Shear Strength Theory

CIVL 354 – Soil Mechanics – Classroom Notes

Introduction

• The shear strength theory is first presented by Mohr (1900), by a theory for rupture in materials. The shear stress was previously defined as a linear function of normal stress by Coulomb (1776).

• Based on Mohr’s theory, there is a critical combination of normal stress and shear stress acting on a plane which creates failure at a point in a soil mass.

Introduction: why do we need to study shear strength ?

• The shear strength of soils are required for the analysis of soil stability problems such as; bearing capacity, slope stability and lateral pressure

• Leaning Tower of Pisa, Transcona Grain Elevator Collapse. (1913)

Introduction: why do we need to study shear strength ?

• Taiwan Landslide 2010.

Introduction: why do we need to study shear strength ?

• Singapore 2004.

Theory

• Any load applied to a soil mass will produce stresses of varying intensity and as a result deformations. The consequences of this mechanism requires consideration under two main topics;

– The possibility of SHEAR FAILURE in the soil.

– The magnitude of deformations (settlement or lateral displacement).

• Shear Failure: since soil is a particulate material, failure is primarily by rolling and slipping of grains (not via tension or compression alone) between two well-defined-by-nature planes or surfaces. These planes are called slip-planes. Soils fail along slip planes in a shearing mode.

Theory

• The shear strength of soils are mobilised by two main mechanisms; – Particle to particle bonding or interparticle friction.

• A siginificant contributor to shear resistance of cohesive soils.

– Particle to particle interlocking and/or interlocking of the soil particulate system. • A significant contributor to shear resistance of cohesionless soils.

• Some possible shear failure modes requiring shear strength evaluation,

Theory

• The shear strength is not a unique value for a soil type but changes significantly with the following factors:

– Soil state (void ratio, grain size, grain shape).

– Soil type (clay, silt, sand, gravel etc.).

– Water content (particularly for clay soils).

– Loading type and rate (rapid loading saturated cohesive soils creates excess pore water pressure and hence shearing takes place in undrained condition).

– Anisotropy (strength normal to bedding plane may be different from that of the parallel).

Theory – two dimensional stress at a point

• Consider the soil element shown in the following sketches, which provide free body diagram of a soil element. The resulting equations developed by Mohr (1882), presents a graphical means of obtaining stresses at a point.

Theory – two dimensional stress at a point

• Assuming that the soil element is 1 unit in thickness, and the length AB is also 1 unit. s1 and s3 are principal stresses acting on principal planes BC and AC respectively at element faces. AB can be considered to be the failure plane inclined with an angle of q corresponding to the horizontal.

Theory – Mohr Circle of Stresses

• Rewriting the above to solve for shear stress and normal stress at a s1 and s3 applied to the soil element gives,

• These are the parametric equations of a circle of stress in the XY plane known as a Mohr-Diagram (created by Otto Mohr in 1882) to obtain the stresses at a point by graphical means.

Theory – Mohr-Coulomb Failure Criterion

• The shear strength of a soil (tf) at a point on a particular slip plane was originally expressed by Coulomb (in the late 1700s) as a linear function of the normal stresses (sf) on the plane at the same point:

𝜏𝑓 = 𝑐 + 𝜎𝑓 tan ∅

where, 𝑐 and ∅ are cohesion intercept and angle of shearing resistance (or internal angle of friction), respectively.

• According to Terzaghi’s fundamental concept that shear stress in a soil can be resisted only by the solid grains (i.e. skeleton of solid particles), shear strength is expressed as a function of effective normal stress:

𝜏𝑓 = 𝑐 + 𝜎𝑓 tan ∅

• When Mohr Circle is applied to represent the state of stress, then tf will be a linearly increasing line tangent to the Mohr Circle at the point where a critical combination of shear stress and effective normal stress develops, also called point of failure.

Theory - Mohr-Coulomb Failure Criterion

• If a number of stress states producing shear failure in soil are known, these can be represented using Mohr Circles, from which a failure envelope can be drawn by a common tangent to these Mohr Circles.

Laboratory Determination of Shear Strength Parameters

• The shear strength parameters based on the Mohr-Coulomb Failure criteria are;

c’: cohesion, f’: internal angle of friction

• The shear strength in cohesive soils is mobilised in undrained condition immediately after the loading, which is termed as undrained shear strength, cu. In the long term, as the effective stress increases with dissipation of excess pore water pressure, the shear strength mobilised converges to the drained shear strength, which can be represented by Mohr-Coulomb Theory using c’ and f’.

• For cohesionless soils, the shear strength mobilisation takes place in the drained state represented with Mohr-Coulomb Theory using c’ and f’.

Laboratory Determination of Shear Strength Parameters

• The most common laboratory tests for the measurement of shear strength characteristics of soils are;

– Unconfined Compression test

– Vane Shear Test

– Direct Shear Box Test

– Triaxial Compression Test

Unconfined Compression Test (Uniaxial Compressive Strength test)

• Only applicable to cohesive soils.

