research article practical soil-shallow foundation model

Research ArticlePractical Soil-Shallow Foundation Model for NonlinearStructural Analysis

Moussa Leblouba1 Salah Al Toubat1

Muhammad Ekhlasur Rahman2 and Omer Mugheida3

1Department of Civil amp Environmental Engineering University of Sharjah PO Box 27272 Sharjah UAE2Department of Civil amp Construction Engineering Faculty of Engineering and Science Curtin University SarawakCDT 250 98009 Miri Sarawak Malaysia3Department of Civil Engineering Abu Dhabi University PO Box 59911 Abu Dhabi UAE

Correspondence should be addressed to Moussa Leblouba mlebloubasharjahacae

Received 13 March 2016 Accepted 15 June 2016

Academic Editor Manuel Pastor

Copyright copy 2016 Moussa Leblouba et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Soil-shallow foundation interaction models that are incorporated into most structural analysis programs generally lack accuracyand efficiency or neglect some aspects of foundation behavior For instance soil-shallow foundation systems have been observedto show both small and large loops under increasing amplitude load reversals This paper presents a practical macroelement modelfor soil-shallow foundation system and its stability under simultaneous horizontal and vertical loads The model comprises threespring elements nonlinear horizontal nonlinear rotational and linear vertical springs The proposed macroelement model wasverified using experimental test results from large-scale model foundations subjected to small and large cyclic loading cases

1 Introduction

Several researchers ([1ndash5] among others) have investigatedextensively the subject of soil-structure interaction (SSI)Rayhani et al [6] showed that soil-structure interactionmightamplify or attenuate the base shear through inertial andkinematic interactions Stewart et al [7] concluded that theeffects of SSI on rigid structures founded on soil are moresignificant when compared to flexible structures Bobet etal [8] showed that the soil-structure system is a functionof the relative rigidity of the structure compared to that ofthe ground SSI is shown to be significant in the presenceof soft soils or when structural mass is very large [9] As aconsequence the SSI problem can be excluded from compu-tations if the soil where the structure is founded is very rigidWhen considering the effect of soil-structure interaction inbase isolated multistoried structures on elastic layered soilSpyrakos et al [10] have found that SSI effects are significantfor squat lightweight buildings on low stiffness soil-stratum

Although their study dealt with harmonic excitations it stillgives an insight on the danger of neglecting SSI in the designof base isolated buildings

In the seismic resistant design of structures we are mostinterested in the strength reduction factors to account for thenonlinear behavior that might be experienced by a structuresubjected to an earthquake ground motion Few researchers[11 12] have recently attempted to assess the effect of SSI onthe strength reduction factors Eser et al [11] have shownthat the presence of soft soils reduces the strength reductionfactors which is primarily controlled by the changes in thestructural period and displacement ductility

Incorporation of SSI requires explicit modelling of soil-foundation system adequately For instance several modelshave been proposed depending on the foundation type itsembedment and its rigidity ([13ndash15] among others) TheFederal Emergency Management Agency [16] required thatthe foundation stiffness should be determinedwith one of thefollowing three methods

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4514152 10 pageshttpdxdoiorg10115520164514152

2 Mathematical Problems in Engineering

ks

Actual soil-foundation system Proposed macroelement model

k

k120579

Figure 1 Proposed macroelement model for soil-shallow foundation system

(i) uncoupled spring model comprising three springelements for shallow foundations that are stiffer thanthe supporting soil

(ii) a finite element formulation of linear (or nonlinear)foundation behavior using Winkler models for shal-low foundations that are less stiff than the supportingsoil

(iii) decoupled Winkler model for shallow foundationsthat are flexible with respect to the supporting soil

However El Ganainy and El Naggar [17] have demonstratedthat the decoupled technique (Beam on a Nonlinear WinklerFoundation BNWF) is not capable of predicting accuratelythe settlement seen in foundations on soft soils Although itcan be used to predict the overall deformation behavior offoundations the BNWF requires a large number of nonlinearsprings which is considered as a major drawback [17]

To address some of the above-mentioned issuesmacroelement formulations have been proposed The firstformulation has been developed by Nova and Montrasio[18] and later modified andor extended by other researchers[19ndash22] and recently the formulations developed by Gajan etal [23] and Shirato et al [3] The major advantages of suchmodels are their simplicity and their ability to capture theglobal response of bearing foundations [22 24 25] Howeveron one hand calibrating macroelement parameters posea constraint on their adoption for practical applicationsAnd on the other hand knowing that most of the availablemacroelement models are based on specified boundingsurfaces poses another problem for their capability to covera wide range of problems [17]

In a completely different modelling approach El Shamyand Zamani [26] proposed a new 3D particle-based tech-nique using the discrete element method (DEM) to analyzethe seismic performance of soil-foundation-structure sys-tems In their model the soil is idealized as a collection ofspherical particles using DEM the footing is considered as arigid block whereas the structure ismodelled using a numberof spherical particles in the form of a column which can beclamped to simulate a rigid structure or bonded to simulate aflexible structure of predefined rigidity

To overcome the difficulties in performing complete non-linear simulations Seylabi et al [27] proposed an equivalentlinearization of nonlinear soil-structure systems consideringboth the effect of SSI and the nonlinear behavior of thestructure on equivalent linear parameters In their model

the structure is modelled as an elastoplastic single-degree-of-freedom system (SDOF) and the soil beneath the structure ismodelled by a discrete model combining different spring anddashpot elements

In this paper a newmacroelementmodel is developed forthe analysis of the nonlinear response of shallow foundationsunder cyclic loading This model may easily be incorpo-rated into available structural analysis programs such asOpenSees [28]The soil-foundation system is simulated usingthree spring elements horizontal and rotational nonlinearsprings and a linear vertical spring The nonlinear springsare assigned appropriate nonlinear model of plasticity withmaterial degrading parameters

2 Proposed Macroelement Model

The problem being studied here is that of a shallow foun-dation of any shape embedded in soil and subjected tosimultaneous axial and lateral forces as shown in Figure 1The foundation is considered to be very stiff The depth ofembedment is 119867 The proposed model incorporates threetypes of springs

(i) vertical translational elastic spring with stiffness 119896V(ii) shear inelastic spring with preyield stiffness 119896

119904

(iii) rotational inelastic spring with preyield stiffness 119896120579

These equivalent springs represent the foundation-soil sys-tem The macroelement model replaces the system soil-shallow foundation thus decreasing considerably both theoverall number of degrees of freedom and the computationeffort required to run large models

21 Constitutive Equations Two material models are consid-ered in this study namely the Bouc-Wen model [29 30] forthe shear and rotational springs and the linear model for thevertical spring

211 Shear and Rotational Springs

Model AssumptionsThe smooth Bouc-Wenmodel of hystere-sis by Bouc [29] and Wen [30] has found many engineeringapplications For instance the use of original Bouc-Wenmodel and its extensions to soil-structure interaction include[31ndash33]

In this study we considered the Baber and Noori [34]extension to the original Bouc-Wen model This version

Mathematical Problems in Engineering 3

includes the degrading behavior observed in many engineer-ing materials

The nonlinear behavior of the soil-shallow foundationsystem is modelled via the nonlinear shear and nonlinearrotational springs A force 119891

119904(119904) is mobilized at the shear

spring and a moment 119891120579(120579) is mobilized at the rotational

spring In what follows we use a general force 119891(119909) to denote119891119904(119904) and119891

120579(120579) depending onwhich springwe aremodelling

The constitutive relationship for 119891 is expressed in theBouc-Wen model as a linear part and a hysteretic part

