THe dynamic response of flexible surface strip-foundations allowed to uplift is numerically obtained for externally applied forces of a transient time variation. The soil medium is represented by an isotropic, homogeneous and linear half-space. The soil is treated by a time domain boundary element method, while the flexible foundation is treated by the finite element method. It was concluded that intermediate relative stiffness leads to moderate deformations when uplift is permitted. Very flexible footings produce higher deformations in unilateral contact compared to bilateral contact, and thus should be considered in their design. Unilateral contact does not significantly increase deformations for stiff footings subjected to concentrated central loading. However, relatively large deformation differences occur when the loading is eccentric, necessitating consideration of uplift in their design

Patel, P. N.

Spyrakos, Constantine C.

English

NASA Technical Reports Server

N88- 13641
Significance of Foundation-Soil Separation in Dynamic
Soil-Structure Interaction
C. C. Spyrakos
P. N. Patel
The dynamic response of flexible surface strip-
foundations allowed to uplift is numerically obtained
for externally applied forces of a transient time
variation. The soil medium is represented by an
isotropic, homogeneous and linear elastic half-space.
The soil is treated by a time domain Boundary Element
Method, while the flexible founadtion is treated by the
Finite Element Method. In order to effectively simulate
soil-foundation separation, thin-layer FEM interface
elements are used at the contact area. The numerical
procedure of determining the area of contact by solving
the nonlinear equations of motion is based on the BEM
and FEM appropriately combined through equilibrium and
compatibility considerations. For various relative
stiffnesses between the foundation and soil the system
is subjected to a concentrated impulse force and/or
moment acting on the surface foundation. It is observed
that separation significantly affects the foundation
response, and should be considered in the analysis for a
range of relative stiffness between the foundation and
the soil.
INTRODUCTION
Most soil-structure interaction problems are treated under the
assumption of complete bond between the foundation and the soil [i-
3]. However, for a given eccentricity and intensity of external
dynamic forces, a foundation will partially separate from the
underlying soil, as tension is incompatible with the constitutive
laws of soils. Recently, some attention has been directed towards
the study of the effects that partial foundation-soil separation
may cause on the structure response [4-6]. These studies have been
initiated from observations during strong ground motions, actual
performance of structures during earthquakes and laboratory tests
[7-10]. Both analytical studies and numerical investigations
demonstrated that uplift may have controversial effects on
structural behavior. Factors such as slenderness ratio, foundation
to superstructure mass ratio, eigen properties of the structure,
495
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type and duration of the exciting disturbance may have either
benevolent or malevolent effects on the structure response.
The methods of non-linear analysis usually employed to obtain the
structure response can be classified into three categories: a.
Employment of discrete systems idealizing the foundation in a
small number, usually two, of elasto-plastic springs and ignoring
both the radiation damping and the coupling between the soil-
foundation contact stresses at the time of separation [11]; b.
Simulation of the soil behavior by either a damped Winkler
foundation or a foundation supported on two elastic spring-dampers
attached at the ends [4,5,12]; c. Employment of a finite
difference [8] or finite element method (FEM) of analysis [13,14]
to model soil media leading to a large system of equations.
Recently, Wolf et al. [15,16] determined the response of a typical
nuclear-reactor building modeled by a single degree of freedom in
the vertical direction supported by a rigid circular foundation
subjected to vertically incident seismic waves. Their formulation
is based on a time domain indirect boundary element formulation
(BEM) employing an inverse Fourier transform on the level of the
individual boundary elements.
In this paper, the dynamic response of massless flexible surface
strip-foundations allowed to uplift is numerically obtained for
externally applied forces of a transient time variation. The soil
medium is represented by a homogeneous and linear elastic half-
,pace. The soil is treated by the BEM, while the foundation and
the interface are treated with the aid of FEM. The numerical
procedure of determining the area of contact by solving the
nonlinear equations of motion is based on the BEM and FEM
appropriately combined through equilibrium and compatibilty
considerations. Thus, the formulation does not require the
adoption of frequency independent compliances needed for the
solution of nonlinear dynamic soil-structure interaction problems.
