(库伦土压力理论)

Chapter 6 Lateral Earth Pressure

6.3 Coulomb’s Earth Pressure Theory (库伦土压力理论) (i)

Coulomb (1776) proposed that a condition of limit equilibrium exists in a soil wedge between a retaining wall and a trial slip plane. (库伦土压力理论假设一个滑动面, 整个滑动块体处于极限平衡状态).

(ii) The force between the wedge and the wall is determined by considering the equilibrium of forces

acting on the wedge. (利用整个滑动块体上静力平衡条件来确定土压力).

(iii) Among these trial slip planes, the critical slip plane is the one which gives the maximum

lateral pressure on the wall (在假定滑动面中, 临界滑动面产生最大的土压力).

(iv) Poncelet (1840) used Coulomb’s limit equilibrium approach to obtain the active and passive earth

pressure coefficients for the following cases:

(a) backfill is dry, homogenous and cohesionless soil with an angle of internal friction φ, (填土

是干, 均质和无粘性土)

(b) backfill is sloping at an angle a to the horizontal, (填土表面与水平面夹角为α)

(c) wall friction φo is present, (墙背与填土之间的摩擦角为φo )

(d) wall face inclined at an angle e to the vertical, (墙背面与竖直线的夹角为ε)

(v) For active failure, the wall moves away from the soil mass. The forces acting on the soil wedge

above the slip plane are shown in Figure 6.11. The forces acting on soil wedge ABC is under equilibrium: its weight [W], the reactions on the slip plane AC [R] and the wall AB [Pa ]. (墙体离开填土方向, 产生主动破坏, 滑动块体上力的分布见图6.11:土体ABC 的重量W, 滑动面A C 上的反力R 与墙背A B 上的反力P a 达至静力平衡) (vi) Consider the sine rule (通过正弦定律)

P a sin (θ-φ)

=

W

sin(90︒+φ+φo +ε-θ)

W =γ⋅∆ABC =γ⋅

1212

12

AB

2

⋅

sin(90︒+α-ε) ⋅sin(90︒+ε-θ)

sin(θ-α)

W =γ⋅H

2

⋅

sin(90︒+α-ε) ⋅sin(90︒+ε-θ)

cos

2

ε⋅sin(θ-α)

P a =

γ⋅H

2

⋅

cos(ε-α) ⋅cos(θ-ε) ⋅sin(θ-φ) cos

2

ε⋅sin(θ-α) ⋅cos(θ-φ-φo -ε)

Differentiating the above expression for Pa w.r.t. θ and equating the derivative to zero, we can obtain the critical value of θ that gives maximum Pa : (将P a 对θ 求导数, 并令其等於零)

P a (max.) =

12γ⋅H

2

2

⋅cos

2

cos (φ-ε)

⎡

ε⋅cos(ε+φo ) ⋅⎢1+

⎢⎣

sin(φ+φo ) ⋅sin(φ-α) ⎤

⎥

cos(ε+φo ) ⋅cos(ε-α) ⎥

⎦

2

P a (max.) =

12

γ⋅H ⋅K a

2

K a =

cos

2

cos (φ-ε)

⎡

ε⋅cos(ε+φo ) ⋅⎢1+

⎢⎣

sin(φ+φo ) ⋅sin(φ-α) ⎤

⎥

cos(ε+φo ) ⋅cos(ε-α) ⎥

⎦

2

2

If the wall is vertical (ε = 0), there is no friction between the wall and the soil (φo = 0) and the surface of the backfill is horizontal (α = 0), the expression of Ka becomes [假定墙背面竖直(ε = 0) 与光滑(φo = 0), 填土面为水平(α = 0)]

K a =

cos

2

φ

[1+sin φ]2

=

1-sin

2

φ

[1+sin φ]2

=

φ⎫2⎛

=tan 45︒-⎪

1+sin φ2⎭⎝

1-sin φ

(vii) For passive failure, the wall is pushed against the soil mass. The forces acting on the soil wedge

above the slip plane are shown in Figure 6.12. (墙体向填土方向挤压, 产生被动破坏, 滑动块体上力的分布见图6.12)

(viii) Consider the sine rule (通过正弦定律)

P p sin (θ+φ)

=

W

sin(90︒+ε-φo -φ-θ)

2

W =

1212

γ⋅H ⋅

sin(90︒+α-ε) ⋅sin(90︒+ε-θ)

cos

2

ε⋅sin(θ-α)

P p =

γ⋅H

2

⋅

cos(ε-α) ⋅cos(θ-ε) ⋅sin(θ+φ) cos

2

ε⋅sin(θ-α) ⋅cos(θ+φ+φo -ε)

(ix) Differentiating the above expression for P p w.r.t. θ and equating the derivative to zero, we can

obtain the critical value of θ that gives maximum Pp : (将P p 对θ 求导数, 并令其等於零)

P p (max.) =

12γ⋅H ⋅

cos

2

2

2

cos (φ+ε)

⎡

ε⋅cos(ε-φo ) ⋅⎢1-

⎢⎣

sin(φ+φo ) ⋅sin(φ+α) ⎤

⎥

cos(ε-φo ) ⋅cos(ε-α) ⎥

⎦

2

P p (max.) =

12

γ⋅H ⋅K p

cos (φ+ε)

2

2

K p =

cos

2

⎡

ε⋅cos(ε-φo ) ⋅⎢1-

⎢⎣sin(φ+φo ) ⋅sin(φ+α) ⎤

⎥

cos(ε-φo ) ⋅cos(ε-α) ⎥

⎦

2

(x) If the wall is vertical (ε = 0), there is no friction between the wall and the soil (φo = 0) and the surface of the backfill is horizontal (α = 0), the expression of Kp becomes [假定墙背面竖直(ε = 0) 与光滑(φo = 0), 填土面为水平(α = 0)]

K p =

cos

2

φ

[1-sin φ]2

=

1-sin

2

φ

[1-sin φ]2

=

1+sin φ

φ⎫2⎛

=tan 45︒+⎪

1-sin φ2⎭⎝

(xi) Rankine’s theory is based on the stress equilibrium of a soil element. It assumes that the wall is

vertical, frictionless and the soil surface is horizontal. In general, it over-estimates the active pressure but under-estimates the passive pressure (朗肯理论基于土单元体的应力平衡条件来建立, 假定墙背面竖直与光滑, 填土面为水平, 朗肯理论高估主动土压力与低估被动土压力)

(xii) C oulomb’s theory is based on limiting equilibrium of forces within a soil wedge for a given slip

surface. It assumes that the failure surface is a plane. In general, it under-estimates the active pressure and over-estimates the passive pressure. (库伦理论基于滑动块体的静力平衡条件来建立, 假定滑动面为平面, 库伦理论低估主动土压力与高估被动土压力)

o -ε

A

Figure 6.11 Force diagrams for active failure

A

Figure 6.12 Force diagrams for passive failure