Investigation of the Dynamic Response of Rigid Footings on Tensionless Winkler Foundation

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Soil Dynamics and Earthquake Engineering 28 (2008) 577–591

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Investigation of the dynamic response of rigid footings on tensionlessWinkler foundation

Ioannis N. Psycharis�

Laboratory for Earthquake Engineering, School of Civil Engineering, National Technical University of Athens, Polytechnic Campus, 15780 Athens, Greece

Received 20 November 2006; received in revised form 23 July 2007; accepted 28 July 2007

Abstract

The dynamic response of a rigid footing resting on an elastic tensionless Winkler foundation is examined. A parametric investigation,

concerning the effect of the main parameters on the response, is performed for harmonic excitation. The parameters examined include the

stiffness and the damping of the foundation, the excitation frequency and the superstructure characteristics and loads. The maximum

rocking response, the minimum length of contact after uplift, the maximum stress developed at the soil and the factor of safety with

respect to the bearing capacity of the soil are used to measure the effect of each dimensionless parameter. An example for earthquake

excitation is also given for a plane frame. The results are compared to the ones of a simplified static approach based on the maximum

values of the applied loads, similarly to the procedure that is usually applied in practice. The results show that the static approach can

predict the response satisfactorily if resonance does not happen, if the stiffness of the foundation is not large compared to the stiffness of

the superstructure and if the dynamic part of the axial force of the column is not large; in these cases, it may underestimate or

overestimate the response significantly, depending on the sign of the dynamic axial force that is considered.

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Keywords: Earthquake response; Harmonic response; Footings; Foundation; Winkler foundation; Uplift; Rocking

1. Introduction

The phenomenon of partial uplift of shallow foundationsis common during strong earthquakes and is due to theinability of the soil to develop tensile stresses. As a result,part of the base of a footing loses contact with the groundwhenever the vertical displacement of either corner of thebase becomes larger than a critical value, which dependson the dimensions of the footing, the stiffness of thefoundation mat and the static loads. It is known thatfoundation uplift affects the dynamic response of thesuperstructure, usually in a beneficial way, since it reducesits response. For this reason, the effect of the base uplift onthe earthquake response of structures has been investigatedthoroughly, e.g. by Meek [1,2], Psycharis [3,4], Yim andChopra [5,6], Celep and Guler [7], Wang and Gould [8],Olivet et al. [9], Harden and Hutchinson [10] among others.

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On the contrary, the effect of partial lift-off on thestresses developed at the soil has not been studied enough.The seismic codes address this problem by static con-siderations, which are based on the forces and momentsapplied to the footing during the seismic design situation(Eurocode 7 [11], FEMA 356 [12], ATC 40 [13]). In theapproach of the American codes (for a thorough investiga-tion see [14]), a relation is established between theoverturning moment and the footing rotation for eachpossible state: elastic behaviour of the soil and full contactof the base; inelastic behaviour of the soil and full contactof the base; elastic behaviour of the soil and partial uplift;inelastic behaviour of the soil and partial uplift. InEurocode 7 (EC7)—Appendix D, the effect of the baseuplift is considered in the calculation of the bearingcapacity of the soil through a reduction of the effectivearea of contact between the footing and the foundationmat. The reduction in the dimensions of the area of contactis proportional to the eccentricity of the loads applied,which is measured by the ratio M/N, where M is theapplied moment and N is the axial force.

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ARTICLE IN PRESSI.N. Psycharis / Soil Dynamics and Earthquake Engineering 28 (2008) 577–591578

In both approaches, the applied loads M and N play animportant role to the design of the foundation. During anearthquake, these loads vary with the dynamic response ofthe superstructure. If a modal analysis is applied for thecalculation of the response of the superstructure, as it iscommon in practice, the maximum values of the dynamiccomponents of M and N can be calculated through acombination rule of the modal responses. However,although new generation combination rules can handlereasonably well the problem of the simultaneous values ofmultiple response quantities, the sign of the axial force islost. Thus, it is not known whether the total axial force atthe base of the columns should be calculated by adding orsubtracting the dynamic component of N to the corre-sponding gravity loads. This dynamic component of theaxial force may be large, especially for columns locatedclose to the perimeter of the building and for coupled walls.Thus, for strong earthquakes, forces larger than the staticones can be developed. In such cases, the application of theseismic codes with the most adverse combination of signsfor M and N may lead to a conservative design andunnecessarily large dimensions of the foundation.

In this analysis, the dynamic response of a rigid footingon elastic tensionless Winkler foundation is studied. Aparametric investigation of the effect of all the parametersinvolved in the problem is performed for harmonicexcitation. Emphasis is given to the aforementionedproblem, concerning the sign of the axial force. Harmonicwave forms are assumed for both the base motion and thedynamic part of the loads imposed to the footing from thesuperstructure. This type of excitation is chosen instead ofearthquake records in an effort to minimize the parametersconcerning the excitation and, also, because it gives a betterinsight to resonance phenomena. However, an example forearthquake excitation is also given.

The results of the dynamic analysis are also comparedwith the ones of a static approach, which is based onthe maximum values of the dynamic components of theloads, as it is the usual procedure in practice during theapplication of the specifications of the codes. In thisway conclusions concerning the accuracy of this approachare drawn and a suggestion, concerning the value of the

b

a

CM

M

k0, c0

h

δ δ

Fig. 1. (a) Footing on elastic Winkler found

axial force that should be used in these calculations, ismade.

