(2) Jim Shiau. 1. INTRODUCTION Road pavement is one of the good examples to explain the shakedown concept in the plasticity design of a continuum. When a road pavement is overloaded by a heavy vehicle repeatedly moving in a single direction, it leads to plastic (irrecoverable) deformations on the road surface. A redistribution of stress in the road pavement thus occurs. The plastic stress present in the road pavement following unloading is known as a “residual stress”. Provided that subsequent loads are less than a certain limit load experienced by the road pavement, this residual stress in the road pavement offers protection against further accumulation of plastic deformations. This load is known as the ‘shakedown limit load’ and the protective residual stresses associated with this shakedown limit load are the optimal residual stresses for the life of the structure. It is thus clear that if a body (for example road pavement ) shakes down after a number of variable repeated loadings, the associated shakedown quantities such as shakedown limit load (Psd), residual stresses (σ r), and permanent deformations (∆ r) are of main concern in the design of structures subjected to moving repeated loadings. A rigorous step-by-step procedure for determining the shakedown limits under complex traffic loads is difficult. Apart from the unknown loading history associated with pavements, such a step-by-step procedure would not be practical in view of high computation costs. Melan’s static shakedown theorem provides a simple alternative for estimating the shakedown limit load and the associated residual stress fields. It enables us to determine the overall behaviour of the pavement under variable repeated traffic loading. Melan’s static shakedown theorem states that ‘‘If the combination of a time independent, self-equilibrated residual stress field σ rij and the elastic stresses field λσ eij can be found which does not violate the yield condition anywhere in the region, then the material will shakedown”. In other words, if no such residual stress fields can be found then the system will not shakedown and plastic deformation will be accumulated at every passage of the load.. λ. Element equilibrium . Shakedown load factor. σ eij. . Elastic stress fields D. D. Displacement boundary. D. +. D. D D. Stress boundary. Element equilibrium. σ rij. Residual stress fields D. D D. Compatibility. f λσ eij + σ rij < 0. Discontinuity equilibrium. Figure 1. Finite element application of Melan’s static shakedown theorem. Figure 1 shows such a graphical representation of Melan’s static shakedown theorem. Supposing that the elastic stresses are proportional to a load factor λ, the combined stresses are therefore σ tij = λσ eij + σ rij. (1).

(3) Jim Shiau. where λ is the shakedown load factor, σ eij are the elastic stresses resulting from a unit pressure application and σ rij are the residual stresses. In conjunction with finite elements and linear programming, this paper will employ this theorem to predict the shakedown capacity as well as the associated residual stress fields for a plane strain rolling and sliding contact problem. A brief discussion of the numerical procedure is shown as follows. 2. FINITE ELEMENT APPLICATION OF STATIC SHAKEDOWN THEOREM Shown in Figure 1 is the illustration of a finite element application of Melan’s static shakedown theorem. As indicated, both elastic stress fields and residual stress fields required by the shakedown theorem are assumed to be linearly distributed across the continua by making use of the displacement and stress finite elements respectively. To ensure that the application of Melan’s static shakedown theorem leads to a linear programming problem, the Tresca yield criterion can be linearised. The calculation of shakedown limits are then considered as a large linear programming problem: the maximisation of the shakedown load factor λ subject to the constraints due to: (1) Element equilibrium; (2) Discontinuity equilibrium; (3) Stress boundary condition; and (4) Linearised Mohr--Coulomb yield constraint. A typical finite mesh for the shakedown analysis is shown in Figure 3. The proposed finite element technique can be used to deal with complicated contact load distribution, layered soil and inhomogeneous material properties which are often encountered in the fields of geotechnical engineering. More details of the formulation can be found in Shiau (2001), Shiau and Yu (2000), and Sloan (1987) and will not be repeated here. 3. JOHNSON’S SOLUTION AND NUMERICAL VERIFICATION In order to compare the numerical shakedown limit obtained by the proposed formulation with the analytical shakedown solution of a free rolling contact derived by Johnson (1962 and1985), the contact stress distribution is assumed to be a form of Hertz contact pressure as shown in Figure 2. The normal pressure p v(x) and the surface shear traction p h(x) at the surface are distributed semielliptically according to p v(x) = p v. 1 − (x∕a)2. p h(x) = mp v. (2). 1 − (x∕a)2. (3) x. Pv. a. Ph. Pv. p v(x) = pv. p h(x) = mpv. Ph Layered material. 1 − (x∕a)2. Y. Figure 2. Hertz load distribution for plane strain model. 1 − (x∕a)2 m=. X. p h(x) pv(x).

