geosciences

Article Numerical Modelling of Structures Adjacent to Retaining Walls Subjected to Earthquake Loading

Xiaoyu Guan and Gopal S. P. Madabhushi *

Schofield Centre, Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK; [email protected] * Correspondence: [email protected]; Tel.: +44-12-2376-8053



Received: 28 September 2020; Accepted: 26 November 2020; Published: 3 December 2020


Abstract: In an urban environment, it is often necessary to locate structures close to existing retaining walls due to congestion in space. When such structures are in seismically active zones, the dynamic loading attracted by the retaining wall can increase. In a novel approach taken in this paper, finite element-based numerical analyses are presented for the case of a flexible, cantilever sheet pile wall with and without a structure on the backfill side. This enables a direct comparison of the influence exerted by the structure on the dynamic behaviour of the retaining wall. In this paper, the initial static bending moments and horizontal stresses prior to application of any earthquake loading are compared to Coulomb’s theory. The dynamic behaviour of the retaining wall is compared in terms of wall-top accelerations and bending moments for different earthquake loadings. The dynamic structural rotation induced by the differential settlements of the foundations is presented. The accelerations generated in the soil body are considered in three zones, i.e., the free field, the active and the passive zones. The differences caused by the presence of the structure are highlighted. Finally, the distribution of horizontal soil pressures generated by the earthquake loading behind the wall, and in front of the wall is compared to the traditional Mononobe-Okabe type analytical solutions.

Keywords: earthquake; finite element modelling; retaining walls; structures

1. Introduction The increasing population in large cities and towns around the world place a high demand on available land. In such situations, excavations are required next to existing buildings to construct new metros, railways, highways or new buildings with basement floors. Retaining walls are often used to create the necessary difference in ground levels with backfills placed behind the retaining walls or excavations carried out in front of them. In the former case, new buildings could be constructed close to the retaining wall, while, in the latter case, where existing buildings are present behind the wall, the retaining walls have to protect from any damage due to ground movement. The location of these structures can often be quite close to the retaining wall, often only a few meters away. An example of a retaining wall close to an existing structure is shown in Figure1. In this figure the retaining wall is anchored using soil nails, but many cases exist where cantilever retaining walls are used. The designer has to evaluate the additional soil-structure interaction effects due to the presence of the structure on the backfill side.

Geosciences 2020, 10, 486 ; doi:10.3390/geosciences10120486 www.mdpi.com/journal/geosciences Geosciences 2020, 10, 486 2 of 18 Geosciences 2020, 10, x FOR PEER REVIEW 2 of 18

Figure 1. AA structure structure in in close close proximity proximity to to a a retaining retaining wall wall [1] [1] (courtesy (courtesy Aarsleff Aarsleff GroundGround Engineering LtdLtd.. ( (Newark,Newark, UK UK)).)).

In seismic regions of the world, additional loading is is expected on on the retaining wall due to earthquake loading. In In these these extreme extreme loading loading cases cases,, it it can can be be anticipated anticipated that additional lateral loads act on on the the retaining retaining wall wall due due to increased to increased horizontal horizontal stresses, stres additionalses, additional inertial inertial loads from loads the from building, the buildingetc. There, etc. is a There significant is a significant dynamic soil-structuredynamic soil- interactionstructure interaction due to the due action to ofthe earthquake action of earthquake loads. It is loads.imperative It is that imperative designers that provide designers adequate provide wall adequate sections for wall the sections retaining for wall the to retaining withstand wall these to withstandadditional these loads. additional Additionally, loads. the Additionally wall movements, the canwall have movements adverse ecanffects have on adverse the foundations effects on of the foundationsstructure causing of theit structure to settle causing and rotate. it to The settle interaction and rotate. between The interaction the retaining between walls the and retaining structures walls is andalso structures important is from also the important bridge engineering from the bridge perspective. engineering For example,perspective. the spandrelFor example walls, the of spandrel masonry wallsbridges, of masonry acting as retainingbridges, acting walls, as interact retaining strongly walls, with interact the backfill strongly soils. with Pel theà et backfill al. [2] andsoils. D’Amato Pelà et al. et [al.2] [and3] have D’Amato investigated et al. [3 the] have seismic investigated behaviour the of seismic masonry behaviour arch bridges. of masonry Similarly, arch the bridges. use of Similarly static and, thedynamic use of tests static on and bridges dynamic was recommendedtests on bridges based was onrecommended the modal analyses based on by the Marcheggiani modal analyses et al. [by4]. MarcheggianiIn all these cases, et al. the [4 consideration]. In all these ofcases the, interactionthe consideration between of the the abutment interaction walls between and the the soil abutment backfill wallsis important, and the particularlysoil backfill duringis important, strong particularly earthquake dur loading.ing strong earthquake loading. There have been examples of damage to to retaining walls in in previous earthquakes earthquakes—for—for example example,, a sea front retaining wall collapsed during the Bhuj earthquake in 2001 [5].]. Zeng and Steedman [ 6] used dynamic centrifuge modelling ttoo investigate flexible,flexible, cantilever retaining walls subjected to earthquake loading with dry and saturated backfills.backfills. Madabhushi Madabhushi and and Zeng [ 7,87,8]] investigated the same retaining walls using the finite finite element method in the time domain using a fully coupled co codede called Swandyne [ 9]].. They They show show that that the the finite finite element element ( (FE)FE) code code SWANDYNE SWANDYNE was able to capture the dynamic response observed in in the centrifuge tests tests both in terms of the wall displacements and dynamic bending moments. Cilingir Cilingir et et al al.. [ [1010]] extended this w workork to investigate anchored retaining walls in dry soils. Their research research shows that SWANDYNE was able to capture the magnitude of the anchor forces recorded in the dynamic centrifuge test well, in addition to the bending moments and wall displacements. More More recent recent improvements in in the numerical modelling of retaining walls were carried out by Callisto and Soccodato [ [11]] and Conti and Viggiani [ 12] with the aim of developing improved design methods. Conti and Viggiani Viggiani [ 12] used used the the “ “thresholdthreshold or or yield yield acc acceleration”eleration” concept concept to estimate estimate the the lateral lateral wall wall displacements displacements and and the accumulation the accumulation of the ofwall the displacements wall displacements over diff erent over different cycles of an earthquake. Yeganeh et al. [13] carried out numerical analyses of a high-rise building behind an anchored wall, observing more damage in the model where the real building was

Geosciences 2020, 10, 486 3 of 18 cycles of an earthquake. Yeganeh et al. [13] carried out numerical analyses of a high-rise building Geosciences 2020, 10, x FOR PEER REVIEW 3 of 18 behind an anchored wall, observing more damage in the model where the real building was simulated insteadsimulated of being instead simplified of being as simplified surcharge asloading. surcharge Although loading the. dynamicAlthough behaviour the dynamic of retaining behaviour walls of wasretaining investigated walls was by manyinvestigated researchers by many previously researchers [6–8], previously very little research [6–8], very exists little in theresearch literature exists that in investigatedthe literature the that dynamic investigated performance the dynamic of the performance retaining walls of the in retaining the presence walls of in a the structure presence on of the a backfillstructure side, on the in close backfill proximity side, in to close the retainingproximity wall. to the retaining wall. InIn many many cases, cases, thethe soil soil onon the the backfill backfill side, side, asas well well asas in in front front ofof the the retainingretaining wall, wall, maymay be be saturated,saturated, suchsuch asas inin thethe casecase ofof water-frontwater-front structures.structures. EarthquakeEarthquake loadingloading cancan leadlead toto fullfull oror partialpartial liquefactionliquefaction ofof suchsuch soils.soils. In this paper, thethe focusfocus isis onon drydry sandysandy soilssoils andand thethe performanceperformance ofof thethe retainingretaining wallwall isis studiedstudied forfor thisthis relativelyrelatively simplesimple soilsoil case,case, so thatthat the eeffectffect ofof thethe structurestructure onon thethe backfillbackfill cancan bebe easilyeasily identified.identified. The presencepresence of pore fluidfluid inin thethe saturatedsaturated soilsoil casecase wouldwould leadlead toto additionaladditional loadingloading duedue toto hydrodynamichydrodynamic actionaction onon thethe retainingretaining wall,wall, asas wellwell asas degradationdegradation inin soilsoil stistiffnessffness duedue toto excessexcess porepore pressurepressure generation.generation. Overall,Overall, thisthis paperpaper adoptsadopts aa novelnovel approachapproach wherewhere anan identicalidentical flexibleflexible retainingretaining wallwall systemsystem willwill bebe analysedanalysed withwith andand withoutwithout aa structurestructure placedplaced onon thethe backfillbackfill side.side. ThisThis allowsallows forfor aa directdirect comparisoncomparison betweenbetween the retaining wall onon itsits ownown andand thethe changeschanges inducedinduced byby thethe presencepresence ofof thethe structurestructure andand itsits foundations.foundations.

