MULTIGRID PRECONDITIONER FOR UNSTRUCTURED NONLINEAR 3D FE MODELS By William G. Davids,1 Student Member, ASCE, and George M. Turkiyyah, 2 Member, ASCE ABSTRACT: Multigrid and multigrid-preconditioned conjugate-gradient solution techniques applicable for unstructured 3D finite-element models that may involve sharp discontinuities in material properties, multiple element types, and contact nonlinearities are developed. Their development is driven by the desire to efficiently solve models of rigid pavement systems that require explicit modeling of spatially varying and discontinuous material properties, bending elements meshed with solid elements, and separation between the slab and subgrade. General definitions for restriction and interpolation operators applicable to models composed of multiple, displacement-based isoparametric finite-element types are proposed. Related operations are used to generate coarse mesh element properties at integration points, allowing coarse-level coefficient matrices to be computed by a simple assembly of element stiffness matrices. The proposed strategy is shown to be effective on problems involving spatially varying material properties, even in the presence of large variations within coarse mesh elements. Techniques for solving problems with nodal contact nonlinearities using the proposed multigrid methods are also described. The performance of the multigrid methods is assessed for model problems incorporating irregular meshes and spatially varying material properties, and for a model of two rigid pavement slabs subjected to thermal and axle loading that incorporates nodal contact conditions and both solid and bending elements. INTRODUCTION This study focuses on the development of robust multigridpreconditioned conjugate-gradient methods suitable for the solution of large-scale 3D finite-element problems in structural mechanics that involve material inhomogeneity, contact nonlinearities, and multiple element types. Our motivation for developing these solution strategies is the desire to efficiently model rigid concrete pavement systems (Fig. 1). Such analyses are of great practical interest to pavement researchers and designers, and exhibit several sources of complexity: separation of the slab from the base layers under temperature gradients, the need to accurately model the dowels by combining bending elements with solid elements, and spatially varying and discontinuous material properties encountered when several soil layers are present. Accurate rigid pavement simulations involve the use of 3D finite-element models, the solution of which requires a large amount of computational effort and storage. Direct solution techniques, based on Gauss elimination, are often not feasible due the large amount of memory required to store the matrix factorizations. Further, as the size of the problem increases, the number of operations required to factor the matrix generally increases at least quadratically with the number of unknowns, making factorization techniques inefficient for large problems. [Note: For example, a regular square mesh of N elements will have a stiffness matrix with a bandwidth of ᏻ(兹N) using natural ordering. Using a banded representation, the factorization requires C = ᏻ(N 2) operations. In 3D, the bandwidth is ᏻ(N 2/3) and the number of operations, C, is ᏻ(N 7/3) for a regular cubical mesh of N elements. The use of sparse factorization techniques with appropriate reorderings (such as minimum degree) can reduce C significantly, but an almost quadratic relationship between C and N is generally observed.] 1 Res. Asst., Dept. of Civ. Engrg., Univ. of Washington, Box 352700, Seattle, WA 98195-2700. E-mail: wdavids@ce.washington.edu 2 Assoc. Prof., Dept. of Civ. Engrg., Univ. of Washington, Box 352700, Seattle, WA. E-mail: george@ce.washington.edu Note. Associate Editor: Sunil Saigal. Discussion open until July 1, 1999. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 19, 1997. This paper is part of the Journal of Engineering Mechanics, Vol. 125, No. 2, February, 1999. 䉷ASCE, ISSN 0733-9399/99/0002-0186 – 0196/ $8.00 ⫹ $.50 per page. Paper No. 17207. 186 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 At present, the most efficient general iterative techniques are based on multigrid methods, which rely on a hierarchy of discretizations of the domain coupled with a standard iterative method (typically Jacobi or Gauss-Seidel) to achieve a solution. Early research on multigrid methods focused on twodimensional (2D) finite-difference approximations to general elliptic boundary value problems using regular grids (Brandt 1977). When used to solve such regular finite-difference discretizations, multigrid methods require an amount of work proportional to the number of unknowns, making them very attractive for large problems (Brandt 1977). Recently, multigrid techniques have begun to find widespread applications in finite-element modeling of structures and solid mechanics problems. Much of the groundwork for the application of multigrid techniques in solid mechanics may be found in Parsons and Hall (1990a,b). In these studies, the convergence properties of the multigrid method were investigated for structured 2D and 3D meshes. Important parameters were identified, for which optimal values were determined; in addition, both Gauss-Seidel and conjugate-gradient smoothers were employed. Another application of multigrid techniques to the 3D finite-element modeling of structures may be found in Muttin and Chenot (1995), where a two grid solver using a conjugate-gradient smoother was developed. The smoother was implemented on an element-by-element basis, eliminating the need to assemble the system stiffness matrices, and making the multigrid method more amenable to parallel processing. More recently, the use of unstructured meshes with poorly conditioned problems arising from the discretization of shell structures was addressed (Fish et al. 1996). To overcome the inherent poor conditioning of shell discretizations, a modified incomplete Choleski factorization smoother was employed, along with weighted acceleration schemes based on a linesearch minimization of potential energy. Other, more indirect applications of the multigrid method may also be found in the literature. For example, in Farhat and Sobh (1989) a conjugate-gradient preconditioner based on a nested coarse mesh approximation was developed that can be interpreted as a multigrid method employing a conjugategradient smoother. Multigrid methods in solid mechanics have also been integrated with adaptive solution techniques, where a series of meshes of increasing refinement are generated automatically based on error estimates; see Kǒcvara (1993), for example. hexahedra, and a triangular plate discretized with quadratic tetrahedra. The effectiveness of the proposed techniques for handling spatially varying material properties is also examined via a model problem. Methods for nodal contact modeling appropriate for use with the proposed multigrid methods are developed. To illustrate the generality of the multigrid methods developed in this study, a model of a rigid pavement system incorporating embedded dowel elements and separation between the slab and subgrade is presented and solved. The final section of this paper gives conclusions. BACKGROUND The system of linear equations arising from a displacementbased, linearly elastic finite-element discretization can be written as: P = KU (1) where K = system stiffness matrix (symmetric, positive definite); U = unknown displacements; and P = applied forces. Letting U* denote the exact (unknown) solution, and defining the error, e = U* ⫺ U, then FIG. 1. r = P ⫺ KU = Ke System of Two Rigid Pavement Slabs Despite the progress in multigrid methods for 3D finiteelement modeling of structures and solid mechanics problems, there are still many issues that must be addressed before these methods can be used routinely and reliably for the solution of complex structures such as the rigid pavement systems presently of interest. • Multigrid methods should be generalized to allow multiple element types and varying numbers of nodal degrees of freedom within the same model. Such scenarios are common, occurring whenever plate or beam elements are meshed with or embedded in solid elements. • The use of unstructured sequences of meshes should be allowed. It is often impractical in 3D to generate nested sequences of meshes, due to constraints on element aspect ratios and limits on the problem size. When local mesh refinement is required or complex geometries are considered, this problem becomes even more difficult, as automatic mesh generation codes do not readily generate nested sequences of meshes. • Multigrid methods should be generalized to handle nonlinear problems with contact constraints. Contact constraints cause difficulties because of the need for defining intergrid restriction and interpolation operators appropriate for them. • There is a need for multigrid methods that can be conveniently integrated with conventional, displacementbased finite-element methods to avoid expensive replication of existing FE codes. This study specifically addresses these needs. After a brief background on multigrid methods and algorithms, general methods for the intergrid transfer of information are proposed, including restriction and interpolation operators and techniques for handling spatially varying material properties. Details of the searching algorithms required to allow the efficient implementation of these techniques are also discussed. A model problem of a thick plate discretized with quadratic brick elements is solved with various degrees of refinement to illustrate the efficiency of the proposed multigrid methods and provide comparison with a state-of-the-art direct sparse solver. The effect of irregular meshes is examined via two model problems: a thick rectangular plate discretized with quadratic (2) To solve (1), the multigrid method relies on a few applications of a standard iterative technique, called smoothing, which is intended to reduce the high frequency components of e, coupled with coarse grid approximations to the smoothed error (Brandt 1977). The general, recursive multigrid algorithm is given in Algorithm 1 (Jesperson 1984). In Algorithm 1, R restricts from the fine mesh to the coarse mesh, and T interpolates the coarse mesh error to the fine mesh. This subroutine is placed in an outer loop, and called repeatedly until an appropriate convergence criterion is met. If the inner loop limit, , is greater than one, the algorithm corresponds to a ‘W-cycle,’ where multiple processes of smoothing/restricting and interpolating/smoothing at coarser levels are implemented. A ‘V-cycle’ is implied by = 1, where smoothing and restricting are applied sequentially from the finest to the coarsest grid followed by sequential interpolation and smoothing back to the finest grid. As shown by Parsons and Hall (1990a), there is not a significant difference in computational effort between the V-cycle and the W-cycle for well- ALGORITHM 1. General Recursive Multigrid Algorithm JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 / 187 conditioned problems without singularities. For this reason, the value of is fixed at one in the present study, although it must be noted that W-cycle may be more appropriate in certain cases. While multigrid methods (MG) are an efficient iterative solution technique, they can be significantly more effective when used to precondition a conjugate-gradient iteration (MG-PCG), which takes advantage of the symmetric positive definiteness of K. This has been demonstrated in Meza and Tuminaro (1996) for solving discretizations of the semiconductor equations, and in Ashby and Falgout (1996) for ground-water flow simulations with spatially varying conductivities. More recently, this idea was extended to the preconditioning of the generalized minimum residual and biconjugate-gradient algorithms (Saad 1996) for the solution of singularly perturbed unsymmetric problems (Oosterlee and Washio 1998). The solution of nonlinear heat conduction problems with large discontinuities in material properties using a multigrid-preconditioned Newton-Krylov method was investigated in Rider and Knoll (1997). Preconditioning (1) may be expressed as solving the equivalent system M⫺1P = M⫺1KU (3) where M is some symmetric positive definite approximation to K (Saad 1996). This preconditioning step is implemented within a conjugate-gradient iteration by solving Mz = q (4) where z = displacement vector and q = vector of residual forces. The effectiveness of the preconditioner depends on how closely M approximates K; using one or more symmetric multigrid iterations to approximately solve (4) can be an effective preconditioner. While more multigrid preconditioning cycles implies that M better approximates K, and hence fewer global conjugate-gradient iterations will be required, the dominant amount of work per iteration lies in the preconditioning step; preliminary studies indicate that the optimal number of multigrid-preconditioning cycles is one for well-conditioned problems. INTERGRID TRANSFER OF INFORMATION: FORMULATION be repeated here. Similarly, by requiring that the fine and coarse mesh strain energies be equal, and using R = TT, it follows that the optimal coarse mesh stiffness, Kc*, is Kc* = TTK f T = RK f RT where the superscripts f and c denote fine and coarse, respectively. The computation of Kc* is expensive for sequences of unstructured meshes, since it involves two (albeit sparse) matrix products. Alternatively, Kc may be assembled from the coarse mesh element stiffness matrices in the usual manner and used in lieu of Kc*. This is the approach used exclusively in this study to simplify incorporation of the proposed multigrid methods in existing finite-element codes; details are presented later in this section. We note here that preliminary studies comparing the performance of V-cycle MG when using Kc as opposed to Kc* indicated little difference in efficiency for well-conditioned problems. The Interpolation Operator A natural way of determining T when using the finite-element method is to use the element shape functions to interpolate the displacements from the coarse to the fine mesh; this also guarantees that the interpolation has the same degree of accuracy as the solution. Defining T for two unnested meshes then involves the following, as illustrated in Fig. 2: • For every fine-mesh node, the coarse-mesh element it is located within is determined. • The natural (local) coarse-mesh element coordinates corresponding to the physical fine-mesh nodal coordinates are computed for each node/element pair. • The element shape functions evaluated at these local coordinates allow interpolation from the coarse-mesh nodal values to the value at the fine-mesh node. Interpolation for each fine-mesh node may be written as: d fi = N cj U cj • R and T meet the virtual work requirement that R = TT. • Both R and T are defined and applied in the same manner for any sequence of meshes, including unstructured and unnested meshes. • R and T permit the use of multiple isoparametric element types within the same model, including the meshing of beam or plate elements with solids. • R and T may be easily computed using subroutines available in existing finite-element codes. Work and Energy Considerations 188 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 (6) where U cj = vector of nodal displacements for coarse-mesh element j; N cj = usual array of shape functions for element j evaluated at fine-mesh node i; and d fi represents the displace- When developing any multigrid method, methods for the intergrid transfer of information, including consistent and computationally efficient definitions of R and T, are of primary importance. In particular, it is desirable that the intergrid operations uniformly handle different element types (volume, surface, and line elements) and different element topologies (tetrahedral, hexahedral, etc.) without any alignment constraints between successive meshes. The definitions presented in this section possess the following characteristics: The requirement that R = TT is easily derived from the consideration that the work done by the external forces be equal on each mesh (Parsons and Hall 1990a); details will not (5) FIG. 2. Interpolation and Restriction ment vector at fine mesh node i. This process may also be written in matrix form for the entire mesh as U f = TUc (7) c j where the element shape function matrices, N , have been appropriately mapped to T, which is of size n f by nc; n f and nc denote the number of nodal unknowns in the fine and coarse meshes. In this study, the matrix representation of T is never formed explicitly, and the process of interpolation is carried out node-by-node as shown in (6) with U f assembled from each d fi. Note that this definition of T is quite general, allowing solid (or planar) elements of different type to be meshed together, as well as permitting the meshing of beam or plate elements with solid elements, as shown in Fig. 2. The Restriction Operator Since the residual is actually a vector of unbalanced forces, R must be chosen so that it converts a fine-mesh force vector to a statically equivalent force vector on the coarse mesh. Energy principles require that a force vector at a fine-mesh node i, f fi, applied at any point within a coarse-mesh element j be converted to element nodal forces as: f p cj = N cT j fi (8) c j where p represents the portion of r corresponding to coarsemesh element j, and N cT is the transpose of the matrix of j coarse-mesh element shape functions evaluated at the location of the point load, f fi (Zienkiewicz and Taylor 1994). It is apparent that the information required to form R is identical to that required for T. Further, restriction may be represented by the following matrix multiplication: rc= Rr f c (9) f In contrast to T,the size of R is n by n . This definition of R is consistent with the energy requirement that R = TT. Searching Algorithm One difficulty with the proposed definitions of R and T is the determination of the node/element pairs for general unstructured meshes. A naı̈ve search would require ᏻ(n fnc) operations and be prohibitively expensive. To maintain the linear proportionality between n f and solution time, this information must be determined with computational effort proportional to n f. To achieve this goal, a searching algorithm similar to that detailed in Löhner and Morgan (1987) is employed. Only elements with straight edges/surfaces will be considered. To illustrate the method with a 2D example, consider Fig. 3, which shows fine and coarse meshes, ᏹf and ᏹc, discretizing the same physical domain. Superimposed over ᏹf and ᏹc is a rectangular grid, Ᏻ defined by the maximum dimensions of the domain. For any element of ᏹc with straight edges, it is a simple matter to determine what cells of Ᏻ it overlaps by checking its maximum and minimum coordinates against the bounds of the cells. Given a list of elements of ᏹc for each cell in Ᏻ, the coarse-mesh element containing any (x, y) point may be found by searching only the list of coarse-mesh elements associated with the cell that the point lies within. The determination of whether or not a point lies within an element involves simple vector operations for an element with straight edges/faces; the inclusion of general curved domains would require extension of the algorithm to handle the more complex geometry. This searching procedure first requires a single loop over all elements in ᏹc, followed by a single loop over all nodes n f within which a small list of coarse-mesh elements must be checked to determine which element n f lies within. Thus, as long as the refinement of Ᏻ follows the refinement of ᏹc, the amount of work required is proportional to n f ⫹ nc. Other geometric search techniques may also be used for building the correspondence between successive meshes (Goodman and O’Rourke 1997). The final issue that must be tackled is the determination of the coarse-mesh element local coordinates corresponding to the fine-mesh nodal locations. For a distorted isoparametric element, the determination of the local element coordinates requires the solution of a system of nonlinear equations, whose