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B-1 APPENDIX B DEVELOPMENT OF ALTERNATIVE M-E JPCP CRACKING MODEL TO INCORPORATE SLAB-BASE EFFECTS A PCC/base interface model adapted in this study is the simplified friction model that is based on the following assumptions (Khazanovich and Gotlif 2002). ï· Friction forces on the interface between two layers do not depend on vertical pressure but are related to the difference in horizontal strains at the bottom surface of the upper layer and top surface of the lower layer (i.e. simplified friction modeling of the interface). ï· Each layer has the same deflection basin (i.e., no separation is allowed). ï· The stress normal to mid-plane, ï³z, is small compared with other stress components and may be neglected. ï· The two-layered slab can be modeled as a system of two modified medium-thick Kirchhoff plates. This means that each layer has a horizontal plane that remains unstrained subsequent to bending and normal after deformations to all cross-sections to which it was normal before deformation. Note that a tradition Kirchhoffâs hypothesis requires that a neutral plane be a mid-plane. The generalized hypothesis does not require that such a plane be even within the plate thickness limits. Given the adoption of the simplified friction model instead of the âall or noneâ approach, the implementation of the simplified friction model adopts a friction coefficient Î as a parameter characterizing a degree of friction at the slab-base interface. The following function form for friction forces form satisfies these conditions: ï¨ ï©)(* 2,1,2,1, tyybyytxxbxxfxxN ï¥ï¥ïï¥ï¥ ïï«ïïï½ 1 ï¨ ï©)(* 2,1,2,1, txxbxxtyybyyfyyN ï¥ï¥ïï¥ï¥ ïï«ïïï½ 2 )(* 2,1, t xy b xyfxyN ï¥ï¥ ïïï½ 3 where Îµxx, Îµyy, and Îµxy are strain tensors in Cartesian coordinates, Î is a friction factor, Nfxx, Nfyy, and Nfxy are components of a vector of frictional forces at the interface, Nf, and the superscripts b,1 and t, 2 denote the bottom surface of the upper layer and top surface of the lower layers, respectively. A very small value of Î refers to low friction and behavior is expected to be close to an unbonded interface. A very high value of Î refers to high friction and behavior is expected to be close to a full-bond interface. If bonded/unbonded interface is assumed, then the two-layered system can be analyzed using the transformed section concept as developed by Ioannides et al (1992). In the transformed section concept, an original two-layered system (a new PCC pavement and a base layer or PCC overlay and existing pavement) is replaced by a fictitious, composite, homogeneous plate that would exhibit the same deflection basin as the original system. The following discussion further illustrates transformed sections. Figure B-1 provides an example of the effect of bond between slab and base interaction on equivalent slabs relative to an original layered system. In Figure B-1, for specified layer elastic moduli, Poisson ratios, and Winkler subgrade stiffness for 9 inches of PCC over 6 inches
B-2 of lean concrete base material, the effective thickness heff of the equivalent slab is 9.22 inches if an unbonded interface is assumed and 11.82 inches if a bonded interface is assumed. Figure B-1. Transformed section concept (from Khazanovich and Gotlif 2002) While this concept has been successfully implemented in the AASHTO M-E PROCEDURE, it only accounts for fully bonded or fully unbonded interfaces. Khazanovich and Gotlif (2002) generalized the Transformed Section concept for the case of a simplified friction interface. This generalization introduced coefficient of friction Î and the Transformed Section concept. For the generalized Transformed Section concept of Khazanovich and Gotlif (2002), consider the equivalent slab thickness heff is expressed in terms of the effective flexural stiffness Deff, Poisson ratio Î¼ for the two layers, and PCC layer elastic modulus E1. 3 1 2 )1(*12 E D h eff eff ïï ï½ 4 The flexural stiffness Deff can be expressed in terms of the respective layer elastic moduli E1 and E2, thicknesses h1 and h2, where subscript 1 designates the PCC layer and subscript 2 designates the base layer. ï¨ ï© ï¨ ï©ï·ï· ï¸ ï¶ ï§ ï§ ï¨ ï¦ ï«ï ï ï«ï«ï ï ï½ï«ï½ 2222 2 2 3 22 22 111 2 1 3 12 1 21 33 1 33 13 1 bhbhh E bhbhh E DDDeff ïï 5 The influence of the friction factor Î in Deff is found in parameters b1 and b2, expressed in Equations 24 and 25 below. ð1 = 0.5ð¸2 ââ2 2 + â1Î Î + ð¸2 â + Î Î ( Î(0.5ð¸1 ââ1 2 + â1Î) â (ð¸1 ââ1 + Î)(â0.5ð¸2 ââ2 2 + â1Î) ð¸1 âð¸2 ââ1â2 + ð¸1 ââ1Î + ð¸2 ââ2Î ) 6 ð2 = Î(0.5ð¸1 ââ1 2 + â1Î) â (ð¸1 ââ1 + Î)(â0.5ð¸2 ââ2 2 + â1Î) ð¸1 âð¸2 ââ1â2 + ð¸1 ââ1Î + ð¸2 ââ2Î 7 Figure B-2 illustrates the relationship between location of the neutral plane in each layer measured from the top of the slab and non-dimensional friction factor, Î*, defined as follows: Original slab thickness = 9.2 in LCB Equivalent plate unbonded interface Equivalent plate bonded interface thickness = 6 in PCC thickness = 9 in thickness = 11.82 in