2. Numerical Modelling 2. Numerical Modelling InIn thisthis paper,paper, finitefinite elementelement modellingmodelling isis carriedcarried outout usingusing thethe SwandyneSwandyne codecode [[99],], whichwhich isis aa general-purposegeneral-purpose FE FE code code for for problems problems in Geomechanics. in Geomechanics. As explained As explained earlier, earliertwo models, two are models analysed are withanalysed and withoutwith and a structurewithout a behind structure the behind retaining the wall. retaining The dimensions wall. The dimensions of the model of withthe model a structure with h area structure shown inare Figure shown2. in The Fig retainingure 2. The wall retaining had a retainedwall had soil a retained height (soil) ofheight 3.6 m, (h and) of 3.6 the m wall, and itself the d h d penetratedwall itself penetrated to a depth (to) ofa depth 5.76 m, (d giving) of 5.76 an m,/ givingratio ofan 0.625. h/d ratio The of soil 0.625. was The modelled soil was as amodelled uniform as dry a sanduniform bed dry on either sand sidebed ofon the either retaining side of wall. the Theretaining structure wall. was The modelled structure as awas sway modelled frame with as a a sway high centreframe ofwit gravity.h a high In centre addition, of gravity. loading In on addition, the roof loading was chosen on the to beroof asymmetric, was chosen with to be larger asymmetric loading, appliedwith large tor the loading columns applied closer to to the the columns retaining clos wall.er to The the columnsretaining of wall. the structureThe columns were of supported the structure on individualwere supported strip footings.on individual The right-hand strip footings. side The structural right-hand footing side was structural placed about footing 2.1 was m from placed the about back edge2.1 m offrom the the retaining back edge wall. of Inthe Swandyne retaining wall. FE code, In Swandyne two distinct FE stepscode, aretwo requireddistinct steps for any are dynamicrequired analyses.for any dynamic Initially, analyses. a static run Initially must, be a static executed run tomust establish be executed the geostatic to establish stresses the in geostatic the soil and stresses for the in modelthe soil to and establish for the equilibrium model to establish of forces. equilibrium In the second of forces. step, the In finalthe second stressstate step, from the final the static stress run state is utilisedfrom the as static the initial run is stress utilised state as tothe start initial the stress dynamic state run. to start the dynamic run.

FigureFigure 2.2. DimensionsDimensions ofof aa retainingretaining wallwall systemsystem (unit:(unit: m).m). 2.1. Finite Element Discretisation 2.1. Finite Element Discretisation The finite element discretisation used in the analyses described in this paper was carried out using The finite element discretisation used in the analyses described in this paper was carried out eight noded isoparametric, quadrilateral elements with nine gauss points for integration. This enabled using eight noded isoparametric, quadrilateral elements with nine gauss points for integration. This accurate modelling of the soil elements, as well as the structural elements forming the retaining wall and the enabled accurate modelling of the soil elements, as well as the structural elements forming the retaining wall and the sway frame structure. Slip elements were utilised next to the retaining wall on both the active and passive sides and below the structure. The material properties used in these analyses are presented in Section 2.2.

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Geosciences 2020, 10, x FOR PEER REVIEW 4 of 18 sway frame structure. Slip elements were utilised next to the retaining wall on both the active and passive sidesAn and overview below the of structure. the FE discretisation The material propertiesis shown in used Figure in these 3a for analyses the case are of presented a retaining in Section wall only 2.2. and FigureAn overview 3b for the of thecase FE ofdiscretisation a retaining wall is shownwith a insway Figure frame3a forstructure the case behind of a retaining it. In each wall of these only FEand meshes Figure3, bthere for the were case 1690 of a retainingand 1716 wallsolid with nodes a sway, respectively, frame structure with behindtwo degrees it. In eachof freedom of these per FE node.meshes, The there number were 1690of elements and 1716 in solideach nodes,of these respectively, meshes were with 781 two and degrees 789, respectively. of freedom per The node. element The sizesnumber and of time elements steps inineach the dynamic of these meshes analyses were were 781 chosen and 789, to respectively.allow for the The transmission element sizes of S andh waves time fromsteps the in the bedrock dynamic towards analyses the were ground chosen surface to allow following for the the transmission recommendations of Sh waves of Semblat from the et bedrock al. [14] andtowards Haigh the et ground al. [15]. surface following the recommendations of Semblat et al. [14] and Haigh et al. [15].

(a) (b)

Figure 3. Finite element (FE) discretization: (a) a retaining wall only and (b) a retaining wall with an Figure 3. Finite element (FE) discretization: (a) a retaining wall only and (b) a retaining wall with an adjacent structure on the backfill. adjacent structure on the backfill. In these FE analyses, the base nodes of the meshes shown in Figure3a,b were fixed in both the x and yIndirections. these FE analyses This was, the necessary, base nodes since of the the earthquake meshes shown loading in Figure was applied 3a,b were as nodal fixed accelerations in both the x and y directions. This was necessary, since the earthquake loading was applied as nodal at all the fixed nodes in the meshes. In these analyses, a simple sinusoidal motions and sine sweep accelerationsmotions were at used all the as fixed the input nodes accelerations in the mesh toes. theIn these base nodes.analyses The, a simple reason sinusoidal for this choice motions of input and sine sweep motions were used as the input accelerations to the base nodes. The reason for this choice motion is to determine the evolution of the peak bending moments at a time instant and the evolution of input motion is to determine the evolution of the peak bending moments at a time instant and the of horizontal earth pressures next to the retaining wall, with and without the structure on the backfill. evolution of horizontal earth pressures next to the retaining wall, with and without the structure on The edge boundaries were modelled as fixed in the x direction and free in the y direction. While these the backfill. The edge boundaries were modelled as fixed in the x direction and free in the y direction. boundary conditions are satisfactory for the static analyses, it is recognised that unwarranted wave While these boundary conditions are satisfactory for the static analyses, it is recognised that reflections may occur when earthquake loading is applied. In previous research, e.g., Madabhushi unwarranted wave reflections may occur when earthquake loading is applied. In previous research, and Zeng [7,8] and Cilingir et al. [10], absorbing boundaries were used in the numerical analyses to e.g., Madabhushi and Zeng [7,8] and Cilingir et al. [10], absorbing boundaries were used in the simulate the boundaries in the dynamic centrifuge tests. In this paper, the lateral extent of the boundary numerical analyses to simulate the boundaries in the dynamic centrifuge tests. In this paper, the value problem was sufficiently far from the active and passive zones of the boundary. The material lateral extent of the boundary value problem was sufficiently far from the active and passive zones and Rayleigh damping in the numerical analyses were considered to be large enough to absorb the of the boundary. The material and Rayleigh damping in the numerical analyses were considered to wave reflections. Nonetheless, it must be pointed that some error will occur in the dynamic analyses be large enough to absorb the wave reflections. Nonetheless, it must be pointed that some error will due to the wave reflections from the vertical boundaries. occur in the dynamic analyses due to the wave reflections from the vertical boundaries. The flexural properties and densities of different components used in these analyses are presented The flexural properties and densities of different components used in these analyses are in Table1. The flexural sti ffness is expressed for the retaining wall and the structural columns and is presented in Table 1. The flexural stiffness is expressed for the retaining wall and the structural expressed as “per m”, as the analyses were carried out with a plane strain formulation. The average columns and is expressed as “per m”, as the analyses were carried out with a plane strain formulation. density of the slab was calculated to allow for the additional mass added to the right-hand edge of the The average density of the slab was calculated to allow for the additional mass added to the right- structure (Figure3b) to provide additional structural loads for the right-hand side columns. This allows hand edge of the structure (Figure 3b) to provide additional structural loads for the right-hand side for the inclusion of a degree of asymmetry in the structure. The resulting bearing pressures on the left columns. This allows for the inclusion of a degree of asymmetry in the structure. The resulting and right footings are also included in Table1. bearing pressures on the left and right footings are also included in Table 1.

