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Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 §2 stress in soil • Stress due to soil weight • Contact stress • Stress due to loading Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 §2.2 Stress due to Self-weight •General situation z z x k0 z Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Three Cases: -For Many Layers of soil, the vertical stress due to self-weight of soil is given as following. n z 1 h1 2 h2 ...... n hn i hi i 1 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 -With uniform surcharge on infinite land surface z z p ( z h) Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 •Effective Vertical Stress due to Self-Weight of Soil - Consider a soil mass having a horizontal surface and with the water table at surface level. The total vertical stress (i.e. the total normal stress on a horizontal plane) at depth z is equal to the weight of all material (solids + water) per unit area above that depth ,i.e. σv = γsat z - The pore water pressure at any depth will be hydrostatic since the void space between the solid particles is continuous, therefore at depth z: u = γw z - Hence the effective vertical stress at depth z will be: σ’v=σv- u =( γsat - γw)z= γ’z where γ’ is the buoyant unit weight of the soil. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 -Under water Table γ1h1 Sand Water table γ1'h2 γwh2 Clay (watertight) σz=γ1h1+γ1'h2+γwh2+γsat3h3 z 1 h1 '1h2 w h2 sat2 h3 General, for sand below water table, the γ’ is used; but for clay below water table, it is difficult to determine which one(γ’ or γsat) is suitable. We often choose the buoyant unit weight when the index of liquid LI>=1; the saturated unit weight when the index of liquid LI<=0. When 0<LI<1, the disadvantageous one is choosen. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Example The stratum’s conditions and the related physical characteristics parameters of a foundation are shown in Fig below. Calculate the stress due to self-weight at a,b,c. Draw the stress distribution. w=15.6% e=0.57 γs=26.6kN/m3 w=22% wL=32% wp=23% γs=27.3kN/m3 Civil Engineering Department of Shanghai University For silver sand, the buoyant unit weight (γ‘)is used : s w 26.6 10 9.9kN / m3 1 e 1 0.67 ' 1 I L For saturated clay, Soil Mechanics Chapter 2 w wp wL w p 22 23 0( Semisolid ) 32 23 So, the clay can be considered the watertight; then the saturated unit weight (γsat)is used : Gs w Sr 1 e e Gs w 27.3 / 10 0.22 0.6 sat 2 s e w 1 e 27.3 0.6 10 20.8kN / m 3 1 0.6 Civil Engineering Department of Shanghai University a σz=0 b σz(upper)=γ’1h1=9.9×2=19.8kPa Soil Mechanics Chapter 2 σz(Down)=γ’1h1+ γw(h1+hw)=9.9×2+10×(2+1.2)=51.8kPa c σz=γ’1h1+ γw(h1+hw)+ γsat2h2 = 9.9×2+10×(2+1.2)+20.8×3=114.2kPa Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Exercise 2-2 The stratum’s conditions and the related physical characteristics parameters of a foundation are shown in Fig below. Calculate the stress due to self-weight at 10m depth. Draw the stress distribution. Note: For saturated clay, both cases (watertight and non-watertight) need to consider. w=8% e=0.7 γs=26.5kN/m3 e=1.5 γs=27.2kN/m3 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 § 2.3 Contact stress 2.3.1 Concept of Contact pressure ♦ A foundation is the interface between a structural load and the ground. The stress p applied by a structure to a foundation is often assumed to be uniform. The actual pressure then applied by the foundation to the soil is a reaction, called the contact pressure p and its distribution beneath the foundation may be far from uniform. ♦ This distribution depends mainly on: · stiffness of the foundation. i.e. flexible → stiff → rigid. · compressibility or stiffness of the soil. · loading conditions – uniform or point loading. 2.3.2 Contact pressure – uniform loading The effects of the stiffness of the foundation (flexible or rigid) and the compressibility of the soil (clay or sand) are illustrated in Figure below.. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Figure The distribution of contact pressure Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.3.3 Stiffness of foundation • A flexible foundation has no resistance to deflection and will deform or bend into a dish-shaped profile when stresses are applied. An earth embankment would comprise a flexible structure and foundation. • A stiff foundation provides some resistance to bending and well deform into a flatter dish-shape so that differential settlements are smaller. This forms the basis of design for a raft foundation placed beneath the whole of a structure. • A rigid foundation has infinite stiffness and will not deform or bend, so it moves downwards uniformly. This would apply to a thick, relatively small reinforced concrete pad foundation. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.3.4 Stiffness of soil • The stiffness of a clay will be the same under all parts of the foundation so for a flexible foundation a fairly uniform contact pressure distribution is obtained with a dish-shaped (sagging) settlement profile. For a rigid foundation the dish-shaped settlement profile must be flattened out so the contact pressure beneath the centre of the foundation is reduced and beneath the edges of the foundation it is increased. Theoretically, the contact pressure increases to a very high value at the edges although yielding of the soil would occur in practice , leading to some redistribution of stress. • The stiffness of a sand increase as the confining pressures around it increase so beneath the centre of the foundation the stiffness will be smaller. A flexible foundation of the sand will, therefore, produce greater strains at the edges than in the centre so the settlement profile will be dishshaped but upside-down (hogging) with a fairly uniform contact pressure. For a rigid foundation this settlement profile must be flattened out so the contact pressure beneath the centre would be increased and beneath the edges it Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.3.5 Contact pressure-point loading • An analysis for contact pressure beneath a circular raft with a point load W at its centre resting on the surface of an incompressible soil (such as clay) has been provided by Borowicka,1939. • This shows that the contact pressure distribution is non-uniform irrespective of the stiffness of the raft or foundation. For a flexible foundation the contact pressure is concentrated beneath the point load which is to be expected and for a stiff foundation it is more uniform. For a rigid foundation the stresses beneath the edges are very considerably increased and a pressure distribution similar to the distribution produced by a uniform pressure on a clay is obtained. This suggests that a point load at the centre of a rigid foundation is comparable to a uniform pressure. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.3.6 Stress distribution • The stresses that already exist within the ground due to self-weight of the soil are discussed in Chapter 4. • When a load or pressure from a foundation or structure is applied at the surface of the soil this pressure is distributes throughout the soil and the original normal stresses and shear stresses ate altered. For most civil engineering applications the changes in vertical stress are required so the methods given below are for increases in vertical stress only. -Central Loading P=P/F P B p=P/F Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 -Eccentric Loading p1 P Pe P 6e (1 ) p2 F W F B L e P (0<e<=B/6) B p2 p1 0<e<B/6 p1 e=B/6 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 e P 2P p' B 3( e) L 2 e P (e>B/6) p2 p1 p1 L e>B/6 B Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 h p pa 2.3.7 Additional pressure on the bottom of foundation ¦ Ãh B pa=p-γh Where pa-additional pressure; p-contact pressure; γ-natural unit weight of soil( bouyant unit weight if below water surface); h- buried depth of the foundation. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Exercise 2-3 A load of 1200kN as well as a moment of 1500 kNm is carried on a rectangular foundation (axb=6x4m) at a shallow depth in a soil mass as shown in Fig below. (1) Determine the maximum & minimum contact stresses on the bottom of foundation. (2) If the stress is tensional, Where does the width of foundation extend assuming the stress at right tip is zero and the position & value of loading, the length of the foundation a do not change? Calculate the contact stress after the extension. M=1500kNm N=1200kN x 4m Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 § 2.4 Stress due to loading 2.4.1 Stresses beneath point load（点荷载） • Boussinesq published in 1885 a solution for the stresses beneath a point load on the surface of a material which had the following properties: • semi-infinite – this means infinite below the surface therefore providing no boundaries of the material apart from the surface • homogeneous – the same properties at all locations • isotropic –the same properties in all directions • elastic –a linear stress-strain relationship. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Linear elastic assumption Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 P X r β x y σz R Y z τzy τyx Z M σy (x,y,z) τyz τzx xz τ τx y σx Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 3P Z 3 z 5 2 R Substitute Z/R for Cos β, 3 p cos 3 z 2 R2 z 3P 2 2Z 1 r 1 Z 2 5 2 3 r 2 1 Z 2 5 2 P Z2 Properties Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 p x z below point a below direction Z along horizontal surface Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 P σz Bulbs of pressure(压力泡) Isoline(等值线)——Lines or Contours of equal stress increase Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 P2 P1 σz2 σz1 σz1 +σz2 Z Z Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 ８ 2.4.2 Stresses due to line load（线荷载） pdy X Y p X ８ Z τzx σx τxz Z M(X,Z) Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 3Z 3 pdy 3 pdy Z3 d z 5 2R 2 X 2 Y 2 Z2 z d z 2 pZ 3 X Z 2 2 1 X 2 1 Z 2 2 2 p Z 5 2 Civil Engineering Department of Shanghai University -polar coordinates Z R X Z , cos R0 2 0 2 2 2p 3 z cos R0 2p x cos sin 2 R0 2p xz sin cos 2 R0 Soil Mechanics Chapter 2 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 ８ 2.4.3 Stresses due to uniform vertical loading on an infinite strip （条形荷载） b/2 b/2 Po dξ Po X(ξ) dξ ξ Z σz τzx σx X Z τxz M(X,Z) Civil Engineering Department of Shanghai University d z z b2 b / 2 2Z 3 p0 d X Z 2 d z b2 b / 2 Soil Mechanics Chapter 2 2 2 2 Z 3 p 0 d X Z 2 2 2 b b b b Z X Z X X X p0 2 2 2 arctg 2 arctg 2 2 Z Z b b 2 2 Z X Z X 2 2 p0 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 -polar coordinates b dx P0 A β1 R0 d dx cos X B 0 β β2 R0 Z β σz dβ Z M R0 d 2 p0 cos 2 p0 dx 2 p0 3 3 d z cos cos cos 2 d R0 R0 Civil Engineering Department of Shanghai University 1 z d z 2 2 p0 1 Soil Mechanics Chapter 2 cos 2 d 2 p0 1 1 sin 2 sin 2 1 1 2 2 2 2 p0 1 1 x sin 2 sin 2 1 1 2 2 2 2 p0 cos 2 2 cos 2 1 xz 2 -Principal stress 2 1 X Z 2 Z X xz 3 2 2 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 1 p0 sin 3 ψ β1 β2 (Out) or tg 2 tg ( 1 2 ), max then ψ β1 β2 1 2 p0 1 1 3 Sin 2 Properties 2 (In) (1) Isoline(等值线) (2) Direction of principal stress (3) The value of maximum shear stress Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 b P0 A Χ B β2 β1 θ ψ/2 ψ Μ σ3 Ζ σ1 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.4.4 Stresses due to Linearly increasing vertical loading on an infinite Strip Y ∝ b ξP0 b -X dξ P0 Ο X dξ ξ Ζ Χ-ξ σΖ X Z Μ( X , Z ) Civil Engineering Department of Shanghai University d z 2Z 3 X Z 2 2Z 3 p0 z b p0 X b p0 b 2 2 p0d b d X 0 Soil Mechanics Chapter 2 2 Z 2 2 X X b Z X b arctg arctg 2 2 Z Z Z Z b Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.4.5 Stresses due to a uniform loaded rectangular area -central point method（中点法） a/2 a/2 dn r b/ 2 o dξ n b/ 2 ξ R z σΖ Y m( x.y.z ) z x Civil Engineering Department of Shanghai University R Z 2 2 Soil Mechanics Chapter 2 2 12 3Z 3 p 0 dd z a 2 b 2 2R 5 2 p0 ab 2abZ a 2 b 2 8Z 2 arctg 2Z a 2 b 2 4Z 2 a 2 4Z 2 b 2 4Z 2 a 2 b 2 4Z 2 a0 p0 a 2 b2 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 -Corner method（角点法） a/2 a/2 po b/ 2 ξ dξ R dn n b/ 2 x POd o C po z σΖ Y M( x.y.z ) z 3Z 3 p0 dd Z a / 2 b / 2 2R 5 p0 abZ a 2 b 2 2 Z 2 ab arctg 2 2 2 2 2 2 2 2 2 2 2 a Z b Z a b Z Z a b Z c p0 a/2 b/2 mn(m 2 2n 2 1) m arctg 2 2 2 2 2 2 2 n m n 1 (m n )(1 n ) m n 1 a z Where, m , n b b 1 c 2 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Principle of superposition For stresses beneath points other than the corner of the loaded area the principle of superposition should be used, as described in Fig below. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Stresses beneath flexible area of any shape Stresses beneath flexible area of any shape (Figure 5.9) Newmark (1942) devised charts to obtain the vertical stress at any depth, beneath any point (inside or outside) of an irregular shape. Use of the charts is explained in Figure 2.9. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Stresses beneath a flexible rectangle ♦ Stresses beneath a flexible rectangle (figure 2.7) The vertical stressσ y at a depth z beneath the corner of a flexible rectangle supporting a uniform pressure q has been determined using: σ v= qI and influence factors I. given by Giroud (1970) are presented in Figure 5.7. they are for an infinite soil thickness. These curves are equivalent to the commonly used charts of Fadum (1948) but are easier to use. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Stresses beneath a rigid rectangle Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Exercise 2-4 A point load of P is applied at the ground surface. Calculate the stress and plot the variation: (1) Vertical stress due to load P along vertical line(r=0) at the different depths Z=1,2,3,4,5m. (2) Vertical stress due to load P from different distances (r=0,1,2,3,4,5m, respectively) along horizontal surface (at the depth Z=2m). (3) Vertical stress due to load P along vertical line(r=1.5m) at the different depths Z=0,0.5,1,2,3,4,5,6m. Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Exercise 2-5 A uniform load of p(=20kpa) is applied at the ground surface as shown in Fig below. Calculate the stress due to load p beneath point A at 5m depth.