size is the number of spatial dimensions of the element (Dracopoulos and Crisfield 1995). Newton’s method is used to solve for the local element coordinates given the (x, y, z) nodal location; the Jacobian matrix needed for the iteration is simply the transpose of the Jacobian used in coordinate transformations during the integration of the element stiffness matrix. Given the element local coordinates, the element shape functions are evaluated as required for each instance of restriction and interpolation with minimal computational overhead. Determination of Coarse-Mesh Stiffness As stated earlier, the coarse mesh stiffness matrices are determined by the usual assembly of the coarse-mesh element matrices. To maintain generality, it is assumed that the material properties can vary over the domain being modeled; this may be due to nonlinearities or simply differing linear materials used in a single model. This variation in material properties is assumed to be defined in the usual fashion, element-by-element, but only on the fine mesh. The information must then be transferred to the coarse-mesh elements to allow the assembly of Kc. Computation of any coarse-mesh element stiffness, k ci , requires that constitutive properties be determined at the element integration points, which depend on the strain increment and the current stress state known only for the fine mesh. The coarse-mesh integration points may be easily located in the fine-mesh elements using the previously described searching algorithms (this is independent of restriction and interpolation, and requires a search over the fine-mesh elements for all integration points on each coarse mesh). The computation of the strain increments and subsequent updating of the stress states at the coarse element integration points is then performed by the appropriate fine-mesh elements for which constitutive relations are defined. One potential problem with this technique must also be noted: if the coarse mesh is not defined appropriately in regions where large changes in material properties occur, Kc may not provide a sufficiently accurate coarse grid approximation. This issue will be explored numerically with a model problem in the next section. INTERGRID TRANSFER OF INFORMATION: NUMERICAL STUDIES FIG. 3. Grid Search The effectiveness of the proposed techniques for the intergrid transfer of information is verified in this section via sevJOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 / 189 eral model problems. Both the MG and multigrid-preconditioned conjugate gradient (MG-PCG) methods are studied, and a point-wise Gauss-Seidel smoother with four pre- and postrelaxation sweeps is employed exclusively. It is important to note that a symmetric preconditioner is necessary for the conjugate gradient method; symmetry can be accomplished by choosing the postsmoother to be the transpose of the presmoother (line 5 of Algorithm 1). A Jacobi iteration has this property if used for pre- and postsmoothing; use of forward Gauss-Seidel for presmoothing and backward Gauss-Seidel for postsmoothing as is done in this study also provides symmetry. The first model problem is designed to provide performance baselines for well-conditioned, linearly elastic problems and allows comparison with effective sparse matrix factorization techniques. Although the model geometry at the finest level is regular, unnested mesh sequences are used at the coarser levels to illustrate the applicability of the proposed definitions of R and T. Following this, the effect of localized mesh refinement is examined through two different models, one discretized with quadratic hexahedra and the other with quadratic tetrahedra. These models demonstrate the generality of the proposed definitions of R and T, and show that MG and MG-PCG suffer no performance losses when mesh sequences consisting of a locally refined fine mesh and regular coarse meshes are employed. The final model problem studies spatially varying material properties and the resulting effects on convergence. It is demonstrated that MG-PCG performs quite well even when sharp material boundaries not captured explicitly by the coarse mesh are present. TABLE 1. Number of Elements Sequence (1) Sequence 1 Sequence 2 Sequence 3 Sequence 4 Sequence 5 Sequence 6 Sequence 7 Sequence 8 Sequence 9 Sequence 10 Thick-Plate Model Problem — Baseline Performance Studies Prior research has shown that achieving a linear relationship between the number of floating point operations (Nops) and the number of unknowns (n) requires adherence to several geometric and algorithmic criteria (Parsons and Hall 1990a). When using sequences of unstructured, unnested meshes such criteria are not easily satisfied; for this reason, the relationship between Nops and n is studied numerically using different discretizations of the domain shown in Fig. 4. Table 1 defines the sequences of meshes used to study convergence. Uniform meshes with reasonable element aspect ratios are used to minimize numerical effects on convergence. The modulus of elasticity, E, is fixed at 100, and = 0.20 for all cases. Note that the number of meshes, m, is fixed at three for each sequence, that the coarsest mesh is identical for each sequence of meshes, and that the level of discretization of the intermediate mesh is approximately midway between the finest and coarsest meshes. Although the discretizations are regular, the mesh sequences are unnested, and general definitions of FIG. 4. Thick Plate Model and Typical Mesh Sequence 190 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 Sequences of Meshes Used to Study Convergence Sequence 11 Mesh (2) nx (3) ny (4) nz (5) n (6) coarse medium fine coarse medium fine coarse medium fine coarse medium fine coarse medium fine coarse medium fine coarse medium fine coarse medium fine coarse medium fine coarse medium fine coarse medium fine 3 4 6 3 5 8 3 6 10 3 8 13 3 10 16 3 11 18 3 13 22 3 15 25 3 17 30 3 19 35 3 19 42 3 4 6 3 5 8 3 6 10 3 8 13 3 10 16 3 11 18 3 13 22 3 15 25 3 17 30 3 19 35 3 19 42 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 288 735 2,037 288 1,080 3,429 288 1,491 5,181 288 2,511 8,484 288 3,795 12,597 288 4,536 15,789 288 6,216 23,253 288 8,160 29,796 288 10,368 42,501 288 12,840 57,456 288 12,840 82,173 the interpolation and restriction operators are required. No attempt has been made to optimize the mesh sequences. Coarsegrid solutions are achieved using SuperLU (Demmel et al. 1997; Li 1996), a state-of-the-art sparse direct solver. Fig. 5 shows the performance of MG and MG-PCG. All runs were done on a Sun UltraSpare 1/200 with 256 MB of RAM, and the solution was completed using only core memory. The number of MG or MG-PCG iterations required to achieve convergence was taken as the first iteration where ε= 㛳r㛳2 ⱕ 10⫺05 㛳P㛳2 (10) P is defined as the vector of applied nodal forces and r is the vector of fine-mesh residual forces. For comparison, the performance of a sparse direct solver is also shown in Fig. 5. SuperLU (Demmel et al. 1997) with minimum-degree reordering was employed, but no pivoting was used as it is not required for well-conditioned symmetric positive definite systems. Memory requirements pose a severe constraint on a direct solver: the problems defined in sequences 6 – 11 cannot be factored in core. The following points can be observed from Fig. 5: 1. MG-PCG clearly outperforms MG for all levels of discretization, and exhibits a nearly linear increase in solution time as the mesh is refined as shown in the bottom diagram of Fig. 5. 2. The amount of computational effort required to build the auxiliary data structures, which include R, T, and the coarse mesh stiffness matrices, increases linearly with n, and requires about half the computational efforts required to assemble K. are the same as those in sequence 6, and are shown in Fig. 6(a). The amount of time required to achieve a solution using MG and MG-PCG methods was 272 seconds and 185 seconds, respectively. Comparing these values with the total solution times in Fig. 5 indicates that there is effectively no loss in efficiency for MG-PCG, and in this case there is an improvement for the MG method, when compared to regular mesh solutions with the same number of unknowns. We note here that regular coarse meshes were used to achieve a solution on an irregularly refined mesh with no difficulty. The second model problem is an equilateral triangular plate discretized as shown in Fig. 6(b). All the meshes in the multigrid hierarchy are irregular (generated using the automatic mesh generator QMG (Mitchell and Vavasis 1992). The boundary conditions consist of vertical supports on one edge and at the opposite vertex, and horizontal supports sufficient to prevent rigid body motion. A point load is applied near the center of the plate where the mesh is refined. The total time required to solve this problem using the MG-PCG method was approximately 202 seconds; a comparison with Fig. 5 indicates that this is approximately three-quarters the time required for a thick rectangular plate model with the same number of degrees of freedom. No generalizations can be made, however — due to the differences in element type, problem, and mesh sequence — between the rectangular and triangular plate models. Performance on Mesh with Spatially Varying Material Properties The proposed method of dealing with spatially varying material properties will be illustrated with a model problem. Recall that material properties are defined element-by-element on the fine mesh, and this information is transferred to coarser meshes only at the coarse-mesh element integration points. This presents a possible source of error when evaluating the coarse-mesh element stiffness matrices, k, by the usual Gaussian integration of FIG. 5. Relative Performance of MG, MG-PCG, and Sparse Direct Solver k= 冕 BTDB dV (11) V 3. MG suffers fairly severe performance losses as the mesh is refined, with a superlinear relationship between solution time and n. The comparison of total solution times shown in Fig. 5 in> 3,000, MG-PCG outperforms direct facdicates that for n ⬇ torization; MG-PCG also requires far less memory. Note that the performance of the direct solver is affected by node ordering. For example, the worst-case scenario for a direct solver would be a cube meshed with equal numbers of elements in all directions, while the meshes used here have significantly fewer elements through the thickness of the plate and optimally numbered nodes. Performance on Locally Refined Unstructured Meshes In order to verify that the proposed intergrid strategies are effective on general unstructured meshes, we examine two model problems in this section. The first is the previously analyzed thick plate, but the fine mesh is refined over the center 750 mm by 750 mm region as shown in Fig. 6(a). The total number of elements is 960, and n = 14,949, which nearly equal the values for the fine mesh in sequence 6 of Table 1. The coarse and medium meshes used in the refined mesh solution where B = matrix of linear operators, and D = constitutive matrix. Gaussian integration is exact for polynomials when an appropriate number of integrations points is used. In this study, 3 ⫻ 3 ⫻ 3 Gaussian integration is used to evaluate (11), which is exact for a quadratic, hexahedral element with uniform, linearly elastic material properties. If a coarse-mesh element covers a region with more than one applicable constitutive law, a sharp ‘‘jump’’ in D may occur, possibly introducing significant error in the evaluation of k. This is studied with the following model problems, which has been purposely constructed to introduce significant errors due to sharp material boundaries. Consider the mesh shown in Fig. 7, which follows the same pattern as the refined mesh presented in the previous section, but with fewer elements. The center 4 ⫻ 4 ⫻ 2 block of rectangular elements is assumed to have a modulus of elasticity, Ec, differing from the remainder of the domain, where E = 100 N/mm2. To solve the problem using MG or MG-PCG, a single coarse mesh consisting of nine elements is used. Fig. 7 shows that if the regular, nine element coarse mesh presented previously (denoted by ᏹc) is used, only three of the 27 center element integration points lie within the center 750 mm ⫻ 750 mm refined region of the fine mesh, which can be expected to introduce significant error in the evaluation of k. However, if the coarse-mesh element boundaries coincide with the material JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 / 191 FIG. 6. Mesh Sequences for Refined Rectangular and Triangular Plates property boundary, the evaluation of k is exact (see Fig. 7); this mesh is denoted by ᏹe. Fig. 8 shows the effect of using ᏹe as opposed to ᏹc. Note that no results are presented for MG with ᏹc and Ec /E > 20, as it diverges (convergence can be achieved for Ec /E > 20, but only by increasing the number of Gauss-Seidel smoothing operations within each MG cycle). The following observations are based on the results presented in Fig. 8: but MG-PCG remained stable and efficient. However, care should be taken when developing coarse meshes where sharp discontinuities in material properties exist; alternatively, more accurate integration techniques could be employed in the evaluation of (11) in these regions. Forming Kc * (5) in lieu of Kc may be advantageous when sharp material discontinuities exist. MODELING NODAL CONTACT CONSTRAINTS • When using ᏹe, the solution times for both MG and MGPCG are essentially constant, verifying that the only source of error is the variation in the coarse-mesh element boundaries. • MG-PCG appears to be stable even if significant errors are introduced by the use of ᏹc, and the increase in solution time is only doubled for Ec /E = 100. • The performance of both MG and MG-PCG is improved by the use of ᏹe as opposed to ᏹc, indicating that errors in integrating the coarse-mesh stiffness can have detrimental effects on solution efficiency. This study’s proposed approach for handling spatially varying materials appears to be viable when MG-PCG solution methods are used and MG, even when nonconvergent itself, is still a very effective preconditioner. A similar conclusion was drawn in Ashby and Falgout (1996) for finite difference simulations of ground-water flow, where increasing degrees of subsurface heterogeneity resulted in nonconvergence of MG, 192 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 Of particular interest in the study of rigid pavement systems is the loss of contact between the slab and subgrade that may occur due to a temperature differential through the slab thickness or under the action of wheel loads near the joint, as described previously (see Fig. 1). In the present study, this condition is modeled assuming frictionless nodal contact, which requires the imposition of nodal inequality constraints. In this section, the formulation of the contact problem is developed, and a solution technique based on Uzawa’s method, coupled with an inner MG-PCG solver, is proposed. Formulation of Nonlinear Nodal Contact Problem For frictionless nodal contact problems, inequality constraints are required. To illustrate, consider the following figure depicting a portion of a larger finite-element mesh. In Fig. 9, two surfaces are separated by a scalar distance ck; the outward normal for surface 1 is given by n. Mathematically the constraint can be expressed as FIG. 9. Nodal Contact s.t. G kTU = ck (15) where ␦U = displacement increment to be solved for, and r k = current residual. The constraint matrix, Gk, and ck are superscripted as they change at each iteration when constraints are updated. Recognizing that the current solution vector, U, may be written k FIG. 7. erties Meshes Used to Study Spatially Varying Material Prop- U = ␦U0 ⫹ ␦U1 ⫹ ⭈ ⭈ ⭈ ⫹ ␦Uk (16) the constraints may be expressed as G kT(Uk⫺1 ⫹ ␦Uk) = c (17) G kT␦Uk = c*k (18) c*k = ck ⫺ G kTUk⫺1 (19) or where The constrained system given by (14) and (15) may now be rewritten with the constraints expressed in terms of the displacement increment. Enforcement of the constraints with Lagrange multipliers leads to 冋 FIG. 8. Spatially Varying Material Properties and Element Integration Error (di ⫺ dj) ⭈ n ⱕ cij (12) where di and dj are nodal displacement vectors. Eq. (12) merely states that surface 1 may not interpenetrate surface 2 at the location of nodes i and j. Inequality constraints require that appropriate criteria for maintaining/releasing constraints be employed. In the case of frictionless contact constraints, the criteria are straight-forward: for a pair of currently unconstrained nodes, if (12) is false, the nodes are constrained; for a pair of constrained nodes, they must be released when n ⭈ ( ⭈ n) > 0.0 (13) where = average of stress tensors at nodes i and j. In the present study, Lagrange multipliers are used to enforce the constraints in lieu of a penalty parameter approach, which causes difficulties for iterative solution techniques due to the resultant poor conditioning of the system stiffness matrix (Bathe 1996). When constraints are imposed and Newton’s method is used to solve the nonlinear system, solution of the following equation is required at any iteration k: K␦U k = r k (14) K G kT 册冋 册 冋 册 Gk 0 ␦Uk ␦k = rk c*k (20) which is expressed in terms of both incremental displacements and Lagrange multipliers. The solution of (20) for ␦Uk and ␦k is analogous to the solution of a linear system within a Newton iteration. Multigrid Preconditioned-CG Methods for Constrained Systems The multigrid methods developed in this study may not be used to solve (20) directly because the coefficient matrix is not positive definite. Further, K itself is singular, since the only vertical support provided to the slab is through contact with the upper base layer and methods based on the Schur complement of K, GTK⫺1G, are not feasible. We use an adaptation of Uzawa’s method as detailed in Zienkiewicz et al. (1985) requiring that (20) be modified by the following perturbation: 冋 K ⫹ K⬘ GkT 册冋 册 冋 Gk 0 ␦Uk ␦k = 册 P ⫹ Gkc*k c*k (21) where K⬘ = GkGkT and K ⫹ K⬘ is nonsingular. Note that once the constraints are satisfied, the perturbation does not modify the original system. The scalar perturbation parameter, , is mathematically equivalent to a penalty parameter, and JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 / 193 may be interpreted as a spring stiffness. However, it is typically chosen to be a relatively small value so as not to cause numerical difficulties. The perturbed system is then solved iteratively as follows: y = GkT␦Uk ⫺ c*k (22) ␦ = ␦ ⫹ y (23) ␦Uk = (K ⫹ K ⬘)⫺1((P ⫹ Gkc*k) ⫺ Gk␦) (24) For convergence, must be sufficiently small (Zienkiewicz et al. 1985); numerical studies indicate that if is the same order as avg(Kii), convergence is not impaired and is fairly rapid. More complex perturbations can be employed in an attempt to speed convergence, but in general the simple approach presented is sufficient. In this study, MG-PCG methods are used in the solution of (24). Since constraint imposition and updating are done only for the fine mesh where a solution is achieved, this requires that K⬘ be restricted from the fine mesh to all coarser meshes. This poses a difficulty: since the constrained fine mesh nodes do not necessarily coincide with the coarse-mesh nodes, a finemesh nodal contribution to the perturbation cannot be merely added to the coarse-mesh stiffness. This may be overcome by forming the coarse-mesh perturbations as Ktc = Kc ⫹ RK⬘R T (25) where Ktc refers to the total coarse mesh stiffness and Kc is assembled from the coarse-mesh stiffness matrices, as detailed previously. The term RK⬘RT represents the coarse-mesh perturbation that is consistent with the virtual work requirement that the fine- and coarse-mesh strain energies be equal; this term will be denoted as K⬘c for the remainder of this discussion. To simplify the calculation of K⬘c, it is computed separately for each pair of constrained nodes. Refer to Fig. 9, which shows the fine-mesh nodes i and j as being constrained; the nodal contribution to the fine-mesh perturbation is represented by a spring, whose stiffness may be expressed as (n 嘸 n) (26) Using the searching algorithm detailed previously, the coarsemesh elements within which these fine nodes