B-3 Îâ = Î ð¸1â1 + ð¸2â2 8 The use of this parameter for friction illustrates the influence that a simplified friction (partial bond) model can have on the analysis of slab-base behavior in rigid pavement systems. To illustrate the behavior of a simplified friction model, the two-layered slab considered in Figure B- was analyzed for friction factors varying from 5 lb/in to 5.0x108 lb/in. Figure B-2. Effect of friction on neutral axis location (Khazanovich and Gotlif 2002). As Figure B-2 depicts clearly, an increase in friction reduces the distances between the neutral axes of the layers. In the case of a fully bonded interface, the slab has only one neutral plane, that of the effective slab itself. Furthermore, Figure B-3 illustrates the effect of the partial bond at the interface (through Î*) on effective slab thickness. Figure B-3. Contribution of bond to effective slab thickness (Khazanovich and Gotlif 2002). Hence, the use of the simplified friction model moves beyond an unbonded or fully bonded condition while retaining the direct analysis of that approach. It should be noted that in spite of its simplicity, the model has the following drawbacks:
B-4 ï· The coefficient of friction Î is not a material parameter. Furthermore, its value depends on layer thicknesses and material properties. ï· The model assumes no separation between pavement layers. ï· It is quite possible that load level and temperature conditions can affect Î adversely. The simplified friction model has been implemented in ISLAB2005, a commercial finite element analysis package for rigid pavements (Khazanovich et al 2000). The use of this model allows for detailed slab-base interaction that has measurable effects on the response of the pavement structure to wheel and thermal loading. However, the main advantage of the model is that it is compatible with the stress calculation procedure implemented in AASHTO M-E B.1 Introduction to modifications to the AASHTO M-E JPCP transverse cracking model The modified JPCP transverse cracking prediction model to account for slab-base interaction considers all of the inputs in a manner that is nearly identical to the existing AASHTO M-E procedure, and where they are treated differently, the difference is due to major revisions to the underlying models to account for slab-base interaction. The modified AASHTO M-E damage calculation and performance prediction process closely follows the existing procedure with some additional steps to account for the gradual deterioration of the slab-base interaction coefficient Î. The modified procedure for JPCP transverse cracking involved major revisions of three main areas: 1. Thermal linearization 2. Stress analysis and damage calculation 3. Cracking prediction (outputs) The final four steps of the development process involved many modification to the transverse cracking procedure. Table B-1 summarizes the difference in the two models in regards to the estimation of the structural response and damage calculation, where ï· Ï is the critical stress, ï· MR is the modulus of rupture (i.e. the strength criteria), ï· n is the applied load repetitions, and ï· N is the allowable number of load repetitions given changes to the input wheel load, climate, and new slab-base interface parameter, where ï· ï* is the dimensionalized slab-base interface parameter, derived from the new input ï that can be used in the new procedure to represent slab-base interaction; ï· P is the load level applied incrementally for a given load type; ï· Load type is the axle type (single, tandem, and tridem for bottom-up cracking; short, medium, and long wheelbase for top-down cracking); ï· Traffic wander is the load location in the wheel path; ï· Hour refers to the age of the slab in terms of changes to PCC flexural strength, PCC elastic modulus, shoulder LTE, and other sensitive parameters; ï· ïTL is the linear temperature difference through the equivalent slab;
