami, M. Role of Subgrade

Reaction Modulus in Soil-

Foundation-Structure Interaction in

Concrete Buildings. Buildings 2022,

12, 540. https://doi.org/10.3390/

buildings12050540

Academic Editors: Chih-Wei Lu and

Meen-Wah Gui

Received: 1 March 2022

Accepted: 19 April 2022

Published: 24 April 2022

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional affil-

iations.

Copyright: © 2022 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

buildings

Article

Role of Subgrade Reaction Modulus inSoil-Foundation-Structure Interaction in Concrete BuildingsAli Khosravifardshirazi 1, Ali Johari 2 , Akbar A. Javadi 1,*, Mohammad Hassan Khanjanpour 1,Behnaz Khosravifardshirazi 1 and Mohammad Akrami 1,*

1 Department of Engineering, University of Exeter, Exeter EX4 4QF, UK; ak723@exeter.ac.uk (A.K.);mk592@exeter.ac.uk (M.H.K.); bk341@exeter.ac.uk (B.K.)

2 Department of Civil and Environment Engineering, Shiraz University of Technology, Shiraz 71557-13876, Iran;johari@sutech.ac.ir

* Correspondence: a.a.javadi@exeter.ac.uk (A.A.J.); m.akrami@exeter.ac.uk (M.A.)

Abstract: One of the key issues in structural and geotechnical engineering is that most parts ofbuildings 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) infoundations of concrete buildings was evaluated using the direct finite element method (DFEM). 3Dmodels were developed and used to analyse concrete buildings with different stories constructedon soft soil. Foundation settlement, deformation of foundation, soil pressure diagram, and weightof reinforcement in the foundation were considered as the main parameters. Deformation of thefoundation 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 valuesof subgrade reaction modulus (Ks) in foundation design, the effects of SSI on tall buildings can beconsidered automatically. The results also show that the soil–structure interaction can cause changesin the pattern of foundation settlement, foundation deformation, and the weight of reinforcementused in foundation design. Furthermore, dishing deformation in foundation appeared in terms ofSSI effects. An equation is provided to simplify considering SSI effects in foundation design. Thismethod is practical for civil (especially structural) engineers, and they can conveniently considerthese effects in foundation design without using DFEM.

Keywords: soil–structure interaction; subgrade reaction modulus; finite element method; concretestructures; foundation analysis

1. 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 onthe behaviour of each of these components as well as the complete system. The relativestiffness of a building structure, its foundation, and the soils that support the foundationinfluences the stresses and displacements of both the structure and the soil. In the designof structural systems, soil–structure interaction (SSI) effects are sometimes neglected bythe use of a structural model supported on a fixed base, while there has been significantprogress in understanding the soil–structure interaction (SSI) effects resulting from externalforces such as winds and earthquakes [1]. Given the lack of information about the effects ofSSI 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 parame-ter that should be taken into account. However, simulating SFSI often requires complexmodels 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. https://doi.org/10.3390/buildings12050540 https://www.mdpi.com/journal/buildings

Buildings 2022, 12, 540 2 of 17

been developed to consider these effects. The SSI effect can be disregarded in structuresconstructed on hard subgrade (rock) [2]. However, in structures that are constructed onsoft foundation soil, the soil–structure interaction is very important [3–6]. The structuraldeformations and the soil displacements are not separate from each other during earth-quakes [7,8]. Therefore, there is a need for improved coordination of both geotechnical andstructural engineering practices. The soil’s response affects the structure’s movement, andthe structure’s movement affects the soil’s response [9]. Once the soil deformation is inves-tigated, it concludes that soil–structure interaction has been considered in evaluating thestructural response. Documenting past seismic events is a significant step in understandingand accurately characterizing dynamic geotechnical earthquake engineering problems [10].With the expansion of the construction industry, civil engineering software has been widelyused for simulating earthquake loads on structures and foundations [11–15]. SSI effects areusually caused by the flexibility of the soil under the foundation as well as relative vibrationbetween the foundation and free field [16]. This implies that the response of a structureconstructed on a site depends on the relationship between the structural characteristicsand properties of the underlying soil layers and free field motion. The standard methodsto evaluate soil structure interaction effects can be classified into two groups: direct andsubstructure methods [17,18]. In the direct analysis method, the structure, foundation, andsoil are included in the one model and analysed all together in one step as a completesystem. Consequently, it is unnecessary to use the principle of superposition, and thesoils can be simulated using the finite element methods (FEM) [19]. In this method, thestructure 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 toconsider the non-linear behaviour of soil. The elastic-perfectly plastic vertical stresses haveregular distributed rather than matching elastic stress values, and the elastic-plastic verticaldisplacements are about 25 percent greater than the corresponding elastic values [20]. Thesubstructure or indirect analysis method is the most common method used to solve theproblems of soil–structure interaction. In this method, soil–structure interaction problemsare sub-divided into separate parts of simpler problems. The problems are analysed sepa-rately (using the common methods), and then the results are combined by the principleof superposition to complete the analysis. In this method, it is necessary to assume linearbehaviour for the structure and the soil. Shakib and Fuladgar [21] studied the effects ofdynamic soil–structure interaction on the seismic response of asymmetric buildings. Theyfound 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 proposedan equation to calculate the spring stiffness in the x, y, and z directions on a shallow foun-dation above the soil surface [22] (x and y axis are in direction of foundation’s width andfoundation’s length, respectively, and the z axis is in direction of foundation’s depth). Anew macro-element (very small elements) for modelling the effects of SSI in the shallowfoundation can be used to numerically investigate the behaviour of a wider variety ofconfigurations that are difficult to study experimentally [23].