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Table 1. Structural properties.

Structural Element Flexural Stiffness (EI) (MNm 2/m) Density (kg/m 3) Bearing Pressure (kPa) structural columns 66.68 2800 - slab Rigid 5906.7 - strip footing (left) Rigid 2800 83.0 strip footing (right) Rigid 2800 117.0 retaining wall 34.02 2800 -

2.2. Constitutive Models In the analyses described in this paper, three types of constitutive models were used. The simple “elastic” model was used for the sway frame structure including the roof slab and the footings. Similarly, the retaining wall was also modelled as an “elastic” material. The constitutive parameters ascribed to these materials are presented in Table2. Interface elements were used between the retaining wall and the soil both on the active and passive sides. These were modelled as “slip elements”, with the properties shown in Table2.

Table 2. Details of constitutive parameters.

Value Parameter Interface Structure/Retaining Definition Dry Sand Elements Wall Constitutive model Mohr-Coulomb V Slip Elastic Type of constitutive model used Young’s modulus 50 MPa 50 MPa 70 GPa Soil stiffness for static equilibrium Young’s modulus Soil stiffness for damping in 50 MPa 50 MPa 70 GPa (dynamic) dynamic analyses Links strains in horizontal and Poisson’s ratio 0.3 0.3 0.15 vertical directions Uniaxial yield 100 Pa - - Cohesion stress Friction angle 30º 16.4º - To obtain critical state failure line (critical state) Dilatancy angle 2º - - To obtain the peak friction angle Work-hardening The slope of the stress vs. yield 100 - - modulus strain For the calculation of material Void ratio 0.8 - - density

For modelling soil behaviour in FE analyses of geotechnical problems, it is important to account for the plastic behaviour of soil. Accordingly, many elasto-plastic models are used in the FE analysis. One of the simplest soil models is the Mohr-Coulomb type model. Such a model has been adapted to handle the elastic-perfectly plastic behaviour of soil, which has been implemented in the FE code SWANDYNE. Various versions of such models are available in this code, with increasing levels of complexity. The analyses presented in this paper were carried out using Mohr-Coulomb V model. The main features of the model are briefly outlined below, and the actual parameters that are used are presented in Table2. Since the primary concern in these analyses is with dry, sandy soil, the nonassociated flow rule must be used. Many researchers have established that sandy soils follow the nonassociative flow rule [16–18]. The Mohr-Coulomb V model in SWANDYNE captures the nonassociative Mohr-Coulomb elastic-perfectly plastic response. The Mohr-Coulomb V constitutive model was used for cyclic loading problems investigated by previous researchers [7,10,19]. The main advantages of this constitutive model are discussed below: The variation of bulk and/or shear modulus is considered with mean confining effective stress. • Geosciences 2020, 10, 486 6 of 18

Cohesion can be included. The stress state will be cut off if the mean effective confining stress • is more negative than the allowable cohesion. In the analyses presented in this, only a nominal cohesion of 100 Pa was used. The plastic potential can have a different slope with the yield surface. • A smooth fit to the triaxial compression and triaxial extension state, so there is no corner or • singularity in the π-plane [20]. Strain hardening of the soil is incorporated in this model, so that the hardening of soil that occurs • cycle by cycle during an earthquake load can be captured.

Some of the limitations of the Mohr-Coulomb model are given below:

As the Mohr-Coulomb model is a simple model, the shapes of the yield surface and plastic • potential surface are similar. This model cannot be used for saturated soil conditions, as it does not capture the volumetric • strains under cyclic loading, and hence, no excess pore pressure build-up occurs.

In the present analyses, a nominal cohesion of 100 Pa was used primarily to achieve numerical stability. Additionally, the interface friction between the retaining wall and the dry sand was chosen to be about 0.5 the peak friction angle of the sand. In this paper, only uniform dry sand was considered. However, in case of waterfront structures, the shallow portions of the backfill above the water table can be under suction. Ng et al. [21] discussed the variation of soil stiffness and its influence on the retaining walls under such conditions. Although it is beyond the scope of this paper, it would be interesting to consider the influence of such suction dependent shear modulus and its variation under cyclic loading.

3. Static Equilibrium of the Retaining Wall As explained earlier, before Swandyne code can be used to run any dynamic analyses, it requires a static run to establish the geostatic stresses. Such runs were carried out for the present analyses, and in this section, the static behaviour of the retaining wall on its own will be compared to the case where a structure is present on the backfill side. It must be pointed out that, in running the static analyses, the retaining wall was wished-in-place, i.e., the construction process of the retaining wall was not taken into account.

3.1. Wall Deflections and Bending Moments The two important parameters in the retaining wall design are the static wall deflections on completion of the wall construction and the bending moments that are generated in the wall section. In Figure4, the normalised static wall deflections are plotted against the normalised depth. Both these parameters are normalised with the total height of the wall Hwall used in these analyses. In Figure4, the normalised wall deflections obtained for the case of the retaining wall only (i.e., “no structure”) are compared with the case when the sway frame structure was present on the backfill side (see Figure3b). Clearly, the wall deflections increased substantially when the structure is in place behind the retaining wall. The wall tip deflection increased nearly by a factor of three. In Figure4, the increased curvature of the wall can also be seen. This implies the bending moments in the wall, which are proportional to the curvature, should also increase. In Figure5, the normalised bending moments are plotted against the normalised depth. The bending moments M are normalised by the unit weight of the soil and wall height (H3) to obtain a nondimensioned variable. The same normalisation will be used for the results from dynamic analyses presented later. In Figure5, it can be seen that the normalised bending moments increased substantially in the presence of the structure. Further, it can also be seen that the location of the peak bending moment was just below the excavation depth for the retaining wall-only case. This location of peak bending moment seems to shift downwards in the presence of the structure in the backfill. In general, the shapes of the bending moment curves are as expected in a static analysis. Geosciences 2020, 10, x FOR PEER REVIEW 6 of 18

 A smooth fit to the triaxial compression and triaxial extension state, so there is no corner or singularity in the π-plane [20].  Strain hardening of the soil is incorporated in this model, so that the hardening of soil that occurs cycle by cycle during an earthquake load can be captured. Some of the limitations of the Mohr-Coulomb model are given below:  As the Mohr-Coulomb model is a simple model, the shapes of the yield surface and plastic potential surface are similar.  This model cannot be used for saturated soil conditions, as it does not capture the volumetric strains under cyclic loading, and hence, no excess pore pressure build-up occurs. In the present analyses, a nominal cohesion of 100 Pa was used primarily to achieve numerical stability. Additionally, the interface friction between the retaining wall and the dry sand was chosen to be about 0.5 the peak friction angle of the sand. In this paper, only uniform dry sand was considered. However, in case of waterfront structures, the shallow portions of the backfill above the water table can be under suction. Ng et al. [21] discussed the variation of soil stiffness and its influence on the retaining walls under such conditions. Although it is beyond the scope of this paper, it would be interesting to consider the influence of such suction dependent shear modulus and its variation under cyclic loading.

3. Static Equilibrium of the Retaining Wall As explained earlier, before Swandyne code can be used to run any dynamic analyses, it requires a static run to establish the geostatic stresses. Such runs were carried out for the present analyses, and in this section, the static behaviour of the retaining wall on its own will be compared to the case where a structure is present on the backfill side. It must be pointed out that, in running the static analyses, the retaining wall was wished-in-place, i.e., the construction process of the retaining wall was not taken into account.

