第五章 土的抗剪强度 Chapter 5 Shear Strength of Soil

In this lecture, we’ll learn

• A simple friction model [摩擦力模型 ]

• Mohr-Coulomb failure criteria [莫尔 -库伦破坏准则 ]

• Principal stresses [主应力 ]

• Mohr’s stress circle [莫尔应力圆 ]

• Shear test[剪切试验 ]

• shear failure plane [剪破面 ]

5.1 Simple Friction Model 摩擦力模型

A wooden block is pushed by a given horizontal force (H) against the surface of a table as shown in figure on the top right.

Friction [ 摩擦力 ] (T) is the reaction [ 反力 ] of H and normal force [ 法向力 ] (N) is the reaction of the weight W.

Figure on the bottom right shows the change of T with displacement [ 位移 ] x.

When T is smaller than Tlimit, there is no displacement of the block. When T reaches a limiting value [ 极 限 值 ] Tlimit, the block starts to slide [ 滑 动 ] on the surface of the table.

Displacement [ 位移 ] x

T

x

Tlimit

This limiting value of friction (Tlimit) is related to N by the following expression:

NT itlim

where is the coefficient of friction [ 摩擦力糸数 ] between the table and the block. is independent of N and is expressed as:

'tan

where ’ is the angle of friction [ 摩擦角 ]. The horizontal surface between the block and the table is called the slip plane [ 滑动面 ].

Similarly, friction [ 摩擦力 ] also exists within the inter-particle contact [ 土粒之间的接触面 ] of a conglomerate （聚集物） of soil particles

When the soil particles start to slide [ 滑动 ] against each other along a given slip plane, what are normal effective stress [ 法向有效应力 ] at failure (’nf), shear stress [ 切向应力 ] at failure (f) and direction of this slip

plane [滑动面方向 ] ?

滑动

滑动

5.2 stress state 应力状态

yx

P ,

zx

P ,

zy

P zz

yy

xx

1. Normal stress [ 法向应力 ]

yx

F

2. Shear stress [ 切向应力 ]

Figure below shows the normal and shear stresses on a cubic element. If the shear stress on a plane is zero, the plane is called a principal plane [ 主 应 力 面 ] and the normal stress is called the principal stress [ 主应力 ].

3. Principal stress [ 主应力 ]

5.3 Mohr’s stress circle [ 莫尔应力圆 ]

Figure below shows a typical Mohr’s stress circle. ’x and ’z are

normal effective stresses [ 法 向 有 效 应 力 ] in x and z direction, respectively. xz and zx are shear stresses [ 切向应力 ]. The state of ’x,

’z, xz and zx can be represented by a stress circle in ’ : plane. (Normal

compressive stress and shear stress, which gives anti-clockwise couple are positive).

’

’

’

’ ’

’

’

’

’’

Stresses ’1 and ’3 are the principal stresses. Since 1 > 3, ’1 is the

major principal stress [ 最大主应力 ] and ’3 is the minor principal stress

[ 最小主应力 ]. The plane on which the major principal stress ’1 acts is

the major principal plane [ 最大主应力面 ]. Minor principal stress ’3 acts

on minor principal plane [ 最 小 主 应 力 面 ] that is perpendicular to the major principal plane. The direction of principal planes are illustrated as follows:

’

’

’ ’

’

’

’

’’

2

’1

’3

If the magnitudes and directions of ’1 and ’3 are known, the normal

stress (’ ) and shear stress ( ) act on a plane at angle with the plane of

major principal stress will be

2cos2

''

2

''' 3131

2sin

2

'' 31

2

’ ’

’’’’

’

’ ’

’

2

'' 31

’

5.4 Mohr-Coulomb failure criteria [ 莫尔 - 库伦破坏准则 ]

The shear strength of a soil [ 土 的 抗 剪 强 度 ] (f) at a point on a

particular plane was originally expressed by Coulomb as a linear function of the normal stress (’f) on the plane at the same point by

'tan''c ff

where c’ is cohesion [ 粘聚力 ] and ’ is internal angle of friction [ 内摩擦角 ].

’

’

c’

At failure the straight line represented by Coulomb’s equation will be tangential [ 切线 ] to the Mohr’s stress circle. f is the angle between the shear failure plane [ 剪破面 ] and the major principal plane [ 最大主应力面 ].