It has been shown that the top-floor displacement in the model with SSI effects is2% to 3% greater than the model without soil–structure interaction [17]. The structure’sresponse to seismic loads is influenced by the relationships between the three associatedsystems: the structure, foundation, and soil under the foundation [24]. The collaborationbetween geotechnical and structural engineers is therefore vital for a successful design.The subgrade reaction modulus (Ks) can be investigated as a suitable interface betweenthe soil and the structure [25]. It has a conceptual relationship between soil pressure andthe structure’s deformation in foundation design. The subgrade reaction modulus is nota soil constant, but depends on many factors such as the dimensions of the foundation,soil characteristics, load level, and superstructure rigidity. Winkler Theory (1867) providesa simple theory to calculate the contact stresses using the modulus of subgrade reaction,

Buildings 2022, 12, 540 3 of 17

assuming elastic behaviour for soil without considering the effect of the soil-structureinteraction (Equation (1)) [26]:

Ks = P/S (1)

where Ks, P, and S are subgrade reaction modulus, pressure, and vertical settlement,respectively. Boussinesq considered a more realistic model than the Winkler model andshowed that the distribution of displacements is continuous [27]. Filonenko-Borodich [28]and Hetenyi [29,30] proposed some modifications to the Winkler model. Farouk showedthat the distribution of Ks is non-uniform and concentrated at the edge elements of thefoundation and the value reduced in the inner parts [25]. They concluded that ignoring SSIeffects always led to overestimation of the inner foundation design and underestimationof the outer foundation design. Therefore, the role of Ks and consideration of SSI in finiteelement method are very important for practical engineers.

A review of the literature shows that the effects of the variable subgrade reactionmodulus (Ks) for the foundation have not been investigated in detail. In this research, thesoil–structure interaction (SSI) effects in the foundation of concrete buildings with variableKs values were analysed using the direct finite element method (DFEM). In the followingsections, the methodology is presented and the results of the application of the method tothree concrete structures are evaluated and discussed.

2. Materials and Methods

In this study, three concrete structures with six-, eight-, and 10-stories with the sameselected plan (Figure 1) were used to investigate the effects of SSI on the foundation ofthe structure. The structures were designed with ETABS software. ANSYS meshing wasused to discretise the structure. A grid sensitivity analysis was carried on by producingten different meshes (from 398,871 cells to 2,951,341) for the initial design using ANSYSmeshing software. Analysis of the results showed that the standard deviation of the valuesof the base support reactions was around 0.78 % for 1,995,321 cells. Therefore, for therest of the FEM analysis, 1,995,321 was chosen due to the fact that the computationaltime of the numerical analysis increased significantly with an increase in the number ofcells. ANSYS structural analysis software was used to simulate and analyse the structures.The structure, foundation, and soil were modelled simultaneously in ANSYS with theabove-mentioned number of elements. Linear elements with three transitional and threerotational degrees of freedom (BEAM4) were used to discretize the beams and columns. Tomodel the slabs and shear walls, surface elements (SHELL181) with three transitional andthree rotational degrees of freedom were used. In this study, three column sizes and onebeam size were considered in all buildings. The roofs were assumed to be concrete slabswith 0.35 m thicknesses (Figure 2). For the foundation, volume elements 3D SOLID45 withthree transitional degrees of freedom were used. The geometry of the 10-story building isshown in Figure 3.

