DYNAMIC ANALYSIS OF CONCRETE PAVEMENTS resting on a two parameter medium.pdf

8/14/2019 DYNAMIC ANALYSIS OF CONCRETE PAVEMENTS resting on a two parameter medium.pdf

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INTERNATIONAL J O U R N A L FOR NUME RI CAL METHODS I N ENGINEERING, VOL. 36, 1465-1486 (1993)

DYNAMIC ANALYSIS OF CONCRETE PAVEMENTSRESTING O N A TWO-PARAMETER ME DIU MMUSH ARRA F ZAMAN,* ARUMU GAM ALVAPPILLAI' A ND MICHAEL R TAHERI'

School of Cioil Engrnerriny and Environmental Science, The University of Oklahoma, Norman, 73019, U.S.A.

SUMMARYAn analysis procedure based on the finite element method is presented to solve moving-load problems ofrigid pavemen ts. Th e algorithm presented consid ers the dyna mic pavement-aircraft interaction effects. Th epavement foundation system is modelled by thin-plate, non-conforming finite elements resting on atwo-parameter elastic medium. The moving aircraft loads are represented by masses supported bya spring-dashpot system moving at a specified initial horizon tal velocity and acc eleration. Th e accuracy ofthe finite element progra m developed is verified by co mparin g the numerical results of a static problem withthe available solution. A parametric study is conducted to determine the effects of the various parameters onthe dynamic response of pavemen ts. Emphasis is placed on identifying the influence of fictitious edge andcorner forces acting on the plate due to the deformation of the soil medium outside the plate.

1. INTRODUCTIONDynamic analysis of pavements to moving loads has become an important areaof research in therecent years due to its significance in the reliable and economic design of pavements. In theconventional methods, the design of rigid pavements is based on the closed-form solutionsobtained from the static analysis of infinitely long plates resting on an elastic foundation. Since aninfinitely long plate is considered in such analyses, the actual discontinuous nature of pavementsystems is disregarded in this approach. Further, the dynamic effects of moving vehicles areaccounted for by indirectly applying an impact factor.The importance of dynamic analysis was first felt in the design of railroad bridges. Manyanalytical and numerical methods have been presented in the past to predict the dynamicbehaviour of bridges to moving Most of these studies provide solutions to one- ortwo-span uniform beams or plates and assume a simplified pavement model to account for thedynamic interaction between the moving loads and the pavement.The dynamic response of beams and plates resting on an elastic foundation and subjected tomoving loads was also studied by several researchersin the past. However, most of these studieswere limited to steady-state analytical solutions for infinitely long beams4 or infinitely longplates5 resting on a Winkler-type elastic foundation. Also, in these studies, the dynamic interac-tion effects were not properly accounted for. Recently, a number of studies6-' have been reportedon the dynamic analysis of pavements. In these studies, an improved algorithm including the*Assoc ia te Professor1University of Shiraz, IranFormer Graduate Research Assistant

OO29-598 1/93/091465-22tS16.00993 by John Wiley & Sons, Ltd. Received 21 August 199Revised 6 April 1992


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1466 M. ZAMAN, A. A LV A PPILLA I AND M. R. TAHERIvehicle-pavement interaction was proposed based on the finite element method. However, in allthe aforementioned studies involving dynamic analysis of pavements, the supporting subgradewas modelled by Winkler-type foundation model, which assumes tha t the pavement is supportedby a series of closely spaced, isolated vertical ,$rings and dashpots. Such an idealization ofsubgrade does not have any mechanism to provide an interaction between adjacent springs or theso-called shear interaction. Obviously, this foundation model is unrealistic because the actual soilmedium is continuous and, therefore, capable of providing shear interaction. This deficiencyofthe Winkler idealization can be improved by modelling the subgrade as a two-parametermedium, which provides shear interaction between individual spring elements. A number ofdifferent two-parameter have been proposed in the past; and a brief review of theavailable models is presented in the next section.In the past, several researcher^'^-'^ have demonstrated the capability of two-parametermodels in representing the soil medium. Although the modelling of subgrade as a continuum ismore accurate, it is difficult to incorporate in the dynamic analysis of pavements due to itscomplexity. Also, in using such models, the computing cost can increase drastically due to anincrease in the size of the FE mesh by incorporation of the subgrade continuum. In such cases, thetwo-parameter foundation models provide a better representation of the underlying soil mediumand a conceptually more appealing approach than the one-parameter (Winkler) foundationmodel.In this study, an analysis procedure based on the finite element method is presentedfor analysing the dynamic response of rigid pavements due to moving-aircraft loads. Thepavement-subgrade system is modelled by rectangular thin plate elements resting on a two-parameter foundation proposed by Vla~ov.'~he dynamic interaction between the movingaircraft and the pavement is taken into account by idealizing the aircraft suspension system bysprings and dashpots and having a specified initial horizontal velocity and acceleration. Thesolution scheme proposed utilizes the position of the vehicle as a pseudotime to define thepavement deflection at any instant. The accuracy of the two-parameter foundation model used torepresent the subgrade is verified by comparing the finite element results obtained from thepresent study with the available solutions. A parametric study is conducted to demonstrate thesignificance of the two-parameter foundation model.

