A new model for predicting the performance of a subgrade during soilcompaction by rollers as a function of contact pressure and the strengthcharacteristics of soils was developed. The model can be used to developimproved methods of pavement design, considering that the accumula-tion of plastic strains may continue under additional load repetitions ormay cease to increase with time, indicating adaptation or shakedownconditions. To develop the mechanistic model of a subgrade, the homo-geneous semi-infinite elastoplastic half-space under repeated loadingwas considered. By using a Mohr-Coulomb failure criterion, the bound-ary loads for which shakedown conditions or the steady state will beattained were determined. The residual horizontal normal stresses in thehalf-space were calculated and were shown to be in agreement with themeasured distribution with depth. The theoretical and experimentalresults as they apply to soil compaction and pavement design problemsare discussed. The required compaction contact pressure and the thick-ness of the compacted zone are estimated. The structural pavementdesign approach, considering residual normal stresses in a subgrade, ispresented.

An adaptation of subgrade soil to repetitive loading has a directbearing on two problems that are of interest to road engineers. First,although a variety of rollers are used to construct earth structures,soil compaction usually continues until no additional densificationoccurs under an additional number of roller passes. At the end of thisprocess the plastic deformation does not increase appreciably withload repetitions and the resilient deformation approaches a constantvalue. Thus, the accomplishment of rolling compaction looks as ifthe soil adapts itself to the roller loading. Second, a pavement andsubgrade are subjected to repetitive loading by moving vehicles, andbecause of the seasonal variation in moisture content, the subgradebearing capacity may significantly be reduced in comparison withthat at the end of a soil compaction process. The performance of thepavement and the subgrade is affected by the pavement structure,load value, and mechanical properties of the soil and pavementmaterials. The accumulation of permanent deformations may con-tinue under an additional number of load repetitions or may cease toincrease. The cessation of plastic strain accumulation looks as if thestructure adapts itself to the repetitive traffic loading.

In terms of plasticity theory (1) the adaptation of an elastoplasticsystem under a cyclic loading is called the shakedown effect. Ac-cording to the shakedown theory, a structure will adapt to repeatedcyclic loads if a self-equilibrated residual stress field could be foundsuch that yield conditions and equilibrium conditions are satisfied ateach point [the static shakedown theorem of Melan (2)]. The shake-down theory was first applied to the case of an elastic, perfectly plas-tic half-space by Johnson (3) and has more recently been applied toplates (4,5) and pavement design problems (6,7). Various numeri-

82 TRANSPORTATION RESEARCH RECORD 1547

cal algorithms have been used to search for a residual stress distrib-ution by the finite-element method (6,7). However, these methodsfall short of satisfying all constraints by the Melan theorem for anarbitrarily chosen distribution of residual stresses. The primary pur-pose of the present study was to assimilate the shakedown theory topredict the subgrade behavior under repeated loads. In this paper theanalytical approach for the application of the shakedown theory tosubgrade soils is presented.

ANALYTICAL STUDY OF SHAKEDOWN

Subgrade soil is considered a homogeneous elastoplastic half-spacethat can be characterized by two elastic constants (E and n) and twoplastic constants (C and w) by a Mohr-Coulomb failure criterion. Anormal load Q and proportional tangential load T, equal to µQ, wereassumed to be applied to the surface of the half-space by a rollingrigid cylinder (the conditions of plane deformation) that moves onthe free surface along the x-axis (Figure 1). A two-dimensionalplane strain scheme is suitable for the roller compaction problem;the axisymmetric assumption is more common for wheel loading.However, in view of the lateral distribution or scatter of wheelpasses over the traffic lane, a two-dimensional plain strain idealiza-tion of three-dimensional stress-strain conditions was adopted as asimplification for shakedown predictions.

For a very small load a passage of the cylinder will cause only elas-tic deformations in the half-space. If the load achieves the firstboundary value, Q*, at the most critical (in the sense of shear) pointsbelonging to the horizontal plane situated at some depth from the sur-face, plastic deformations will occur. If the load is greater than thefirst boundary value (Q . Q*), plastic deformations will occur insome zone bounded by horizontal planes parallel to the free surface.In this zone the residual stress field will arise. If this load, however,is not greater than the second load, Q** (Q* , Q # Q**), after somenumber of repeated load applications the increased residual stresseswill prevent the further plastic deformation and the elastoplastic half-space will begin to behave elastically. Under this condition the half-space material adapts to repetitive loads. If the load is greater thanthe second boundary value (Q . Q**), plastic deformation willaccompany every passage of the load and will continue indefinitely.

