Chapter 2 Permeability of Soil2.1 IntroductionSoils are assemblages of solid particles with interconnected voids through which water can flow from a point of high energy to a point of low energy. The study of the flow of water through porous soil media is important in soil mechanics. This is necessary for estimating the quantity of underground seepage under various hydraulic conditions; for investigating problems involving the pumping of water for underground constructions; and for making stability analyses of earth dams and earth-retaining structures that are subjected to seepage forces. (i) Hydrostatic Groundwater Condition (静水条件)Pore-water pressure (u, 孔隙水压力) at a depth z below the water table is defined as z u w ⋅γ=(3.1)where γw is unit weight of the water (水的重度) and z is depth (深度) of the water below groundwatertable. (ii) Total Head (总水头)The total head (h) is related to the pore-water pressure (u) and elevation head (z) by the followingequation:z uh w+γ=(3.2)z of equation (3.2) is defined with respect to a chosen datum.(iii) Groundwater FlowWater will flow through the soil (say from point A to B) if hydraulic gradient (or a difference in total head) exists between points A and B.2.2 Darcy ’s Law (达西定律)In 1856, Darcy published a simple equation for the discharge velocity of water through saturated soils, which may be expressed asA i k A Lhk q ⋅⋅=⋅⋅= (3.3)whereq = flow rate (m 3/s), 渗流量 i = hydraulic gradient 水力梯度 h = total head difference (m) 水头差L = length of flow (m) 渗径长度k = coefficient of permeability (m/s) 渗透系数 A = cross-sectional area of the specimen (m 2) 横截面积Equation (3.3) implies that the flow rate (q) bears a linear relationship to the hydraulic gradient (i). However, a non-linear relationship between q and i is found for clay [叁考: 土力学 p.69 图 3-3]. For practical reason, the following relationship between q and i is assumed for clay:()A i i k q b ⋅-⋅=(3.3a)where i b is starting hydraulic gradient (起始水力梯度). For gravelly soil (砾土), linear relationship between q and i appears at small hydraulic gradient only. As hydraulic gradient increases beyond a critical value, the relationship becomes non-linear as the flow becomes turbulent (紊流) [叁考: 土力学 p.69 图 3-3].Darcy ’s law is valid for laminar flow (层流) condition where Reynolds numbers is smaller than or equal to 1. Reynolds numbers (雷诺数) is defined as follows:1dv Re ≤η⋅⋅ρ=(3.4)where ρ is density of water (水的密度), v is velocity of water (流速), η is viscosity of water (水的粘滞系数) and d is average diameter of soil grains (土粒子平均粒径). As ρ = 1 g/cm 3, v = 0.25 cm/s, η = 0.0131 g/s ⋅cm at temperature of 10︒C, d is calculated by equation (3.4)m m 52.0v0.1d ≤⋅ρη⋅≤Thus Darcy ’s law can be applied to soils which are finer than coarse sand.2.3 Determination of Coefficient of Permeability (渗透系数的测定)2.3.1Constant head test (常水头测试)A typical arrangement of the constant head permeability test is shown in Fig.2.1. In the setup, the total water head is always maintained constant during the period of the test. The constant head test is suitable for coarse-grained soils. The coefficient of permeability is calculated by the following equation:t h A LV Lh t A V i v k ⋅∆⋅⋅=∆⋅== (3.5)where V is volume of water collected, L is length of the specimen, A is cross-sectional area of the specimen, ∆h is total head difference and t is duration of the test.2.3.2 Falling Head Test (变水头测试)A typical arrangement of the falling head permeability test is shown in Fig.2.2. It is suitable for fine-grained soils. The rate of flow of water through the specimen at any time t is given bydtdhA A L h k dt dV 21⋅-=⋅⋅= (3.6a)where V is volume of water collected, L is length of the specimen, A 1 is cross-sectional area of the specimen, A 2 is cross-sectional area