3. Stress Distribution in Soil 55
3.4 Contact Pressure between the Foundation
(a) (b)
Fig. 3.6. Example 3.2. (a) Geological section and (b)σ,u,σ distribution.
The pore water pressure at any depth will be hydrostatic since the void space between the solid particles is continuous, so at depth C, we have
u=γwh2. (3.13)
Hence, from the Terzaghi’s effective stress principle, the effective vertical stress at depth C will be
σv =σv−u=γh1+ (γsat−γw)h2 =γh1+γh2, (3.14) where γ is the buoyant unit weight of the soil.
Example 3.2. A foundation is composed of multiple layers of soils, which is shown in the geological section, Fig. 3.6(a). Calculate and draw the total normal stress distribution, pore water pressure distribution, and the effective normal stress distribution of the soils along the depth direction.
Solution. Take three points A, B, and C as the reference points to calculate σ,u, andσ, as shown in Table 3.1. The distribution of the total normal stress, pore water pressure, and effective normal stress of the soils along the depth direction are shown in Fig. 3.6(b).
3.4 Contact Pressure between the Foundation and
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Table 3.1. Calculation of Example 3.2.
Reference
points σ(kN/m2) u(kN/m2) σ(kN/m2)
A 3×17 = 51.0 0 51
B (3×17) + (2×20) = 91.0 2×9.8 = 19.6 71.4 C (3×17) + (2×20) + (4×19) = 167.0 6×9.8 = 58.8 108.2
the interaction between a structural foundation and the supporting ground soil media is of primary importance to both structural and geotechnical engineering.
Contact pressure is the intensity of loading transmitted from the underside of a foundation to the ground soil (Whitlow, 2001). The magnitude and the distribution pattern of the contact pressure have an important impact on the stress increment induced in the ground.
The magnitude and the distribution pattern of the contact pressure depend on many factors such as the magnitude and distribution of the structure load applied, the rigidity and embankment depth of the foundation, and the soil properties.
It has been found from tests that for a foundation with a very low rigidity or for a flexible foundation, the magnitude and the distribution pattern of the contact pressure are the same as those of the load applied on the foundation. This is because the foundation is compatible to the deformation of the ground soil. When the load on the foundation is uniformly distributed, the contact pressure (normally denoted as the reaction force on the underside of a foundation, ditto) is also uniformly distributed, as shown in Fig. 3.7(a). When the load distribution is trapezoidal, the contact pressure distribution is also trapezoidal, as shown in Fig. 3.7(b).
For a rigid foundation that cannot be compatible to the ground deformation due to a significant difference in rigidity, the distribution of the contact pressure varies with the magnitude of the applied load, the embedment depth of the foundation, and the properties of the ground soil. For instance, when a centric load is applied to the rigid strip foundation founded on the surface of a sandy ground, the contact pressure at the centerline of the foundation is maximum, the contact pressure at the edge of the foundation is zero, and its
(a) (b)
Fig. 3.7. Contact pressure distribution beneath a flexible foundation.
(a) (b)
Fig. 3.8. Contact pressure distribution beneath a rigid foundation.
distribution looks like a parabolic curve, as shown in Fig. 3.8(a).
This is because no cohesion is available among sand particles. When a centric load is applied to the rigid strip foundation founded on the surface of a clayey ground, some loads can be carried on the edge of the foundation due to cohesion of the clayey soils. Therefore, when the applied load is relatively small, high contact pressure will be imposed on the edge of the foundation and low contact pressure will be imposed on the center of the foundation. The distribution curve is shaped like a saddle. When the load is increased gradually to failure load, the distribution curve of the contact pressure becomes higher at the center and lower at the edge of the foundation and is shaped similar to a bowl, as shown in Fig. 3.8(b).
Empirically, when the width of the rigid foundation is small and the applied load is relatively small, the contact pressure distribution follows approximately a linear distribution assumption. The error
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induced between the assumption and the reality would be acceptable, according to St. Venant’s principle. If forces acting on a small portion of the surface of an elastic body are replaced by another statically equivalent system of forces acting on the same portion of the surface, this redistribution of loading produces substantial changes in the stresses locally but has a negligible effect on the stresses at distances that are large in comparison with the linear dimensions of the surface on which the forces are changed. (Timoshenko and Goodier, 1951).
Introduced below is the simplified calculation method normally used in engineering practice for computing the contact pressure based on the linear distribution assumption.
3.4.2 Contact pressure due to a vertical centric load
The length and width of a rectangular foundation are l and b, respectively, as shown in Figs. 3.9(a) and 3.9(b). A vertical centric load P is applied on the foundation. According to the linear distribution assumption, the value of the contact pressure is
p= F+G
A = F+G
l×b , (3.15)
where lowercaseprepresents the contact pressure (kPa);F represents the vertical load on the upside of the foundation (kN); Grepresents the self-weight of the foundation and the soil weight on the steps of the foundation, and generally, the value 20 kN/m3 is adopted as the average unit weight; and A = l×b represents the area of the foundation (m2). l and b represent the length and width of the foundation.
