Journal Fea

16th ASCE Engineering Mechanics Conference July 16-18, 2003, University of Washington, Seattle

FINITE ELEMENT ANALYSIS OF CURLING OF SLABS ON PASTERNAK FOUNDATION Lev Khazanovich1, Member ASCE ABSTRACT Winkler or dense liquid (DL) and elastic solid (ES) subgrade models (figures 1 and 2, respectively) are used most often in pavement modeling. However, neither the DL nor the ES idealization is entirely adequate when applied to real soils and the predictions from both exhibit discrepancies with observed, in situ behavior. The DL model assumes that shear resistance of the subgrade is negligible compared to shear capacity of the subgrade. It models the foundation as a set of independent springs. On the other hand, the ES model attributes to the foundation a higher degree of shear interaction than is usually available in the field. Use of the ES model results in infinite stress predictions under the edges and corners of a plate resting on it. Moreover, both models fail to describe subgrade behavior beyond the slab edge; soil deflections vanish more rapidly than predicted by the ES but not nearly as fast as assumed in the DL formulation.

k

Figure 1. Slab on Winkler (DL) foundation.

Es

Figure 2. Slab on elastic solid (ES) foundation. Also known as the Vlasov model, the Pasternak foundation offers an attractive alternative to the ES continuum in providing a degree of shear interaction between adjacent soil elements (Pasternak 1954, Vlasov and Leontev 1960). One of the possible visualizations of this model is a combination of a shear layer resting on top of the spring layer (figure 3). The deflection profile predicted by the Pasternak foundation vanishes much faster than the corresponding ES basin and may be a better approximation of the deflections observed in a real foundation of finite depth. In the recommendations of the 4th International Symposium on Theoretical Modeling of Concrete Pavements and the 8th International Symposium on Concrete Roads, the Pasternak model was named the preferable option for subgrade modeling. Comparisons between DL and Pasternak 1

Associate Professor, University of Minnesota, Department of Civil Engineering, 500 Pillsbury Drive SE, Minneapolis, MN 55455, email: [email protected]

models presented by Pronk suggest that the latter can be considered a logical improvement of DL (Pronk 1993). G k

Figure 3. Slab on Pasternak foundation. One important generalization of the Pasternak model was proposed by Vlasov and later expanded by Kerr (Vlasov and Leontev 1960, Kerr 1964). Vlasov suggested that if a soft layer is placed on a stiffer subgrade, then a higher order than Pasternak’s model idealization is required to match the “exact” elastic solution accurately. Kerr proposed another physical interpretation of this model as a spring layer placed above a Pasternak foundation (see figure 4). We will refer this idealization as the Kerr model. Kerr and his colleagues obtained several analytical solutions for practically important boundary value problems of the Kerr model.

ku G kL Figure 4. Slab on Kerr foundation. In the past several years, significant progress has been made in the development of analytical tools for analysis of slabs on a Pasternak foundation and application of these tools to pavement problems. Several analytical solutions were obtained for analysis of infinite and semi-infinite slab resting on Pasternak foundations (Cauwelaert 1997, Cauwelaert et al. 2002). Stet and his colleagues used those solutions to develop a backcalculation procedure for determination of Pasternak’s foundation parameters. However, these closed form solutions can analyze only the effect of wheel loading. Moreover, they assume full contact between the slab and the foundation; this condition may not be present in the field due to temperature curling and moisture warping of the PCC slab. A finite element formulation of analysis of a slab on the Pasternak foundation was first developed by Ioannides (Ioannides et al. 1985) and implemented into a finite element program, ILLI-SLAB. This formulation divides the foundation beneath the slab on the rectangular 4noded, 12 degrees-of-freedom elements using the same nodes that are used for decartelization of the plate. Also, the same deflection shape functions that are used for the plate element are used for the Pasternak subgrade. That formulation, however, failed to account properly for the behavior of the subgrade beyond the slab edge. Therefore, it was limited only to the case of interior loading of a sufficiently large slab where the influence of subgrade deformation beyond the slab edges could be neglected. To address this limitation, Khazanovich and Ioannides (1993) introduced semiinfinite elements in one or two horizontal dimensions that properly describe the model behavior near slab edges. The formulation was implemented into a finite element program ILSL2. The finite element formulation proposed by Khazanovich and Ioannides is based on the assumption proposed by Vlasov and Leont'ev (1960) for the soil surface beyond the slab. This

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approximation assumes that deflections beyond the slab are a function of the deflection of the nearest point at the slab edge and the subgrade parameters. ILSL2 can also analyze a slab resting on a Kerr foundation. In deriving the stiffness matrix for the Kerr model, a special 8-noded, 24 degree-of-freedom element was introduced in the present study (see figure 5). The first four nodes are placed at the top of the upper (DL) springs, while the other four nodes are positioned at the top of the Pasternak foundation (i.e., at the bottom of the upper springs). The stiffness matrix for this element is:

 [K] =  

[ K DLU ] - [ K DLU ]

 [ K DLU ] + [ K P ]  - [ K DLU ]

(1)

where [KDLU] is the stiffness matrix of the upper (DL) springs and [KP] is the stiffness matrix for Pasternak foundation. Shear layer 2 Upper spring layer

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1 Lower spring layer

5

4 3 7

Figure 5. 8-noded finite element for Kerr model. Although ILSL2 can analyze the behavior of a slab resting on a Pasternak or Kerr foundation, it has the following limitations: • •

Only wheel loading can be considered if the Pasternak or Kerr model is used for subgrade characterization Only a single slab can be considered if the Pasternak or Kerr model is used for subgrade characterization

These limitations significantly reduce the applicability of ILSL2 and these subgrade models to analysis of portland cement concrete (PCC) pavements. Indeed, curling stresses significantly affect performance of PCC pavements and single slab analysis does not permit evaluation of the effect of joint load transfer on pavement behavior. The recently developed finite element program ISLAB2000 (Khazanovich et al. 1999) retains all the positive features of ILSL2 but is free of some unnecessary limitations. One of the improvements made during development of ISLAB2000 was enabling curling analysis of slabs on the Pasternak and Kerr foundations. ISLAB2000 curling analysis assumes that the slab is separated from the subgrade if the contact between the slab and the subgrade is in tension. Estimation of the contact pressure is easy for the Kerr model because it is equal to the stress in the upper spring layer. Separation between the plate and the Kerr model is easy to implement by cutting springs in the upper layer.

