Coupling elastic models through interfacial conditions with application to concrete pavement overlays

Applied Mathematics and Computation 123 (2001) 187±204 www.elsevier.com/locate/amc

Coupling elastic models through interfacial conditions with application to concrete pavement overlays Luther W. White a,*, Mushara€ Zaman b a

b

Department of Mathematics, The University of Oklahoma, Norman, OK 73019, USA Department of Civil and Environmental Engineering, The University of Oklahoma, Norman, OK 73019, USA

Abstract A simple mathematical model accompanied with a validation is presented to capture the deformation behavior observed in laboratory tests involving overlayed concrete samples. The model presented is based on linear elasticity in which the layers are conceptually envisioned as Mindlin plates overlaying one another while resting of a Winkler-type foundation. The interfacial conditions involve a coupling between the plates through elastic conditions satis®ed by their relative displacements. Experimental validation is conducted on samples using analogous Timoshenko beam models. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Parameter estimation; Mindlin±Timoshenko plates; Pavement overlays

1. Introduction The study in this paper is motivated by a practical problem arising in e€orts to repair and strengthen existing roadway pavement. A current technique commonly used to extend the life and performance of concrete pavement is to overlay existing concrete with a layer of new concrete. The e€ectiveness of such

*

Corresponding author. E-mail address: [email protected] (L.W. White).

0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 0 ) 0 0 0 6 1 - 8

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a rehabilitation strategy depends on several factors among which bonding between the old and the new layers is probably most critical. Indeed a topic of current research involves the study of the in¯uence of surface preparations on the performance of concrete overlays [2]. There is, thus, a need for mathematical and computational models to facilitate the analysis, prediction, and design of these transportation structures. We present a mathematical model accompanied with a validation in an e€ort to determine a simple model capturing the deformation behavior observed in laboratory tests. The model presented in this paper is based on linear elasticity in which the layers are conceptually envisioned as Mindlin plates overlaying one another while resting on a Winkler-type foundation. The interfacial conditions involve a coupling between the plates through conditions satis®ed by their relative displacements. Concrete is a composite and a very complicated material involving di€erent types of nonlinearities. Hence, in posing a simple model one ®rst questions the validity of using an underlying linear elasticity theory. Our use of a linear model is based on laboratory investigations demonstrating that, within the range and scale of interest, the material comprising the pavement exhibits a linear stress±strain relationship [2]. Indeed, Figs. 1±3 indicate a linear stress± strain relationship for tests of several samples. Further simpli®cations are made under the geometric assumptions for thin plate models. The plate assumptions are imposed with the goal toward eventually producing a model to test against laboratory beam experiments. Consequently, in this work we study the use of a linearly based theory along with simplifying geometric plate assumptions to produce mathematical models for the purpose of validating against laboratory measurements made on the concrete overlayed beams with interfaces. The behavior and durability of the interface between the elastic plates is of practical importance as normal displacements give rise to shearing across the interface. Thus, the Mindlin plate model is used, as it is the simplest model that is based on linear elasticity and plate theory incorporating linear shearing. It is then further restricted to that of a coupled beam system for the purpose of comparing calculated deformations with those from laboratory measurements. The validation consists in showing that by adjusting what amounts to the interfacial sti€ness coecients, it is possible to produce displacements comparing favorably with those observed in the laboratory. In the following we present the derivation of the model based on energy considerations for the underlying models of Mindlin plates and Timoshenko beams. A brief discussion is included showing the well-posedness of the model that results from the beam system with an interface and foundation. The ®nite element approximating problem is posed in Section 3 and the experimental results are discussed in Section 4. Several conclusions are presented in Section 5.

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189

Fig. 1. Compressive stress vs. longitudinal strain.

Fig. 2. Compressive stress vs. longitudinal strain.

Fig. 3. Compressive stress vs. longitudinal strain.