The analytical part of the present study has two main objectives:the determination of boundary loads Q* and Q** and the determi-nation of the residual stresses that occur as a result of plastic defor-mation. The value of Q* is determined as a load at which the maxi-mum reduced shear stress,

(1)

reaches the cohesion at the most critical point of the half-space

τ τ σ ϕmn

n n= +( )max tan

Shakedown of Subgrade Soil UnderRepeated Loading

BORIS S. RADOVSKY AND NATALIE V. MURASHINA

B. S. Radovsky, 921 North Sierra Bonita Avenue #5, Los Angeles, Calif.90046. N.V. Murashina, Department of Road Building Materials, Trans-portation University of Ukraine, 1 Suvorov Street, Kiev, 252010, Ukraine.

(2)

where

sn, τn 5 normal and shear stresses, respectively, on the plain withnormal n;

w 5 angle of internal friction;C 5 cohesion;τm 5 maximum reduced shear stress (active shear stress); andτM 5 maximum reduced shear stress in the most critical (from

the shear strength) point of a half-space; the compressivestresses are assumed to be negative, and this is reflectedin Equation 1.

For the moving load, Q 5 Q*, the limiting shear state will beachieved in points along a plane at some depth ZM from the free sur-face. At other points the material deforms elastically, allowing theuse of elastic theory for stress determination. The expressions forstresses in elastic half-space, subjected to a normal load Q and tan-gential load T 5 µQ by the rigid cylinder, can be found in Equations4.49 and 7.05 in the book by Johnson (8). If only normal load isapplied, after maximizing τm by using Equation 2, the followingequations are obtained:

p0* 5 Ch1 ⁄2M [hM 1 (1 1 h2

M)1 ⁄2][hM 1 2(1 1 h2M)1 ⁄2]1 ⁄2 (3)

Q* 5 v(p0*/C)2 (4)

where

hM 5 [(5 1 4 sin w)1 ⁄2 2 1] {4 2 [(5 1 4 sin w)1 ⁄2 2 1]2}21⁄2

,p0* 5 first boundary maximum contact pressure,Q* 5 first boundary normal load per unit length of a cylinder,hM 5 ZM/b 5 dimensions of depth,ZM 5 depth of the most critical points,

b 5 2Q/pp0,p0 5 [QE/pR(1 2 n2)]1 ⁄2,v 5 pC2R(1 2 n2)/E,b 5 semicontact width,R 5 radius of a cylinder,

τ τMx z

m C= =max,

Radovsky and Murashina 83

p0 5 maximum contact pressure under normal load Q,E 5 modulus of elasticity, andn 5 Poisson’s ratio.

Under a normal load only and for internal friction angle values inthe range from 0 to 45 degrees, the first boundary contact pressurevaries between p0* equal to 3.33C and 15.98C and the first boundaryload varies between Q* equal to 11.09v and 255.3v.

The second boundary load Q** was determined by using the sta-tic shakedown Melan theorem (2). At steady state the residualstresses introduced during early cycles of rolling load, in combina-tion with the stresses in the elastic half-space due to the applied load,must satisfy a yield condition, equilibrium conditions, and bound-ary conditions. A study of the state of stress by Johnson (3) showedthat because the system of self-equilibrated residual stresses can betaken only as horizontal normal stresses, which remain after loadapplications and which are functions of depth, other components ofthe residual stress tensor are lacking:

(sx)r 5 fx(z) (sy)r 5 fx(z)(sz)r 5 (τxy)r 5 (τ yz)r 5 (τzx)r 5 0 (5)

From Equations 5 the residual stresses automatically satisfy theequilibrium and boundary conditions.

In the analysis of a subgrade soil a Mohr-Coulomb failure crite-rion seems to be useful. The shakedown condition of an elastoplas-tic material is given by

(s1)t 2 (s3)t 1 [(s1)t 1 (s3)t]sin w # 2C cos w (6)

where (s1)t and (s3)t are equal to total major and minor principalstresses, respectively.