of the standpipe and h is total head difference. Re-arranging equation (3.6a) into⎪⎭⎫⎝⎛-⋅⋅=h dh k A L A dt 12(3.6b)Integrating equation (3.6b)⎪⎭⎫⎝⎛-⋅⋅⋅=⎰⎰h dh k A L A dt 21h h 12T⎪⎪⎭⎫ ⎝⎛⋅⋅⋅=2112h h ln k A L A T ⎪⎪⎭⎫ ⎝⎛⋅⋅⋅=2112h h ln T A L A k(3.6c)The values of k for different types of soil are typically within the ranges shown in Table 2.1.2.4 Seepage and Flow Nets (渗流和流网)2.4.1 Laplace ’s equationLaplace differential equation of continuity is used to describe the two-dimensional steady flow condition for a given point in the soil mass. Let us consider a soil element as shown in Fig. 2.3. The element has dimension of dx and dz in x and z direction, respectively. Water flows through the element due to a hydraulic head difference between upstream and downstream side. Let v x and v z be the discharge velocity in the x and z direction, respectively. The rates of flow of water into the element in the x and z direction are1dz v q x )in (x ⋅⋅= (3.7a) 1dx v q z )in (z ⋅⋅=(3.7b)The rates of flow of water out of the element in the x and z direction are1dz dx x v v q x x )out (x ⋅⋅⎪⎭⎫⎝⎛∂∂+=(3.7c)1dx dz z v v q z z )out (z ⋅⋅⎪⎭⎫⎝⎛∂∂+=(3.7d)Assuming water is incompressible and no volume change in the soil mass occurs, the total rate of inflow should be equal to the total rate of outflow1dx v 1dz v 1dx dz z v v 1dz dx x v v z x z z x x ⋅⋅+⋅⋅=⋅⋅⎪⎭⎫⎝⎛∂∂++⋅⋅⎪⎭⎫ ⎝⎛∂∂+0zv x v z x =∂∂+∂∂(3.7e)Using Darcy ’s law, v x and v z can be expressed asx hk i k v x x x x ∂∂⋅=⋅= (3.7f) z h k i k v z z z z ∂∂⋅=⋅=(3.7g)Substituting equations (3.7f) and (3.7g) into equation (3.7e)0zhk x h k 22z 22x =∂∂⋅+∂∂⋅(3.7h)If the soil is isotropic k x = k z , equation (3.7h) becomes0z h x h 2222=∂∂+∂∂ (3.7i)2.4.2 Flow NetsThe continuity equation [equation(3.7i)] in isotropic medium represents two orthogonal families of curves – that is, the flow lines (流线) and equipotential lines (等势线).A flow line is a line along which a water particle will travel from upstream to the downstream side in the permeable soil medium. An equipotential line is a line along which the total head (总水头) at all points is the same. If piezometers (测压计) are placed at different points along an equipotential line, the height of water will rise to the same elevation in all of them. A combination of a number of flow lines and equipotential lines is called a flow net.To complete the graphical construction of a flow net, the flow and equipotential lines are drawn in such a way that (i) the equipotential lines intersect the flow lines at right angle, (ii) the flow elements are approximate squares, (iii) the upstream and downstream surfaces of the permeable layer are equipotential lines and (iv) the boundary of the impervous layer is a flow line.Based on the flow net, the total flow rate (q) can be estimated bydfN N H k q ⋅⋅= (3.8)where H is the total head difference between the upstream and downstream sides, N f is the number of flow channels and N d is the number of equipotential drops.In addition, the pore-water pressure (u) at a given point A determined from the flow net is⎪⎪⎭⎫ ⎝⎛⋅-=d dt Nn H H h (3.9a)()w e t h h u γ⋅-=(3.9b)where H is total head difference, N d is total number of equipotential drops, n d is number of equipotential drops at point A, h t is total head at point A and h e is elevation head at point A.2.5 Effective stress (有效应力)2.5.1 Effective stress principleEffective stress (σ’) is defined asAN 's∑=σ (3.10a)where ∑N s is the sum of inter-contact forces and A is the cross-sectional area of the soil mass under consideration.Total stress (σ) is defined as()u A A N A s s ⋅-+=⋅σ∑()u AA A AN s s⋅-+=σ∑(3.10b)where A s is cross-sectional area of the soil mass occupied by solid-to-solid contacts. As the ratio A s /A is small and can