EQUIVALENT AXLE LOAD FACTORS FOR DESIGN OF RIGID PAVEMENTS DERIVED FROM FATIGUE CRITERIA

This paper presents determination of equivalent axle load factors for rigid pavements on the basis of fatigue criteria used in mechanistic-empirical methods of pavement design. Fatigue criteria for selected methods of rigid pavement design and analysis are presented. These criteria were used to derive the formulas to calculate equivalent axle load factors. It was found that formulas derived for rigid pavements are much more complex than the well known “fourth power equation”. The equivalency factors depend on the slab thickness and bending strength of cement concrete and other pavement parameters. There are considerable differences between the equivalent axle load factors derived from different fatigue criteria used in various design methods. The presented analysis indicates that traffic analysis and fatigue criteria should be interrelated.


Introduction
The equivalent axle load factors are the basis for determining the design traffic and, as a consequence, for designing of road pavements. The most frequently used formula to calculate equivalent axle load factor has the following exponential form: , where F j -equivalent axle load factor; Q s -standard axle load, kN; Q j -actual axle load, kN. In most cases the exponent n = 4 is used in practical design, and therefore the Eq (1) is called "fourth power equation". For rigid pavements the exponent n varies from 4 to 12 in different design methods. The German catalogue of typical pavement structures (RStO 01 -Richtlinien für die Standardisierung des Oberbaues von Verkehrsflächen [Guidelines for the Standardization of the Pavement Structures of Trafficked Areas]) uses the standard axle load of 100 kN and the same uniform exponent n = 4 for all types of pavement structures (flexible, semi-rigid and rigid). French Design Manual for Pavement Structures (Corté, Goux 1996) uses standard axle load of 130 kN and the formula for traffic aggressiveness in form similar to Eq (1) with exponent n = 12 for rigid and semi-rigid pavements, and n = 5 for flexible pavements. The Polish catalogues of flexible, semi-rigid and rigid pavements issued in 1997 and 2001 use stand-ard axle load of 100 kN or 115 kN and exponent n = 8 for rigid and n = 4 for flexible and semi-rigid pavements. The AASHTO Guide for Design of Pavement Structures (1993) uses standard axle load of 80.1 kN (18-kip) and presents a set of tables which define the equivalent axle load factors for rigid pavements in relation to actual axle load, axle configuration, concrete slab thickness and terminal value of p t -Present Serviceability Index. It can be derived from the AASHTO Guide 1993 for rigid pavements that exponent n is in a range of 4. For example, for a concrete slab h = 280 mm and terminal value of p t = 2.5 the exponent n varies from 4.13 to 4.36. In Lithuania, according to Sivilevičius and Šukevičius (2007), the impact of traffic load on pavement structures is expressed by the formula same as Eq (1) presented above with exponent n = 4, and with standard axle load either 80 kN or 100 kN. Load axle equivalency for rigid pavements was a subject of several studies. Hajek and Agarwal (1990) analyzed main factors affecting load equivalency in relation to spacing of dual and triple axles. Gillespie et al. (1993) analyzed effects of heavy vehicles in relation to axle load, axle suspension, axle spacing, tire inflation pressure, tire configuration, type of pavement (rigid and flexible) and environmental factors. Hajek (1995) proposed simplified approach for load equivalency in form independent of pavement related variables. Lin et al. (1996) performed full-scale test on a section of rigid pavement in Taiwan and found that the AASHTO 1993 method yields reliable prediction of the equivalency factors. Kuo and Lin (2001) used finite element model and developed regression formulas for equivalent axle load factors for rigid pavements. These factors differed significantly from the AASHTO 1993 method and were strongly related to concrete slab thickness, slab support and material strength. Chatti et al. (2004) used the laboratory determined fatigue equations to develop load equivalency factors for multiple axle loading. Lukanen (2005) described field tests on instrumented flexible and rigid pavement sections in Minnesota and mechanistic analysis which was used to determine axle load factors. In case of flexible pavements they were able to predict axle load factors on the basis of measured strains but in case of rigid pavements the analysis was more problematic and obtained results were not clear due to erratic strain behaviour in concrete slabs. Smith and Diefenderfer (2009) developed trucks equivalent standard axle factors for flexible and rigid pavements on the basis of weigh-in-motion data. Modern analysis used in regulation policy of permitting super heavy vehicles with multiple axles is based on fatigue cracking life of pavement structures (Chen et al. 2005;Salama et al. 2006). Wardęga (2010) used weighing in motion data on roads in Poland and determined truck equivalency factors for flexible and rigid pavements.
This paper presents analytical determination of axle load equivalency factors for rigid pavements based on fatigue criteria applied in modern mechanistic-empirical methods of pavement design. It was proven in this paper that the equivalency between actual and standard axle load is not constant but is strongly dependent on the fatigue law which is applied in pavement design. For rigid pavements axle load equivalency is much more complex than exponential Eq (1) and depends on several variables, including material properties and pavement structure characterization. The similar approach was used by the author earlier for flexible and semi-rigid pavements (Judycki 2010).