Total stresses are brought about by the residual stresses intro-duced during previous cycles of loading, in combination with thestresses due to applied load in the half-space as if it is elastic. Thesetotal stresses can be written as

(sx)t 5 sx 1 (sx)r (sy) t 5 sy 1 (sy)r

(sz) t 5 sz (τ zx)t 5 τzx (7)

where sx, sy, sz, and τzx are the stresses in the elastic half-space dueto the applied load, and (sx)r and (sy)r are the residual horizontalstresses introduced during previous cycles of loading. Then the useof Equations 7 leads to the following equations for total principalstresses:

(s1) t 5 1 ⁄2[sx 1 (sx)r 1 sz] 2 1 ⁄2{[sx 1 (sx)r 2 sz]2 1 4τ 2zx}

1 ⁄2 (8)

(s2)t 5 n(sx 1 sz) 1 (sy)r (9)

(s3)t 5 1 ⁄2[sx 1 (sx)r 1 sz] 2 1 ⁄2{[sx 1 (sx)r 2 sz]2 1 4τ2zx}

1 ⁄2 (10)

It follows from Equations 8 and 10 that the principal stresses (s1)t

and (s3)t remain unaffected because of variation in residual stress(sy)r. Therefore, from Equation 9 the residual stress (sy)r could bechosen such that (s2)t should be an intermediate principal stress.Equations 8 and 10 can be substituted into Inequality 6,

{[sx 1 (sx)r 1 sz]2 1 4τ 2zx}

1 ⁄2

1 [sx 1 (sx)r 1 sz]sin w # 2C cos w (11)

FIGURE 1 Representation of elastoplastic half-space underrolling cylinder.

Inequality 11 is the shakedown condition and can be expressed inthe form

(12)

where

S 5 1 ⁄2[sx 1 (sx)r 1 sz] (13)

(14)

(15)

Inequality 12 is fulfilled only when D is #0. From Equation 15for compressive (negative) vertical stress sz, this latter requirementimplies that

(16)

It follows from Inequalities 12 and 16 that the shakedown conditionfor an elastoplastic half-space is fulfilled if the Mohr-Coulombstrength condition is satisfied for an elastic half-space on horizontalplanes at all points.

This result has a clear sense. For an elastoplastic half-space theresidual horizontal normal compressive stress promotes the increasein the shear resistance at all planes except the horizontal plane. There-fore, shakedown is controlled by the maximum reduced shear stresson the horizontal plane in the elastic half-space due to applied load

(17)

The stresses tzx and sz vary in proportion to contact pressure p0.Thus, Equation 17 could be written in the form

(18)

Then, from Equations 16 and 18, for the second boundary maximumcontact pressure one obtains

p0** 5 C/maxx,z

(|τzx|/p0 1 sz/p0 tan w) (19)

Here tzx and sz are the stresses in the elastic half-space due to appliedloads Q and T. The second boundary normal load per unit length ofa cylinder is then given by

Q** 5 v(p0**/C)2 (20)

At a normal load only and for internal friction angle in the range offrom 0 to 45 degrees, the second boundary contact pressure variesbetween p0** 5 4.0C and p0** 5 20.4C and the second boundary loadvaries between Q** 5 16v and Q** 5 416v. Under the actions ofboth loads Q and T 5 0.2Q, p0** and Q** decrease. If w 5 0 degrees,then p0** 5 3.2C and Q** 5 10.2v. If w 5 45 degrees, then p0** 512.9C and Q** 5 166v.

During primary cycles of loading, plastic strains occur at suchpoints of the elastoplastic half-space where an active shear stress

(21)

is in excess of the cohesion

(22)τm x z C,( ) >

τ σ σ τ σ σ ϕ ϕm x z zx x z= −( ) +[ ] + +( ){ }2 2 1 24 2sin cos/

τ τ σ ϕzx z

zx zp p p= +( )0 0 0max / / tan,

τ τ σ ϕzx z

zx z= +( )max tan,

τ σ ϕzx z C+ ≤tan

D Czx z= +( ) − −( )[ ]1 2 2 2tan tanϕ τ σ ϕ

P C z= −( )σ ϕ ϕtan tan

S P D+( ) + ≤2 0

84 TRANSPORTATION RESEARCH RECORD 1547

Here the stresses sx, sz, and tzx are determined as for the elastic half-space. For a load moving along the x-axis the coordinate x of a point(x,z) varies. Therefore, during primary cycles of loading, plasticstrains take place at the depth z if

(23)

A zone of plastic deformation, z1 # z # z2, is bounded by the planesz 5 z1 and z 5 z2, which are parallel to the free surface. The valuesof z1 and z2 may be estimated from the following equation:

(24)

where z1,2 is the depth (z1 or z2) of plastic zone boundaries from thefree surface.