2. TWO-PARAMETER FOUNDATION MODELSThe development of the two-parameter models has been approached in two distinct ways. In thefirst case, the subgrade model is based on the Winkler idealization and improves its discontinuousbehaviour by providing some kind of mechanical interaction between the individual springs. Thetwo-parameter model developed by Filonenko-Borodichg provides shear interaction between theindividual springs by connecting them by a thin elastic membrane under a constant tension.Hetenyi considered a similar mechanism but, instead of using an elastic membrane, used anelastic plate in bending. The model developed by Pasternak assumes the existence of shearinteraction between spring elements by connecting them to an incompressible shear layer whichdeforms only in transverse shear.The second type of the two-parameter model has been developed based on the elasticcontinuum approach and imposing constraints or simplifying assumptions with respect to theexpected displacements and/or stresses. In this study, a two-parameter model developed byV l a ~ o v ' ~ased on this approach (Figure 1) is used to idealize the underlying subgrade. Thesubgrade is assumed to be a single-layer elastic foundation of thicknessH . The load-displacement


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DYNA MIC ANALYSIS OF CONCRETE PAVEMENTS 1467

Figure 1 . (a) Vlasov two-parameter foundation model; (b) rate of decrease of displacement and its derivative ina single-layer two-parameter medium

relationship for the Vlasov two-parameter foundation model can be obtained as4(x , Y = k w x , Y - 2tV2w(x, Y 1

where Vz is the Laplacian operator in the rectangular Cartesian co-ordinates, w(x, y ) is thevertical deflection at point (x, y), and k and t are the two independent parameters needed to definethe elastic medium.In the model development, it is assumed that the displacements and the normal stresses withinthe elastic medium decrease with depth (Figure l(b)). The rate of decrease is determined bya function ( z ) given by the following equation:sinh [y(H Z)/rn]

sinh ( H/m)4 =


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1468 M. ZAMAN, A. ALVAPPILLAI AND M. R. TAHERIwhere y is a constant determining the rate of decrease of displacements with depth, and m is theshort half-length of the plate (Figure l(a)). Based on the variational principle, the parametersk and t can be obtained as13

where

and

Eoy sinh(yH/m) cosh(yH/m)+ yH/mk =2m(l ) sinh2(yH/m)Eom sinh(yH/m)cosh(yH/m) yH/mt = 8YU - v o ) sinh2(yH/m)

In the above expressions, Is and v, are, respectively, the modulus of t.uticity antratio of the foundation, and H is the foundation thickness.

(5b)Poissons

3. ANALYSIS OF RECTANGULAR PLATES RESTING ONA TWO-PARAMETER ELASTIC FOUNDATIONConsider a rectangular plate, with all four edges free, resting on a single-layer two-parameterelastic foundation and traversed by a moving load. The governing differential equation for thissystem can be expressed as