2. Basic models The mathematical models underlying our analysis are brie¯y described in this section. The model underlying our analysis is that of the Mindlin plate [4] since it represents the simplest plate model coupling the norm displacements with shearing displacements. The starting point is the equations of three-dimensional linear elasticity. Under the small deformation gradient assumption [3], we assume the components of the stress and the strain tensors are related by the equations:

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E …1 ‡ l†…1 E r22 ˆ …1 ‡ l†…1 E r33 ˆ …1 ‡ l†…1

r11 ˆ

r12 ˆ G12 ;

2l† 2l† 2l†

‰…1

l†11 ‡ l22 ‡ l33 Š;

‰l11 ‡ …1

l†22 ‡ l33 Š;

‰l11 ‡ l22 ‡ …1

r13 ˆ G13 ;

and

…2:1†

l†33 Š;

r23 ˆ G23 ;

where E is Young's modulus, l is Poisson's ratio, and G ˆ 2E=…1 ‡ l† is the shear modulus. Denoting displacements in the x; y; and z directions by functions of x; y; and z by U ˆ U …x; y; z†; V ˆ V …x; y; z†; and W ˆ W …x; y; z†, respectively, the strain±displacement relations are     oU 1 oU oV 1 oU oW ; 12 ˆ 11 ˆ ‡ ; 13 ˆ ‡ ; ox 2 oy ox 2 oz ox   …2:2† oV 1 oV oW oW ; 23 ˆ : ‡ ; 33 ˆ 22 ˆ oy 2 oz oy oz From Eqs. (2.1) and (2.2) various mathematical models can be derived that provide theories approximating the behavior of elastic bodies depending on the geometry and the displacement assumptions. Realizing the limitations of the resulting theory, we nevertheless assume the hypotheses of linear plate theory, in particular that of Mindlin plates. Hence, the body is represented by a domain X ˆ D  … h; h† where D is a bounded domain in R2 with a Lipschitz boundary and h is suciently small that the surface force normal to the plane containing D is absorbed into the body force. Moreover, the z component of the displacement, W , is a function of only x and y and not a function of z. Thus, the displacement normal to the plane containing D is uniform throughout the thickness of X. On the other hand, in-plane displacements are assumed to support linear shearing. Setting r33 ˆ 0; 33 may be expressed in terms of 11 and 22 and 33 may be eliminated in the above equations. It then follows that r11 ˆ

E …1

r12 ˆ G12 ;

l2 †

‰11 ‡ l22 Š;

r13 ˆ G13 ;

and and

r22 ˆ

E …1

l2 †

‰l11 ‡ 22 Š;

…2:3†

r23 ˆ G23 :

The displacement assumptions associated with the Mindlin plate take the form U …x; y; z† ˆ z/…x; y†;

…2:4†

V …x; y; z† ˆ zw…x; y†;

…2:5†

and

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while the normal displacement is given by W …x; y; z† ˆ w…x; y†:

…2:6†

Following the development in Mindlin [4], we introduce the strain energy V…U ; V ; W † Z Z Z h 1 ˆ fr11 11 ‡ r12 12 ‡ r13 13 ‡ r22 22 ‡ r23 23 g dz dx dy; 2 D h …2:7† and the Lagrangian

Z Z

W…U ; V ; W † ˆ V…U ; V ; W †

D

f …x; y†W …x; y† dx dy:

…2:8†

The displacement assumed by the body X is the minimizer of the Lagrangian over suitable spaces of displacement functions for which the work functional is de®ned and that satisfy appropriate boundary conditions. With the strain±displacement relations (2.2), the stress±strain relations (2.3), and the displacement assumptions (2.4)±(2.6), the Lagrangian is expressed as Z Z  3 h i 1 hE 2 2 2 W…U ; V ; W † ˆ …1 l†/ ‡ l…/ ‡ w † ‡ …1 l†w x y x y 2 1 l2 D  h i hE 2 2 2 …/y ‡ wx † ‡ …/ ‡ wx † ‡ …w ‡ wy † ‡ dx dy 2…1 ‡ l† Z Z fw dx dy: …2:9† D