B-5 ï· TNL is the nonlinear top or bottom surface temperature; and ï· TNLï±a is the nonlinear surface temperature at characteristic length, a, away from the top or bottom surface of the slab. Thus Table B-1 illustrates the sensitivity of the structure response and damage parameters to the aggregate input parameters in the original JPCP transverse cracking model and in the modified cracking model to be discussed in the sections below. In addition, note that the use of an exclamation point in Table B-1 indicates partial or qualified sensitivity of the response/damage calculation to the input parameter. That is, the slab-base interface influences the structural response calculation of sigma in the 1-37A model only in terms of loss-of-friction; furthermore, the nonlinear top and bottom surface temperatures used in 1-37A are exact only for 18 kip loads. Table B-1. Summary of differences in the original JPCP transverse cracking procedure and the modified procedure developed to account for slab-base interaction Original NCRHP 1-37A Procedure Alternative Procedure Ï MR n N Ï MR n N Slab-base interface (ï*) ï¡ ï¼ ï¼ P ï¼ ï¼ ï¼ ï¼ ï¼ ï¼ ï¼ Load type ï¼ ï¼ ï¼ ï¼ ï¼ ï¼ ï¼ Traffic wander ï¼ ï¼ ï¼ ï¼ ï¼ ï¼ ï¼ Hour ï¼ ï¼ ïTL ï¼ ï¼ ï¼ ï¼ ï¼ ï¼ ï¼ TNL ï¡ ï¡ ï¡ ï¼ ï¼ ï¼ ï¼ TNLï±a ï¼ ï¼ ï¼ Finally, a very important feature of the modified procedure to be described in the following sections is that the developed procedure can be reduced entirely to the original AASHTO M-E procedure if the user so desires. This of course ignores special arbitrary âtuningâ coefficients that are present in the existing AASHTO M-E procedure and not in the newly developed, modified procedure. This is to say that the fundamental M-E modeling framework of the modified model can reproduce the original model given certain limitations. B.2 Major modifications to the processing of EICM temperature data for the modified JPCP transverse cracking model The AASHTO M-E procedure â and likewise the new cracking model â considers climate inputs from EICM project files in two main areas: ï· Monthly relative humidity data is used to estimate monthly differential shrinkage; and ï· Hourly pavement temperature profile data is converted to distributions of equivalent linear temperature differences by calendar month. The modified cracking model incorporating slab-base interaction does not modify the first of these two focuses. The reader is referred to the AASHTO M-E documentation for more information on that area. However, the new cracking procedure includes a completely revised
B-6 treatment of the use of hourly pavement data to develop monthly linear temperature difference tables. Previously, under the procedure developed in NCHRP 1-37A and adopted for the AASHTO M-E design procedure, the thermal linearization was conducted for bonded and unbonded interfaces for the 12 calendar months in a representative design year (meaning the single year typified any given year in the entire design life). The thermal linearization in the previous scheme depends on effective thickness, which in turn depended on friction (i.e. bonded or unbonded). The new thermal linearization process, developed in this research, is conducted for given the interface (or value of ï) associated with every month in the design life. The following steps in the use of temperature profile data are detailed below, including preliminary processing involving ï ahead of the thermal linearization. B.3. Modified stress calculation procedure The modified stress calculation procedure for AASHTO M-E performance modeling accounts for the following factors: ï· Interface conditions between the PCC slab and the base ï· Coefficient of subgrade reaction ï· Joint spacing ï· Load transfer efficiency of a PCC/shoulder joint ï· Axle type ï· Axle weight ï· Tire pressure ï· Wheel spacing ï· PCC thickness ï· PCC modulus of elasticity ï· PCC Poissonâs ratio ï· PCC unit weight ï· PCC coefficient of thermal expansion ï· Base thickness ï· Base modulus of elasticity ï· Base unit weight ï· Base Poissonâs ratio ï· PCC temperature at 11 evenly spaced nodes in the PCC layer The AASHTO M-E procedure employs a computationally efficient procedure for computing the critical stresses using rapid solution. For bottom-up damage, it employs two rapid solutions: ï· NNA predict stresses in single layer single slab subjected to the combined effect of the slab curling and axle loading. The size of in-plane dimensions of the slab are the same as in the designed PCC pavement ï· NNB predict stresses in a single layer two-slab system subjected to an axle loading only. The load transfer efficiency between two slabs is equal to the load transfer efficiency of the PCC slab/shoulder joint. Using the combination of these two solutions and the similarity concept the stresses in the design PCC pavement are computed. Currently, AASHTO M-E allows only for full bond and full slip conditions for the interface. The partial friction model permits accounting for intermediate conditions and in extreme cases of very low and very high friction yields unbonded and bonded conditions, respectively. Below the modification of the AASHTO M-E stress calculation procedure for the simplified interface model are presented in the format closely followed NCHRP 1-51 report Apppendex QQ (http://onlinepubs.trb.org/onlinepubs/archive/mepdg/2appendices_QQ.pdf).