DV4W(X,y) 2tV2w(x,y) + kw(x, y) = 4(x, y) (6)where D is the plate rigidity and 4(x, y) is the dynamic subgrade-pavement-vehicle interactionforce transmitted to the plate, as discussed in the next section.The governing differential equation (equation (6)) has to be solved by applying appropriateboundary conditions. For a plate resting on a Winkler foundation and with its edges free to thesubgrade, the boundary conditions are identical to that of the Kirchhoff boundary conditions foran unsupported plate. However, for a plate resting on a two-parameter medium, these free-endboundary conditions are no longer valid due to the deformation of the soil medium beyond theplate edges. In such cases, based on the deflection pattern of the soil medium outside the plate, it isnecessary to apply distributed forces along the plate edges and concentrated forces on all the platecorners. However, the determination of exact edge and corner reactions is difficult since the actualfoundation deformation beyond the plate edges is rather complicated in nature. Vlasov andLeontevI3 presented an approximate method to evaluate the edge and corner reactions based onthe virtual-work principle by assuming the following surface deflection of the soil medium outsidethe plate (Figure 2):

w,(x, y) = wl(y)e-a(x-m) in the positive direction of the x-axisw,(x, y) = wm(x)e-a(J-) n the positive direction of the y-axis

(74(7b)

w,(x, y) = w,e-a(x-m)e-a(Y-l) n the region 1x1 > m and l y ( > 1 (74and


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DYNAMIC ANALYSIS OF CONCRETE PAVEMENTS 1469

r w e - a@-0mYFigure 2. Deflections of the plate and the surface displacements of the elastic medium

Figure 3. Aircraft-pavement model


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1470 M . ZAMAN, A. ALVAPPILLAI AND M. R. TAHERIwhere w, nd w are the edge deflections alon g the plate edges x = f m and y = & I respectively,w, is the corner deflection, and c1 = Jk/2r.The distributed edge forces Ql and Qm along plates edges x = & in and y = rt_ 1, respectively,can be obtained as

andQ m = 2 t [ W, (g m&(29m]

where the subscripts 1 and m denote t hat the deflections and derivatives are obtained at x = rnor y = kl. The corner reactions R to be applied at the plate corners are given byK = + t w , 9)

where w s the corner deflection.

4. FI NI TE ELEMENT FORMULATIONThe non-conforming thin plate elements are used in this study to model the plate. The four-noded, isoparametric plate element having three degrees of freedom at each node, namely, thevertical displacement w, the rotation a bo ut x-axis, O x , and the rotation ab ou t y-axis, Q , , are usedin this study to model the pavement. The stiffness and the mass m atrices of this element a re welldocumented in the literature.Th e gov erning differential equation (equation (6)) for a rectangular plate element resting ona two-parameter foundation can be transformed into the following stiffness matrix form byadopting the variational principle:

( C k o l - Chi+ Ik2lI dI = {Q} (10)where [ k o ] is the element plate stiffness matrix, [ k l ] and [ k 2 ] are the foundation stiffnessmatrices corresponding to the foundation parameters t and k , respectively, {Q> is the elementnodal force vector and ( d } is the element nodal displacement vector. The stiffness matrices andthe force vector defined in equation (10) can be expressed as

and


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DYNAMIC ANALYSIS OF CONCRETE PAVEMENTS

where1471

(12a)

Land [ N ] s the vector of interpolation functions. In equation s (12a) and (12b), the subscripts x andxx denote, respectively, the first- and the second-order partial derivatives with respect to x Inequation (12c), E and I are the elastic mod ulus an d the Poissons ratio of the plate, respectively,and h is the plate thickness.Th e force vector ( Q ) in equation ( 1 Id) can be determined by evaluating the dynamic interac-tion force q ( x ,j) ransmitted to the plate due to m oving loads.

5. DYNA MIC PAVEMENT-AIRCRAFT INTERACTIONTh e dynamic pavement-aircraft interaction is accounted for in the formu lation by treating thepavement and the moving aircraft as a single integrated system as shown in Figure 3. The mainlanding gear of the aircraft is modelled by a set of independent discrete units consisting ofa lumped mass ( M i ) upported by a spring k i )and a dashpot ci). hc moving aircraft is assumedto have specified values for velocity, v(t), and acceleration, a t).The following discussion is based o n th e formulation of a dyna mic pavemen t-aircraft interac-tion model presented in Reference 7. The dynamic interaction force F i x, y ) , transmitted to theplate due to the moving aircraft is given by

Fi(X, y ) = M i ( S i i ( t ) ) d ( X ti,y i ) (13)where u i is the vertical displacement of the suspended mass M i ,( t i , i ) s the position vector of theith suspended unit, 6 is the Dirac delta function and the over dot ( . ) denotes derivative withrespect to time t. By considering the inertia of the plate and the damping of subgrade, the totaldynamic interaction force q x, y ) can be obtained as