2. System considered

The system considered is shown in Fig. 1(a). It consistsof a rigid footing connected to the superstructure by acolumn. The footing is shown rectangular, with dimensionsb� a but the following analysis can also be applied to anyother shape that is symmetric about a vertical axis throughthe middle point of the base, M, by assuming that b is thewidth of the base, h is the height at which the centre ofgravity (CM) of the footing is located and a is the totalheight of the footing, equal to the distance between thebottom of the column and the foundation mat.The soil is modelled as a tensionless Winkler foundation

that does not yield. For simplicity, the soil flexibility in thehorizontal direction is not considered, because it does notaffect significantly the vertical stresses that are developed atthe soil, which are of main concern in this analysis anddepend on the axial force and the rotation of the footingonly. It should be noted that, for tall buildings and slenderstructures, the effect of the horizontal translation of thebase can be neglected in comparison to the effect of therocking even for the response of the superstructure [15].Thus, the footing is allowed to rotate and move verticallyonly.The Winkler foundation is characterized by two para-

meters: the subgrade modulus, k0 and the correspondingdamping, c0. This simple model is widely used insoil–structure interaction (SSI) problems, although itcannot model precisely the dynamic response of footingsin which rocking and vertical vibrations occur simulta-neously. This is due to the fact that these modes are notindependent from each other. In order to uncouple thevibrations, a higher-order modelling should be applied, asthe one proposed by Hall et al. [16]. The present analysis,however, aims to the qualitative investigation of thedynamic response and not to the exact calculation of therotations and displacements. Taking under considerationthat the governing equations of motion after lift-off are

ah

b

k0, c0

CM

Kr, sup

x

z

M

ϕ

ation, (b) equivalent model of analysis.

ARTICLE IN PRESSI.N. Psycharis / Soil Dynamics and Earthquake Engineering 28 (2008) 577–591 579

already very complicated, it was decided to use the simpleWinkler foundation mentioned above.

Fig. 1(b) shows the corresponding model that is used inthe analysis, which differs from the one of Fig. 1(a) only inthe absence of the column, which has been substituted by arotational spring placed at the centre of the base (point M).This spring, which models the stiffness of the super-structure, has stiffness Kr,sup ¼ Kr,col+a2 Kx,col, with Kr,col

and Kx,col being the rotational and the horizontal stiffnessof the column (at its base), respectively and a being theheight of the footing. The spring is placed at point M andnot at the centre of gravity, CM, because the equations ofmotion are derived about that point. There is no need toadd a horizontal spring at point M modelling thehorizontal stiffness of the column, because the base is notallowed to move in this direction.

The motion of the footing is considered only on theplane of the cross section shown in Fig. 1. In the directionnormal to this plane, a unit width is assumed for thefooting. The loads and the stiffness are considered per unitlength in this direction.

Since the horizontal movement of the base is restricted, themotion can be described by two independent variables, namelythe rotation j and the vertical displacement z of the centre ofmass [see Fig. 1(b)]. Note that, although there is no horizontalmotion at the base, the rotation produces horizontal displace-ments at all other points, which, however, can be expressed interms of j. For example, for small rotations, the horizontaldisplacement, x, of CM is equal to hj.

The superstructure loads, which in the real system act onthe footing through the column, are also transferred topoint M and are considered as external loads to thefooting, as shown in Fig. 2. If Mcol, Ncol and Vcol are themoment, the axial force and the shear force at the base ofthe column, respectively, the moment and the axial force atpoint M are: M ¼Mcol+a Vcol and N ¼ Ncol. The axialforce N is considered positive if it is compressive. No shearforce is considered at point M, because this point is notmoving in the horizontal direction.

a

h

b

k0, c0

CM

Kr,sup

x

z

M

NG+NE

MG+ME

..zg

xg

..

ϕδ

Fig. 2. Base accelerations and superstructure loads.

In this analysis, the response of the foundation isuncoupled from the response of the superstructure. Thecolumn loads Mcol, Ncol and Vcol can be approximatelycalculated from the response of the superstructure assum-ing fixed conditions at the base. However, when these loadsare applied to the footing, a rotation j occurs and thus, theactual moment that acts at the base of the column issmaller than Mcol, because it is relaxed by the termKr,sup �j. In this way, SSI effects are considered in anindirect way. For a more accurate solution, the shear forcesof the columns could be calculated for the seismic loads ofthe superstructure that correspond to the dynamic char-acteristics of the system on a flexible foundation, while themoments at the base of the columns are calculated fromthese forces assuming that the base is fixed. Such aprocedure, however, is not easily applied and, in general,would not lead to the exact solution either, since theredistribution of the seismic loads among the columns,caused by the variation of their stiffness due to the rotationat their base, is not considered. As an approximation, thecolumn loads can be directly calculated from the responseof the superstructure assuming fixed base conditions. Inthis case, the approximation concerns: (i) the earthquakeloads applied to the superstructure, because the flexibilityof the foundation is not considered and (ii) the distributionof these loads among the columns. The present analysisfocuses on the investigation of the response of the footing,assuming ‘‘known’’ loads from the superstructure and,thus, this matter is not examined further.The superstructure loads consist of two terms: the static

term, which is due to the gravity loads of the superstructureand will be denoted by the subscript ‘‘G’’ and theearthquake term, which is caused by the dynamic responseof the superstructure and will be denoted by the subscript‘‘E’’. The dynamic part of each load is time dependent.Therefore, one can write:

MðtÞ ¼MG þMEðtÞ, (1)

NðtÞ ¼ NG þNEðtÞ. (2)

3. Dynamic analysis

The equations of motion can be derived from thekinematics of the rocking block shown in Fig. 2. It isassumed that the moment of inertia of the superstructuredoes not affect the rocking response, because, in general,the rotations of the footings produce only local effects andare not related to the rotation of the structure as a whole.Also, the mass of the superstructure that is moving withthe footing affects the vertical vibrations and must beconsidered in the equations. Since a small part of the wholestructure is examined, the exact calculation of this mass isnot possible. For simplicity, it is assumed that this mass isequal to NG/g, with g being the acceleration of gravity. Inother words, it is assumed that this mass is equal to the one


that corresponds to the static axial force at the base of thecolumn, although this is not generally true. Under theseassumptions, the following equations of motion can bederived for small rotations (sinjEj and cosjE1) and forj40:

Full contact:

IM €jþ 112

c0b3 _jþ 1

12k0b

3jþ K r;supj

¼ �mh €xg þMG þME, ð3aÞ

mþNG=g� �

€zþ c0b_zþ k0bz

¼ � mþNG=g� �

€zg �NE. ð3bÞ

With uplift:

IM €jþ1

3c0

b3

8þ

d� z

j

� �3" #

_j�1

2c0

b2

4

��

d� z

j

� �2#_z

þ1

3k0

b3

8þ

d� z

j

� �3" #

jþ K r;supj

þ1

2k0

b2

4�

d� z

j

� �2" #

d� zð Þ

¼ �mh €xg þMG þME, ð4aÞ

mþNG=g� �

€zþ c0b

2þ

d� z

j

� �_z�

1

2c0

b2

4

��

d� z

j

� �2#j

þ1

2k0bz�

1

2k0

b2

4

�þ

d� z

j

� �2#j ¼ �

1

2mgþNGð Þ

� mþNG=g� �

€zg �NE ð4bÞ

in which m is the mass of the footing, IM is the massmoment of inertia about point M, d is the static deflectiondue to the gravity loads, given by the relation d ¼ ðNG þ

mgÞ=k0b and €xg; €zg are the base accelerations in thehorizontal and the vertical directions, respectively.

In the above equations of motion, second-order effects arenot included. These second-order effects concern thehorizontal displacement of the CM and the verticaldisplacement of the corner of the base, caused by therotation of the footing. The former displacement producesan extra moment of the vertical loads with respect to theCM and the latter one an extra moment of the horizontalloads. However, since the superstructure loads, which arethe most important external loads applied to the footing, areconsidered in an approximate way, it is reasonable to neglectsecond-order effects, which are of small importance.

Note that the above equations of motion can be seen asan extension of the equations that govern the motion of afree-rocking block [17], if the superstructure stiffness andloads are included and the above-mentioned second-orderterms of the inertia forces are neglected.

The equations of motion for the full-contact regime [Eqs.(3a) and (3b)] are valid for positive (j40) and negative(jo0) angles of rotation. For the uplift regime, Eqs. (4a)and (4b) are valid for j40, only. The equations for jo0are similar to (4a) and (4b) but some terms appear withopposite sign. One way to calculate the response for jo0 isby using Eqs. (4a) and (4b) for this regime also, byassuming that j is temporarily considered positive counter-clockwise and the x-axis is positive to the left. In this case,the sign of the terms €xg, MG and ME in (4a) and (4b), thatare affected by this change, should be reversed. Aftercalculating the angle of rotation, one should change itssign, in order to return to the initial notation of clockwisepositive angles (Fig. 1b).Due to the gravity loads, an initial angle of rotation, j0,

exists in the system which must be considered in the solutionof the equations of motion. If Kr,tot is the total rotationalstiffness, the initial rotation is equal to MG/Kr,tot. The totalrotational stiffness is equal to the sum of the rocking stiffnessof the Winkler foundation, Kr,f and the stiffness of thesuperstructure, Kr,sup, i.e. Kr,tot ¼ Kr,sup+Kr,f. Assuming thatuplift does not occur for gravity loads, Kr,f ¼ k0b

3/12. Notethat z is measured from the equilibrium position and,therefore, the initial value of z is zero.The value of the static vertical displacement, d, should be

used to determine when uplift occurs or full contact is re-established. Actually, uplift starts when the verticaldisplacement of any corner of the base becomes greaterthan d and full contact is re-established when thisdisplacement becomes smaller than d. Assuming that z ispositive upwards (Fig. 2), the criteria for the initiation andthe end of uplift can be written as:

Initiation of uplift:

zþ 12b j�� Xd. (5a)

End of uplift:

zþ 12b j�� pd. (5b)

All the results of the present analysis were obtained byintegrating the equations of motion using Runge–Kutta ofthe fourth order and a small time step, equal to 0.0002 s.This time step is equal to 5% of the smallest eigenperiodduring full contact among all the cases of the parametricinvestigation, while in most of them it is much smaller. Thecriteria for uplift or contact were checked at each time step.For all the cases examined, the maximum angle of rotation,jmax, the minimum length of contact, smin, and the maxi-mum corner stress developed at the soil, sc,max, (Fig. 3)were determined from the corresponding time historiesusing the following equations:

Full contact:

sðtÞ ¼ b, (6a)

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h

CM

M

Mf

Nf

s

z

s

z

Mf

σc

σc

CM

h

M

Nf

δϕ

δϕ

Fig. 3. Length of contact and soil stresses: (a) full contact, (b) during uplift.