B-7 B.3.1. Calculate the Equivalent Slab Thickness Compute equivalent slab thickness using Equation 4. B.3.2. Calculate Unit Weight of the Equivalent Slab The AASHTO M-E equivalent slab unit weight is determined as follows: eff PCCPCC eff h hï§ ï§ ï½ 9 where ï§eff = effective unit weight hPCC = PCC slab thickness ï§PCC. = PCC unit weight heff = effective thickness Equation 9 implies that the self-weight of the equivalent slab is equal to the weight of the PCC slab and the base if the interface is bonded. If the interface is unbonded then the weight of the equivalent slab is equal to the weight of the PCC slab only to account for a possible separation between the PCC slab and base. To ensure that the developed solution yields the results intermediate between the full bond and full slip, the following equation of the effective unit weight is proposed: ð¾ððð = âðð¶ð¶ âððð ð¾ðð¶ð¶ + âððð ð âððð ð¾ððð ð â ( âððð â âð¢ðð âðððð â âððð ) 10 B.3.3. Calculate Radius of Relative Stiffness and Thus, the radius of relative stiffness, â, uses the modified effective thickness heff that considers Î and is â = â ð¸ðð¶ð¶(âððð) 3 12ð [1 â (ðððð) 2 ] 4 11 where ï¬ = radius of relative stiffness heff = effective thickness EPCC = PCC elastic modulus ïPCC= PCC Poissonâs ratio k = coefficient of subgrade reaction
B-8 B.3.4. Calculate Effective Temperature Differential The equivalent temperature difference should produce the same effective bending moment as original temperature distribution. Accounting for the fact that the temperature of the base layer is assumed to be constant throughout the base depth and equal to the temperature at the bottom of the PCC slab, ð0. This leads to the effective linear temperature is has the following form: Îðððð = 12(1 â ð2) ð¸1âððð 2 (â â« ð¸1 1 â ð2 ð¼1(ð1(ð§1) â ð0)ð§1 dz1 â1âð1 âð1 + ð¼1ð¸1â1(0.5â1 â ð1) + ð¼2ð¸2â2(0.5â2 â ð2)) 12 Where T1(z) is the temperature in the PCC layer at the distance z from the neutral axis. B.3.5. Compute Korenevâs Non-dimensional Temperature Gradient eff effe 2 2 PCCPCC T k h ) + (1 2 = ï ï§ ïï¡ ï¦ ï¬ 13 Where ï¦= non-dimensional temperature gradient hPCC = PCC slab thickness ï¡PCC = PCC coefficient of thermal expansion ïPCC = Poisson's ratio for PCC ï§eff = effective unit weight k = modulus of subgrade reaction (k-value) ï = radius of relative stiffness ïTeff = effective temperature gradient B.3.6. Compute Adjusted Load/Pavement Weigh Ratio (Normalized Load) hWL P = q effeffï§ * 14 where q* = adjusted load/pavement weigh ratio P = axle weight. hPCC = PCC slab thickness ï§PCC = PCC unit weight L = slab length W = Slab width
B-9 B.3.7. Determine effective slab thickness The AASHTO M-E design procedure define the effective slab thickness is a thickness of the slab with the modulus of elasticity and Possionâs ratio equal to 4,000,000 psi and 0.15, respectively, resting on the Winkler foundation with the coefficient of subgrade reaction equal to 100 psi/in, and having the same radius of relative stiffness as the equivalent slab. The effective slab is determined using the following equation: =heq 3 4 3410 ï¬ 15 Where heq = equivalent slab thickness, in ï¬ = radius of relative thickness, in B.3.8. Compute Curling-Related Stresses in the Effective Slab Using NNA, compute stresses in the effective plate which has the same ratio of radius of relative stiffness to joint spacing, joint spacing, traffic offset and appropriate Korenevâs nondimensional temperature gradient, ï¦, and normalized load ratio q*. The following cases should be considered: ï· Case I â Combined effect of load and temperature: Korenevâs nondimensional temperature gradient, ï¦, is equal to the nondimensional temperature gradient determined in Step 5; normalized load ratio q* is equal to normalized load ratio determined in Step 6. ï· Case II â Temperature only: Korenevâs nondimensional temperature gradient, ï¦, is equal to the nondimensional temperature gradient determined in Step 5; normalized load ratio q* is equal 0. ï· Case III â Axle load only: Korenevâs nondimensional temperature gradient, ï¦, is equal to 0; normalized load ratio q* is equal to normalized load ratio determined in Step 6. B.3.9. Compute Curling-Related Stresses in the Equivalent Structure The stresses obtained in step 8 represent stresses the slab with the modulus of elasticity and Possionâs ratio equal to 4,000,000 psi and 0.15, respectively, resting on the Winkler foundation with the coefficient of subgrade reaction equal to 100 psi/in, and having the same radius of relative stiffness as the equivalent slab. The stresses in the equivalent slab are determined using the following equations: ),(),( TP h h TP A effeq eqeffA eff ïï½ï ï³ ï§ ï§ ï³ 16 ),0(),0( T h h T A effeq eqeffA eff ïï½ï ï³ ï§ ï§ ï³ 17
