J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(3): 329-333

DOI: 10.1007/s12204-010-1012-4

Influence of Material Properties on Analysis and Design ofPavements Using Shakedown Theory

SUN Yang∗ (� �), SHEN Shui-long (���), ZHAO Jian-li (���)(School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China)

© Shanghai Jiaotong University and Springer-Verlag Berlin Heidelberg 2010

Abstract: Currently, traditional analysis and design methods are unreasonable in evaluating the performance andsafety of pavements. The characteristic of shakedown theory lends itself particularly to the analysis of pavements.Based on the lower bound shakedown theory, a simple and practical procedure for shakedown analysis of pavementshas been developed using plastic strain energy as shakedown criteria via considering a special loading path. Theproposed method is verified by using a homogeneous isotropic half space. Then, the method is applied to analyzea two-layered pavement. The dependence of shakedown limit on material properties of pavements is investigatedin details. According to the analysis results, some valuable references are suggested for the selection of materialsfor accurate evaluation of the bearing performance of pavement under traffic loading.Key words: pavements, shakedown, material properties, yield criterion, finite elementCLC number: U 416.02 Document code: A

1 Introduction

Till now, most of the pavement design models usedin different road authorities and organizations in vari-ous countries concentrate on the elastic behavior of thevarious pavement layers[1]. Nevertheless, the elastic re-sponse by its very definition cannot represent failurebehavior and reflect the substantial reserve of strengthof pavement. In engineering practice, it is economicallydesirable to construct pavements that can sustain stresslevels well beyond the elastic limit of their constituentmaterials. Moreover, the traffic loading is consideredas static loading in many design methods. Obviously,this does not meet the realistic condition. The staticload assumption results in 20%—40% more deflectionscompared to field deflections[2].

Consequently, it is desirable to develop an approach,which takes account of the permanent strain that occursin pavement structures and attempt to model the actualfailure mechanisms observed in pavements. Accord-ingly, shakedown theory is the best tool for aforemen-tioned situation[3]. The critical load limit, below whichshakedown can occur, is named as shakedown limit.The static shakedown theorem proposed by Melan[4],

Received date: 2009-06-30Foundation item: the National High Technology Re-

search and Development Program (863) of China(No. 2006AA09A103) and the Shanghai Leading Aca-demic Discipline Project (No. B208)

∗E-mail: yangsun mail@126.com

together with the kinematic shakedown theorem pro-posed by Koiter[5], constitutes the cornerstone of shake-down theory for elastic-plastic structures under cyclicloading.

Sharp and Booker[6] did the pioneering work in useof shakedown concepts to analyze pavement and in re-cent years shakedown theory has been applied in thedesign of road pavements[7-11]. Mathematical program-ming method which exhibits dimension limitation andcomputation complexity is the basic idea of aforemen-tioned literature. To seek efficient and accurate methodof numerical is an important development direction ofshakedown analysis. The proposed procedure in thispaper will solve effectively the problems of using math-ematical programming method.

2 Formulation of the Problem

2.1 Pavement Structural ModelIn a vertical plane along the travel direction, Sharp

and Booker[6] considered trapezoidal load distributionsfor both the vertical and horizontal loading with a planestrain deformation normal to the travel direction by re-placing the wheel load as a roller with an infinite width.The plane strain model of a half-space is assumed fora pavement under moving loads, as shown in Fig. 1,where a is bottom width of trapezoidal load, b is topwidth of trapezoidal load, h is depth of layer 1, pV isnormal load with trapezoidal load distribution appliedto the pavement from a repeated loading, p0 is the peakvalue of pV, and pH is shear force due to the friction

330 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(3): 329-333

between moving wheel and the pavement. Therefore,the relation between pH and pV can be defined as

μ = pH/pV, (1)

where μ is the frictional coefficient. E, ν, c and ϕ rep-resent elastic modulus, Poisson’s ration, cohesion andfriction angle of material respectively, and the subscriptdenotes layer number, as shown in Fig. 1.

b

h

pVpH

x

yL=20a

H=

10a

Travel direction

a

Layer 1

Layer 2

E1, 1,c1, 1

E2, 2,c2, 2

Fig. 1 Simplified computation model of pavement

The length L of the simulated region is 20a, andheight H is 10a. This range is discretized with 3 600eight-node quadrilateral elements. The mesh used inthe analysis is the finest at the contact width 2a wherelarge stress and strain gradients are anticipated.2.2 Yield Criterion

Here, pavements are assumed to behave like rate-independent elastic-plastic solids. The equivalent circleyield criterion[10] with associated flow rules is used tomodel the plastic behavior of the material. The equiv-alent circle yield criterion can be expressed as followsfor a plane strain problem

f(σx, σy, σxy) = αI1 +√

J2 − k = 0, (2)

α =2√

3 sin ϕ√

2√

3π(9 − sin2 ϕ), (3)

k =6√

3c cosϕ√

2√

3π(9 − sin2 ϕ), (4)

where, f is yield function; σx, σy and σxy are stresscomponents of plane strain problem; I1 and J2 are thefirst invariant and the second invariant of deviatoricstress.