Now suppose that there are two bodies that we view as plates 1 and 2, see Fig. 4. We suppose that plate 1 is situated such that it occupies the set X1 ˆ D  …0; 2h1 † while plate 2 occupies the set X2 ˆ D  … 2h2 ; 0†: The interface between the plates consists of the set of points D0 ˆ f…x; y; 0†: …x; y† 2 Dg: We may assign local coordinate systems 1 and 2 such that coordinate system 1 is related to the global coordinate system x; y; and z by x1 ˆ x; y1 ˆ y; and z1 ˆ z h1 : On the other hand, coordinate system 2 is related to the global coordinate system by the relations x2 ˆ x; y2 ˆ y; and z2 ˆ z ‡ h2 : The displacement functions for the ith plate are given by Ui …x; y; zi † ˆ zi /i …x; y†; Vi …x; y; zi † ˆ zi wi …x; y†; and Wi …x; y; zi † ˆ wi …x; y†:

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Fig. 4. Orientation of plates.

Following Eq. (2.9) the strain potential for the ith plate is expressed as Z Z  3 1 h i Ei 2 ‰…1 li †/2ix ‡ li …/ix ‡ wiy † Vi …/i ; wi ; wi † ˆ 2 2 1 l D i hi E i h 2 …/iy ‡ wix † ‡ …1 li †w2iy Š ‡ 2…1 ‡ li † i ‡ …/i ‡ wix †2 ‡ …wi ‡ wiy †2 dx dy: …2:10† The total potential energy of the two plates is now given as the sum of the individual strain energy functionals as V ˆ V1 ‡ V2 ; and the Lagrangian is given by Z W0 ˆ V f1 …x; y†w1 …x; y† dx dy D

Z D

f2 …x; y†w2 …x; y† dx dy:

The Lagrangian W0 is formulated such that displacements of the two plates are not related. The interface conditions are imposed to relate the displacement of the two plates. Towards this end, we introduce functions Kx ; Ky ; and Kz from D into R‡ for the purpose of capturing the elastic properties of the interface. For a linear theory we introduce the integrals

L.W. White, M. Zaman / Appl. Math. Comput. 123 (2001) 187±204

Vx ˆ

1 2

Vy ˆ

1 2

Vz ˆ

1 2

and

Z Z

193

2

D

Z Z D

Kx …x; y†‰h1 /1 …x; y† ‡ h2 /2 …x; y†Š dx dy; Ky …x; y†‰h1 w1 …x; y† ‡ h2 w2 …x; y†Š2 dx dy;

Z Z D

Kz …x; y†‰w1 …x; y†

w2 …x; y†Š2 dx dy

as the potential energy due to relative displacement between plates 1 and 2 in the x; y; and z directions, respectively. Finally, to account for the interaction with the supporting foundation, we de®ne the functionals Z Z 1 VFz ˆ KFz w22 …x; y† dx dy; 2 D Z Z 1 VFx ˆ KFx /22 …x; y† dx dy; 2 D Z Z 1 VFy ˆ KFy w22 …x; y† dx dy: 2 D The Lagrangian for the layered system with elastic coupling through the interface and with a foundation is given by W ˆ W0 ‡ Vx ‡ Vy ‡ Vz ‡ VFx ‡ VFy ‡ VFz : Remark 2.1. In the present work we only consider linear models since it is our goal here to provide a class of models along with a validation against experimental data. Terms arising from contact forces, friction, or nonquadratic expressions of the interface and foundation conditions are not included. 3. The beam system A specialized system is posed as two Timoshenko beams coupled through an interface with a foundation. For the purposes of this derivation the domain D is given as D ˆ … k; k†  …0; L† and Eqs. (2.4)±(2.6) are replaced with U …x; y; z† ˆ z/…x†;

…3:1†

V …x; y; z† ˆ 0;

…3:2†

W …x; y; z† ˆ w…x†:

…3:3†

and

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The Lagrangian corresponding to (2.9) is  Z  1 L 2kh3 E 2 4khE 2 …/ ‡ w / † Wˆ ‡ dx x 2 0 1 l2 x 1 ‡ l

Z

L 0

fw dx:

This description is applied to a two-beam system in which beam 1 occupies the set X1 ˆ …0; L†  … k; k†  …0; 2h1 † with local coordinates x1 ˆ x; y1 ˆ y; and z1 ˆ z hand, beam 2 occupies the set

h1 , see Fig. 5. On the other

X2 ˆ …0; L†  … k; k†  … 2h2 ; 0† with local coordinates given by x2 ˆ x; y2 ˆ y; and z2 ˆ z ‡ h2 : Displacement functions for the ith beam are Ui …x; y; zi † ˆ zi /i …x†; Vi …x; y; zi † ˆ 0; and Wi …x; y; zi † ˆ wi …x†: The Lagrangian for the ith beam can be expressed as  Z L Z L 2kh3i Ei 2 4khi Ei 2 / …/ ‡ w † fi wi dx Wi ˆ ‡ dx ix 1 l2i ix 1 ‡ li i 0 0 and that for the two-beam system W ˆ W1 ‡ W2 :

Fig. 5. Orientation of the beams.

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195

As in the case of the plates, we model the elastic interface between the beams by introducing the functionals giving the elastic potential for relative displacements between the beams Z 1 L 2 2 VI ˆ fKS …x†…h1 /1 …x† ‡ h2 /2 …x†† ‡ KN …x†…w1 …x† w2 …x†† g dx 2 0 and for the foundation by introducing the functional Z 1 L 2 2 VF ˆ fKFS …x†/2 …x† ‡ KFN …x†w2 …x† g dx: 2 0 For the applications we have in mind there are two forces acting on the beam system. The ®rst is the body force due to gravity. The work done by this force is given by Z L WB ˆ 4gk …q1 h1 w1 …x† ‡ q2 h2 w2 …x†† dx: 0

The second is an applied force Z L WA ˆ fw1 dx: 0

The Lagrangian of the two-beam system with an interface and foundation is thus expressed as L…/1 ; w1 ; /2 ; w2 † ˆ V1 …/1 ; w1 † ‡ V2 …/2 ; w2 † ‡ VI ‡ VF

2kh3i Ei ; …1 l2i †

bi ˆ

4khi Ei : …1 ‡ li †

and

We also set 2

3 /1 6 w1 7 7 uˆ6 4 /2 5; w2  Eˆ

E0 0

WA : …3:4†

For ease we assign ai ˆ

WB

 E0 ˆ

 0 ; E0

0 1

 0 ; 0 2

a1 60 Aˆ6 40 0

0 b1 0 0

0 0 a2 0

3 0 07 7; 05 b2

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L.W. White, M. Zaman / Appl. Math. Comput. 123 (2001) 187±204

2

h21 6 0 gS ˆ 6 4 h1 h2 0 2 0 0 60 0 gFS ˆ 6 40 0 0 0

0 h1 h2 0 0 0 0 0 0 3 0 0 0 07 7; 1 05 0 0

3 0 07 7; h22 5 0 and

2

0 60 gN ˆ 6 40 0 2 gFN

0 60 ˆ6 40 0

0 1 0 1

0 0 0 0

0 0 0 0

0 0 0 0

3 0 17 7; 0 5 1 3

0 07 7: 05 1

Finally, de®ne the vector of displacement functions 2 3 /1 6 w1 7 7 uˆ6 4 /2 5; w2 and the forcing vectors 2 3 0 6 2gkq1 h1 7 7; FB ˆ 6 4 5 0 2gkq2 h2

and

2 3 0 6f 7 7 FA ˆ 6 4 0 5: 0

De®ne the Hilbert spaces with the usual norms and inner products H ˆ L2 …0; L; R4 †; V ˆ H 1 …0; L; R4 †; and V0 ˆ fV: w1 …0† ˆ w1 …L† ˆ 0g; and the bilinear form on V  V Z 1 L T …ux ‡ Eu† A…vx ‡ Ev† a…u; v† ˆ 2 0 ‡ uT …KS gS ‡ KN gN ‡ KFN gFN ‡ KFS gFS †v dx: With these assignments the functional of (3.4) takes the form Z L L…u† ˆ a…u; u† …FA ‡ FB †T u dx: 0