(14)( x , y ) = Fi x, ) + (my mG ,w)where m is the mass density/unit area of the plate material and c, is the damping coefficient of thesubgrade.Using equations (lld), (13) and (14) and expressing the plate deflection M in terms of nodalvariables i d } , equation (10) becomes

d 2{dl[NlTm[N) dA ~{[ko] k , ] + [ k z ] [ d = d t2[NITc , [N] dA d J d } + [I?( , v i ) l T M i ( g 2) 15)

A d t


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1472 M. ZAMAN. A. ALVAPPILLAI AND M. R. TAHERII L/-7/L

0 969 - _ _ - - -0.970.9790.992)

/0.892 0.7600.906) 0 .780)0.953 /0.966)

Figure 4 a). Comparison of dimensionless plate deflections (results presented by Yangi6 are given in parentheses)

Figure qb). Comparison of dimensionless edge and corner forces (results presented by Yanglb are given in parentheses)

where the tilde ( - ) above [ N ] indicates that the shape functions are evaluated for a specificelement where the mass M is located.By assembling equation (15), the following equilibrium equation for the total system can beobtained.

where


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DYNAM IC ANALYSIS OF CONCRETE PAVEMENTS 1473

and { e l is the system displacement vector. The sum mation signthe assembly of the individual element matrices.in the above equations denotes

5.1 Equation of motion of the aircraft maSseSApplying D'Alembert's principle to the aircraft suspension system,

Mib(t) + c i [&( t) 5 7i)] + k i [ u i f )- ( 5 7i)] = M i 8 (1 )By expressing w in terms of nodal variables, equation (18) can be transformed into the form

dL-dldti & + c i J i+ kiui = Mig + c i [ f i 5 , i ] ci[G


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1474 M. ZAMAN. A. ALVAPPILLAI AND M. R. TAHERI

j - 1

j 1

(21)whcre Y and #I are the constants associated with the Newmark-beta integration scheme, h is theincrement in the aircraft position along the x-axis, and subscripts j and j 1 refer to the timesteps j and f j- 1, respectively. F or convenience, equ ation (20) and (21) can be combined into thefollowing matrix equation

iwhere

+ ( 2)e)2 I j - 1

(23d)


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DYNAMIC' ANA LYSIS OF C ON C R ETE PA V EM EN TS 14756. SOLUTION SCHEME

It ca n be observed that the stiffness matrix in equation (22) is not sym metric since [ S1 2 ] is notequal to [ Sz 1 I T . owcver, the submatrix [S,,] is symmetric and banded, and [S,,] is a diagonalmatrix. Also, the stiffness matrix an d the right-h and side force vector ar e time-dependent an dhave to be evaluated at each time stepj.Fo r convenience in solving, equation (22) can be separated into two m atrix equations as

[ S l l I j e j + [ S lz l j ( u i ) j = IIR11jCSzlIjej + C S ~ ~ l j ( u i } jIRzIj

( 2 4 4(24b)

From equations (24a)an d (24b), the vector {ui can be eliminated by static condensation, a nd thefollowing equation for ej can be obtained by rearranging the resulting expression. Thus,[ S l l l j e j = [ R ~ l j - S12IjCS22l~'CR2lj- C ~ , 2 l j C s , , l J 1 C ~ ~ l l j e j (25)

All terms on the right-hand side of equation (25) are kn own except for the vector c j . TOsolvethis equation, an iterative scheme is adop ted, The unknown nodal displacement vector e j on theright-hand side of the equa tion is first approxim ated by e , j - and the system of equation is solvedby a direct elimination procedure. The resulting displacement vector is then substituted into theright-hand side of the equation, and a modified system of nodal displacement vector e j iscomputed. This process is repeated until the displacements converge to a specified tolerance.Knowing the nodal displacement vector ej, the vehicular displacements [ui l j can be computedfrom equation (24a) or equation (24b).The solution scheme outlined above to solve equation (25) is applicable only to the Winklerfoundation model. For two-parameter foundation, as discussed before, the fictitious edge andcorner reactions have to be determined and inco rporated before solving equation (25). Evaluationof these fictitious forces at time step j requires that the deflected shape of the plate be knowna priari, as seen from equations (8) and (9). On the other hand, the plate deflections cannot bedetermined until the fictitious forces are known. To solvc this problem, an iterative procedure,similar to the one prop osed by Yang,16 is used here. The stepwise proced ure is outlined in thefollowing.

