Spring and Piston Model of Consolidation.
model is a pressure gage that can measure the pressure of the water at any time in the cylinder.
If an external load P is applied to the piston with the valve closed, one would discover (just as in a laboratory consolidation test on a saturated soil sample) that immediately after loading, no settlement results. Instead, the force applied to the piston P must be balanced by the force supporting the piston. The force supporting the piston is the combination of the force in the spring Ps and the force in the water Pw. If the total area inside the cylinder is A, the area of the wire in the spring is As, and the area of the piston covered by water is Aw, then equilibrium of forces requires that
Figure 3.8 Piston-spring model of consolidating soil.
where S is the compression of the spring (equal to the settlement of the piston).
The bulk modulus of the water is defined as the change in volume of the water that results from the water pressure u resulting from the application of the external force P
The bulk modulus of the water is defined as the change in volume of the water that results from the water pressure u resulting from the application of the external force P
where AVw is the change in volume of water, Vw is the original volume of water, and u is the pressure developed in the water. If the area of the cylinder is uniform, then the change in the volume of water is equal to the settlement times the area of the water:
An expression for the water pressure due to the change in volume of the water under the applied external force is obtained by inserting Eq. 3.22 into Eq. 3.21 and rearranging:
The force carried by the water is equal to the water pressure acting over the area of the piston:
Equilibrium of forces requires the externally applied force to equal the sum of forces carried in the spring and water. This is expressed as
Stresses are found by dividing the forces by the area of the cylinder. Thus, total stress applied to the cylinder is
The stress carried by the spring is
The porewater pressure is
because As is much less than Aw. In this model, As is much less than Aw by choice, but studies of real soils have found that this is an excellent approximation. Thus, after dividing both sides by the area of the piston, Eq. 3.25b reduces to
Usually this equation is written as
Equation 3.30 is called Terzaghi’s effective stress equation, and is called the effective stress. Other notations for effective stress in common use include the symbols and This text will use to denote effective stress in most p. situations and use the prime ( ) notation applied to other symbols to indicate
conditions under which stresses are effective stresses.
conditions under which stresses are effective stresses.
When the spring and piston model is applied to real soils, the spring represents the compressible framework of soil particles and Ps is the force carried by this framework. The effective stress is then the ratio of the force carried in the particle framework to the total area of the soil.
The settlement of the piston S appears in both Eq. 3.20 and Eq. 3.23. Solving both equations for S, one obtains
The settlement of the piston S appears in both Eq. 3.20 and Eq. 3.23. Solving both equations for S, one obtains
Again, dividing both sides by the piston area A and multiplying both sides by the spring constant k, one obtains an expression for effective stress:
If one inserts values of k obtained by testing real soils and the appropriate values for the bulk modulus of water Cw, the volume of water Vw, the piston area A, and the water area Aw, one finds that
In other words, when the piston is loaded and the valve is closed, approxi- mately 99.9% of the load goes into the water and none into the spring.
Laboratory measurement of the consolidation properties of saturated soils has found that if a pressure is applied to the soil and drainage is not allowed, then the resulting porewater pressure u equals the applied pressure . Insertion of typical values into Eq. 3.31 demonstrates why the settlement is essentially
zero when a saturated soil is loaded without drainage.
Laboratory measurement of the consolidation properties of saturated soils has found that if a pressure is applied to the soil and drainage is not allowed, then the resulting porewater pressure u equals the applied pressure . Insertion of typical values into Eq. 3.31 demonstrates why the settlement is essentially
zero when a saturated soil is loaded without drainage.
If the valve is opened, the water in the cylinder escapes, settlement occurs, and the spring compresses under the external force. Settlement causes the spring (representing the soil particle framework) to compress and to take on an increasing fraction of the externally applied load. Eventually, enough water will escape so that the spring carries the full externally applied load and the porewater pressure is zero. The spring and piston model of consolidating soil is then in equilibrium. One can now calculate the spring constant of the model by dividing P by the measured settlement S.
With the known value of the spring constant, piston settlements can be computed for any given applied load. In addition, measurement of the time rate of settlement in the laboratory and computing the soil property that characterizes the time rate of settlement (represented by the opening in the valve in the model) would allow the computation of the time rate of settlement in the field using an appropriate theory.
With the known value of the spring constant, piston settlements can be computed for any given applied load. In addition, measurement of the time rate of settlement in the laboratory and computing the soil property that characterizes the time rate of settlement (represented by the opening in the valve in the model) would allow the computation of the time rate of settlement in the field using an appropriate theory.
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Soil
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