lie may easily be determined. Defining N i and N j as the usual arrays of shape functions of the coarse-mesh elements that the fine-mesh nodes i and j lie in, the coarse-mesh perturbation corresponding to the constraint at nodes i and j is easily determined using the principle of virtual work as k⬘c = 冋 册 N Ti ⫺N Tj (n 嘸 n)[Ni ⫺ Nj] (27) Adding the contribution of each k⬘c gives K⬘c. At this point, it is worth detailing the complete nonlinear solution algorithm. The rigid pavement systems of interest in this study will be modeled with linearly elastic materials, and the only source of nonlinearity will be the nodal inequality constraints required to model slab lift-off. The solution technique used in this study (Algorithm 2) employs three levels of iteration: (1) at the outer level, the nodal constraints and tangent system stiffness matrices (appropriate for use in a Newton iteration) are updated; (2) the constrained problem is solved using Uzawa’s method as detailed previously; and (3) the innermost kernel is the solution of the resultant linear system. RIGID PAVEMENT MODEL PROBLEM In this section, a typical model of a rigid pavement system will be solved to illustrate the applicability of the proposed multigrid methods to nonlinear problems involving multiple element types and nodal contact. Such simulations are of great practical interest to pavement designers and researchers, and have been the subject of recent studies; see Channakeshava et al. (1993) and Kuo et al. (1996) for example. Presently, the writers are developing finite-element tools for the modeling of rigid pavement systems. The multigrid methods presented in this study have allowed us to model realistic design conditions and explore the effect of various parameters on the response of pavement systems. Model Description ALGORITHM 2. Solution of Constrained System 194 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 The system to be modeled is typical, consisting of two doweled, 254 mm thick rigid pavement slabs resting on two 300 mm thick base layers above the natural subgrade — similar to the system illustrated in Fig. 1. The slabs are 3350 mm long ⫻ 3660 mm wide and are separated by a transverse construction joint skewed at an angle of 10⬚. Three 485 mm long, 32 mm diameter dowels spaced at 300 mm are located in each wheelpath at the midthickness of the slabs. The slabs and base layers are meshed with 20-noded quadratic brick elements. The natural soil below the solid elements is treated as a Winkler foundation (bed of springs) and is meshed with 8-noded quadratic elements that share nodes with the bottom faces of the quadratic hexahedra discretizing the lower base layer. The dowels are modeled using an embedded bending element that permits the discretization of doweled slabs without requiring the beam nodes to coincide with those of the solid elements (Davids and Turkiyyah 1997). We note that the inclusion of nodes having rotational degrees of freedom poses no difficulty for the intergrid transfer of information using the element-based methods developed in this study; in fact, the embedded bending element was originally developed to permit the generation of unnested sequences of regular meshes of doweled slab systems. The number of degrees of freedom of a given node indicates whether it lies within a solid or a bending element, and the evaluation of element shape functions is straightforward in either case. The boundary conditions are the minimum necessary to prevent rigid body motion. Surface plots of the meshes used in the multigrid solution are shown in Fig. 10; note the dowel elements shown as thick lines at the skewed joint. The numbers of solid elements in the coarse, medium, and fine models are 36, 216, and 1,152, respectively; the corresponding numbers of degrees of freedom are 1,314, 4,929, and 21,285. The slab and base layers are treated as linearly elastic with material properties as given in Table 2, where ␣ is the coefficient of thermal expansion and is the mass density. The values of E for the top and bottom base layers represent typical values for a cement-treated soil and a strong gravel, respectively (Darter et al. 1995). The Winkler foundation is assumed to have a modulus of 0.06 MPa/mm, typical for a natural soil deposit. A sharp material boundary between the two base layers exists and is not captured by the coarsest mesh. The left-most slab is subjected to a single axle load of 80 kN near the joint and centered transversely on the slab with a wheel spacing of 1830 mm; each wheel load is distributed over a 500 mm ⫻ 250 mm area. In addition, the slabs are subjected to a linear temperature gradient through their thickness corresponding to a ⫺5⬚C temperature change at the top of the slab and a ⫹5⬚C temperature change at the bottom. This temperature differential required that the loss of contact between the unbonded upper base layer and the slab be modeled using the nodal contact approach presented in the previous sections. This dictated that the slab and upper base layers be meshed FIG. 11. Appropriate Details for Nodal Contact Modeling FIG. 12. Displaced Shape of Rigid Pavement System independently; details near the transverse joint are depicted in Fig. 11. The value of required for the Uzawa iteration was computed at run-time as the average value for all constrained nodes of Kii corresponding to the constraint direction, n. To illustrate the analysis results, a 3D perspective of the displaced shape is shown in Fig. 12; note the slab lifting off the base layer due to the temperature gradient. Performance of Solver The iterative nonlinear solution strategy presented previously was used to solve the model problem. Both the global convergence tolerance, ε, and the constraint tolerance, , were fixed at 10⫺05. The solution required 12 global iterations and 2,731 seconds of CPU time to solve on a Sun UltraSparc 1/200 workstation, and was achieved using only core memory. Table 3 gives a breakdown of the solution components. Note that the total number of Uzawa iterations required to achieve conTABLE 3. sults FIG. 10. Mesh Sequence Used for Rigid Pavement Model Problem TABLE 2. Material Properties of Slab and Subgrade Portion of system (1) Slabs Upper base layer Lower base layer E MPa (2) (3) ␣ ⴗC⫺1 (4) kg/m3 (5) 28,000 3,500 150 0.25 0.20 0.20 1.1 ⫻ 10⫺05 — — 2,400 — — Breakdown of RIgid Pavement Model Solution Re- Global iteration (1) Uzawa iterations (2) Total MG-PCG iterations (3) 1 2 3 4 5 6 7 8 9 10 11 12 4 3 3 3 3 3 3 3 2 2 2 2 12 30 30 29 32 30 27 29 22 22 22 14 Number of constructions updated (4) 264 155 70 89 61 40 29 23 19 15 3 0 Residual (5) 8.22 7.32 6.71 6.50 5.13 4.44 3.15 2.41 1.58 7.80 2.35 1.11 ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ 10⫺03 10⫺03 10⫺03 10⫺03 10⫺03 10⫺03 10⫺03 10⫺03 10⫺03 10⫺04 10⫺04 10⫺07 JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 / 195 vergence for the constrained system never exceeds four, and averages about three. While at each Uzawa iteration the MGPCG method is used to solve the resulting linear system, the number of MG-PCG iterations per linear solve varies significantly, with the largest number required for the first solution. To put these results in perspective, comparisons were made with the solution when a sparse direct solver is used in lieu of MG-PCG. This was done only for the medium mesh (4923 DOF), as a fine mesh solution would require far too much memory with a direct solver. Note that when using a direct solver, K may be factored outside the Uzawa iteration, and each linear solve then becomes a back substitution. This advantage for the direct solver is offset by the fact that the auxiliary data structures required for the MG-PCG method need be constructed only once. In fact, even on the medium mesh using a two mesh sequence, the total solution time for MGPCG is 407 seconds, less than the 503 seconds required when a sparse solver is used. Further, the MG-PCG solver required 21 MB of RAM, while the sparse direct solver required about 85 MB (with a minimum degree reordering). CONCLUSIONS Multigrid (MG) and multigrid-preconditioned conjugate gradient (MG-PCG) methods applicable to large-scale unstructured 3D finite-element models of structures have been developed as part of an effort to simulate efficiently the structural response of rigid pavement systems. These methods are quite general, relying on definitions for the interpolation and restriction operators that allow multiple element-types and the meshing of bending members with solid elements within a single model. Spatially varying material properties are handled in a manner that relies on principles similar to those used to develop the restriction and interpolation operators. The MG-PCG method has been implemented in a general nonlinear solution strategy, permitting the solution of contact problems common in rigid pavement systems and other structures. A model problem of a rigid pavement system was solved that incorporated multiple element types, spatially varying materials, and nodal contact conditions to illustrate the generality of the proposed multigrid methods. Baseline performance studies were conducted, providing comparisons between MG and MG-PCG methods versus stateof-the-art sparse direct solution methods. The MG-PCG method was shown, as expected, to be significantly more efficient than direct solution methods for large problems, and exhibits a nearly linear increase in solution time with the number of unknowns. Memory requirements beyond the storage of the stiffness matrix (or the element stiffness matrices) are small. The effectiveness of the proposed intergrid transfer techniques was demonstrated in the presence of mesh irregularities, local refinement, spatially varying material properties, and sharp material boundaries that may not be captured by all grids in the multigrid hierarchy. Further, the ability to precondition problems involving inequality constraints via restriction of the required fine-mesh perturbation to unnested coarse meshes was shown to be effective in the general nonlinear solution strategy. Finally, we note that the MG-PCG method has the potential for being parallelized efficiently, perhaps with a block Jacobi smoother, a characteristic that is becoming increasingly important with the availability of relatively inexpensive distributed-memory high-performance computers. ACKNOWLEDGMENTS This work was supported in part by WSDOT under grant T9903-54 and by fellowships from the Valle Scholarship and Scandinavian 196 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 1999 Exchange Program and the Osberg Family Trust at the University of Washington. APPENDIX. REFERENCES Ashby, S., and Falgout, R. (1996). ‘‘A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations.’’ Nuclear Sci. and Engrg., 124, 145 – 159. Bathe, K.-J. (1996). Finite element procedures. Prentice-Hall, Englewood Cliffs, N.J. Brandt, A. (1977). Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation, 31(138), 333 – 390. Channakeshava, C., Barzegar, F., and Voyiadjis, G. (1993). ‘‘Nonlinear FE analysis of plain concrete pavement with doweled joints.’’ J. Transp. Engrg., 119(5), 763 – 781. Darter, M., Hall, K., and Kuo, C.-M. (1995). ‘‘Support under Portland cement concrete pavements.’’ NCHRP report No. 372 National Cooperative Highway Research Program, Washington, D.C. Davids, W., and Turkiyyah, G. (1997). ‘‘Development of embedded bending member to model dowel action.’’ J. Struct. Engrg., 123(10), 1312 – 1320. Demmel, J. W., Gilbert, J. R., and Li, X. S. (1997). SuperLU users’ guide. Comp. Sci. Div., University of California, Berkeley, Berkeley, Calif. Dracopoulos, M., and Crisfield, M. (1995). ‘‘Coarse/fine mesh preconditioners for the iterative solution of finite element problems.’’ Int. J. Numer. Methods in Engrg., 38, 3297 – 3313. Farhat, C., and Sobh, N. (1989). ‘‘A coarse/fine mesh preconditioner for very ill-conditioned finite element problems.’’ Int. J. Numer. Methods in Engrg., 28, 1715 – 1723. Fish, J., Pan, L., Belsky, V., and Gomaa, S. (1996). ‘‘Unstructured multigrid method for shells.’’ Int. J. Numer. Methods in Engrg., 39, 1181 – 1197. Goodman, J. E., and O’Rourke, J. (1997). Handbook of discrete and computational geometry. CRC, Boca Raton, Fla. Jesperson, D. (1984). ‘‘Multigrid methods for partial differential equations.’’ MAA studies in mathematics, G. Golub, ed., Vol. 24, Mathematical Assoc. of America, Washington, D.C., 270 – 317. Kǒcvara, M. (1993). ‘‘An adaptive multigrid technique for three-dimensional elasticity.’’ Int. J. Numer. Methods in Engrg., 36, 1703 – 1716. Kuo, C.-M., Hall, K., and Darter, M. (1996). ‘‘Three-dimensional finite element model for analysis of concrete pavement support.’’ Transp. Res. Rec. 1505, TRB, National Research Council, Washington, D.C., 119 – 127. Li, X. S. (1996). Sparse Gaussian elimination on high performance computers. Comp. Sci. Div., University of California, Berkeley, Berkeley, Calif. Löhner, R., and Morgan, K. (1987). ‘‘An unstructured multigrid method for elliptic problems.’’ Int. J. Numer. Methods in Engrg., 24, 101 – 115. Meza, J., and Tuminaro, R. (1996). ‘‘A multigrid preconditioner for the semiconductor equations.’’ SIAM J., 17(1), 118 – 132. Mitchell, S., and Vavasis, S. (1992). ‘‘Quality mesh generation in three dimensions.’’ Proc., ACM Computational Geometry Conf., Assoc. for Computing Machinery, New York, 212 – 221. Muttin, F., and Chenot, J.-L. (1995). ‘‘On a conjugate-gradient two-grid method for three-dimensional elasticity.’’ Engrg. Computations, 12, 3 – 20. Oosterlee, C., and Washio, Y. (1998). An evaluation of parallel multigrid as a solver and a preconditioner for singularly perturbed equations. SIAM J., 19(1), 87 – 110. Parsons, I., and Hall, J. (1990a). ‘‘The multigrid method in solid mechanics: Part I — algorithm description and behavior.’’ Int. J. Numer. Methods in Engrg., 29, 719 – 737. Parsons, I., and Hall, J. (1990b). ‘‘The multigrid method in solid mechanics: Part II — practical applications.’’ Int. J. Numer. Methods in Engrg., 29, 739 – 753. Rider, W., and Knoll, D. (1997). ‘‘Solving nonlinear heat conduction problems with multigrid preconditioned Newton-Krylov methods.’’ Tech. Rep. LA-UR-97-2929, Los Alamos National Lab, Los Alamos, N.M. Saad, Y. (1996). Iterative methods for sparse linear systems. PWS Publishing, Boston. Zienkiewicz, O., and Taylor, R. (1994). The finite element method — volume I. McGraw-Hill, New York. Zienkiewicz, O., Vilotte, J., Toyoshima, S., and Nakazawa, S. (1985). ‘‘Iterative methods for constrained and mixed approximation. An Inexpensive Improvement of FEM Performance.’’ Computational Methods of Appl. Mech. and Engrg., 51, 3 – 29. EFFECT OF DOWEL LOOSENESS ON RESPONSE CONCRETE PAVEMENTS OF JOINTED By William G. Davids,1 Member, ASCE (Reviewed by the Highway Division) ABSTRACT: This paper examines the effect of dowel looseness on the structural response of jointed concrete pavements. A technique for modeling dowels in 3D finite-element analyses of rigid pavement systems is presented that relies on an embedded formulation of a quadratic beam element. This embedded element permits the efficient modeling of dowel looseness using a nodal contact approach and allows the dowels to be exactly located irrespective of the slab mesh lines. The embedded bending element is extended to include a general bond-slip law, and the practical case of a Winkler foundation sandwiched between the dowel and slab is implemented. The results of parametric studies examining the significance of dowel looseness on the response of rigid jointed pavements to both axle and combined axle and thermal loadings are presented. These studies indicate that significant increases in both slab and soil stresses can be expected due to small gaps (#0.24 mm) between the dowels and the slabs. The importance of explicitly modeling nonlinear load transfer arising from dowel looseness is also examined, and it is shown that equivalent, back-calculated dowel support moduli should be used with caution when dowel looseness exists. INTRODUCTION The realistic modeling of dowel load transfer across rigid pavement joints is necessary for the accurate prediction of pavement response to applied axle and thermal loadings. The accelerated pavement distress and joint faulting commonly attributed to poor joint load transfer (Ioannides et al. 1992; Huang 1993) underscores this need. Unfortunately, the complexities inherent in the mechanics of dowel-slab interaction make achieving this requirement difficult. In particular, dowel looseness—resulting from poor construction practices or cumulative damage to the slab concrete—has been shown to significantly reduce dowel load transfers (Snyder 1989; Buch and Zollinger 1996) and should not be neglected in simulations of rigid pavement response. Most investigators have modeled dowel-slab interaction by assuming a Winkler foundation sandwiched between the dowel and surrounding slab; the Winkler foundation constant is typically referred to as the ‘‘modulus of dowel support.’’ Indeed, this fundamental approach was in use prior to the development of finite-element analysis, with early models based on the explicit solution for an infinitely long beam resting on an elastic foundation (Friberg 1940). Investigators employing 2D finiteelement programs that rely on medium-thick plate elements to discretize the slabs have typically modeled dowel bars as discrete beam elements spanning between adjacent slabs (Tabatabaie and Barenberg 1980; Tayabji and Colley 1986). To account for the effect of dowel-slab interaction, springs are placed between the dowels and the slabs; the spring constants are derived by considering the embedded portion of the dowel as an infinitely long beam on a Winkler foundation. Guo et al. (1995) critically examined this approach and developed a component dowel bar element that couples the embedded portion of the dowel—modeled as a finite length beam on elastic foundation—with a shear beam spanning the joint. 1 Asst. Prof., Dept. of Civ. and Envir. Engrg., Univ. of Maine, 5711 Boardman Hall, Orono, ME 04469-5711. E-mail: wdavids@umeciv. maine.edu Note. Discussion open until July 1, 2000. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 1, 1998. This paper is part of the Journal of Transportation Engineering, Vol. 126, No. 1, January/February, 2000. qASCE, ISSN 0733-947X/00/0001-0050–0057/$8.00 1 $.50 per page. Paper No. 19745. Also using 2D finite-element analysis, Ioannides and Korovesis (1992) developed a technique where a uniform joint stiffness is used to replace the individual dowels. The joint stiffness is a function of the effective modulus of dowel support, which can be back-calculated from measured load transfer efficiencies. However, such a back-calculated value is effectively a secant stiffness, strictly valid for one loading, geometry, and set of material properties. One study employing dynamic 2D finite-element analysis explicitly considered dowel looseness (Zaman and Alvappillai 1995), and small gaps (#0.2 mm) were found to have a large effect on pavement response. In recent years, nonlinear 3D finite-element modeling of rigid pavements has become more common in research settings (Channakeshava et al. 1993; Zaghloul et al. 1994; Kuo et al. 1996; Davids et al. 1998a). However, even though most investigators have modeled dowels with beam elements meshed within the solid slab, dowel-slab interaction has generally not been addressed. One notable exception is the study of Channakeshava et al. (1993), where the interaction between the dowels and the concrete was modeled with discrete nonlinear springs connecting the ends of the dowels to the slabs. The nonlinear spring force-displacement relations were determined from separate finite-element models of a single dowel bar embedded in a portion of the slab, allowing the effects of local stress concentrations and concrete nonlinearities around the dowel to be incorporated. Hammons (1997) employed the effective joint stiffness concept of Ioannides and Korovesis (1992) in 3D finite-element models of rigid pavements, alleviating the difficulty of meshing bending and solid elements. However, a uniform dowel spacing is implicit in this approach, which is common for newly constructed pavements but rarely the case for retrofitted joints where dowels are typically located only in the wheelpaths (Hall et al. 1993). This brief review of dowel modeling techniques indicates that even though dowel looseness has been explicitly treated by Zaman and Alvappillai (1995), it has not been considered in prior research efforts employing 3D analysis. This study attempts to address the need for a more complete treatment of dowel-slab interactions in 3D finite-element analyses of jointed concrete pavements with an emphasis on the effect of gaps between the dowels and slabs on pavement response. Following an overview of the 3D finite-element models and solution strategies used in this study, a recently developed technique for the finite-element modeling of dowel load transfer is briefly presented that relies on an embedded formulation 50 / JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 of a quadratic beam element (Davids and Turkiyyah 1997) and allows the efficient and rigorous consideration of dowel looseness. The element formulation is then extended to permit the inclusion of a nonlinear, 3D bond-slip law between the dowels and the slabs. The special case of a Winkler foundation sandwiched between the dowels and slabs is implemented to permit explicit comparisons between this traditional approach to modeling dowel-slab interaction and the rigorous consideration of dowel looseness. Parametric studies on the response of a typical, dowel-retrofitted pavement system subjected to axle and combined axle and thermal loadings are conducted to examine the effect of dowel looseness on system response. The importance of explicitly modeling dowel looseness as opposed to using an equivalent modulus of dowel reaction is also examined, and conclusions and recommendations for future research are addressed. MODELING TECHNIQUES AND SOLUTION STRATEGY FIG. 2. Embedded Dowel Element EverFE also incorporates a highly efficient multigrid-preconditioned conjugate gradient solver that permits the solution of large, nonlinear 3D finite-element models such as that of Fig. 1 on desktop computers (Davids and Turkiyyah 1999). DOWEL MODELING The finite-element models employed in this study are similar to that shown in Fig. 1. The slabs and base are linearly elastic and are discretized with 20-noded quadratic hexahedra (Zienkiewicz and Taylor 1994). A Winkler foundation is used to represent the subgrade below the bottommost base layer and is incorporated in the model with an eight-noded quadratic interface element that displaces compatibly with the quadratic hexahedra above it. Loss of contact between the slabs and upper base layer is captured using a frictionless nodal contact approach where neither tension nor interpenetration is permitted at the slab/base interface (Fig. 1). The model boundary conditions are the minimum required to prevent rigid-body displacement and rotation of the slabs and base in the horizontal plane; vertical support is provided through the Winkler foundation. The basic models are generated with EverFE, a recently developed user-friendly finite-element analysis tool for rigid pavement analysis. Details and features of EverFE may be found in Davids et al. (1998a); verification of the finite-element modeling techniques through comparison with experimental results has been presented in Davids et al. (1998b). The dowel modeling technique used in this study relies on an embedded finite-element formulation for the dowels having the following features: • The dowels can be located exactly without regard to meshing of the slabs as shown in Fig. 2. • The dowels can be debonded relative to the slabs. • Gaps between the dowels and surrounding slabs (dowel looseness) can be explicitly modeled using a nodal contact approach. • In lieu of explicitly modeling dowel looseness, a Winkler foundation can be specified between the dowel and surrounding slabs to model dowel-slab interaction. Three-noded, 18-degree-of-freedom quadratic beam elements are used to represent the portions of the dowel embedded in the slabs, which ensures that the beam displaces compatibly with the quadratic solid elements used to discretize the slab. A conventional two-noded shear beam is used to span the joint. Embedded Formulation of Dowel Element The embedded formulation of the dowel element relies on expressing the nodal displacements of the dowel as functions of the nodal displacements of the solid element that it is embedded within, debonding conditions, and any gaps that may exist. Consider a single (unembedded) dowel element having a nodal displacement vector Ud and stiffness matrix Kd. The displacement vector Ud may be expanded to include the displacement vector of the solid element the dowel lies within, Ue Ude = FG Ud Ue (1) The new displacement vector for the embedded dowel element may be transformed back to Ud as follows: Ud = TUde (2) The transformation matrix T incorporates debonding and gap information for each dowel node. It follows from the principle of virtual work that the stiffness matrix of the embedded dowel Kde can be computed as Kde = T T Kd T FIG. 1. eling FE Mesh of Rigid Pavement System and Contact Mod- (3) Because nonlinearities arising from nodal contact and debonding information are encapsulated in the tangent stiffness matrix Kde, this formulation is particularly amenable to inclusion in a general nonlinear solver. [Full details of the element deri- JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 / 51 vation and implementation, including the constraint updating strategy that is necessary when modeling dowel looseness, can be found in Davids and Turkiyyah (1997).] where h = half the length of the dowel l. The stiffness contribution due to bond-slip Kdb is given by Inclusion of Bond-Slip Law between Dowel and Slab This section details the extension of the embedded dowel element to permit the specification of general bond-slip constitutive relations between the dowels and slabs. Using this approach, a Winkler foundation sandwiched between the dowels and slabs can be readily incorporated in finite-element models of rigid pavement systems. Consider the case where no dowel looseness exists. A general (possibly nonlinear) constitutive law relates incremental interface stress (force per unit length) between the dowel and the surrounding slab df9 with the incremental relative displacement of the dowel and the embedding element dD9 as follows: df9 = DdD9 (4) The vectors df9 and dD9 are of size 3 3 1; the prime indicates that they are defined as having components in the dowel local coordinate directions defined by the unit vectors q, s, and t (Fig. 2). For this case, constraint of the dowel exists only through stresses developed at the dowel-concrete interface, and the dowel element retains its independent displacement degrees of freedom and contributes no stiffness terms to the embedding element. However, it is convenient to consider the dowel as embedded with a corresponding displacement vector Ude [i.e., (2) is still valid]. The vector of incremental relative displacements dD may then be expressed as dD = BdUde (5) in the global Cartesian coordinates. The matrix operator B is analogous to the matrix of linear differential operators common in continuum elements and is defined as: B = [N1 0 N2 0 N3 0 2Ne] Ni = F Ni 0 0 0 Ni 0 (6) G 0 0 Ni Ni = quadratic shape function for dowel, (7) (i = 1, . . . , 3) 0 = 3 3 3 array of zeros (8) (9) Ne = usual array of embedding element shape functions (10) E 1 Kdb = BT(QTDQ)Bh dh (14) 21 which may be readily evaluated using conventional numerical integration techniques (three-point Gaussian quadrature is used here). The total dowel stiffness matrix Kdt is then Kdt = Kde 1 Kdb (15) The vector of nodal forces at the dowel-concrete interface F b is computed as E 1 Fb = BTf h dh (16) 21 For the case of a Winkler foundation sandwiched between the dowel and surrounding slabs, D reverts to a constant, diagonal 3 3 3 matrix. In the present study, the local t direction corresponds to the global z (vertical) direction of the finiteelement model. Correspondingly, if k is defined as the modulus of dowel reaction (N/mm3) and d is the dowel diameter (mm), D(3,3) = kd. A small nonzero value is used for D(1,1) to avoid rigid-body motion of the dowels, and D(2,2) is set to a large value, preventing relative horizontal displacements between the dowels and slabs. PARAMETRIC STUDY: MODEL DESCRIPTION The parametric studies employ the same basic system as shown in Fig. 3. For simplicity, the joint is not skewed, and the boundaries of the subgrade have not been extended beyond the edges of the slabs. The 150-mm-thick base layer is assumed to be a compacted gravel, below which 300 mm of natural soil are modeled as a linearly elastic continuum. A dense liquid is used to represent the remaining natural soil with a modulus of 0.054 MPa/mm. The elastic material properties of the slab and soil are given in Table 1, where a is the coefficient of thermal expansion and r is the mass density. Fig. 4 shows the layout of the 32-mm-diameter steel dowels, which is consistent with a typical retrofit design used by the Washington State Department of Transportation (WS DOT) and similar to those used by other agencies (Hall et al. 1993). The dowels are assumed to be debonded along their entire length, except at a single node as required to prevent rigidbody motion. The horizontal boundary conditions used to pre- Transforming to the dowel local coordinate system where the incremental stress-strain relations given by D are assumed to be defined may be accomplished by dD9 = QdD (11) where Q = 3 3 3 matrix whose rows are the dowel element local unit vectors q, s, and t. The stiffness contribution due to bond-slip is formulated using virtual work principles, where the incremental internal virtual work dPe is given by dPe = E FIG. 3. df9 ? dD9 dl TABLE 1. l Using (4), (5), and (11) and transforming the variable of integration to the local element coordinate h, (12) can be written as follows: e dP = dU deT SE 1 21 T T Model of Two Rigid Pavement Slabs (12) D B (Q DQ)Bh dh dU de (13) Portion of system (1) Slabs Gravel base layer Natural soil 52 / JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 Material Properties of Slab and Subgrade E (MPa) (2) n (3) a (&C21) (4) r (kg/m3) (5) 28,000 150 75 0.25 0.20 0.20 1.1 3 10205 — — 2,400 — — FIG. 4. Plan View of Slabs Showing Dowels and Wheel Loads FIG. 6. Dowel FIG. 5. Finite-Element Mesh Used in Parametric Studies (29,721 Degrees of Freedom) vent rigid-body motion of the slabs are also indicated in Fig. 4. The finite-element mesh used in the analyses is shown in Fig. 5. Three elements are used through the slab thickness, and both the base and subgrade are meshed with a single layer of elements. A single 80-kN axle load—the usual equivalent single axle load assumed in pavement design—is applied near the joint as shown in Fig. 4. The axle has dual wheels, with each wheel idealized as a uniform pressure acting over a 180-mm-long by 180-mm-wide rectangular area. The wheel spacing of the axle is typical, and the tire contact region was selected to give a realistic uniform pressure of approximately 0.62 MPa. The transverse axle position was chosen based on a design case used by the WSDOT. To study the combined effect of nighttime temperature curling and axle loading, a second load case consisting of a linear temperature gradient through the slab thickness with a temperature of 247C on top of the slab and 47C on the bottom superimposed with the axle loading is also considered. The slab self-weight is included in all of the analyses. No aggregate interlock shear transfer is assumed across the joint, which is reasonable for pavements that have been in service for a number of years and require joint retrofit. The single parameter considered in the study on dowel looseness is the value of the gap between the dowel and surrounding slab at the joint face. As discussed earlier, dowel looseness can arise from poor construction techniques (Snyder 1989) as well as damage to the surrounding concrete under cyclic loading (Buch and Zollinger 1996), and magnitudes of the gap have been measured up to 0.6 mm. Fig. 6 details the assumed gap g, which tapers parabolically from a maximum value to zero over half the embedded length of the dowel. The 12.7-mm joint opening is artificial and is used only to ensure that the dowel element spanning the joint is not so stiff that it causes numerical conditioning problems. The magnitude of the gap at the joint face is varied from 0 mm up to the value beyond Cross-Sectional Detail at Joint and Discretization of which there is no significant effect on the pavement response. The number of elements in the region of the gap is fixed at three, giving six potential points of contact—note the nodal locations shown in Fig. 6. Prior convergence studies indicate that this is sufficient refinement to give convergent displacements and dowel shears (Davids et al. 1998b). It must be noted that different shapes for the gap profile (i.e., linear, cubic, etc.) could be expected to produce different analysis results. However, assuming that the formation of the gaps is due to degradation of the concrete surrounding the dowels under cyclic loading and that this degradation is a function of the magnitude of the compressive stresses between the dowel and the slab the assumption of a parabolic gap is not unreasonable. PARAMETRIC STUDY: AXLE LOADING Displacement Response As expected, the relative vertical displacements of the two slabs near the joint grow with an increase in g. Fig. 7 shows the displaced shape of the top surface of the slabs for a gap of 0.12 mm; the vertical displacement component has been scaled by a factor 1,000. Note the larger vertical displacements of the slab on the side where the wheel loads are closer to the edge. Fig. 8 shows the variation in displacement load transfer efficiency (LTE) with the magnitude of the gap. The LTE is defined as du /dl, where du is the displacement of the loaded slab and dl is the displacement of the unloaded slab, and has been computed after subtracting the vertical displacement of the slabs due to their self-weight of 0.10 mm. Note that since the axle and dowel locations are unsymmetric, the slab displacements vary significantly across the joint; for this reason, the LTE has been measured at (x, y, z) = (3,660, 21,266.9, 2230) and (3,660, 563.1, 2230), which correspond to the fi- FIG. 7. Displaced Shape of Top of Slab for Gap of 0.12 mm— Axle Loading JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 / 53 FIG. 8. LTE versus Gap—Axle Loading by the loaded slab varies from 107 to 180 kPa as the gap varies from 0.00 to 0.12 mm—an increase of 68%. A similar drop in stresses applied to the base under the unloaded slab also occurs. It must be noted that these stress values may be somewhat unrealistic due to the possibility of soil yielding, which is not captured by the linearly elastic model. However, the fact that they increase dramatically with the presence of small values of g is significant: loss of base support and strength due to pumping action and base deterioration under high stresses is suspected of being a significant component of many joint failures (Ioannides et al. 1992). It is also illuminating to examine the principal tensile stresses in the bottom of the slab. The maximum principal stresses occur under each dual wheel load at the joint face and vary from 667 to 1,037 kPa with gaps of between 0.00 and 0.12 mm—an increase of 55%. The orientation of the maximum principal stress is in the transverse y direction. Effectively, as g grows and the joint load transfer decreases, the stresses in the loaded slab approach those due to an unprotected edge loading condition. PARAMETRIC STUDY: AXLE AND TEMPERATURE LOADING Displacement Response FIG. 9. Dowel Shear versus Gap—Axle Loading nite-element model nodes closest to the center of each dual wheel load at the joint face. The largest gap considered in the analyses is only 0.12 mm, because larger values produce no significant change in the model response. However, the reduction in LTE corresponding to g = 0.12 mm is on the average 37%—a significant decrease. As for the case of axle loading only, the relative vertical displacements of the two slabs near the joint grow with increasing g. The maximum value assumed for g was 0.24, beyond which there was not a significant change in the model response. When computing the LTE, the displacements due to temperature and self-weight along were determined from separate analyses and subtracted from the displacements due to self-weight, temperature, and axle loading. The LTE has been plotted at (x, y, z) = (3,660, 21,266.9, 2230) and (3,660, 563.1, 2230), which correspond to the nodes closest to the center of each dual wheel load (Fig. 10). Note that the maximum value of g that affects the system response is twice that for axle loading alone. This may be explained by the loss of base support near the joint due to the temperature gradient, which effectively softens the system, resulting in larger displacements for a given load. Dowel Shears The dowel shears follow much the same pattern with respect to dowel location observed for axle loading only but with increased magnitudes as shown in Fig. 11. This increase in Dowel Shears As shown in Fig. 9, decreasing LTE with gap is paralleled by decreasing dowel load transfer. The most heavily loaded dowel (at y = 610 mm, centered between two wheel loads) transfers about 9.6 kN when g = 0; when g has reached 0.12 mm, however, the same dowel has a shear of only 1.5 kN. Fig. 9 also confirms the localized nature of dowel load transfer noted by other researchers (Ioannides and Korovesis 1992; Guo et al. 1995). It is interesting to note, however, that as g grows, the individual dowel shears converge to nearly constant values ranging between about 0.7 and 2 kN. Slab and Base Layer Stresses Given that increasing g significantly decreases the dowel shears, we would expect an increase in stresses applied to the base layer under the loaded slab adjacent to the joint with a concomitant decrease in applied stresses under the unloaded slab. In fact, the maximum vertical stress applied to the base FIG. 10. 