All the beams and columns were of a rectangular shape and each story was 3 m inheight. Beams, columns, and shear walls were concrete with isotropic materials suchas Ex = 2.52 × 1010 Pa, ϑ = 0.2, density (ρ) = 2400 Kg/m3, where Ex and ϑ are themodulus of elasticity and Poisson’s ratio, respectively. Ceilings, material properties areEx = 2 × 1010 Pa, ϑ = 0.2, density (ρ) = 2060 Kg/m3. Table 1 shows the details of theused sections, Sh-W represents the shear walls in this table.

Buildings 2022, 12, 540 4 of 17

Buildings 2022, 12, x FOR PEER REVIEW 4 of 17

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

Figure 2. Details of the floor slabs.

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

Buildings 2022, 12, x FOR PEER REVIEW 4 of 17

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

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

Buildings 2022, 12, 540 5 of 17

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

6-Story Building 8-Story Building 10-Story Building

Stories Beams Columns Sh-W Stories Beams Columns Sh-W Stories Beams Columns Sh-W

1–2 30 cm × 30 cm 40 cm × 40 cm 20 cm 1–3 40 cm × 40 cm 45 cm × 45 cm 20 cm 1–4 45 cm × 45 cm 55 cm × 55 cm 25 cm

3–4 30 cm × 30 cm 35 cm × 35 cm 20 cm 4–6 40 cm × 40 cm 40 cm × 40 cm 20 cm 5–7 45 cm × 45 cm 50 cm × 50 cm 25 cm

5–6 30 cm × 30 cm 30 cm × 30 cm 20 cm 7–8 40 cm × 40 cm 35 cm × 35 cm 20 cm 8–10 45 cm × 45 cm 45 cm × 45 cm 25 cm

Buildings 2022, 12, 540 6 of 17

Buildings 2022, 12, x FOR PEER REVIEW 5 of 17

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

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=2.52×1010 Pa, ϑ = 0.2, density (r) = 2400 Kg m�⁄ , where Ex and ϑ are the modulus of

elasticity and Poisson’s ratio, respectively. Ceilings, material properties are Ex=2×1010 Pa,

ϑ = 0.2, density (r) = 2060 Kg m�⁄ . 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.

6-Story Building 8-Story Building 10-Story Building

Stories Beams Columns Sh-W Stories Beams Columns Sh-W Stories Beams Columns Sh-W

1–2 30 cm × 30 cm 40 cm × 40 cm 20 cm 1–3 40 cm × 40 cm 45 cm × 45 cm 20 cm 1–4 45 cm × 45 cm 55 cm × 55 cm 25 cm

3–4 30 cm × 30 cm 35 cm × 35 cm 20 cm 4–6 40 cm × 40 cm 40 cm × 40 cm 20 cm 5–7 45 cm × 45 cm 50 cm × 50 cm 25 cm

5–6 30 cm × 30 cm 30 cm × 30 cm 20 cm 7–8 40 cm × 40 cm 35 cm × 35 cm 20 cm 8–10 45 cm × 45 cm 45 cm × 45 cm 25 cm

For all buildings, the foundation considered was 1 m thick with the modulus of elas-

ticity, E = 2.52×1010 Pa, Poisson’s ratio, � = 0.2, and density � = 2400 kg/m�. Three-dimen-

sional 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) = 4×107 Pa, ϑ =0.35, density = 1800 Kg/m3, cohesion (C) = 1×105 Pa, angle

of shearing resistance (∅) = 25°, and dilation angle (ψ) = −7°. Modulus of elasticity (Ex) =

1×108 Pa, ϑ = 0.30, density = 1800 Kg/m3, cohesion (C) = 1×105 Pa, angle of shearing re-

sistance (∅) = 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 combi-

nation of dead, live, and earthquake loads, and type 2 is a combination of dead and live