I.N. Psycharis / Soil Dynamics and Earthquake Engineering 28 (2008) 577–591 581

scðtÞ ¼ k0 d� zðtÞ þ 12b jðtÞ��

. (6b)

With uplift:

sðtÞ ¼b

2þ

djðtÞ�� zðtÞ

jðtÞ�� , (7a)

scðtÞ ¼ k0sðtÞjðtÞ. (7b)

Note that complete separation of the footing from thefoundation is possible to occur for stiff foundations. If thishappens, s becomes zero and the analysis stops. In suchcases, the maximum value of jmax and sc,max up to the timeof complete separation was used in the plots that arepresented in the following.

In each case, the bearing capacity of the soil, Rd,according to EC7 (Appendix D) was also determined andcompared to the total maximum axial force, Nf, at thefoundation level. Thus, the factor of safety was derivedfrom the ratio Rd/Nf. The bearing capacity depends on thesoil properties, the inclination of the load and the effectivelength of the base, b0. Here, the effective length wasassumed equal to 2

3smin, as explained in the following

section. The calculation of the bearing capacity was basedon the maximum values of the loads, with the dynamic partof the axial force obtaining positive (for compression) ornegative (for tension) values.

It should be noted that, in general, the factor of safetyshould be interpreted differently for static and for dynamicloading. For static loading, values of factor of safety belowunity are not allowed, while, for dynamic loading, values ofsafety factor below 1.0 do not necessarily imply failure, asthey merely suggest initiation of plastic flow at thefoundation. However, in the static approach of thisanalysis, the loads are pseudo-static, in the sense that theysimulate the maximum dynamic loads and, thus, theiraction is not considered permanent. In this way, the factorsof safety between the dynamic analysis and the staticapproach can be directly compared.

4. Static approach

The results of the dynamic analysis were compared to theones of a simplified static approach. In this procedure, themaximum values of the forces applied to the footing wereused to calculate the angle of rotation, the length ofcontact, the soil stress and the safety factor.First, the total axial force and the total moment applied

to the footing were calculated by adding the weight of thefooting and the inertia forces to the loads from thesuperstructure. Assuming that the superstructure is movingin phase with the base, these forces are:

Mtot;max ¼MG þME;max þmAgh, (8)

N tot;max=min ¼ NG �NE;max þmg, (9)

where m is the mass of the footing, Ag is the amplitude ofthe base acceleration and MG, ME,max, NG, NE,max, are thestatic and the maximum dynamic components of themoment and the axial force of the superstructure,respectively. In Eq. (9), the plus sign in front of NE,max

corresponds to a compressive dynamic axial force leadingto the maximum value of Ntot, while the minus signcorresponds to the case of a tensile dynamic force leadingto the minimum value of Ntot. If the seismic response of thesuperstructure creates significant axial forces to thecolumns, it is possible that Ntot,min becomes negative,meaning that the total axial force applied to the footing isupward.If uplift does not occur, the maximum angle of rotation

would be:

jmax ¼Mtot;max

K r;tot. (10)

If uplift occurs, the total rotational stiffness should becalculated using a reduced value for Kr,f, in order to takeunder consideration the reduced length of contact. Thiswould require an iterative procedure, since this lengthdepends on jmax. In practice, SSI is usually applied usingthe stiffness of the foundation that corresponds to the full-contact state, since consideration of uplift would require a

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Table 1

Typical values of the dimensionless parameters considered in the analysis

Parameter Value

oj/p 30.00

Kr,sup/Kr,f 2.00

zj (zz) 0.15 (0.05)

a/b 0.20

eG/b 0.05

eE/b 1.00

NG/WF 7.50

NE,max/NG 0.20

o/oj (o/oz) 0.50 (2.50)

Ag/g �0.50

b (m) 4.00

g (kN/m3) 25.00

I.N. Psycharis / Soil Dynamics and Earthquake Engineering 28 (2008) 577–591582

non-linear analysis. In this way, the loads applied to thefoundation are calculated considering the full-contactregime only and any unfavourable effect of uplift is takenunder consideration later, during the calculation of thebearing capacity of the soil. Following this procedure, jmax

was calculated without reducing Kr,tot in Eq. (10). Thisapproximation leads to an underestimation of the max-imum rotation.

After the maximum angle of rotation was estimated, themoment, Mf, applied to the foundation was calculated bythe relation: Mf ¼Mtot,max�Kr,sup �jmax. Since the corre-sponding axial force is Nf,max/min ¼ Ntot,max/min, the mini-mum length of contact and the maximum soil stress weredetermined by static equilibrium (see Fig. 3) using thefollowing relations:

Full contact:

smin ¼ b, (11a)

sc;max ¼N f

bþ

6M f

b2. (11b)

With uplift:

smin ¼3b

2�

3M f

N f, (12a)

sc;max ¼2N f

smin. (12b)

If the minimum value of the axial force, Nf,min, whichcorresponds to the minus sign of NE,max in Eq. (9), is usedin the equations after uplift, Eq. (12a) may lead to verysmall or even negative values of the contact length, smin,implying that complete separation occurs. In such cases,large stresses are developed at the soil, as can be seen fromEq. (12b).