54 / JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 LTE versus Gap—Axle and Temperature Loading FIG. 11. Loading Dowel Shear versus Gap—Axle and Temperature dowel shears is due to the fact that under temperature loading, the slab and base separate and less axle load is transferred to the subgrade in the immediate vicinity of the wheels. Slab and Base Layer Stresses As g increases, the loaded slab tends to ‘‘tip’’ increasingly toward the joint due to the axle loading, resulting in different contact regions between the slab and base and larger contact stresses between the slab and base layer. In fact, the maximum stress applied to the base layer varies from 34 to 104 kPa as g increases from 0.00 and 0.24 mm, an increase of 206%. These stresses are still significantly lower than those observed for axle loading only, however. Nighttime temperature curling generally constitutes a critical load case for tensile stresses on top of the slab. Fig. 12 shows maximum principal stress on top of the slab for this condition. Note that the maximum principal stress in each slab is nearly equal when g = 0.00 mm—with good load transfer and separation of the slab and base, each slab carries a nearly equal share of the wheel loads. As the gap increases, the maximum principal tensile stress in the loaded slab increases from 890 to 943 kPa with a concomitant decrease in tensile stress in the unloaded slab from 924 to 763 kPa. These stresses are less than the maximum principal tensile stresses observed in the bottom of the loaded slab for axle loading only and exhibit relatively little variation with g. To better put these values in perspective, the maximum principal stresses in the loaded and unloaded slab for temperature loading alone are 527 and 504 kPa for g = 0.00 and g = 0.24 mm, respectively. SIGNIFICANCE OF NONLINEAR JOINT LOAD TRANSFER This section illustrates the importance of modeling nonlinearities in joint load transfer arising from dowel looseness. The same system employed in the previous parametric studies is modeled, but in lieu of gaps between the dowels and slabs, a Winkler foundation is sandwiched between the dowels and surrounding slabs. This allows a quantitative comparison of pavement response when modeling dowel-slab interaction using the conventional modulus of dowel support and explicitly modeling dowel looseness. In addition, the importance of modeling nonlinear joint load transfer is studied by examining system response under different applied load levels assuming either a fixed value of dowel looseness or an equivalent modulus of dowel support. FIG. 12. Maximum Principal Stress on Top of Slab—Axle and Temperature Loading Effect of Modulus of Dowel Support on System Response As noted by Tabatabaie-Raissi (1978), resported values for the modulus of dowel support k vary from 80 to 8,600 MPa/ mm, with a typical value being 400 MPa. To study the effect of this variation in k on model response, the model was analyzed assuming k to vary between 80 and 6,000 MPa/mm. Note that these values must be multiplied by the dowel diameter d to get a distributed stiffness per unit length along the dowel. Fig. 13(a) shows the variation in LTE with k; for comparison, the variation in LTE with g is reproduced on the same plot. Note the sharp transition in the curve over the range 200 < k < 1,200, indicating that small variations in k over this region could have relatively large effects on load transfer and hence slab and base layer stresses. Fig. 13(a) can be used to correlate dowel looseness and k for the specified pavement system and loading. For example, a gap of 0.02 mm produces the same LTE (89.2%) as a typical value of k = 400 MPa/mm. JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 / 55 FIG. 14. FIG. 13. LTE and Tensile Edge Stress versus Gap and Modulus of Dowel Support: (a) Variation in LTE with Dowel Looseness and Modulus k; (b) Variation in Tensile Edge Stress with Dowel Looseness and Support Modulus k However, this equivalence holds only for this system and load level; this issue will be explored later in this section. Fig. 13(b) portrays the change in the maximum tensile edge stresses syy at the base of the loaded slab with k and g. Note that syy is very nearly equal to the maximum principal tensile stress and is a critical design value for this loading condition. Using Fig. 13, dowel looseness can be correlated to k in terms of edge stress in the same manner as LTE. Assuming a gap of 0.02 mm, the corresponding value of syy is 740 MPa; the value of k that produces the same stress level is 1,000 MPa. Note that 1,000 MPa is 250% larger than the value of 400 MPa required to produce equivalent LTE. This is not surprising, as there is not a 1:1 correspondence between stress and LTE. However, it illustrates that if measured LTE is used to backcalculate an equivalent value of k, the stresses predicted by the model employing k may be quite different from the actual conditions if there is dowel looseness. Effect of Load Level on System Response In practice, we would like to be able to correlate field-determined response with modeling parameters to allow better simulations of pavement behavior. Perhaps the most commonly measured system response parameter is joint LTE. However, depending on the in-situ testing method used (i.e., Benkelman beam, falling weight deflectometer, etc.), the effective load level used in the field determination of LTE may be quite different from the actual service loads seen by the pavement structure. The remainder of this section examines Variation in LTE with Load Level the importance of load level and its relation to nonlinear joint load transfer through the analysis of two models subjected to a range of axle loadings between 20 and 80 kN. The first model has a fixed value of g = 0.02 mm, and the second assumes a value for k of 400 MPa/mm as back-calculated previously. Fig. 14 shows the variation in LTE with load level. For lower load levels, there is a large difference between the two models, with the dowel looseness model predicting a 16% smaller value of LTE for a load of 20 kN. The low predicted LTE at small loads for g = 0.02 mm is explained by the lack of contact between the dowels and slabs at lower load levels. The model with a constant k, on the other hand, predicts relatively constant but decreasing LTE with increased loads. The implication of this analysis is clear: A back-calculated linear spring stiffness should be used with caution when considering different loadings. As a practical case, if a modulus of dowel support k is back-calculated using data from falling weight deflectometer readings or other nondestructive testing methods, analyses employing this value of k are only strictly valid for the load level used in the nondestructive test. SUMMARY AND CONCLUSIONS This study examines the effect of dowel looseness on the response of jointed rigid pavements. A dowel modeling technique that permits the explicit modeling of dowel looseness was presented and extended to allow the sandwiching of a Winkler foundation between the dowels and slabs. Small gaps between the dowels and slabs (#0.24 mm) were shown to significantly increase principal tensile stresses in concrete pavement slabs subjected to both wheel and temperature loading, as well as to increase the vertical stresses applied to the base layer. The effect of the modulus of dowel support on pavement response was also examined. Studies comparing this traditional approach to modeling dowel-slab interaction with the explicit modeling of dowel looseness were performed. Dowel looseness and the modulus of dowel support correlated on the basis of equivalent LTE were shown to produce significantly different maximum tensile edge stresses in the same pavement system. It was also demonstrated that a modulus of dowel support back-calculated on the basis of measured LTE should be used with caution when different load levels are considered. Although the modeling techniques verified in this paper represent significant advances in quantifying dowel looseness and dowel joint load transfer, several issues need to be addressed in future research. 56 / JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 • More controlled laboratory studies incorporating highcycle loadings typical of pavements should be conducted to better quantify dowel looseness. • A mechanistic-based model for predicting the occurrence of dowel looseness, calibrated to the laboratory data, should be developed. • Field verification is necessary to provide a higher degree of confidence in the model predictions. Achieving these future research objectives would lead to more reliable and accurate quantification of dowel load transfer. Integration of a predictive model with a reliable, efficient, and user-friendly 3D finite-element package such as EverFE (Davids et al. 1998a) and its use in conjunction with appropriate nondestructive joint evaluation techniques would aid researchers, designers, and planners in determining pavement retrofit and maintenance schedules. ACKNOWLEDGMENTS The writer would like to thank Associate Prof. George Turkiyyah and Prof. Joe Mahoney of the Department of Civil and Environmental Engineering at the University of Washington for their advice. APPENDIX. REFERENCES Buch, N., and Zollinger, D. G. (1996). ‘‘Development of dowel looseness prediction model for jointed concrete pavements.’’ Transp. Res. Rec. 1525, Transportation Research Board, Washington, D.C., 21–27. Channakeshava, C., Barzegar, F., and Voyiadjis, G. (1993). ‘‘Nonlinear FE analysis of plain concrete pavement with doweled joints.’’ J. Transp. Engrg., ASCE, 119(5), 763–781. Davids, W., and Turkiyyah, G. (1997). ‘‘Development of embedded bending member to model dowel action.’’ J. Struct. Engrg., ASCE, 123(10), 1312–1320. Davids, W., and Turkiyyah, G. (1999). ‘‘Multigrid preconditioner for unstructured nonlinear 3D FE models.’’ J. Engrg. Mech., ASCE, 125(2), 186–196. Davids, W., Turkiyyah, G., and Mahoney, J. (1998a). ‘‘EverFE—Rigid pavement 3D finite element analysis tool.’’ Transp. Res. Rec. 1629, Transportation Research Board, Washington, D.C., 41–49. Davids, W., Turkiyyah, G., and Mahoney, J. (1998b). ‘‘Modeling of rigid pavements: Joint shear transfer mechanisms and finite element solution strategies.’’ Tech. Rep. WA-RD 455.1, Washington State Department of Transportation. Friberg, B. (1940). ‘‘Design of dowels in transverse joints of concrete pavements.’’ Trans. ASCE, 105, 1076–1116. Guo, H., Sherwood, J. A., and Snyder, M. B. (1995). ‘‘Component dowelbar model for load-transfer systems in PCC pavements.’’ J. Transp. Engrg., ASCE, 121(3), 289–298. Hall, K. T., Darter, M. I., and Armaghani, J. M. (1993). ‘‘Performance modeling of joint load transfer restoration.’’ Transp. Res. Rec. 1388, Transportation Research Board, Washington, D.C., 129–139. Hammons, M. I. (1997). ‘‘Development of an analysis system for discontinuities in rigid airfield pavements.’’ Tech. Rep. GL-97-3, U.S. Army Corps. of Engineers. Huang, Y. (1993). Pavement analysis and design. Prentice-Hall, Englewood Cliffs, N.J. Ioannides, A. M., and Korovesis, G. T. (1992). ‘‘Analysis and design of doweled slab-on-grade pavement systems.’’ J. Transp. Engrg., ASCE, 118(6), 745–768. Ioannides, A., Lee, Y.-H., and Darter, M. (1992). ‘‘Control of faulting through joint load transfer design.’’ Transp. Res. Rec. 1286, Transportation Research Board, Washington, D.C., 49–56. Kuo, C., Hall, K., and Darter, M. (1996). ‘‘Three-dimensional finite element model for analysis of concrete pavement support.’’ Transp. Res. Rec. 1505, Transportation Research Board, Washington, D.C., 119– 127. Snyder, M. B. (1989). ‘‘Cyclic shear load testing of dowels in PCC pavement repairs.’’ Transportation Res. Rec. 1215, Transportation Research Board, Washington, D.C., 246–257. Tabatabaie-Raissi, A. (1978). ‘‘Structural analysis of concrete pavement joints,’’ PhD thesis, University of Illinois, Urbana, Ill. Tabatabaie, A., and Barenberg, E. (1980). ‘‘Structural analysis of concrete pavements.’’ J. Transp. Engrg. of ASCE, 106(5), 493–506. Tayabji, S., and Colley, B. (1986). ‘‘Analysis of jointed concrete pavements.’’ Tech. Rep. FHWS-RD-86-041, Federal Highway Administration, Washington, D.C. Zaghloul, S., White, T., and Kuczek, T. (1994). ‘‘Evaluation of heavy load damage effect on concrete pavements using three-dimensional, nonlinear dynamic analysis.’’ Transp. Res. Rec. 1449, Transportation Research Board, Washington, D.C., 123–133. Zaman, M., and Alvappillai, A. (1995). ‘‘Contact-element model for dynamic analysis of jointed concrete pavements.’’ J. Transp. Engrg., ASCE, 121(5), 425–433. Zienkiewicz, O., and Taylor, R. (1994). The finite element method—Volume I. McGraw-Hill, New York. JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 / 57 The International Journal of Geomechanics Volume 1, Number 3, 309–323 (2001) 3D Finite Element Study on Load Transfer at Doweled Joints in Flat and Curled Rigid Pavements William G. Davids Department of Civil and Environmental Engineering, 5711 Boardman Hall, University of Maine, Orono, ME 04469–5711 e-mail: wdavids@umeciv.maine.edu ABSTRACT. This study examines load transfer across doweled joints in rigid pavements using 3D finite element analysis. A recently developed dowel modeling strategy is employed that allows the efficient and rigorous consideration of dowel/slab interaction. Parametric studies on the response of a typical, dowelretrofitted pavement system subjected to axle loads and varying degrees of slab curling are conducted. To examine the effect of slab support on pavement response, the studies consider two different foundation types: layered elastic with an asphalt-treated base and a dense liquid foundation. The results of the studies are discussed with emphasis on the effect of slab curling and foundation type on joint load transfer and the potential for joint distress. While there are significant differences in response for the ATB-supported slabs and the slabs founded on a dense liquid, slab curling does generally increase dowel shears and dowel/slab bearing stresses. However, further examination of the parametric study results that accounts for compressive fatigue of the concrete at the dowel/slab interface indicates that slab curling may not significantly increase the potential for damage to the slab concrete surrounding the dowels. I. Introduction The accurate three-dimensional finite element analysis (3D FEA) of the response of doweled, jointed concrete pavements to environmental and mechanical loading requires that the mechanics of dowel load transfer be accurately captured. The accelerated pavement distress and joint faulting commonly attributed to poor joint load transfer [1, 2] underscores this need. However, realistic modeling of dowel load transfer is difficult: in addition to the complex mechanics of dowel-slab interaction, simply generating realistic, solvable 3D models can be extremely laborious. Despite these complexities, several investigators employing 3D FEA have explicitly modeled dowels using conventional beam elements meshed within the solid slab [3, 4]. However, dowelslab interaction has not always been addressed rigorously in the 3D FEA of rigid pavements. Key Words and Phrases. 3D finite elements, rigid pavement, dowel load transfer, dowel/slab-bearing stress. © 2003 ASCE DOI: 10.1061/(ASCE)1532-3641(2001)1:3(309) ISSN 1532-3641 310 William G. Davids One notable exception is the study of Channakeshava et al. [5], where the interaction between the dowels and the concrete was modeled with discrete nonlinear springs connecting the ends of the dowels to the slabs. Hammons [6] employed the effective joint stiffness concept of Ioannides and Korovesis [7] in 3D FEA of rigid pavements, alleviating the difficulty of meshing bending and solid elements. However, uniform dowel spacing is implicit in this approach, which is common for newly constructed pavements but rarely the case for retrofitted joints where dowels are typically located only in the wheel-paths [8]. To overcome this limitation, Davids [9] developed an embedded bending element that permits dowels to be precisely located within the finite element mesh irrespective of the slab mesh lines. This embedded bending element also permits the rigorous treatment of dowel looseness, which has been shown to detrimentally affect joint load transfer both numerically [9, 10] and experimentally [11, 12]. This brief review of dowel modeling techniques indicates that while significant progress has been made in modeling dowel load transfer, there are still issues that need to be addressed. In particular, few studies employing 3D FEA have considered the combined effects of slab curling and mechanical loading in conjunction with a rigorous treatment of load transfer at doweled joints. Also of great interest is the potential for damage to the concrete surrounding the dowels in pavements subjected to realistic loading conditions with various foundation properties. This study directly explores these issues using the program EverFE, which has been developed specifically for the 3D FEA of rigid pavements [13]. A brief overview of the modeling strategies employed by EverFE is presented that includes a discussion of a recently developed embedded bending element for modeling dowel load transfer [9]. Following this, parametric studies on the response of a typical, dowel-retrofitted pavement system subjected to axle loads and varying degrees of slab curling are conducted. The results of the parametric studies are discussed with emphasis on the effect of joint properties and foundation type on joint load transfer and the potential for joint distress. II. Modeling strategies A. Overview of modeling strategies The finite element models employed in this study are generated with EverFE, a recently developed user-friendly finite element analysis tool for rigid pavement analysis. Details and features of EverFE may be found in Davids et al. [13]; verification of EverFE’s joint load transfer modeling techniques through comparisons with experimental results has been presented by Davids and Mahoney [14]. Only a brief overview of the modeling strategies employed by EverFE is presented here. EverFE permits the simulation of 1, 2, or 3 slab systems with straight or skewed joints. The slabs and base are linearly elastic and are discretized with 20-noded quadratic hexahedra; up to three distinct base and soil layers can be considered in the analysis. A Winkler foundation is used to represent the subgrade below the bottom-most foundation layer and is incorporated in the model with an 8-noded quadratic interface element that displaces compatibly with the quadratic hexahedra above it. Loss of contact between the slabs and base layer is captured using a frictionless nodal contact approach where neither tension nor interpenetration is permitted at the slab/base interface (Figure 1); this feature is crucial to accurately capturing joint load transfer when the foundation is modeled as an elastic solid. The model boundary conditions are the minimum required to prevent rigid-body displacement and rotation of the slabs and base in the horizontal plane. It should also be noted that EverFE incorporates a highly efficient multigrid-preconditioned conjugate gradient solver that permits the solution of large, nonlinear 3D finite element models on 3D Finite Element Study on Load Transfer at Doweled Joints in Flat and Curled Rigid Pavements FIGURE 1 311 Detail of slab/base separation near joint. desktop computers [15]. A typical finite element model employed in this study has over 40,000 degrees of freedom and is nonlinear due to the loss of contact between the slab and base; solutions are generally achieved in less than one hour using a 400 MHz Pentium II computer with 256 MB of RAM. B. Dowel modeling techniques The dowel modeling technique used in this study relies on an embedded finite element formulation for the dowels having the following features: • • • Dowels can be located exactly irrespective of the slab mesh lines as shown in Figure 2(a,b). Gaps between the dowels and surrounding slabs (dowel looseness) can be explicitly modeled using a nodal contact approach. In lieu of explicitly modeling dowel looseness, a Winkler foundation can be specified between the dowel and surrounding slabs to model dowel-slab interaction [see Figure 2(c)]. Three-noded, 18 degree-of-freedom quadratic beam elements are used to represent the portions of the dowels embedded in the slabs, which ensures that the dowels displace compatibly with the quadratic solid elements used to discretize the slab. A conventional two-noded shear beam is used to span the joint. Only a brief overview of the dowel modeling strategies is given here, as full details are presented elsewhere [9, 16]. The formulation of the embedded element relies on expressing the nodal displacements of the dowel as functions of the nodal displacements of the solid element it is embedded within. The three-noded quadratic beam element (that originally has 18 degrees of freedom in its element displacement vector, Ud ) takes on the degrees of freedom of the solid element it is embedded within, Ue . The element displacement vector of the embedded bending element, Ude , becomes: Ude = Ud . Ue (1) As shown in Davids and Turkiyyah [16], Ud can be recovered through the following matrix 312 FIGURE 2 William G. Davids Embedded dowel element. transformation: Ud = TUde . (2) The matrix T contains shape functions of the embedding element and constrains the dowel to embedding element. It follows that the stiffness matrix of the embedded element, Kde , can be determined from the original dowel element stiffness, K, as follows: Kde = TT KT . (3) With the foregoing formulation, the embedded bending element can be extended to permit the inclusion of general bond-slip relations between the dowel and surrounding slab. We can denote the incremental vector of relative displacements between the slab and dowel as dδ, and assume that the corresponding incremental force vector, df can be computed as: df = Dd1 . (4) In equation (4), D is a 3 × 3 constitutive matrix. It has been shown [9] that the stiffness matrix of the embedded dowel element with general bond-slip relations, Kdt , can then be computed as: Z 1 dt de BT DBhdη . (5) K =K + −1 3D Finite Element Study on Load Transfer at Doweled Joints in Flat and Curled Rigid Pavements 313 In equation (5), B is a matrix operator containing shape functions of both the embedding solid element and the dowel element, and h is the length of the dowel; integration is performed with respect to the dowel element local coordinate, η. Physically, this formulation of the embedded element with a general bond-slip relationship between the dowel and slab is analogous to the classic beam on elastic foundation [17], but differs in that forces in all directions can exist between the dowel and the slab. The magnitude of the forces depends on the relative displacement between the dowel and the slab and the constitutive relations of the dowel/slab interface incorporated in the matrix, D. In the present study, we will consider the case where the dowel is considered as supported by a dense liquid foundation sandwiched between the dowel and slab as shown in Figure 2(c). This approach has been widely used to capture dowel/slab interaction in the 2D FEA of doweled pavement systems [18]. The diagonal term of D corresponding to the vertical coordinate—D(3, 3) in the present case, since the z direction is vertical for the models of this study—then takes on the value of Kd, where K is the modulus of dowel reaction [MPa/mm], and d is the dowel diameter [mm]. Small non-zero values are used for the remaining diagonal terms of D to prevent rigid-body motion of the dowel. This implies that significant transverse (y direction) and longitudinal (x direction) forces do not exist between the dowel and the slab. III. Parametric study: model description The parametric study focuses on joint load transfer under applied axle loads and varying degrees of slab curling. The model used for all simulations consists of two 4600-mm-long by 3660-mm-wide by 254-mm-thick concrete slabs having an elastic modulus, E, of 28,000 MPa, Poisson’s ratio, ν, of 0.20, and a density of 2400 kg/m3 . Prior studies have demonstrated that accurate modeling of the foundation type and properties is important to accurately predict load transfer demands [4, 19]. To account for the effect of foundation type on joint load transfer, two basic support configurations that bound expected foundation properties are considered in the parametric studies: An unbonded, 150-mm-thick asphalt treated base (E = 3500 MPa) is explicitly modeled as an elastic continuum; this is founded on 1200 mm of natural soil (E = 50 MPa), which in turn rests on a dense liquid foundation with a modulus, k, of 0.027 MPa/mm. The base and subgrade are both extended 1000 mm beyond both sides and the ends of the slabs to more realistically capture actual foundation extents. 