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Analysis of a plate on the Kerr model with applied external wheel and temperature loads is performed in ISLAB2000 using the following iterative scheme: 1. Formulate individually the stiffness matrices for the plate, the spring interlayer, and the Pasternak subgrade and combine these matrices to form the global stiffness matrix. 2. Formulate individually the load vectors for the external applied load (upper slab), selfweight (both slabs) and temperature differentials (Przemieniecki 1968). 3. Combine these load vectors into the corresponding global load vector. 4. Solve the resulting system of equations to determine the nodal displacements and rotations. 5. Check the vertical displacement at each node to determine whether contact is maintained between the slab and the Kerr subgrade. 6. If the difference of displacements at corresponding nodes of the upper spring layer of the Kerr subgrade is negative, the plate and the subgrade are no longer in contact and the corresponding upper layer spring stiffness is set to zero. 7. Return to (1) and perform another iteration. The process should continue until the contact condition at all nodes remains unchanged during two consecutive iterations. Separation of the plate from the Pasternak foundation is a more complex contact problem. Moreover, the finite element formulation presented for the Pasternak foundation uses the same set of node for plate layer and for the Pasternak subgrade, which makes analysis of the slab separation impossible. On the other hand, it was shown by Khazanovich and Ioannides (1993) that the Kerr model produces practically the same results as the Pasternak model if the stiffness of the upper spring layer is sufficiently high. Therefore, curling analysis for the Pasternak model can be performed by converting the Pasternak foundation into the corresponding Kerr foundation with a very stiff upper spring layer. Selection of the fictitious upper string stiffness, however, is not a trivial task. On the one hand, this stiffness should be as high as possible to introduce as little discrepancy between the Pasternak and Kerr models as possible. On the other hand, too high a stiffness causes numerical instability and non-convergence of the finite element solution. The author’s experience shows that the following upper spring stiffness provides good convergence:

ku ≤ G s 2 where

k u is upper layer spring stiffness G is shear layer stiffness s is a minimum finite element side size More research is needed to establish more comprehensive recommendations. Several examples demonstrated that the program produces reasonable results and converges to known solutions in limited cases. The importance of accounting for slab separation for accurate determination of the slab stresses was also shown. KEY WORDS: Pavement, Pasternak foundation, finite element analysis.

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REFERENCES Cauwelaert, F. van, 1997. TheLlimited Rectangular Slab on a Limited Pasternak-Kerr Foundation. Internal report of the CROW Workgroup UEC, Ede, The Netherlands. Cauwelaert, Van F., Stet, M.J.A., Jasienski, A., 2002. The General Solution For a Slab Submitted to Centre and Edge Loads and Resting on a Kerr foundation, The International Journal of Pavement Engineering, Vol. 3, No. 1, pp. 1-18, ISSN 1029-8436. Ioannides, A.M., Thompson, M..R., and Barenberg, E.J. (1985). Finite Element Analysis of Slabs-On-Grade Using a Variety of Support Models. Proc., Third Int. Conference on Concrete Pavement Design and Rehabilitation, Purdue University, Apr., 309-324. Kerr, A.D., 1964. Elastic and visco-elastic foundations models. Journal of Applied Mechanics, ASME Vol. 31, pp. 491-498. Khazanovich, L. (1994). Structural Analysis of Multi-Layered Concrete Pavement Systems. Ph.D. Thesis, University of Illinois, Urbana, IL. Khazanovich, L., and A.M. Ioannides. (1993). Finite Element Analysis of SlabsOn-Grade Using Improved Subgrade Soil Models. Proceedings, ASCE Specialty Conference - Airport Pavement Innovations:Theory to Practice. Waterways Experiment Station, Vicksburg, MS, pp. 16-30. Khazanovich, L., H.T. Yu, S. Rao, K. Galasova, E. Shats, and R. Jones. (2000). ISLAB2000—Finite Element Analysis Program for Rigid and Composite Pavements. User’s Guide. Champaign, IL: ERES Consultants. Vlasov, V.Z., and Leontev, N.N. (1960). Beams, Plates and Shells on Elastic Foundations. NASA-NSF, NASA TT F-357, TT 65-50135, Israel Program for Scientific Translations (translation date: 1966). Pasternak, P.L., 1954. On a new method of analysis of an elastic foundation by means of two foundations constants (in Russian). Gasudarstvennoe Izdatelstvo Literaturi po Stroitelstvui Arkhitekure, Moscow, USSR. Pronk, A.C., 1993. The Pasternak foundation – An attractive alternative for the Winkler foundation. Proceedings of the 5th International Conference on Concrete Pavement Design and Rehabilitation, Vol. 1, Purdue University, West Lafayette, USA. Pronk, A.C., 1994. The wedge-segment (pie) load case in the Westergaard model. Proceedings of the 3rd International Workshop on Theoretical Design of Concrete Pavements, pp. 61-70, Krumbach, Austria. Przemieniecki, J.S., 1968. Theory of Matrix Structural Analysis, McGraw Hill, New York, N.Y.

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