The primary contributions of this work are the development of a
methodology that allows a rigorous treatment of the separation
effects on soil-structure interaction problems as well as a
thorough investigation of the influence of uplift on the response
of flexible surface strip-foundations.
METHODOLOGY
Consider the soil-structure system of figure l, which is allowed
to oscillate with unilateral contact. The foundation and the
interface are treated with the aid of FEM, while the soil is
treated by the BEM. The two domains are appropriately combined
through equilibrium and compatibility considerations at the soil-
foundation interface. The interface is modeled with thin-layer
elements of negligible influence on the system response. The
treatment of the thin-layer elements simulating the interface
behavior is discussed in the next section. In the following, the
treatment of the soil and the foundation is briefly discussed.
Under the assumptions of zero initial conditions and zero body
forces, the BEM formulation is developed through a numerical
treatment of the integral equation governing the soil motion at
the soil-foundation interface having the form [17,18]
496
F_u_(_,t)=J{v _[x,t-_It(_)_ (x,t)] -
s
-_(_)_ [x,t;_/_ (x,t)]}ds(x), (I)
where s denotes the soil-foundation interface as well as a portion
of the free surface around it, and the tensors v _ and
_(_)_ represent the fundamental solution palr of the infinite
space under conditions of plane strain.
Es,Ys
xI
Fig. 1 Soil-structure modelling
The numerical treatment of the boundary integral equation (I)
involves both time and spatial discretization. Thus, the time
variation of t ....(x,t) is approximated as a sequence of
rectangular impul_ of equal duration At. The soil-foundation
interface, as well as a part of the surrounding free soil surface
are dicretized into Q elements of equal length L. The foundation
response at time t=N_t due to a sequence of impulses initiating at
time mat can be determined from
([IGl%s]{ tN-l+_ _
q=l n=m as [_ N-I+1- Flqds] {u }),
&s
where G lq and F lq are the discretized kernel functions
(2)
v _[x,t;£/t(_)_ (x,t)] and _(6)_ [x,t;_/_ (x,t)]
respectively, n=l,2 .... ,N, l=N+n-l, q=l,2,...,Q, and p=I,2,...,Q.
The other component of the system, the flexible foundation is
analyzed through standard finite element procedures. The
discretization is carried out using four node rectangular
isoparametric plane-strain finite elements. The dynamic equation
of the foundation motion is given by
f
[Mf]{qt}+[ctf]{qt}+[K t ]{qt}:{Rt}-{P t} (3)
where [Mr] is the mass matrix, [ctf] and [Ktf] are the time
497
dependent damping and stiffness matrices respectively, the vectors
{q }, {q } and {q } are the nodal acceleration, velocity and
t t t
displacement vectors, respectively, the vectors {R t} and {P_} are
the nodal external forces and n_dal force_ associated wi_h the
contact stresses. The matrices [C t ] and [kt ] are time dependent
because they contain the terms pertaining to thin-layer interface
elements. The propertiese of the thin-layer elements are dependent
on the contact area which is a function of time.
Equation (2) relates the average vertical displacements at the
center of each element to the contact stresses developed over the
elements of the soil-foundation interface. Equation (3), relates
the vertical nodal displacement to the nodal forces associated
with the contact stresses developed at the ends of the FEM
elements at the interface. In order to introduce compatibility
between the deflection of the foundation and the soil motion at
the interface, the average displacement over an element q is
approximated by the mean value of the nodal displacements at the
ends of the element q. Similarly, compatibility of forces can be
established if each contact force Pt applied at a node i is
approximated by the mean value of _he two resultant forces R t
associated with the contact stresses that develop over two
successive elements joined at the common node i. Thus, for the
whole interface region the compatibility relationships can be
expressed as
{qt}=[T] {ut }
and (4 )
{Pt }= [T]T{Rt }
where the entries of matrix [T] are either 0 or 1/2. The order of
matrix [T] is Qx(Q+I).