…3:5†

We pose the minimization problem as follows. Find u 2 V0 such that L…u† ˆ minimum L…v† for any v 2 V0 :

…3:6†

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197

The existence of a unique solution u of (3.6) is a consequence of the following. Theorem 3.1. Let KS ; KN ; KFN ; and KSN P j > 0: There exists a positive constant j0 such that 2

a…u; u† P j0 kukV : Proof. Let c ˆ minfa1 ; b1 ; a2 ; b2 g: Then by Cauchy's inequality, we obtain     Z L 1 1 a…u; u† P c /21x ‡ /22x ‡ 1 w21x ‡ 1 w22x ‡ ‰…1 1 † 1 2 0   1 ‡ KS h21 …1 3 †Š/21 ‡ ‰…1 2 † ‡ KS h22 1 3    1 2 2 2 2 ‡ 4h2 KFS Š/2 ‡ KN …1 4 †w1 ‡ ‰KN 1 ‡ KFN Šw2 : 4 Setting 1 ˆ 1 ‡

2KFS KS h21 ; c…KS ‡ 4KFS †

2 ˆ 1 ‡

2KFS h22 ; c

and choosing KS ‡ 2KFS KS > 3 > ; KS ‡ 4KFS KS ‡ 2KFS and 1 > 4 >

KN ; KN ‡ KFN

it is possible to select a positive constant j0 as indicated above.



Remark 3.2. The displacement vector u is a minimizer of (3.5) if and only if Z L a…u; v† ˆ …FA ‡ FB †v dx: …3:7† 0

We now view the displacement u as a function of the interface elastic coecient KS which is a function de®ned on …0; L†: Thus, u ˆ u…KS †: The following gives the continuity properties of this mapping. The proof is straight forward and may be found in [6].

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L.W. White, M. Zaman / Appl. Math. Comput. 123 (2001) 187±204 …n†

Corollary 3.3. Suppose KS > j > 0 is a sequence in L1 …0; L† such that …n† …n† KS ! KS in L1 as n ! 1: Then u…KS † ! u…KS † in V as n ! 0: 4. Numerical model and validation It is our purpose to use the above model along with measurements of normal displacements of concrete samples from laboratory measurements to validate our model. A total of six beam specimens (3.25 ft. long, 1 ft. wide, and 9 in. thick) were prepared maintaining dry and ``no grout'' interface conditions [2]. Three of these specimens had 3 in. (7.5 cm) thick overlay, while the other tree had a 4 in. (10 cm) thick overlay. The samples were stored and cured at a temperature of approximately 70°F (21°C.) with a relative humidity of 65±70%. Mechanical properties, especially compressive and split tensile strength for each batch of overlay concrete were determined to ascertain the quality of prepared specimens and to identify any variability between di€erent mixes. The instrumentation consisted of a series of LVDTs mounted on the top surface (Figure). All data were collected, controlled, and stored by an HP 3497A data acquisition unit coupled with a personal computer. A computer code ``COMPRS'' was used for this purpose. The simply supported beams were tested under a third-point loading in accordance with ASTM Test Method C 78 (ASTM, 1995) [1]. De¯ections along the centerline of each point were obtained by averaging the de¯ections of LVDTs place on opposites sides. The objective is to compare the normal displacements observed experimentally with the displacements predicted from our model. The only parameters to be adjusted are the interface sti€ness coecients. By adjusting the shearing coecient we can account for nonsymmetric displacements of the system observed in the laboratory. In order to proceed, we solve numerically the ®nite element formulation of the beam system. To give the discrete version of (3.6) and (3.7), we partition the interval …0; L† into n subintervals of equal length. Let bi ; i ˆ 1; . . . ; n ‡ 1 be piecewise linear functions [5] de®ned with respect to the mesh such that bi …x† takes the value 1 at the point x ˆ …i 1†L=n and is zero for any x ˆ jL=n with j 6ˆ i 1: Designate by b…x† ˆ ‰b1 …x†; b2 …x†; . . . ; bn‡1 …x†Š the vector valued function of …0; L† to Rn‡1 and the 4  …4…n ‡ 1†† matrix valued function 2 3 b…x† 0 0 0 6 0 b…x† 0 0 7 7 B…x† ˆ 6 4 0 0 b…x† 0 5 0 0 0 b…x† for any x 2 …0; L†: Let c be a 4…n ‡ 1† column vector then we let u…n† …x† ˆ B…x†c