Figure 2 provides the values of depths z1 and z2 for a normal loadvalue, which is the second shakedown boundary in pure rolling (µ 5 0, Q 5 Q**, and p0 5 p0**). The lower boundary depth of theplastic zone increases with the angle of internal friction.

Residual horizontal normal stresses accompany plastic strainsand remain after the removal of the load. If a load value is in theshakedown range, Q* , Q , Q**, the residual stresses increasewith the number of repeated load applications until the momentwhen the character of deformation changes from elastoplastic toelastic. The condition of plastic strain cessation at a point (x,z) canbe written as

(25)

where [tm(x,z)]t is the total active shear stress with allowance forresidual stress is given by

(26)

Residual compressive stress is then necessary to suppress a furtherplastic strain at this point and may be obtained from Equations 25and 26 as

τ σ σ σ τ

σ σ σ ϕ ϕ

m t x x r z zx

x x r z

x z,

sin cos

( )[ ] = + ( ) −( ) +[ ]{+ + ( ) +[ ] }

2 2 1 24

2 /

τm tx z C,( )[ ] =

max , ,x

m x z Cτ 1 2( ) =

max ,x

m x z Cτ ( ) >

FIGURE 2 Zone of plasticity due tonormal load, where µ 5 0, Q 5 Q**,and P0 5 P0**.

(27)

The moving of load in the positive x-direction has the same effectas the variation of x from ∞ to 2∞. It follows that a residual com-pressive (negative) stress at depth z that remains after the removalof load may be expressed as

(28)

where [sx(z)]r is equal to residual stress at depth z, and [sx(x, z)]r isequal to residual compressive stresses calculated from Equation 27for the variable x and for a given value of z. If the angle of internalfriction equals zero, Equation 28 reduces to

(29)

The distribution of residual stresses in the elastoplastic half-spacewith depth for w 5 0, µ 5 0, and p0 5 p0** 5 4C is presented in Fig-ure 3. A residual stress approaches zero for small and large depthsand attains a single maximum value. For w 5 0 the maximum resid-ual stress occurs at depth z 5 0.5 and is equal to 20.134 p0**. Thisresult coincides with that determined by Johnson (3,8) from theTresca yield criterion.

EXPERIMENTAL RESULTS

Experimental studies were conducted to determine the applicabilityof the shakedown theory for an elastoplastic medium for soil behav-ior under repetitive loads.

A laboratory test was set up to assess qualitatively the characterof a residual stress increasing with the number of load repetitions.Under laboratory conditions sandy loam specimens of 100 mm indiameter by 130 mm in height were tested in a rigid cylindricalmold. A pressure cell of the diaphragm type of 26 mm in diameter

σ σ σ τx rx

z x zxz x z x z C x z( )[ ] = ( ) − ( ) + − ( )[ ]{min , , ,2 2 2 1 2

σ σx rx

x rz x z( )[ ] = ( )[ ]min ,

σ σ σ σ ϕ ϕ

ϕ σ ϕ τ

x r z x z

z zx

x z C

C

, tan tan

tan tan

( )[ ] = − − −( )

+ +( ) −( ) −[ ]2

2 1 2 1 2 2 2 1 2

Radovsky and Murashina 85

and 2.5 mm thick with a wire strain gauge transducer was used tomeasure horizontal residual stresses. For the measurement of thehorizontal stress the pressure cell should not be cross sensitive. Asubstantial guard ring increased the radial stiffness to reduce the output from a stress applied along a diameter to 1 ⁄20 of that for thesame stress applied to the cell face. Before installation the pressurecells were each calibrated under isotropic gas pressure, as a checkof the linearity of output, and in the same soil. The cells were cali-brated in the pressure range of 5 to 250 kPa. After a tentative com-paction, the pressure cell was installed 45, 60, or 110 mm below thesurface. Repeated loads were applied to the specimen through a rigidcircular plate 98 mm in diameter with a compaction rammer with a2500-g sliding weight with a free fall of 300 mm. The residual nor-mal stress acting on the vertical plane was measured during eachunloading.