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

For all buildings, the foundation considered was 1 m thick with the modulus ofelasticity, E = 2.52 × 1010 Pa, Poisson’s ratio, v = 0.2, and density ρ = 2400 kg/m3. Three-dimensional elements in SOLID45 were also employed to model the soil. The thickness ofthe soil was considered to be 20 m in two layers. The Drucker–Prager model was used todescribe the behaviour of the soil. Material properties for the top soil layer are modulusof elasticity (Ex) = 4 × 107 Pa, ϑ = 0.35, density = 1800 Kg/m3, cohesion (C) = 1 × 105 Pa,angle of shearing resistance (∅) = 25◦, and dilation angle (ψ) = −7◦. Modulus of elasticity(Ex) = 1 × 108 Pa, ϑ = 0.30, density = 1800 Kg/m3, cohesion (C) = 1 × 105 Pa, angle of shear-ing 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 betweenthe soil and the foundation. This element was used in these analyses to represent contactand 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 elementis located on the surfaces of 3-D solid or shell elements. Furthermore, parameters such asFKN (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 type2) of loads were applied to analyse the behaviour of the structure. Type 1 is a combinationof 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, whichwere assigned as a pressure load with a value of 200 Kg/m2 for each floor and 150 Kg/m2

for the top floor. The earthquake loads were considered as an inertia acceleration with avalue = 0.981 m/s2 in the buildings’ width and length directions. The base support reac-tions of the three types of buildings were calculated by ANSYS. The results of the numericalanalysis using ANSYS were validated against ETABS software. Figure 4 shows the reactionof the 8-story building in the x-direction (Fx) under earthquake loading applied in thex-direction with ANSYS and ETABS. The static model of the sole structure was carried outto analyse the static response of the 3D fixed-base model under both load combinations. Ascan be seen, ANSYS results were in very good agreement with those obtained from a staticanalysis performed with the ETABS software.

Buildings 2022, 12, 540 7 of 17

Buildings 2022, 12, x FOR PEER REVIEW 6 of 17

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/m2 for each floor and 150

Kg/m2 for the top floor. The earthquake loads were considered as an inertia acceleration

with a value = 0.981 m/s2 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 (Fx) under earthquake loading ap-

plied 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.

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 soft-

ware. The foundation was divided into 1 × 1 m2 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 [34–38]. A trial and error ap-

proach 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.

Figure 4. Comparison of the predicted Fx reactions for the 8-story building under earthquake load-

ing in ANSYS and ETABS.

3. 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 load-

ing. 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 and d show the deformation of the foundation in the z-direction with

-700,000

-600,000

-500,000

-400,000

-300,000

-200,000

-100,000

0

100,000

0 5 10 15 20 25

Fx

(N)

Nodes

ANSYS ETABS

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

SAFE software was used to analyse and design different foundations. The obtainedvalues of base support reactions from ANSYS software were inserted in the SAFE software.The foundation was divided into 1 × 1 m2 elements in SAFE. In this software, the foun-dation length was divided into 20 sections and the width of the foundation was split into14 sections (identified with letters A–O). The values of subgrade reaction modulus (Ks) fordifferent elements were considered to be variable [34–38]. A trial and error approach wasused by changing the values of the subgrade reaction modulus (Ks) for each element untilthe deformations of the foundation in ANSYS matched those of the SAFE.

3. Results

The analysis was repeated for three tall (six, eight, and 10-story) buildings foundedon the selected soil. For an analysis of the results, the deformation of the buildings waselaborated in this section. The six, eight, and 10-story buildings were analysed based ontype 1 and 2 loadings. The behaviour of the structure under type 1 and type 2 loadings ispresented and compared in Figures 5 and 6, respectively. All units in Figures 5 and 6 are inmeters. Figure 5a shows the deformations of the eight-story structure with type 1 loading. Itcan be seen that the total deformation was in the horizontal direction, which is the directionof the applied earthquake load. Figure 5b shows the foundation of the selected model andits deformation based on applied type 1 loading. The middle of the foundation experienceda 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 Ksand with variable Ks, respectively. It can be seen that the maximum deformation of themodel with constant Ks occurred in the corner of the foundation; however, in the modelwith variable KS, it was observed in the middle of the foundation. Deformation of theeight-story structure with type 2 loading is shown in Figure 6a. It can be seen that themaximum deformation occurred at the centre of the foundation. According to Figure 6b,the minimum deformation was at the corner of the foundation. Finally, the deformationof the foundation with constant Ks (Figure 6c) was compared with the foundation withvariable Ks (Figure 6d). The difference between the maximum and minimum deformationsin the constant Ks model was 14 mm, while for the variable Ks model, it was 8 mm. Thisclearly shows that the consideration of variable Ks can lead to different deformations in thefoundations. Moreover, the deformation of the foundation was in good agreement with

Buildings 2022, 12, 540 8 of 17

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 totaldeformation of the foundation considering the SSI effects.