be neglected for pressure ranges encountered in practical problems. Thus, equation (3.10b) becomesu '+σ=σ(3.11)2.5.2 Pore-water pressure and effective stress under hydrostatic conditionThe total stress at depth z isz sat ⋅γ=σ(3.12a)The pore-water pressure at depth z isz u w ⋅γ=(3.12b)The effective stress at depth z is()z 'z u 'w sat ⋅γ=⋅γ-γ=-σ=σ(3.12c)where γsat is saturated unit weight of soil, γw is unit weight of water and γ’ is submerged unit weight ofsoil.2.5.3 Pore-water pressure and effective stress under seepage(i) Downward seepageThe pore-water pressure at depth z is reduced because of the downward flow of waterh z u w w ⋅γ-⋅γ=(3.13a)where h is the hydraulic head difference between ground surface and depth z. The effective stress at depth z becomesh z 'u 'w ⋅γ+⋅γ=-σ=σ(3.13b)(ii) Upward seepageThe pore-water pressure at depth z is increased because of the upward flow of waterh z u w w ⋅γ+⋅γ=(3.14a)where h is the hydraulic head difference between ground surface and depth z. The effective stress at depth z becomesh z 'u 'w ⋅γ-⋅γ=-σ=σ(3.14b)2.5.4 Critical hydraulic gradient (临界水力梯度)At boiling or quick condition under a critical hydraulic gradient (i c ), the weight of the soil will be balanced by the upward seepage force such that the effective stress becomes zero. From equation (3.14b)0h z 'c w =⋅γ-⋅γ()()()n 11G e11G 'z h i s s w c c -⋅-=+-=γγ==(3.15)where e is void ratio and n is porosity.Table 2.1 Typical values of coefficients of permeability (k)Key wordsClay粘土Constant head test 常水头测试Cross-sectional area 横截面积Darcy’s Law 达西定律Density 密度Diameter of soil grains 土粒子粒径Discharge velocity 渗透速度Effective Stress有效应力Equipotential line 等势线Falling head test 变水头测试Flow line 流线Flow net 流网Flow rate 渗流量Gravel 砾石Hydraulic Head 水头Hydraulic head difference 水头差Hydraulic gradient 水力梯度Breaking hydraulic gradient起始水力梯度Critical hydraulic gradient 临界水力梯度Hydrostatic condition静水条件Laminar flow 层流Laplace Equation 拉普拉斯方程Mixture混合物Permeability 渗透性Coefficient of permeability 渗透系数Pore-water pressure孔隙水压力Porosity 孔隙率Reynolds numbers 雷诺数Sand 砂土Seepage 渗流Seepage Force渗流力Silt 粉土Total Stress总应力Turbulent flow 紊流Viscosity 粘滞系数V oid ratio 孔隙比Fig. 2.1 Setup of constant head permeability test Fig. 2.2 Setup of falling head permeability test Fig. 2.3 Seepage through a soil elementChapter 3 Stresses in soil (土体应力)3.1 IntroductionSoil mass can bear loadings from structures as well as its self-weight. Stresses are produced due to these loadings. The stresses are used to evaluate the deformation of the structures founded on the soil mass and the stability of the soil mass. This chapter presents the methods to determine the self weight and additional stresses acted on a soil mass.3.2 In-situ stresses (原位应力)3.2.1 Effective vertical stress (竖向应力)The effective vertical stress (σ’z ) is calculated from the unit weight of the soil (土的重度),γ.(i) No ground water table (地下水位) exists [叁考: 土力学 p.39 图 2-1]z 'z ⋅γ=σ(3.1)where γ (kN/m 3) is unit weight of the soil and z (m) is depth of the soil.(ii) Ground water table locates at the ground surface()z 'z 'w z ⋅γ=⋅γ-γ=σ(3.2)where γ’ (kN/m 3) is submerged unit weight of the soil (土的浮重度) and γw (9.81 kN/m 3) is unit weightof water (水的浮重度).