The results of one of the tests for residual horizontal stresses arepresented in Figure 4. Under repeated vertical loading, the residualhorizontal compressive stress rose. In this test, after a few dozen rep-etitions, the residual stress reached a constant value equal to 220kPa. In Figure 4 the curve represents the best-fit line through thedatum points, having a horizontal tangent line at the point that cor-responds to some finite number of load applications. This number isequal to 64 for the curve in Figure 4. The residual stress thatincreased with the number of loadings looks like the vertical strainaccumulation. When the sum of the horizontal residual stress andthe horizontal impact lateral stress approaches the vertical impactstress value, permanent deformations ceased. In the qualitativesense, the described residual stress variation with the number of loadapplications is consistent with the adaptation of an elastoplastic sys-tem under a cyclic loading, the shakedown (1,2).

The full-scale experiment was designed to determine the generalform of a residual horizontal stress distribution within soil. Theexperimental section was situated on the road between Kiev andKharkov in Ukraine during road reconstruction. A pneumatic-tiredroller was used for silty loam compaction. This semitrailer roller hadfive tires with a wheel weight of 14.8 kN. The tires had a size of14.00–20, and the inflation pressure was 310 kPa. The soil had a liq-

FIGURE 3 Distribution withdepth z of the residual horizontalcompressive stress (wx)r where w 5 0, µ 5 0, and p0 5 p0** 5 4C.

FIGURE 4 Variation of residual horizontal normal stresswithin sandy loam due to repeated-shock load compactionwith number of load applications.

uid limit of 30 and a plastic limit of 21. The percentages of sand, silt,and clay were 10, 77, and 13, respectively. The soil had a moisturecontent of 15.0 percent. The initial and final dry weights were 1.52and 1.73 mg/m3, respectively. Pressure cells of the diaphragm type(9) with a soil strain gauge transducer were used to measure hori-zontal residual compressive stresses. The pressure cell had an over-all diameter of 38.5 mm and an overall thickness of 9.5 mm. Beforeand after installation the soil pressure cells were individually cali-brated under isotropic gas pressure to check that the output was alinear function of pressure in the pressure range of 1 to 50 kPa. Thenthe soil pressure cells were calibrated in the rigid cylindrical moldwith the same soil as the subgrade to account for the influence ofstress concentration factor, dry density, and water content. Afterprecompaction of the soil, pressure cells were installed in variouslateral and vertical positions, shown in Figure 5 along the centralline of a rolling strip with a diaphragm perpendicular to the rollingdirection. For installing the cells, vertical slots were cut with a pairof parallel semicircular blades separated by pins, and the soil wasremoved from between the blades after cutting. When the cell wasinserted, a good fit resulted. The readings were taken after each passof the roller. After 12 passes of the roller the residual stresses ceasedto increase, and this coincided in time with the termination of soilcompaction.

The measured residual stresses for the steady state are presentedin Figure 5. The curve represents the distribution of residual stressesin the elastoplastic half-space with depth calculated from Equation28. The stresses were computed for the coefficient of friction in thecontact of µ equal to 0.1 that corresponds to the towed wheel condi-tion at the end of soil compaction. The contact pressure of 310 kPa

86 TRANSPORTATION RESEARCH RECORD 1547

was taken to be equal to the inflation tire pressure. The semicontactwidth b of 13.5 cm was estimated to be half of the length of the tire’scontact area. The angle of internal friction of 20 degrees was deter-mined from the shear box text results.

It is well known that cohesion is not constant but varies withchanges in the rate and time of loading, which are difficult to eval-uate for a moving load. Therefore, the cohesion was estimated in thefollowing way. From Equations 4.49 and 7.05 of Johnson (8), thestresses τzx and sz were computed. For w equal to 20 degrees themaximum reduced shear stress on the horizontal plane in the elastichalf-space due to an applied load occurs at the point x 5 1.06b andz 5 0.40b. This is equal to

Termination of the rolling compaction process coincides with theshakedown of soil to the repetitive loads. Thus, the second bound-ary shakedown pressure can be taken to be equal to the contact pres-sure p0** equal to 310 kPa. Then, from Equation 19 one obtains thefollowing for cohesion: C 5 310 ? 0.1548 5 48 kPa.