Buildings 2022, 12, x FOR PEER REVIEW 7 of 17

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 and 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 defor-

mation of the model with constant Ks occurred in the corner of the foundation; however,

in the model with variable 𝐾𝑆, it was observed in the middle of the foundation. Defor-

mation 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 foun-

dation 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 defor-

mations 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 re-

sults from total deformation of the foundation considering the SSI effects.

(a)

Buildings 2022, 12, x FOR PEER REVIEW 8 of 17

(b)

Figure 5. Cont.

Buildings 2022, 12, 540 9 of 17

Buildings 2022, 12, x FOR PEER REVIEW 9 of 17

(c)

Figure 5. Cont.

Buildings 2022, 12, 540 10 of 17

Buildings 2022, 12, x FOR PEER REVIEW 10 of 17

(d)

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

Then, the model with variable Ks was analysed and the weight of the used reinforce-ments was calculated. The results are presented in Table 2. In another case, the foundationwas modelled with a constant subgrade reaction modulus of Ks = 1 × 104 kN/m3 and theweight of the used reinforcements was obtained (Table 2). It can be seen the value of thesame Ks in the first row and the domain of variable values of Ks in the second row relatedto six-, eight-, and 10-story buildings in this table. The last column of this table shows theused weight of reinforcement for buildings with different numbers of stories, with the sameKs and with variable Ks. It can be seen that by using variable Ks, the weight of the requiredreinforcement of the foundation was reduced by 32%, 37%, and 38% for six-, eight-, and10-story buildings respectively and it is economical for engineers [40]. Nowadays reliabilityanalysis is considered for uncertainty in exact values of soil parameters [41–43] and it willbe considered in next study in combination with soil-structure interaction effects.

Buildings 2022, 12, 540 11 of 17Buildings 2022, 12, x FOR PEER REVIEW 11 of 17

(a)

(b)

Figure 6. Cont.

Buildings 2022, 12, 540 12 of 17

Buildings 2022, 12, x FOR PEER REVIEW 12 of 17

(c)

Figure 6. Cont.

Buildings 2022, 12, 540 13 of 17

Buildings 2022, 12, x FOR PEER REVIEW 13 of 17

(d)

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.

Then, the model with variable Ks was analysed and the weight of the used reinforce-ments was calculated. The results are presented in Table 2. In another case, the foundation was modelled with a constant subgrade reaction modulus of Ks = 1×104 kN/m3 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]. Nowa-days reliability analysis is considered for uncertainty in exact values of soil parameters

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

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

Story KS Wr

6 1.0 × 104 kN/m3 2693.95 kg6 0.8 × 103 − 5 × 103 kN/m3 1831.66 kg8 1.0 × 104 kN/m3 4152.69 kg8 1.1 × 103 − 5.8 × 103 kN/m3 2588.38 kg10 1.0 × 104 kN/m3 6271.96 kg10 1.5 × 103 − 6.9 × 103 kN/m3 3836.78 kg

There is a correlation between soil elasticity modulus and modulus of subgradereaction 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

Buildings 2022, 12, 540 14 of 17

SSI effects in foundation design. Regions and variable values of subgrade reaction modulusare obtained from Equation (2).

KSNew = a × b × Ks kN/m3 (2)

The dimensionless parameters a and b are respectively the coefficient of stories andthe 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) (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 thesafe side.

Buildings 2022, 12, x FOR PEER REVIEW 14 of 17

[41–43] and it will be considered in next study in combination with soil-structure interac-

tion effects.

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

Story �� Wr

6 1.0 × 104 kN m�⁄ 2693.95 kg

6 0.8 × 103− 5 × 103 kN m�⁄ 1831.66 kg

8 1.0 × 104 kN m�⁄ 4152.69 kg

8 1.1 × 103− 5.8 × 103 kN m�⁄ 2588.38 kg

10 1.0 × 104 kN m3⁄ 6271.96 kg

10 1.5 × 103− 6.9 × 103 kN m3⁄ 3836.78 kg

There is a correlation between soil elasticity modulus and modulus of subgrade re-

action 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 modu-

lus are obtained from Equation (2).

����� = a × b × �� kN m�⁄ (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(3� + 3) �� a = 0.30(3� + 3) (3)

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.

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

Table 3. Values of the b coefficient in Equation (2) based on different zones in Figure 7.

Position Based on the Figure 7 Values of b Coefficient

A1 0.18–0.23

A2 0.20–0.35

A3 0.30–0.40

A4 0.75–1.00

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

Table 3. Values of the b coefficient in Equation (2) based on different zones in Figure 7.