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199

be a vector that is to approximate a minimizer u of (3.6). By substituting u…n† for u; the functional L…u† becomes a function c 7! L…c† of R4…n‡1† into R: De®ne the matrices Z L T …Bx …x† ‡ EB…x†† A…Bx …x† ‡ EB…x†† dx; GA ˆ 0

Z GS ˆ and

L

T

KS …x†B…x† gS B…x† dx;

0

Z GN ˆ

L 0

T

KN B…x† gN B…x† dx:

Finally, de®ne the vectors Z L fA ˆ FAT B…x† dx; 0

and

Z fB ˆ

L 0

FBT B…x† dx:

The functional is now given as L…c† ˆ cT fGA ‡ GS ‡ GN gc

T

…f A ‡ f B † c:

…4:1†

In order to examine the character of the interface, we allow the shearing interface coecient KS to be a function of the spatial variable. For the numerical model, we introduce the functions Ni ; i ˆ 1; . . . ; m; where  1 if x 2 ……i 1†L=m; iL=m†; Ni …x† ˆ 0 otherwise assume that KS …x† ˆ

m X

ji Ni …x†:

kˆ1

Introducing the component matrices for i ˆ 1; . . . ; m; Z L …i† T Ni …x†B…x† gS B…x† dx; GS ˆ 0

it follows that for KS of the form (4.2), we may express m X …i† GS ˆ ji GS : iˆ1

…4:2†

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We may now rewrite Eq. (4.1) as ( ) m X …i† T L…c† ˆ c GA ‡ ji GS ‡ GN c iˆ1

T

…f A ‡ f B † c:

…4:3†

From Hamilton's principle the displacement resulting from the body and applied forces minimizes function (4.3) with respect to the set of admissible displacement functions. It follows that the displacement must satisfy the equation ( ) n X …i† GA ‡ ji GS ‡ GN c ˆ f A ‡ f B : …4:4† iˆ1

Designating the vector 2 3 j1 6 .. 7 jˆ4 . 5 jn

the solution of (4.4) is a function j 7! c…j†: The calculated displacement is given by u…x† ˆ B…x†c…j†:

…4:5†

Now suppose that data vector d 0 2 Rnobs is obtained experimentally. In the nobs present the data corresponds to the measurement of w1 at locations fxdi giˆ1 : Hence, to formally compare the model prediction with data, we introduce the linear operator C : R4…n‡1† 7! Rnobs as 2 3 w1 …xd1 † 6 7 .. Cc…j† ˆ 4 5: . w1 …xdnobs †

Hence, we see that 2 3 b…xd1 † 6 7 .. Cˆ4 5 . b…xdnobs † describing the mapping from the solution space to the data space. The length of the beam is L ˆ 37 in: As noted earlier, loading occurs at points L=3 and 2L=3 on the beam. Measurement locations in inches are given by the vector