It should be noted that the length of contact after uplift,as defined by Eq. (12a), is different than the effective lengthof contact, b0, used for the calculation of the bearingcapacity of the soil according to EC7. The latter is definedas b0 ¼ b� 2M f=N f . Using Eq. (12a), the effective lengthaccording to Eurocode can be written as: b0 ¼ 2

3smin. This

relation was also used in the dynamic analysis for thecalculation of the bearing capacity.

5. Parametric investigation for harmonic excitation

The parametric investigation was performed for anorthogonal footing with base width b ¼ 4.00m, made ofconcrete with specific weight g ¼ 25 kN/m3. Nine dimen-sionless parameters, related to the stiffness and thedamping of the foundation, the stiffness of the super-structure, the dimensions of the footing, the frequency ofthe excitation and the loads from the superstructure wereexamined. The effect of each parameter was investigated byvarying its value while all the other parameters were kept

constant. The dimensionless parameters examined andtheir typical values that they attained when the effect ofanother parameter was investigated are given in Table 1.The terms that appear in these quantities are: the rockingeigenfrequency of the footing (Fig. 2) for the full-contact

state, which can be expressed as oj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK r;tot=IM

p; the

corresponding eigenfrequency for vertical vibrations dur-

ing full contact, defined as oz ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik0b=ðmþNG=gÞ

p; the

characteristic frequency p, which depends on the geometry

of the footing and is defined by the relation p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimgh=IM

psimilarly to the one for a block rocking on a rigidfoundation [18], with the difference that the pole ofrotation is located at the middle of the base and not atthe corner; the coefficient of critical damping of thefoundation, which is measured either by the dampingcorresponding to the rocking vibrations during full

contact: zj ¼ 124

c0b3=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIMK r;tot

por by the damping corre-

sponding to the vertical vibrations: zz ¼12c0b=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðmþNG=gÞk0bp

; the weight of the footing WF ¼ mg;

the static eccentricity eG ¼MG= NG þmgð Þ and the dy-namic eccentricity eE ¼ME;max=ðNG þmgÞ. Note that oj

and oz are partially coupled, since both depend on thestiffness of the soil. However, oz changes with the axialload of the column, while oj is independent of this loadand also depends on the stiffness of the superstructure. Therelation between oj and oz for an orthogonal footing isgiven in the following (see Eq. (13)).As mentioned above, the effect of each parameter on the

response of the system was measured by means of themaximum angle of rotation, jmax, the minimum length ofcontact, smin, the maximum corner stress of the soil, sc,max

and the factor of safety with respect to the bearing capacityof the soil. The length of contact was normalized withrespect to the width of the base, b and the stress of the soilwas normalized with respect to the corresponding stress forgravity loads only, sc,G, which is equal to ðNG þmgÞ=bþ 6MG=b2. For the calculation of the bearing capacity,drained conditions were assumed for the soil and themethod proposed in Section D4 of Appendix D of EC7 wasapplied. The following effective properties of the soil were


assumed: internal friction j0 ¼ 301; cohesion c0 ¼ 40 kPa;specific weight g0 ¼ 18 kN/m3 and overload pressure atfoundation level q0 ¼ 30 kPa. These soil properties do notrepresent any particular soil type and are used here just as anumerical example. A strip foundation was considered byassuming that the length of the footing in the normaldirection is equal to 10.00m. Also, the shear force actingon the footing from the superstructure was found assumingthat the column is clamped at both ends and the base-storey height is 5.00m.

The parametric study was performed for the horizontalcomponent of the base excitation only. It was assumed thatthe ground motion, €xg and the dynamic components of thesuperstructure loads, ME(t) and NE(t) are harmonicfunctions of frequency o, specifically, €xg ¼ �Ag sin ðotÞ,ME(t) ¼ME,max sin (ot) and NE(t) ¼ NE,max sin (ot). Thisis an approximation, since it is assumed that the super-structure is moving with the same frequency with the baseexcitation, which is valid for the steady-state response only.Also, the assumed 1801 phase difference is not, in general,true. It should be noted, however, that the base excitationaffects the impulsive forces of the footing only, which aregenerally small compared to the loads from the super-structure. In this sense, these approximations affect slightlythe results and are adopted here in order to minimize thenumber of the parameters concerning the excitation.

In Fig. 4, the effect of the ratio of the stiffness of thesuperstructure to the stiffness of the foundation, Kr,sup/Kr,f,is illustrated. On each plot, the results obtained from thedynamic analysis and the static approach are shown forpositive and negative values of NE,max. It is evident that theratio Kr,sup/Kr,f affects significantly the length of contact,

Fig. 4. Effect of the ratio of the stiffness of the superstructure to

the soil stress and the factor of safety. Note that theseresults were obtained for constant total rotational stiffness,Kr,tot, since the ratio oj/p remains constant (see Table 1).Thus, when Kr,sup/Kr,f decreases, the superstructure be-comes softer and the foundation stiffer. For a givenrotation, a reduction in Kr,sup leads to a larger portion ofthe total moment transferred to the foundation and,therefore, for small values of Kr,sup/Kr,f the length ofcontact obtains small values and the corner stress of thesoil becomes large.The angle of rotation, jmax, depends on the total