119891 = 120572119896119909 + (1 minus 120572) 119891119910119911 (1)

where 119909 is the deformation 119896 is the preyield stiffness 119891119910

is the mobilized force at the beginning of yielding 120572 is thepost-to-preyield stiffness ratio and 119911 is a hysteretic quantitycontrolling the nonlinear behavior The latter is governed bythe following differential equation with respect to time 119905

=

119860 minus [120573 || 119911 |119911|119899minus1

+ 120574 |119911|120578

] ]120578

(2)

Equation (2) represents the Baber andNoori [34] formulationof the rate of hysteretic deformation which accommodatesdegradation In (2) 120573 120574 and 119899 are parameters that control theshape and transition from elastic to inelastic regions of thehysteretic loop while119860 ] and 120578 are variables that control thestiffness degradation and material deterioration Note thatthe yield deformation of the spring (119909

119910) does not appear in

(2) unlike the original Bouc-Wenmodel In fact the adoptionof the Baber-Noori [34] formulation in calculating forces iseasy by considering that 120573 and 120574 are expressed as

120573 =1205731015840

119909120578

119910

120574 =1205741015840

119909120578

119910

(3)

inwhich1205731015840 and 1205741015840 can take values as in the original Bouc-Wenmodel of hysteresis In this case 119911 is a dimensional quantityand is bounded between minus119909

119910and +119909

119910 The evolution of

material degradation is governed by the following equations[34]

119860 = 1198600minus 120575119860119890 (4a)

] = 1 minus 120575]119890 (4b)

120578 = 1 + 120575120578119890 (4c)

where 119890 is defined by the following rate equation

119890 = (1 minus 120572) 119891119910119911 (5)

And 120575119860 120575] and 120575

120578are the parameters that control the

degradation

Incremental Response Equations The tangent of 119891(119909) (ie120597119891(119909)120597119909) is computed using (1)

120597119891 (119909)

120597119909= 120572119896 + (1 minus 120572) 119891

119910

120597119911

120597119909 (6)

Note that this is the continuum tangent and not the algo-rithmically consistent tangent It is clear from (6) that thederivative of 119911 with respect to 119909 is needed To get it (2) canbe rewritten as

=119860 minus |119911|

119899

[120573 sgn (119911) + 120574] ]120578

=120597119911

120597119909

120597119909

120597119905 (7)

Hence

120597119911

120597119909=119860 minus |119911|

120578

[120573 sgn (119911) + 120574] ]120578

(8)

From this point we will derive incremental response equa-tions to obtain computer implementable equations In whatfollows the same procedure developed by Haukaas and DerKiureghian [35] is used The force at time 119905119894+1 is obtained as

119891119894+1

= 120572119896119909119894+1

+ (1 minus 120572) 119891119910119911119894+1

(9)

The variable 119911 is next discretized by a Backward-Eulersolution scheme

119911119894+1

= 119911119894+1

+ Δ119905

119860119894+1

minus10038161003816100381610038161003816119911119894+110038161003816100381610038161003816

119899

[120573 sgn (((119909119894+1 minus 119909119894) Δ119905) 119911119894+1) + 120574] ]119894+1

120578119894+1(119909119894+1

minus 119909119894

Δ119905) (10)

It is seen that Δ119905 cancels from the equation yielding anonlinear equation in 119911

119894+1 A Newton scheme of the form119909119894+1

= 119909119894

minus 119891(119909119898

)1198911015840

(119909119898

) to solve the general nonlinearequation of the form 119891(119909) = 0 is employed to solve for 119911119894+1 in(10) [35]

The equations describing the degradation behavior aredescribed as follows

119860119894+1

= 1198600minus 120575119860119890119894+1

(11a)

]119894+1 = 1 minus 120575]119890119894+1

(11b)

120578119894+1

= 1 + 120575120578119890119894+1

(11c)

where 119890119894+1 is found by discretization of the rate equation in(5) using the Backward-Euler solution scheme

119890119894+1

= 119890119894+1

+ Δ119905 (1 minus 120572) 119891119910(119909119894+1

minus 119909119894

Δ119905) 119911119894+1

(12)


U

V

x

ks s

y

H

P

F

k

k120579 120579

Undeformed state Deformed state

Figure 2 Undeformed and deformed configurations of the proposed macroelement model

where Δ119905 again cancels Note that 119911119894 119909119894 and 119890119894 are historyvariables that must be stored at each converged step

For the complete implementable procedure to obtain119891119894+1 and the algorithmically consistent tangent 120597119891119894+1120597119909119894+1

(referred to here as the Local Newton Routine) the readeris referred to [35]

212 Vertical Spring The vertical force 119891V mobilized in thevertical translational spring is given by

119891V = 119896V119881 (13)

in which 119896V is the elastic vertical stiffness of the soil-foundation system and 119881 is the vertical displacement of thespring Note that an appropriate constitutive model could beassigned to the vertical spring however in this paper thespring is considered linear elastic but the effect of axial loadsis present on the soil-shallow foundation global behavior

22 Equilibrium Equations and Model Implementation Themacroelement model described in the previous subsectionsis used to simulate the soil-shallow foundation interaction

Initially we assume that the shear and rotational springsare linear then we replace them by the general forces 119891

119904(119904)

and 119891120579(120579) respectively The total potential energy Π of the

model with reference to Figure 2 is expressed as follows

Π =1

21198961205791205792

+1

21198961199041199042

+1

2119896VV2

minus 119875119881 minus 119865119880 (14)

where 119880 and 119881 are the foundationrsquos top horizontal andvertical displacements respectively and are dependent on thedepth of embedment119867 and the springrsquos displacements

119880 = (119867 minus V) sin (120579) + 119904 cos (120579) (15a)

119881 = 119867 (1 minus cos (120579)) + 119904 sin (120579) + V cos (120579) (15b)

In (15b) 119881 positive in case of compression is the sumof the vertical component of the displacement originatingfrom the axial flexibility of the soil and the additionalvertical displacement that happens in the laterally deformedconfiguration shown in Figure 2

The above kinematic equations (see (15a) (15b)) whichrelate the total displacement 119880 and the vertical displacement119881 to the internal displacements 119904 V and 120579 assume largedisplacements and large rotations

Considering the model subjected to the axial load 119875

with the resulting horizontal displacement 119880 the vector ofunknowns is then x = ⟨119904 V 120579 119865⟩119879 Using Castiglianorsquos secondtheory by imposing the stationary of Π with respect to 119904 Vand 120579 yields the following governing equations

1198921= 119896119904119904 minus 119865 cos (120579) minus 119875 cos (120579) = 0 (16a)

1198922= 119896VV + 119865 sin (120579) minus 119875 cos (120579) = 0 (16b)

1198923= 119896120579120579 minus 119875 [(119867 minus V) sin (120579) + 119904 cos (120579)]

minus 119865 [(119867 minus V) cos (120579) minus 119904 sin (120579)] = 0(16c)

The following equation should also be considered to permitthe displacement-controlled cyclic analysis by imposing thetop lateral displacement 119880

1198924= 119880 minus [(119867 minus V) sin (120579) + 119904 cos (120579)] = 0 (16d)

However when the lateral force-controlled cyclic response isdesired 119865 is to be imposed Hence the system of governingequations to be solved is reduced to the three equilibriumequations (see (16a) (16b) and (16c))The lateral and verticaldisplacements (119880 119881) may be calculated at the end of theanalysis using the kinematic equations (see (15a) (15b))

Considering the case of lateral displacement-controlledanalysis (16a) (16b) (16c) and (16d) are rewritten in thefollowing form

g =

1198921

1198922

1198923

1198924

=

119891119904(119904) minus 119865 cos (120579) minus 119875 sin (120579)

119896VV + 119865 sin (120579) minus 119875 cos (120579)119891120579(120579) minus 119875 [(119867 minus V) sin (120579) + 119904 cos (120579)] minus 119865 [(119867 minus V) cos (120579) minus 119904 sin (120579)]