’

’

f

’1f’3f ’

f

c’

’f

’1f

’1f

’3f’3f f

’f

f

2

'' f3f1

2

'' f3f1

’

’

f

’1f’3f ’

f

c’

’f

’1f

’1f

’3f’3f f

’f

f

2

'' f3f1

2

'' f3f1

2cos''2

1''

2

1' 3131f 2sin''

2

131f

2

'45f

’

’

’1f’3f ’

c’

2

'' f3f1

2

'' f3f1

'cos'c2'sin'''' f3f1f3f1

2

'45tan'c2

2

'45tan'' 2

f3f1

2

'45tan'c2

2

'45tan'' 2

f1f3

Failure

Failure

Failure

• Direct shear test [ 直接剪切试验 ]

• Unconfined compression test [ 无侧限抗压试验 ]

• Triaxial compression test [ 三轴压缩试验 ]

5.5 Shear strength test [ 剪切强度试验 ]

Three types of test will be introduced

The test involves shearing a soil sample [ 样 本 ] along a horizontal slip plane [ 水平滑动面 ].

The sample is contained within the shear box, which can be split horizontally to two halves.

Vertical force is applied through a metal platen resting on top of the soil sample.

Horizontal force is applied to slide apart the top and bottom halves of the box.

Methods [ 方法 ]

1. The Direct Shere Test 直接剪切试验

Vertical and horizontal loads are measured using proving rings [ 量力钢环 ].

Horizontal and vertical displacements of the top half of the box are recorded using dial gauges [ 百分表 ] to obtain shear and volumetric strains [ 剪应变与体变 ].

Measurements [ 量测 ]

Drainage condition cannot be controlled [ 不能控制排水 ]

Pore-water pressure cannot be measured [ 不 能 测 量 孔隙水压力 ].

Slip plane is horizontal.

Limitations [ 限制 ]

Effective stress is used provided that the test is conducted at slow rate so that water is allowed to drain away.

Interpretation [ 分析 ]

’

’

c’’1 ’2 ’3

1

2

3

The photo of a typical triaxial apparatus [ 三轴压力仪 ] is shown in figure on the right.

The device consists of a loading frame, a triaxial cell [ 三 轴 压 力室 ], and some measurement devices [ 测 量 系 统 ], such as proving ring, dial gauge, pressure transducers [ 压力传感器 ].

Equipment [ 仪器 ]

2. Triaxial compression test [ 三轴压缩试验 ]

A cylindrical sample is enclosed in a rubber membrane [ 橡皮薄膜 ], placed inside a water-filled cell and stressed under conditions of axial symmetry [ 轴向对称 ].

It is subjected to an all-round cell pressure [ 围 压 ], consolidation [ 固结 ] is allowed to take place and then the axial stress [ 轴 向 应 力 ] is gradually increased until failure of the specimen.

Axial stress is the major principal stress [ 最大主应力 ] and radial stress is the minor principal stress [ 最小主应力 ].

Methods [ 方法 ]

Two types of test: consolidated drained (CD) test [ 固结排水试验 ] and consolidated undrained (CU) test [ 固结不排水试验 ].

Both the CD and the CU tests subject the soil sample to an initial consolidation.

After consolidation, the soil sample is sheared without allowing water to drain out for CU test and pore-water pressure is measured. In the contrary, water is allowed to drain out for CD test.

Tests [ 试验 ]

Usually, three or more soil samples are tested at different values of consolidation pressure.

The shear strength parameters (c’ and ’) can be obtained by drawing Mohr’s stress circles.

Interpretation [ 分析 ]

three or more soil samples are tested

Failure stress got from curve in different confine pressure

Conmon tangential line

A cylindrical [ 圆 柱 体 ] soil sample is subjected to an axial compressive load [ 轴 向 压 力 ] between two metal plates.

There is no confinement of the sample in the radial direction [ 侧向应力为零 ].

Since there is no arrangement to control drainage, the soil sample is sheared at a fast rate to ensure undrained condition [ 不 排 水 条件 ].

Methods [ 方法 ]

3. Unconfined compression (UU) test [ 无侧限抗压试验 ]

Vertical load is recorded using a proving ring

Axial deformation of the soil sample is recorded using a dial gauge.