(4) Jim Shiau. where pv is the maximum contact pressure and a is the semicontact width. This implies that a constant coefficient of surface friction m is adopted, where m = p h(x)∕p v(x). As shown in Figure 2, a loading cycle on the structure surface consists of the loading patch moving from x = − ∞ to x = + ∞. Application of Melan’s quasi-static shakedown theorem requires, in addition to the elastic stresses, a system of self--equilibrium residual stresses. Johnson’s analytical approach to this problem will be discussed as follows. Johnson (1985) stated that because the same loading history is experienced for all points on the pavement surface, the residual stress distribution must be independent of the travel direction. It follows that the only residual stresses are σ rx and σ rz for such a plane strain condition and are uniform over any horizontal plane. If σ rz is chosen as the intermediate stress, we can write the following equation for using the Tresca yield criterion. 1 (σ e + σ r ) − σ e 2 + (τ e ) 2 ≦ c 2 (4) x y xy 4 x This equation cannot be satisfied if τ exy exceeds c. However, if we choose σ rx = σey − σex , it can just be satisfied with τ exy equal to c. Thus, the limiting conditions for shakedown to occur in the solid is possible when the value of τ exy is a maximum. The maximum elastic shear stress under the unit Hertz contact stress distribution is given in Johnson (1962) where (τ exy)max = 0.25p v occurs at x = 0.87a and y = 0.5a. It therefore gives a lower bound to the shakedown limits such that p v∕c ≧ 4.00. The analytical residual stress distribution for a uniform half space can be obtained by rearranging equation (4) which gives σ rx(y) = (σ ey − σ ex) + 2c 2 − (τ exy) 2. 1∕2. (5). As shown before, the maximum value of elastic shear stress under the Hertz contact stress distribution is (τ exy)max = 0.25p v and the corresponding value of (σ ey − σ ex) = 0.134pv (Johnson 1962, 1985). By putting these values into (5), the following equation can be established.. . σ rx(y) = 0.134p v + 2 c 2 − 1 (p v) 2 16. . 1∕2. (6). Given values of c and p v on the right hand side of equation (6), the maximum analytical residual stress (σ rx)max can be determined accordingly. A typical rectangular type of finite element mesh used for both elastic stress field and residual stress field is shown in Figure 3. It should be noted that a full mesh is required when the surface shear stress is applied. The displacement finite element mesh consists of 576 quadratic elements and 1201 nodes while the stress finite element mesh consists of 576 linear stress elements and a total of 1728 nodes. The total number of discontinuities for the stress-based mesh is 840. The shakedown limit obtained in this study for a purely cohesive material with p h∕pv = 0 is 3.891 which is 2.7% less than 4.0 that is the analytical solution derived by Johnson (1962 and 1985)..

(5) Jim Shiau. L B Number of nodes = 1201 Number of elements = 576. PV. PH. Displacement finite element mesh. Number of nodes = 1728 Number of independent coordinates = 313 Number of elements = 576 Number of discontinuities = 840 σrn = τ r = 0. Stress-based finite element mesh. Figure 3. Finite element full mesh for shakedown analysis (rectangular type). Given p v = 3.891 and c = 1, equation (6) yields the maximum analytical residual stress (σ ) = 0.985. This value is about 2% less than the numerically determined one which gives 1.007c for a mesh with 576 finite elements (Figure 6 ). More discussions on the residual stress distribution will be made in a later section. r x max. 4. EFFECT OF SURFACE FRICTION AND RESIDUAL STRESS DISTRIBUTION The effect of surface friction on the shakedown limits for isotropic, homogeneous cohesive soil is presented in Figure 4. It can be seen that the dimensionless shakedown limit decreases dramatically with the increase in the coefficient of surface friction m. The elastic limit loads, as shown in this figure, are obtained by insisting that no residual stresses exist in the media by using the shakedown formulation. The differences between these two curves thus indicates the benefit of shakedown phenomenon under repeated loadings. It was previously shown that, for shakedown to occur, the orthogonal shear stress (τ exy)max must not exceed the shear strength c at any point in the stress field. The position and magnitude of (τ exy)max will have to be evaluated for different values of surface friction. These have been done by using displacement finite element analysis and are presented in Figure 5 where stress contours of (τ exy)max are plotted for different values of m. Further examination on the elastic shear stresses from these figures shows that the location of maximum shear stress moves from a depth of y=0.5a when m = 0 to the pavement surface when m is approximately equal to 0.37. This indicates a type of transfer from subsurface failure to the surface failure. When m = 0 , the direct elastic stress components are symmetric and the elastic shear components are antisymmetric about the central axis. These figures also show that the effect of a surface tangential stress ( m > 0 ) is to increase one of these peak values (τ exy)max and, at the same time, to decrease the other. It is well known that the first yield (elastic limit) is reached at a point beneath the surface (Johnson, 1985). The existence of the surface shear forces will introduce a new state of stress.