Position Based on the Figure 7 Values of b Coefficient

A1 0.18–0.23

A2 0.20–0.35

A3 0.30–0.40

A4 0.75–1.00

4. Conclusions

In this research, the effects of SSI on the foundation of concrete buildings were investi-gated 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 foun-dation based on Equation (2) is a convenient method for considering soil–structureinteraction in foundation design without using the direct finite element method inmodelling.

(2) By using variable Ks, the weight of the required reinforcement of the foundation canbe reduced considerably.

(3) The results of load combinations illustrate that the total deformations were in thedirection of the applied earthquake load. In particular, consideration of the full in-teraction led to an overall increase in deformability of the soil–foundation–structuresystem compared to the fixed-base structure. In addition, the middle of the founda-tion experienced a significant deformation in the z-direction (in the direction of thefoundation depth).

Buildings 2022, 12, 540 15 of 17

(4) By considering the soil–structure interaction, a dishing effect was observed. Thedeformation of the foundation was in good agreement with the pattern of Ks providedin Chapter 4 of ATC FEMA273.

(5) The results presented in this paper improved the current understanding of the SSI inthe foundations of concrete buildings.

This suggested method can be used for concrete buildings but for steel buildings, itshould 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.

Author Contributions: Conceptualization, A.J. and A.K.; methodology, A.J. and A.K.; Software, A.K.;Validation, A.K., A.J., B.K., M.H.K. and M.A.; Formal analysis, A.K.; Investigation, A.K.; Resources,A.K. and B.K.; Data curation, A.J.; Writing—original draft preparation, A.J. and A.K.; Writing—review and editing, A.A.J., A.K., B.K., M.H.K. and M.A.; Visualization, A.J.; Supervision, A.J. andA.A.J.; Project administration, A.J. All authors have read and agreed to the published version of themanuscript.

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: This is available upon request.

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

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

Nomenclature

E Modulus of elasticityϑ Poisson’s ratioρ DensityC CohesionΦ Internal friction angleψ Dilation angleFKN Normal penalty stiffness factorFTOLN Penetration tolerance factorICONT Initial contact closureKs Subgrade reaction modulusWr Weight of used reinforcementst Thicknessm MeterH Hourglass factorDL Dead loadLL Live loadEx Earthquake load in X axisEy Earthquake load in Y axisS SettlementSUB Sub step of requested resultsRSYS Results coordinate systemDMX Maximum displacementSMN Minimum stressSMX Maximum stressa Coefficient of storiesb Coefficient of different zones

Buildings 2022, 12, 540 16 of 17

References1. Stewart, J.P.; Seed, R.B.; Fenves, G.L. Seismic soil-structure interaction in buildings. II: Empirical findings. J. Geotech. Geoenviron.

Eng. 1999, 125, 38–48. [CrossRef]2. Iqbal, A.; Al-Hussaini, T. Soil-structure interaction effects on tall buildings with mat foundation. SESOC J. 2014, 27, 48–60.3. Hassan, I.; Abeku, D.; Salihu, C.; Maxwell, S. Buildings Collapse and Socio-Economic Development of People in the Federal

Capital Territory, Abuja, Nigeria. Int. J. Econ. Dev. Res. Investig. 2016, 7, 51–64.4. Selvadurai, A.P. Elastic Analysis of Soil-Foundation Interaction; Elsevier: Amsterdam, The Netherlands, 2013.5. Spyrakos, C.; Beskos, D. Dynamic response of flexible strip-foundations by boundary and finite elements. Soil Dyn. Earthq. Eng.

1986, 5, 84–96. [CrossRef]6. Bielak, J. Dynamic response of non-linear building-foundation systems. Earthq. Eng. Struct. Dyn. 1978, 6, 17–30. [CrossRef]7. Datta, D. Effect of Non-Linear Soil Structure Interaction on the Seismic Response of Critical Infrastructure Facilities; Purdue University

Graduate School: West Lafayette, IN, USA, 2020.8. Fathizadeh, S.; Vosoughi, A.; Banan, M.R. Considering soil–structure interaction effects on performance-based design optimization

of moment-resisting steel frames by an engineered cluster-based genetic algorithm. Eng. Optim. 2020, 53, 440–460. [CrossRef]9. El Kahi, E.; Deck, O.; Khouri, M.; Mehdizadeh, R.; Rahmé, P. A new simplified meta-model to evaluate the transmission of ground

movements to structures integrating the elastoplastic soil behavior. Structures 2020, 23, 324–334. [CrossRef]10. Kramer, S.L. Geotechnical Earthquake Engineering; Pearson Education; Prentice-Hall: New Delhi, India, 1996.11. Králik, J. Probabilistic safety assessment to determine the seismic fragility of NPP. MATEC Web Conf. 2020, 313, 00015. [CrossRef]12. Králik, J. Comparison of Probabilistic Methods to Solve the Reliability of Structures in ANSYS. In Proceedings of the Conference