3.1. Wall Deflections and Bending Moments The two important parameters in the retaining wall design are the static wall deflections on completion of the wall construction and the bending moments that are generated in the wall section. In Figure 4, the normalised static wall deflections are plotted against the normalised depth. Both these parameters are normalised with the total height of the wall Hwall used in these analyses. In Figure 4, the normalised wall deflections obtained for the case of the retaining wall only (i.e., “no structure”) are compared with the case when the sway frame structure was present on the backfill side (see GeosciencesFigure2020 3b)., 10, Clearly 486 , the wall deflections increased substantially when the structure is in place behind7 of 18 the retaining wall. The wall tip deflection increased nearly by a factor of three.

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In Figure 4, the increased curvature of the wall can also be seen. This implies the bending moments in the wall, which are proportional to the curvature, should also increase. In Figure 5, the normalised bending moments are plotted against the normalised depth. The bending moments M are normalised by the unit weight of the soil and wall height (H3) to obtain a nondimensioned variable. The same normalisation will be used for the results from dynamic analyses presented later. In Figure 5, it can be seen that the normalised bending moments increased substantially in the presence of the structure. Further, it can also be seen that the location of the peak bending moment was just below the excavation depth for the retaining wall-only case. This location of peak bending moment seems to shift downwards in the presence of the structure in the backfill. In general, the shapes of the FigureFigure 4. Static4. Static deflection deflectionof of thethe retaining wall wall.. bending moment curves are as expected in a static analysis.

FigureFigure 5. Normalised 5. Normalised static static bending bending momentsmoments in in the the retaining retaining wall. wall.

3.2. Horizontal3.2. Horizontal Stresses Stresses and and Earth Earth Pressures Pressures It wouldIt would be interesting be interesting to compareto compare the the horizontal horizontal stressesstresses mobilised mobilised in inthe the soil soil for forthe thetwo twocases. cases. In FigureIn Figure6a, the 6a horizontal, the horizontal stress stress contours contours in in the the soil soil body body are plotted for for the the “retaining “retaining wall wall-only”-only” case incase the in unit the unit of kPa. of kPa. In thisIn this figure, figure it, it can can be be seen seen that,that, generally generally,, the the horizontal horizontal stresses stresses increase increase with depth. However, there is a bulge of increased horizontal stress in front of the retaining wall on with depth. However, there is a bulge of increased horizontal stress in front of the retaining wall the passive side, which is required to establish the equilibrium of the retaining wall. There are also on thesmall passive drops side, in the which horizontal is required stresses toon establish the active theside. equilibrium In Figure 6b,of an theequivalent retaining contour wall. plot There for are also smallthe horizontal drops in stresses the horizontal in the case stresses with the on structure the active on the side. backfill In Figureside is 6presented.b, an equivalent In this figure contour, plot forthe the horizontal horizontal contours stresses are not in thevery case uniform with on the the structure active side on due the to backfillthe presence side of is the presented. two footings In this figure,of the the horizontal structure. The contours horizontal are not stresses very below uniform the footings on the active are expected side due to increaseto the presence due to the of the two footingsincreased of bearing the structure. pressure The of the horizontal footings, causing stresses horizontal below the stress footings bulbs areto form expected in this to region. increase It is due to thealso increased interesting bearing to see pressure that, on of the the passive footings, side causing, the horizontal horizontal stresses stress increase bulbs just to form in front in this of the region. It is alsowall. interesting A much larger to see pressure that, on bulb the is passiveformed in side, this the region horizontal than in the stresses “retaining increase wall- justonly in” case. front of the wall.Again, A muchthis larger larger pressure pressure bulb bulb is required is formed to establish in this equilibrium region than in inthe the “structure “retaining on backfill wall-only”” case.case. Again, thisIn larger addition pressure to the bulb general is required distribution to establish of the ho equilibriumrizontal stresses in the shown “structure in Figure on backfill”6a,b, it is case. Inpossible addition to obtain to the generalthe horizontal distribution earth pressures of the horizontal acting at each stresses gauss shown point inof Figurethe soil6 elementsa,b, it is possiblenext to obtainto the the retaining horizontal walls earthfrom the pressures FE analyses acting. In addition, at each the gauss Coulomb’s point ofearth the pressure soil elements coefficients next can to the retainingbe calculated walls from using the the FE following analyses. well In-known addition, Equations the Coulomb’s (1) and (2) earthfor active pressure and passive coeffi cases:cients can be calculated using the following well-known Equationssin(α (1)− φ and) (2) for active and passive cases: ⎛ sinα ⎞ K = , (1)  sin(φ + δ)sin(φ2− β) sin(α + δ) +sin(α ϕ)  ⎝  sin−α sin(α − β) ⎠ Ka =  r  , (1)  sin(ϕ+δ) sin(ϕ β)   (α+δ)+ −   √sin sin(α β)  −

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Geosciences 2020, 10, x FOR PEER REVIEW 8 of 18  2 Geosciences 2020, 10, x FOR PEER REVIEW  sin(α+ϕ)  8 of 18  sin α  Kp =  sinr(α + φ)  . (2)  (ϕ δ) (ϕ β)   (α δ) sin + sin +  ⎛  √sin sinα (α β)  ⎞ K = − − sin . (2) sinsin(α(+φφ+)δ−)sin(φ + β) sin(α − δ) − ⎛ sinα sin(α − β) ⎞ In Figure7, the static distributionsK =⎝ of the earth pressures acting⎠ on. the retaining wall for(2) the sin(φ + δ)sin(φ + β) “wall-only” case and the “structure onsin the(α backfill”− δ) − case are plotted, both on the active side and the In Figure 7, the static distributions⎝ of the earth pressuressin(α − actingβ) on⎠ the retaining wall for the passive“wall side.-only The” case theoretical and the “ earthstructure pressure on the distributions backfill” case given are plotted, by Equations both on (1)the andactive (2) side are and also the plotted In Figure 7, the static distributions of the earth pressures acting on the retaining wall for the in thispassive figure. side. On The the theoreticalactive side, earth the pressureearth pressures distributions for the given “wall-only” by Equations case (1 are) and pretty (2) are close also to the “wall-only” case and the “structure on the backfill” case are plotted, both on the active side and the Coulomb’splotted activein this figure. earth pressureOn the active line. side However,, the earth forpressures the “structure for the “wall on-only the backfill”” case are pretty case, theclose active passive side. The theoretical earth pressure distributions given by Equations (1) and (2) are also earthto pressure the Coulomb’s shows active a pressure earth bulgepressure between line. However, 2–4-m depth. for the This“structure is due on to the the backfill increased” case, horizontal the plotted in this figure. On the active side, the earth pressures for the “wall-only” case are pretty close active earth pressure shows a pressure bulge between 2–4-m depth. This is due to the increased pressuresto the coming Coulomb’s from active the foundations earth pressure of line. thestructure. However, Onfor the “ passivestructure side, on the for backfill the “wall-only” case, the case”, horizontal pressures coming from the foundations of the structure. On the passive side, for the “wall- it canactive be seen earth that press the passiveure shows pressures a pressure do bulgenot fully between mobilise 2–4- comparedm depth. This to the is due Coulomb’s to the increased distribution. only case”, it can be seen that the passive pressures do not fully mobilise compared to the Coulomb’s This ishorizontal well-known, pressures and coming only partial from the passive foundations pressures of the arestructure. mobilised On the to passive attain side, equilibrium. for the “wall The- FE distribution. This is well-known, and only partial passive pressures are mobilised to attain analysesonly also case”, show it can that be seen the that passive the passive earth pressures pressure do mobilisation not fully mobilise depends compared on the to vertical the Coulomb’s stress and equilibrium. The FE analyses also show that the passive earth pressure mobilisation depends on the distribution. This is well-known, and only partial passive pressures are mobilised to attain the lateralvertical strain. stress Passiveand the earthlateral pressures strain. Passive are larger earth pressureswhen larger are strains larger when are mobilised larger strains due are to wall equilibrium. The FE analyses also show that the passive earth pressure mobilisation depends on the deflectionmobilised and due reach to wall a peak deflection at about and a depthreach a of peak 5.5 at m. about Below a depth this depth, of 5.5 m. they Below start this to depth, decrease they as the vertical stress and the lateral strain. Passive earth pressures are larger when larger strains are mobilisedstart to strains decrease reduce. as the mobilised strains reduce. mobilised due to wall deflection and reach a peak at about a depth of 5.5 m. Below this depth, they start to decrease as the mobilised strains reduce.