If the foundation is oblong (theoretically, when l/b approaches infinity, it is called a strip foundation; practically, whenl/bis greater than or equal to 10, it can be considered as a strip foundation), a free body of 1 m unit length can be truncated in the longitudinal direction of the foundation for the calculation, as shown in Fig. 3.9(c). In this scenario, the contact pressure is given as
p= F +G
b , (3.16)
wherebrepresents the width of the foundation (m); the other symbols have the same meaning as presented before.
(a) (b) (c) Fig. 3.9. Contact pressure distribution due to a vertical centric load.
3.4.3 Contact pressure due to a one-way vertical eccentric load
When a one-way eccentric load is applied to a rectangular foundation (as shown in Fig. 3.10), the contact pressure at any arbitrary point can be calculated using the formula of eccentric compression in mechanics of materials, as given by
pmax
min = F +G
A + M
W = F+G A
1±6e
l
, (3.17)
where pmax and pmin represent the maximum and minimum contact pressures on both sides of the underside of the foundation (kPa); M represents the moment of the eccentric load about the Y −Y axis;
W represents the resisting moment of the underside of foundation, and W = bl62 if the area is rectangular (m3); and e is the offsetting of the eccentric load line to the Y–Y axis.
It can be seen from Eq. (3.17) that when the resultant offsetting e is less than l/b, the distribution curve of the contact pressure is trapezoidal. When the resultant offsetting e is equal to l/6, pmin is zero and the distribution curve of the contact pressure is triangular shape. When the resultant offsetting e is greater than l/6, pmin is less than zero and tension force would appear on one side of the underside of the foundation, as shown in Figs. 3.10(a)–
3.10(c). Generally speaking, the tension force on the underside of the foundation is not allowed in the engineering practice; therefore, when designing the size of the foundation, the resultant offsetting should satisfy a criterion of eless than l/6, for the sake of safety. Because
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(a)
(b)
(c)
(d)
Fig. 3.10. Contact pressure distribution due to a one-way vertical eccentric load.
soil cannot bear tension force, the contact pressure is adjusted. The calculation principle is that the composite pressure of foundation base pressure is identical to the total load (as shown in Fig. 3.10(d)), and the formula of the maximum contact pressurepmaxis given by
pmax= 2(F+G)
3ba , (3.18)
wherearepresents the distance between the action point of eccentric load and the edge of the pmax,a= 2l −e(m).
Similarly, for a strip foundation, the maximum and minimum contact pressures of the underside of the foundation are given as
pmax
min = F+G A
1± 6e
b
. (3.19)
3.4.4 Contact pressure due to a two-way vertical eccentric load
When a two-way eccentric load is applied to a rectangular foundation (as shown in Fig. 3.11), the contact pressure at any arbitrary point can be calculated using the formula of eccentric compression in mechanics of materials, as given by
pmax
min = F+G
A ±Mxy
Ix ±Myx
Iy , (3.20)
where Mx= (F +G)ey represents the moment of the eccentric load about the X–X axis (ey is the offsetting of the eccentric load line to the X–X axis); My = (F +G)ex represents the moment of the eccentric load about theY–Y axis (exis the offsetting of the eccentric load line to the Y–Y axis); Ix = bl3/12 represents the moment of inertia of the area of the underside of the foundation about theX–X axis; Iy=lb3/12 represents the moment of inertia of the area of the underside of the foundation about the Y–Y axis.
Fig. 3.11. Contact pressure distribution due to a two-way vertical eccentric load.
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3.4.5 Additional stress on the underside of the foundation Additional stress is defined as the increased pressure in the founda- tion due to building the architecture, as shown in Fig. 3.12.
(1) When the foundation is constructed above the ground surface (Fig. 3.12(a)), the additional stress on the underside of the foundation p0 is the contact pressure of the underside of the foundationp, that is
p0 =p. (3.21)
(2) When the foundation is constructed at some depth under the ground surface (Fig. 3.12(b)), the additional stress on the underside of the foundation p0 is calculated by the following equation:
p0=p−σc =p−γ0d, (3.22) wherepis the contact pressure of the underside of the foundation (kPa), σc is the overburden pressure at the foundation base (kPa),dis the depth from the ground surface to the underside of the foundation (m), andγ0 is the weighted average unit weight of the soil layers above the foundation base (kPa) and is given as follows:
γ0= γihi d .
(a) (b)
Fig. 3.12. Diagram for calculating the additional stress on the underside of the foundation.
The increase in pressure is triggered by constructing the structure after the earth is excavated, so it is the contact pressure of the underside of the foundation after subtracting the original overburden pressure.