ANSYS, Bratislava, Slovakia; 2009. Available online: https://www.researchgate.net/profile/Juraj-Kralik-2/publication/267949875_Comparison_of_Probabilistic_Methods_to_Solve_the_Reliability_of_Structures_in_ANSYS/links/54beaeea0cf2f6bf4e0420a7/Comparison-of-Probabilistic-Methods-to-Solve-the-Reliability-of-Structures-in-ANSYS.pdf (accessed on 28 February 2022).

13. Băetu, S.; Ciongradi, I.-P. Nonlinear Finite Element Analysis of Reinforced Concrete Slit Walls with ANSYS (I); Faculty of CivilEngineering and Building Services, “Gheorghe Asachi” Technical University of Iasi: Ias, i, Romania, 2011; Volume LVII, pp. 31–45.

14. Talukdar, S. Design, Analysis and Soil-Structure Interaction of an Intze Tank using ETABS, ANSYS and SAFE. Int. J. Eng. Res.Technol. 2018, 07, IJERTV7IS050231.

15. Zhang, Y.; He, X.; Yan, K.; Feng, M. Fatigue Analysis of Single Lap Self-piercing Riveted Joints based on ANSYS/FE-SAFE. NewTechnol. New Process 2011, 9, 29.

16. Hayir, A.; Todorovska, M.I.; Trifunac, M.D. Antiplane response of a dike with flexible soil-structure interface to incident SHwaves. Soil Dyn. Earthq. Eng. 2001, 21, 603–613. [CrossRef]

17. Sudarsana, I.; Widiarsa, I.; Predana, I. Analysis the effects of soil-structure interaction on the seismic behaviours of a 5-storiesreinforced concrete building with basements. IOP Conf. Ser. Earth Environ. Sci. 2020, 419, 012025. [CrossRef]

18. Gutierrez, J.A.; Chopra, A.K. A substructure method for earthquake analysis of structures including structure-soil interaction.Earthq. Eng. Struct. Dyn. 1978, 6, 51–69. [CrossRef]

19. Aydinoglu, M.N. Consistent formulation of direct and substructure methods in nonlinear soil-structure interaction. Soil Dyn.Earthq. Eng. 1993, 12, 403–410. [CrossRef]

20. Alonge, J.L. Modeling the effects of soil-structure interaction on a tall building bearing on a mat foundation. Civil Eng. Pract.2005, 20, 51–68.

21. Shakib, H.; Fuladgar, A. Dynamic soil–structure interaction effects on the seismic response of asymmetric buildings. Soil Dyn.Earthq. Eng. 2004, 24, 379–388. [CrossRef]

22. Gazetas, G. Formulas and charts for impedances of surface and embedded foundations. J. Geotech. Eng. 1991, 117, 1363–1381.[CrossRef]

23. Huynh, V.Q.; Nguyen, X.H.; Nguyen, T.K. A Macro-element for Modeling the Non-linear Interaction of Soil-shallow Foundationunder Seismic Loading. Civ. Eng. J. 2020, 6, 714–723. [CrossRef]

24. NIST. GCR 12-917-21 (2012) Soil-Structure Interaction for Building Structures; National Institute of Standards and Technology:Gaithersburg, MD, USA, 2012.

25. Farouk, H.; Farouk, M. Effect of elastic soil structure interaction on modulus of subgrade reaction. In Proceedings of theGeo-Hubei 2014 International Conference on Sustainable Civil Infrastructure, Yichang, China, 20–22 July 2014; pp. 111–118.

26. Winkler, E. Die Lehre von der Elasticitaet und Festigkeit Besonderer Rucksicht auf ihre Anwendung in der Technik fur Politech-niche Schulen, Bauakademien, Ingenieure, Maschinenbauer, Architecten, etc. 1867. Available online: https://archive.org/details/bub_gb_25E5AAAAcAAJ (accessed on 26 February 2022).