rotational stiffness and not on the ratio Kr,sup/Kr,f. For theplots of Fig. 4, the rotational stiffness during full contact,Kr,tot, is not changing and thus jmax is constant according tothe static analysis and slightly changing according to thedynamic analysis (in this case, it is affected by the rotationalstiffness after uplift, which is smaller than Kr,tot). In thecontrary, jmax is changing significantly when Kr,tot isvarying. This is shown in Fig. 5, which plots the variationof the rotation and the soil stress with the ratio oj/p, whichis proportional to the square root of Kr,tot. These resultscorrespond to a constant value of Kr,sup/Kr,f (Table 1) andthus the stiffness of the superstructure is varying similarlywith the stiffness of the foundation; the latter depends on thesubgrade modulus, k0, the variation of which is also shownon the plots of Fig. 5 (top horizontal axis). It is evident thatjmax decreases exponentially as the stiffness of the soilincreases, while the soil stress is practically unaffected.The results depicted on Figs. 4 and 5 show that the sign

of NE,max is not important according to the dynamicanalysis. In the contrary, the static approach is significantlyaffected by the sign of NE,max. Thus, unrealistically small

the stiffness of the foundation on the response of the footing.

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Fig. 5. Effect of the ratio oj/p on the response of the footing.


values for the length of contact and the safety factor andquite large ones for the soil stress were obtained forNE,maxo0, while for NE,max40 the soil stress was close tothe one of the dynamic analysis, in spite of the fact that theapproximate method overestimates the length of contactand the bearing capacity, especially for medium to largevalues of Kr,sup/Kr,f (soft foundation). Note that the valueof NE,max used for the derivation of these results wasrelatively small, equal to only 20% of NG (see Table 1).

The foundation damping does not seem to affect theresults significantly, as it is shown on the top-row diagramsof Fig. 6. In these plots, the coefficient of damping forrocking vibrations during full contact, zj, was varyingfrom 0.05 to 0.50 and the results show a small decrease inthe rotation and the soil stress as the damping increases, asexpected.

Similarly, the dimensions of the footing are notimportant to the stresses that are developed at the soil, asillustrated on the bottom-row diagrams of Fig. 6. Note thatin all the results, the width of the footing, b, was assumedconstant, equal to 4.00m and thus, the height of thefooting was varying from 0.4 to 2.0m. The small influenceof the ratio a/b on the stresses of the soil was expected,since the dimensions of the footing affect the inertia forces,which are usually small compared to the loads from thesuperstructure. Note that the angle of rotation is decreas-ing exponentially as a increases. This happens because therotational stiffness increases with a2 in order oj/p toremain constant.

It is known that, for a free rocking block on an elasticfoundation, the size of the block is important to theresponse (e.g. [19]) with smaller blocks experiencing largerrotations than ones with larger dimensions of the sameaspect ratio. For the problem that is examined here, thefooting is not free, since it is connected to the super-structure. The size enters in the analysis through thecharacteristic frequency p. In order to check whether it isimportant, the plots of Fig. 5 were reproduced for the samevalues of the dimensionless parameters and for one half of

the width of the base, i.e. for b ¼ 2.00m instead of 4.00m.The results obtained were exactly the same with the onesshown in Fig. 5 and thus, it can be concluded that theparametric investigation presented here is practicallyindependent of the absolute size of the footing.The effect of the frequency of the excitation, o (same for

the base motion and the dynamic loading from thesuperstructure) on the response is illustrated on Fig. 7.The frequency o is shown normalized with respect to therocking eigenfrequency during full contact, oj (bottomhorizontal axis) and with respect to the correspondingvertical eigenfrequency, oz (top horizontal axis). This ispossible because, for an orthogonal footing, oj and oz arerelated according to the equation:

oj

oz

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þNG=mgÞð1þ K r;sup=K r;f Þ

1þ 4ða=bÞ3

s, (13)

which leads to a constant relation between oj and oz forconstant values of the dimensionless terms of Table 1.It is evident from the plots of Fig. 7 that resonance

phenomena occur when oEoj and oEoz, which result tocomplete separation of the footing from the soil (the lengthof contact and the factor of safety become zero in this case)and a significant increase in the rotation and the soil stress.The maximum values of jmax and sc,max are observed forvalues of oj/o and oz/o a little greater than unity, becausethe effective rocking frequency and the effective verticalfrequency after uplift are shorter than oj and oz,respectively. The term ‘‘effective’’ is used here to denotethe ‘‘apparent’’ frequencies of the response, since upliftingsystems do not possess natural frequencies in the classicalsense.The response of the system when o is close to oj is

further examined in Fig. 8, in which the time-histories ofthe maximum soil stress (at the most unfavourable corner),the rotation and the vertical displacement of CM are givenfor oj/o ¼ 0.8, 1.0 and 1.1 and for NE,max40 andNE,maxo0. Note that a significant increase in the vertical

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Fig. 6. Effect of the foundation damping, zj, (top row) and the aspect ratio of the footing’s dimensions (bottom row) on the response.

Fig. 7. Effect of the excitation frequency, o, (same for the base motion and the dynamic loads from the superstructure) on the response of the footing.


oscillations occurs for oj/o ¼ 1.0 and 1.1, compared to thecase of oj/o ¼ 0.8, which is larger than the correspondingincrease in the rotations, although the resonance concerns

the rotational vibrations. As a result, complete separationhappens in these cases. This shows that a transfer of energyfrom the rocking mode to the vertical mode occurs. This

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Fig. 8. Time-histories of the maximum soil stress (top row), the rotation (middle row) and the vertical displacement of CM (bottom row) for oj/o ¼ 0.8,

1.0 and 1.1 and for NE,max40 and NE,maxo0.


phenomenon, which happens after uplift, has also beenobserved in the case of a free rocking block on elasticfoundation [19].