Concept for calculation of equivalent axle load factors based on the fatigue criteria
The following formula is used to convert the actual, varying axle loads to equivalent standard axles: where N s -number of equivalent standard axles corresponding to the N j number of actual axles; N j -number of actual axles; F j -equivalent axle load factor. Axle equivalency means that N s number of standard axles Q s has the same damaging effect to a specific pavement structure as N j number of actual axles of Q j load.
The equivalent axle load factor F j defines the damage to a pavement by pass of the actual axle in question relative to the damage by pass of a standard axle load (Huang 2003). In this paper the following practical definition of equivalent axle load factor F j will be used: "The equivalent axle load factor F j is the ratio between the total number of passes to failure N fs of a standard axles of defined load Q s to the total number of passes to failure N fj of the actual axles of Q j load, for the same pavement structure": . (3) In the mechanistic-empirical methods of design for rigid pavements the fatigue cracking of cement concrete slab caused by repeated axle loading is taken into account. The fatigue laws, which determine the number of axle load applications which can be carried out by the pavement until the failure occurs, are described by the following general function: , where s -bending stresses generated in a concrete slab under a specific axle load; K -group of parameters characterizing a pavement layer, accounted for by the fatigue law, which are not related to the axle load. When combined, the formulas (3) and (4) yield the following formula for calculating the equivalent axle load factor: , where the stresses s depend on the axle load and pavement design parameters. Subscripts j and s denote the values for the actual and standard axles respectively. From the form of fatigue laws it can be seen that functions f(s, K) may be expressed as a product of two independent functions: For this reason the parameters K in Eq (5) will be reduced and the formula for equivalent axle load factor will take a simpler form: . ( The further analysis in this paper is based on the assumption that the traffic induced stresses s in cement concrete slabs are linearly proportional to the axle load value Q: where β -proportionality coefficient, depending on the characteristics of a pavement structure and the loading parameters. This assumption is justified by the Westergaard formula for stresses in concrete slabs (Eq (24)).

Derivation of equivalent axle load factors for rigid pavements
Fatigue criteria for rigid pavements were selected from different design guides. These are: Portland Cement Association (PCA), Zero-Maintenance Fatigue Model (USA), AASHTO 2004 and French design method. For the sake of simplicity the same symbols will be used across different criteria in further analyses. However, sometimes the symbols used here differ from those used in the original formulas of the respective design methods. The symbols used below are: Q -axle load; Q s -standard axle load; Q j -actual axle load in question; N -number of load applications; N f -fatigue life, number of load applications to fatigue cracking of concrete slab; -stress ratio; s -tensile stress in bending, induced in concrete slab by wheel load; R b -bending strength of cement concrete slab, named also as modulus of rupture; β -proportionality factor between traffic induced stress s in concrete slab and axle load Q, as in Eq (8).
Portland Cement Association (PCA) fatigue criteria. The PCA design method uses the following fatigue criteria for cement concrete slabs (Huang 2003): for 0.45 < SR < 0.55, (10) The first of the above formulas, for SR ≥ 0.55, is held in most practical cases and will be considered further. It can be rearranged to obtain the following formula for calculating the equivalent axle load factor: , Assuming linear relation between stress and axle load in form s = βQ it was obtained that: for SR ≥ 0.55.
Zero-Maintenance Design Fatigue Model (USA). It is expressed by the following formula, developed by Darter and Barenberg (1977), with assumption of failure probability equal to 24%: , In a similar way as before it was obtained that: .
(18) Ker et al. (2008) found that the AASHTO 2004 mechanistic-empirical model was not adequate and proposed a new model on the bases of Long-Term Pavement Performance data-base.
French Design Manual (1996). The French criterion in design of rigid pavements is tensile stress induced in cement concrete slab. The value of admissible tensile stress is given by the relation (Corté, Goux 1996) where s t,ad -admissible tensile strength at bending at N load applications; s t (N) -stress for which failure on a 360day concrete sample is obtained for N load applications; k c -factor to account for the observed behaviour of the same type of pavement; k d -factor to account for discontinuities in the base layer; k r -factor to account for variability of test results and calculation risk; k s -factor to account for errors in preparation of base underneath concrete slab. The fatigue law for the cement treated materials is represented by the following expression: , where N -number of load applications inducing stress s(N); γ, β -empirical coefficients; s 0 -initial strength. If actual axle Q j induces in concrete slab stress σj and standard axle load Q s induces stress s s , than: and . (21) After rearrangement the following formula for equivalency load factor is obtained: .
For cement concrete slab in the French method , thus: .
(23) Table 1 presents the derived formulas for equivalency factors for cement concrete pavements.

