[静止土压力].doc

Chapter 6 Lateral Earth Pressure (Part 1)

6.1 Earth Pressure at rest [静止土压力] (Ko)

(i) Consider the wall shown in Figure 6.1. Assume there is no ground water table. If no movement of the wall takes place, the soil is at rest.
(ii) The lateral earth pressure at-rest [静止土压力] (eo) acting on a soil element located at a depth of z is:
(6.1)
where Ko is coefficient of earth pressure at-rest [静止土压力系数] and g is unit weight of soil. Ko is the ratio of horizontal and vertical effective stresses when a retaining wall does not move at all or “at rest”.
(iii) Jaky (1944) proposed the following empirical expression to estimate Ko for normally consolidated soil [正常固结土]:
(6.2)
where f’ is effective angle of internal friction [有效內摩擦角].
(iv) The lateral earth pressure varies linearly with depth. The total lateral force per unit length (Eo) imposed on a retaining wall is:
(6.3)
where H is height of the wall and Eo is acting at H/3 from the bottom of the wall.

6.2 Rankine’s Earth Pressure Theory [朗肯土压力理论]

The following assumptions are made to develop the Rankine’s earth pressure theory:

(i) The wall is rigid, vertical, frictionless and extends to an infinite depth
(ii) The backfill is a dry, homogeneous, isotropic and cohesionless soil mass
(iii) The soil surface is horizontal and there are no shear stresses on horizontal and vertical planes, i.e. the horizontal and vertical stresses are principal stresses.

6.2.1
Active Earth Pressure [主动土压力]

(i) If the wall is moved away from the backfill (AB to A’B in Figure 6.2), the effective horizontal stress will reduce but the effective vertical stress will remain constant.
(ii) Therefore, the Mohr’s circle will expand as shown in Figure 6.3 until touching the Mohr-Coulomb failure surface and the soil elements will undergo active failure [主动破坏].
(iii) The slip planes for active failure are inclined at 45°+f’/2 with respect to the horizontal.
(iii) The active earth pressure [主动土压力] (ea) acting on a soil element located at a depth of z is:
(6.4)
where Ka is coefficient of active earth pressure [主动土压力系数] and g is unit weight of soil. Ka is the ratio of horizontal and vertical effective stresses when a retaining wall moves away (by a small amount) from the retained soil and is defined by the following expression:
(6.5)
(v) The active earth pressure varies linearly with depth (see Figure 6.4). The total lateral force per unit length (Ea) imposed on a retaining wall is:
(6.6)
where H is height of the wall and Ea is acting at H/3 from the bottom of the wall.

6.2.2 Passive Earth Pressure [被动土压力]

(i) If the wall is moved towards the backfill (AB to A’B in Figure 6.5), the effective horizontal stress will increase but the effective vertical stress will remain constant.
(ii) Thus, the Mohr’s circle will first contract and then expand until touching the failure surface as shown in Figure 6.6 and the soil elements will undergo passive failure.
(iii) The slip planes for passive failure are inclined at 45°-f’/2 with respect to the horizontal.
(iv) The passive earth pressure [被动土压力] (ep) acting on a soil element located at a depth of z is:
(6.7)
where Kp is coefficient of passive earth pressure [被动土压力系数] and g is unit weight of soil. Kp is the ratio of horizontal and vertical effective stresses when a retaining wall moves against a soil mass and is defined by the following expression:
(6.8)
(v) The passive earth pressure varies linearly with depth (see Figure 6.7). The total lateral force per unit length (Ep) imposed on a retaining wall is:
(6.9)
where H is height of the wall and Ep is acting at H/3 from the bottom of the wall.

6.2.3 Some key points for Rankine’s Earth Pressure Theory

(i) The lateral earth pressures on retaining walls are related directly to the vertical effective stress through the active (Ka), at-rest (Ko) and passive (Kp) earth pressure coefficients.
(ii) Ka < Ko < Kp and Ka = 1 / Kp
(iii) Only a small movement of the wall away from the soil is required to mobilise full active earth pressure in the soil mass (see Figure 6.8).
(iv) Substantial wall movement towards the soil is required to mobilise full passive earth pressure in the soil mass.
(v) Rankine’s earth pressure coefficients are only applicable to a frictionless, rigid, vertical wall retaining homogeneous soil with a horizontal ground surface.

6.2.4 Effect of ground water table on earth pressure

(i) If groundwater is present, the pore water pressure must be added to the lateral earth pressure.
(ii) DO NOT multiply the pore water pressure with a coefficient of earth pressure.
(iii) If the groundwater table is located at a height H2 from the bottom of the wall (see Figure 6.9), the hydrostatic pore water pressure is gwH2 and the total imposed lateral force due to groundwater is ½gwHw2
(iv) Below ground water table, the submerged unit weight (g’) is used to evaluate the earth pressure.

6.2.5 Backfill is cohesive soil

(i) If the backfill of the retaining wall is a cohesive soil, the cohesion (c) is not zero.
(ii) At a depth z from the top of the wall (see Figure 6.10), the active (ea) and passive (ep) earth pressure are as follows:
(6.10)

(6.11)
(6.12)

Key words
Adhesion 粘著力

Backfill 回填土

Coefficient of earth pressure 土压力系数
Coefficient of active earth pressure 主动土压力系数
Coefficient of earth pressure at-rest 静止土压力系数
Coefficient of passive earth pressure 被动土压力系数
Cohesion 粘聚力
Cohesionless soils 无粘性土
Cohesive soils 粘性土
Culmann Construction (库尔曼图解法)
Earth pressure 土压力
Active earth pressure 主动土压力
Earth pressure at-rest 静止土压力
Passive earth pressure 被动土压力
Earth pressure theory土压力理论
Coulomb’s earth pressure theory库伦土压力理论
Rankine’s Earth Pressure Theory 朗肯土压力理论
Earth retaining structures挡土结构物
Effective angle of internal friction 有效內摩擦角
Effective stress 有效应力
Failure 破坏
Active failure 主动破坏
Passive failure 被动破坏
Mohr’s circle Homogeneous soil 均匀土
Limiting equilibrium 极限平衡
Mohr’s circle莫尔圆

Normally consolidated soil 正常固结土
Principal effective stress有效主应力
Shear strength 抗剪强度
Slip plane 破裂面
Static Equilibrium静力平衡
Surcharge 荷载

Figure 6.1 Pressure diagram for earth pressure at-rest

Figure 6.2 Slip plane for active failure

Figure 6.3 Mohr’s stress circle for active failure

Figure 6.4 Active earth pressure diagram

Figure 6.5 Slip plane for passive failure
Figure 6.6 Mohr’s stress circle for passive failure

Figure 6.7 Passive pressure diagram

Figure 6.8 Relationship between earth pressure and wall displacement

Figure 6.9 Effects of water table on earth pressure
Figure 6.10 Earth pressures for cohesive soils