119880 minus [(119867 minus V) sin (120579) + 119904 cos (120579)]

(17)


Table 1 TRISEE parameters of the numerical model (units kN m)

Phase Shear spring Rotational spring Verticalspring

119896103

119891119910

120572 () 119896103

119891119910

120572 () 119896V103

HDI 1322 125 60 586 10 60 120III 70 99 9 35 111 2 80

LDI 54 38 47 254 38 4 65III 35 404 1 8 333 1 27

Note that the resultant moment 119872 (119872 = 119865(119867 minus 119881) +

119875119880) found by the equilibrium of the macroelement in thedeformed configuration needs also to be applied at the topof the foundation Also note that in (17) the terms 119896

119904119904 and

119896120579120579 were replaced by the general forces 119891

119904(119904) and 119891

120579(120579)

respectively These forces are assigned the constitutive model119891(119909) developed in Section 211

The system g(x) can be solved using Newtonrsquos methodfollowing the pseudocode provided below (referred to hereas the Global Newton Routine)

Previous converged solution x = ⟨119904 V 120579 119865⟩119879

(1) While (g(x) lt tolerance)(2) Update119891

119904119891120579 120597119891119904120597119904 and 120597119891

120579120597120579using the Local

Newton Routine(3) Compute g(x)(4) Compute the Jacobian matrix J(x)(5) 119889x = J(x)minus1g(x)(6) x = 119909 + 119889x

end

where the Jacobianmatrix J(x) is defined as 119869119894119895= 120597119892119894120597119909119895The

tangents 120597119891119904120597119904 and 120597119891

120579120597120579 required in the Jacobian matrix

should be algorithmically consistent tangents (Section 211)From the current estimation of the top lateral displace-

ment 119880 and axial load 119875 the code returns the vector x =

⟨119904 V 120579 119865⟩119879 that satisfies equilibrium and kinematics (see(17)) In each global Newton iteration the current values of119891119904119891120579 120597119891119904120597119904 and 120597119891

120579120597120579 based on the current values of 119904 and

120579 are updated using the Local Newton Routine (Section 211)The above pseudocode (Global Newton Routine) along

with the Local Newton Routine (that implements the Bouc-Wen-Baber-Noori model of hysteresis) has been imple-mented numerically in MATLAB (Mathworks Inc)

3 Verification with Experimental Results

To verify the validity of the proposedmacroelementmodel inpredicting the cyclic behavior of the soil-shallow foundationsystem its predictions are compared with experimentalresults In the framework of the TRISEE Project (3D SiteEffects and Soil-Foundation Interaction in Earthquake and

Vibration Risk Evaluation) a program of large-scale 1 gmodelhas been tested to investigate the response of soil-shallowfoundation under cyclic and dynamic loads The experimentwas carried out at ELSA (European Laboratory for StructuralAssessment) in Ispra Italy test results are reported in manyreferences ([36] among others)

The TRISEE experiment consists of three phases onlyPhases I and III are considered in the comparison becausethe proposedmacroelementmodel is restricted to 2D loadingconditions only However the model may be extended toinclude the 3D case

The experimental setup consists of a square steel shallowfoundation (1m times 1m) mounted inside a stiff concretecaisson (46m times 46m times 3m) filled with Ticino sand Theembedment of the foundation was about 1m with a steelframework placed around the foundation to retain the sandTwo different soil relative densities have been used119863

119903= 85

and119863119903= 45 representing high (HD) and low density (LD)

soil conditions respectivelyThe HD and LD specimens were loaded vertically by

300 kN and 100 kN respectively The static safety factorwas about 5 in both tests In the HD test the load wasapplied at 09m above the foundation and in the LD test at0935m In Phase I a series of unidirectional force-controlledsmall amplitude cycles was applied to identify the signifi-cance of nonlinear soil behavior In Phase III displacement-controlled unidirectional increasing amplitude cycles wereimposed to the top of the foundation For further informationon experimental setup and results refer to [36]

The new macroelement model is used to simulate thefoundation The parameters of the numerical model arepresented in Table 1 These parameters have been calibratedusing experimental moment-rotation and horizontal force-horizontal displacement curves Parameters for the Bouc-Wen-Baber-Noori model of hysteresis are given in Tables 2and 3 for HD and LD tests respectivelyThe parameters wereagain calibrated with experimental results

Figures 3ndash6 compare the experimental and the numericalhysteretic horizontal force-horizontal displacement curvesand moment-rotation curves for Phases I and III Thenumerical model reproduces correctly the overall behaviorof the foundation verifying the ability of the proposedmacroelement model to simulate the cyclic behavior ofshallow foundations


Table 2 Parameters of the Bouc-Wen-Baber-Noori model HD test

Phase Shear spring Rotational spring119860 120573

1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578

I 1 05 05 1 0 minus001 0 1 05 05 1 0 minus001 0III 1 05 05 07 0 0 01 1 01 09 07 0 0 01

Table 3 Parameters of the Bouc-Wen-Baber-Noori model LD test


1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578

I 1 05 05 1 0 minus001 0 1 05 05 1 0 minus001 0III 1 05 05 03 0 minus001 01 1 033 067 02 0 minus001 01

times10minus4

times10minus4

minus20

minus10

0

10

20

Forc

e (kN

)

0 1 2minus1minus2

Displacement (m) times10minus40 1 2minus1minus2

Displacement (m)

minus20

minus10

0

10

20

minus20

minus15

minus10

minus5

0

5

10

15

minus20

minus15

minus10

minus5

0

5

10

15

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

0 2 4minus2minus4

Rotation (rad) times10minus4minus2minus4 2 40

Rotation (rad)

Numerical Experimental


Figure 3 Comparison of experimental and numerical results HD test Phase I

The uplift is important for the HD Phase III test and thiscan be observed from the S shaped moment-rotation curve(see Figure 5) For the LD sand only plasticity is developedand the uplift is not present

4 Foundation Stiffness Matrix

In the previous section large rotations have been consideredin formulating the macroelement model However assuming


times10minus4 times10minus4


minus1 0 1 2minus2

Displacement (m)minus1 0 1 2minus2

Displacement (m)

minus2 0 2 4minus4

Rotation (rad)0 2 4minus2minus4

Rotation (rad)

minus8

minus6

minus4

minus2

0

2

4

6Fo

rce (

kN)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)

minus8

minus6

minus4

minus2

0

2

4

6

Forc

e (kN

)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)



Figure 4 Comparison of experimental and numerical results LD test Phase I

that small rotations permits the construction of the founda-tion stiffness matrix In this case (17) can be rewritten as

g =

1198921

1198922

1198923

1198924

=

119891119904(119904) minus 119865 minus 119875120579

119896VV minus 119875

119891120579(120579) minus 119875 (119867120579 + 119904) minus 119865119867

119880 minus119867120579 minus 119904

(18)

Ryan et al [37 38] studied the stability of bearing isolatorsand constructed the lead-rubber bearing stiffness matrixrelating a change in the isolator forces to the change indisplacements Following the same idea the foundationmatrix k

119891 relating a change in foundation forces 119889f

119891=

⟨119889119865 119889119875⟩119879 to the change in displacements 119889U

119891= ⟨119889119880 119889119881⟩

119879is derived in three steps

(1) Differentiate the equations of equilibrium (in (18) 1198921

1198922and 119892

3) resulting in

keq119889V = Γ119889F119891

(19)

in which 119889V = ⟨119889119904 119889119881 119889120579⟩119879 and thematrices keq and

Γ are given by

keq =[[[[[[

[

120597119891 (119904)

1205971199040 minus119875

0 119896V 0

minus119875120597119891120579

120597120579minus119875119867

]]]]]]