Step 1: Neglect the edge a nd th e corner reactions an d solve equation (25) for e i } j .Step 2: Based on the no da l deflections obtained in step 1, determine the edge and the cornerforces from eq uations (8) and (9). Use the stan da rd forward finite difference techniqueto ob tain the first- and second -order derivatives appearing in e qua tions (8)and (9).F orsimplicity, the sam e finite element mesh is used a s the finite difference grid t o o bta in th ederivatives.Step 3: Apply the fictitious edge and the corner reactions ob tained in step 2 before re-solvingequation (25).Step 4: Repeat steps 2 and 3 until the desired convergence is achieved.No te that the stiffness matrix IS,,], appearing in equation (25) is kept unchanged during thisiterative procedure, and it is inverted only once to obtain the solution to the moving-massproblem at a particular time step j .

6.1. oving-force solutionTh e governing equation of motion or this case is obtained from equ ation (20) asThe moving-force solutions a re ob tained by neglecting the inertia effects of the moving mass.


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1476 M ZA M A N . A . A LV A PPILLA l A N D M . R. TAHERI

+ [IWI + [$(t, ri)l Mig (26)The ab ove equation can be solved along with the iterative scheme outlined above to determinethe edge and the corner reactions.

7. NUM ER I CAL RESULTS AND DISCUSSION7.I . Algorithm t~eriJcat ion

T o verify the accuracy of the finite element algorithm developed, a static problem of a rectangu -lar plate resting on a two-parameter founda tion is analysed an d the numerical results obtainedare comp ared w ith the results from Reference 16. Th e following plate and foundation propertiesare assumed in the analysis:

length (2l)lwidth 2m) of the rectangular plate = 2.07LEo12m-flexibility index of the plate Y = 2 1.0D(1 o )

y = the parameter determining the rate of decrease of displacement within the foundationy o = __- = 0.4 where v 5 is the Poisson's ratio of the foundationv = Poisson's ratio of the plate material = 0.0.

A vertical, concentrated force ( P ) s assumed to act at the centre of the plate.Fo r the abo ve parameters, the displacements an d the fictitious edge and corner reactions actingon the plate are numerically obtained by using the computer program developed in this study,and the results are com pared in F igure 4(a) and 4(b) for selected nod al points. Th e comparison ofnon-dimensional plate deflections, M, = 10wEorn/P, is shown in Figure 4(a). The results pre-sented in Reference 16 are given in parentheses. The non-dimensionalized distributed forces,Ql = 10Qlm/Pand Q,,, = lOQ,m/P acting along the plate edges x = m and y = 1, respectively, andthe corn er reaction R = 10R/P are show n in Figure 4(b). Th e numerical results obtained from thepresent study agree well with those presented in Reference 16.The accuracy of the present algorithm is also verified for a number of moving-force andmoving-mass problems. The analytical and the finite element results available for simply sup-ported beams and pla tes'7 7 an d for a n infinitely long plate resting on a Winkler foundationmod el' are used in this verification. Since the dyna mic analysis of rigid pavem ents resting o na two-parameter medium and subjected to moving loads is not available in the literature, theverification is limited to the above problems. An excellent agreement between the analyti-cal/experimental results and the finite element so lution s from the present study is observed for allthe cases considered. Herein, only the resu lts obtaine d for a moving-mass problem t hat con sidersa simply supported beam under a moving load are presented. Th e experimental results presentedby Ayre et al.' are used for comparison. The experimental results were obtained for a simply

depth = 1.51 ,

1 1


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~v /Lw, 0.5nV/LW? 025I0 .2 .4 .6 a 1.02

Experirnenta (Ayre et. al..1051)Experimental (Ayre et al.. 1951). . Experimental (Ayre e t al. 1951)

Present StudyPresent Study

Figure 5. Moving mass over a simply supported beam M / M , = 0.5)

supported beam model having the following properties:beam length L)= 60 in,Young's modulus E ) = 30 x lo6psimass density p )= 7-346 x

width (6) = 4 in, thickness (f) = 3/16 inlb s2/in4

The fundamental frequency of this beam ol)s 4.7 cycles/s. For a comparison of the results, themass ratio (hf/Mhj is taken as 0.5, while the velocity ratio nu/Lwl) s considered to be 0.25, 0.5and 0 7 5 . Th e following definitions ar e used: M = mass of the moving load; k f h = mass of thebeam; u = velocity of the moving load. A com parison of the results are presented in Figure 5. It isseen that the finite element solution s agree well with the experimental observations. A maximumof 10 per cent error is observed between the finite element and the experimental results.7.2. Parametric study