(a) (b)

FigureFigure 6. Horizontal 6. Horizontal stresses stresses(a) in thein the soil soil for: for: (a ()a the) the “‘retaining “‘retaining wall-only”wall-only” case case(b and) and (b (b) )the the “structure “structure on the backfill”on the backfill” case. case. Figure 6. Horizontal stresses in the soil for: (a) the “‘retaining wall-only” case and (b) the “structure on the backfill” case.

Figure 7. Horizontal earth pressures acting on the retaining wall.

FigureFigure 7. Horizontal 7. Horizontal earth earth pressures pressures acting on on the the retaining retaining wall. wall.

Geosciences 2020, 10, x FOR PEER REVIEW 9 of 18 Geosciences 2020, 10, 486 9 of 18 In the case of “structure on the backfill”, much larger passive pressures are mobilised, as requiredIn the to case establish of “structure the horizontal on the backfill”, static equilibrium. much larger This passive is consistent pressures with are mobilised, the bending as requiredmoments toobserved establish earlier the horizontal in Figure static 5. As equilibrium. seen in Figure This 7, is even consistent in this withcase, the the bending passive momentspressures observed are much earliersmaller in than Figure the5. Coulomb’s As seen in Figurepassive7, pressure even in this line case, given the by passive Equation pressures (2). are much smaller than the Coulomb’s passive pressure line given by Equation (2). 4. Dynamic Response of the Retaining Wall 4. Dynamic Response of the Retaining Wall Following the static runs, dynamic analyses were carried out using the same constitutive models andFollowing parameters the shown static runs,in Table dynamic 2. The analyses final stress were state carried from out the using static the run same was constitutive used as the models initial andstress parameters state in the shown dynamic in Table runs.2. The Several final stressdifferent state earthquake from the static loadings run was were used considered as the initial, such stress as a statesine in sweep the dynamic motion runs.and sinusoidal Several di ffmotionserent earthquake of different loadings amplitudes were, considered,to replicate suchmoderate as a sine and sweep strong motionearthquakes. and sinusoidal motions of different amplitudes, to replicate moderate and strong earthquakes.

4.1.4.1. Acceleration Acceleration Time Time Histories Histories InIn Figure Figure8, 8 the, the accelerations accelerations recorded recorded at at the the top top of of the the retaining retaining wall wall are are presented, presented along, along with with thethe input input motion motion applied applied atat thethe basebase nodes.nodes. This This input input motion motion was was a a 1 1-Hz-Hz sinusoidal sinusoidal motion motion wi withth 10 10cycles cycles of of loading. loading. It It has has a a peak peak acceleration acceleration of of about 0.23 g.g. The inputinput motionmotion waswas obtainedobtained from from a a dynamicdynamic centrifuge centrifuge test test conducted conducted earlier earlier at at 60 60 g g using using the the servo-hydraulic servo-hydraulic earthquake earthquake actuator actuator [ 16[16].]. InIn this this figure, figure the, the wall wall top top accelerations accelerations are are presented presented for for the the cases cases of “wall-only” of “wall-only and” “structureand “structure on the on backfill”.the backfill As” expected,. As expected the peak, the accelerationspeak accelerations are larger are forlarger the casefor the with case the with structure the structure on the backfill. on the Additionally,backfill. Additionally in both cases,, in both the inputcases, accelerationsthe input accelerations are amplified are amplified by the time by they the reachtime they the wallreach top. the Thewall amplification top. The amplification is larger when is larger the structure when the is structure present on is thepresent backfill. on the backfill.

FigureFigure 8. 8.Wall Wall top top accelerations. accelerations. 4.2. Dynamic Bending Moments 4.2. Dynamic Bending Moments The additional loading applied by the earthquake causes dynamic bending moments in the The additional loading applied by the earthquake causes dynamic bending moments in the retaining wall. In Figure9, the envelopes of the maximum normalised bending moments are plotted retaining wall. In Figure 9, the envelopes of the maximum normalised bending moments are plotted against normalised depth for the cases of “wall-only” and “structure on the backfill” subjected to the against normalised depth for the cases of “wall-only” and “structure on the backfill” subjected to the same input motion shown in Figure8. In Figure9, it can be seen that the dynamic bending moments same input motion shown in Figure 8. In Figure 9, it can be seen that the dynamic bending moments are much larger for the case of the retaining wall with a structure on the backfill. It is also possible to are much larger for the case of the retaining wall with a structure on the backfill. It is also possible to compare this figure with the static bending moments shown in Figure5 and conclude that the dynamic compare this figure with the static bending moments shown in Figure 5 and conclude that the bending moments are much larger. In fact, they increased by a factor of almost 2.5 during this 0.23-g dynamic bending moments are much larger. In fact, they increased by a factor of almost 2.5 during earthquake for the case with a structure on the backfill. In addition, the location of the peak bending this 0.23-g earthquake for the case with a structure on the backfill. In addition, the location of the peak moment appears to have shifted deeper relative to the static case. bending moment appears to have shifted deeper relative to the static case.

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Figure 9. Envelopes of a maximum, normalised dynamic bending moment. FigureFigure 9. 9.Envelopes Envelopes ofof aamaximum, maximum, normalised normalised dynamic dynamic bending bending moment. moment.

However,However,However, the the peak peak bending bending moments momentsmoments may maymay not notnot occur occuroccur at atat the thethe same samesame time timetime at atat all all elevations elevations of ofof the the retainingretainingretaining wall. wall. wall. Therefore,Therefore, Therefore, in in Figure in Fig Figure ure10 ,10 the10, , normalisedthe the normalised normalised bending bending bending moment moment moment profiles profiles profiles are plotted are are at plotted plotted different at at differenttimedifferent instants time time markedinstants instants onmarked marked the input on on the the motion input input plottedmotion motion atplotted plotted the bottom at at the the ofbottom bottom the figure. of of the the Timefigure figure. instants. Time Time instants 1instant and 3 s 1were1 and and chosen3 3 were were atchosen chosen 0 acceleration at at 0 0 acceleration acceleration (maximum (maximum (maximum+ve and +ve +veve and and velocity), − −veve velocity) velocity) and time, and, and instants t imetime instants instants 2 and 4 2 were2 and and 4 chosen 4 were were − chosenatchosen maximum at at maximum maximum+ve and +ve +veve and and accelerations − −veve accelerations accelerations (zero velocity). (zero (zero velocity). velocity). −

0 0 0 0 No structure No structure No structure No structure With a structure With a structure 0.2 0.2 0.2 With a structure 0.2 With a structure l l l l l l l 0.4 0.4 l a a 0.4 0.4 a a w w w w H H H / / H / / z z

0.6 0.6 z z 0.6 0.6

0.8 0.8 0.8 0.8 At time point 1 At time point 2 At time point 5 At time point 6 1 1 1 1 0 0.02 0.04 0 0.02 0.04 0 0.02 0.04 0 0.02 0.04 3 3 M/ H M/ H M/ H3 M/ H3

0 0 0 0 No structure No structure No structure No structure With a structure With a structure 0.2 0.2 0.2 With a structure 0.2 With a structure l l l l l l l

0.4 0.4 l a a 0.4 0.4 a a w w w w H H H H / / / / z z z

0.6 0.6 z 0.6 0.6

0.8 0.8 0.8 0.8 At time point 3 At time point 4 At time point 7 At time point 8 1 1 1 1 0 0.02 0.04 0 0.02 0.04 0 0.02 0.04 0 0.02 0.04 3 3 M/ H M/ H M/ H3 M/ H3

