Soil mechanics – a short introduction Hillel, pp. 341-382 Soil mechanics – a short introduction Picture: Tennessee Department of Transportation Picture: Ch. Salm, Terre AG

Why mechanics of soils? Force Like other solid materials (e.g. metals, rock), soils deform when they are exposed to forces. Force Unlike many other materials in our environment , soils show a wide range of possible mechanical behavior which influences considerably their use for …

Why mechanics of soils? foundation of buildings… …or agricultural production

Why mechanics of soils? Understanding soil deformation behavior is crucial to: design slopes and retaining walls build tunnels in ‘soft’ rock assess hazards due to land slides prevent soil from compaction optimize soil management techniques …

Definition of stress and strain The reaction of a solid body to a force F or a combination of forces acting upon or within it can be characterized in terms of its relative deformation or strain. The ratio of force to area where it acts is called stress. normal stress s = Fn / A shear stress t = Fs / A                                 normal strain e = dz / zo shear strain g = dh / zo                           Note that compressive stresses and strains are positive and counter-clockwise shear stresses and strains are positive.

Total vs. effective stresses When a load is applied to soil, it is carried by the water in the pores as well as the solid grains. The increase in pressure within the pore water causes drainage (flow out of the soil), and the load is transferred to the solid grains. The rate of drainage depends on the permeability of the soil. The strength and compressibility of the soil depend on the stresses within the solid granular fabric. These are called effective stresses.

Stress in homogeneous soil The total vertical stress acting on a soil element below the ground surface is due to the weight of everything lying above: soil, water, and surface loading. In a homogeneous soil, the total vertical stress sv on an element with distance z from the surface is determined by the weight of the overlying soil and can be calculated as: With rb the soil bulk density, Qm the gravimetric water content and g the gravity constant. Typical values of rb are 1000 – 1800 kg m-3.

Stress in homogeneous soil Any change in total vertical stress sv may also result in a change of total horizontal stress sh on the same soil element. There is no simple relationship between horizontal and vertical stress. In a homogeneous soil, the total horizontal stress sh on an element with distance z from the surface can be estimated as : sv is the vertical stress and n soil Poisson’s ration. Typical values for Poisson’s ratio are between 0.25 and 0.4. For practical purposes a ratio of sh /sv = 0.5 provides a good first estimation.

Stress in a multi-layer soil The total stress sv at depth z is the sum of the weights of soil in each layer above. For example the total vertical stress sv at a depth z in layer 3 is where the bulk density of the layers 1 to 3 the gravimetric water content of the layers 1 to 3 the thickness of the layers 1 to 3

Stress in soil with a ‘wide’ surface load The addition of a surface load will increase the total stresses below it. If the surcharge loading is extensively wide, the increase in vertical total stress below it may be considered constant with depth and equal to the magnitude of the surcharge q. The vertical total stress at depth z under a wide load q becomes then

Stress in soil with a ‘narrow’ surface load For narrow loads, e.g. stresses at the soil surface under a strip footing or a wheel, the induced total vertical stresses will decrease both with depth and horizontal distance from the center of the load. In such cases, it is necessary to use a suitable model to estimate the stress distribution in the soil under the surface load (Boussinesq (1885), Froehlich (1934) ). For a vertical load q, homogeneously distributed over a circular area of radius R, the vertical stress sv,q(z) in depth z in the soil can be calculated as R q z sv,q

Stress in soil with a ‘narrow’ surface load The total vertical stress sv in the depth z due to a homogeneous surface load q on a circular area and the overlying soil can therefore be calculated as with rb the soil bulk density, Qm the gravimetric water content, g the gravity constant, q the surface load and R the radius of the contact area.

Uniaxial stress and strain – Hook’s law Steel wire Young’s modulus  Poisson's ratio  sa Young's modulus and Poisson's ratio are measured directly in uniaxial compression or extension tests, i.e. tests with constant (or zero) stress on the horizontal surfaces. Leonardo da Vinci’s (1452-1519) uniaxial tension test

Shear stress and strain As the shear stress t’ increases materials distort (change shape). This change in shape can be expressed as an angular shear strain g. The shear modulus G' relates the change in shear stress dt’ to the change in shear strain dg. Shear modulus 

Isotropic compression As the isotropic stress s’ increases, materials compress (reduce in volume). The bulk modulus K' relates the change in volumetric strain dev=dV/V to the change in isotropic stress s’. Bulk modulus 

Stiffness of soil material The relationship between a strain and stress is termed stiffness OA: linear and recoverable ABC: non-linear and irrecoverable BCD: recoverable with hysteresis DE: continuous shearing The stress-strain curve of a soil has features which are characteristic for different material behavior. Soils show elastic, plastic and viscous deformation when exposed to stresses.