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201

xd ˆ ‰0; 9:25; 18:5; 27:75; 37Š: We present two examples one with a 3 in. overlay and another with a 4 in. overlay. For the 3 in. overlay, we ®nd a displacement vector of 2 3 0 6 0:0014 7 6 7 7 d0 ˆ 6 6 0:00245 7: 4 0:0017 5 0 As a second case with the 4 in. overlay, we have a displacement vector 2 3 0 6 0:0033 7 6 7 7 d0 ˆ 6 6 0:0044 7: 4 0:0036 5 0 Our objective is to determine how well we can ®t the beam data observed in the laboratory by using the model we have derived. We assume that the beam is supported at x ˆ 0 and x ˆ L so that displacement normal to the beam is zero at those locations. Hence, in our laboratory tests there is no foundation term. Moreover, it is assumed that the elastic coecients corresponding to the normal displacements across the interface are large. We wish to determine the elastic interfacial shearing coecients so that our model matches the observations. Essentially at this step we would like to see if it is possible to use the present model to represent observed displacements, by adjusting only the interfacial shearing coecients. For numerical approximations, we used N ˆ 10 as the number of subintervals each of length ˆ L=10 to de®ne the piecewise linear basis functions to approximate displacements. The same mesh is used to approximate the interfacial shearing sti€ness. We de®ne a ®t-to-data functional that is the square of the Euclidean norm of the error J …j† ˆ jCc…j††

d 0 j2 :

…4:6†

This functional is minimized by an iterative Newton method with an initial guess of ji ˆ 2:0

for i ˆ 1; . . . ; 5;

ji ˆ 7:0

for i ˆ 6; . . . ; 10:

We calculate a relative error, RE, given by 1=2

RE…j† ˆ

J …j† : jd 0 j

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In the case of the 3 in. overlay Fig. 6, we obtain RE ˆ 0:0168: The corresponding interface sti€ness vector j is given by j ˆ ‰13 39 50 8503 20 66 000 36 9Š: In the case of the 4 in. overlay Fig. 7, we obtain RE ˆ 0:0250: The corresponding interface sti€ness vector j is given by j ˆ ‰3 54 58 49 27 62 554 402 7Š: For comparison a relative error of RE ˆ 0:0976 with ji ˆ 5:0

for i ˆ 1; . . . ; 10

is obtained as the optimal value among constant sti€nesses. In the case of a piecewise constant sti€ness function that may take on two values.

Fig. 6. Displacement for 3 in. overlay.

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203

Fig. 7. Displacement for 4 in. overlay.

RE ˆ 0:0868 for the optimal j vector with components ji ˆ 7:0

for i ˆ 1; . . . ; 5

ji ˆ 2:0

for i ˆ 6; . . . ; 10:

5. Conclusions We have presented a derivation of a mathematical model to describe the deformation of a plate system with a coupling at an interface. The boundary value problem thus obtained is based on Mindlin±Timoshenko models with the relative displacement between the plates viewed as an elastic stretching of the interface coupling the plates. The model presented is thus based on linear elasticity and plate/beam geometric approximations. A validation against laboratory data is presented. It is observed that agreement with measurements of deformations of beams obtained as concrete overlays is excellent and improves as the mesh for the ®nite element approximating systems is re®ned.

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References [1] Annual Book of ASTM Standards, Vol. 04.02, American Society for Testing and Materials, 1997. [2] Z. Hossain, Behavior of thin bonded overlay on rigid pavements: an experimental approach, Masters Thesis, University of Oklahoma, 1998. [3] S.C. Hunter, Mechanics of Continuous Media, Ellis Horwood, New York, 1976. [4] R.D. Mindlin, In¯uence of rotatory inertia and shear on ¯exural motions of isotropic elastic plates, J. Appl. Mech. (1951) 31±38. [5] M. Schultz, Spline Analysis, Prentice-Hall, Englewood Cli€s, NJ, 1973. [6] L. White, J. Zhou, Continuity and uniqueness of regularized output least squares optimal estimators, J. Math. Anal. Appl. 196 (1995) 53±83.