B-10 )0,()0,( P h h P A effeq eqeffA eff ï³ ï§ ï§ ï³ ï½ 18 Where Aï³ = stress in the equivalent structure A eff ï³ = stress in the effective structure (obtained using NNs0 effh = effective slab thickness eqh = equivalent slab thickness effï§ = equivalent slab unit weight eqï§ = effective slab unit weight B.3.10. Using NNB Compute load-only caused stresses in the effective structure from the wheels located at the mid-slab In the case of a single axle loading, compute stresses from all wheels in the axle. Two sub-steps are required: 1. Compute stresses in the effective structure assuming that there is no load transfer between the slabs in the system B (LTE=0). If the axle consists from dual tires, they should be treated as two sub-axles. Calculate stresses separately from these sub-axles and superimpose the resulting stresses to obtain )0(1B eff ï³ . 2. Compute stresses in the effective structure assuming that the load transfer efficiency between two slabs in the system B is equal to shoulder LTE. If the axle consists from dual tires, they should be treated as two sub-axles. Calculate stresses separately from these sub-axles and superimpose the resulting stresses to obtain to obtain )(1 sh B LTE eff ï³ . B.3.11. Determine load-only caused stresses in the effective structure from the entire axle )()( )0()0( 1 1 sh B sh B BB LTELTE effeff effeff ï³ï³ ï³ï³ ï½ ï½ 19 B.3.12. Determine Load-only Caused Stresses in the Equivalent Structure The load-only causing stresses in the equivalent structure can be determined using the following expressions: )0()0( 2 2 B eff eff B eq eff h h p p ï³ï³ ï½ 20
B-11 )()( 2 2 sh B eff eff sh B LTE h h p p LTE eq eff ï³ï³ ï½ 21 where )0(Beffï³ = stresses in the effective structure if there is no load transfer between the slabs in the system B (LTE=0) )( sh B eff LTEï³ = stresses in effective structure if the load transfer efficiency between two slabs in the system B is equal to shoulder LTE. )0(Bï³ = stresses in the equivalent structure if there is no load transfer between the slabs in the system B (LTE=0) )( sh B LTEï³ = stresses in equivalent structure if the load transfer efficiency between two slabs in the system B is equal to shoulder LTE. eff h = effective slab thickness eq h = equivalent slab thickness eff p = wheel pressure in the effective system =100 psi p = actual wheel pressure B.3.13. Find stress load transfer efficiency for the given axle load configuration and the axle load position. )0( )( B sh B stress LTE LTE ï³ ï³ ï½ 22 B.3.14. Find axle loading induced component of bending stresses Find axle loading induced component of bending stresses (stress in the slab caused by the action of axle loading on top of the temperature curling) in the equivalent structure if the shoulder provides no edge support to the traffic lane slab. )0()0,(),0(),(, BAAA shouldernoload PTTP ï³ï³ï³ï³ï³ ï«ïïïïï½ 23 Also determine the axle loading induced component of bending stresses (stress in the slab caused by the action of axle loading on top of the temperature curling) accounting for the shoulder edge support to the traffic lane slab. stressshouldernoloadshoulderload LTE*,, ï³ï³ ï½ 24 B.3.15. Find combined stress in the equivalent system curlshoulderloadcomb ï³ï³ï³ ï«ï½ , 25
B-12 B.3.16. Find bending PCC stresses ðððð(ð) = â1ð¾2â2 2 â2ð¾1â1 2 ððð(ð) 26 B.3.17. Find total PCC stresses NLTbendPCCPCC ï³ï³ï³ ï«ï½ , 27 Where PCCï³ is the total stress at the bottom of the PCC slab and where ÏNLT is defined as ððð¿(ð§) = ð¸1ð¼1 1 + ð (ð(ð§) â ð01) 28 where T01 is the slab reference temperature and the nonlinear temperature distribution is TN(z) TN(z) is expressed as a function of position, z, through the layer thickness ðð(ð§) = ð(ð§) â ðð1 â ðð¿1(ð§) + 2ð01 29 and where TC1 is a constant strain temperature through the layer and TL1 is the linear strain temperature through the layer. For the stress calculation of Equation 26, the analysis would assume a value of h1 for z (corresponding to the bottom of the slab, i.e. the interface location) in Equations 28 and 29. It can be observed that implementation of the simplified friction model required minimal changes in the stress calculation