Input Input 0.2 0.2 4 8 0.1

) 0.1 ) g g ( (

. 0 . c 0 c

c 3 1 c 5 7 A -0.1 A -0.1

-0.2 -0.2 6 2 -0.3 -0.3 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Time (s) Time (s) (a) (a) (b(b) ) Figure 10. Dynamic bending moment profiles at different time instants: (a) during an earlier cycle and Figure 10. Dynamic bending moment profiles at different time instants: (a) during an earlier cycle and (bFigure) during 10. aDynamic later cycle. bending moment profiles at different time instants: (a) during an earlier cycle and (b) during a later cycle. (b) during a later cycle. In Figure 10a, the bending moment variations through time instants 1 to 4 during an earlier cycle for both cases of “wall-only” and “structure on the backfill” are presented. There is no great reversal of InIn Figure Figure 10a 10a, the, the bending bending moment moment variations variations through through time time instants instants 1 1 to to 4 4 during during an an earlier earlier cycle cycle bending moments, as the retaining wall is locked into a particular curvature due to the soil. However, forfor both both cases cases of of “ “wallwall-only-only”” and and “ “structurestructure on on the the backfill backfill”” are are presented. presented. There There is is no no great great reversal reversal the peak bending moment does increase from time point 1 to 4, with small drops in value at time ofof bending bending moments moments, ,as as the the retaining retaining wall wall is is locked locked into into a a particula particular rcurvature curvature due due to to the the soil. soil. However,However, the the peak peak bending bending moment moment does does increase increase from from time time point point 1 1to to 4, 4, with with small small drops drops in in value value

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at time points 2 and 3. This is due to the direction of soil acceleration and ensuing changes in dynamic points 2 and 3. This is due to the direction of soil acceleration and ensuing changes in dynamic earth earth pressure. The accumulation of bending moments can continue with the increasing number of pressure. The accumulation of bending moments can continue with the increasing number of loading loading cycles throughout the earthquake loading. This is illustrated in Figure 10b, in which the cycles throughout the earthquake loading. This is illustrated in Figure 10b, in which the bending bending moment variations are plotted for a later loading cycle at time points 5–8. In this figure, it moment variations are plotted for a later loading cycle at time points 5–8. In this figure, it can be seen can be seen that the peak bending moments for both cases (i.e., with and without the structure on the that the peak bending moments for both cases (i.e., with and without the structure on the backfill) backfill) increased relative to the earlier cycle presented in Figure 10a. The retaining wall can accrue increased relative to the earlier cycle presented in Figure 10a. The retaining wall can accrue more more deflections away from the backfill during the dynamic loading but cannot move back into the deflections away from the backfill during the dynamic loading but cannot move back into the soil. This soil. This is why the bending moments can either increase when deflection of the wall increases (and, is why the bending moments can either increase when deflection of the wall increases (and, hence, the hence, the curvature of the wall increases) or can remain the same but can never decrease by a curvature of the wall increases) or can remain the same but can never decrease by a significant amount. significant amount. 5. Soil Response 5. Soil Response Accelerations can be obtained at any of the solid nodes in the FE discretisation shown in Figure3a,b. Accelerations can be obtained at any of the solid nodes in the FE discretisation shown in Figure In Figure 11, the acceleration-time histories obtained in a soil column in the free field, far away from 3a,b. In Figure 11, the acceleration-time histories obtained in a soil column in the free field, far away the retaining wall, are presented. The depth of the node below the ground surface is shown in each from the retaining wall, are presented. The depth of the node below the ground surface is shown in plot. It can be seen that the input accelerations amplify as they propagate upwards to the ground each plot. It can be seen that the input accelerations amplify as they propagate upwards to the ground surface. Additionally, there is very little difference between the “wall-only” case and the “structure on surface. Additionally, there is very little difference between the “wall-only” case and the “structure the backfill” case, supporting that this column represents the far-field assumption reasonably well. on the backfill” case, supporting that this column represents the far-field assumption reasonably well. Similar acceleration-time history plots can be obtained for the nodes on the active side and passive Similar acceleration-time history plots can be obtained for the nodes on the active side and passive side. It may be more instructive to plot the amplification factor defined as the peak acceleration at a side. It may be more instructive to plot the amplification factor defined as the peak acceleration at a node normalised by the peak input acceleration. Such a plot is presented in Figure 12 for the free-field, node normalised by the peak input acceleration. Such a plot is presented in Figure 12 for the free- active and passive sides. field, active and passive sides.

FigureFigure 11.11. Acceleration time histories in the free-fieldfree-field soil.soil.

ReferringReferring to Figure 1111,, the the amplification amplification factor factor increases increases from from 1 at 1 atthe the bottom bottom to about to about 1.5 1.5at the at theground ground surface surface in thein the free free field field.. Such Such amplification amplification in in dry, dry, relatively relatively loose loose sand sand layers has been been previouslypreviously observedobserved both both in in earthquake earthquake case case histories, histories as, as well well as otheras other numerical numerical analyses analyses [7,8 ].[7,8] In the. In activethe active zone zone behind behind the the retaining retaining wall, wall a, dia ffdifferenceerence in in amplification amplification factors factors is is noticed noticed forfor thethe casecase ofof “structure“structure onon thethe backfill”.backfill”. The amplificationamplification that occurs towards thethe groundground surfacesurface isis almostalmost completelycompletely stopped stopped by by the the presence presence of the of structure.the structure. This This may be may due be to due the additional to the additional vertical stresses vertical appliedstresses byapplied the footings by the increasingfootings increas the confininging the confining pressures pressures in this zone, in this forcing zone, the forcing soil to the act moresoil to like act amore rigid like body. a rigid Similarly, body. onSimilarly the passive, on the side, passive a small side, attenuation a small attenuation of the amplification of the amplification factors is noticed. factors Itis isnoticed. not clear It whyis not the clear presence why ofthe structure presence should of structure influence should the soil influence response the on soil the passiveresponse side, on andthe thispassive aspect side may, and require this aspect further may investigation. require further investigation.

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Figure 12. Amplification factor in the free-field, active and passive sides. FigureFigure12. 12. AmplificationAmplification factorfactor inin thethe free-field,free-field,active activeand and passive passive sides. sides. 6. Dynamic Response of the Structure 6. Dynamic Response of the Structure 6. DynamicOne of the Response objectives of theof this Structure paper is to investigate the response of the structure placed adjacent to a retainingOneOne of of the the wall objectives objectives on the ofbackfill. of this this paper paper The is response tois to investigate investigate of the the structure the response response was of the investigatedof structurethe structure placedfirst placed by adjacent applying adjacent to a retainingsineto a sweepretaining wall input on wall the motion on backfill. the inbackfill. The the responserange The responseof of 0.1 the–2 structure Hz.of the The structure was peak investigated acceleration was investigated first magnitude by applying first byof a sineapplyingthis sweepinput a inputmotionsine sweep motion is only input in the0.05 rangemotion g. As of inin 0.1–2 thethe Hz. caserange The of ofthe peak 0.1 sinusoidal acceleration–2 Hz. The motion, magnitudepeak accelerationthis input of this motion input magnitude motion was taken isof only this from 0.05input g.a Aspreviousmotion in the is casedynamic only of the0.05 centrifuge sinusoidal g. As in testthe motion, ofcase Madabhushi thisof the input sinusoidal motion et al. [22] wasmotion,. In taken Figure this from 13input a, the previous motioninput motion dynamic was taken and centrifuge its from Fast a testFourierprevious of Madabhushi transform dynamic et( centrifugeFFT al. [)22 are]. Inplotted test Figure of atMadabhushi 13 the, the bottom. input motionet The al. [22] acceleration and. In its Figure Fast Fourierresponse 13, the transform input of the motion roof (FFT) of and are the plottedits sway Fast atframeFourier thebottom. and transform its FFT The accelerationare (FFT plotted) are plottedat response the top at of thein the Figure bottom. roof 13. of theThe Due sway acceleration to the frame differential and response its FFT settlement areof the plotted roof of the atof the thefootings top sway in Figureasframe a consequence 13and. Due its FFT to theof are the differential plotted lateral at displacement settlementthe top in Figure of of the the footings 13. retaining Due asto athewall consequence differential described of earlier,settlement the lateral the ofstructure displacement the footings also ofexperienceas the a consequence retainings vertical wall of describedmovementsthe lateral earlier, displacement, which the are structure also of shownthe also retaining experiences in Figure wall 13 described vertical. It can be movements, earlier,seen that the the which structure magnitude are alsoalso shownofexperience these in vertical Figures vertical accelerations13. Itmovements can be seenis quite, which that small the are magnitude alsoand shownmore oftowards in these Figure vertical the 13 higher. It accelerationscan- befrequency seen that is end. quite the Moreover,magnitude small and moreitof can these towardsbe verticalseen thethat accelerations higher-frequency the frequencies is quite end.above small Moreover, 0.25 and Hz more itare can significantly towards be seen thatthe amplifiedhigher the frequencies-frequency by the above time end. 0.25they Moreover, Hz reach are significantlytheit can roof be slab. seen amplified There that theis by also frequencies the a time sharp they spike above reach in the0.25 the roof Hzamplification slab.are significantly There isat also 1 Hz, aamplified sharp which spike is bythe in the thenatural time amplification frequencythey reach at 1ofthe Hz, the roof which sway slab. isframe. theThere natural is also frequency a sharpof spike the swayin the frame. amplification at 1 Hz, which is the natural frequency of the sway frame.