The static approach is independent of the frequency of theexcitation. The results shown in Fig. 7 were obtained forconstant superstructure loads and thus, the curves corre-sponding to the static approach are horizontal lines. Notethat the static approach shows a significant dependence of allthe quantities, except of the rotation, on the sign of NE,max.Thus, for NE,maxo0 the length of contact is about one thirdthe value corresponding to NE,max40, the soil stress is almostdouble and the bearing capacity according to EC7 is nil,giving a safety factor equal to zero. In the contrary, thedynamic analysis is much less dependent on the sign ofNE,max, as it is evident from the plots of Figs. 7 and 8.

The last parameters examined are related to the loadsfrom the superstructure. The results concerning the staticloads (not presented here) show that jmax and sc,max

increase as MG increases, while smin and the factor of safetydecrease, as expected. The dynamic part of the appliedmoment was measured by the eccentricity eE ¼ME;max=ðNG þmgÞ, which was normalized with respect to thewidth of the base, b. Note that eE is not the real eccentricityof the dynamic part of the moment, because the dynamic

part of the axial force, NE,max, is not included in thedenominator. This is done in order to normalize ME,max

with respect to a quantity that is independent of the otherdynamic loads of the superstructure. The dynamic axialforce was measured by the ratio NE,max/NG, which attainsboth positive and negative values, since NE,max can becompressive or tensile.The results concerning the dynamic eccentricity, eE, are

given in Fig. 9. The dynamic analysis shows that the angleof rotation and the soil stress increase almost linearly witheE while the length of contact and the safety factor decreaserapidly and become zero for values of eE greater thanapproximately 1.5b. The sign of the axial force does notseem to be important, especially for small values of eE. Thestatic approach gives relatively good results for NE,max40and for eE/bo1.25, except of the expected underestimationin the angle of rotation. For eE/b41.25, however, the soilstress increases exponentially. Much worse are the resultsfor NE,maxo0: in this case, the static approach largelyunderestimates the length of contact and the factor ofsafety and overestimates the soil stress, even for eE/b lessthan unity. It should be noted that complete separationoccurs for eE/b greater than about 1.75 for both thedynamic and the static analyses.

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Fig. 9. Effect of the dynamic eccentricity of the superstructure loads on the response of the footing.

Fig. 10. Effect of the dynamic axial force from the superstructure on the response of the footing.


A thorough investigation of the effect of the axial force isillustrated on the plots of Fig. 10, on which the response isplotted versus the ratio NE,max/NG. It is evident that thestatic approach is extremely conservative for NE,maxo0,leading to zero values of the safety factor (which,

practically, means that the footing dimensions are insuffi-cient) while it is against safety for NE,max40, especially forlarge values of NE,max/NG. In the contrary, the dynamicanalysis is practically independent of NE,max/NG, even forlarge tensile values of NE,max (e.g. for NE,max/NG ¼ �0.8).


The response is further examined in Fig. 11, in which thetime histories of the soil stress and the vertical displacementof CM are given for NE,max/NG ¼ 0.5 and NE,max/NG ¼ �0.5: although the change of sign of NE,max causesa significant phase shift to the vertical displacement, itseffect on the soil stress is small.

In practice, the correct sign of NE,max that correspondsto ME,max is not known, since these values are usuallyderived from a modal analysis after an SRSS or a CQCcombination of the modal responses. For this reason, a

Fig. 11. Time-histories of the maximum soil stress (top row) and the

vertical displacement of CM (bottom row) for NE,max/NG ¼ 0.5 and

NE,max/NG ¼ �0.5.

7.00

4.0

0

2.00

0.6

0

B 30/60

C 40/4C 40/40

2.00

Fig. 12. (a) Geometrical data of the plane frame considered in

negative value of NE,max (tension) is usually used in favourof safety. This procedure, however, may lead to a veryconservative design according to the aforementionedresults. In the contrary, the use of a positive value forNE,max (compression) may be against safety. The results forharmonic excitations show that neglecting the dynamicpart of the axial force would be more realistic. Thisproposal is further investigated in the following sectionthrough an example for earthquake excitation.

6. Example for earthquake excitation

An example for the plane frame shown on Fig. 12(a),which is subjected to an earthquake excitation, is presentedhere. The width of the footings in the normal direction is2.00m. In Fig. 12(b), the theoretical model for fixed baseconditions is given. First, the static loads (self-weight andload q on the beam) were applied to the fixed base model,in order to determine MG and NG; also, a unit rotation atpoint A in order to determine Kr,sup. Next, this model wassubjected to the El Centro, 1940 earthquake magnified 1.20times and ME(t) and NE(t) were calculated.Using these forces and moments as the superstructure

loads, the equations of motion (3) and (4) were solved andjmax, sc,max and smin were derived for each footing. Themaximum values of ME(t) and NE(t) were used to calculatethe response according to the static approach. Theprocedure was repeated for eight values of the subgrademodulus, ranging from 2.5 to 20MN/m3. In all cases, zjwas taken equal to 0.15.An example of the results obtained is given in Fig. 13, in

which the time histories of the maximum soil stress (at themost unfavourable corner of the footing) and the length ofcontact for k0 ¼ 15MN/m3 are shown for footings A andB. On the right column, a zoom for t ¼ 2.0–4.0 s ispresented. Note that when one footing is in full contactthe other is rocking in the uplift regime, since the dynamicaxial forces from the superstructure that act on them haveopposite signs. Another interesting observation is that themaximum soil stress occurs at t ¼ 2.63 s for footing A(right top diagram), when this footing is in the full-contact

7.00

4.0

0

0

q=80 KN/m

A B

the analysis, (b) equivalent model for fixed base conditions.

