ACUERDO DEL CONSEJO GENERAL DEL INSTITUTO ELECTORAL DEL ESTADO DE QUERÉTARO, POR EL QUE AUTORIZA AL PERSONAL DEL INSTITUTO PARA QUE EFECTÚE LA ENTREGA DE LAS BOLETAS ELECTORALES

Odemark’s method of equivalent layer thickness (MET) has been widely used for pavement response analyses (Ullidtz, 1987) and FWD backcalculation (Ullidtz et al., 2006). Ullidtz (1987) showed that the pavement responses (stresses, strains and deflections) calculated by the method of equivalent thickness utilising Boussinesq’s equations were comparable to those calculated for the same structure with the CHEVRON (Elsym 5) computer program. The results obtained with this method varied between 89% and 92% of the values obtained from the theory of elasticity.

In a recent and extensive study – based on a two-layer system with the first layer thickness varying from 2in (50mm) to 15in (380mm), with 5 different modular ratios of E1/E2 ranging from ~3 to ~70 and with a Poisson’s ratio of

0.35 for all layers – El-Badawy & Kamel (2011) concluded that:

 A good agreement existed between the vertical stresses at the interface between the two layers, in a two-layer system, calculated using the theory of elasticity and Odemark’s concept when using a correction factor (f) in the range of 0.8 to 0.9 which agrees with the other literature studies.

 However, at any other depth within each layer, this correction factor is not a constant value but was found to be a function of the layer thickness, depth and modular ratio.

Lu et al. (2008) investigated the relative accuracy of backcalculated moduli using three response models (i.e., to calculate deflections) – Odemark, the linear elastic program WESLEA developed by the U.S. Army Corps of Engineers Waterways Experiment Station, and a layered elastic program called LEAP developed by Symplectic Engineering, Inc. for the University of California – on a three-layer pavement structure consisting of a hot-mix asphalt layer over aggregate base and subgrade (AC/AB/SG) with various known moduli assumed. Results of their analysis showed that the backcalculated moduli from the three models are all within 7 percent of the assumed values. Calculated deflections with MET, from 30 tests sites, were found generally to be within 10% of the measured (from FWD) centreline deflections (Pologruto, 2001).

As reported in Ullidtz and Zhang (2002), strains under the FWD load predicted by using three methods (Odemark method (MET), the linear elastic theory (LET) and the finite element method (FEM)) reproduce with reasonable accuracy real strains measured in pavements, even if in most cases the best agreement (and the most reasonable values for layer moduli) was found using the method of equivalent thicknesses. Similar conclusions were drawn by Zhang & Macdonald (2000). For the horizontal strains at the bottom of the asphalt layer, the calculated values with all three methods could be seen to match the measured values. The calculated vertical strains with MET were within the range of the measured values. Similarly, analysis of the backcalculation results from five 200 m long experimental sites in Tuscany (Italy) demonstrated that a good correlation could be established between the modulus and strain values obtained with the different calculation methods adopted (MET, LET and FEM) (Losa et al., 2008). They noted that all three calculation methods provided fairly similar results (with variations of about 10%) for the horizontal strain at the bottom of the asphalt concrete layer while FEM and LET underestimate by an average of 14% the vertical strain on top of subgrade when compared to the value obtained by the MET approach.

4.5.2.5 Discussion

This literature review shows that elastic multi-layer theory can be used to obtain horizontal strains at the bottom of the asphalt layer. However linear elastic models cannot correctly predict the vertical compressive strain at the top of the subgrade. To overcome this problem two main options are proposed, as indicated in the AMADEUS report (2000):

 One is to make use of a very simple model relying on Boussinesq’s equations, modified for non-linear material behaviour, and Odemark’s transformation of a layered system. Because of its simplicity, this method lends itself quite well to an incremental procedure.

 The other option is to make use of the Finite Element Method (FEM) which could treat material non-linearity mathematically exactly and which also allowed visco-elastic, elastoplastic and anisotropic materials. The main drawback of this method is that it is reported as being very computer intensive, particularly if 3D problems had to be treated.

Although MET (Method of Equivalent Thickness) is not a precise method from a mathematical point of view, it can predict the strains and stresses in pavement layers reasonably well. This should be considered as a simple and efficient method for practical purposes (AMADEUS, 2000). Pearson (2011) also states clear advantages with the use of Odemark’s method:

 It is very fast and simple, can be included in a spreadsheet or used in a pavement management system where stresses and strains must be calculated millions of times. Therefore, this method makes structural analysis of pavements a more practical and convenient tool for engineers.  For most practical purposes, the accuracy of approximate methods such

as the MET should be quite sufficient.

 A non-linear elastic subgrade may be easily included.

Based on the results of the literature review and on the wide use of the BISAR program in the UK for analytical design it was decided to make a comparison between the MET and the BISAR approaches, see chapter 8. The main focus of this research is not in the absolute accuracy in the calculations of stresses

and strains in pavements (and consequent pavement lives). The main objective is rather the development of a practical approach to pavement design where the effect of variations in material properties on pavement performance can be easily assessed and statistically expressed. More precise but more complicated design formulae would necessitate sophisticated computer programs or laborious calculations which are outside the scope of this work.