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In this work, two numerical methodologies are proposed for the solution of unilateral contact problems between a structural member (beam or arch) and an elastic foundation. In the first approach, the finite element method is used to discretize the structure and elastic foundation and the contact problem is formulated as a constrained optimization problem. Only the original variables of the problem are used, subjected to inequality constraints, and the relevant equations are written as a linear complementary problem (LCP). The second approach is based on the Ritz method, where the coordinates defining the limits of the contact regions are considered as additional variables of the problem. The contact problem here is treated as an unconstrained optimum design problem. These proposed methodologies are then tested and compared using results from specific problems involving structures under unilateral contact constraints.

Foundations of structures can be divided into shallow and deep
foundations. In the modeling of shallow foundations, structural elements such
as beams, arches, plates, and shells are supported by a continuous substrate.
The primary difficulty in the analysis of the structure-substrate system lies
in the determination of the contact pressure in the interface. Usually the
substrate is modeled as an elastic foundation. Most constitutive models
consider that the foundation reacts both under tension and compression.
However, certain types of soil and most liquids only react under compression.
In these circumstances, the structure may lose contact with the foundation in
certain regions, leading to unexpected contact pressure concentration with a
consequent variation of the structure’s internal forces. Problems where the
structure can enter in or lose contact with other bodies, or even slide on its
support, are usually found in the literature under the denomination of “unilateral
contact problems” [

The first step in obtaining the numerical solutions of contact problems involving contin-uous systems generally consists in reformulating the problems in approximation spaces. To this end, numerical
techniques such as the Ritz, finite element, or boundary element methods are
employed. After discretization, a proper methodology to adequately treat the
unilateral contact constraints must be selected. This usually requires the
problem to have a finite dimension. Among the options found in the literature, two
are noteworthy.

Transformation of the contact
problem into a minimization problem without restriction by applying usual
formulations of the structural mechanics—differenti-able
functional and bilateral constraints—to the case of unilateral contact
constraints. There is no guarantee of convergence of these procedures, which by
their nature are unavoidably incremental-iterative. These procedures, however, introduce
no new concepts. So, existing codes for nonlinear analyses that can be adapted to
this particular case, resulting in efficient computational time, granted that there
is no change in the contact region between two load steps [

Use of mathematical programming techniques.
This approach allows the solution of the contact problem with or without
explicit elimination of unilateral constraints. Methods such as Lagrange's
multipliers or penalties allow the elimination of the unilateral constraints.
Usually these methods are based on the use of special finite elements derived
to simulate the impenetrability condition between two surfaces [

Since the seventies, these two
general approaches have been employed for the numerical treatment of different
types of contact problems. Following the first approach, Stadter and Weiss [

Numerical simulation of contact
problems based on mathematical programming can also be easily found in
literature. Fundamentals of the unilateral contact boundary value problem,
including friction, together with finite element applications to the solution
of the variational inequalities arising in static and dynamic structural
contact problems can be found in Panagiotopoulos’ articles [

These two general
approaches are employed in this work for the treatment of unilateral contact
constraints. Since 1990, the authors have analyzed several contact problems
involving a deformable structure and an elastic foundation [

The basic structural unilateral contact equations and inequations are stated in the next
section. Two numerical methodologies are then proposed to solve the contact
problem. In the first methodology, the contact problem is treated as a
constrained optimization problem and the finite element method is used for the
structure and for the Winkler-type elastic foundation. Two alternative linear
complementary problems (LCPs) are derived and solved by Lemke’s algorithm [

Consider the
structural contact problem shown in Figure _{c}, _{u}, and _{f}. The displacements
are prescribed in _{u} and the
forces in _{f}. _{c} is the region where boundary conditions
are “ambiguous.” For the structure, the equilibrium equations, the cinematic
relations, and the constitutive equations are given by_{ij} are the Cauchy stress components,
_{i} are the displacements, and

Structure under unilateral contact constraints imposed by an elastic foundation.

Engineering problem

Engineering model

Deformation pattern

If the elastic foundation is described by Winkler’s model, then the following
constitutive relation can be written:_{b} and _{b} are the displacement and reaction, respectively, of the
elastic foundation and _{b} is the foundation modulus.

For the structural system studied here, the following boundary conditions must be
satisfied:
_{i} is the deflection of the structure
orthogonal to the foundation and _{c}. Inequality (

As the elastic foundation reacts only to compression, the
following inequality must also be satisfied on _{c}:
_{b} should be verified:

These restrictions define in a
complete way the contact as being unilateral. Figure

Domain of validity of the contact constraints.