]

Γ =[[

[

1 120579

0 1

119867 119867120579 + 119904

]]

]

(20)

(2) Differentiate the equations of kinematics (see (15a)and (15b))

119889U119891= Γ119879

119889V (21)

(3) Substitute 119889V from (19) into (21)

119889U119891= (Γ119879 feq Γ) 119889F119891 (22)


minus150

minus100

minus50

0

50

100

150Fo

rce (

kN)

minus150

minus100

minus50

0

50

100

150

minus150

minus100

minus50

0

50

100

150

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

0 002 004minus002minus004

Rotation (rad)minus002 0 002 004minus004

Rotation (rad)

0 5 10minus5minus10

Displacement (mm)0 5 10minus5minus10

Displacement (mm)

minus150

minus100

minus50

0

50

100

150



Figure 5 Comparison of experimental and numerical results HD test Phase III

where feq = kminus1eq The resultant flexibility matrix f119891=

Γ119879feqΓ relates the displacement increment 119889U

119891to

the force increment 119889F119891and is given by

f119891=

[[[[[

[

120597119891120579120597120579 + 119875119867 + (120597119891

119904120597119904)119867

2

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

(120597119891119904120597119904) (119867120579 + 119904)

2

+ 1198751198671205792

+ 2119875119904120579

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

+1

119896V

]]]]]

]

(23)

Then foundation stiffness matrix k119891is the inverse of the

flexibility matrix f119891 The stiffness matrix is computed after

calculation of displacements and computation of forces tosatisfy the governing equations (see (17)) In a structuralanalysis program the routine to compute the foundationstiffness matrix may be incorporated into existing structuralanalysis software and used for analysis of structures onshallow foundations

5 Conclusions

A practical macroelement model is presented in this paper tosimulate the response of soil-shallow foundation systemsTheproposedmodel was verified against experimental test resultsof large-scale model foundations subjected to small and largeloading cycles A summary of the main points presented inthis paper is given below




minus60

minus40

minus20

0

20

40

60Fo

rce (

kN)

minus60

minus40

minus20

0

20

40

60

0 001minus001minus002

Displacement (m)minus001 0 001minus002

Displacement (m)

minus50

minus25

0

25

50

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

minus50

minus25

0

25

50



Rotation (rad)

Figure 6 Comparison of experimental and numerical results LD test Phase III

(1) The proposedmacroelementmodel can simulate witha good accuracy the lateral response and rocking ofshallow foundations under quasistatic cyclic loadings

(2) The uplift could be simulated adequately using well-chosen parameters of the Bouc-Wen-Baber-Noorimodel of hysteresis

(3) The soil squeeze-out phenomenon observed by ElGanainy and ElNaggar [17] can be included by properchoice of degradation parameters of the Baber-Noorimodel

(4) The proposed model does not take into considerationfull coupling between the different springs Never-theless the model represents a first step for futureimprovements

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the Sustainable ConstructionMaterial and Structural Systems (SCMASS) Research Groupof the University of Sharjah UAE

References

[1] A S Veletsos and A M Prasad ldquoSeismic interaction of struc-tures and soils stochastic approachrdquo Journal of StructuralEngineering vol 115 no 4 pp 935ndash956 1989

[2] T Balendra and A C Heidebrecht ldquoInfluence of different siteson seismic base shear of buildingsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 4 pp 623ndash642 1986

[3] M Shirato T Kouno S Nakatani and R Paolucci ldquoMostimpacting conventional design parameters on the seismicpermanent displacement of shallow foundationsrdquo Soils andFoundations vol 52 no 2 pp 346ndash355 2012

[4] M Naseri and E S Hosseininia ldquoElastic settlement of ringfoundationsrdquo Soils and Foundations vol 55 no 2 pp 284ndash2952015

[5] N Zamani and U El Shamy ldquoA microscale approach for theseismic response of mdof structures including soil-foundation-structure interactionrdquo Journal of Earthquake Engineering vol18 no 5 pp 785ndash815 2014

[6] M H T Rayhani M H El Naggar and S H Tabatabaei ldquoNon-linear analysis of local site effects on seismic ground responsein the bam earthquakerdquo Geotechnical and Geological Engineer-ing vol 26 no 1 pp 91ndash100 2008

[7] J P Stewart S Kim J Bielak R Dobry and M S PowerldquoRevisions to soil-structure interaction procedures in NEHRP


design provisionsrdquo Earthquake Spectra vol 19 no 3 pp 677ndash696 2003

[8] A Bobet R Salgado and D Loukidis ldquoSeismic design ofdeep foundationsrdquo Publication FHWAINJTRP-200022 JointTransportation Research Program Indiana Department ofTransportation and Purdue University West Lafayette IndUSA 2001

[9] Q Gao J H Lin W X Zhong W P Howson and F WWilliams ldquoIsotropic layered soil-structure interaction caused bystationary random excitationsrdquo International Journal of Solidsand Structures vol 46 no 3-4 pp 455ndash463 2009

[10] C C Spyrakos Ch A Maniatakis and I A KoutromanosldquoSoil-structure interaction effects on base-isolated buildingsfounded on soil stratumrdquo Engineering Structures vol 31 no 3pp 729ndash737 2009

[11] M Eser C Aydemir and I Ekiz ldquoSoil structure interactioneffects on strength reduction factorsrdquo Structural Engineeringand Mechanics vol 41 no 3 pp 365ndash378 2012

[12] B Ganjavi and H Hao ldquoStrength reduction factor for MDOFsoil-structure systemsrdquo Structural Design of Tall and SpecialBuildings vol 23 no 3 pp 161ndash180 2014

[13] M Ornek M Laman A Demir and A Yildiz ldquoPrediction ofbearing capacity of circular footings on soft clay stabilized withgranular soilrdquo Soils and Foundations vol 52 no 1 pp 69ndash802012

[14] Z Y Ai Z X Li and Y C Cheng ldquoBEM analysis of elasticfoundation beams on multilayered isotropic soilsrdquo Soils andFoundations vol 54 no 4 pp 667ndash674 2014

[15] S C Dutta K Bhattacharya and R Roy ldquoEffect of flexibilityof foundations on its seismic stress distributionrdquo Journal ofEarthquake Engineering vol 13 no 1 pp 22ndash49 2009

[16] American Society of Civil Engineers FEMA 356-Prestandardand Commentary for the Seismic Rehabilitation of BuildingsFederal Emergency Management Agency Washington DCUSA 2000

[17] H El Ganainy and M H El Naggar ldquoEfficient 3D nonlinearWinkler model for shallow foundationsrdquo Soil Dynamics andEarthquake Engineering vol 29 no 8 pp 1236ndash1248 2009

[18] R Nova and L Montrasio ldquoSettlements of shallow foundationson sandrdquo Geotechnique vol 41 no 2 pp 243ndash256 1991

[19] R Paolucci ldquoSimplified evaluation of earthquake-inducedpermanent displacements of shallow foundationsrdquo Journal ofEarthquake Engineering vol 1 no 3 pp 563ndash579 1997

[20] S Pedretti Nonlinear seismic soil-structure interaction analysisand modeling method [PhD thesis] Politecnico di MilanoMilano Italy 1998

[21] C Cremer A Pecker and L Davenne ldquoCyclic macro-elementfor soil-structure interaction material and geometrical non-linearitiesrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 25 no 13 pp 1257ndash1284 2001

[22] C Cremer A Pecker and L Davenne ldquoModelling of nonlineardynamic behaviour of a shallow strip foundation with macro-elementrdquo Journal of Earthquake Engineering vol 6 no 2 pp175ndash211 2002