algorithm. The original AASHTO M-E rapid solutions could be used in this process; the parameters that would require modifications could be derived using closed form analytical solutions. B.4. Revised thermal linearization procedure The previous thermal linearization process used by the AASHTO M-E JPCP transverse cracking procedure has some drawbacks that directly influence analysis dealing with slab-base interaction. The main disadvantage to the linearization process in the existing cracking model is that it assumes that the component of stress due to the interaction between nonlinear temperature and traffic is constant for all traffic loads. While this assumption has been overlooked until now, it is obviously a simplification that should be called into question when investigating the effect of slab-base interaction on the damage accumulation process in the AASHTO M-E procedure. For this reason, the new cracking model reconsiders and revises the thermal linearization process significantly. The AASHTO M-E linearization process eliminates the need to compute the number of loads as a function of both linear and nonlinear temperature gradients by equating stresses due to nonlinear gradients with those due to linear gradients. The first step in this process is to compute the monthly PCC stresses frequency distribution in the pavement at critical locations for linear temperature gradients, ÎTL, non-linear temperature gradients, TNL, and standard axle loading.
B-13 For bottom-up damage accumulation, an 18-kip single axle load is placed at the mid-slab edge, where it will produce the maximum stress, as shown in Figure B-4 on the left. For top-down damage accumulation, a 12-kip single axle load and a 34-kip tandem axle load with a medium wheel base is placed at the critical loading location, as shown in Figure B-4 on the right. Figure B-4. Critical load and stress locations for bottom-up cracking (at left) and top-down cracking (at right) [from ARA (2004)] The second step in the linearization process involves finding the frequency distribution of linear temperature gradients, in increments of 2Â°F, which produces the PCC bending stress frequency distribution (without non-linear temperature stresses) that is the same as the stress distribution from the previous step. Though the temperature frequency distribution for each month is developed only for the standard load and wheel offset conditions, it is then used in the fatigue analysis for all axle loads and offsets conditions. This process drastically reduces the amount of computing required to estimate stresses, which was a major concern in the development of the original AASHTO M-E procedure. While the frequency tables that result from this methodology are very large, we examine a small subset of those tables to better illustrate the concept. In the example of Table B-2 and Table B-3, we have a given project whose thermal gradients have been linearized according to the revised method developed in this research. Table B-2 presents the probability of different combinations of ïTL and TNL in the pavement system for the first two hours of any day (12-1 AM and 1-2 AM) in the the first month (January). For the sake of illustration, let us consider the likelihood of ïTL = -22ï°F and TNL= -2ï°F (at the top surface) in both hours of a January day. Table B-2 outlines and highlights the probabilities of that combination of thermal characteristics in the first two hours of that day (the values being 0.0435 and 0.0734, respectively). The new thermal linearization process adds an additional dimension to this array of probabilities, in that it also determines the likelihood of TNLï±a within a specific combination of ïTL and TNL. Thus, Table B-3 illustrates the probabilities of TNLï±a taking on different values for the specific case when ïTL = -22ï°F and TNL= -2ï°F in the first two hours of a January day. As the thermal linearization method was implemented and tested extensively, there are simple mathematical self-consistency checks that can be illustrated to reassure the reader of the implementation. For instance, in Table B-2, all probabilities associated with Hour 1 in the lookup Table B-sum to a value of 1; the same is true for all probabilities associated with Hour 2. Likewise, in Table B-3, the expressed probabilities for each value of TNLï±a sum to the probability associated with the ïTL = -22ï°F and TNL= -2ï°F for the hour indicated in Table B-2.