Combination of equations (2), (3) and (4) results in a system of
nonlinear equations of motion
[M]{qt}+[Ct]{qt}+[Kt]{qt}={Ft}-{Pt} (5)
All quantities in equation (5) are known at a given time. Equation
(5) is solved iteratively to satisfy the time dependent boundary
conditions at the soil-foundation interface. The contact area at
the beginning of each time step is known from the iterative
solution of the previous time step. Thus equation (5) at time
t+_t, where _t is a small time increment, can be written as
[M]{_ti+l}+[Ct]{_qt i+l} +
+l}={ARt}+{Rt n'+ [kt]{sqti i} (6)
where {qt+at}={qt}+{Aqt },
[Kt+_t]=[Kt]+[AKt], etc.
{ARt n'i} is the unknown nonlinear load vectorand corresponding to
the time increment At to be determined by iteration and i is the
number of iteration within the same time step. The vector {ARt n'l}
498
is given by
' " t5{ 5{_Rt n i}=_[_Ct 1]{qt+nt qt+'_t "
An unconditionally stable scheme of direct integration based on
Wilson 0 method is used in the time domain. At the desired time
t+At the accelerations, velocities and displacements are given by
the linear acceleration assumptions:
{ _t +,"-t } =( 1-I/0 )_t+( I/e )$t+_ (8)
{qt+At }=_+ (At/2) (_t+_t+_t) (9)
{qt+_t } =_ +'_ t_lt+ ( _t:/6 ) ( _ t+_t+2qt ) ( I0 )
where T is given by _=e_t. When e=l.0 the algorithm reduces to the
standard linear acceleration method. A stability analysis reported
by Wilson, Farhoomand and Bathe [19] shows that the scheme is
unconditionally stable provided e!1.37.
THIN-LAYER INTERFACE ELEMENTS
In order to simulate unilateral contact at the soil foundation
interface, the interface is modeled with the aid of FEM thin-layer
elements of negligible influence on the system response. The
interface element can undergo four basic modes of deformation. (I)
Stick or no-slip, (2) slip or sliding, (3) separation or
debonding; and (4) rebonding. An interface element is in stick
mode when there is no relative motion between the adjoining
bodies. If a relative movement takes place while maintaining the
contact between the adjoining bodies, the slip or sliding is said
to occur. Separation or debonding takes place when the bodies open
up due to contraints of unilateral contact. If the interface
element in separation mode returns to stick mode in subsequent
loading, rebonding takes place.
The interface element described above has been successfully used
for solution of a number of static as well as dynamic two-
demensional problems where all domains are discretized with the
aid of FEM [14,20]. In this study, the equations of the interface
elements are derived separately and then added to those of the
foundation prior to establishing the compatibility and equilibrium
criteria with the soil BEM modeling described in the previous
section.
The primary reason for resorting to interface elements at the
interface is to facilitate the computation of the contact area
prior to each time step. The interface element when in stick mode
is essentially treated like any other plane strain element with
the soil elastic modulus, Es, and Poisson ratio, vs . In the
present study, the concept of sliding is not addressed. In
debonding mode of a given interface element, the elastic modulus
is assigned a value of zero. This in essence creates a void
element with no stiffness. Within a given cycle of iterations in a
time step, if rebonding is detected through interpenetration, the
forces associated with the contact stresses are applied to the
499
penetrating node. Thus stick mode of a previously defined void
element can be stimulated without modification to the global
stiffness matrix [K_ ]. The interface element thickness plays an
important role in _he convergence of the solution as reported by
several researchers [14,20]. In this study the ratio between the
thickness of the interface element, and the thickness of its
neighbouring FEM element is taken as 0.01.
