New concepts have been developed to determine the long term mechanical behaviour of unbound materials under repeated loadings. All these concepts are presented in a special issue (Yu, 2005) of the International Journal of Road Materials and Pavements Design.
The shakedown concept applied to pavements was introduced first by Sharp and Booker (1984). The various possible responses of an elastic-plastic structure to a cyclical load history are indicated schematically in Fig. 9.8. If the load level is sufficiently small, the response is purely elastic, no permanent strains are induced
|
Fig. 9.8 Classical elastic/plastic shakedown behaviour under repeated cyclic tension and compression. Reprinted from Wong et al. (1997), @ 1997, with permission from Elsevier |
and the structure returns to its original configuration after each load application. However, if the load level exceeds the elastic limit load, permanent plastic strains occur and the response of the structure to a second and subsequent loading cycle is different from the first. When the load exceeds the elastic limit the structure can exhibit three long term responses depending on the load level (Fig. 9.8). After a finite number of load applications, the build up of residual stresses and changing of material properties can be such that the structure’s response is purely elastic, so that no further permanent strain occurs. When this happens, the structure is said to have shaken down: it is in the elastic shakedown region. In a pavement this could mean that some rutting, subsurface deterioration, or cracking occurs but that, after a certain time, this deterioration ceases and no further structural damage occurs.
At still higher load levels however, shakedown does not occur, and either the permanent strains settle into a closed cycle (plastic shakedown behaviour) or they go on, increasing indefinitely (ratcheting behaviour). Contributions of Yu & Hossain (1998), Collins and Boulbibane (2000), Arnold et al. (2003) and Maier et al. (2003) are based on the fact that if either of these latter situations occurs, the structure will fail. The critical load level below which the structure shakes down and above which it fails is called the shakedown load and it is this parameter that is the key design load. The essence of shakedown analysis is to determine the critical shakedown load for a given pavement. Pavements operating above this load are predicted to exhibit increased accumulation of plastic strains under long-term repeated loading conditions that eventually lead to incremental collapse. Those pavements operating at loads below this critical level may exhibit some initial distress, but will eventually settle down to a steady state in which no further mechanical deterioration occurs.
The direct calculation of the shakedown load is difficult. Indeed lower and upper bounds are usually calculated using Melan’s static or Koiter’s kinematic theorems, respectively. These procedures are similar to the familiar limit analysis techniques for failure under monotonic loading, except that now the elastic stress field needs to be known and included in the calculation. Finally, the material is assumed to be perfectly plastic with an associated flow rule.
Shakedown models require an elasticity framework and parameters, for example as provided by the Universal model, and the knowledge of the rupture parameters of the Drucker-Prager or Mohr-Coulomb surfaces. 2D finite element plane strain calculations of pavements have been carried out in this way.
Parameters needed are:
• Elastic behaviour: k1, k2, k3, v; and
• Plastic behaviour: c, y.
Contributions of Habiballah and Chazallon (2005) and Allou et al. (2007) are based on the theory developed by Zarka and Casier (1979) for metallic structures submitted to cyclic loadings. Zarka defines the plastic strains at elastic shakedown with the Melan’s static theorem extended to kinematic hardening materials. The evaluation of the plastic strains when plastic shakedown occurs is based on his simplified method. Habiballah has extended the previous results to unbound granular materials with a non-associated elasto-plastic model. The Drucker-Prager yield surface is used with a von Mise plastic potential.
This approach requires the elasticity parameters of the k — 0 or other model, the rupture line and the law describing the development of the kinematic hardening modulus. This model requires a “multi-stage” procedure, developed by Gidel et al., (2001) which consists, in each permanent deformation test, of performing, successively, several cyclic load sequences, following the same stress path, with the same q/p ratio, but with increasing stress amplitudes. Finite element calculations have been carried out under axi-symmetric condition and 3D. The initial state of stress is determined with the k— 0 or other model, then the plastic strains are calculated.
Parameters required are:
• Elastic behaviour: k1, k2, v (assuming that the k—0 model was selected); and
• Plastic behaviour: c, y, H(p, q).
