1. 第三章 土体应力 Chapter 3 Stresses in Soil
第一节 简介 3.1 Introduction
第二节 地基中的上覆有效应力
3.2 Effective overburden pressure in the ground
Effective overburden pressure represents the at-rest in situ stress state due to the effective self-weight of the ground soils.
Ground is assumed to be a semi-infinite，homogeneous，
linear，isotropic and elastic material.

2. sz (1) Effective vertical stress (竖向有效应力 )
(1-a) No groundwater table (不存在地下水 ) 地面 z sz
where
 (kN/m3) is unit weight of the soil (土的重度)
z (m) is depth of the soil (深度) M

3. (1-b) Groundwater table at the ground surface (地下水位在地面)z sz M
where
’ (kN/m3) is submerged unit weight of the soil (土的浮重度)
w (9.81 kN/m3) is unit weight of water (水的重度)

4. sz sx z 地面 Water table at ground surface No water table
where  is unit weight of the soil (土的重度) w is unit weight of water (水的重度) and Ko is coefficient of earth pressure at rest (静止侧压力系数)

5. If groundwater table locates at z1 from the ground surface, what is sz at M?地面 z1 g
地下水位 sz z2 g’ M

6. sz (1-c) n Layers of stratum 成层土 g1 z1 g2 z2 where地面 g1 z1 g2 z2
where
i is unit weight of the soil of ith stratum (ith土层的重度)
zi is depth of the soil of ith stratum
(ith土层的深度) gn sz zn M

7. sz sx (2) Effective horizontal stress 侧向有效应力 z where地面 z sz sx
where
Ko is coefficient of earth pressure at rest (静止侧压力系数) M

8. Typical values of Ko (静止侧压力系数)
Soil type Ko
Loose sands (松砂)
0.40 – 0.45
Dense sands (密砂)
0.45 – 0.50
Well-compacted fills (压实填土)
0.8 – 1.5
Normally consolidated clays (正常固结粘土)
0.5 – 0.6
Overconsolidated clays (超固结粘土)
1.0 – 4.0

9. 第三节 基底压力 3.2 Contact pressure between Foundation and found
(1) Flexible Foundation 柔性基础
Low flexural rigidity (小刚度)
e.g. oil tank (油罐) and earth-fill dam (土坝) 荷载 反力
变形地面
Pressure distribution at the bottom of a flexible foundation (柔性基础基底压力分布)

10. Pressure distribution at the bottom of a rigid foundation (刚性基础基底压力分布)
High flexural rigidity (大刚度)
e.g. box foundation (箱形基础) and concrete dam (混凝土坝) 荷载
变形地面 反力
Pressure distribution at the bottom of a rigid foundation (刚性基础基底压力分布)

11. Assume linear pressure distribution (假定线性压力分布)
(3) Contact pressure due to vertical centric load
中心荷载下的基底压力 Pv
where
L is length of the foundation (基础长度)
B is breadth of the foundation (基础宽度)
Pv is the sum of applied vertical load (竖向荷载) and weight of the foundation (基础和回填土的总重) p L

12. (4) Contact pressure due to vertical eccentric load 偏心竖向荷载
中心线 Pv e 基础 基底
where L is length of the foundation, B is breadth of the foundation and e is eccentricity of the total vertical load
pmin
pmax L

13. If e > L/6, pmin < 0, i.e. Tension
中心线 Fv e
pmin
pmax
最小压力
最大压力 L

14. As the foundation cannot resist tension, pressure re-distribute
If e > L/6
As the foundation cannot resist tension, pressure re-distribute
因为基础不能抵抗拉力, 基底压力重新分布
中心线 Fv e
pmax Lp

15. (5) Net stress 基底净压力 g z0 p A A At A-A Initial Stress Final Stress
A-A is bottom of foundation (基底) g z0 p A A
At A-A Initial Stress
Final Stress
Net stress increase

16. 第四节 地基中的附加应力 3.4 Stress Increases in the Ground基础 基底
Stress increases (附加应力)
How does stress increase change with depth (附加应力与深度关系)? We start with the Three-dimensional problems (三维问题).

17. 1、Stress increase in spatial problems 空间问题的附加应力
Consider a homogeneous (均质), elastic (弹性) and isotropic (各向同性) semi-infinite half space (半无限空间体) P x r y q
dsz x R
dtzx z
dtzy
dtxz y
dtyz
dsx
dtxy M
dtyx
dsy z

18. (1) Stress increases due to vertical point load
竖向集中力作用下附加应力
Boussinesq’s solution (布辛内斯克解答) for normal stress increases at point A due to the point load F are
where  is Poisson’s ratio (泊松比)

19. Boussinesq’s solution for shear stress increases at point A due to the point load F are
where  is Poisson’s ratio (泊松比)

20. Boussinesq’s solution for stress increases in vertical direction can be re-arranged as follows:
where

21. (1-a) Example
A square foundation 2 m x 2 m carries a point load of 1500 kN at its centre. Determine the vertical stress at a point 5 m below the centre of the foundation.
P = 1500 kN, z = 5 m & r = 0 m
1500kN
5 m

22. (2) Stress increases due to several vertical point loadsFi
Fi+1 x
Fi-1 ri
ri+1
ri-1 z y M z

23. By the principle of superposition (叠加原理), stress increases in vertical direction (z) at point M due to several point loads Fi ( i = 1,2,…n) can be presented by the following expression:

24. (3) Vertical stress increases under the corners of a rectangular foundation underside due to vertical uniform load 竖直矩形均布荷载 角点下 x p B L z
Integration method y M z

25. Ks show in Page103 table 3-1
where m = L/B and n = z/B, L is length of the rectangle, B is breadth of the rectangle, z is depth of point M from the bottom of the foundation and p is net pressure on the foundation.

26. Complex corner point method
In the side
In the plane
out the plane
B is always the shorter line, L is always the longer line

27. (3-a) Example
A rectangular foundation 6 m x 3 m carries a uniform pressure of 300 kN/m2 near the surface of a soil mass. Determine the vertical stress at a depth of 3 m below corner A of the foundation. 6m 3m A
p = 300 kN/m2
m = L/B = 6/3 = 2
n = z/B = 3/3 = 1
土力学 表 2-2 (p.52)
Ks =  0.2

28. (4) Vertical stress increases under the corners of a rectangular foundation underside due to vertical triangular load 竖直三角形荷载角点下
m=L/B，n=z/B，
B is the variational line , L is the unchangeable line
Kt show in page 112 table 3-4

30. (5) Vertical stress increases under the center of a vertical and uniformly loaded circular foundation
竖直圆形均布荷载中心点下 ro p O z M z

31. where r0 is the radius of the circular foundation, z is depth of point M from the center of the foundation and p is net pressure on the foundation.

32. 2、Stress increase in Two-dimensional plane problem 两维平面问题x z R1 x z M

33. (1) Vertical stress increases due to a line load 线荷载dy x p y x R z R1 y M z

34. It is a plane strain problem (平面问题), i.e. y = 0
z
xz
y = 0
x M

35. where

36. (2) Vertical stress increases beneath a vertical and uniformly loaded strip foundation 条形荷载x B z R1 x z M

37. where m = x/B and n = z/B, B is breadth of the strip foundation and p is net pressure on the foundation. Values of Kzs can be found in table 3-6 (p.121)