2. The slabs are supported directly with a tensionless dense liquid having a modulus, k, of 0.027 MPa/mm. This model is more applicable for slabs founded on a soft or cohesive material with no base layers. 1. Three 460-mm-long, 38-mm-diameter steel dowels spaced at 300 mm are located in each wheel path, which is typical for a dowel-retrofitted pavement system. To permit the effect of dowel-slab interaction to be examined, three different values of the modulus of dowel reaction are employed in the analyses: K = 40 MPa/mm (soft); K = 400 MPa/mm (intermediate); and K = 4,000 MPa/mm (stiff). Prior work has shown K to be an important parameter [7, 9], and we note that K = 400 MPa/mm has been considered a typical value in past studies [6, 20]. A single, transversely centered axle load consisting of two 40 kN wheel loads spaced at 1830 mm and applied at the joint is considered for all analyses. This loading is the usual Equivalent Single Axle Load (ESAL) used in the pavement design [21] and is positioned to be critical with respect to joint load transfer demand. To simulate slab curling and the resultant loss of contact 314 William G. Davids between the slab and base near the slab edges, negative linear temperature gradients, 1T , of 0◦ C/mm, −0.008◦ C/mm, −0.016◦ C/mm, and −0.031◦ C/mm are considered simultaneously with the axle load. It is important to note that curling of this magnitude could be due to temperature differentials and/or differential slab shrinkage; however, for simplicity, we will generally refer to equivalent temperature gradients, 1T , throughout the remainder of this study. A typical finite element mesh used in the analyses has 14,227 nodes and 43,365 degrees of freedom and is shown in Figure 3. Each slab has 432 brick elements, with three elements through the slab thickness and a maximum element aspect ratio of 4.5:1. To verify the adequacy of the level of slab discretization, preliminary analyses were conducted for a single slab on a dense liquid and the predicted stresses compared with Westergaard’s solutions. For the case of an interior loading of 40 kN acting over a circle with a 150 mm radius, the finite element model and Westergaard solutions varied by less that 0.1%. For an edge loading case, the two solutions were within 1%. Both the base and subgrade layers are discretized with two elements over their depth as shown in Figure 3. FIGURE 3 Finite element mesh used in parametric studies (dowels not shown). IV. Parametric study: results A. Slabs founded on ATB and elastic solid subgrade Table 1 presents the displacement load transfer efficiencies (LT Eδ ) and maximum dowel shears for the simulations employing an asphalt-treated base (ATB) and an elastic solid subgrade. Interestingly, LT Eδ remains fairly constant with 1T for K = 4000 MPa/mm, while it decreases significantly with increasing 1T for K = 40 MPa/mm. Note that while the increase in LT Eδ with increasing K is relatively small, there are large increases in dowel shear with increasing K. Further, larger negative temperature gradients (1T ) dramatically increase dowel shears; this increase is due to the loss of base layer support at the joint as the slab edges curl upward. 3D Finite Element Study on Load Transfer at Doweled Joints in Flat and Curled Rigid Pavements 315 TABLE 1 Displacement LTE and Dowel Shears for Model with ATB K = 40 MPa/mm K = 400 MPa/mm K = 4,0000 MPa/mm 1T LT Eδ Shear LT Eδ Shear LT Eδ Shear (%) (kN) (%) (kN) (%) (kN) (◦ C/mm) 0 90.7 0.77 93.4 2.00 95.4 3.13 −0.008 85.5 1.39 92.5 2.49 95.8 3.15 −0.016 81.6 2.02 91.1 3.59 94.8 4.88 −0.024 79.3 2.78 90.6 4.65 95.1 5.97 −0.031 78.6 3.50 91.2 5.23 95.6 6.37 Figures 4 and 5 show the nodal contact patterns predicted under the smallest non-zero temperature gradient (1T = −0.008◦ C/mm) and the largest gradient (1T = −0.031◦ C/mm). Note that when 1T = −0.031◦ C/mm, the row of nodes in the loaded slab at the joint remain separated from the ATB, while when 1T = −0.008◦ C/mm, slab/base contact is regained near the joint under the application of the axle load. Figures 6 and 7 present the maximum principal stresses on the top of the loaded slabs for the same conditions. As expected, top-slab tension increases dramatically with the increasing 1T . FIGURE 4 Nodal contact pattern for 1T = −0.008◦ C/mm + axle load. Of great interest in the present simulations are the dowel/concrete bearing stresses. Figure 8 shows the predicted dowel/slab bearing stresses when K = 400 MPa/mm for 1T equal to 0◦ C/mm and −0.031◦ C/mm. In Figure 8, bearing stress between the top of the dowel and the slab is plotted as positive and bearing stress between the bottom of the dowel and the slab is plotted as negative. The increase in maximum bearing stress with increasing temperature gradient is over 100%. Further, when 1T = −0.031 ◦ C/mm, the maximum dowel/slab bearing stress occurs under the unloaded slab as opposed to the loaded slab when 1T = 0 ◦ C/mm. This can be explained as follows: under the action of 1T alone, the maximum dowel/slab bearing stress occurs at the bottom of the dowel in both slabs. However, the axle load causes opposing dowel/slab bearing stresses at the top of the dowel in the loaded slab and increases bearing stresses at the bottom of the dowel in the unloaded slab. 316 William G. Davids FIGURE 5 Nodal contact pattern for 1T = −0.031◦ C/mm + axle load. FIGURE 6 Max. principal stresses on top of slab for 1T = −0.008◦ C/mm + axle load. Figure 9 depicts the variation in maximum dowel/slab bearing stress with 1T . As expected, the bearing stresses increase significantly with increasing K. Interestingly, the larger the value for K, the more sensitive the dowel bearing stress is to 1T . We also note that as expected there is a strong—although not direct—correlation between dowel shear and dowel/slab bearing stresses. For example, when K = 400 MPa/mm and 1T = 0◦ C/mm, the maximum shear and bearing stress are 2.00 kN and 4.08 MPa, respectively; when K = 400 MPa/mm and 1T = −0.031◦ C/mm, the maximum shear and bearing stress are 5.23 kN and 9.16 MPa/mm, respectively. 3D Finite Element Study on Load Transfer at Doweled Joints in Flat and Curled Rigid Pavements FIGURE 7 Max. principal stresses on top of slab for 1T = −0.031◦ C/mm + axle load. FIGURE 8 Bearing stress distribution along dowel: ATB, K = 400 MPa/mm. 317 318 William G. Davids FIGURE 9 Maximum dowel bearing stress vs. 1T for model with ATB. B. Slabs founded on dense liquid Table 2 gives LT Eδ and maximum dowel shears for the case where the slabs are founded directly on a dense liquid. In contrast to the slabs founded on an ATB, LT Eδ increases significantly with increasing K. Further, although dowel shears increase with increasing K, they remain nearly constant with increasing 1T . This can be attributed to the mechanics of joint deformation for the dense liquid-supported slabs. Even under large temperature gradients, the foundation remains almost entirely in contact with the slabs, and thus nearly the same fraction of the axle load is transferred to the foundation under the loaded slab irrespective of 1T . Equilibrium of the loaded slab requires that the remainder of the axle load transferred to the unloaded slab via dowel action also stay largely unchanged. Figure 10 shows the dowel/slab bearing stress distribution along the dowel length for K = 400 MPa/mm and 1T of 0◦ C/mm and −0.031◦ C/mm. Similar to the bearing stresses predicted for the model with the slabs resting on an ATB, the maximum dowel/slab bearing stress occurs in the unloaded slab at higher temperature gradients. Figure 11 illustrates maximum dowel bearing stress vs. 1T . Note that for all values of K, the maximum bearing stress initially decreases with increasing 1T , and then begins to increase. As with the slabs resting on an ATB, this is due to the switch in location of the maximum bearing stress from the loaded to the unloaded slab as 1T increases. However, unlike the slabs founded on an ATB, there is no correlation between dowel shears and the dowel/slab bearing stresses for larger values of K. Further, the magnitude of the maximum bearing stress is relatively insensitive to temperature gradient compared the slabs founded on an ATB, especially for smaller values of K. 3D Finite Element Study on Load Transfer at Doweled Joints in Flat and Curled Rigid Pavements 319 TABLE 2 Displacement LTE and Dowel Shears for Slabs on Dense Liquid K = 40 MPa/mm K = 400 MPa/mm K = 4,0000 MPa/mm 1T LT Eδ Shear LT Eδ Shear LT Eδ Shear (%) (kN) (%) (kN) (%) (kN) (◦ C/mm) 0 64.2 4.87 86.7 6.05 93.5 6.75 −0.008 64.2 4.87 86.9 6.04 93.7 6.75 −0.016 64.7 4.88 86.9 6.05 93.8 6.76 −0.024 66.3 4.89 87.6 6.06 94.1 6.76 −0.031 69.1 4.91 88.7 6.06 94.6 6.76 FIGURE 10 Bearing stress distribution along dowel: dense liquid, K = 400 MPa/mm. V. Potential for loss of load transfer effectiveness The results presented in the last section indicate that dowel/slab bearing stresses are significant. In fact, assuming that the slabs are constructed from concrete having fc0 = 28 MPa (consistent with the assumed modulus of elasticity of 28,000 MPa), the maximum predicted dowel/slab bearings stresses can exceed the concrete compressive strength. Perhaps of more concern than the short-term loads, however, is the potential for cumulative fatigue-induced damage under repeated loading. This study has considered a quasi-static 80 kN axle load—the usual ESAL assumed in design. For the purposes of determining required pavement thickness, major highways must commonly be considered to support 10 million or more ESALs 320 FIGURE 11 William G. Davids Maximum dowel bearing stress vs. 1T for slabs on dense liquid. over their lifetime [21]. Experimental studies on doweled joints [11, 12] have shown that dowel looseness can develop under similar repeated loading, which leads to significant losses in load transfer. This is not surprising, since the dowel/concrete bearing stresses are largely compressive, and a large body of research indicates that potential for damage to concrete under repeated compressive stresses [22, 23]. At this point, it might seem obvious to conclude that given the fairly dramatic increases in dowel/slab-bearing stresses with temperature gradient when a stiff base is present, the potential for fatigue damage to the concrete surrounding the dowels increases significantly with increased slab curling. However, due to the relative deformations of the slabs and dowels under slab curling alone and combined curling and axle loading, this conclusion cannot be immediately drawn. To further examine this issue, we will attempt to more rigorously quantify the potential for fatigue damage. A number of relationships predicting the sustained uniaxial compressive strength of concrete under large numbers of load cycles are available in the literature; see Hsu [24] for an overview. This study uses the relationship proposed by Tepfers and Kutti [25], which is given as: fcrit = 1 − 0.0685(1 − R) log(N ) . fc0 (6) In equation (6), fcrit is the largest compressive stress that can be applied for N cycles, at which point the concrete fails due to compressive fatigue. The parameter R is the ratio of the minimum applied compressive stress to the maximum applied compressive stress (i.e., (1 − R)fmax is the fatigue stress range, where fmax is the maximum repeatedly applied compressive stress). 3D Finite Element Study on Load Transfer at Doweled Joints in Flat and Curled Rigid Pavements 321 The value of R is significant in the current situation: if 1T = 0 ◦ C/mm, R = 0, since there is effectively no bearing stress until the axle passes over the joint. If 1T > 0◦ C/mm, however, R may take on a positive, non-zero value depending on whether the dowel/slab-bearing stresses under temperature gradient alone occur on the same side of the dowel as those under the combined temperature/axle loading. We will consider the following cases: 1T = 0◦ C/mm and 1T = −0.031◦ C/mm for K = 400 MPa/mm and K = 4000 MPa/mm for both the ATBsupported and Winkler foundation-supported slabs. For the purposes of comparing the potential for fatigue damage under different loading conditions, consider a dimensionless damage parameter, D, defined as follows: D= fmax . fcrit (7) In equation (7), fcrit is computed according to equation (6) assuming that N = 10 million cycles. Clearly, larger values for D indicate a larger potential for damage. The results of the required calculations are presented in Tables 3 and 4. Note that both sides of the joint—loaded and unloaded—are considered in the analysis, since the direction of the dowel/slab-bearing stresses is significant. For clarity, all bearing stresses are given: f1 is the bearing stress under the loaded slab, and fu is the bearing stress under the unloaded slab. The sign convention is the same as that used in Figures 6 and 8, where stresses arising from contact between the top of the dowel and the slab are positive. TABLE 3 Bearing Stress Fatigue Calculations for Slabs on ATB 1T Only 1T + Axle K 1T f1 fu f1 fu (MPa/mm) (◦ C/mm) (MPa) (MPa) (MPa) (MPa) 400 0 0 0 4.08 −1.56 −0.031 −4.32 −4.32 5.76 −9.15 4,000 0 0 0 15.4 −3.83 −0.031 −19.5 −19.5 12.1 −27.3 Loaded Slab R D 0 0.28 0 0.40 0 1.1 0 0.83 Unloaded Slab R D 0 0.11 0.47 0.44 0 0.26 0.71 1.1 Loaded Slab R D 0 0.74 0 0.44 0 2.1 0 0.77 Unloaded Slab R D 0 0.44 0.43 0.53 0 0.77 0.68 1.3 TABLE 4 Bearing Stress Fatigue Calculations for Slabs on Dense Liquid 1T Only 1T + Axle K 1T f1 fu f1 fu (MPa/mm) (◦ C/mm) (MPa) (MPa) (MPa) (MPa) 400 0 0 0 10.72 −6.44 −0.031 −4.61 −4.61 6.46 −10.73 4,000 0 0 0 30.6 −11.2 −0.031 −20.8 −20.8 11.2 −30.5 Two conclusions can be drawn from the results presented in Tables 3 and 4: 322 William G. Davids 1. The slabs founded on the ATB are generally less prone to fatigue damage at the dowel/concrete interface than the slabs founded on a dense liquid. This holds irrespective of the magnitude of K, even though the worst-case dowel/slab bearing stresses for the slabs founded on the ATB are nearly as large as those experienced by the slabs founded on a dense liquid. 2. Damage to the slab concrete surrounding the dowels appears to be only slightly more likely for curled slabs vs. flat slabs when they are founded on a stiff base layer. For the slabs founded on the dense liquid, the potential for degradation in dowel/slab interaction may actually be decreased by slab curling. It must be noted that this analysis has been somewhat simplistic: in real pavement systems, the degree of dowel/slab interaction decreases with increasing fatigue damage to the surrounding concrete. Further, the actual three-dimensional state of stress in the concrete surrounding the dowel is quite complex and is not accounted for in the foregoing discussion. Despite the uncertainties in the analysis, however, it does seem that there is potential for damage to the slab concrete surrounding the dowels. Contrary to intuitive expectations, however, it appears that slab curling may not exacerbate the potential for the occurrence of this damage. VI. Summary and conclusions This study has examined load transfer at doweled joints in plain concrete pavement for both curled and flat slabs. The study employed 3D finite element models developed with EverFE [13]. Recent extensions to EverFE’s dowel modeling capabilities were outlined, with an emphasis on modeling dowel/slab interaction. Parametric studies were completed that considered various degrees of dowel/slab interaction, realistic levels of slab curling, and two foundation models (layered elastic with an ATB and a dense liquid) expected to bound realistic levels of response. Results of the parametric studies indicated that foundation type and the presence of slab curling have a large effect on predicted dowel shears and dowel/slab concrete bearing stresses. This can largely be explained by the mechanics of joint deformation, which are quite different for the two foundation models considered in this study. The potential for fatigue damage to the concrete surrounding the dowels was quantified, and led to two important conclusions. First, rigid pavements founded on a stiff base are likely less prone to experience dowel looseness and damage to the concrete surrounding the dowels than slabs founded on a relatively soft soil. Secondly, slab curling—while it increases dowel shears and dowel/slab bearing stresses significantly—may not increase the potential for damage to the concrete surrounding the dowels. However, it must be noted that the analyses conducted here are by no means comprehensive, and the issue of dowel/slab interaction and the progressive loss of load transfer effectiveness under repeated loading warrants further investigation, both simulation based and experimental. References [1] A.M. Ioannides, Y.-H Lee and M. Darter, Control of Faulting through Joint Load Transfer Design, Transportation Research Record No. 1286, TRB, National Research Council, Washington, DC, pp. 49–56, (1992). [2] Y. Huang, Pavement Analysis and Design, Prentice-Hall, Englewood Cliffs, NJ, 1993. [3] W. Uddin, R.M. Hackett, A. Joseph, Z. Pan, and A.B. Crawley, Three-Dimensional Finite-Element Analysis of Jointed Concrete Pavement Having Discontinuities, Transportation Research Record 1482, TRB, National Research Council, Washington, DC, pp. 26–32, (1992). [4] C. Kuo, K. Hall, and M. Darter, Three-Dimensional Finite Element Model for Analysis of Concrete Pavement Support, Transportation Research Record No. 1505, TRB, National Research Council, Washington, DC, pp. 119– 127, (1996). 