Relation between stresses induced in concrete slab and axle load
The single axle load with single wheels is considered further. For rigid pavements stresses induced by a single wheel load are calculated from the Westergaard equations at 3 points of the slab: centre, edge and corner. The most appropriate point for designing the jointed concrete pavement is the transverse slab edge, which is most frequently loaded by each passing wheel. The following formula is applied for the slab edge: , where s -tensile stresses in bending, kN/m 2 ; wheel load (half of a single axle load Q, kN); E -modulus of elasticity of cement concrete slab, kN/m 2 ; ν -Poisson's ratio of cement concrete slab; h -thickness of concrete slab, m; k -modulus of subgrade reaction, kN/m 3 ; m, for a < 1.724h, b = a, for a > 1.724h, a -radius of contact area between tire and pavement, m. Tensile stresses in bending of concrete slabs are linearly proportional to the wheel load P, and consequently to the axle load Q. The proportionality coefficient in relation s = βQ is equal to: .
Units are important in this relation. Axle load Q is always expressed in kN, thus s should be expressed in kN/m 2 and β in 1/m 2 , other values are in units given above as explanation to Eq (24). Table 2 presents calculated values of the proportionality coefficient β for the following input data: diameter of tire contact area a = 0.15 m, modulus of elasticity of cement concrete E = 35 × 10 6 kN/m 2 , Poisson's ratio of cement concrete n = 0.15, modulus of subgrade reaction k = 100 × 10 3 kN/m 3 and k = 200 × 10 3 kN/m 3 , thickness of concrete slab h = from 0.16 to 0.30 m. It can be seen that the β coefficient is strongly related to thickness of the slab h and less to modulus of subgrade reaction k.

Calculated values of equivalent axle load factors F j
Calculated values of factors F j depend on the fatigue law and are related to actual and standard axle load, bending strength of concrete and coefficient β, which in turn depends on thickness of a concrete slab h, modulus of subgrade reaction k and other pavement parameters. The formulas presented in Table 1 were used and calculations were performed for the following data: thickness of concrete slab h from 0.16 m to 0.30 m, modulus of subgrade reaction k = 100 × 10 3 kN/m 3 and k = 200 × 10 3 kN/m 3 , bending strength R b = 5.5 × 10 3 kN/m 2 , standard axle load Q s = 100 kN, actual axle load Q j varied from 20 to 160 kN. It is clearly seen from Fig. 1 that the form of fatigue criteria have significant effect on equivalent axle load factor. The exponent n is constant in the French method (n = 16) and nearly constant in the AASHTO 1993 method (n ≈ 4).
In other methods n increases with the increase of the actual axle load Q j . The exponent n derived from the Zero Maintenance and PCA models increases from 4 and 6 at Q j = 20 kN to 10 and 14 at Q j = 160 kN, respectively. The n value from the new AASHTO 2004 method, not shown in Fig. 1, is exceptionally high (n > 20). Fig. 2 indicates that the effect of slab thickness on F j and on the exponent n is very significant. Axle load aggressiveness, indicated by exponent n, is greater for thinner slabs. The effect of slab thickness should be considered in practical analysis. Fig. 3 shows that the effect of subgrade reaction k on equivalency factor is much less significant than the effect of slab thickness h and this effect can be omitted in practical analysis.

Conclusions
The equations for calculating the equivalent axle load factors F j for rigid pavements derived from the analysed fatigue criteria are much more complex than "fourth power equation" which is often used in practice. Equivalency factors F j obtained for rigid pavements deviate strongly from the "fourth power equation". The value of exponent n determined from some fatigue criteria depends on several factors, such as the form of applied fatigue criterion, thickness of concrete slab, standard and actual axle loads, and for most methods varies from 4 to 16. The French criterion gives constant exponent n = 16. The new AASHTO 2004 criterion gives n > 20. The important factor that influences axle load equivalency is the form of the fatigue criterion itself. The second important factor is thickness of the slab. Axle load aggressiveness, indicated by exponent n, is greater for thinner slabs. The effect of the modulus of subgrade reaction is less significant, however greater traffic aggressiveness was noticed for lower slab support.
The main conclusion from this study is that the axle load equivalency factors should be related to applied fatigue criteria. If the design traffic is calculated with use of the "fourth power equation" and after that any selected fatigue criterion is applied in mechanistic-empirical analysis of rigid pavement the results of the design could be erroneous. The axle factors and consequently the number of equivalent axle loads change for the same real traffic if different fatigue law is applied in analysis. The proper approach in the design should be such that after selection of the fatigue criterion for rigid pavement design the method of calculation of equivalency axle load factors should be respectively adjusted. It is important that the fatigue criteria and the load equivalency factors should be interrelated. The presented method of analysis can be used for any other fatigue criteria applied in design of rigid pavements.