NUMERICALEXAMPLES
The combined time domain BEM-FEM technique described above is
employed here to determine the dynamic response of a flexible
massless strip-footing subjected to externally applied loads. The
dynamic behavior of undamped flexible footing depends on the
special distribution of the externally applied forces and by the
material properties of the elastic footing. Therefore, the footing
and the supporting elastic medium are analyzed in this work for
three sets of elastic contstants and two types of exernal forces
(figure 2). The parameter characterizing the flexibility of the
soil-foundation system is the relative stiffness defined by
Kr=Df.D s (II)
where Df=Eftf3/(l-vf 2) and Ds=2(l-vs)/(Es b3) (12)
and where the subscript f and s denotes the footing and the soil,
respectively, E and v represent
poisson's ratio, respectively, and
footing.
P(t)
1
I !
(a) Central concentrated load
modulus of elasticity and
tf is the thickness of the
PI t P(t]f
(b) Force-couple load
Fig. 2 Loadings considered
The soil is discretised into 16 BEM elements and the foundation is
discretised into 40 FEM elements as shown in figure i. Figure 2
shows the two types of external loadings considered, the point
force and a moment applied as a force couple of two equal,
opposite _int forces. The duration of both impulse forces is
At=0.16xl0 sec, and the relative stiffness considered are
K =0.3, K =3.0 and K =30.0. The response of the center point A,
a_d edge rpoint B ofrfigure 1 are plotted in figure 3 through 5.
All responses are compared with the corresponding solution of the
complete bond case, i.e. uplift not permitted.
500
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The responses at the midpoint A and the edge point B of figure 1
due to a vertcal concentrated rectangular impulse load at midpoint
A are plotted for unilateral and bilateral contact for various
relative stiffnesses (K =0.3, 3.0, 30.0). As seen from figure 3,
r
the response at the center for the unilateral contact is higher
then the corresponding bilateral contact case. Figure 4 shows that
the deformations at the edge point to be significantly higher for
the case of unilateral contact then that for the bilateral case.
The deformations are in the opposite sense because the foundation
is not held back as tension is incompatible with the assumed
constitutive laws of the soil (unilateral contact). At both, the
center point and edge point locations the differences between the
unilateral and bilateral contact conditions decreases with
increasing foundation stiffness.
In the case of force couple loading, the softer foundation K =0.3
r .
and stiffer foundation K =30.0 undergo higher deformatlon
• r .differences than the xntermedlate stiffness K =3.0 as shown in
figure 5. The deformations become identical with the passage of
time as seen for the concentrated load case.
CONCLUSIONS
It can be concluded that intermediate relative stiffness leads to
moderate deformations when uplift is permitted. Very flexible
footings produces higher deformations in unilateral contact
compared to bilateral contact, and thus should be considered in
their design. Unilateral contact does not significantly increase
deformations for stiff footings subjected to concentrated central
loading. However, relatively large deformation differences occur
when the loading is eccentric necessitating consideration of
uplift in their design.
REFERENCES
I. Wolf, J.P., Dynamic Soil-Structure Interaction, Prentice Hall,
Englewood Cliffs, New Jersey, 1985.
2. Spyrakos, C.C.,
Analysis of Flexible
Proc. 3rd Int. Cony.
1986.
Patel, P.N. and Beskos, D.E., "Dynamic
Embedded Foundations: Plane Strain Case,"
Compt. Meth. Exper. Meas., Porto Corras,
3. Spyrakos, C.C. and Antes, H.m "Time Domain Boundary Element
Approaches in Elastodynamics: A Comparative Study," Comp. Struct.,
Vol. 24, No.4, pp 529-535, 1986.
4. Yim, S. C.-S. and Chopra A., "Simplified Earthquake Analysis
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6. Spyrakos, C.C., "Dynamic Behavior of Foundations in Bilateral
504
and Unilateral Contact, Shock and Vibration Digest, (submitted).
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5O5
20. Chen, R.-J., "Finite Element Modeling of Interface Behavior in
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requirements for the degree of Doctor of Philosophy.
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Significance of foundation-soil separation in dynamic soil-structure interaction

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