3D Finite Element Study on Load Transfer at Doweled Joints in Flat and Curled Rigid Pavements 323 [5] C. Channakeshava, F. Barzegar, and G. Voyiadjis, Nonlinear FE Analysis of Plain Concrete Pavement with Doweled Joints, J. Transport. Eng., ASCE, 119(5), pp. 763–781, (1993). [6] M.I. Hammons, Development of an Analysis System for Discontinuities in Rigid Airfield Pavements, Tech. Report GL-97-3, U.S. Army Corps of Engineers, 1997. [7] A.M. Ioannides and G.T. Korovesis, Analysis and Design of Doweled Slab-on-grade Pavement Systems, J. Transport. Eng., ASCE, 118(6), pp. 745–768, (1992). [8] K.T. Hall, M.I. Darter, and J.M. Armaghani, Performance Modeling of Joint Load Transfer Restoration, Transportation Research Record No. 1388, TRB, National Research Council, Washington, DC, pp. 129–139, (1993). [9] W.G. Davids, Effect of Dowel Looseness on Response of Jointed Concrete Pavements, J. Transport. Eng., ASCE, 126(1), pp. 50–57, (2000). [10] M. Zaman and A. Alvappillai, Contact-element Model for Dynamic Analysis of Jointed Concrete Pavements, J. Transport. Eng., ASCE, 121(5), pp. 425–433, (1994). [11] M.B. Snyder, Cyclic Shear Load Testing of Dowels in PCC Pavement Repairs, Transportation Research Record No. 1215, TRB, National Research Council, Washington, DC, pp. 49–56, (1989). [12] N. Buch and D.G. Zollinger, Development of Dowel Looseness Prediction Model for Jointed Concrete Pavements, Transportation Research Record No. 1525, TRB, National Research Council, Washington, DC, pp. 21–27, (1996). [13] W. Davids, G. Turkiyyah, and J. Mahoney, EverFE: Rigid Pavement 3D Finite Element Analysis Tool, Transportation Research Record No. 1629, TRB, National Research Council, Washington, DC, pp. 41–49, (1998). [14] W. Davids and J. Mahoney, Experimental Verification of Rigid Pavement Joint Load Transfer Modeling with EverFE, Transportation Research Record No. 1684, TRB, National Research Council, Washington, DC, pp. 81– 89, (1999). [15] W. Davids and G. Turkiyyah, Multigrid Preconditioner for Unstructured Nonlinear 3D FE Models, J. Eng. Mech., ASCE, 125(2), pp. 186–196, (1999). [16] W. Davids and G. Turkiyyah, Development of Embedded Bending Member to Model Dowel Action, J. Struct. Eng., ASCE, 123(10), pp. 1312–1320, (1997). [17] A.C. Ugural and S.K. Fenster, Advanced Strength and Applied Elasticity, 2nd ed., Prentice-Hall, Englewood Cliffs, New Jersey, 1987. [18] H. Guo, J.A. Sherwood, and M.B. Snyder, Component Dowel-bar Model for Load-transfer Systems in PCC Pavements, J. Transport. Eng., ASCE, 121(3), pp. 289–298, (1995). [19] W. Davids, Foundation Modeling for Jointed Concrete Pavements, Proceedings of the Transportation Research Board’s 79th Annual Meeting, (2000). [20] A. Tabatabaie and E.J. Barenberg, Finite-Element Analysis of Jointed or Cracked Concrete Pavements, Transportation Research Record No. 671, TRB, National Research Council, Washington, DC, pp. 11–19, (1978). [21] AASHTO Guide for the Design of Pavement Structures. American Association of State Highway and Transportation Officials, Washington, DC, 1993. [22] S. Mindess and J.F. Young, Concrete, Prentice-Hall, Englewood Cliffs, NJ, 1981. [23] A.M. Neville, Properties of Concrete, 4th ed., John Wiley & Sons, New York, 1996. [24] T.T.C. Hsu, Fatigue of Plain Concrete, ACI J., 78(4), pp. 292–305, (1981). [25] R. Tepfers and T. Kutti, Fatigue Strength of Plain, Ordinary, and Lightweight Concrete, ACI J., 76(5), pp. 635–652, (1979). 92 ■ Transportation Research Record 1853 Paper No. 03- 2223 Three-Dimensional Finite Element Analysis of Jointed Plain Concrete Pavement with EverFE2.2 William G. Davids, Zongmu Wang, George Turkiyyah, Joe P. Mahoney, and David Bush The features and concepts underlying EverFE2.2, a freely available three-dimensional finite element program for the analysis of jointed plain concrete pavements, are detailed. The functionality of EverFE has been greatly extended since its original release: multiple tied slab or shoulder units can be modeled, dowel misalignment or mislocation can be specified per dowel, nonlinear thermal or shrinkage gradients can be treated, and nonlinear horizontal shear stress transfer between the slabs and base can be simulated. Improvements have been made to the user interface, including easier load creation, user-specified mesh refinement, and expanded visualization capabilities. These new features are detailed, and the concepts behind the implementation of EverFE2.2 are explained. In addition, the results of two parametric studies are reported. The first study considers the effects of dowel locking and slab–base shear transfer and demonstrates that these factors can significantly affect the stresses in slabs subjected to both uniform shrinkage and thermal gradients. The second study examines transverse joint mislocation and dowel looseness on joint load transfer. As expected, joint load transfer is greatly reduced by dowel looseness. However, while transverse joint mislocation can significantly reduce peak dowel shears, it has relatively little effect on total load transferred across the joint for the models considered. The use of three-dimensional finite element (FE) methods for analyzing rigid pavements subjected to mechanical and environmental loadings has grown significantly in the last decade. The increased use of three-dimensional FE analysis has given pavement researchers and designers a better understanding of critical aspects of pavement response that cannot be captured with analytical solutions, such as joint load transfer (1, 2), the effect of slab support on stresses (3), and pavement response under dynamic loads (4, 5). However, many aspects of rigid pavement behavior have not been thoroughly studied with three-dimensional FE analysis. This can be attributed to several factors, including the complexity of concrete pavement structures (especially joint load transfer mechanisms), the need to consider both environmental and mechanical load effects, the difficulty of model generation and result interpretation, and the relatively long solution times required for large three-dimensional FE analyses. These factors become especially challenging for the analyst when general-purpose FE programs are used. To circumvent these issues, three-dimensional FE analysis packages have been developed specifically for analyzing rigid pavements (6, 7). EverFE1.02, which was first made available in 1998 (7), addressed these difficulties through the use of an interactive graphical user interface for easy model definition and visualization of results, specialized techniques for modeling both dowel and aggregate interlock joint load transfer (2, 8), and fast iterative solution strategies for inclusion of inequality constraints for modeling slab–base separation and material nonlinearity (9). Recently, EverFE2.2 has been developed, which retains the original capabilities of EverFE1.02 while incorporating the following features that substantially extend its usefulness: • The ability to model tied adjacent slabs and shoulders. Multiple slab or shoulder systems can be modeled, and transverse tie bars are explicitly incorporated. • Extended dowel modeling capabilities. Dowel–slab interaction can be captured via either the specification of dowel looseness or springs sandwiched between the dowels and slabs, and the effect of dowel misalignment or mislocation can be simulated. • Modeling of nonlinear thermal gradients. Bilinear or trilinear thermal gradients through the pavement thickness can be specified. • Simulation of slab–base interaction. Separation of the base and slab under tension is handled via inequality constraints, and intermediate degrees of horizontal slab–base shear transfer can be captured. • Expanded postprocessing capabilities. In addition to visualizing slab stresses and displacements—as well as retrieving precise stress and displacement values at specific coordinates—the user can view shears and moments in individual dowels. • Expanded library of axle loads. Loads ranging from single wheels to dual-wheel, tandem axles can be quickly created, positioned, and deleted, as shown in Figure 1a. This manuscript details the features of EverFE2.2 and the concepts underlying implementation, with a primary focus on modeling of the dowels and ties, treatment of nonlinear thermal gradients, and simulation of slab–base interaction. The results of parametric studies that consider the effects of dowel locking, slab–base shear transfer, and transverse joint mislocation on pavement response are reported to illustrate the flexibility and modeling capabilities of EverFE2.2. FEATURES OF EverFE2.2 W. G. Davids and Z. Wang, Department of Civil and Environmental Engineering, University of Maine, 5711 Boardman Hall, Orono, ME 04469-5711. G. Turkiyyah and J. P. Mahoney, Department of Civil and Environmental Engineering, University of Washington, Box 352700, Seattle, WA 98195-2700. D. Bush, Dynatest Consulting, Inc., 165 South Chestnut Street, Ventura, CA 93001. EverFE2.2 employs several element types to discretize concrete pavement systems that have from one to nine slab or shoulder units. Up to three elastic base layers can be specified below the slab, and the subgrade is idealized as either a tensionless or a tension- Davids et al. Paper No. 03- 2223 93 (a) (b) FIGURE 1 Model generation with EverFE2.2: (a) axle and thermal load specification, (b) typical model illustrating discretization and element types. supporting dense liquid foundation. Twenty-noded quadratic hexahedral elements are used to discretize the slabs and elastic base layers (10), and the dense liquid foundation is incorporated via numerically integrated, eight-noded quadratic elements that are meshed with the bottommost layer of solid elements. Linear or nonlinear aggregate interlock joint load transfer as well as dowel load transfer can be modeled at transverse joints. Load transfer across longitudinal joints via transverse tie bars can also be modeled. Figure 1b is a screen shot of the EverFE2.2 meshing panel, showing many of the basic elements. (The user can selectively refine the number of elements used to discretize the slabs and base or subgrade layers.) The remainder of this section highlights significant features that are new to EverFE2.2; detailed discussions of the basic components, including the nonlinear aggregate interlock modeling 94 Paper No. 03- 2223 Transportation Research Record 1853 capabilities, are available in Davids and Mahoney (2), Davids (8), and Davids and Turkiyyah (11). Nonlinear Thermal Gradients Previous studies noted that thermal gradients through the depth of concrete pavements are often nonlinear (12, 13). EverFE2.2 allows the consideration of this important effect by the specification of a bilinear or trilinear approximation to a nonlinear gradient, which is easily defined in the loading panel (Figure 1a). The temperature changes are converted to equivalent element prestrains via the slab coefficient of thermal expansion, and these strains are numerically integrated over the element volume to generate equivalent nodal forces (10). The 20-noded quadratic element employed by EverFE2.2 can accurately capture strains that vary linearly over its volume. This implies that multiple elements through the pavement thickness should be used to accurately model bilinear or trilinear thermal gradients. The effect of uniform or nonuniform shrinkage strains can be simulated through their conversion to equivalent temperature changes for input to EverFE2.2. Dowel and Transverse Tie-Bar Modeling EverFE2.2 models dowels and transverse tie bars explicitly with embedded flexural finite elements (8, 11), which has the advantage of allowing the dowels and tie bars to be precisely located irrespective of the slab mesh lines, as shown in Figure 1b. This embedded element formulation also permits significant savings in computation time by allowing a range of load transfer efficiencies to be simulated without requiring a highly refined mesh at the joints. Dowel–slab interaction can be captured either by specifying a length and magnitude of gap between the dowels and the slabs or by specifying dowel support moduli in the dowel local coordinates, which translate into springs sandwiched between the dowels and slabs (Figure 2a). The latter approach was not available in EverFE1.02, and it permits varying degrees of dowel–slab interaction to be modeled while avoiding the contact nonlinearity inherent in the modeling of dowel looseness. However, this approach is a simplification of a complex phenomenon (8). The localized stresses in the concrete surrounding the dowels may not be accurately predicted when the embedded element formulation is used. Tie–slab interaction is captured via user-specified tie-bar support moduli in the tie-bar local coordinates. Once the dowels have been located within the model, the user can specify four misalignment or mislocation parameters (∆ x, ∆z, α, β) that shift an individual dowel along the x- and z-axes and define its angular misalignment in the horizontal and vertical planes (see Figure 2b and 2c). The dowel support moduli coincide with the local dowel coordinate axes (q, r, s), which are rotated from the global (x, y, z) axes by the angles α and β. The meshing algorithm precisely locates individual flexural elements within the mesh of solid elements by first solving for the intersection of each dowel with solid element faces and then subdividing each dowel into at least 20 individual quadratic embedded flexural elements. Simulation of Slab–Base Interaction Modeling interaction of the slab and base is crucial for accurately predicting pavement response to axle loads near joints and thermal or shrinkage gradients. EverFE2.2 allows the specification of either perfect bond between the slab and base (no slip and no separation) or free separation of the slab and base under tension. In both cases, the slab and base do not share nodes, and constraints are used to satisfy the required contact conditions (Figure 3). The solution algorithm relies on a perturbed Lagrangian formulation and a constraint updating scheme based on the current normal stress between the slab and base. Shear transfer between the slab and base can be important when analyzing pavements subject to uniform thermal expansion or contraction or shrinkage strains. Rasmussen and Rozycki (14) overviewed the factors governing slab–base shear transfer, noting that both friction and interlock between the slab and base play a role. x ∆x y Original position α r Misaligned position q Plan View Gap between dowel and slab Slab x C.L. Original position z ∆z β s Dowel-slab springs (a) FIGURE 2 Misaligned position Elevation (b) (a) Dowel–slab interaction and (b) dowel misalignment parameters. q Davids et al. Paper No. 03- 2223 95 Pairs of nodes vertically constrained if compression at interface τ Slab element τo δzz kSB δx or δy δo (a) Interface Base element elements transfer shear stress δx or δy (b) FIGURE 3 Modeling of (a) slab–base interaction and (b) interface shear transfer. In addition, a bilinear, elastic-plastic shear transfer model was calibrated on the basis of push tests of slabs on various bases. The study concluded that the effect of slab–base shear transfer should be incorporated in three-dimensional analyses of pavement systems. A study by Zhang and Li (15) focused on developing a one-dimensional analytical model for predicting shrinkage-induced stresses in concrete pavements that accounts for slab–base shear transfer. Like the model developed by Rasmussen and Rozycki, that model ultimately relied on a bilinear, elastic-plastic shear transfer model. Zhang and Li concluded that the type of supporting base—and thus the degree to which it restrains slab shrinkage—significantly affects slab stresses. To capture slab–base shear transfer, EverFE2.2 employs a 16-noded, zero-thickness quadratic interface element that is meshed between the slab and the base (Figure 3). The element constitutive relationship is based on that given by Rasmussen and Rozycki (14) and Zhang and Li (15). The bilinear constitutive relationship, defining the relationship between the shear stress (τ) and the relative slip between the slab and base is shown in Figure 3. This relationship is characterized by an initial distributed stiffness kSB (MPa/mm) and slip displacement δ0. (While kSB has the same units as the wellknown modulus of subgrade reaction, kSB is a distributed stiffness in the horizontal direction, and the shear stresses developed at the slab–base interface depend on the relative horizontal displacements between the slab and the base layer.) This constitutive relationship is assumed to apply independently in both the x and y directions if the slab and base remain in contact, which implies that a compressive normal stress exists at the slab–base interface. That there will be little or no shear transfer when slab–base separation occurs is accommodated by setting the interface stiffness and shear stress to zero whenever δz > 0. Modeling this loss of shear transfer with loss of slab–base contact is important, especially when thermal gradients are simulated. The interface element stiffness matrix and nodal force vector are computed numerically via 3 × 3 Gauss point integration. For very large values of kSB, this model approaches Coulomb friction with a very large friction coefficient, and for very small values of kSB, it is equivalent to a frictionless interface. An advantage of this modeling scheme is that the symmetry of the system stiffness equations is maintained, which allows the use of the existing, highly efficient preconditioned conjugate-gradient solver. Idealizing slab–base interaction with conventional Coulomb friction would destroy this symmetry, requiring the use of more complex (and likely less efficient) solution techniques. EFFECT OF DOWEL LOCKING AND SLAB–BASE SHEAR TRANSFER ON THERMAL STRESSES The potential detrimental effects of dowel locking—where the dowels become effectively bonded to the slabs—on pavement response to thermal loads are well recognized. Dowel locking is commonly attributed to dowel misalignment, which can cause flexure of the dowels and large frictional forces to develop at locations of dowel–slab contact, or corrosion of the dowels, which can result in bond between the dowels and slabs. In addition, one study suggested that friction between properly aligned dowels and slabs can provide significant axial restraint and increased stresses in slabs that are simultaneously subjected to a uniform temperature change and a negative thermal gradient (1). Other studies (14, 15) also concluded that shear transfer at the slab–base interface can significantly affect slab stresses. Here, EverFE2.2 is used to simulate the effect of dowel locking on a rigid pavement system subjected to a variety of thermal and selfweight loadings. The degree of slab–base interaction also is varied to study the effect of this important parameter on response. Model Description A three-slab system was modeled to capture the effect of the restraint provided by adjacent slabs. The 250-mm-thick slabs were 4,600 mm long and 3,600 mm wide, with a modulus of elasticity E of 28,000 MPa, a Poisson ratio ν of 0.20, a coefficient of thermal expansion of 1.1 × 10−5 per °C, and a density of 2,400 kg/m3. The slabs were founded on a 150-mm-thick asphalt-treated base with E of 3,500 MPa, ν of 0.20, and density of 2,000 kg/m3. The dense liquid foundation was assumed to have a modulus of subgrade reaction of 0.03 MPa/mm. Each transverse joint had 11 dowels 32 mm in diameter and 450 mm long, spaced at 300 mm on center. The FE mesh, shown in Figure 4, had 3,024 solid elements. The center slab was meshed with 18 × 18 elements in plan, and the outer slabs were meshed more coarsely, as they are of secondary interest. The analyses considered dowels that were both locked and unbonded (free slip). In all cases, the locked and unbonded dowels were assumed to have no looseness (i.e., they provided maximum vertical joint load transfer). No tensile bond stresses were allowed between the slab and the base, but three levels of slab–base shear transfer were considered in the analyses to capture the effect of this important parameter. The low degree of slab–base interaction corresponded to a slab–base interface shear stiffness kSB of 0.0001 MPa/mm, which is the minimum value used by EverFE2.2; this value might 96 Paper No. 03- 2223 Transportation Research Record 1853 TABLE 1 Maximum Principal Stresses Caused by Temperature Curling or Shrinkage or Both Degree of Slab-Base Interaction Dowel Type Locked FIGURE 4 Finite element mesh used in parametric study on dowel locking. Free Slip be expected when a bond-breaker such as polyethylene