DYNAMIC ANALYSIS OF RIGID PAVEMENTS CONSIDERING VEHICLE-PAVEMENT INTERACTION

Abstract

The study of the dynamic response of rigid pavements to moving vehicle/aircraft loads has received significant attention in the recent years because of its relevance to the design of highways and airport runways. The analysis procedure can be broadly grouped under three heads as 1-D, 2-D and 3-D analysis based on the idealization of the rigid pavement as a beam, as a plate and as a 3-D continuum respectively. Various subgrade models like Winkler's soil medium, two-parameter soil medium, 3-D elastic continuum were found suitable for simulating the supporting soil foundation. Although the importance of more accurate dynamic analysis of pavements has been realized, analytical solutions are available for simple cases partly due to mathematical complexities involved in the analysis. Most of the available analytical solutions represent the pavement-soil foundation system by an infinitely long beam or plate, resting on either the Winkler's soil medium or viscoelastic foundation. One of the major drawbacks in these dynamic studies is that the inertia effects of the traveling vehicles/aircrafls are neglected. The dynamic interaction between the moving load and pavement are not fully accounted for in the analysis. In spite of its simplicity, the Winkler's(1864) assumption leads to satisfactory results in the stress analysis of beams on an elastic foundation but restricts its applicability to the soil media which possess the slightest amount of cohesion or transmissibility of applied forces. In such a case the soil media have often been idealized as 3-D continuous elastic solids or elastic continua. Although the modeling of subgrade as a continuum is more accurate, it leads to incorporate the complex and tedious mathematical calculations in the analysis of pavements. Bridging the gap between these two extreme cases, the two parameter foundation models are found to provide a better representation of the underlying soil medium and a conceptually more appealing approach than the one-parameter (Winkler) foundation model. In the recent years, limitations of analytical solutions have been overcame with the development of high-speed computers and efficient numerical techniques such as Finite Element Method. In the analysis of rigid pavements the supporting soil medium is generally assumed to be homogeneous, which is not true. Stresses calculated with this assumption may be erratic when base material is not homogeneous. Adopting the finite element method the non-homogeneous nature of the base materials can be properly accounted in the analysis. In FEA, the structures underneath the soil surface are generally assumed to be surrounded by infinite soil medium, while structures on and near the soil surface are assumed to lie on a semi infinite half space. Fixed boundary conditions can be applied at some distance from the region of interest in case of static analysis. However in dynamic analysis, such boundary conditions will reflect outward propagating waves back into the model. By taking advantage of the material damping a larger model can minimize this problem. However, the increase in model size implies an unwanted, and probably excessive, increase in computational time. A rational and logical approach for modeling such an infinite medium is to divide it into two parts; the near field, and the far field, separated by an artificially truncated boundary. The structure and near field can be effectively and efficiently modeled using the finite element method. Based on the concept of absorption of the wave energy at the artificially truncated boundary, various kinds of artificial boundary conditions [Lysmer and Kuhlemyer, (1969); Kausel et al., (1975); White et al., (1977); Cundall et al., (1978); Liao et al., (1984); Novak and Mitwally, (1988); Hagstrom and Hariharan (1998); Thompson et al. (2001); Zhao, C. and Liu, T.,(2002, 2003)] are proposed in the past, to simulate the effect of the far field on the dynamic response of the structure and near field. Bettess (1977), Zhao and Valliappan (1993a, 1993b, 1994, 1998) developed infinite elements as an. alternative technique to represent the far field of an infinite medium. The development of the infinite elements has helped in proper physical modeling of the far field behavior and also reduction in number of elements and hence computational cost. Kelvin elements are also found to be effective in simulating such boundaries involved in both static and dynamic analyses. It is a well known fact that soil is not a linear material and so to model it as a linear material will lead to considerable error in the analysis. Kondner (1963) has described the non-linear stress-strain relationship in the form of a hyperbolic curve. Very little research has been extended to consider nonlinear behaviour of soil in the rigid pavement analysis. Based on a detailed review of literature pertaining to the above aspects, the solution algorithms based on Finite Element Method are developed to analyze the dynamic response of rigid pavement under moving vehicular or aircraft load. The concrete pavement is discretized by beamlplate/3-D elements. The underlying soil medium is modeled either by Pasternak model or by 3-D continuum. In the Pasternak model the soft subgrade soil (existing soil) is modeled by spring elements and the sub-base course (if needed) is modeled by Pasternak layer. Infnite elements are incorporated to treat the infinite domain in all the above mentioned approaches. In