(iii) If the ground consists of n different layers of stratum (土层)i n1i i z z '⋅γ=σ∑=(3.3)where γi and z i are unit weight and depth of the soil of the i th stratum, respectively.3.2.2 Effective horizontal stress (侧向应力)The effective horizontal stress (σ’h ) is much more difficult to be evaluated. Normally, the horizontal stress is expressed as a function vertical stress by the following expression:v o h 'K 'σ⋅=σ(3.4)where K o is coefficient of earth pressure at rest (静止侧压力系数). Typical values of K o are listed inTable 3.1.3.3 Contact pressures at the bottom of the foundation (基底压力)3.3.1 Flexible and rigid foundationsThe pressure distribution beneath a foundation depends on the type of load, the flexural rigidity of the foundation and the type of material on which it is resting.A flexible foundation (柔性基础) has zero stiffness but a rigid foundation (刚性基础) has infinite stiffness. A uniformly loaded flexible foundations (e.g. oil tank and earth dam) resting on an elastic material such as saturated clay will take a sagging profile and the contact pressure is also uniformly distributed [叁考: 土力学 p.41 图 2-4].A rigid foundation (e.g. concrete dam, box foundation) resting on saturated clay will undergo uniform settlement and the contact pressure distribution looks like a saddle, i.e. large at the edge but small in the middle [叁考: 土力学 p.42 图 2-5].If the size of the foundation is not too large and the applied load is small, linear distribution of the contact pressure can be used.3.3.2 Pressure distribution at the bottom of the rigid foundation(i) Vertical load at the center of the foundation (中心竖向荷载)BL F p v⋅=(3.5)where F v is the sum of the applied vertical load (竖向荷载) and the weight of the foundation and backfill (基础和回填土的总重), L is the length of the foundation and B is breadth of the foundation.(ii) Eccentric and vertical load (偏心竖向荷载)⎪⎭⎫⎝⎛±⋅=L e 61B L F }p p v min max (3.6)where e is the eccentricity of the total vertical load.For e < L/6, a trapezoidal pressure distribution, For e = L/6, a triangular pressure distribution,For e > L/6, p min < 0, part of the foundation is separated from the soil and the stress redistributes to a triangular pressure distribution [叁考: 土力学 p.43 图 2-7e] with p max equals to()e L 5.0B 3F 2p vmax -⋅⋅=(3.7)(iii) Net pressure at the bottom of the foundation (基底净压力)Before the construction of a foundation, the soil exhibits an overburden pressure due to its self weight. During construction, soil is excavated to the level of foundation. The final (or net) pressure exerted on the foundation is the applied pressure minus the weight of the excavated soil.o net z p p ⋅γ-=(3.8)where z o is depth of foundation.3.4 Stress increase due to surface load (地基中的附加应力)3.4.1 Stresses due to a vertical point load (竖向集中力作用下附加应力)Boussinesq (1883) solved the problem of stresses produced at any point in a homogeneous (均质), elastic (弹性) and isotropic (各向同性) medium as the result of a point load applied on the surface of a semi-infinite half space (半无限空间体). Boussinesq ’s solution for stresses at a point M [叁考: 土力学 p.47 图 2-10] due to the point load F is()()()⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡-⋅+⋅+-+⋅μ-+⋅π=δσ35252x R z R z R x z R 2z R R 1321R z x 2F 3(3.9a)()()()⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡-+⋅+-+⋅μ-+⋅π=δσ35252y R z R z R y z R 2z R R 1321R z y 2F 3 (3.9b)5332z R z 2F 3cos R2F 3π=θ⋅π=δσ(3.9c)()()⎥⎦⎤⎢⎣⎡⋅++μ--⋅⋅π=δτ=δτ325y x xy R y x z R z R 2321R z y x 2F 3 (3.9d)52zxxz R z x 2F 3⋅π=δτ=δτ(3.9e)52zyy z R z y 2F 3⋅π=δτ=δτ (3.9f)22y x r +=(3.9g) 22222z r z y x R +=++=(3.9h)where μ is Poisson ’s ratio. Equation (3.9c) can be re-arranged as follows:225225253z z F K z r 11z 2F 3R z z 2F 3R z 2F 3⋅=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛+⋅π=⎪⎭⎫⎝⎛⋅π=π=δσ (3.9i)where252z r 123K ⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛+⋅π=(3.9j)K is a function of r/z.By using the principle of superposition (叠加原理), the vertical stress produced at point M due to several point loads F i ( i = 1,2,…n) [叁考: 土力学 p.47 图 2-12] can be deduced from equation (3.9j) and presented as follows:∑∑==⋅=⋅=δσn1i i i2n1i 2i i z F Kz 1z F K(3.9k)3.4.2 Stress increase in 3-dimensional problems (空间条件下的附加应力)(i) Vertical stress beneath the corner of a vertical and uniformly loaded (竖直均布荷载) rectangularfoundationLet the net pressure on the rectangular area be equal to p [叁考: 土力学 p.48 图 2-13]. The total load on the elementary area is pdxdy. The vertical stress (d σz ) at point M due to the load on this elementary area can be obtained from equation (3.9c)dy dx p R 2z 3d 53z ⋅⋅⋅π=σ(3.10a)The increase of stress at M due to the entire rectangular loaded area can be found by integratingequation (3.10a)()⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛+⎪⎭⎫ ⎝⎛+++++⋅π=++⋅π⋅⋅⋅=⋅π⋅⋅⋅=δσ-⎰⎰⎰⎰n m tan n 11n m 1n m 1nm 2p z y x 2dydx p z 3R 2dy dx p z 3122222B 0L052223B 0L 053z (3.10b)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 A from the bottom of the foundation. Equation (3.10b) can be re-written as follows:p K s z ⋅=σ(3.10c)K s is a function of m and n or L. B and z. Values of K s can be obtained from 土力学 p.52表2-2.(ii) Vertical stress beneath the center of a vertical and uniformly loaded circular foundationLet the net pressure on the circular foundation of radius r o be equal to p [叁考: 土力学 p.52 图 2-18]. The total load on the elementary area is prdrd θ. The vertical stress (d σz ) at point A belowthe center due to the load on this elementary area can be obtained from equation (3.9c) The increase of stress at A due to the entire circular area can be found by integrating d σz()p K r /z 1111p z r 2prdrd z 3d r 2320220r 0522320r 0zz 00⋅=⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛+-=+πθ=σ=σ⎰⎰⎰⎰ππ (3.11)K r is a function of z/r 0 and values of K r are summarized in 土力学 p.55表2-5.3.4.3 Stress increase in 2-dimensional problems (平面问题条件下的附加应力)(i) Stresses due to a line load (线荷载)The figure [叁考: 土力学 p.56 图 2-19] shows a line load of infinite length (y direction) having a pressure p/unit length on the surface of a semi-infinite soil mass. The total load on the elementary length is pdy. The vertical stress (d σz ), horizontal stress (d σx ) and shear stress (τxz ) at point A due to the load on this elementary length can be obtained from equations (3.9c), (3.9a) and (3.9e), respectively. The increase of stresses at M due to the entire line load can be found by integrating d σz , d σx and τxz ,β⋅π=⋅π⋅=⋅π⋅⋅=σ⎰+∞∞-3141353z cos R p 2R z p 2R2dy p z 3 (3.12a)β⋅β⋅π=σ21x sin cos R p2(3.12b)β⋅β⋅π=⋅π⋅⋅=τ=τsin cos R p2R z x p 221412zx xz (3.12c)where221z x R += (3.12d) zx tan =β(3.12e)This is a plane stress problem and from theory of elasticity()z x y σ+σ⋅μ=σ(3.12f)(ii) Vertical stress beneath a vertical and uniformly loaded strip foundation (条形基础竖直均布荷载)Equation (3.12a) derived for the vertical stress increases as the result of a line load can be used to determine the stress increases at a point due to a strip load of width B [叁考: 土力学 p.56 图 2-20]. Let the pressure acting on the foundation be p n and consider an elementary strip of of width d ζ. The load per unit length of this strip is p n d ζ. This elementary strip can be treated as a line load. The vertical stress increase (d σz ) is calculated from equation (3.12a) by replacing p by p n d ζ. and xby (x - ζ). The increase of stresses at A due to the entire strip load can be found by integrating d σz ,()[]p K z x d z p 2s z 2223Bz ⋅=+ζ-ζ⋅⋅π=σ⎰(3.13a)where()()⎥⎦⎤⎢⎣⎡-+-⋅-+-+⎪⎭⎫ ⎝⎛--⎪⎭⎫ ⎝⎛⋅π=--222211sz1m n 1m n m n n m n 1m tan n m tan 1K(3.13b)where m = x/B and n = z/B and values of K z s are tabulated in 土力学 p.59表2-6.Table 3.1 Typical values of K oKey wordsClay 粘土Normally consolidated clay 正常固结粘土Overconsolidated clay 超固结粘土Coefficient of earth pressure at rest (K o) 静止侧压力系数Earth dam 土坝Eccentric 偏心Elastic 弹性Fill 填土Backfill 回填土Well-compacted fill 压实填土Foundation 基础Bottom of foundation 基底Flexible foundation 柔性基础Rigid foundation 刚性基础Strip foundation 条形基础Ground water table 地下水位Homogeneous 均质In-situ stresses 