[23] S Gajan JMThomas and B L Kutter ldquoPhysical and analyticalmodeling of cyclic load deformation behavior of shallowfoundationsrdquo in Proceedings of the of the 57th Annual Meetingof Earthquake Engineering Research Institute Ixtapa Mexico2005

[24] C T Chatzigogos A Pecker and J Salencon ldquoMacroelementmodeling of shallow foundationsrdquo Soil Dynamics and Earth-quake Engineering vol 29 no 5 pp 765ndash781 2009

[25] C T Chatzigogos R Figini A Pecker and J Salencon ldquoAmacro-element formulation for shallow foundations on cohe-sive and frictional soilsrdquo International Journal forNumerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 902ndash9312011

[26] U El Shamy and N Zamani ldquoDiscrete element method simula-tions of the seismic response of shallow foundations includingsoil-foundation-structure interactionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 36 no10 pp 1303ndash1329 2012

[27] E Esmaeilzadeh Seylabi H Jahankhah and M Ali GhannadldquoEquivalent linearization of non-linear soil-structure systemsrdquoEarthquake Engineering and Structural Dynamics vol 41 no 13pp 1775ndash1792 2012

[28] F McKenna G L Fenves M H Scott and B Jeremic OpenSystem for Earthquake Engineering Simulation (OpenSees)Pacific Earthquake Engineering Research Center University ofCalifornia Berkeley Calif USA 2000

[29] R Bouc ldquoModele mathematique drsquohysteresis application auxsystemes a un degre de liberterdquo Acustica vol 21 pp 16ndash25 1971(French)

[30] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of EngineeringMechanics-ASCE vol 102 no 2 pp 249ndash263 1976

[31] A M Trochanis J Bielak and P Christiano ldquoSimplified modelfor analysis of one or two pilesrdquo Journal of GeotechnicalEngineering-ASCE vol 117 no 3 pp 448ndash466 1991

[32] D Badoni and N Makris ldquoNonlinear response of single pilesunder lateral inertial and seismic loadsrdquo Soil Dynamics andEarthquake Engineering vol 15 no 1 pp 29ndash43 1996

[33] N Gerolymos andG Gazetas ldquoDevelopment ofWinkler modelfor static and dynamic response of caisson foundations withsoil and interface nonlinearitiesrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 5 pp 363ndash376 2006

[34] T T Baber and M N Noori ldquoRandom vibration of degradingpinching systemsrdquo Journal of EngineeringMechanics vol 111 no8 pp 1010ndash1026 1985

[35] T Haukaas and A Der Kiureghian ldquoFinite element reliabilityand sensitivity methods for performance-based earthquakeengineeringrdquo PEER 200314 Pacific Earthquake EngineeringResearch Center College of Engineering University of Califor-nia Berkeley Calif USA 2004

[36] P Negro R Paolucci S Pedretti and E Faccioli ldquoLarge scalesoil-structure interaction experiments on sand under cyclicloadrdquo in Proceedings of the 12thWorld Conference on EarthquakeEngineering Auckland New Zealand 2000

[37] K L Ryan J M Kelly and A K Chopra ldquoExperimentalobservation of axial load effects in isolation bearingsrdquo inProceedings of the of the 13th World Conference on EarthquakeEngineering Vancouver Canada August 2004

[38] K L Ryan J M Kelly and A K Chopra ldquoNonlinear modelfor lead-rubber bearings including axial-load effectsrdquo Journal ofEngineering Mechanics vol 131 no 12 pp 1270ndash1278 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


ks

Actual soil-foundation system Proposed macroelement model

k

k120579

Figure 1 Proposed macroelement model for soil-shallow foundation system

(i) uncoupled spring model comprising three springelements for shallow foundations that are stiffer thanthe supporting soil

(ii) a finite element formulation of linear (or nonlinear)foundation behavior using Winkler models for shal-low foundations that are less stiff than the supportingsoil

(iii) decoupled Winkler model for shallow foundationsthat are flexible with respect to the supporting soil

However El Ganainy and El Naggar [17] have demonstratedthat the decoupled technique (Beam on a Nonlinear WinklerFoundation BNWF) is not capable of predicting accuratelythe settlement seen in foundations on soft soils Although itcan be used to predict the overall deformation behavior offoundations the BNWF requires a large number of nonlinearsprings which is considered as a major drawback [17]

To address some of the above-mentioned issuesmacroelement formulations have been proposed The firstformulation has been developed by Nova and Montrasio[18] and later modified andor extended by other researchers[19ndash22] and recently the formulations developed by Gajan etal [23] and Shirato et al [3] The major advantages of suchmodels are their simplicity and their ability to capture theglobal response of bearing foundations [22 24 25] Howeveron one hand calibrating macroelement parameters posea constraint on their adoption for practical applicationsAnd on the other hand knowing that most of the availablemacroelement models are based on specified boundingsurfaces poses another problem for their capability to covera wide range of problems [17]

In a completely different modelling approach El Shamyand Zamani [26] proposed a new 3D particle-based tech-nique using the discrete element method (DEM) to analyzethe seismic performance of soil-foundation-structure sys-tems In their model the soil is idealized as a collection ofspherical particles using DEM the footing is considered as arigid block whereas the structure ismodelled using a numberof spherical particles in the form of a column which can beclamped to simulate a rigid structure or bonded to simulate aflexible structure of predefined rigidity

To overcome the difficulties in performing complete non-linear simulations Seylabi et al [27] proposed an equivalentlinearization of nonlinear soil-structure systems consideringboth the effect of SSI and the nonlinear behavior of thestructure on equivalent linear parameters In their model

the structure is modelled as an elastoplastic single-degree-of-freedom system (SDOF) and the soil beneath the structure ismodelled by a discrete model combining different spring anddashpot elements

In this paper a newmacroelementmodel is developed forthe analysis of the nonlinear response of shallow foundationsunder cyclic loading This model may easily be incorpo-rated into available structural analysis programs such asOpenSees [28]The soil-foundation system is simulated usingthree spring elements horizontal and rotational nonlinearsprings and a linear vertical spring The nonlinear springsare assigned appropriate nonlinear model of plasticity withmaterial degrading parameters

2 Proposed Macroelement Model

The problem being studied here is that of a shallow foun-dation of any shape embedded in soil and subjected tosimultaneous axial and lateral forces as shown in Figure 1The foundation is considered to be very stiff The depth ofembedment is 119867 The proposed model incorporates threetypes of springs

(i) vertical translational elastic spring with stiffness 119896V(ii) shear inelastic spring with preyield stiffness 119896

119904

(iii) rotational inelastic spring with preyield stiffness 119896120579

These equivalent springs represent the foundation-soil sys-tem The macroelement model replaces the system soil-shallow foundation thus decreasing considerably both theoverall number of degrees of freedom and the computationeffort required to run large models

21 Constitutive Equations Two material models are consid-ered in this study namely the Bouc-Wen model [29 30] forthe shear and rotational springs and the linear model for thevertical spring

211 Shear and Rotational Springs

Model AssumptionsThe smooth Bouc-Wenmodel of hystere-sis by Bouc [29] and Wen [30] has found many engineeringapplications For instance the use of original Bouc-Wenmodel and its extensions to soil-structure interaction include[31ndash33]

In this study we considered the Baber and Noori [34]extension to the original Bouc-Wen model This version









119891 = 120572119896119909 + (1 minus 120572) 119891119910119911 (1)



=

119860 minus [120573 || 119911 |119911|119899minus1

+ 120574 |119911|120578

] ]120578

(2)




120573 =1205731015840

119909120578

119910

120574 =1205741015840

119909120578

119910

(3)


119910and +119909



119860 = 1198600minus 120575119860119890 (4a)

] = 1 minus 120575]119890 (4b)

120578 = 1 + 120575120578119890 (4c)


119890 = (1 minus 120572) 119891119910119911 (5)

