Shear strength (soil)

Shear strength in reference to soil is a term used to describe the maximum strength of soil at which point significant plastic deformation or yielding occurs due to an applied shear stress. There is no definitive "shear strength" of a soil as it depends on a number of factors affecting the soil at any given time and on the frame of reference, in particular the rate at which the shearing occurs.

Two theories are commonly used to estimate the shear strength of a soil depending on the rate of shearing as a frame of reference. These are Tresca theory for short term loading of a soil, commonly referred to as the undrained strength or the total stress condition and Mohr–Coulomb theory combined with the principle of effective stress for the long term loading of a soil, commonly referred to as the drained strength or the effective stress condition.

In modern soil mechanics, both these classical approaches (Tresca and Mohr–Coulomb) may be superseded by critical state theory or by steady state theory either of which can be considered in both undrained and drained terms and also cases involving partial drainage. The classical approaches are still in common usage; however, both are taught in undergraduate civil engineering programmes, and consequently, they are also used in practice.

Shear strength of a soil is also of importance in designing for earthquakes where the concept of the soil's steady state shear strength is used.

The shear strength of soils is taught in detail in specialist masters degree programs. Such programs usually include the use of modern numerical modeling techniques such as finite element analysis coupled with a model for shear strength such as critical state soil mechanics.

Factors Controlling Shear Strength of Soils

The stress-strain relationship of soils, and therefore the shearing strength, is affected Harv|Poulos|1989 by:

# soil composition (basic soil material): mineralogy, grain size and grain size distribution, shape of particles, pore fluid type and content, ions on grain and in pore fluid.
# state (initial): Define by the initial void ratio, effective normal stress and shear stress (stress history). State can be describe by terms such as: loose, dense, overconsolidated, normally consolidated, stiff, soft, contractive, dilative, etc.
# structure: Refers to the arrangement of particles within the soil mass; the manner the particles are packed or distributed. Features such as layers, joints, fissures, slickensides, voids, pockets, cementation, etc, are part of the structure. Structure of soils is described by terms such as: undisturbed, disturbed, remolded, compacted, cemented; flocculent, honey-combed, single-grained; flocculated, deflocculated; stratified, layered, laminated; isotropic and anisotropic.
# Loading conditions: Effective stress path, i.e., drained, and undrained; and type of loading, i.e., magnitude, rate (static, dynamic), and time history (monotonic, cyclic)).

Undrained strength

This term describes a type of shear strength in soil mechanics as distinct from drained strength.

Conceptually, there is no such thing as "the" undrained strength of a soil. It depends on a number of factors, the main ones being:
*Orientation of stresses
*Stress path
*Rate of shearing
*Volume of material (like for fissured clays or rock mass)

Undrained strength is typically defined by Tresca theory, based on Mohr's circle as:

"σ₁ - σ₃ = 2 S_u"

Where:

"σ₁" is the major principal stress

"σ₃" is the minor principal stress

$au$ is the shear strength "(σ₁ - σ₃)/2"

hence, $au$ = "S_u" (or sometimes "c_u"), the undrained strength.

It is commonly adopted in limit equilibrium analyses where the rate of loading is very much greater than the rate at which pore water pressures, that are generated due to the action of shearing the soil, may dissipate. An example of this is rapid loading of sands during an earthquake, or the failure of a clay slope during heavy rain, and applies to most failures that occur during construction.

As an implication of undrained condition, no elastic volumetric strains occur, and thus Poisson's ratio is assumed to remain 0.5 throughout shearing. The Tresca soil model also assumes no plastic volumetric strains occur. This is of significance in more advanced analyses such as in finite element analysis. In these advanced analysis methods, soil models other than Tresca may be used to model the undrained condition including Mohr-Coulomb and critical state soil models such as the modified Cam-clay model, provided Poisson's ratio is maintained at 0.5.

One important empirical relationship used extensively by practicing engineers is the empirical SHANSEP (stress history and normalized soil engineering properties) relationship.Harv|Ladd|Foott|1974 This is based on the observation that the logarithm of the undrained shear strength S_u normalized by the vertical consolidation stress σ_vc plots linearly against the logarithm of the over consolidation ratio or OCR, that is, S_u/σ_vc=K*OCR^N where K and N are constants that depend on the soil and the loading used to shear the soil. To date, no physical model has been proposed that explains this empirical observation.

Drained strength

This term describes a type of shear strength in soil mechanics as distinct from undrained strength.

The drained strength is the strength of the soil when pore water pressures, generated during the course of shearing the soil, are able to dissipate rapidly. It also applies where no pore water exists in the soil (the soil is dry). It is commonly defined using Mohr-Coulomb theory (it was called "Coulomb's equation" by Karl von Terzaghi in 1942 harv|Terzaghi|1942 combined with the principle of effective stress.

Drained strength is defined as:

$au$ = "σ' tan(φ') + c"'

Where "σ' =(σ - u)", known as the principle of effective stress. "σ" is the total stress applied normal to the shear plane, and "u" is the pore water pressure acting on the same plane.