B-14 Table B-2. Example of frequency distribution tables providing probability of a given combination of ïTL and TNL for a given hour of a specific calendar month Table B-3. Example of frequency distribution Table B-expanding probability of total frequencies from Table B-2 with respect to TNLï±a
B-15 B.5. Major modifications to the damage calculation in the modified JPCP transverse cracking prediction model This step involves the calculation of accumulated top-down and bottom-up damage in the slab given the structure response (i.e. critical stress calculation) according to Minerâs hypothesis, where ð¹ð· = â ðð,ð,ð,ð,ð,ð,ð ðð,ð,ð,ð,ð,ð,ð 30 where n is the applied number of load applications; indices i through o refer to age (i), month (j), axle type (k), load level (l), equivalent temperature difference (m), traffic offset (n), and hourly truck traffic (o) (respectively); and N is the allowable number of load applications such that log(ðð,ð,ð,ð,ð,ð) = ð¶ð â ( ðð ð ðð,ð,ð,ð,ð,ð ) ð¶ð â log (1 â ð¶ð ðð,ð,ð 0 + ððð¿ ðð¡ðð¡ ) + ð´3 31 where ðð,ð,ð 0 is the bending stress under a thermal load with no applied load for the same equivalent temperature gradient, ððð¿ is the stress due to the non-linear temperature gradient, ðð¡ðð¡ is the total stress (the sum of the bending and non-linear stresses); ðð ð is the PCC modulus of rupture at age i in psi; ðð,ð,ð,ð,ð,ð is the applied critical stress for associated indices in psi; A3 is a term equivalent to 0.4371 to match the existing AASHTO M-E procedure (note that this value had been zero in previous versions of the AASHTO model); and Ca and Cb are calibration constants (AASHTO, 2008). In addition, a calibration term, associated with calibration constant Cc, has been added to better account for curling stresses in the system, where: ï· if the curling stress is zero, then allowable number of repetitions is calculated the same as in the existing AASHTO M-E procedure; and ï· if a curling stress exists, then the number of allowable load repetitions will be higher to account for a reduction in the stress range during the axle load repetitions. Finally, it should be noted that the estimation of the modulus of rupture, MRi, which constitutes the failure criteria for the equivalent system, has been modified for the new cracking model. This modification was made due to the fact that it was found during the re-creation of the AASHTO M-E procedure for transverse cracking that the existing model was calculating unrealistically high stresses due to high nonlinear temperatures at either the top or bottom surface. To reduce the severity of these stresses, it was found that the existing AASHTO M-E model introduced and arbitrary calibration factor of roughly 0.4 to act as a multiplier of the calculated stress. In developing the modified procedure, a more rational approach developed was developed and adopted. That approach involves the use of a fictitious modulus of rupture that is a function of stress at the surface and stress at a location that is a characteristic length away from surface. This determines a sort of gradient of stresses in the slab, and the higher this gradient is, the higher the apparent strength of the slab. Section B.4.5.1 more closely details modifications to the approximation of MRi in the damage criteria.