view of the studies reported by Zaman(1993) and Kukreti et al. (1992) highlighting the significance of the dynamic vehicle-pavement interaction (VPI), the inertia effects of the traveling vehicles/aircrafts are properly accounted for in the solution algorithm. Emphasis is also needed to consider the nonlinear behaviour of soil at higher strain levels. It is proposed to consider the following four mathematical approaches/models in the present study. • Pavement as Beam resting on Pasternak's two-parameter model • Pavement as Plate resting on Pasternak's two-parameter model • Pavement descritized by 3-D brick elements resting on 3-D continuum soil medium • Pavement as Beam resting on Pasternak's two-parameter model (Nonlinear analysis) In one-dimensional analysis, the concrete pavement is discretized by finite and infinite beam elements. The exponential decay functions (Bettess, 1977) are used to model the infinite domain. The underlying soil medium is modeled by Pasternak model in which the soft subgrade soil (existing soil) is modeled by spring elements and the base and sub-base courses are modeled by Pasternak shear layer. Analysis is further extended to incorporate nonlinearity of the soil medium using hyperbolic relationship. An iterative incremental approach is employed to account for material nonlinearity. Two-dimensional analysis is based on the classical theory of thick plates resting on two-parameter soil medium that accounts for the transverse shear deformation and bending of the plate. The concrete pavement is discretized by rectangular finite and infinite thick plate elements. Four node isoparametric rectangular thick plate elements with three degree of freedom per node (w, 9x and 9y) are used to model finite domain. The two dimensional infinite elements with reciprocal decay function are used to simulate the infinite domain. Pasternak's two-parameter model is used to model the underlying soil medium To simulate the 3-D nature of pavement-soil system full three-dimensional geometric models were developed. Taking the advantage of symmetry, only one-half of the actual models were built. The pavement-soil system is idealized as an assemblage of 20-node isoparametric continuum elements. These elements are suitable for modeling the response of a system dominated by bending deformations. For the analysis purpose three continuous pavement slabs of dimensions (Lp x Bp) each are considered. The supporting soil medium domain is divided into two regions as near field and far field. The near field is fixed within the range of 3 times the width of pavement slab (Bp) from the centre of the base of the central slab along the longitudinal and vertical direction. To the exterior boundary nodes of the near field Kelvin Elements are attached in both the directions which are supposed to absorb the energy waves propagating in horizontal and vertical directions and thus not allowing them to reflect back into the near field. Also to simulate the far field in the vehicle traversing direction the 16 node 3-D infinite elements are attached to this transmitting boundary. Before solving the dynamic force equilibrium equations for pavement and vehicle in all the above mentioned approaches, the time variable 't' is transformed into a pseudo-time variable 'x' representing the position of the vehicle. The Newmark-Beta integration method is adopted to evaluate the displacements at each time step. In each case a parametric study is conducted to investigate the effect of moving load and some selected soil and pavement parameters on the dynamic response of pavement. A comparison between the analysis with VPI (moving mass case) and analysis without VPI (moving force case) is made to bring out the significance of the dynamic vehicle-pavement interaction effects. Based on the parametric analysis carried out in each case of analysis mentioned above, the major conclusions drawn may be summarized as: 1) The plots of variations in maximum deflection with velocity show a behaviour similar to resonance phenomenon in dynamic problems having several peaks. 2) Critical velocities (velocities corresponding to the peak values of maximum deflections) increases with the increase in soil and shear modulus, while they decreases with increase in thickness of the pavement. Reverse trend is observed in case of maximum deflections. 3) The dynamic vehicle-pavement interaction (VPI) force has got a significant impact on the pavement response. The comparison between the moving mass (analysis (iv) considering VPI) and moving force (analysis without considering VPI) analyses cases revealed that the dynamic vehicle-pavement interaction effects have significantly increased the maximum deflections of the pavement. 4) The 1-D nonlinear analysis has revealed the impact of the material nonlinearity of the soil medium on pavement response showing increase in the magnitude of maximum deflection of the pavements as compared to the corresponding linear analysis. The ultimate stress parameters also have shown significant impact on the pavement response. 5) Emperical relationships are suggested in the non-dimensional form to predict critical velocity and the corresponding maximum deflection at prominent peaks. Predicted values using these realationships are in good agreement with the actual values. 6) From the comparative study of 1-D, 2-D and 3-D analyses, it is noticed that the results obtained from 2-D and 3-D analysis are in close agreement whereas considerable deviation is being observed in the results from 1-D analysis.