原位应力Isotropic 各向同性Load 荷载Horizontal load 水平荷载Inclined load 倾斜荷载Line load 线荷载Triangular load 三角形分布荷载Uniform load 均布荷载Vertical load 竖向荷载Vertical point load 竖向集中力Moment 力矩Pressure 压力Net pressure 净压力Principle of superposition 叠加原理Right-angled trapezoidal load 直角梯形分布荷载Sand 砂土Loose sand 松砂Dense sand 密砂Semi-infinite half space 半无限空间体Stratum 土层Stress 应力Horizontal stress 侧向应力Stress increase 附加应力Vertical stress 竖向应力Two-dimensional problem 平面问题Unit weight of soil 土的重度Submerged unit weight of soil 土的浮重度Chapter 4 Compressibility and Consolidation of Soil [土的压缩与固结]4.1 IntroductionThe process of consolidation [固结] is often confused with the process of compaction [压实]. Compaction increases the density of an unsaturated soil by reducing the volume of air in the voids (see Fig.4.1). However, consolidation is a time-related process of increasing the density of a saturated soil by draining some of the water out of the voids (see Fig.4.1).Consolidation is generally related to fine-grained soils [幼粒土] such as silts and clays. Coarse-grained soils [粗粒土], such as sands and gravels, also undergo consolidation but at a much faster rate due to their high permeability. Saturated clays consolidate at a much slower rate due to their low permeability.Consolidation theory is required for the prediction of both the magnitude and the rate of consolidation settlements [沉降量与沉降速度] to ensure the serviceability of structures founded on a compressible soil layer.4.2 A simple one-dimensional consolidation model [单向固结模型]Since water can flow out of a saturated soil in any direction, the process of consolidation is essentially three-dimensional [三维]. However, in most field situations, water will not be able to flow out of the soil by flowing horizontally because of the vast expanse of the soil in horizontal direction. Therefore, the direction of flow of water is primarily vertical [竖向] or one-dimensional. As a result, the soil layer undergoes one-dimensional (1-D) consolidation settlement [单向固结沉降] in the vertical direction.Fig.4.2 shows a simple model for 1-D consolidation. The spring [弹簧] is analogous to the soil skeleton [土骨架]. The stiffer the spring, the less it will compress. Therefore, a stiff soil will undergo less compression than a soft soil. The stiffness of a soil [土的硬度] influences the magnitude of its consolidation settlement. The valve [阀门] opening size is analogous to the permeability of the soil [土的渗透性]. The smaller the opening, the longer it will take for the water to flow out and dissipate its pressure. Therefore, consolidation of a fine-grained soil takes longer to complete than that of a coarse-grained soil. Permeability of a soil influences the rate of its consolidation.4.3 One-dimensional consolidation test [单向固结试验]The one-dimensional (1-D) consolidation test is performed in an oedometer [固结仪]. An oedometer is shown in Fig.4.3. The soil specimen [土样本] is in a form of a disc (normally 20 mm in height and 80 mm in diameter), which is confined in a steel ring and immersed in a water bath. A vertical load [竖向荷载] is applied to compress the specimen and water is permitted to drain away through porous stones [透水石] placed at the top and bottom of the specimen.4.3.1 Time-related consolidationFor each vertical load increment [竖向荷载增量], the vertical settlement [竖向沉降] of the soil specimen is recorded using a dial gauge [百分表]. Fig.4.4 shows the time relationships of verticalsettlement, vertical total stress, excess pore-water pressure [超孔隙水压力] and vertical effective stress. Initially, 100 % of the vertical load is taken by pore water because, due to low permeability of the soil specimen, the pore water is unable to flow out of the voids quickly. Therefore, there is very little settlement of the soil