Eventually in Equations 27 and 28 µ is set equal to 0.1, p0 is 310kPa, b is 13.5 cm, w is 20 degrees, and C is 48 kPa. The data in Fig-ure 5 demonstrate that the maximum residual horizontal stress doesnot occur immediately below the loaded area but occurs at a depth-to-semicontact width ratio of 0.4. The agreement between theoreti-cal predictions and experimental measurements of residual stressesshows the possibility of applying the shakedown theory to describethe behaviors of subgrade soils.

APPLICATIONS

The results obtained in the present study were applied to provide abetter understanding of the response of soil to repeated movingloads with reference to subgrade compaction and pavement designproblems.

For convenience, the calculated first and second boundary con-tact pressures under the action of a normal load were closely approx-imated by the following expressions:

p0* 5 C(3.33 1 4.6 tan w 1 8 tan2 w) (30)

p0** 5 C(4 1 5.9 tan w 1 10.4 tan2 w) (31)

For a pressure p0 5 p0** the depth of the lowest plastic zone bound-ary (Figure 2) was approximated by

z2 5 (bp0**/2.5C) tan (45 2 w/2) (32)

The deformation of soil as a function of the number of compactiveloads may be divided into two distinctive phases: in phase 1 plasticstrains are dominant over elastic strains, and in phase 2 plasticstrains are smaller than elastic strains. In line with the shakedowntheory, at the second phase of the rolling compaction process, theresidual stresses increase with the number of load applications untilno additional densification occurs. Thus, according to the foregoingtheory and test results, there exists an unique relationship betweenthe ground contact pressure, s, and the strength characteristics ofcompacted soil

s 5 p**(C,f) (33)

max tan .,x z

zx zp pτ σ ϕ/ /0 0 0 1548+( ) =

FIGURE 5 (a) Test setup for determination of distributionof residual horizontal stresses within a soil; (b) comparisonbetween computed and measured residual stresses afterrolling compaction.

The function p**(C,f) may be determined approximately fromEquation 31. Considering Equations 31 and 33, the required groundcontact pressure can be expressed as

(34)

where the angle of internal friction, wd, and a cohesion, Cd, shouldconform to the desired compacted dry weight of soil and its mois-ture content during compaction.

For instance, assume that for a desired compacted dry weight atan optimum moisture content the angle of internal friction of the soilis equal to 32 degrees and its cohesion is equal to 0.04 MPa. Then,to the end of the rolling compaction, the required ground contactpressure (controlled by tire inflation pressure) is

which is easily met by a 20-in. rim with 16- or 18-ply tires in accordwith current practice.

The depth of accumulation of residual stresses (Figure 4) approx-imately corresponds to the active zone of compaction according toKharkhuta (10). It looks like the active zone of compaction is thezone of small plastic strains and residual stresses caused by thosestrains. A compacted thickness of the subgrade soil, h, can be esti-mated as the depth of the lowest boundary of a plastic zone:

(35)

Use of Equations 32 and 35 leads to the following expression for thecompacted thickness of soil:

(36)

Here b1 is the semilength of the tire’s contact area (controlled byweight of a roller).

For instance, take an angle of internal friction of 32 degrees (fd),a tire outside diameter (D1) of 1.32 m, and of the tire sinkage (d) of0.02 m. The semilength of the tire’s contact area may be estimatedfrom (8)

Then the expected maximal depth of the compacted zone is to be

It is noteworthy that according to Equation 36 the maximal depth ofthe compacted zone remains unaffected because of variation incohesion and increases with the angle of internal friction. The latteris in accord with experimental data from the field (10). Note thatalthough there are no empirical constants in Equations 34 and 36,these equations nevertheless lead to reasonable results.

The proposed analytical approach can be applied to pavementdesign problems for the consideration of the multilayered structureof pavement systems. A structural failure is generally associated withcracking or permanent displacements in either pavement layers or theunderlying subgrade (rutting). Krivissky (11) proposed three pave-ment design schemes according to the main types of subgrade behav-ior with load repetition, as illustrated in Figure 6. The results pre-sented in this paper can be used to predict a pavement behavior

h = + +( ) −( )=

0 16 1 6 2 36 32 4 16 32 45 32 20 42

2. . . tan . tan tan.