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Fig. 13. (a) Time histories of the maximum soil stress (at the most unfavourable corner of the footing) and the length of contact for k0 ¼ 15MN/m3, (b)

zoom for t ¼ 2.0–4.0 s (El Centro, 1940 earthquake).

Fig. 14. Variation of the normalized minimum length of contact and the maximum soil stress with the subgrade modulus according to the dynamic

analysis and the static approach (El Centro, 1940 earthquake).


regime (right bottom diagram) and not during uplift, asone would expect. This happens because the soil stressdepends not only on the rotation but also on the verticaldisplacement of the footing.

The length of contact according to the static approach isalso shown on the plots of Fig. 13 for NE,max40 andNE,maxo0. The latter assumption leads to a very smallvalue for smin/b, equal to 0.11, which would not beacceptable in a design procedure. The corresponding soil

stress is very large (1266 kPa) which exceeds the scale of thevertical axis on the top-row diagrams and for this reason itis not shown. In the contrary, the assumption NE,max40leads to an underestimation of uplift to about one half ofits value; the soil stress, however, is predicted reasonablywell, except of a couple of spikes, in the case of footing A,that exceed the value of the static approach.The variation of the minimum length of contact and

the maximum soil stress with the value of the subgrade


modulus is shown in Fig. 14 for both the dynamic analysisand the static approach. According to the dynamicanalysis, almost nil values of smin occur for k0 ¼ 5MN/m3 showing some kind of resonance. The approximatemethod cannot capture this phenomenon and predictssignificantly smaller uplift (larger values of smin than thedynamic analysis) for subgrade modulus less than 10MN/m3; the soil stress, however, is not affected by this decreasein the length of contact and both methods give similarresults.

For stiffer soils with k0 greater than 10MN/m3, the staticapproach leads to unrealistically small values for the lengthof contact and extremely high soil stresses, if the dynamicpart of the axial force from the superstructure is consideredtensile (NE,maxo0). In the contrary, for NE,max40 (com-pressive), the soil stress obtained is slightly underestimatedin spite of the fact that uplift of the footings is significantlyunderestimated. The results corresponding to NE,max ¼ 0are also plotted on these diagrams, for comparison:concerning the soil stress, the results are similar to theones for NE40 and rather on the safety side; concerningthe minimum length of contact, the values obtained forNE,max ¼ 0 are close to the average values for NE,max40and NE,maxo0.

7. Conclusions

The dynamic response of footings sitting on an elastic,tensionless Winkler foundation is examined in this paper.The superstructure stiffness is considered by an equivalentrotational spring, modelling the stiffness of the column.The equations of motion are derived for each regime ofthe response (full contact and uplift) and are integratednumerically.

A parametric investigation of the effect of a number ofdimensionless terms, involving all the main parameters, asthe foundation stiffness, the foundation damping, thefrequency of the excitation and the superstructure loads, isperformed for harmonic excitation. The investigationconcerns the effect of each dimensionless parameter tothe maximum angle of rotation, the minimum length ofcontact after uplift, the maximum stress developed at thesoil and the factor of safety with respect to the bearingcapacity of the soil according to Eurocode 7, assuming thatall the other parameters remain constant. The results of thedynamic analysis are compared to the ones of a simplifiedstatic approach, which is based on the maximum values ofthe loads. An example for a plane frame subjected to the ElCentro, 1940 earthquake is also given for several values ofthe subgrade modulus. The most important conclusionscan be summarized as follows:

The investigation showed that the stiffness of thefoundation practically does not affect the length of contactand the soil stress, as far as resonance does not occur.Similarly, the damping of the foundation does not seem tobe significant. In the contrary, the ratio of the super-structure stiffness to the foundation stiffness is important:

when this ratio decreases the amount of uplift and thestresses that are developed at the soil increase while thefactor of safety decreases.The static approach can, in general, predict the response

satisfactorily. Its accuracy, however, decreases significantlyin two cases: (a) if resonance happens and (b) if thedynamic part of the axial force of the superstructurebecomes large. In case (a), which can happen when theexcitation frequency is close to either the rotational or thevertical eigenfrequency of the system for the full-contactregime, the static analysis underestimates, in general, theresponse. In case (b), the static analysis greatly depends onthe sign of the dynamic part of the axial force of thesuperstructure: for tensile forces, it overestimates theresponse and can be conservative; for compressive forcesit may give results against safety. It seems that neglectingthe dynamic axial force of the superstructure is not, ingeneral, against safety while it leads to significantly moreeconomical design. These results are verified for theearthquake response of the frame, but more research isneeded on this subject.

Acknowledgements

The work presented in this paper was performed in theframework of the research program ‘‘Ultimate bearingcapacity of shallow foundations under seismic actions,’’coordinated by Professor M. Kavvadas of the NationalTechnical University of Athens and financed by theGreek Earthquake Planning and Protection Organization(OASP).

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Documents

Investigation of the Dynamic Response of Rigid Footings on Tensionless Winkler Foundation