specimen immediately after placing the vertical load. The settlement of soil is possible only when there is an increase in effective stress, which in turn requires that the void ratio of the soil be reduced by the expulsion of pore water. After a few seconds, the pore water begins to flow out of the voids. This results in a decrease in excessive pore-water pressure and void ratio of the soil specimen and an increase in effective stress. Finally, all excess pore-water pressure dissipates and the vertical effective stress equals to the vertical total stress.4.3.2 Compression Curve [压缩曲线]Several increments of vertical stress are applied in an oedometer test. For each increment, the final settlement of the soil specimen is recorded. The end points from number of loading and unloading increments [荷载与卸载增量] of an oedometer test may be plotted as a conventional stress-strain curve [应力-应变曲线] as shown in Fig.4.5. The increment of vertical strain [竖向应变增量] (∆εv ) for each loading increment is given byv H h∆=ε∆ (4.1)where ∆h is the final settlement for the loading increment (i.e. the change in specimen height) and H 0 is the initial specimen height [样本初始高度] before loading increment is applied.(i) e : σ’v curveAs the soil specimen is not allowed to deform in horizontal direction, there is only change in void ratio. Thus, ∆εv can be expressed in terms of the void ratio [孔隙比] (e) by0v e 1eH h +∆=∆=ε∆ (4.2)where ∆e is the change in void ratio due to the loading increment and e 0 is the initial void ratio of the specimen [初始孔隙比] before loading increment is applied. The test results are plotted in terms of vertical effective stress (σ’v ) and void ratio (e) and shown in Fig.4.6.The coefficient of volume compressibility [体积压缩系数] (m v ) is defined as the ratio of change in volumetric strain (∆εvol ) over change in vertical effective stress (∆σ’v ) and shown as follows:v o v vol v 'ee 11'm σ∆∆⋅+-=σ∆ε∆= (4.3) The unit for m v is m 2/kN and its value depends on the stress range over which it is calculated. Example: Consider the second loading increment shown in Fig.4.7,kN /m 1082.11002002.18.02.111'e e 11m 23v o v -⋅=--⋅+-=σ∆∆⋅+-=(ii) e : log σ’v curveThe e : σ’v curve becomes almost linear if σ’v is plotted on a log scale as shown in Fig.4.8. The slope of the loading curve is called the Compression Index [压缩指数] (C c ) and is dimensionless. It isdefined as:⎪⎪⎭⎫⎝⎛σσ∆+σ∆-=σ-σ-=vo v vo vo1v o1c '''log e 'log 'log e e C (4.4)The negative sign is used because the void ratio decreases when the effective stress is increased. The slope of the unloading curve is called the Expansion Index [回弹指数] (C e ) and is calculated using the same procedure.(iii) Other compressibility parametersOedometric modulus [侧限压缩模量] (E s ) is defined as the ratio of vertical effective stress (σ’z ) over vertical strain (εz ) and shown as follows:z z s 'E εσ=(4.5)From generalised Hooke ’s law [广义虎克定律], elastic strains [弹性应变] in x, y and z directions areexpressed as follows: ()z y x x EE σ+σμ-σ=ε (4.6a) ()z x y y EEσ+σμ-σ=ε (4.6b) ()y x z z EE σ+σμ-σ=ε(4.6c)where E is Young ’s modulus [弹性模量]. In 1-D consolidation, εx = εy = 0, thus Equations (4.6b) and (4.6c) become ()z y x σ+σ⋅μ=σ (4.6d) ()z x y σ+σ⋅μ=σ(4.6e)Substitute Equation (4.6e) into Equation (4.6d)μ-μ=σσ1z x (4.6f)Since σx = σy = σz K 0, where K 0 is the coefficient of earth pressure at rest [静止土压力系数], thereforeμ-μ=1K 0 (4.6g)In one-dimensional consolidation, substitute Equations (4.5) and (4.6f) into Equation (4.6c) and re-arranged to the following expression: ()⎪⎪⎭⎫ ⎝⎛μ-μ-⋅=⎪⎪⎭⎫ ⎝⎛μ-μ-⋅εσ=σ⋅μ-σ⋅μ-σε=121E 1211E 2s 2zzy x z z (4.6h)4.4 Effect of stress history [应力历史]Fig.4.9 shows the stress history of a clay deposit. Due to the melting of ice and ground erosion,。