Figure 13. Structural response to a sine sweep motion.

Figure 13. Structural response to a sine sweep motion.

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The dynamic response of the structure is also investigated during a stronger 1-Hz sinusoidal earthquakeThe dynamic motion response introduced ofthe earlier. structure In Figure is also 14 investigated, the input acceleration during a stronger and the 1-Hz horizontal sinusoidal and earthquakevertical accelerations motion introduced recorded on earlier. the roof In Figureslab are 14 presented., the input The acceleration input acceleration and the horizontalin this case and has verticala maximum accelerations value of 0.23 recorded g. However, on the roof as this slab is areapplied presented. at 1 Hz, The which input is accelerationthe natural frequency in this case of hasthe asway maximum frame, valuelarge ofaccelerations 0.23 g. However, were registered as this is appliedby the roof at 1 slab. Hz, whichIn the ishorizontal the natural direction frequency, these of theaccelerations sway frame, reach large a magnitude accelerations of werenearly registered 0.4 g, while by, thein the roof vertical slab. In direction the horizontal, a peak direction, acceleration these of accelerations0.1 g was registered. reach a magnitudeThe FFT plots of nearly in Figure 0.4 g, 14 while, also confirm in the vertical the amplification direction, a peakof accelerations acceleration at of 1 0.1Hz. g was registered. The FFT plots in Figure 14 also confirm the amplification of accelerations at 1 Hz.

FigureFigure 14.14. Structural responseresponse toto aa sinusoidalsinusoidal motion.motion.

AnotherAnother interestinginteresting observationobservation thatthat waswas obtainedobtained fromfrom thesethese dynamicdynamic analysesanalyses waswas thethe rotationrotation susufferedffered by by the the structure. structure. The The structure structure suff ered suffered rotation rotation due to due the outward to the outward movement movement of the retaining of the wallretaining creating wall a settlementcreating a settlement trough in the trough backfill in andthe backfill a consequent and a diconsequentfferential settlement differential between settlement the leftbetween and right the left footings. and right In Figure footings. 15, theIn Figure rotation 15 su, theffered rotation by the suffer structureed by obtained the structure from theobtained differential from settlementthe differential of the settlement footings isof plotted,the footings along is withplotted the, along sinusoidal with inputthe sinusoidal motion. input In this motion. figure, In it canthis befigure seen, it that can the be structureseen that accumulates the structure rotation accumulates in each rotation cycle of in the each input cycle motion of the and input accrues motion a total, and residualaccrues a settlement total, residual of about settlement 4◦. It is of also about interesting 4°. It is toalso see interesting that the rotation to see that ceases the to rotation accrue ceases after the to endaccrue of theafter strong the end cycles of the in thestrong input cycles motion. in the input motion.

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Figure 15.15. Rotation of the structure due to a sinusoidal input motion.motion. 7. Dynamic Earth Pressures 7. Dynamic Earth Pressures The earthquake loading will induce additional dynamic earth pressures on the retaining wall. The earthquake loading will induce additional dynamic earth pressures on the retaining wall. Traditionally, limit equilibrium methods such as the Mononobe-Okabe (M-O) method developed Traditionally, limit equilibrium methods such as the Mononobe-Okabe (M-O) method developed by by Mononobe [23] and Okabe [24] are used to calculate the dynamic lateral earth pressures. In the Mononobe [23] and Okabe [24] are used to calculate the dynamic lateral earth pressures. In the M-O M-O method, the dynamic inertial forces are considered as additional static forces applied to the soil method, the dynamic inertial forces are considered as additional static forces applied to the soil wedge. The dynamic active and passive earth pressure coefficients can be calculated by modifying the wedge. The dynamic active and passive earth pressure coefficients can be calculated by modifying Coulomb’s equations given earlier in Equations (1) and (2), as follows: the Coulomb’s equations given earlier in Equations (1) and (2), as follows: coscos2(2ϕ(φ −α α −θ)θ) Kae = − − , (3) Kae =  q 2 2, sin(ϕ+δ) sin(ϕ β θ) cos(θ) cos2(α) cos(δ + α + θ) 1 + sin(φ + δ)sin(φ−−−β − θ) (3) cos(θ)cos2(α)cos(δ + α + θ) [1 + √ cos(δ+α+θ) cos(β α) ] cos(δ + α + θ)cos−(β − α) cos2(ϕ + α θ) K = − , (4) pe  q 2 cos2(φ + α − θsin) (ϕ+δ) sin(ϕ+β θ) cos(θ) cos2(α) cos(δ α + θ) 1 + − Kpe = cos(δ α+θ) cos(β α) 2 , − − − sin(φ + δ)sin(φ + β − θ) (4) and noting that cos(θ)cos2(α)cos(δ − α + θ) [1 + √ ] k cos(δ − α + θ)cos(β − α) tan θ = h . (5) (1 kv) and noting that − Using the above equations, it is possible to have the limiting dynamic earth pressure lines as kh shown in Figure 16, along with the maximumtan dynamicθ = earth. pressures obtained from the dynamic(5) analyses for the sinusoidal input motion with a peak(1 acceleration− kv) of 0.23 g. It should be pointed out that theUsing M-O the method above willequations, result init anis possible increased to active have pressurethe limiting line dynamic and a reduced earth pressure passive pressure lines as lineshown relative in Figure to the 16 static, along earth with pressures the maximum predicted dynamic by Coulomb’s earth pressures equations obtained shown from in Figure the 7dynamic. It can beanalyses seen that for thethe dynamicsinusoidal passive input motion earth pressures with a peak are muchaccelerati largeron of in 0.23 the caseg. It ofshould the structure be pointed on theout backfillthat thecompared M-O method to the will “wall-only” result in an case. increased These areactive also pressure larger than line the and static a reduced earth pressures passive pressure for both cases,line relative with the to peakthe static horizontal earth pressures earth pressure predicted reaching by Coulomb’s nearly 200 equations kPa for the shown case with in Figure a structure 7. It can on thebe seen backfill. that Thisthe dynamic is an increase passive of earth nearly pressures 50 kPa comparedare much larger to the staticin the earthcase of pressure the structure for this on case. the Forbackfill both compared cases, the dynamicto the “wall passive-only earth” case. pressures These are loweralso larger than thosethan predictedthe static byearth the pressures M-O method, for althoughboth cases, this with line the on thepeak passive horizontal side is earth closer pressure to this limitingreaching earth nearly pressure 200 kPa line, for compared the case with to the a structure on the backfill. This is an increase of nearly 50 kPa compared to the static earth pressure for

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this case. For both cases, the dynamic passive earth pressures are lower than those predicted by the Geosciences 2020, 10, 486 15 of 18 M-O method, although this line on the passive side is closer to this limiting earth pressure line, compared to the static earth pressures to the Coulomb passive pressure line shown in Figure 7. staticHowever, earth the pressures dynamic to theactive Coulomb earth pressures passive pressure on the backfill line shown side inobtained Figure7 by. However, the numerical the dynamic analyses activeare larger earth than pressures those on predicted the backfill by theside M obtained-O line byfor the both numerical the “wall analyses-only” and are larger“structure than on those the predictedbackfill” bycases. the The M-O bulge line for in both active the earth “wall-only” pressure and at about “structure 2.75- onm depth the backfill” attributed cases. earlier The bulge to the inpresence active earth of the pressure structure at still about persist 2.75-ms in depth the dynamic attributed analyses. earlier to the presence of the structure still persists in the dynamic analyses.