(6) Jim Shiau. at the surface. When the coefficient of surface friction exceeds a certain value, yield may begin at the surface rather than beneath it. It has been demonstrated that this critical value of m is approximately equal to 0.37 from our displacement finite element study. It is also noted that for cases m > 0.37, the critical condition for shakedown has moved to the surface and is then controlled by the surface stresses. Thus, it may be concluded that the shakedown limit load is not significantly different from the elastic limit load for high values of m. Figure 4 showed such a state where elastic limits are rather close to the shakedown limits at high coefficient of surface friction m. This may imply that the protective residual stresses may not be developed in the case where high surface shear stresses exist. The study of these protective residual stresses with varying m will be discussed next.. 4.5 4.0. Subsurface Failure Surface Failure. 3.5. λp v c 3.0. Shakedown Limit. 2.5 2.0 Elastic Limit. 1.5 1.0 0.5 0.0 0.0. 0.37 0.2. 0.4. 1.0 m Coefficient of Surface Friction m 0.6. 0.8. Figure 4. A Shakedown Map indicating the effect of the coefficient of surface friction m upon dimensionless shakedown limits.

(7) Jim Shiau. (τ exy)max = 0.253. -.252849 -.196661 -.140472 -.084283 -.028094 .028094 .084283 .140472 .196661 .252849 (a). m = 0 , λp v∕c = 3.953 (τ exy)max = 0.317. -.192467 -.135853 -.079239 -.022626 .033988 .090602 .147216 .203829 .260443 .317057 (b). m = 0.2 , λp v∕c = 3.143 (τ exy)max = 0.365. -.157461 -.099386 -.041311 .016763 .074838 .132913 .190988 .249063 .307138 .365212 (c). m = 0.37 , λp v∕c = 2.731 (τ exy)max = 0.381. -.145793 -.087231 -.028669 .029893 .088455 .147017 .205579 .264141 .322702 .381264. (d). m = 0.4 , λp v∕c = 2.617. Figure 5. Elastic shear stress contour τ exy for various surface frictions.

(8) Jim Shiau B. pv. m = 0 , λp v∕c = 3.891. ph. 0.00 D/B. m = p h∕pv = 0. D. - 0.25 - 0.50 - 0.75 - 1.00 0.0. 0.5. 1.0. r 1.5 σ x∕c. Figure 6. Horizontal residual stress distribution for m = 0 , λp v∕c = 3.891. Figure 6 shows the numerically determined distribution of horizontal residual stresses at shakedown state for the case of m = 0 . The value of σ rx∕c reaches a maximum at a depth of D/B=0.25 with σ rx = 1.007c. This value converged to zero approximately at a depth of D/B=0.75. The principal stress vectors beneath the loaded area are also shown on the right hand side of this figure. As shown, only σ rx exists in the media. In such a plane strain analysis of line contact, residual stresses vary with depth only and are uniform over any horizontal plane (Johnson 1985). B. m = 0.2 , λp v∕c = 3.270. pv. m = p h∕pv = 0.2. ph. 0.00 D/B. - 0.25. D. - 0.50 - 0.75 - 1.00 0.0. 0.5. 1.0. r 1.5 σ x∕c. Figure 7. Horizontal residual stress distribution for m = 0.2 , λp v∕c = 3.270. Residual stress distributions for different values of m are shown from Figure 7 to Figure 9. Figure 10 collects the residual stress distributions for all values of m. It is found that as m is increased, the location of maximum σ rx∕c moves to the pavement surface and the value of maximum σ rx∕c is significantly decreased. This means that very little protective residual stresses is introduced when the value of m is high. It further implies that the shakedown limit load will.