B-16 B.5.1. Energy of elastic deformation strength criteria The following attempts to develop a criteria for fracture initiation in a quasi-brittle material under a non-uniform uniaxial tensile stress field. The criteria hypothesis is that a characteristic volume, when subjected to an elastic deformation with an energy equal to the energy of an uniform deformation when the stress is equal to tensile strength, i.e. ð = ð3 ðð¡ 2 2ð¸ 32 where ð is energy, ð is the characteristic length, ðð¡ is the tensile strength, and ð¸ is the elastic modulus. This characteristic volume is illustrated in Figure B-5a. (a) (b) Figure B-5. At left, volume (energy) associated with characteristic length in a beam under bending; at right, illustrating changes in flexural strength relative to tensile strength as characteristic length ð changes We now consider a stress state in a beam under bending with a constant moment. In this configuration, the stress, ð, at any point of the beam is ð(ð§) = 2ð§ â ð0 33 where â is the thickness of the beam, ð§ is the distance from the neutral axis, and the stress at the surface is ð0 = ð(ð§ = â 2 ), which is assumed to be tensile (i.e. positive). The energy of elastic deformation of a representative cube with side ð is equal to ð = ð2 2ð¸ â« 2ð§ â ð0dz â 2 â 2âð = ð3ð0 2 2ð¸ [1 â 2 ð â + 4 3 ( ð â ) 2 ] 34
B-17 By equating the expressions for energy in Equations 32 and 34, one obtains an expression for the modulus of rupture, ðð , i.e the value of ð0 at which the beam fails. ðð , also referred to as the flexural strength, is ðð = ðð¡ â1 â 2 ð â + 4 3 ( ð â ) 2 35 Thus, according to Equation 35, if one has a purely brittle material (in which characteristic length ð = 0), then ðð = ðð¡. However, if the beam thickness, â, is comparable with the characteristic length, the flexural strength will be higher than the tensile strength ðð¡, as illustrated in Figure B-5b. Consider a case when the stress varies linearly through depth ð§ within a characteristic length ð, but the manner of variation is stress is not proportional to ð§. This may happen when a beam is subjected to bending and a non-linear temperature distribution. Let ð1 = ð (ð§ = â 2 â ð), which allows ð(ð§) to be written as ð(ð§) = 1 ð [ð0 (ð§ â â 2 + ð) + ð1 ( â 2 â ð§)] 36 Given Equation 5, the expression for the energy of elastic deformation in Equation 34 becomes ð = ð3 ð0 2 6ð¸ [1 + ð1 ð0 + ( ð1 ð0 ) 2 ] 37 Equating the two expressions for the energy of elastic deformation above â Equations 34 and 37 â one can obtain a critical value for ð0, the stress at the surface of the beam, henceforth ð0 â. Thus, the criteria for failure becomes ð0 â ðð = â3 â 6 ð â + 4 ( ð â) 2 â1 + ð1 ð0 â + ( ð1 ð0 â) 2 38 where values of ðð are the flexural strength for a given ratio â ð . Similarly, this criteria can be considered in terms of ðð¡, in which case it becomes ð0 â ðð¡ = â3 1 â 2 ð â + 4 3 ( ð â) 2 â1 + ð1 ð0 â + ( ð1 ð0 â) 2 39
B-18 B.6. Transverse cracking prediction in the modified model (outputs) This step is unmodified from the original procedure and uses the sinusoidal relationship ð¶ð = ð¶1 ð¶2 + ð¶3 â ð¹ð·ð¶4 40 to predict transverse cracking CR in the JPCP project, where Ci are global calibration constants. The calculation of cracking has been detailed elsewhere in the report, and the reader is referred to the AASHTO M-E reporting for more information (ARA, 2004; AASHTO, 2008). References Khazanovich, L., and Gotlif, A. (2002). âISLAB2000 Simplified Friction Model,â Paper 02- 3529. Proceedings of the 81st Transportation Research Board Annual Meeting, Transportation Research Board, National Research Council, Washington, DC. AASHTO (2008). Mechanistic-Empirical Pavement Design Guide, Interim Edition: A Manual of Practice. American Association of State Highway and Transportation Officials, Washington, D.C. Applied Research Associates, Inc. (2004). Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures. Final Report, Project 1-37A. National Cooperative Highway Research Program, Transportation Research Board, National Research Council, Washington, D.C.