• A cylindrical soil sample is loaded with a compressive force until failure is obtained.

• The force applied at failure is noted as well as the axial deformation of the sample.

P

qu = s1 = compressive stress at failure = P/Ac

where, Ac = area corrected = A/(1-ea) and, ea = DH / H0

Vane Shear Test

Vane Shear Test

• Only applicable to cohesive soils. The test can be applied in the laboratory on undisturbed samples or insitu.

• A cylindrical soil sample contained in a sampling mold is used.

• A vane with four blades is inserted into the sample and a torque is applied until failure is obtained. The applied torque is then equated to the undrained shear strength using circumferential shear stress developed along the shaft formed by the rotation of the vane.

Direct Shear Box Test

• Shear failure of the soil sample along a thin, pre-determined slip plane is measured. The test is carried out under a constant normal stress applied vertically to the sample. The bottom half of the box is moved laterally with respect to the top half, which is connected to a load ring for measurement of the shear forces developed along the slip plane.

Direct Shear Box Test

• During testing, the shear resistance, horizontal displacement and vertical displacement of the sample is measured.

Direct Shear Box Test

• The horizontal displacement measurement at failure is used to estimate the area of the slip plane, 𝐴𝑓, and considering the shear force at failure, 𝑇, one can also calculate the shear stress at failure as

𝜏𝑓 =𝑇

𝐴𝑓, and also, normal stress is calculated as, 𝜎𝑛 =

𝑁

𝐴𝑓

where, 𝐴𝑓 =𝐴0

1+∆𝑙𝑓

𝐿0

Carrying out the test three times under various normal stresses, the failure envelope can be plotted as;

∆𝑙𝑓 𝐿0

𝜏𝑓

𝜎𝑛

∅

𝑐

Direct Shear Box Test

• Typical behaviour obtained in the test results are plotted in the following figures,

Triaxial Test

• Cassagrande (1930), was the first the develop triaxial testing equipment which is capable of testing soils in three dimensional stress state. Detailed testing methods were later published by Bishop and Henkel (1962). Triaxial test is widely used as it is suitable for all types of soils and have the advantage of simulating the insitu sress state best compared to other shear strength tests.

• In the most common types of triaxial test, a cylindrical soil sample is loaded with an all round pressure (confining stress) to simulate the insitu effective stress, and once the insitu stress state is attained the shear strength behaviour is tested by applying a compressive stress. The test can be carried out by an effective control of the drainage conditions, hence both undrained and drained behaviour can be measured. During the test, pore water pressure and/or volume change measurements can be carried out.

Triaxial Test

s3: All-round pressure

Ds: Deviator stress (compressive stress)

s1: major principal stress = s3 + Ds

Triaxial Test

• In a triaxial compression test, the all-round pressure applied is the minor principal stress s3, and the major principal stress then becomes the total of the all-round pressure and the compressive stress applied;

𝜎2 = 𝜎3 in a standard triaxial test 𝜎1 = 𝜎3 + ∆𝜎

• The maximum shear stress at failure occurs along the weakest plane in the sample, which is inclined with an angle of from the horizontal, where = 45 + f/2 (slip plane, failure plane). If the test is carried out three times under various all-round pressures, then a failure envelope can be drawn by the help of Mohr Circles.

• In the triaxial test, the drainage conditions can be controlled. Hence, if drainage is allowed during the test, i.e. the test is carried out without allowing for excess pore water pressures to develop, then the effective shear strength parameters can be measured.

Triaxial Test

• The following standard triaxial compression tests can be carried out.

Unconsolidated Undrained (UU, quick shear test) • All round pressure is

applied. • Then the axial compressive

stress is applied without allowing drainage.

• Pore water pressure is not measured.

• Undrained shear strength, cu, is measured.

𝐴𝑓 =𝐴0

1 −∆𝑯𝑓

𝑯0

Consolidated Undrained (CU) • In Stage 1, all round

pressure is applied. • Drainage is allowed to

enable consolidation of the sample.

• In Stage 2, drainage lines are closed and axial compressive stress is applied in the undrained condition.

• Pore water pressure measurements are recorded.

• Both, total and effective stress parameters are measured, i.e. cu, fu and c, f .

Stage 1 requires volume and axial deformation correction, Stage 2 requires axial deformation correction only.

Consolidated Drained (CD) • Stage 1 same as CU test, • However, in Stage 2,

drainage lines are kept open, and the axial compression is applied at a slow rate such that excess pore water pressures are not developed.

• effective stress parameters are measured, c, and f.

𝐴𝑓 =𝐴0 1 −

∆𝑽𝑓𝑽0

1−∆𝑯𝑓

𝑯0

Triaxial Test

• Typical behaviour measured in standard triaxial compression tests are presented in the following.

Triaxial Test

• Typical behaviour measured in standard triaxial compression tests are presented in the following.

Triaxial Test

Triaxial Test

Triaxial Test

References • R.F. Craig, “Soil mechanics”.

• B.M. Das, “Fundamentals of Geotechnical Engineering”.