And 120575119860 120575] and 120575


degradation


120597119891 (119909)

120597119909= 120572119896 + (1 minus 120572) 119891

119910

120597119911

120597119909 (6)


=119860 minus |119911|

119899

[120573 sgn (119911) + 120574] ]120578

=120597119911

120597119909

120597119909

120597119905 (7)

Hence

120597119911

120597119909=119860 minus |119911|

120578

[120573 sgn (119911) + 120574] ]120578

(8)


119891119894+1

= 120572119896119909119894+1

+ (1 minus 120572) 119891119910119911119894+1

(9)


119911119894+1

= 119911119894+1

+ Δ119905

119860119894+1

minus10038161003816100381610038161003816119911119894+110038161003816100381610038161003816

119899

[120573 sgn (((119909119894+1 minus 119909119894) Δ119905) 119911119894+1) + 120574] ]119894+1

120578119894+1(119909119894+1

minus 119909119894

Δ119905) (10)



= 119909119894

minus 119891(119909119898

)1198911015840

(119909119898



119860119894+1

= 1198600minus 120575119860119890119894+1

(11a)

]119894+1 = 1 minus 120575]119890119894+1

(11b)

120578119894+1

= 1 + 120575120578119890119894+1

(11c)


119890119894+1

= 119890119894+1

+ Δ119905 (1 minus 120572) 119891119910(119909119894+1

minus 119909119894

Δ119905) 119911119894+1

(12)


U

V

x

ks s

y

H

P

F

k

k120579 120579







119891V = 119896V119881 (13)




119904(119904)



Π =1

21198961205791205792

+1

21198961199041199042

+1

2119896VV2

minus 119875119881 minus 119865119880 (14)


119880 = (119867 minus V) sin (120579) + 119904 cos (120579) (15a)







1198922= 119896VV + 119865 sin (120579) minus 119875 cos (120579) = 0 (16b)







g =

1198921

1198922

1198923

1198924

=




(17)




119896103

119891119910

120572 () 119896103

119891119910

120572 () 119896V103

HDI 1322 125 60 586 10 60 120III 70 99 9 35 111 2 80

LDI 54 38 47 254 38 4 65III 35 404 1 8 333 1 27



119904119904 and


119904(119904) and 119891

120579(120579)





119904119891120579 120597119891119904120597119904 and 120597119891

120579120597120579using the Local


end


tangents 120597119891119904120597119904 and 120597119891













119903= 85









1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578

I 1 05 05 1 0 minus001 0 1 05 05 1 0 minus001 0III 1 05 05 07 0 0 01 1 01 09 07 0 0 01



1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578


times10minus4

times10minus4

minus20

minus10

0

10

20

Forc

e (kN

)

0 1 2minus1minus2


Displacement (m)

minus20

minus10

0

10

20

minus20

minus15

minus10

minus5

0

5

10

15

minus20

minus15

minus10

minus5

0

5

10

15

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

0 2 4minus2minus4


Rotation (rad)










minus1 0 1 2minus2


Displacement (m)

minus2 0 2 4minus4


Rotation (rad)

minus8

minus6

minus4

minus2

0

2

4

6Fo

rce (

kN)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)

minus8

minus6

minus4

minus2

0

2

4

6

Forc

e (kN

)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)





g =

1198921

1198922

1198923

1198924

=

119891119904(119904) minus 119865 minus 119875120579

119896VV minus 119875

119891120579(120579) minus 119875 (119867120579 + 119904) minus 119865119867

119880 minus119867120579 minus 119904

(18)



119891=


119891= ⟨119889119880 119889119881⟩



1198922and 119892

3) resulting in

keq119889V = Γ119889F119891

(19)


Γ are given by

keq =[[[[[[

[

120597119891 (119904)

1205971199040 minus119875

0 119896V 0

minus119875120597119891120579

120597120579minus119875119867

]]]]]]

]

Γ =[[

[

1 120579

0 1

119867 119867120579 + 119904

]]

]

(20)


119889U119891= Γ119879

119889V (21)


119889U119891= (Γ119879 feq Γ) 119889F119891 (22)


minus150

minus100

minus50

0

50

100

150Fo

rce (

kN)

minus150

minus100

minus50

0

50

100

150

minus150

minus100

minus50

0

50

100

150

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)



Rotation (rad)

0 5 10minus5minus10


Displacement (mm)

minus150

minus100

minus50

0

50

100

150






119891to


f119891=

[[[[[

[

120597119891120579120597120579 + 119875119867 + (120597119891

119904120597119904)119867

2

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

(120597119891119904120597119904) (119867120579 + 119904)

2

+ 1198751198671205792

+ 2119875119904120579

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

+1

119896V

]]]]]

]

(23)




5 Conclusions





minus60

minus40

minus20

0

20

40

60Fo

rce (

kN)

minus60

minus40

minus20

0

20

40

60



Displacement (m)

minus50

minus25

0

25

50

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

minus50

minus25

0

25

50



Rotation (rad)






Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119891 = 120572119896119909 + (1 minus 120572) 119891119910119911 (1)



=

119860 minus [120573 || 119911 |119911|119899minus1

+ 120574 |119911|120578

] ]120578

(2)




120573 =1205731015840

119909120578

119910

120574 =1205741015840

119909120578

119910

(3)


119910and +119909



119860 = 1198600minus 120575119860119890 (4a)

] = 1 minus 120575]119890 (4b)

120578 = 1 + 120575120578119890 (4c)


119890 = (1 minus 120572) 119891119910119911 (5)

And 120575119860 120575] and 120575


degradation


120597119891 (119909)

120597119909= 120572119896 + (1 minus 120572) 119891

119910

120597119911

120597119909 (6)


=119860 minus |119911|

119899

[120573 sgn (119911) + 120574] ]120578

=120597119911

120597119909

120597119909

120597119905 (7)

Hence

120597119911

120597119909=119860 minus |119911|

120578

[120573 sgn (119911) + 120574] ]120578

(8)


119891119894+1

= 120572119896119909119894+1

+ (1 minus 120572) 119891119910119911119894+1

(9)


119911119894+1

= 119911119894+1

+ Δ119905

119860119894+1

minus10038161003816100381610038161003816119911119894+110038161003816100381610038161003816

119899

[120573 sgn (((119909119894+1 minus 119909119894) Δ119905) 119911119894+1) + 120574] ]119894+1

120578119894+1(119909119894+1

minus 119909119894

Δ119905) (10)



= 119909119894

minus 119891(119909119898

)1198911015840

(119909119898



119860119894+1

= 1198600minus 120575119860119890119894+1

(11a)

]119894+1 = 1 minus 120575]119890119894+1

(11b)

120578119894+1

= 1 + 120575120578119890119894+1

(11c)


119890119894+1

= 119890119894+1

+ Δ119905 (1 minus 120572) 119891119910(119909119894+1

minus 119909119894

Δ119905) 119911119894+1

(12)


U

V

x

ks s

y

H

P

F

k

k120579 120579







119891V = 119896V119881 (13)




119904(119904)



Π =1

21198961205791205792

+1

21198961199041199042

+1

2119896VV2

minus 119875119881 minus 119865119880 (14)


119880 = (119867 minus V) sin (120579) + 119904 cos (120579) (15a)







1198922= 119896VV + 119865 sin (120579) minus 119875 cos (120579) = 0 (16b)







g =

1198921

1198922

1198923

1198924

=




(17)




119896103

119891119910

120572 () 119896103

119891119910

120572 () 119896V103

HDI 1322 125 60 586 10 60 120III 70 99 9 35 111 2 80

LDI 54 38 47 254 38 4 65III 35 404 1 8 333 1 27



119904119904 and


119904(119904) and 119891

120579(120579)





