Product: ABAQUS/Standard
This one-dimensional problem has a well-known linear solution (see Terzaghi and Peck, 1948) and, thus, provides a simple verification of the consolidation capability in ABAQUS. The analysis of saturated soils requires solution of coupled stress-diffusion equations, and the formulation used in ABAQUS is described in detail in “Analysis of porous media,” Section 2.8 of the ABAQUS Theory Manual, and “Plasticity for non-metals,” Section 4.4 of the ABAQUS Theory Manual. The coupling is approximated by the effective stress principle, which treats the saturated soil as a continuum, assuming that the total stress at each point is the sum of an “effective stress” carried by the soil skeleton and a pore pressure in the fluid permeating the soil. This fluid pore pressure can change with time (if external conditions change, such as the addition of a load to the soil), and the gradient of the pressure through the soil that is not balanced by the weight of fluid between the points in question will cause the fluid to flow: the flow velocity is proportional to the pressure gradient in the fluid according to Darcy's law. A typical case is a consolidation problem. Here the addition of a load (usually an overburden) to a body of soil causes pore pressure to rise initially; then, as the soil skeleton takes up the extra stress, the pore pressures decay as the soil consolidates. The Terzaghi problem is the simplest example of such a process. For illustration purposes, the problem is treated with and without finite-strain effects. The small-strain version is the classical case discussed by Terzaghi and Peck (1948), and the finite-strain version has been analyzed numerically by a number of authors, including Carter et al. (1979).
The problem is shown in Figure 1.14.1–1. A body of soil 2.54 m (100 in) high is confined by impermeable, smooth, rigid walls on all but the top surface. On that surface perfect drainage is possible, and a load is applied suddenly. Gravity is neglected. Because of the boundary conditions, the problem is one-dimensional, the only gradient being in the vertical direction. The purpose of the analysis is to predict the evolution of displacement, effective stress, and pore pressure throughout the soil mass as a function of time following the load application.
ABAQUS contains no one-dimensional elements for effective stress calculations. Therefore, we use a two-dimensional plane strain mesh, with one element only in the 
-direction. Element type CPE4P is chosen to perform the finite-strain analysis, and element type CPE8P is chosen for the small-strain analysis. We recommend the use of linear elements for applications involving finite strain, impact, or complex contact conditions and second-order elements for problems where stress concentrations must be captured accurately or where geometric features such as curved surfaces must be modeled. In this particular example the linear and second-order elements yield almost identical results.
The soil is assumed to be linear elastic, with a Young's modulus of 689.5 GPa (108 lb/in2) and Poisson's ratio of 0.3. The specific weight of the pore fluid is assumed to be 276.8 × 103 N/m3 (1 lb/in3). The permeability is assumed to vary linearly with the void ratio, with a value of 8.47 × 10–8 m/sec (2.0 × 10–4 in/min) at a void ratio of 1.5 and a value of 8.47 × 10–9 m/sec (2.0 × 10–5 in/min) at a void ratio of 1.0. The void ratio is assumed to be 1.5 initially throughout the sample. ABAQUS uses effective permeability, which is permeability divided by the specific weight of the pore fluid. Therefore, the fluid in this problem is assigned the value 276.8 × 103 N/m3 (1 lb/in3) for the specific weight (water, for example, has a specific weight of 9965 N/m3, 0.036 lb/in3) and the permeability is scaled accordingly.
The boundary conditions are as follows. On the bottom and two vertical sides, the normal component of displacement is fixed (
0 on the bottom and 
0 on the sides), and no flow of pore fluid through the walls is permitted. This latter is the natural boundary condition in the fluid mass conservation equation, so no explicit specifications need to be made (as with zero tractions in the equilibrium equation). On the top surface a uniform downward load (an overburden) is applied suddenly. The magnitude of this load is taken to be 689.5 GPa (108 lb/in2). This large load will cause considerable deformation, thus illustrating the difference between the small- and large-strain solutions. This surface allows perfect drainage so that the excess pore pressure is always zero on this surface.
The problem is run in two steps. The first step is a single increment of a *SOILS, CONSOLIDATION analysis with an arbitrary time step, with no drainage allowed across the top surface (the natural boundary condition in the mass conservation equation governing the pore fluid flow). This establishes the initial solution: uniform pore pressure equal to the load throughout the body, with no stress carried by the soil skeleton (zero effective stress). The actual consolidation is then done with a second *SOILS, CONSOLIDATION step, using automatic time stepping.
The accuracy of the time integration for the second *SOILS, CONSOLIDATION procedure, during which drainage is occurring, is controlled by the UTOL parameter. This parameter specifies the allowable pore pressure change per time step. Even in a linear problem UTOL controls the accuracy of the solution, because the time integration operator is not exact (the backward difference rule is used). In this case UTOL is chosen as 344.8 GPa (5.0 × 107 lb/in2), which is a relatively large value and, so, should only give moderate accuracy: this is considered to be adequate for the purposes of the example.