Measurements [ 量测 ]

Undrained shear strength [ 不排水抗剪强度 ] (cu) of the soil sample is

given by:

Interpretation [ 分析 ]

f1uf 2

1c

cu

4. Vane shear test 旁压试验

Saturated soil

Non-saturated soilDry soil

Compreesion

Expansion

5.6 Pore pressure coefficients in Triaxial test [ 三轴剪切试验中的孔隙水压力系数 ]

In this lecture, we’ll learn

Shear behaviour of sands ( 砂 土 的 剪 切 性状 )

Density: Loose and dense sands

Shear behaviour of clays ( 粘土的剪切性状 )

Stress history: NC and OC clays

5.7 The shear strength characteristics of soil 土的剪切性状

Stress – strain curve [ 应力 - 应变曲线 ]

Dense sands and OC clays

Residual Strength [残余强度 ]

Loose sands and NC clays

Peak Strength [峰值强度 ]

Sands under the same confining pressure [ 相同的周围应力 ]

1. Shear behaviour of sand [ 砂土的剪切性状 ]

Mohr-Coulomb strength envelope [ 莫尔 - 库伦强度包线 ]

Type 1 soil (loose sand)

'tan'

Residual value [ 残余值 ]

Peak value [ 峰值 ]

’

Linear relationship between and ’ (OCA)

Strength envelope passes through origin O

Mohr-Coulomb strength envelope [ 莫尔 - 库伦强度包线 ]

Type 2 soil (dense sand)

'tan''c

Residual value [ 残余值 ]

Peak value [ 峰值 ]

’

Non-linear relationship between and ’ (OBC)

Is it correct to define a best-fit straight line for peak strength data?

What is meaning of c’?c’

Volumetric- shear strain relationship [ 体变 - 剪应变关系 ]

Loose sands and NC clays

Dense sands and OC clays

剪缩

剪胀

Volumetric- shear strain relationship [ 体变 - 剪应变关系 ]

Loose sands and NC clays

Dense sands and OC clays

Type 1 soil (loose sand)

Sliding would be initiated on the horizontal plane a-a

Soil particles would tend to move into the void spaces

Direction of motion would have a downward component, indicating volumetric contraction [ 体缩 ]

Volumetric- shear strain relationship [ 体变 - 剪应变关系 ]

Loose sands and NC clays

Dense sands and OC clays

Type 2 soil (dense sand)

Relative horizontal sliding of row 2 with respect to row 1 is restrained by interlocking [ 咬 合 作 用 ] of the particles

Sliding can only be initiated on an inclined plane

Particles must ride up over each other or be pushed aside or both

The direction of motion would have an upward component indicating volumetric dilation [ 体胀 ]

Critical void ratio [ 临界孔隙比 ]

Initial void ratio (e0) of type 1 soils (loose sand) is fairly high and that of type 2 soils (dense sand) is fairly low

During shearing, loose sand contracts while dense sand dilates

Therefore, void ratio of loose sand decreases and that of dense sand increases during shearing

There exists a void ratio (ecs) where no volume change occurs during shear. This void ratio is called critical void ratio (临界孔隙比 )

Liquefaction of sand [砂土的液化 ]

Liquefaction means the soil does not possess any shear strength and behaves like a liquid

From shear strength equation, there will be no shear strength if effective normal stress becomes zero

From principle of effective stress, if pore water pressure equals to the total stress, effective stress will be vanish

For example, excess pore-water pressure builds up during earthquake ( 地 震 ) which trigger liquefaction of loose sand

'tan' u'

Niigata Earthquake (1964)

Undrained shear strength [ 不排水强度 ]

The shear strength of clay under undrained condition is called the undrained shear strength [ 不 排 水 强度 ] (cu) and is equal to the radius of

Mohr’s circle

cu depends only on the initial void

ratio or the initial water content of the soil.

For a given initial void ratio, cu is a

constant, thus the total stress strength envelope ( 总 应 力 强 度 包 线 ) is a horizontal line ( 水平线 ) and u = 0.

cu

2

''

2c f3f1f3f1

u

u = 0

2. Shear behaviour of clay [ 粘土的剪切性状 ]

Undrained shear strength [ 不排水强度 ]

An increase in effective confining stress causes a decrease in void ratio and an increase in cu as shown in

figure on the right

cu is used evaluate shear strength

under short term condition, e.g. immediate after excavation in clay

cu2

cu1

Drained shear strength [ 排水强度 ]

The shear behaviour of clay is governed by the stress history ( 应力历史 )

During shear, normally consolidated ( 正 常 固 结 ) [NC] clay behaves like loose sand and overconsolidated ( 超固结 ) [OC] clay behaves like dense sand