119904119891120579 120597119891119904120597119904 and 120597119891

120579120597120579using the Local


end


tangents 120597119891119904120597119904 and 120597119891













119903= 85









1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578

I 1 05 05 1 0 minus001 0 1 05 05 1 0 minus001 0III 1 05 05 07 0 0 01 1 01 09 07 0 0 01



1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578


times10minus4

times10minus4

minus20

minus10

0

10

20

Forc

e (kN

)

0 1 2minus1minus2


Displacement (m)

minus20

minus10

0

10

20

minus20

minus15

minus10

minus5

0

5

10

15

minus20

minus15

minus10

minus5

0

5

10

15

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

0 2 4minus2minus4


Rotation (rad)










minus1 0 1 2minus2


Displacement (m)

minus2 0 2 4minus4


Rotation (rad)

minus8

minus6

minus4

minus2

0

2

4

6Fo

rce (

kN)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)

minus8

minus6

minus4

minus2

0

2

4

6

Forc

e (kN

)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)





g =

1198921

1198922

1198923

1198924

=

119891119904(119904) minus 119865 minus 119875120579

119896VV minus 119875

119891120579(120579) minus 119875 (119867120579 + 119904) minus 119865119867

119880 minus119867120579 minus 119904

(18)



119891=


119891= ⟨119889119880 119889119881⟩



1198922and 119892

3) resulting in

keq119889V = Γ119889F119891

(19)


Γ are given by

keq =[[[[[[

[

120597119891 (119904)

1205971199040 minus119875

0 119896V 0

minus119875120597119891120579

120597120579minus119875119867

]]]]]]

]

Γ =[[

[

1 120579

0 1

119867 119867120579 + 119904

]]

]

(20)


119889U119891= Γ119879

119889V (21)


119889U119891= (Γ119879 feq Γ) 119889F119891 (22)


minus150

minus100

minus50

0

50

100

150Fo

rce (

kN)

minus150

minus100

minus50

0

50

100

150

minus150

minus100

minus50

0

50

100

150

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)



Rotation (rad)

0 5 10minus5minus10


Displacement (mm)

minus150

minus100

minus50

0

50

100

150






119891to


f119891=

[[[[[

[

120597119891120579120597120579 + 119875119867 + (120597119891

119904120597119904)119867

2

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

(120597119891119904120597119904) (119867120579 + 119904)

2

+ 1198751198671205792

+ 2119875119904120579

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

+1

119896V

]]]]]

]

(23)




5 Conclusions





minus60

minus40

minus20

0

20

40

60Fo

rce (

kN)

minus60

minus40

minus20

0

20

40

60



Displacement (m)

minus50

minus25

0

25

50

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

minus50

minus25

0

25

50



Rotation (rad)






Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










U

V

x

ks s

y

H

P

F

k

k120579 120579







119891V = 119896V119881 (13)




119904(119904)



Π =1

21198961205791205792

+1

21198961199041199042

+1

2119896VV2

minus 119875119881 minus 119865119880 (14)


119880 = (119867 minus V) sin (120579) + 119904 cos (120579) (15a)







1198922= 119896VV + 119865 sin (120579) minus 119875 cos (120579) = 0 (16b)







g =

1198921

1198922

1198923

1198924

=




(17)




119896103

119891119910

120572 () 119896103

119891119910

120572 () 119896V103

HDI 1322 125 60 586 10 60 120III 70 99 9 35 111 2 80

LDI 54 38 47 254 38 4 65III 35 404 1 8 333 1 27



119904119904 and


119904(119904) and 119891

120579(120579)





119904119891120579 120597119891119904120597119904 and 120597119891

120579120597120579using the Local


end


tangents 120597119891119904120597119904 and 120597119891













119903= 85









1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578

I 1 05 05 1 0 minus001 0 1 05 05 1 0 minus001 0III 1 05 05 07 0 0 01 1 01 09 07 0 0 01



1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578


times10minus4

times10minus4

minus20

minus10

0

10

20

Forc

e (kN

)

0 1 2minus1minus2


Displacement (m)

minus20

minus10

0

10

20

minus20

minus15

minus10

minus5

0

5

10

15

minus20

minus15

minus10

minus5

0

5

10

15

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

0 2 4minus2minus4


Rotation (rad)










minus1 0 1 2minus2


Displacement (m)

minus2 0 2 4minus4


Rotation (rad)

minus8

minus6

minus4

minus2

0

2

4

6Fo

rce (

kN)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)

minus8

minus6

minus4

minus2

0

2

4

6

Forc

e (kN

)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)





g =

1198921

1198922

1198923

1198924

=

119891119904(119904) minus 119865 minus 119875120579

119896VV minus 119875

119891120579(120579) minus 119875 (119867120579 + 119904) minus 119865119867

119880 minus119867120579 minus 119904

(18)



119891=


119891= ⟨119889119880 119889119881⟩



1198922and 119892

3) resulting in

keq119889V = Γ119889F119891

(19)


Γ are given by

keq =[[[[[[

[

120597119891 (119904)

1205971199040 minus119875

0 119896V 0

minus119875120597119891120579

120597120579minus119875119867

]]]]]]

]

Γ =[[

[

1 120579

0 1

119867 119867120579 + 119904

]]

]

(20)


119889U119891= Γ119879

119889V (21)


119889U119891= (Γ119879 feq Γ) 119889F119891 (22)


minus150

minus100

minus50

0

50

100

150Fo

rce (

kN)

minus150

minus100

minus50

0

50

100

150

minus150

minus100

minus50

0

50

100

150

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)



Rotation (rad)

0 5 10minus5minus10


Displacement (mm)

minus150

minus100

minus50

0

50

100

150






119891to


f119891=

[[[[[

[

120597119891120579120597120579 + 119875119867 + (120597119891

119904120597119904)119867

2

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

(120597119891119904120597119904) (119867120579 + 119904)

2

+ 1198751198671205792

+ 2119875119904120579

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

+1

119896V

]]]]]

]

(23)




5 Conclusions





minus60

minus40

minus20

0

20

40

60Fo

rce (

kN)

minus60

minus40

minus20

0

20

40

60



Displacement (m)

minus50

minus25

0

25

50

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

minus50

minus25

0

25

50



Rotation (rad)






Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119896103

119891119910

120572 () 119896103

119891119910

120572 () 119896V103

HDI 1322 125 60 586 10 60 120III 70 99 9 35 111 2 80

LDI 54 38 47 254 38 4 65III 35 404 1 8 333 1 27



119904119904 and


119904(119904) and 119891

120579(120579)





119904119891120579 120597119891119904120597119904 and 120597119891

120579120597120579using the Local


end


tangents 120597119891119904120597119904 and 120597119891













119903= 85









1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578

I 1 05 05 1 0 minus001 0 1 05 05 1 0 minus001 0III 1 05 05 07 0 0 01 1 01 09 07 0 0 01



1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578


times10minus4

times10minus4

minus20

minus10

0

10

20

Forc

e (kN

)

0 1 2minus1minus2


Displacement (m)

minus20

minus10

0

10

20

minus20

minus15

minus10

minus5

0

5

10

15

minus20

minus15

minus10

minus5

0

5

10

15

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

0 2 4minus2minus4


Rotation (rad)










minus1 0 1 2minus2


Displacement (m)

minus2 0 2 4minus4


Rotation (rad)

minus8

minus6

minus4

minus2

0

2

4

6Fo

rce (

kN)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)

minus8

minus6

minus4

minus2

0

2

4

6

Forc

e (kN

)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)





g =

1198921

1198922

1198923

1198924

=

119891119904(119904) minus 119865 minus 119875120579

119896VV minus 119875

119891120579(120579) minus 119875 (119867120579 + 119904) minus 119865119867

119880 minus119867120579 minus 119904

(18)