A param etric stud y is cond ucted using thc proposed finite element algorithm to determine theeffects of various p arame ters on the dynam ic response of rigid pavem ents. Emphasis is placed onidentifying th e effects of fictitious edge a nd corner reactions, tha t occur d ue t o the d eforma tion ofthe soil medium beyond the plate edges, on the pavemen t response. F or this purpose, results arepresented for three different founda tion idealizations: (1) Winkler idealization; (2) two-parameterfoundation model without the fictitious edge and corner reactions; and 3 ) two-parameterfoundation model with the fictitious edge and corner reactions. Tn this study, the response ofa single slab due to a moving suspension unit representing an aircraft wheel load is considered.


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1478 M. ZAMAN, A. ALVAPPILLAI AND M . R. TAHERIA critical aircraft of B-727, which has twin landing gear with a maximum gross weight of 170 kip.is assumed for the aircraft model. It is also assumed that 90 per cent of the aircraft weight iscarried by the main landing gear. Hence, each gear in the twin assembly carried approximately77 kip. The following properties for the pavement subgrade system and the moving mass-suspension system are assumed:

length of the pavement (21) = 300 inwidth of the pavement 2m)= 150 inpavement thickness = 12 inmass density ( p )= 0.0002174 lb sz/in4Youngs modulus of pavement material = 3.6 x lo6 psiPoissons ratio of pavement material = 0.15Foundation damping = 5 per centPoissons ratio of the foundation material v,) = 0.3H = thickness of the single-layered foundation = oy = 1.5

Properties of the suspension system:weight of the moving mass = 77 kspring stiffness = 100 k/indamping ratio = 0.5 per centTh c non-dimensionalized static deflections w E o m / P ( l v i along th e pavement centre line fordifferent flexibility index values are prese nted in Figures 6 an d 7. The deflections are obtaine d forthe case when the moving suspension unit is at the centre of the plate. On ly one-half of the plateresponse is plotted due to its symmetry. Th e foundation parameters k and t are calculated fromequations (3) and (4). The foundation modulus E , ) needed for the determination of these

Two Parameter with fictitious forcesTwo Parameter without fictitious forces

10

- 4

0 2 4 6 8X l l

Figure 6 Static deflection of a centrally loaded pavement for r = 20


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D Y N A M I C ANALYSIS OF CONCRETk. P AVE ME NT S 14792

x/lFigurc 7. Static deflection of a centrally loaded pavement for r = 100

param eters is ob tained from the flexibility index r = ~ ~ n E , 1 ~ r n / D ( l g ) , where all the parame tersare fixed except E , . In this study, results are reported for r = 20 and 100 which fall within therange of practical importance.Figure 6 presents the p avement deflection for r = 20. It is observed th at the central deflectionfor thc Winklcr idealization foundation is larger than the other two cases considered. Thedeflections obtained fo r using the Winkler model and the two-param eter model w ithout edge an dcorner reactions do not show much variation near the point of load application, but two-parameter idealization has the tendency to cause an uplift near the transverse plate edge. Theapplication of corner and edge forces with two-parameter idealization causes significant reduc-tion in the maximum pavement deflection. Fo r the case considered here, the maximum reductionof 19 per cent is observ ed a t the centre of the plate. However, when the value of the plate flexibilityindex is increased to 100, this reduction in centre deflection is reduced to 12 per cent, as can beseen in Figure 7. This means tha t when the founda tion becomes relatively stiffer, the impo rtanceof edge and corner reactions in the analysis is reduced.Figures 8 and 9 present the dynamic response of the pavement due to a moving massrepresenting an aircraft wheel load which moves from one transverse edge of the pavement toanother along the pavement centre line. The results are presented for the situation when themoving mass is located at the pavement centre. Th e velocity rati o v/v,,) is taken as 0.5, where thecritical velocity uc r is defined by the following expre~sion:~

