CLOSED-FORM SETTLEMENT SOLUTION FOR NORMALLY CONSOLIDATED SOILS

Transcription

1 LOSED-FORM SETTLEMENT SOLUTION FOR NORMALLY ONSOLIDATED SOILS Assume a homogeneous, comressible soil layer of thickness H that will consolidate under the weight of a surface load of infinite lateral extent. If we subdiide the comressible layer into an infinite number of sublayers of height d, the total settlement is gien by the integral c This will be easier to integrate if we rewrite it as x, then H d to c c H d d to If we let dx d d since the in situ stress increases with deth in direct roortion to the buoyant unit weight of the soil (). Substituting x and d dx, we can rewrite the settlement equation as to c c H x dx x dx to to Now x dx x x x so H c to to to c to to to Since, eerywhere, this reduces to c H to to to to which is the final solution for the consolidation settlement. The equation aboe can be used as long as the unit weight of the soil is constant throughout the comressible layer and the initial ( ) and final ( ) stresses h increase with deth at the rate d/d =. In ractice, this means that the surface load has a large lateral extent (more like an engineered fill than a footing).

2 LOSED-FORM SETTLEMENT SOLUTION FOR OVERONSOLIDATED SOILS The settlement equation for an oerconsolidated soil is as follows: s H H r o c o If we subdiide the comressible layer into sublayers of height H i then the settlement in each sublayer is i si rhi chi,, i, i, i In the limit, if we subdiide the comressible layer into an infinite number of sublayers of height d, the settlement is gien by the integral H r dc d to This will be easier to integrate if we rewrite it as H d d d d r r c c to to to to This can be consolidated slightly to r c r c H d d d to to to to If we assume that the oerconsolidation margin is constant with deth within the layer and the difference between and is constant with deth (which assumes a surface load of infinite lateral extent), then all three stresses increase with deth in direct roortion to the unit weight of the soil. This lets us write x dx d d Substituting x and d dx, we can write the settlement equation as r c r c H x dx x dx x dx to to to

3 Now x dx x x x so the settlement equation can be written as r c H r c to to to to to to to to to If we isolate only the terms that don t inle a arithm: Rearranging: r c r c H to to to r H to to c to to Since we e assumed that the oerconsolidation margin is constant with deth within the layer and the difference between and is constant with deth, then all three stresses increase with deth by the same amount,, so to to to which means that all of those terms that don t inle a arithm cancel each other out, leaing r c H r c to to to to to to as the closed-form equation for settlement of oerconsolidated soils with a constant OM within each layer and a constant added at all deths due to a broad surface load. What haens if the reconsolidation stress increases with deth at a different rate than the in situ stress? In other words, what haens if the OM is not constant within the soil layer? Let s back u to the equation r c r c H d d d to to to

4 If changes with deth as s o and changes with deth as s f and changes with deth as then we can make the following substitutions: s xo and d dx o s o x and d dx f s f Now the settlement equation can be written as f x and d dx s r c r c o o f f s to s to to o s f H x dx x dx x dx As before, x dx x x x so the settlement equation can be written as r c H s r so c s f to to to to to to to to to This is as far as we can simlify this equation because we cannot write to to to as we did before to get half of the terms to cancel out. So the calculations become a bit more comlicated, but still doable in closed form as long as the stresses change linearly with deth.

5 A NOTE FOR SOIL LAYERS AT THE SURFAE to If the soil layer being consolidated starts at the ground surface, 0, which oses a roblem because (0) is undefined. If you lot x (x) as x aroaches ero, you find that the equation does not aroach ero monotonically. Instead, it crosses the x-axis at x = 1, then dis below the axis before it asymtotically aroaches ero again. to to to Therefore, assuming 1 roides the correct alue of while all other small alues only aroximate the correct answer.