# Alternative Analytical Solution of Consolidation at Constant Rate of Deformation

- 27 Downloads

## Abstract

In this paper, we present an alternative solution of consolidation at constant rate of deformation (CRD). Governing equation of CRD consolidation is formulated considering excess pore water pressure as primary variable. Formulation of the governing equation is done for soil as linear as well as nonlinear materials. An analytical solution to the governing equation is derived which consists of transient state and steady state components. The solution derived herein yields profile of excess pore water pressure distribution with depth at any time during the test. The present paper analyzes the excess pore water pressure distribution profile during CRD consolidation and factors affecting it in detail. Analysis of the obtained solution indicates the existence of moving boundary condition within the test specimen at early stage of the test where transient state condition prevails. It also indicates that the pore water pressure ratio affects significantly the excess pore water distribution profile, in turn, average value of pore water pressure and average effective stress in the specimen. An expression for consolidation parameters for steady state condition are derived using the present solution. Further, evolution of excess pore water pressure at the base of CRD consolidation test specimen during the test is predicted for normally consolidated as well as for over consolidated soils at different rates of deformation.

## Keywords

Clays Consolidation Compressibility CRD consolidation CRS consolidation## Notation

- CRD
Constant rate of deformation

- IL
Incremental loading

- \(c_{v}\)
Coefficient of consolidation

*e*Void ratio

*F*Dimensionless factor from Eq. 13

- \(H_{0}\)
Initial total depth of the test specimen

*H*Current total depth of the test specimen

*k*Coefficient of permeability

- \(P_{a}\)
Atmospheric pressure

- \(R_{u}\)
Pore water pressure ratio

*t*Time

- \(T_{v}\)
Time factor

*u*Pore water pressure within the specimen

- \(u_{b}\)
Pore water pressure at the base of the test specimen

- \(u_{\text {avg}}\)
Average pore water pressure

- \(U_{\text {lin}} = u/P_{a}\)
Non dimensional pore water pressure factor for linear material

- \(U_{\text {ln}} = \ln (1-u/\sigma )\)
Non dimensional pore water pressure factor for nonlinear material

*U*Non dimensional pore water pressure factor from Eq. 11

*V*Non dimensional term from Eq. 31

*W*Non dimensional term from Eq. 31

*z*Depth

*Z*Non dimensional depth factor

- \(\alpha\)
Ratio of \(u_{\text {avg}}\) and \(u_{b}\)

- \(\gamma\)
A constant term defined in Eq. 11

- \(\gamma _{\text {lin}}\)
A constant term defined in Eq. 9

- \(\gamma _{\text {nl}}\)
A constant term defined in Eq. 10

- \(\gamma _{w}\)
Unit weight of water

- \(\sigma ^{'}\)
Vertical effective stress

- \(\sigma ^{'}_{\text {avg}}\)
Average effective stress

- \((\sigma ^{'}_{\text {lin}})_{\text {avg}}\)
Average effective stress for linear material

- \((\sigma ^{'}_{\text {nl}})_{\text {avg}}\)
Average effective stress for nonlinear material

## Notes

### Acknowledgements

Financial Support from Department of Science and Technology (DST) India through Grants SERB/FTP/ETA-0041/2014 is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of authors and do not necessarily reflect the views of DST.

## References

- Al-Tabbaa A, Wood DM (1987) Some measurements of the permeability of kaolin. Geotechnique 37(4):499–503CrossRefGoogle Scholar
- ASTM:D4186-06 (2008) Standard test method for one-dimensional consolidation properties of soils using incremental loading. ASTM International, West Conshohocken, PA, USAGoogle Scholar
- Armour DW, Drnevich VP (1986) Improved techniques for the constant-rate-of-strain consolidation test. In: Consolidation of soils, STP 892. American Society for Testing and Materials, Philadelphia, pp 170–183Google Scholar
- Crawford CB (1988) On the importance of rate of strain in the consolidation test. Geotech Test J 11(1):60–62CrossRefGoogle Scholar
- Davis EH, Raymond GP (1965) A non-linear theory of consolidation. Geotechnique 15(2):161–173CrossRefGoogle Scholar
- Fox PJ, Pu H, Christian JT (2014) Evaluation of data analysis methods for the CRS consolidation test. J Geotech Geoenviron Eng ASCE 140(6):04014020CrossRefGoogle Scholar
- Gorman CT, Hopkins TC, Dean RC, Drnevich VP (1978) Constant rate of strain and controlled gradient consolidation testing. Geotech Test J 1(1):3–15CrossRefGoogle Scholar
- Hamilton JJ, Crawford CB (1959) Improved determination of preconsolidation pressure of a sensitive clay. Special Technical Publication No. 254, American Society for Testing and Materials, Philadelphia, pp 254–270Google Scholar
- Lee K, Chao V, Lee SH, Queek SH (1993) Constant rate of strain consolidation of Singapore marine clay. Geotechnique 43(3):471–488CrossRefGoogle Scholar
- Moozhikkal R, Sridhar G, Robinson RG (2018) Constant rate of strain consolidation test using conventional fixed ring consolidation cell. Indian Geotechn J. https://doi.org/10.1007/s40098-018-0299-1
- Ozer AT, Lawton EC, Bartlett SF (2012) New method to determine proper strain rate for constant rate of strain consolidation tests. Can Geotech J 49:18–26CrossRefGoogle Scholar
- Pu H, Fox PJ (2016) Numerical investigation of strain rate effect for CRS consolidation of normally consolidated soil. Geotech Test J 39(1):80–90. https://doi.org/10.1520/GTJ20150002 CrossRefGoogle Scholar
- Sheahan TC, Watters PJ (1997) Experimental verification of CRS consolidation theory. J Geotech Geoenviron Eng 123(5):430–437. https://doi.org/10.1061/(ASCE)1090-0241(1997)123:5(430) CrossRefGoogle Scholar
- Smith RE, Wahls HE (1969) Consolidation under constant rates of strain. J Soil Mech Found Div ASCE 95(SM2):519–539Google Scholar
- Wissa AEZ, Christian JT, Davis EH, Heiberg S (1971) Consolidation at constant rate of strain. J Soil Mech Found Div ASCE 97(SM10):1393–1413Google Scholar