sheeting is placed on the base before the slab pour. (Note that kSB cannot be taken as zero because the slabs would be horizontally unrestrained, giving an unstable model.) The intermediate slab–base shear transfer parameters were kSB = 0.035 MPa/mm and δ0 = 0.60 mm, which correspond to an asphalt-treated base (15). The high slab–base shear transfer parameters of kSB = 0.416 MPa/mm and δ0 = 0.25 mm were reported by Zhang and Li for a hot-mix asphalt concrete base (15). Five load cases were considered: Load Case Low Intermediate High DL – T DL + ∆ T DL + ∆ T – T DL – ∆ T DL – ∆ T – T DL – T DL + ∆ T DL + ∆ T – T DL – ∆ T DL – ∆ T – T 0 0.871 (B) 0.870 (B) 0.689 (T) 0.688 (T) 0 0.871 (B) 0.872 (B) 0.689 (T) 0.689 (T) 0.159 (B) 0.886 (B) 0.973 (B) 0.705 (T) 0.818 (T) 0.118 (B) 0.880 (B) 0.906 (B) 0.703 (T) 0.669 (T) 0.594 (B) 0.945 (B) 1.18 (B) 0.815 (T) 0.991 (T) 0.591 (B) 0.938 (B) 1.51 (B) 0.785 (T) 0.547 (T) *All values in MPa; letter in parentheses indicates either top (T) or bottom (B) of slab. Parametric Study Results and Significance Table 1 shows the maximum principal stresses predicted in the center slab for all parameter combinations and loadings. These stresses occurred at either the top center or the bottom center of the middle slab. Note that when there is full bond between the dowels and slabs, the model may predict higher tensile stresses around the dowels; however, these stresses are not reliable because of insufficient mesh refinement at the joints. Figure 5 shows a colormap of principal stresses and the deformed shape of the system under DL − ∆T − T assuming high slab–base shear transfer. When an intermediate level of slab–base shear transfer is assumed, dowel locking increases stresses due to a uniform shrinkage load (DL − T ) by 35%. When high slab–base shear transfer exists, dowel locking has a much less marked effect for this load • A uniform temperature change of −10°C (DL − T); • A positive thermal gradient of 0.032°C/mm (DL + ∆T); • A negative thermal gradient of −0.032°C/mm (DL − ∆T); • A positive thermal gradient of 0.032°C/mm plus a uniform temperature drop of −10°C (DL + ∆T − T); and • A negative thermal gradient of −0.032°C/mm plus a uniform temperature change of −10°C (DL − ∆T − T). The term DL refers to model self-weight. The uniform temperature change is equivalent to a uniform slab shrinkage of 110 µ. (a) (b) FIGURE 5 (a) Tensile stresses on top of slabs and (b) displaced shape (scale factor 500) (DL T T, high slab–base shear transfer). Davids et al. 97 1.6 Intermediate Slab-Base Shear High Slab-Base Shear 1.5 Maximum Principal Stress (MPa) case because of the significant reduction in overall slab shortening. Dowel locking increases stresses for the DL − ∆T − T load case for both intermediate slab–base shear transfer (16% increase) and high slab–base shear transfer (81% increase). However, stress increases caused by dowel locking are only 7% for the DL + ∆T − T load case with intermediate slab–base shear transfer (although the overall stresses are higher than for DL + ∆T − T ). This difference can be attributed to the fact that under DL + ∆T − T, the bottom of the slab is shrinking under a net temperature change of −14°C. This causes shear stresses at the slab–base interface that are concentrated near the edges of the slab and act away from the slab center, which tends to increase tensile stresses in the bottom of the slab significantly more than dowel locking alone. In contrast, under DL − ∆T − T, the net temperature change at the bottom of the slab is only −6°C, and the resulting shear stresses, which are concentrated near the center of the slab, tend to reduce the peak tensile stress that occurs at the top of the slab. Further, the ends of the slab are not in contact with the base because of slab liftoff (see Figure 5). As a result, the increase in tensile stress arising from the restraining effect of the dowels is more pronounced. A counterintuitive result is the 28% decrease in slab stresses caused by dowel locking under DL + ∆T − T with a high degree of slab–base shear transfer. This can be explained by the fact that dowel locking tends to prevent contraction of the bottom of the slab, reducing the relative displacements between the slab and base and thus the shear at the slab–base interface near the transverse joints. In fact, the maximum relative x-direction displacement between the central slab and the base predicted by EverFE, which occurs at the slab ends, is 0.078 mm when dowel locking exists, giving τ = 0.032 MPa. In contrast, when there is no dowel locking, the x-direction relative displacements at the slab ends are 0.25 mm, implying that the peak value of τ = τ0 = 0.104 MPa. As discussed, this reduction in slab stress with dowel locking was not observed for the intermediate degree of slab–base interaction, where the reduced stiffness kSB of the slab–base interface allows a relative x-direction displacement of 0.208 mm when the dowels are locked and 0.308 mm when the dowels are unbonded. These values result in relatively low slab–base interface shear stresses of 0.0073 MPa and 0.011 MPa, respectively. This explanation was further verified by running of simulations in which the effect of the degree of bond between the dowels and slabs on slab stress was simulated by varying the dowel–slab axial restraint modulus. Figure 6 shows the results of these analyses for models with both high and intermediate degrees of slab–base shear transfer subjected to DL + ∆T − T. Note the increase in slab stresses with increasing dowel–slab restraint modulus for the case of intermediate slab–base shear transfer. Conversely, slab stresses decrease with increasing dowel–slab restraint modulus, assuming a high degree of slab–base shear transfer. For both degrees of slab–base shear transfer, the limiting stresses given in Table 1 bound the results shown in Figure 6. Increasing slab–base shear transfer tends to increase slab stresses significantly for most loadings. As expected, when kSB = 0.0001 MPa/ mm, there are no slab stresses for the DL − T load case, as shrinkage is effectively unrestrained; however, significant tensile stresses are observed for DL − T loading with intermediate slab–base shear transfer for both locked and unlocked dowels. The effect of increasing slab–base shear transfer is also dramatic for the model with locked dowels subjected to DL −∆T − T, where slab stresses increase 44% as slab–base shear transfer increases from low to high. Only the DL − ∆T − T loading with unbonded dowels shows a decrease in slab stresses with increasing slab–base shear transfer. This de- Paper No. 03- 2223 1.4 1.3 1.2 1.1 1 0.9 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Dowel Axial Restraint Modulus (MPa) FIGURE 6 Variation in peak slab stress caused by dowel axial restraint (DL T T ). crease results from the increased shear stresses between the slab and the base under uniform temperature shrinkage that tend to reduce the peak tensile stress at the top of the slab. In general, the results of the simulations indicate that there is a complex interaction among dowel locking, slab–base interaction, and thermal loading. The need for three-dimensional analysis (instead of one- or two-dimensional) when simulating both thermal gradients and shrinkage is evident: even under uniform shrinkage, the slab–base shear stresses acting at the bottom of the slab result in slab–base separation and a nonuniform distribution of stresses over the slab thickness because of the eccentricity of the shear stress respective to the center of gravity of the slab. However, creep of both slab and base, which is not considered by EverFE2.2, will mitigate these stress increases. EFFECT OF TRANSVERSE JOINT LOCATION ON JOINT LOAD TRANSFER Poor construction can lead to effective misalignment or mislocation of dowels at transverse contraction joints (16 ). While dowel misalignment or mislocation is suspected to decrease load transfer (16), this topic has not been extensively studied either experimentally or numerically. Here, the effect of transverse joint location is examined by using the dowel mislocation–misalignment feature of EverFE2.2. Model Description The FE model used in this parametric study has the same slab dimensions and material properties assumed in the previous parametric study. However, only two slabs are modeled, because the focus is joint load transfer, and the slab–base shear transfer parameters were fixed at kSB = 0.035 MPa/mm and δ0 = 0.60 mm. The only load case considered is an 80-kN dual-wheel axle located at the joint and centered transversely on the left-hand slab combined with a negative thermal gradient of −0.032°C/mm. Each slab was discretized with 18 × 18 elements in plan, and the slab and base each had two elements through their thickness. Paper No. 03- 2223 Transportation Research Record 1853 Two primary parameters are considered in the analyses: dowel mislocation (simulated through specification of ∆x, as shown in Figure 2a) and dowel looseness. Values of ∆x ranged from −100 mm to 100 mm, where ∆x = 0 corresponds to a perfectly located sawed joint; note that a negative value of ∆x corresponds to a joint sawed too far to the right (i.e., more of the dowel is located in the loaded slab than in the unloaded slab). Dowel looseness was simulated by explicitly modeling gaps of 0 to 0.2 mm between the dowels and slabs, which can have significant effects on joint load transfer (8, 17, 18). The gaps were assumed to vary parabolically along the embedded portions of each dowel, with no gap at the dowel start or end and with the maximum gap at the joint. To ensure sufficient potential points of nodal contact between the dowels and slabs, 24 three-noded flexural elements were used to discretize each dowel. 10 9 8 Dowel Shear (kN) 98 Gap = 0 mm, ∆x = 0 Gap = 0.10 mm, ∆x = 0 Gap = 0.10 mm, ∆x = 100 mm Gap = 0.10 mm, ∆x = -100 mm 7 6 5 4 3 2 1 0 -1500 Parametric Study Results and Significance Peak Dowel Shear (kN) Figure 7 shows the variation in peak dowel shear and total shear transferred across the joint with gap for ∆x = −100 mm, 0 mm, and 100 mm. The peak dowel shear occurs at the third dowel in from the pavement edge, which is centered between two wheels on one side of the axle. As expected, both peak and total shear decrease rapidly with increasing dowel looseness; when ∆ x = 0, total shear transferred across the joint decreases by 73% as the gap increases from 0 mm to 0.2 mm. In addition, the effect of transverse joint location on peak dowel shear is pronounced for intermediate values of dowel looseness (0.05 to 0.10 mm). However, joint location has a small effect on total load transferred across the joint. This can be explained by the equalization of shear between dowels that grows both with increasing gaps and with increasing ∆x. Figure 8 shows the variation in dowel shear across the joint for selected values of dowel looseness and ∆ x, highlighting this equalization of dowel shear. 10 ∆x = -100 mm ∆x = 0 ∆x = 100 mm 8 6 4 2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Magnitude of Gap (mm) 0.2 -1000 -500 0 500 1000 1500 Dowel Location Across Joint (mm) FIGURE 8 Variation in dowel shear across joint with x (gap fixed at 0.10 mm). However, while this equalization of dowel shear can be expected to lead to lower peak dowel–slab bearing stresses, it cannot be concluded from this that dowel mislocation is beneficial. This equalization of dowel shear implies less effective dowel load transfer and higher slab stresses caused by edge loading. In fact, as dowel looseness increases from 0 to 0.2 mm with no joint mislocation, the jointdisplacement load transfer computed between the two wheels on each side of the axle decreases from 99% to 45%, and the peak tensile stress on the slab bottom under the wheel load increases from 0.401 to 0.522 MPa. The results of the analyses indicate that fairly small shifts in joint location can have a large effect on peak dowel shears. Further, dowel looseness has a large effect on joint load transfer. However, total shear transferred across the joint remains relatively constant with joint location, even at shifts in joint location approaching half the embedded length of the dowel. The results of this study cannot be considered conclusive, because only a single load case, system geometry, and set of material properties were considered. Further, dowel mislocation may produce high, localized stresses in the concrete surrounding the dowels that the models employed here cannot capture. However, the need for three-dimensional analysis with which to simulate these effects is evident. (a) SUMMARY AND CONCLUSIONS Total Shear (kN) 40 ∆x = -100 mm ∆x = 0 ∆x = 100 mm 30 20 10 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Magnitude of Gap (mm) (b) FIGURE 7 Variation in dowel shear with joint location and dowel looseness: (a) peak dowel shear and (b) total shear transferred across joint. 0.2 This paper highlighted the features of the program EverFE2.2, which was developed specifically for the three-dimensional FE analysis of jointed plain concrete pavements. EverFE2.2 allows the modeling of one to nine slab–shoulder units with tied adjacent slabs and shoulders and the rigorous treatment of joint load transfer via dowels, aggregate interlock, and transverse tie bars. Dowel misalignment or mislocation can be specified per dowel. In addition, nonlinear thermal and shrinkage gradients can be treated, and slab–base interaction—including separation and horizontal shear stress transfer between the slab and base—can be incorporated in the analyses. The interactive, userfriendly interface of EverFE2.2 eases model generation and result interpretation through simple creation or deletion of a variety of axle Davids et al. types, automatic mesh generation, and efficient visualization of slab stresses and displaced shapes. The use of specialized solvers targeted to the model geometry and mechanics allows solutions to be obtained rapidly on modern desktop machines. Two parametric studies were completed that illustrated the features of EverFE2.2. These studies examined the effect of dowel locking and slab–base shear transfer on pavement stresses due to thermal gradients and uniform slab shrinkage, as well as the effect of dowel misalignment and looseness on pavement response. From these studies, the following conclusions can be drawn: • Slab stresses can be highly affected by shear transfer between the slab and base. In turn, the degree of slab–base shear depends on base type and the particular environmental loading (combination of temperature gradient and uniform shrinkage) considered in an analysis. The complex interaction between the effect of slab–base shear transfer and dowel locking is best captured with three-dimensional FE analysis. • Dowel locking can have an effect on pavement stresses. The effect of dowel locking on stresses caused by shrinkage and by combined shrinkage and thermal gradients is significant for the range of slab–base shear transfer values considered here. The effect of dowel locking was most pronounced for a combined negative thermal gradient and shrinkage, producing an increase in peak tensile stress of 81% when there is a high degree of slab–base shear transfer. • Mislocation of transverse doweled joints can affect joint load transfer. When moderate degrees of dowel looseness exist (0.05 to 0.10 mm), peak dowel shears can be reduced significantly by joint mislocation. However, because of equalization of dowel shears, the total load transferred across the joint remains relatively constant even with a mislocation of the transverse joint approaching half the embedded length of the dowel. • Dowel looseness has a large effect on joint load transfer. Parabolically varying gaps around the dowels as small as 0.20 mm can reduce joint load transfer by as much as 73% under a combined 80-kN axle load and negative thermal gradient. These parametric studies have not fully explored the features of EverFE2.2, and it is expected to be a valuable tool for a wide range of problems in the forensic analysis of pavements as well as pavement design. EverFE2.2 is freely available, and documentation and details for obtaining EverFE2.2 can be found at cae4.ce.washington. edu/everfe/. ACKNOWLEDGMENTS EverFE2.2 was developed with financial support from the Washington and California Departments of Transportation. The authors thank Linda Pierce of the Washington State Department of Transportation and John Harvey of the University of California at Davis for their valuable advice and input during the development of EverFE2.2. Paper No. 03- 2223 99 REFERENCES 1. William, G. W., and S. N. Shoukry. 3D Finite Element Analysis of Temperature-Induced Stresses in Dowel Jointed Concrete Pavements. International Journal of Geomechanics, Vol. 1, No. 3, 2001, pp. 291–308. 2. Davids, W. G., and J. P. Mahoney. Experimental Verification of Rigid Pavement Joint Load Transfer Modeling with EverFE. In Transportation Research Record: Journal of the Transportation Research Board, No. 1684, TRB, National Research Council, Washington, D.C., 1999, pp. 81–89. 3. Kuo, C.-M., K. T. Hall, and M. I. Darter. Three-Dimensional Finite Element Model for Analysis of Concrete Pavement Support. In Transportation Research Record 1505, TRB, National Research Council, Washington, D.C., 1995, pp. 119–127. 4. Shoukry, S. N. Backcalculation of Thermally Deformed Concrete Pavements. In Transportation Research Record: Journal of the Transportation Research Board, No. 1716, TRB, National Research Council, Washington, D.C., 2000, pp. 64–72. 5. Vepa, T. S., and K. P. George. Deflection Response Models for Cracked Rigid Pavements. Journal of Transportation Engineering, Vol. 123, No. 5, 1997, pp. 377–384. 6. Brill, D. R., and I. D. Parsons. Three-Dimensional Finite Element Analysis in Airport Pavement Design. International Journal of Geomechanics, Vol. 1, No. 3, 2001, pp. 273–290. 7. Davids, W. G., G. M. Turkiyyah, and J. P. Mahoney. EverFE: Rigid Pavement Three-Dimensional Finite Element Analysis Tool. In Transportation Research Record 1629, TRB, National Research Council, Washington, D.C., 1998, pp. 41–49. 8. Davids, W. G. Effect of Dowel Looseness on Response of Jointed Concrete Pavements. Journal of Transportation Engineering, Vol. 126, No. 1, 2000, pp. 50–57. 9. Davids, W., and G. Turkiyyah. Multigrid Preconditioner for Unstructured Nonlinear 3D FE Models. Journal of Engineering Mechanics, Vol. 125, No. 2, 1999, pp. 186–196. 10. Zienkiewicz, O. C., and R. L. Taylor. The Finite Element Method, Vol. 1, 4th ed. McGraw-Hill, London, 1994. 11. Davids, W., and G. Turkiyyah. Development of Embedded Bending Member to Model Dowel Action. Journal of Structural Engineering, Vol. 123, No. 10, 1997, pp. 1312–1320. 12. Masad, E., R. Taha, and B. Muhunthan. Finite Element Analysis of Temperature Effects in Plain-Jointed Concrete Pavements. Journal of Transportation Engineering, Vol. 122, No. 5, 1997, pp. 388–398. 13. Pane, I., W. Hansen, and A. R. Mohamed. Three-Dimensional Finite Element Study on Effects of Nonlinear Temperature Gradients in Concrete Pavements. In Transportation Research Record 1629, TRB, National Research Council, Washington, D.C., 1998, pp. 58–66. 14. Rasmussen, R. O., and D. K. Rozycki. Characterization and Modeling of Axial Slab-Support Restraint. In Transportation Research Record: Journal of the Transportation Research Board, No. 1778, TRB, National Research Council, Washington, D.C., 2001, pp. 26–32. 15. Zhang, J., and V. C. Li. Influence of Supporting Base Characteristics on Shrinkage-Induced Stresses in Concrete Pavements. Journal of Transportation Engineering, Vol. 127, No. 6, 2001, pp. 455–642. 16. Guide to Developing Performance-Related Specifications. FHWA-RD98-155, FHWA-RD-98-156, FHWA-RD-98-171, Vol. III, Appendix C. www.tfhrc.gov/pavement/pccp/pavespec/. Accessed March 5, 2003. 17. Zaman, M., and A. Alvappillai. Contact-Element Model for Dynamic Analysis of Jointed Concrete Pavements. Journal of Transportation Engineering, Vol. 121, No. 5, 1995, pp. 425–433. 18. Guo, H., R. M. Larson, and M. B. Snyder. A Nonlinear Mechanistic Model for Dowel Looseness in PCC Pavements. Proc., Fifth International Conference on Concrete Pavement Design and Rehabilitation, West Lafayette, Ind., 1993. Publication of this paper sponsored by Committee on Rigid Pavement Design.