119891=


119891= ⟨119889119880 119889119881⟩



1198922and 119892

3) resulting in

keq119889V = Γ119889F119891

(19)


Γ are given by

keq =[[[[[[

[

120597119891 (119904)

1205971199040 minus119875

0 119896V 0

minus119875120597119891120579

120597120579minus119875119867

]]]]]]

]

Γ =[[

[

1 120579

0 1

119867 119867120579 + 119904

]]

]

(20)


119889U119891= Γ119879

119889V (21)


119889U119891= (Γ119879 feq Γ) 119889F119891 (22)


minus150

minus100

minus50

0

50

100

150Fo

rce (

kN)

minus150

minus100

minus50

0

50

100

150

minus150

minus100

minus50

0

50

100

150

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)



Rotation (rad)

0 5 10minus5minus10


Displacement (mm)

minus150

minus100

minus50

0

50

100

150






119891to


f119891=

[[[[[

[

120597119891120579120597120579 + 119875119867 + (120597119891

119904120597119904)119867

2

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

(120597119891119904120597119904) (119867120579 + 119904)

2

+ 1198751198671205792

+ 2119875119904120579

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

+1

119896V

]]]]]

]

(23)




5 Conclusions





minus60

minus40

minus20

0

20

40

60Fo

rce (

kN)

minus60

minus40

minus20

0

20

40

60



Displacement (m)

minus50

minus25

0

25

50

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

minus50

minus25

0

25

50



Rotation (rad)






Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578

I 1 05 05 1 0 minus001 0 1 05 05 1 0 minus001 0III 1 05 05 07 0 0 01 1 01 09 07 0 0 01



1015840

1205741015840

119899 120575119860

120575V 120575120578

119860 1205731015840

1205741015840

119899 120575119860

120575V 120575120578


times10minus4

times10minus4

minus20

minus10

0

10

20

Forc

e (kN

)

0 1 2minus1minus2


Displacement (m)

minus20

minus10

0

10

20

minus20

minus15

minus10

minus5

0

5

10

15

minus20

minus15

minus10

minus5

0

5

10

15

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

0 2 4minus2minus4


Rotation (rad)










minus1 0 1 2minus2


Displacement (m)

minus2 0 2 4minus4


Rotation (rad)

minus8

minus6

minus4

minus2

0

2

4

6Fo

rce (

kN)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)

minus8

minus6

minus4

minus2

0

2

4

6

Forc

e (kN

)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)





g =

1198921

1198922

1198923

1198924

=

119891119904(119904) minus 119865 minus 119875120579

119896VV minus 119875

119891120579(120579) minus 119875 (119867120579 + 119904) minus 119865119867

119880 minus119867120579 minus 119904

(18)



119891=


119891= ⟨119889119880 119889119881⟩



1198922and 119892

3) resulting in

keq119889V = Γ119889F119891

(19)


Γ are given by

keq =[[[[[[

[

120597119891 (119904)

1205971199040 minus119875

0 119896V 0

minus119875120597119891120579

120597120579minus119875119867

]]]]]]

]

Γ =[[

[

1 120579

0 1

119867 119867120579 + 119904

]]

]

(20)


119889U119891= Γ119879

119889V (21)


119889U119891= (Γ119879 feq Γ) 119889F119891 (22)


minus150

minus100

minus50

0

50

100

150Fo

rce (

kN)

minus150

minus100

minus50

0

50

100

150

minus150

minus100

minus50

0

50

100

150

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)



Rotation (rad)

0 5 10minus5minus10


Displacement (mm)

minus150

minus100

minus50

0

50

100

150






119891to


f119891=

[[[[[

[

120597119891120579120597120579 + 119875119867 + (120597119891

119904120597119904)119867

2

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

(120597119891119904120597119904) (119867120579 + 119904)

2

+ 1198751198671205792

+ 2119875119904120579

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

+1

119896V

]]]]]

]

(23)




5 Conclusions





minus60

minus40

minus20

0

20

40

60Fo

rce (

kN)

minus60

minus40

minus20

0

20

40

60



Displacement (m)

minus50

minus25

0

25

50

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

minus50

minus25

0

25

50



Rotation (rad)






Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












minus1 0 1 2minus2


Displacement (m)

minus2 0 2 4minus4


Rotation (rad)

minus8

minus6

minus4

minus2

0

2

4

6Fo

rce (

kN)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)

minus8

minus6

minus4

minus2

0

2

4

6

Forc

e (kN

)

minus6

minus4

minus2

0

2

4

6

Mom

ent (

kNmiddotm

)





g =

1198921

1198922

1198923

1198924

=

119891119904(119904) minus 119865 minus 119875120579

119896VV minus 119875

119891120579(120579) minus 119875 (119867120579 + 119904) minus 119865119867

119880 minus119867120579 minus 119904

(18)



119891=


119891= ⟨119889119880 119889119881⟩



1198922and 119892

3) resulting in

keq119889V = Γ119889F119891

(19)


Γ are given by

keq =[[[[[[

[

120597119891 (119904)

1205971199040 minus119875

0 119896V 0

minus119875120597119891120579

120597120579minus119875119867

]]]]]]

]

Γ =[[

[

1 120579

0 1

119867 119867120579 + 119904

]]

]

(20)


119889U119891= Γ119879

119889V (21)


119889U119891= (Γ119879 feq Γ) 119889F119891 (22)


minus150

minus100

minus50

0

50

100

150Fo

rce (

kN)

minus150

minus100

minus50

0

50

100

150

minus150

minus100

minus50

0

50

100

150

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)



Rotation (rad)

0 5 10minus5minus10


Displacement (mm)

minus150

minus100

minus50

0

50

100

150






119891to


f119891=

[[[[[

[

120597119891120579120597120579 + 119875119867 + (120597119891

119904120597119904)119867

2

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

(120597119891119904120597119904) (119867120579 + 119904)

2

+ 1198751198671205792

+ 2119875119904120579

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

+1

119896V

]]]]]

]

(23)




5 Conclusions





minus60

minus40

minus20

0

20

40

60Fo

rce (

kN)

minus60

minus40

minus20

0

20

40

60



Displacement (m)

minus50

minus25

0

25

50

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

minus50

minus25

0

25

50



Rotation (rad)






Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus150

minus100

minus50

0

50

100

150Fo

rce (

kN)

minus150

minus100

minus50

0

50

100

150

minus150

minus100

minus50

0

50

100

150

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)



Rotation (rad)

0 5 10minus5minus10


Displacement (mm)

minus150

minus100

minus50

0

50

100

150






119891to


f119891=

[[[[[

[

120597119891120579120597120579 + 119875119867 + (120597119891

119904120597119904)119867

2

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

((120597119891119904120597119904)119867 + 119875) (119867120579 + 119904)

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

(120597119891119904120597119904) (119867120579 + 119904)

2

+ 1198751198671205792

+ 2119875119904120579

(120597119891119904120597119904) (120597119891

120579120597120579 minus 119875119867) minus 1198752

+1

119896V

]]]]]

]

(23)




5 Conclusions





minus60

minus40

minus20

0

20

40

60Fo

rce (

kN)

minus60

minus40

minus20

0

20

40

60



Displacement (m)

minus50

minus25

0

25

50

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

minus50

minus25

0

25

50



Rotation (rad)






Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












minus60

minus40

minus20

0

20

40

60Fo

rce (

kN)

minus60

minus40

minus20

0

20

40

60



Displacement (m)

minus50

minus25

0

25

50

Mom

ent (

kNmiddotm

)

Forc

e (kN

)M

omen

t (kN

middotm)

minus50

minus25

0

25

50



Rotation (rad)






Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article practical soil-shallow foundation model

Documents