/m

b D m1 11 2 0 16= −( )[ ] =δ δ .

h b d d d= + +( ) −( )121 6 2 36 4 16 45 2. . tan . tan tanϕ ϕ ϕ /

h z bd= ( )2 ϕ ,

σ = + +( ) =0 04 4 5 9 32 10 4 32 0 472. . tan . tan . MPa

σ ϕ ϕ= + +( )Cd d d4 5 9 10 4 2. tan . tan

Radovsky and Murashina 87

depending on the stress level of the subgrade as an elastoplastic solid.This prediction can be obtained within the scope of using the tradi-tional elastic layered theory to determine the stresses in the subgrade.

1. The scheme of recoverable deformations is realized under thefollowing condition:

(37)

where τM is the maximum reduced (active) shear stress in the sub-grade, as in the elastic solid due to applied load.

2. The scheme of small elastoplastic deformations is realizedunder the following conditions:

(38)

(39)

where (τM)t is the total maximum reduced shear stress determined withallowance for both residual stresses and the stresses in the elastic soliddue to the applied load. If a road structure is designed such that thelimit of Equation 39 is never reached, then the rate of permanentdeformations in the structure will decrease with the number of loads.

3. The scheme of large elastoplastic deformations (incrementalcollapse) is valid under the following conditions:

(40)

If a road structure is designed such that the limit of Equation 39 isexceeded (i.e., the inequality in Equation 40 is fulfilled), then therate of permanent deformations in the structure may increase withthe number of loads and the permanent deformations developingshortly after the opening of the road may be large.

Within the scope of plane strain analysis, the maximum reducedshear stress due to applied load, τM, and that with allowance for theresidual horizontal compressive stress, (τM)t, may be obtained fromEquation 11 as

(41)

τ ϕ σ σ τ

σ σ ϕM

x zx z zx

x x

x z x z x z

x z x z

= ( ) ( ) − ( )( ) + ( )[ ]{+ ( ) + ( )[ ] }

−2 41 2 2 1 2cos max , , ,

, , sin,

τM t C( ) >

τM t C( ) ≤

τM C≥

τM C<

FIGURE 6 Three schemes of deformationwith load repetitions for pavement design.

(42)

Here [sx(z)]r is the residual stress at depth z expressed by Equation28. The elastic stresses sx, sz, and tzx could be calculated by usingseveral programs that have been developed for elastic layered struc-tures. Note that for a given pavement thickness and subgradestrength, the mode of subgrade behavior with load repetitionsdepends on the magnitude of the load. If Q* , Q , Q**, permanentdeformations (rutting) of the subgrade will cease to increase, indi-cating an adaptation. The ratio

5

increases only from 1.44 to 1.63 for internal friction angles rangingfrom 0 to 45 degrees. These values may be useful for determiningthe legal load limits to be posted on roads.

Although the proposed analytical approach is rather simple, fur-ther research is needed to improve and extend shakedown predic-tions of pavement structures since a plane strain approximation ofthe actual three-dimensional problem was used. In the overall struc-tural design system of flexible pavements, the distribution of plasticstrains in the lateral y-direction and all components of displacementare of relevance to the rutting problem.

SUMMARY AND CONCLUSIONS

The investigation described here was undertaken because of theinterest in shakedown phenomenon for soils and the need for a pro-cedure by which subgrade and pavement performances underrepeated loads could be predicted.

The results of the analytical part of the study are as follows:

1. The boundary loads and contact pressures in which the shake-down condition or steady state can be attained under repeated load-ing were determined. These boundary shakedown loads and pres-sures increase with an increase in subgrade shear strength. The ratioof the largest to the minimum boundary shakedown load dependsonly on the angle of internal friction and ranges from 1.44 to 1.63.

2. The distribution with depth of the residual horizontal stress gen-erated in the half-space by repeated moving loads was determined. Aresidual compressive stress approaches zero at small and large depthsand attains a maximum value, depending on the shear strength prop-erties of the soil. A zone of plastic deformation is bounded by theplanes that are parallel to the free surface. The residual stresses arisewithin this zone. The accumulated residual stresses can prevent fur-ther plastic deformation for additional load repetitions.