Elastic deformation In linear-elastic behavior (OA) the stress-strain is a straight line and strains are fully recovered on unloading, i.e. there is no hysteresis. The elastic parameters are the gradients of the appropriate stress-strain curves and are constant. Young’s modulus  Poisson's ratio  Shear modulus  Bulk modulus 

Typical values of elastic moduli E’ and n’ Typical E’ Unweathered overconsolidated clays 20 ~ 50 MPa Boulder clay 10 ~ 20 MPa Keuper Marl (unweathered) >150 MPa Keuper Marl (moderately weathered) 30 ~ 150 MPa Weathered overconsolidated clays 3 ~ 10 MPa Organic alluvial clays and peats 0.1 ~ 0.6 MPa Normally consolidated clays 0.2 ~ 4 MPa Steel 205 MPa Concrete 30 MPa   Typical n’ Soil 0.25-0.4 Rock 0.3 Steel 0.28 Concrete 0.17

Relationships between elastic moduli In bodies of isotropic elastic material the three stiffness moduli E', K' and G' and Poisson’s ratio (n') are related as: Therefore the deformation behavior of an isotropic elastic material can be described by only two material constants.

Plastic deformation With increasing stress the material behavior goes over from elastic to plastic. This transition is called yield (A). Plastic strains (AB) are not recovered on unloading (BC). Unloading (BC) and reloading (CD) show a hysteresis. With increasing strain (at constant stress) the material eventually fails if brittle or flows if ductile (E). Soils material behavior is often simplified as elastic-perfectly plastic. During perfectly plastic straining (AB), plastic strains continue indefinitely at constant stress. In a brittle perfectly plastic material, the yield stress at point A this is the same as the failure stress at a point B. yield

Viscous deformation Change in volume and shape of soils are generally time-dependent. One way to capture this time-dependency is to model soil as a viscous solids. For the case of simple shear for example, this means that the shear stress t’ is proportional to the shear strain rate dg/dt. The viscosity h relates the change in shear stress dt’ to the change in shear strain rate dg/dt. Shear strain rate Shear stress . Viscosity Linear viscosity

Soil strength – precompression stress The yield stress of a soil under compression is called precompression stress. It separates the stress ranges where elastic and plastic deformation can be expected. If a soil is not loaded above the precompression stress, deformation is elastic (no irrecoverable deformation) and rather small. elastic Precompression stress Pv If a soil is loaded above the precompression stress, deformation is large and plastic ( irrecoverable deformation). plastic

Soil strength – undrained shear The maximum value of stress that may be sustained by a material is termed strength. The strength is independent of the normal stress  since the response to loading simple increases the pore water pressure and not the effective stress. The shear strength tf is a material parameter which is known as the undrained shear strength su. tf = (sa - sr) = constant

Soil strength – the angle of friction The strength increases linearly with increasing normal stress and is zero when the normal stress is zero. t'f = s'n tanf' f' is the angle of friction In the Mohr-Coulomb criterion the material parameter is the angle of friction f’ and materials which meet this criterion are known as frictional. In soils, the Mohr-Coulomb criterion applies when the normal stress is an effective normal stress.

Soil strength - cohesion .. The strength increases linearly with increasing normal stress and is positive when the normal stress is zero. t'f = c' + s'n tanf' f' is the angle of friction c' is the 'cohesion' intercept In soils, the Mohr-Coulomb criterion applies when the normal stress is an effective normal stress.

Typical values of f‘, c’ and su Undrained shear strength Hard soil su > 150 kPa Stiff soil su = 75 ~ 150 kPa Firm soil su = 40 ~ 75 kPa Soft soil su = 20 ~ 40kPa Very soft soil su < 20 kPa Drained shear strength c´ (kPa) f´ (deg) Sands 30° - 45° Clays 0 - 30 kPa 0 - 20° Precompression stress Pv soft 0-50 kPa firm 50-150 kPa stiff > 150 kPa

Summary The aim of this class was to introduce the concept of stress and strain in solid material, especially soils. give you an idea what stiffness and strength of soil materials are and provide you a short overview over the mechanics of one of the most complex solid materials known.

Acknowledgment I would like to acknowledge Prof. John Atkinson, City University, London, and Prof. Sarah Springman, ETH, Zurich, who provided me with many of the graphs I used for this presentation, Prof. Dani Or, UConn, Storrs, who gave me the opportunity to talk about soil mechanics and you, the audience, for coming and staying with me.