3 Computational Method

3.1 MethodFor m independent loads, the load domain, which

represents an m dimensional polyhedron, can be definedby

D ={p

∣∣p = λipi, λi ∈

[λ−

i , λ+i

]}, (5)

i = 1, 2, · · · , m,

where p denotes the vector of generalized loads, λi arescalar multipliers with upper and lower bounds λ+

i andλ−

i respectively, and pi are m fixed and independentgeneralized loads.

As for load of model in Fig. 1, Eq. (5) may be rewrit-ten as

p = λ[pV pH]T = λi[p0 μp0]T = p(λi), (6)

where the peak value p0 may be conveniently set asthe unit pressure in the actual calculation, and λi isa dimensionless scale parameter, then all the externalloads are proportional to λi. The largest value of λi

will give the actual shakedown limit

p(λs) = λs[p0 μp0]T,

where λs is shakedown load multiplier.The increments of plastic strain at the end of one

loading cycle over the time interval [nT, (n + 1)T ] canbe defined as

Δεpij =

∫ (n+1)T

nT

ε̇pijdt, (7)

where T is the period of load cycle, and ε̇pij is plastic

strain rate. After each load cycle of a body of volume V ,let ξ be the generalized displacement increment, then,total work done by external load can be obtained as

∫ (n+1)T

nT

|p|Aξ̇dt = We + Wp =

∫ (n+1)T

nT

dt

∫

V

σij ε̇eijdV +

∫ (n+1)T

nT

dt

∫

V

σij ε̇pijdV, (8)

where A represents pressure area, ξ̇ denotes the gen-eralized displacement rate under load of structure, σij

is total stresses induced by cyclic external loads, ε̇eij is

elastic strain rate, We and Wp are elastic work and plas-tic work, respectively. If ε̇p

ij = 0, after unloading, thefirst term of the right-hand side of Eq. (8) will vanishdue to recovery of elastic response, and the second termis also equal to zero. In this case, residual stress field istime-independent, and determined by plastic deforma-tion in early loading cycle�the response of structureis elastic and Melan’s lower bound theorem is satisfied.Therefore, shakedown will occur in the given load range.

By substituting Eq. (7) into the second term of theright-hand side of Eq. (8), we obtain plastic work of theload cycle

Wp =N∑

i=1

∫

V eσijΔεp

ijdV e, (9)

where V e is volume of each element and N is total ele-ment number of model.

J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(3): 329-333 331

Loading path in Fig. 2 is used to simulate cyclic traf-fic loading. Solution procedures for shakedown limitare given as follows.

(1) At the beginning, the load applied on pavement isadded from 0 to p(λ1) as shown in Fig. 2, then unloadedto t1, and observed whether there exists plastic zones.If there exists plastic zones�the model is loaded top(λ2) after some load cycles (0 → p(λ1) → t1 · · · 0 →p(λ1) → t1) until stable residual stress field is obtained;if not, loaded directly to p(λ2).

(2) Over a number of load steps, assumed that plasticzones is founded in the structure when the load step isp(λi). In the case, if stable residual stress field cannotbe reached, then λs = λi−1; if not, continue to loaduntil p(λi+1).

(3) The load applied to pavement incremental in-creases until reaching the limit load p(λs) under whichstable residual stress field will not be achieved, then,the corresponding shakedown load multiplier is λs.

p

p( i+1)

p( i–1)p( i)

p( 1)

tti+1ti–1t1 ti0

Fig. 2 Loading path

Under a certain load step p(λi), plastic work is cal-culated using Eq. (9) at the end of each cycle (e.g. tiin cycle 0 → p(λi) → ti · · · 0 → p(λi) → ti), which iscumulative plastic work after the load step p(λi). If cu-mulative plastic work of two successive is equal, ε̇p

ij = 0,then the plastic work increment can be zero at each cy-cle of the structure, and the structure has shakedown.Otherwise, when ε̇p

ij �= 0, it indicates that the cyclicgrowth of plastic strain occurs at each cycle, finally, thestructure fails because no structure can bear an infiniteamount of plastic strain. The process mentioned aboveis the criterion for residual stress field reaching stableor not. The response of structure from elastic to plasticfurther to reach stable residual stress field needs a pro-cess under cycle loading. As for a specific structure, thecycle number needed for reaching stable residual stressfield can be determined by trial calculation.3.2 Comparison with Existing Methods

Because of using plastic strain energy as shakedowncriterion, shakedown analysis and elasto-plastic analy-sis are completed simultaneously with no need of anal-ysis of pure elastic field σe

ij . So the use of the methodcan enable the long-term behavior of a pavement to bedetermined without resorting to computationally ex-pensive step-by-step analyses. Moreover, the analy-

sis step is simplified and easier than method commonused based on the lower bound shakedown theory[9-10],models the cyclic nature of the pavement loading, andthereby increases the reliability of the predictions andsolutions in the design process for pavements. Fur-thermore, the method solves effectively the problemsof using mathematical programming method[12-13], andhas advantages of clear physical conception, strongapplicability.

