Acta Geotechnica

, Volume 6, Issue 1, pp 13–20 | Cite as

One-dimensional consolidation under haversine repeated loading with rest period

Research Paper


Presented in this paper is a study of the one-dimensional consolidation process under haversine repeated loading with and without rest period. The analysis was carried out using a hybrid coupled, analytical and numerical implicit finite difference technique. The rate of imposition of excess pore-water pressure was determined analytically, and the remaining part of the governing differential equation was solved numerically. The clay deposit considered was a homogeneous clay layer with permeable top and/or impermeable bottom hydraulic boundary conditions with constant coefficients of permeability and of consolidation. The study reveals that although the loading function is positive for all times, the excess pore-water pressure at the base of the clay deposit with permeable top and impermeable bottom oscillates with time reaching a ‘steady state’ after few cycles of loading depending on whether there is a rest period or not. An increase in the rest period causes a decrease in the number of cycles required to achieve the steady state. The paper shows also that the rest period in the haversine repeated loading decelerates the consolidation process. Similarly, the paper reveals that the effective stress at the bottom of the clay layer with permeable top and impermeable bottom increases with time but showing mild fluctuations that do not change the sign. The maximum positive effective stress achieved depends on the rest period of the haversine repeated loading. A haversine repeated loading without a rest period gives the highest value for the positive effective stress.


Consolidation Haversine repeated loading Hybrid analytical and numerical solution 



Coefficient matrix


Radius of circular tire-pavement contact area; constant quantity


Column vector of unknown quantities


Column vector of known quantities


Coefficient of consolidation in vertical direction


Period of haversine repeated loading without rest period or the duration of loading/unloading phase of the haversine repeated loading with rest period


Thickness of clay layer with permeable top and impermeable bottom or half the thickness of a clay layer with permeable top and bottom


Number of uniform intervals ∆z in H


Loading function (haversine repeated loading with or without rest period)




Number of uniform intervals ∆t within (d)


Clay layer with permeable top and impermeable bottom


Clay layer with permeable top and bottom


Amplitude of haversine repeated loading


Rest period


Function of time


Speed of vehicle for highways or airports


Dimensionless time factor


Actual time


Excess pore-water pressure


Imposed excess pore-water pressure


Vertical coordinate measured from top surface of the clay layer downward positive


Dimensionless coefficient in implicit finite difference equation


Effective stress



The first author thanks German Academic Exchange Service (DAAD) for funding his stay as a guest professor with the second author in 2009/2010.


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental Engineering Chair for Foundation Engineering, Soil & Rock Mechanics, Ruhr-UniversitätBochumGermany

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