Figure 8 illustrates the moving-mass solution for r = 20. The deflections obtained with theWinkler model and the two-parameter model including the edge and corner forces showmaximum difference in the ccntral region of the plate. When the edge and corner forces aredisregarded, the W inkler a nd the two -param eter idealizations yield very close results, particularlyin the centre region. For a small flexibility index of 20, the transverse plate edge ahead of themoving mass tends t o uplift, but th e other edge does not exhibit this behaviour (Figure 8).It can


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1480 M. ZAMAN, A. ALVAPPILLAI AND M. R . TAHERI


10

- a-10-10 - 5 0 5

x / lFigure 8. Moving-mass solution for r = 20 when the m oving load is at the centre of the pavement ( u i u , , = 0.5)

10a642

N0 - 0E m - 4

- a

au. 2w3 - 6

-10-12-14


-10 - 5 0 .5 1.0xll

Figure 9. Moving-mass solution for r = 100 when the moving load is at the centre of the pavement u /u c r = 0.5)

also be noted that the maximum pavement deflection occurs behind the moving mass. However,when the flexibility index increases to 100 (Figure 9), both transverse edges show the upliftcharacteristics due to the increasing foundation stiffness. In addition, the maximum pavementdeflection shifts to the right, near to the location of the moving mass. In all the cases, thetwo-para meter founda tion model with the edge an d corner forces gives significantly less centraldeflection, as expected. As for the static case, the percentage of reduction decreases when theflexibility index increases.


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-.8

A similar set of results, as presented for the mov ing-mass problem, is given in Figures 10and 11for the case of a moving force in w hich the inertial effects of the m oving m ass a re neglected.Fr om the prcceding study, it is understo od tha t the effect of fictitious edge a nd corner reactionsdue to the deformation of the soil medium beyond the plate edges on the pavement responsedecreases for increasing foundation stiffness. In that study, the plate dimensions and materialproperties were kept con sta nt while the foundation stiffness was varied. Another factor which cansubstantially influence these fictitious forces is the plate dimensions. It is obvious that if the plateedges are located far away from the point of application of the load, the magnitude of these

. Winklerv Two Parameter without fictitious forcesTwo Parameter with fictitious forces

Two Parameter with fictitious forcesTwo Parameter without fictitious forces.6

-10 -.5 0 .5 1.0x / l

Figure 10. Moving-force solution for r = 20 when the moving load i s at the centre of the pavement u/u,, = 0.5)


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4 * Two Parameter with fictitious forcesTwo Parameter without fictitious forces67 Oa;Eow-

- 4

- a-10 - 5 0 5 10X l l

Figure 14. Effect of pavement length o n the moving-force solution r = 100, ( v / u c r = 0.5)

Two Parameter with fictitious forcesTwo Parameter without fictitious forcesWinkler

Figure 15. Effect of velocity ratio o n the maximum centre deflection (moving-mass solution, r = 100)

The effect of the velocity ratio on the maximum centre deflection of the plate is depicted inFigure 15 for moving-m ass loading conditions. T he flexibility index of the plate is assumed to be100. It can be observed that the two-parameter idealization without the edge and corner forcesgives more deflection than the Winkler medium.Figure 16 presents the effect of subgrade damping on the maximum centre deflection of thepavement due to a moving mass a nd a moving force, respectively. The two-parameter idealizationwithout considering the edge and corner reactions yields more deflection than the Winkler


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1484 M . ZAMAN, A. ALVAPPILLAI AND M. R. TAHERI

X

1.0


150

2

0 0 25 50 75 100 125Damping Ratio

Figure 16. Effect of subgrade damping on the maximum centre deflection (moving-mass solution, Y = 100, (t>/ccr= 0.5)

x/210 30 0.13 0.25 0.38 0 50 0.63 0.75 0 880.50

0.38

0 25x

0.13

1 / I A / / j0.00 m n l b 0.000.00 0.13 0.25 0.38 0 50 0.63 0.75 0.88 1.00x/21

Figure 17. Maximum principal stresses for moving-mass loading conditions

medium for the values of subgrade damping less than 40 per cent. Also, the two-parameteridealization with the edge and corner reactions gives less deflection than the other two ideal-izations considered. This behaviour is observed for the entire range of 0-1 50 per cent damping.The maximum and m inimum principal stresses a12/P) cting on the pavement due to movingmass are depicted in Figures 17 and 18, respectively. Th e results are presented for two-param eterfoundation model with consideration of the edge and corner reaction. The higher stress concen-trations are observed along the wheel path of the moving load.