4 Results and Discussion

4.1 Verification of Proposed ProcedureThe proposed numerical procedure is used to a single

layered pavement system as shown in Fig. 1 and the sur-face friction is not considered. The results of the dimen-sionless shakedown limit λsp0/c1 with the variation ofinternal friction angle are presented in Fig. 3. As shownin Fig. 3, it can be seen that the dimensionless shake-down limit obtained by the proposed method has thesimilar results as those obtained by lower bounds[6-7].The reason for discrepancies is due to the difference ofanalytical methods and yield criterion.

0 5 10 15 20 25 302

4

6

8

10

12

1/(°)

| sp

0/c 1

|

Yu[7]

Sharp and Booker[6]

Present analysis

b/a=0.5 |pV|≠0, |pH|=0E1=100 MPa, 1=0.3

Fig. 3 Influence of friction angle on shakedown limit of asingle layer pavement model

4.2 Effect of Material Properties on Shake-down Behavior

The proposed analytical procedure is employed toanalyze a two-layered pavement system, and to in-vestigate the effect of material properties on shake-down performance. Figure 4 illustrates the analyti-cal result of effect of cohesion ration between two lay-ers c1/c2 and the internal frictional angle ϕ1 of layer1 of pavement on the dimensionless shakedown limitλsp0/c2. As shown in Fig. 4, the shakedown limitincreases both with the increase of cohesion ratio andthe internal friction angle. However, the rate of increasevaries with the value of c1/c2, i.e., there is a turn pointaround c1/c2 = 2; when c1/c2 < 2, λsp0/c2 increasesfast, and when c1/c2 > 2, λsp0/c2 increases slowly. Atgiven values of relative cohesion ratio, it is noted that

332 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(3): 329-333

the shakedown limit increases with the increase of mate-rial friction angle ϕ1, but the shakedown limit is foundto be nearly independent on a higher value of ϕ1, e.g.,ϕ1 > 35◦.

1 2 3 4 50

2

4

6

8

10

12

| sp

0/c 2

|

1=45°

1=40°

1=35°

1=30°

1=20°

c1/c2

h=a, b/a=0.5, |pH/pV|=0.4E1/E2=1, 1=0.3

2=0.3, 2=30°ν

Fig. 4 Influence of cohesion ratio and friction angle onshakedown limit of a two-layered pavement model

Figure 5 plots the variation of shakedown limit withthe stiffness ratio and relative cohesion ratio of pave-ment. As shown in Fig. 5, E1/E2 = 1 is a turnpoint; when E1/E2 < 1, the shakedown limit increaseswith an increase in the stiffness ratio, however, whenE1/E2 > 1, shakedown limit decreases with the in-crease of the stiffness ratio. Therefore, E1/E2 = 1 isan optimal value of relative stiffness ratio, at which theshakedown has a maximum value.

2 4 6 8 100

4

8

12

16

20

E1/E2

| sp

0/c 2

|

c1/c2=5c1/c2=2c1/c2=1c1/c2=0.5c1/c2=0.2

h=a, b/a=0.5, |pH/pV|=0.4

1=30°, 1=0.32=30°, 2=0.3

Fig. 5 Influence of stiffness ratio and friction angle onshakedown limit of a two-layered pavement model

The interactive effects of internal frictional angle ϕ1

of layer 1 and relative stiffness ratio of pavement onthe shakedown limit are shown in Fig. 6. It can beconcluded that the shakedown limit presents differenttendency with the increase of ϕ1 for different E1/E2.

For E1/E2 � 1, the shakedown limit increase appre-ciably with the rising of ϕ1, while for E1/E2 > 1, itdecreases appreciably with the rising of ϕ1. For a givenvalue of ϕ2, the role of ϕ1 is less significant in the shake-down analysis than relative cohesion ratio and relativestiffness ratio.

20 25 30 35 400

2

4

6

8

10

12E1/E2=1E1/E2=0.2

E1/E2=0.1E1/E2=0.05

E1/E2=5E1/E2=10

E1/E2=20E1/E2=50

| sp

0/c 2

|

h=a, b/a=0.5|pH/pV|=0.4c1=20 kPa, 1=0.3c2=20 kPa, 2=0.3

2=30°

1/(°)

Fig. 6 Influence of stiffness ratio and strength ratio onshakedown limit of a two-layered pavement model

5 Conclusion

The proposed numerical approach based on the lowerbound shakedown theory provides an efficient way toconduct shakedown analysis of pavements. It is a morepowerful tool for the design of pavement. By comparingwith the results from other literatures, it is shown thatthe numerical results from the proposed procedures arevalid.

By using the proposed procedure, the effect of ma-terial properties on the shakedown behavior has beenanalyzed. The results show that the material proper-ties (i.e., c, ϕ, E) have obvious influence on shakedownlimit of pavement. Among these properties, ϕ has rela-tively small influence on shakedown limit and c has thelargest effect on shakedown limit.

The interactive effects of material parameters shouldbe comprehensively considered in the process of pave-ment design, and improvement of a single material in-dex will often cause design deviation.

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