兾 )
3.5 Settlement of Shallow Foundations in Cohesive Soils
When foundations are subjected to vertical loads, there will be settlement. Depending on whether the underlying soils are cohesive or granular, the settlement pattern can be quite
e Q
qmin
B
(a)
x y
B L
(b) B/6 L/6
load eB
eL qmax
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different. In saturated cohesive soils, the settlements consist of three components: immediate settlement (si), consolidation settlement (sc), and secondary compression (ss). Immediate settle- ment occurs immediately after the load is applied and is instantaneous. Consolidation settle- ment occurs due to the expulsion of water from the soil and dissipation of excess pore water pressure. This can take place over a period of several years. Secondary compression settlement, also known as creep, occurs after the consolidation is completed. Therefore, there will be no excess pore water pressure during the secondary compression stage.
3.5.1 Immediate Settlement
Immediate settlement, also known as distortion settlement, initial settlement, or elastic settlement, occurs immediately upon the application of the load, due to lateral distortion of the soil beneath the footing. In clays, where drainage is poor, it is reasonable to assume that immediate settlement takes place under undrained conditions where there is no volume change (i.e., v = 0.5). The average immediate settlement under a flexible footing generally is estimated using the theory of elasticity, using the following equation, originally proposed by Janbu et al. (1956):
s qB
i E
u
= µ µ0 1 (3.73)
The values of µ1 and µ2, originally suggested by Janbu et al. (1956), were modified later by Christian and Carrier (1978), based on the work by Burland (1970) and Giroud (1972). The values of µ0 and µ1, assuming ν = 0.5, are given in Figure 3.12. Obtaining a reliable estimate of the undrained Young’s modulus (Eu) of clays through
laboratory or in situ tests is quite difficult. It can be esti- mated using Figure 3.13, proposed by Duncan and Buchignani (1976) and the U.S. Army (1994). Eu兾cu can vary from 100 for very soft clays to 1500 for very stiff clays.
Typical values of the elastic modulus for different types of clays are given in Table 3.4. Immediate settlement gener- ally is a small fraction of the total settlement, and there- fore a rough estimate often is adequate.
3.5.2 Consolidation Settlement
Consolidation is a time-dependent process in saturated clays, where the foundation load is gradually transferred from the pore water to the soil skeleton. Immediately after loading, the entire applied normal stress is carried by the water in the voids, in the form of excess pore water pressure. With time, the pore water drains out into the more porous granular soils at the boundaries, thus dissipating the excess pore water pressure and increasing the effective stresses.
Depending on the thickness of the clay layer, and its consolidation characteristics, this process can take from a few days to several years.
Consolidation settlement generally is computed assuming one-dimensional consolidation, and then a correction factor is applied for three-dimensional effects (Skempton and Bjerrum 1957). In one-dimensional consolidation, the normal strains and drainage are assumed to take place only in the vertical direction. This situation arises when the applied pressure at the ground level is uniform and is of a very large lateral extent, as shown in Figure 3.14.
TABLE 3.4 Typical Values of Elastic Modulus for Clays
Clay E (MPa)
Very soft clay 0.5–5
Soft clay 5–20
Medium clay 20–50
Stiff clay, silty clay 50–100
Sandy clay 25–200
Clay shale 100–200
After U.S. Army (1994).
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FIGURE 3.12 Values of µ0 (top) and µ1 (bottom) for immediate settlement computation (after Christian and Carrier 1978).
In a clay layer with an initial thickness of H and a void ratio of e0, the final consolidation settlement sc due to the applied pressure q can be estimated from
s e
e H
c =
+
∆
1 0 (3.74)
where ∆e is the change in the void ratio due to the applied pressure q. H and e0 can be obtained from the soil data, and ∆e has to be computed as follows.
0.8 0.85 0.9 0.95 1
0 2 4 6 8 10 12 14 16 18 20 µ0
Df/ B
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.1 1 10 100 1000
µ1
H/B
Circle Square (B /L = 1)
B /L = 0.5 B /L = 0.2 B /L = 0.1 Strip (B / L = 0)
For H/B = ∞ & B /L = 0, µ1 = 3 Df
B × L H q
stiff stratum GL
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Three different cases, as shown in Figure 3.15, are discussed here. Point I corresponds to the initial state of the clay, where the void ratio and the vertical stress are e0 and σ′vo, respectively. With the vertical stress increase of ∆σv, consolidation takes place, and the void ratio decreases by ∆e. Point F corresponds to the final state, at the end of consolidation. Point P corresponds to the preconsolidation pressure (σ′p) on the virgin consolidation line.
Case I. If the clay is normally consolidated, ∆e can be computed from:
∆ ∆
e Cc vo v
vo
= ′ +
′
log σ σ
σ (3.75)
FIGURE 3.14 One-dimensional consolidation settlement within a clay layer.
FIGURE 3.13 Eu兾cu values (after Duncan and Buchignani 1976;
U.S. Army 1994).
Clay H
q
Consolidation settlement
sc
Time t1
s(t1)
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Case II. If the clay is overconsolidated and σ′vo + ∆σv ≤ σ′p (i.e., the clay remains overconsolidated at the end of consolidation), ∆e can be computed from:
∆ ∆
e Cr vo v
vo
= ′ +
′
log σ σ
σ (3.76)
Case III. If the clay is overconsolidated and σ′vo + ∆σv ≥ σ′p (i.e., the clay becomes normally consolidated at the end of consolidation), ∆e can be computed from:
∆ ∆
e Cr p C
vo
c vo v
p
= ′
′
+ ′ +
′
log σ log
σ
σ σ
σ (3.77)
In one-dimensional consolidation, assuming the pressure at the ground level is applied over a large lateral extent, ∆σv= q at any depth. In the case of footings where the loading is not one-dimensional, ∆σv can be significantly less than the footing pressure q and can be estimated using the methods discussed in Section 3.2.
Another but less desirable method to compute the consolidation settlement is to use the coefficient of volume compressibility (mv). The final consolidation settlement can be written as:
sc = m qHv (3.78)
The main problem with this apparently simple method is that mv is stress dependent, and therefore a value appropriate to the stress level must be used. The consolidation settlement s (t1) at a specific time t1 can be determined from the Uavg-T plot in Figure 1.17.
3.5.3 Secondary Compression Settlement
Secondary compression settlement takes place at constant effective stress, when there is no more dissipation of excess pore water pressure. For simplicity, it is assumed to start occurring FIGURE 3.15 ∆e calculations from e vs. log σ′v plot.
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when the primary consolidation is completed at time tp (see Figure 3.16), and the settlement increases linearly with the logarithm of time. Secondary compression settlement can be estimated using the following equation:
s C H
e
t
t t t
s a
p p
p
for
= +
>
1 log (3.79)
Here, ep is the void ratio at the end of primary consolidation and Cα is the coefficient of secondary compression or the secondary compression index, which can be determined from a consolidation test or estimated empirically. Assuming that the void ratio decreases linearly with the logarithm of time, Cα is defined as:
C e
α = ∆ t
∆log (3.80)
Mesri and Godlewski (1977) reported that Cα兾Cc is a constant for a specific soil and suggested typical values. In the absence of consolidation test data, Cα can be assumed to be 0.03–0.08 times Cc. While the upper end of the range applies to organic and highly plastic clays, the lower end of the range is suitable for inorganic clays. Secondary compression settlement can be quite significant in organic clays, especially in peat.