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D Y N A M I C ANALYSIS OF CONCRETE PAVEMENTS 1485x/21

x/21Figure 18. Minimum principal stresses for moving-mass loading conditions

8. CONCLUSIONSAn improved finite element (FE) algorithm is presented to analyse the response of concretepavements to moving aircraft loads. The underlying subgrade is idealized by a two-parametermodel which accounts for the continuous behaviour of the soil medium. The moving aircraftloa ds are modelled by masses each supported by a spring an d a da shp ot system. Th e influence offictitious edge and corner forces acting on the plate due to the deformation of the soil mediumbeyond the plate edges is studied by comparing the result obtained with different foundationidealizations. From the parametric study conducted, it is observed that the influence of thefictitious forces decreases when the foundation stiffness increases or when the plate lengthincreases. The proposed FE algorithm can be effectively used to analyse dynamic response ofairpo rt pavements.

1.2.3.4.5 .6.7.8.9.

10.11.

REFERENCESR. S . Ayre, L. S. Jacobson and C. S. Hsu, Transverse vibration of one and two-span beam under thc action ofa moving mass load, Proc. 1st National Congress ofApplied Mechanics, U.S.A., 1951, pp. 81-90.E. C. Ting, J. Genin and J. H. Cinsbert, A general algorithm for moving mass problcms, J Sound Vib.,33, 49-58(1974).D. M. Yoshida and W. Weaver, Finite element analysis of beams and plates with moving loads, Publ. In t . Assoc.Bridge Struct. Enq., 31, 179-195 (1971).J. D. Achenbach and C. T. Sun, Moving load on a flexibly supported Timoshenko beam, Int. j solids struct., 1.353-370 (19651.W. E. Thompson, Analysisof dynamic behavior of roads subject to longitudinally moving loads, H R B 39, pp. 1-24(1963).R. H. Ledesma, Vehicle-guideway interaction in rigid airport pavements with discontinuities, Master Thesis,University of Oklahoma, 1988.M. R. Taheri and E. C . Ting, Dynamic response of plate to moving loads: finite element method, Comp. Srruct., 34509-521 (1990).M. Zaman, M. R. Taheri and A. Alvappillai, Dynamic analysis of a thick plate on viscoelastic foundation to movingloads, Int. j. numer. anal. methods geomech., 15, 627-647 (1991).M. M. Filmenko-Borodich, Some Approximate Theories of the Elas tic Foundation, Uchenyie Zapiski MoskovskogoGosudarstuenogo Universiteta Mechanika (in Russian), U.S.S.R.,vol. 46, 1940 pp. 3-1 8.M. Hetenyi, Beams an Elustic Foundation, University of Michigan Press, Ann Arbor, MI, 1946.A. D. Kerr, Elastic and viscoelastic foundation models, J . Appl . Mech. ASM E, 31, 491-498 (1964).


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1486 M. ZAMAN, A. ALVAPPILLAI AND M. R. TAHERI12. P. L. Pasternak. On a new method analysis of an elastic foundation b y means of two foundation constants, Gosudar-13. V. L. Vlasov and N . H. Leontev, Beams, plates and shells on elastic foundation (Translated from Russian), Israel14. M. E. Harr, J. L. Davidson, Hoss, Da-Min, L. E. Pombo, S. W. Ramaswamy and J. C . Rosner, Euler beams on15. T. Nogami and Y. C . Lam, Two-parameter layer model for analysis of slab on elastic foundation, J . Eng. Mech.16. T. Y. Yang, A finite element analysis of plates on a two parameter foundation model, Cornp. Struct., 2, 593-61417. F. Zhashua and R.D. Cook, Beam elements on two parameter elastic foundation, J . Eng. Mech. ASCE, 109(3),18. 0.C. Zienkiewicz and Y . Cheung, The finite element method for analysisof elastic isotropic and orthotropic slabs,

stvennoe lzadatelstvo Literaturi PO Stroitelstvu Arkhitekture, (in Russian), Moscow, 1954.Program of Scientific Translation, Jerusalem, 1966.a two-parameter elastic foundation model, J . soil mech.foun d. diu. A SC E , 95, 933-948 (1965).A S C E , 113, 1279-1291 (1987).(1972).1390-1401 (1983).Proc. ins t. c i d eng ., 1964, Vol. 28, pp. 471-488.

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