FigureFigure 16.16. DistributionDistribution of of maximum maximum horizontal horizontal earth earth pressures pressures acting acting on the on retaining the retaining wall. wall.M-O: M-O:Mononobe Mononobe-Okabe.-Okabe. In addition to the maximum horizontal earth pressures, it is possible to trace the evolution of these In addition to the maximum horizontal earth pressures, it is possible to trace the evolution of earth pressures through a cycle of earthquake loading. In Figure 17a, the horizontal earth pressures these earth pressures through a cycle of earthquake loading. In Figure 17a, the horizontal earth are plotted on the active and passive sides at four different time instants, 1–4, for an early cycle. pressures are plotted on the active and passive sides at four different time instants, 1–4, for an early These time instants are noted on the input acceleration at the bottom and were explained earlier with cycle. These time instants are noted on the input acceleration at the bottom and were explained earlier respect to Figure 10. Unlike the bending moments, the horizontal earth pressures can increase and with respect to Figure 10. Unlike the bending moments, the horizontal earth pressures can increase decrease depending on the earthquake loading direction. For example, following the passive earth and decrease depending on the earthquake loading direction. For example, following the passive pressure for the “structure on the backfill” case, it can be seen that, between time instants 1 and 2, earth pressure for the “structure on the backfill” case, it can be seen that, between time instants 1 and the peak horizontal earth pressure increased from 100 to 140 kPa. However, between time instants 2, the peak horizontal earth pressure increased from 100 to 140 kPa. However, between time instants 3 and 4, this earth pressure decreases from 130 to 105 kPa. Similar observations can be made for the 3 and 4, this earth pressure decreases from 130 to 105 kPa. Similar observations can be made for the “wall-only” case. Additionally, similar increases in active earth pressure can be seen on the backfill “wall-only” case. Additionally, similar increases in active earth pressure can be seen on the backfill side in Figure 17a. For comparison, the horizontal earth pressures are plotted for a later cycle at time side in Figure 17a. For comparison, the horizontal earth pressures are plotted for a later cycle at time instants 5–8, as shown in Figure 17b. It is interesting to note that the magnitude of earth pressures is instants 5–8, as shown in Figure 17b. It is interesting to note that the magnitude of earth pressures is much larger for this later cycle of earthquake loading, particularly on the passive side. For example, much larger for this later cycle of earthquake loading, particularly on the passive side. For example, for the structure in the backfill case, the magnitude of peak horizontal earth pressure increased from for the structure in the backfill case, the magnitude of peak horizontal earth pressure increased from 125 kPa to 180 kPa for time instants 2 and 6, respectively (i.e., peak negative acceleration points). This 125 kPa to 180 kPa for time instants 2 and 6, respectively (i.e., peak negative acceleration points). This increase in horizontal earth pressure may be due to the accumulation of locked-in stresses cycle by increase in horizontal earth pressure may be due to the accumulation of locked-in stresses cycle by cycle and partly due to the increase in soil density resulting from earthquake-induced settlements. The cycle and partly due to the increase in soil density resulting from earthquake-induced settlements. location of the peak horizontal earth pressure shifted to a deeper elevation in this later cycle. Similar The location of the peak horizontal earth pressure shifted to a deeper elevation in this later cycle. observations can be made for other time instants as well. On the active side, the increases are much less Similar observations can be made for other time instants as well. On the active side, the increases are pronounced. However, the active earth pressures for the “no structure” case seem to have increased much less pronounced. However, the active earth pressures for the “no structure” case seem to have more than the “with structure” case in this later cycle, especially at deeper elevations. This aspect increased more than the “with structure” case in this later cycle, especially at deeper elevations. This requires further investigation. It should be noted that, in all cases, the M-O predictions underestimate aspect requires further investigation. It should be noted that, in all cases, the M-O predictions the horizontal earth pressure on the active side. underestimate the horizontal earth pressure on the active side.

GeosciencesGeosciences 20202020,, 1010,, x 486 FOR PEER REVIEW 1616 of of 18 18

(a) (b)

Figure 17. Evolution of the horizontal earth pressures through: (a) an earlier earthquake loading cycle Figure 17. Evolution of the horizontal earth pressures through: (a) an earlier earthquake loading cycle and (b) a later earthquake loading cycle. and (b) a later earthquake loading cycle. 8. Conclusions 8. Conclusions In this paper, static and dynamic analyses were carried out on a retaining wall in dry sand with an h/Ind ratiothis paper of 0.625., static These and were dynamic compared analyses to a were case wherecarried a out sway on framea retaining structure wall was in dry placed sand on with the anbackfill h/d ratio adjacent of 0.625. to the These retaining were compared wall. Themain to a case objective where of a this sway research frame wasstructure to investigate was placed the on strong the backfillsoil-structure adjacent interaction to the retaining that may wall. occur The due main to objective the presence of this of theresea structure.rch was Usingto investigate the static the analyses, strong soilit was-structure shown interaction that the deflections that may and occur bending due to moments the presence in the retaining of the structure. wall increased Using due the tostatic the analyses,presence it of was the shown structure that on the the deflections backfill. In and addition, bending the moments horizontal in the pressure retaining bulb wall that increased developed due in tofront the of presence the retaining of the wall st alsoructure increased on the in sizebackfill. when In the addition structure, the was horizontal present on pressure the backfill. bulb Further, that developedthe horizontal in front earth of pressures the retaining obtained wall fromalso increased these analyses in size were when compared the structure to Coulomb’s was present predictions. on the backfill.Using Further the dynamic, the horizontal analyses, earth the response pressures of the obtained top of the from retaining these analyses wall was werfounde compared to increase to in Coulomb’sthe presence predictions. of the structure on the backfill. The maximum bending moments due to the earthquake loadingUsing increased the dynamic in magnitude. analyses, the By response considering of the the top bending of the retaining moments wall at diwasfferent found time to increase instants, init was the presenceshown that of these the structure bending onmoments the backfill. locked The in and maximum did not bending reduce in moments value. The due dynamic to the earthquakeresponse of loading the soil increased was considered in magnitude. next, and By itconsidering was shown the that bending the amplification moments at ratio different below time the instants,structure it on was the shownbackfill that was smallerthese bending than for moments the “wall-only” locked case, in and suggesting did not more reduce “rigid in value. body”-like The dynamicaccelerations response below of thethe footingssoil was considered owing to the next increased, and it was confining shown stress. that the The amplification structural response ratio below was theconsidered structure next, on the and backfill the large was amplification smaller than of for accelerations the “wall-only of the” case, roof suggesting slab was explained. more “rigid Finally, body the”- likedynamic accelerations earth pressures below the acting footings on the owing retaining to the wall increased were considered confining stress. both in The terms structural of their response maxima, wasas well considered as their evolution next, and with the large time. amplification Unlike the bending of accelerations moments, of it the was roof shown slab that was the explained. dynamic Finallyearth pressures, the dynamic can both earth increase pressures and acting decrease on dependingthe retaining on wall the directionwere considered of the input both accelerations. in terms of theirThe maximummaxima, as dynamic well as their earth evolution pressures with were time. compared Unlike to the the bending traditional moments, M-O method. it was shown It was seenthat the dynamic earth pressures can both increase and decrease depending on the direction of the input accelerations. The maximum dynamic earth pressures were compared to the traditional M-O method.

Geosciences 2020, 10, 486 17 of 18 that the M-O method overpredicts the dynamic passive earth pressures, while it underpredicts the active earth pressures.

Author Contributions: X.G. carried out the numerical analyses reported in this paper, with help from the second author. G.S.P.M. helped prepare the first draft of this manuscript, while the first author created all the figures shown in this paper. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The first author would like to thank the financial support from China Scholarship Council and Cambridge Trust for her PhD studies. Conflicts of Interest: The authors declare no conflict of interest.

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