An important issue in such consolidation problems is the choice of initial time step. As the governing equations are parabolic, the initial solution (immediately after the sudden change in load) is a local, “skin effect,” solution. In this one-dimensional case the form of the initial solution is sketched in Figure 1.14.1–2 for illustration purposes. With a finite element mesh of reasonable size for modeling the solution at a later time (when the changes in pore pressure have diffused into the bulk of the body soil), this initial solution will be modeled poorly. With smaller initial time steps the difficulty becomes more pronounced, as sketched in Figure 1.14.1–2. As in any transient problem, the spatial element size and the time step are related to the extent that time steps smaller than a certain size give no useful information. This coupling of the spatial and temporal approximations is always most obvious at the start of diffusion problems, immediately after prescribed changes in the boundary values. For this particular case the issue has been discussed in detail by Vermeer and Verruijt (1981), who suggest the simple criterion
is a characteristic element size near the disturbance (that is, near the draining surface in our case), 
is the elastic modulus of the soil skeleton, 
is the soil permeability, and 
is the specific weight of the permeating fluid. For our model we choose 
254 mm (10 in); and we have 
689.5 GPa (108 lb/in2), 
8.47 × 10–8 m/s (2.0 × 10–4 in/min), 
2.768 × 105 N/m3 (1.0 lb/in3), which gives 
.05 s (0.833 × 10–3 min). Based on this calculation, an initial time step of .06 sec (0.001 min) is used. This gives an initial solution with no “overshoot” at all, as expected.In this case we wish to continue the analysis to steady-state conditions. This is defined by asking ABAQUS to stop when all pore pressure change rates fall below 11.5 KN/m2/s (100 lb/in2/min).
In the small-strain analysis the “steady-state” condition (rate of change of pore pressure with time below the prescribed value) is reached after 20 increments, the last time increment taken being 491 seconds (8.19 min)—about 8000 times the initial time increment. This very large change in time increment size is typical of such diffusion systems and points out the value of using automatic time stepping with an unconditionally stable integration operator for such problems.
The results of the small-strain analysis are summarized in Figure 1.14.1–3 to Figure 1.14.1–5. Figure 1.14.1–3 shows pore pressure profiles (pore pressure as a function of elevation) at various times in the solution. As we would expect, the solution begins by rapid drainage at the top of the sample and loss of pore pressure in that region. This effect propagates down the sample until the entire sample is steadily losing pore pressure throughout its length. At steady state the solution has zero pore pressure everywhere, with the load being carried as a uniform effective vertical stress. Figure 1.14.1–4 shows this transfer of load from the fluid to the skeleton at the 1.905 m (75 in) elevation as a function of time. Figure 1.14.1–5 compares these numerical results with the solution quoted in Terzaghi and Peck (1948). Here the downward displacement of the top surface of the soil, as a fraction of its steady-state value (the “degree of consolidation”), is plotted as a function of normalized time, defined as
is the permeability of the soil, is the Young's modulus of the soil, is the specific weight of the pore fluid, 
is the height of the soil sample, and 
is time.Figure 1.14.1–5 shows that the numerical solution agrees reasonably well with the analytical solution, with some loss of accuracy at later times. This latter effect is attributable to the coarse time stepping tolerance chosen. Higher accuracy could be obtained with a tighter tolerance on the allowable pore pressure stress change parameter (UTOL). However, the solution is clearly adequate for design use.
In the finite-strain analysis of soils, changes in the void ratio can lead to large changes in permeability, therefore affecting the transient response in a consolidation analysis. Typical soils show a strong dependence of permeability on the void ratio (as soil compacts, it becomes increasingly harder for fluid to pass through it), with the consequence that “plugging” may result. This means that a soil that was relatively permeable in its original state becomes less permeable as it consolidates.
In this example the permeability of the soil is assumed to decrease by an order of magnitude as the void ratio decreases from its initial value of 1.5 to a value of 1.0. Such logarithmic dependence of permeability on the void ratio is not uncommon in fully saturated clays. Two finite-strain analyses are run, one with permeability treated as a constant and a second with this variation in permeability. The results are shown in Figure 1.14.1–6, together with the results of the small-strain analysis under similar load. The “plugging” effect of void ratio dependence of permeability is clearly seen in this figure. Since the permeability decreases with the consolidation of the soil, the time required for all excess pore pressure to dissipate increases. The final value of displacement under the applied load is not a function of permeability and is correctly predicted by both large-strain analyses. (The exact solution for this displacement is very easily calculated.) It is interesting to observe that, if the permeability is not dependent on the void ratio, the finite-strain results show more rapid initial consolidation than the corresponding small-strain analysis.
A separate suite of files (terzaghi_cpe8p_rigid.inp, terzaghi_cpe4p_rigid.inp, and terzaghi_cpe8p_ss_rigid.inp) is provided to illustrate the use of the *CONTACT PAIR option in problems involving pore pressure elements. Three rigid surfaces are used to model the three impermeable sides of the specimen shown in Figure 1.14.1–1, thus replacing the boundary conditions used in terzaghi_cpe8p.inp, terzaghi_cpe4p.inp, and terzaghi_cpe8p_ss.inp.
Small-strain analysis (element type CPE8P).
Finite-strain case with permeability depending on the void ratio (element type CPE4P).
Small-strain steady-state solution (element type CPE8P).
Small-strain case with velocity-dependent permeability (Forchheimer flow) and velocity coefficient depending on the void ratio.
*POST OUTPUT postprocessing of terzaghi_cpe8p.inp.
*POST OUTPUT postprocessing of terzaghi_cpe8p_perm.inp.
Identical to terzaghi_cpe8p.inp except that rigid surfaces are used to impose the boundary conditions.
Identical to terzaghi_cpe4p.inp except that rigid surfaces are used to impose the boundary conditions.
Identical to terzaghi_cpe8p_ss.inp except that rigid surfaces are used to impose the boundary conditions.
Carter, J. P., J. R. Booker, and J. C. Small, “The Analysis of Finite Elasto-Plastic Consolidation,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 3, pp. 107–129, 1979.
Terzaghi, K., and R. B. Peck, Soil Mechanics in Engineering Practice, John Wiley and Sons, New York, 2nd, 1948.
Vermeer, P. A., and A. Verruijt, “An Accuracy Condition for Consolidation by Finite Elements,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 5, pp. 1–14, 1981.