27. Boussinesq, J. Application des Potentiels à l’étude de l’équilibre et du Mouvement des solides élastiques: Principalement au Calcul desDéformations et des Pressions que Produisent, dans ces Solides, des Efforts Quelconques Exercés sur une Petite Partie de Leur Surface oude leur Intérieur: Mémoire Suivi de Notes étendues sur Divers Points de Physique, Mathematique et D’analyse; Gauthier-Villars: Paris,France, 1885. (In French)

28. Filonenko-Borodich, M. Some Approximate Theories of Elastic Foundation; Uchenyie Zapiski Moskovkogo GosudarstuennogoUniversiteta Mekhanika: Moscow, Russia, 1940; Volume 46, pp. 3–18.

29. Hetenyi, M. A general solution for the bending of beams on an elastic foundation of arbitrary continuity. J. Appl. Phys. 1950, 21,55–58. [CrossRef]

Buildings 2022, 12, 540 17 of 17

30. Hetényi, M.; Hetbenyi, M.I. Beams on Elastic Foundation: Theory with Applications in the Fields of Civil and Mechanical Engineering;University of Michigan Press: Ann Arbor, MI, USA, 1946; Volume 16.

31. Khizer, M.; Chen, J.-W.; Yang, G.-W.; Wu, Q.-S.; Song, Y.; Zhang, Y. Seismic and stress qualification of LMFR fuel rod and simplemethod for the determination of LBE added mass effect. Nucl. Sci. Tech. 2020, 31, 5. [CrossRef]

32. Hawileh, R.A. Computational Investigation of the Behavior of Precast Hybrid Beam-Column Connections; The University of Wisconsin-Milwaukee: Milwaukee, WI, USA, 2005.

33. Galik, K. The Effect of Design Variations on Stresses in Total Ankle Arthroplasty; University of Pittsburgh: Pittsburgh, PA, USA, 2002.34. Salari, P.; Hafezi Moghaddas, N.; Reza Lashkaripour, G.; Ghafoori, M. Presenting new models to determine subgrade reaction

modulus (Ks) for optimizing foundation calculations in coarse grained soils. Rev. Construcción 2020, 19, 235–243.35. Salari, P.; Moghaddas, N.H.; Lashkaripour, G.R.; Ghafoori, M. Evaluating the Subgrade Reaction Modulus Variations with Soil

Grains Shape in Coarse-Grained Soils Using Genetic Algorithm. Open J. Geol. 2020, 10, 111–123. [CrossRef]36. Ahmad, H.; Mahboubi, A.; Noorzad, A. Scale effect study on the modulus of subgrade reaction of geogrid-reinforced soil. SN

Appl. Sci. 2020, 2, 394. [CrossRef]37. Naeini, S.A.; Moayed, R.Z.; Kordnaeij, A.; Mola-Abasi, H. Prediction of subgrade reaction modulus of clayey soil using group

method of data handling. Sci. Iran. 2020, 27, 1740–1750. [CrossRef]38. Alafeena, M.A.; Aljorany, A.N. Experimental Study on the Modulus of Subgrade Reaction for Sandy Soil Improved by Grouting

Materials. Key Eng. Mater. 2020, 857, 358–366. [CrossRef]39. FEMA. NEHRP Guidelines for the Siesmic Rehabilitation of Buildings; FEMA 274; Federal Emergency Management Agency: Washing-

ton, DC, USA, 1996.40. Larkela, A.; Mengelt, M.; Stapelfeldt, T. Determination of distribution of modulus of subgrade reaction. In Proceedings of the 18th

International Conference on Soil Mechanics and Geotechnical Engineering, French Society for Soil Mechanics and GeotechnicalEngineering (CFMS), Paris, France, 2–6 September 2013; pp. 1313–1315.

41. Johari, A.; Sabzi, A. Reliability analysis of foundation settlement by stochastic response surface and random finite elementmethod. Sci. Iran. 2017, 24, 2741–2751. [CrossRef]

42. Johari, A.; Sabzi, A.; Gholaminejad, A. Reliability analysis of differential settlement of strip footings by stochastic response surfacemethod. Iran. J. Sci. Technol. Trans. Civ. Eng. 2019, 43, 37–48. [CrossRef]

43. Johari, A.; Hosseini, S.; Keshavarz, A. Reliability analysis of seismic bearing capacity of strip footing by stochastic slip linesmethod. Comput. Geotech. 2017, 91, 203–217. [CrossRef]

44. Putri, E.E.; Rao, N.; Mannan, M. Evaluation of modulus of elasticity and modulus of subgrade reaction of soils using CBR test. J.Civ. Eng. Res. 2012, 2, 34–40. [CrossRef]