The main results of the experimental part of the study are as follows:

1. The residual horizontal compressive stresses were measuredin the repeated-shock-load soil compaction test. The residual stressincreased with the number of loadings, and after a few dozen repe-titions it reached a constant value.

2. Under field conditions horizontal compressive stresses weremeasured in soil during soil compaction with a pneumatic tire roller.

(4 1 5.9tanw 1 10.4 tan2 w)2

}}}(3.33 1 4.6tanw 1 8 tan2 w)2

Q**}Q*

τ ϕ σ σ σ

τ σ σ σ ϕM t

x zx x r z

zx x x r

x z z x z

x z x z z x z

( ) = ( ) ( ) + ( )( ) − ( )( )[{+ ( )] + ( ) + ( )( ) + ( )[ ] }

−2

4

1 2

2 1 2

cos max , ,

, , , sin,

88 TRANSPORTATION RESEARCH RECORD 1547

The residual stresses increased with the number of roller passes andgradually ceased to increase by the termination of compaction. Theresidual stress distribution with depth computed by the proposedmethod was in good agreement with the residual stresses measuredat various depths.

Two applications were presented:

1. The final phase of the soil compaction process is interpreted interms of plasticity theory as the elastoplastic-to-elastic shakedowntransition. Thus, according to the present analytical results, therequired contact pressure and thickness of the compacted zone canbe estimated as functions of the desired soil properties.

2. The structural design approach based on determination of theshear stresses in the subgrade in view of residual stresses was pre-sented. If a road structure is designed so that a reduced shear stressdetermined with allowance for both residual stresses due to previ-ous loading and elastic stresses due to applied load never reaches acohesion, then the rate of permanent deformations in the structurewill decrease with the number of traffic loadings.

Further research by using shakedown theory for a three-dimensional system is needed to improve and extend shakedownpredictions of pavement structures.

ACKNOWLEDGMENT

The authors thank K. L. Johnson of the University of Cambridge forvaluable discussions.

REFERENCES

1. Koiter, W. T. A New General Theorem on Shakedown of ElasticplasticStructures. Proc., Koninklishe Nederlandse Akademie van Weterschap-pen, Series B, Vol. 59, No. 1, 1956, pp. 24–34.

2. Melan, E. Theorie Statisch Unbestimmter Systeme aus Ideal-PlastischenBaustoff. Sitzungsberichte der Akademie der Wissenschaften in Wien,Vol. 2a, 1936, pp. 145–195.

3. Johnson, K. L. A Shakedown Limit in Rolling Contact. Proc., 4th U.S. National Congress of Applied Mechanics, Berkeley, Calif., Vol 2,1962, pp. 971–975.

4. Alwis, W. A. M., and P. Grundy. Shakedown of Plates Under MovingLoads. Proc., 7th Australian Conference on Mechanics of Structuresand Materials, 1980, pp. 191–196.

5. Konig, J. A. Shakedown Theory of Plates. Archiwum MechanikiStosowanej, Vol. 21, No. 5, 1969, pp. 623–637.

6. Sharp, R. W., and J. R. Booker. Shakedown of Pavements Under Mov-ing Surface Loads. Journal of the Transportation Engineering Division,ASCE, Vol. 110, No. TE1, 1984, pp. 1–14.

7. Raad, L., D. Weichert, and W. Najm. Stability of Multilayer SystemsUnder Repeated Loads. In Transportation Research Record 1207, TRB,National Research Council, Washington, D.C., 1990, pp. 181–186.

8. Johnson, K. L. Contact Mechanics. Cambridge University Press, Cam-bridge, 1987.

9. McMahon, T. F., and E. J. Yoder. Design of a Pressure Sensitive Cellsand Model Studies of Pressures on a Flexible Pavement Subgrade. HRBProc., Vol. 39, 1960, pp. 650–682.

10. Kharkhuta, N. Y. Compaction Equipment for Soils (in Russian).Machinery Moscow, 1973.

11. Krivissky, A. M. New Flexible Pavement Design Schemes (in Russian).Transport Moscow, 1961.

Publication of this paper sponsored by Committee on Soil and Rock Properties.