"φ"' = the effective angle of shearing resistance. Formerly termed 'angle of internal friction' after Coulomb friction, where the coefficient of friction $mu$ is equal to tan(φ), which is proportional to the normal force on a plane but independent of its area. It is now regarded to have little to do with friction, and more to do with the micro-mechanical interaction of soil particles. It has sometimes been referred to as the "angle of repose" as a dry granular material will form a pile at this angle but no steeper. It is further described as either peak φ'_p, critical state φ'_cv or residual φ'_r. Note that φ'_p is only adopted in relation to Terzaghi's misunderstanding of the nature of "true" cohesion.harv|Schofield|1998 Nowadays, critical state φ'_cv values should be prescribed.

c' = apparent cohesion. Allows the soil to possess some shear strength at no confining stress, or even under tensile stress. Commonly ascribed to temporary negative pore water pressures (suction), that dissipate over time. It may also be due to diagenetic affects caused by soil aging such as chemical bonding, cementation of grains and the effects of creep; indeed Coulomb identified that soil possessed no cohesion when newly remoulded,Harvnb|Heyman|1972 as these diagenetic effects had been destroyed. When shear tests are conducted on an overconsolidated or dense soil, and peak strengths are plotted on a $au$/$sigma$ plot, it appears that cohesion exists as the y-intercept is non-zero. Some feel that this is not due to true cohesion, but is the effect of interlocking of particles.

In any case, the long term loading condition must rely on the soil properties expected to exist and contribute to the shear strength of the soil over the long term, and for these reasons it is generally not considered a reliable soil mechanical property unlike "φ"'.

Critical state theory

A more advanced understanding of the behaviour of soil undergoing shearing lead to the development of the critical state theory of soil mechanics harv|Roscoe|Schofield|Wroth|1958. In critical state soil mechanics, a distinct shear strength is identified where the soil undergoing shear does so at a constant volume, also called the 'critical state'. Thus there are three commonly identified shear strengths for a soil undergoing shear:

*Peak strength $au$_p
*Critical state or constant volume strength $au$_cv
*Residual strength $au$_r

The peak strength may occur before or at critical state, depending on the initial state of the soil particles being sheared:
*A loose soil will contract in volume on shearing, and may not develop any peak strength above critical state. In this case 'peak' strength will coincide with the critical state shear strength, once the soil has ceased contracting in volume. It may be stated that such soils do not exhibit a distinct 'peak strength'.
*A dense soil may contract slightly before granular interlock prevents further contraction (granular interlock is dependent on the shape of the grains and their initial packing arrangement). In order to continue shearing once granular interlock has occurred, the soil must dilate (expand in volume). As additional shear force is required to dilate the soil, a 'peak' strength occurs. Once this peak strength caused by dilation has been overcome through continued shearing, the resistance provided by the soil to the applied shear stress reduces (termed "strain softening"). Strain softening will continue until no further changes in volume of the soil occur on continued shearing. Peak strengths are also observed in overconsolidated clays where the natural fabric of the soil must be destroyed prior to reaching constant volume shearing. Other affects that result in peak strengths include cementation and bonding of particles.

The constant volume (or critical state) shear strength is said to be intrinsic to the soil, and independent of the initial density or packing arrangement of the soil grains. In this state the grains being sheared are said to be 'tumbling' over one another, with no significant granular interlock or sliding plane development affecting the resistance to shearing. At this point, no inherited fabric or bonding of the soil grains affects the soil strength.

The residual strength occurs for some soils where the shape of the particles that make up the soil become aligned during shearing (forming a slickenside), resulting in reduced resistance to continued shearing (further strain softening). This is particularly true for most clays that comprise plate-like minerals, but is also observed in some granular soils with more elongate shaped grains. Clays that do not have plate-like minerals (like allophanic clays) do not tend to exhibit residual strengths.

Use in practice: If one is to adopt critical state theory and take c' = 0; $au$_p may be used, provided the level of anticipated strains are taken into account, and the effects of potential rupture or strain softening to critical state strengths are considered. For large strain deformation, the potential to form slickensided surface with a φ'_r should be considered (such as pile driving).

The Critical State occurs at the quasi-static strain rate. It does not allow for differences in shear strength based on different strain rates. Also at the critical state, there is no particle alignment or specific soil structure.

Steady state theory

The steady state strength is defined as the shear strength of the soil when it is at the steady state condition. The steady state condition is defined as "that state in which the mass is continuously deforming at constant volume, constant normal effective stress, constant shear stress, and constant velocity." Harv|Poulos|1981 Steve Poulos built off a hypothesis that Arthur Casagrande was formulating towards the end of his career.Harv|Poulos|1981 Steady state based soil mechanics is sometimes called "Harvard soil mechanics".

The steady state occurs only after all particle breakage if any is complete and all the particles are oriented in a statistically steady state condition and so that the shear stress needed to continue deformation at a constant velocity of deformation does not change.

The steady state has a slightly different value depending on the strain rate at which it is measured. Thus the steady state shear strength at the quasi-static strain rate (the strain rate at which the critical state is defined to occur at) would seem to correspond to the critical state shear strength. However there is an additional difference between the two states. This is that at the steady state condition the grains align in the direction of shear, whereas no such oriented structure occurs for the critical state. In this sense the steady state corresponds to the "residual" condition.

The steady state applies to both the drained condition and to the undrained condition. Geotechnical engineers use the steady state shear strength when designing for earthquakes.

Harvard University's soil mechanics department shut down in 1970 after Casagrande retired but even today, the concept of the steady state condition remains powerfully relevant, a reflection of the stagnation of research in soil shear.

A primer on the Steady State theory can be found in a report by Poulos Harv|Poulos|1971. Its use in earthquake engineering is described in detail in another publication by Poulos Harv|Poulos|1989.

For a good review of "Harvard Soil Mechanics" and a comparison between the critical state and the steady state see [http://soilmechanics.wordpress.com/2007/10/22/disturbed-soil-properties-and-geotechnical-design/ this review] of Prof. Andrew Schofield's book "Disturbed Soil Properties and Geotechnical Design"

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