Contact law

The solution of the unilateral contact problem can be
obtained by solving (

According to Ascione and Grimaldi [_{b} are the
elements of_{b} and

The first variation of _{b} from the previous equation
by way of relation (

Elimination
of _{b} only, which corresponds to the first variation
of the following functional:_{b} must be obtained in such a way that the first variation of the functional

Using the finite element method, one can assume that for a
generic structure and a foundation finite element the displacement and reaction
fields within the element, _{b}, are related to the nodal
displacements _{b}
by
_{b}
are the matrices that contain the interpolation functions
that describe, respectively, the behavior of the structure and elastic base.

From these definitions and adding the contributions of each
finite element, one arrives at the discretized functional of the problem in the
global form_{e} is the nodal load vector, _{c} is the number of elements of the contact
region.

After the first variation of (

Equation (

Using these new variables,
it is possible to rewrite (

If the stiffness matrix

Substituting
(

Equation (

Thus,
considering the Kuhn-Tucker conditions of this QPP, one can derive a standard
LCP similar to the one described by (

A different strategy of solution is now proposed. This strategy assumes that the contact
constraints (_{c}_{k}) as
additional variables of the problem (see Figure _{k}). Note that the length of each contact
region is a function of the system parameters and is not known a priori.

If
the Ritz method is applied, the following displacement field, written in matrix
form, can be used to approximate _{u} and the vector

Thus, substituting (_{k}.

Algorithm

a ^{0})
are necessary;

a

(1) Initialize: ^{0} (

(2) Iterations (

(i) Compute: ^{(k-1)}

(ii) Compute: ^{(k-1)}

(iii) Check convergence:

(3) Print the variables and stop

Figure

Structural elements under unilateral contact constraints and corresponding deformation patterns.

Consider as a first problem a beam of length ^{4}/_{i} are the modal amplitudes. Excellent agreement is observed between these three different
solution strategies. Note also that the contact region (and the corresponding
displacements) decreases steadily as

Deformed shape of the beam (Figure

Now, consider the beam shown in Figure _{i} and _{f} constitute the additional variables of the problem.
Figure ^{4}/

Deformed shape of the beam (Figure

The last
example, illustrated in Figure ^{°}, _{i} and _{j} are the modal
amplitudes.

Deformed shape of the arch (Figure ^{4}/EI =^{6}.

Elastic foundation reaction ^{4}/EI =^{6}.

In
Figure ^{4}/^{6}. The conventional foundation model (bilateral contact) was
also considered. One can observe that under unilateral contact constraints the
arch displacement ^{°}is the same for all formulations.
Figure

Two numerical approaches to solve the unilateral contact problem between a structure and a tensionless elastic foundation are proposed in this work. The first proposed formulation is based on the finite element method and mathematical programming techniques. Two alternative linear complementary problems (LCPs) are derived—primal and dual—and solved by Lemke’s algorithm. These formulations consider explicitly the inequality constraints that characterize the unilateral contact problem. The primal formulation can be used without restrictions in several structural problems while the dual formulation is restricted to structural systems in which the structure’s stiffness matrix is positive definite. However, the primal formulation leads to higher-order matrices and, consequently, to lower-computational efficiency and more processing time and sometimes present numerical instability.

The second formulation, named
here unconstrained formulation, is based on the Ritz method and an iterative
solution strategy. This methodology is particularly
suited for the analysis of simple problems where the number and location of the
contact regions are known a priori but not its length. This methodology can
substitute in these cases large and time-consuming finite element packages. It
may also be used as a benchmark for more general and complex formulations. However,
this methodology leads to highly nonlinear equilibrium equations (due to the
existence of unknown boundaries). So efficient iterative strategies, as
proposed in Algorithm

The examples involving beams and arches under contact constraints show that there is an excellent agreement among the results obtained from the different formulations, corroborating the effectiveness of all procedures in the analysis of unilateral contact problems. Finally, the results clarify the influence of the unilateral contact on the behavior of the structure.

The authors are grateful for the financial support from the Brazilian National Council for Scientific and Technological Development (CNPq/MCT). They also acknowledge the support from Brazilian Steel Company USIMINAS, Foundation Gorceix, FAPEMIG and FAPERJ-CNE. Special thanks are also due to Professor Michael Engelhardt and his department at the University of Texas for their hospitality during the manuscript’s preparation.