(9) Jim Shiau. eventually be the same as the first yield load when m = 1.0 and there will be no protective residual stresses developed in the media. B. pv. m = 0.4 , λp v∕c = 2.624. m = p h∕pv = 0.4. ph. 0.00 D/B - 0.25. D. - 0.50 - 0.75 - 1.00 0.0. 0.5. 1.0. r 1.5 σ x∕c. Figure 8. Horizontal residual stress distribution for m = 0.4 , λp v∕c = 2.624. B. pv. m = 1.0 , λp v∕c = 1.053. m = p h∕pv = 1.0. ph. 0.00 D/B. - 0.25. D. - 0.50 - 0.75 - 1.00 0.0. 0.5. 1.0. 1.5. σ rx∕c. Figure 9. Horizontal residual stress distribution for m = 1.0 , λp v∕c = 1.053. CONCLUSION Residual horizontal normal stresses remain after the removal of the load. If a load value is in the shakedown range, the residual stresses increase with the number of repeated load applications until the moment when the character of deformation changes from elastoplastic to elastic. This is the state when shakedown occurs. Those residual stresses can then be referred to as the.

(10) Jim Shiau. optimum values for the life of the proposed strucure. It is thus important to determine these optimum residual stresses at shakedown state. 0.00. - 0.25. D/B. - 0.50 m=0 m = 0.2. - 0.75. m = 0.4 m = 1.0. - 1.00 0 00. 0 25. 0 50. 0 75. 1 00. 1 25. σ rx∕c. Figure 10. Horizontal residual stress distribution for various values of m. A detailed numerical investigation has been carried out for residual stresses at shakedown state under the actions of combined normal and surface shear forces. It is concluded that reliable residual stresses can be obtained using the current lower bound shakedown procedure, which is an extension of the lower bound limit analysis formulation using finite elements and linear programming (Sloan 1988). ACKNOWLEDGEMENT The author would like to acknowledge Professor Scott Sloan and Professor Hai-Sui Yu for their guidance during the period 1998--2003 at the University of Newcastle, Australia. Professor Ian Collins’s helpful discussion on the lower bound shakedown solution is greatly appreciated. REFERENCES [1] [2] [3] [4]. Collins, I. F. and Cliffe, P. F. (1987) “Shakedown in frictional materials under moving surface loads,” International Journal for Numerical and Analytical Methods in Geomechanics, 11, 409--420. Johnson, K. L. (1962) “A shakedown limit in rolling contact”, in Proc. 4th Natl. Conf. on Apllied Mechanics, Berkeley, CA, 971--975. Johnson, K. L. (1985) Contact Mechanics, Cambridge University Press. Johnson, K. L. (1992) “The application of shakedown principles in rolling and sliding contact”, Eur. J. Mech., A/Solids, 11, 155-172..

(11) Jim Shiau. [5]. Ponter, A. R. S., Hearle, A. D. and Johnson, K. L. (1985) “Application of the kinematical shakedown theorem to rolling and sliding point contacts”, Journal of the Mechanics and Physics of Solids, 33(4), 339--362. [6] Shiau, J. S. and Yu, H. S. (2000) ‘Load and displacement prediction for shakedown analysis of layered pavements’, Transportation Research Board, No 1730, 117-124. [7] Shiau, J. S. (2001) Numerical Methods for Shakedown Analysis of Pavements under Moving Surface Loads, Ph.D. Thesis, The University of Newcastle, NSW, Australia. [8] Shiau, J. S., Lyamin, A. V. and Sloan, S. W. (2003). “Bearing capacity of a sand layer on clay by finite element limit analysis.” Canadian Geotechnical Journal, 40, 900--915. [9] Shiau, J. S., Augarde, C. E., Lyamin, A. V. and Sloan, S. W. (2006). “Passive earth resistance in cohesionless soil.” Canadian Geotechnical Journal. (Accepted) [10] Sloan, S. W. (1988) “Lower bound limit analysis using finite elements and linear programming”, International Journal for Numerical and Analytical Methods in Geomechanics, 12, 61-67. [11] Hossain, M. Z. and Yu, H. S. (1997) “Lower bound shakedown analysis of layered pavements using discontinuous stress fields”, Research report No. 147.07.1997, The Univesity of Newcastle, NSW, Australia.

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