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Therefore , the moment M in elastic beam isM =-D d2ωd x2. (3)2. 2 Equivalence of Elastic Beam on the FoundationIn order to calculate the bending stress in eachcourse w ith dif ferent set tl ing g radient ratio and set-t l ing leng th , the equivalence between layer- elasticst ructure and monolayer elastic footing beam is as -sumed . Furthermore , the equivalent bending stif fnessof elast ic beam is derived considering the combinat ionconditions in inter-stories of asphalt pavement[ 9-13].S layers are assumed in asphalt pavement , andare equiv alent as a sing le elastic footing beam.The distance betw een neut ral layer and surface ofpavement is also assumed as h0.As ∑Si=1σ idz =0 , so σ i =- Eiz1-μ 2d2ωd x2 .Thenh0=∑Si=1E 1h 21 +2E2 h1 h 2 +2E3 h3( h 1 +h2)+2E 4 h4 ∑Si=1hi +… +2EShS ∑S- 1i=1hi2∑Si=1Eihi. (4) The equivalent bending stiffness is derived basedon the relationship between bending moment andst ress.DZ =E 1[ ( h 1 -h0) 3+h30] +E 2[ ( h1 +h2 -h0) 3-( h 1 -h0) 3] +… +ES ∑Si=1hi -h03- ∑S- 1i=1hi -h 033( 1 -μ 2) . As enti re bending moment MZ =-DZω ″ , thenw″ =2 β2eβk( C2cos( βx) +C1sin( βx) )+2β2e-βk(-C4cos( βx)+C3s in( βx) ) +2 u0L2 .So , bending moment and st ress in i layer areMi =DiDZMZ , Di = E ih3i12( 1 -μ 2),σ i = EiziDz( 1-μ 2)MZ ,(5)w here zi is the distance betw een tensile plane andneut ral plane , and it is negative when pavement is inpressure.2. 3 Influence of Disproportional Settlement onPavementAs illustrated in Eqs. ( 2)-( 5), maximum dis-tortional set tlement μ 0 and characteristic length L ofw idened subg rade are tw o main parameters w hichhave significant influence on the bending stress ofpavement w ith specified material of pavement .A-mong them , characteristic leng th , i. e. length of dis-proportional set tlement , is represented by the dis -tance betw een the combination point of existing sub-grade and the one to be widened and the border of theroad . The sett l ing ratio is defined for the convenienceof calculation.Therefore , the inf luence analy s is of u0 and Lcan be t ransfer to the inf luenc e analy s is of Δi and L .As a result , the allow able sett l ing ratio Δi can be gotby assumption of differential set tlement length L .Furthermore , it is the basis fo r subg rade w idening ofhighw ay const ruct ion and designing.3 Standard Calculation of Dif feren-tial Settlement of an Expressway The leng th of the expressway discus sed is37. 2 km , and design speed is 100 km/h , with thesubg rade width 23. 0 m.Since the use in 1994 , thet raff ic load has increased at a fast speed , with the av -erage increase by 12. 6 %per year and up to 5 076 ve -hicles per day . The expressw ay discussed is not closed and t raf-f ic accidents took place f requently.Middle -and-larges ized vehicles account fo r large proportion of traf ficflow . The pract ical speed is much low er than the de-sign speed. It is expected that the service level will belowered to class E ( i. e. the fourth level stipulated bynational standard); by 2 023 , the daily average traf ficflow will arrive at 47 442 vehicles.The whole sec -tions of the distance of widened is 21. 67 km.Thew idth of i ts subg rade is 26. 0 m , which has 4 drive-w ay s in bi -directions. The desig n speed of whole roadis 100 km/h.A reasonable and ef ficient construction projectand technology is neces sary for subg rade widen pro -ject of the expressw ay for the sake of the l imitation oft ime and const ruct ion field as well as the maintenanceof existent t raffic load.The structures of pavement of the expressw ayare illustrated in Fig.
1 , w hose parameters are alsostated in Tab. 1. Discuss ions on the inf luence of dif-ferential parameters Δi and L of w idened subg rade onbending st res s of st ructural courses of the pavement ,w hich are based on the above equat ions. w hich based on the analysis of set tlement curve ofsubg rade after w idened.The height , h0 =0. 364 m , which is f rom neu-t ral plane to the surface of pavement , is yield af tercombinat ion parameters of Tab. 1 and Eq. ( 4). Someconclus ions are yielded based on this analysis that theadditional st res s in the pavement is tensi le w hich dueto differential set tlement of ex isting subgrade and theone to be w idened , tensile addi tional st res s also can befound in the cement stable g ravel , the neutral planelies in reg ion near interface betw een base and sub-basecourse , and less additional st ress can be found in thisregion .