39 - NPTELStress Isobar (or Pressure bulb) Stress contour or a line which connects all points below...

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

39


Module 3:

Lecture -1 on Compressibility and Consolidation


Stresses in soil from surface loads; Terzaghi’s 1-D consolidation theory; Application in different boundary conditions; Ramp loading; Determination of Coefficient of consolidation; Normally and Over-consolidated soils; Compression curves; Secondary consolidation; Radial consolidation; Settlement of compressible soil layers and Methods for accelerating consolidation settlements.

Contents


Stresses in soil from surface loads

An important function in the study of soil mechanics isto predict the stresses and strains imposed at a givenpoint in a soil mass due to certain loading conditions.

This helps to estimate settlement and to conductstability analysis of earth and earth-retainingstructures, as well as to determine stress conditions onunderground and earth-retaining structures


Idealized stress-strain diagram

Stre

ssStrain

Elastic Plastic

Idealized stress-strain diagram

a

b c

At low stress levels the strainincreases linearly with stress(branch ab), which is the elasticrange of the material.

Beyond a certain stress level thematerial reaches a plastic state,and the strain increases with nofurther increase in stress(branch bc).


dxdy

x

y

z

dz

σxxτxyτxz

σzz

τzyτzx

σyyτyx

τyz

Normal and shear stresses in Cartesian coordinate system

An elemental soil mass with sides measuring dx, dy, and dz: Parameters σxx σyy, and σzz

are the normal stressesacting on the planes normalto the x, y, and z axes (Considered positive when they

are directed onto the surface)(If τij is a shear stress, it means thestress is acting on a plane normal to thei axis, and its direction is parallel to thej axis.)

For equilibrium:


Equations of static equilibrium

dxdy

x

y

z

dz

τxy +


Equations of static equilibriumAlong x-direction:

Along y-direction:


Equations of static equilibrium

The last term of the preceding equation is the self-weight of the soil mass.

Thus,

Along z-direction:

These equations are written in terms of total stresses


Equations of static equilibrium (in-terms of effective stresses)

Thus,

where is the γ′ submergedunit weight of soil.

Note that the shearstresses will not be affectedby the pore water pressure.


Equations of static equilibrium (2-D)In soil mechanics, a number of problems can be solved by two dimensional stress analysis.

For a weight-less medium (i.e., γ = 0) the equationsof equilibrium are:


Idealization of the stress–strain relationship

Typical stress–strain relationship

Linearly elastic behaviourbeing assumed between Oand Y′ (the assumed yieldpoint) followed byunrestricted plastic strain (orflow) Y′P at constant stress

⇒ In general, soils are non-homogeneous, exhibit anisotropyand have non-linear stress–strain relationships which aredependent on stress history and the particular stress pathfollowed.


Elastic–perfectly plastic model

Non-hardening behaviour


Rigid–perfectly plastic model

If only the collapse condition in a practical problemis of interest then the elastic phase can be omittedand the rigid–perfectly plastic model

Non-hardening behaviour


Elastic–strain hardening and softening plastic models

In which plastic strain beyond the yield point necessitates further stress increase.

If unloading and reloadingwere to take placesubsequent to yielding inthe strain hardening model(i.e., at stress at new yieldpoint Y′′ > Y′)

An increase in yield stress is acharacteristic of strain hardening. A further idealization is theelastic–strain softening plasticmodel, represented by OY′P′

The plasticstrain beyond the yield point is accompanied by stress decrease.


In practice the most widely used solutions are those forthe vertical stress at a point below a loaded area onthe surface of a soil mass.

The vertical stress increment at a given point belowthe surface due to foundation loading is insensitive toa relatively wide range of soil characteristics such as:

heterogeneity,

anisotropy and

non-linearity of the stress–strain relationship.

Stresses in soil from surface loads


Stresses in soil from surface loads Accordingly, solutions from linear elastic theory, in

which the soil is assumed to be homogeneous andisotropic, are sufficiently accurate for use in mostcases.

The main exceptions are loose sands and soft clays,particularly where they are overlain by a relativelydense or stiff stratum.

However, that increments of horizontal stress and ofshear stress are relatively sensitive to soilcharacteristics


Stresses in soil from surface loads Displacement solutions from elastic theory can be usedat relatively low stress levels.

Requires a knowledge of the values of Young’s modulus(E) and Poisson’s ratio (ν) for the soil, either for un-drainedconditions or in terms of effective stress.It should be noted that the shear modulus (G), where

G is independent of the drainage conditions, assumingthat the soil is isotropic.


Stresses in soil from surface loads The volumetric strain of an element of linearly elastic material under three principal stresses is given by:

If this expression is applied to soils over the initial part of the stress-strain curve, then for un-drained conditions ∆V/V = 0, hence ν =0.5

If consolidation takes place then ∆V/V > 0 and ν < 0:5 for drainedor partially drained conditions.

(E = 3G)


Stresses in soil from surface loads The stresses within a semi-infinite, homogeneous, isotropicmass, with a linear stress-strain relationship, due to a pointload on the surface, were determined by Boussinesq in1885. The stresses due to surface loads distributed over a particular area

can be obtained by integration from the point load solutions.

The stresses at a point due to more than one surface load areobtained by superposition.

In practice, loads are not usually applied directly on the surface butthe results for surface loading can be applied conservatively inproblems concerning loads at a shallow depth.


Stresses due to point load

Variation of vertical stressdue to point load ⇒

Variationof σz with z on the vertical through the point of application of the load Q

Variation of σz with r for three different values of z

Bell shapede.g., Vertical load transferred to the soil from an electrical power line


Stresses due to point loadStresses at X due to a point load Q on the surface are as follows:

It should be noted that when v = 0.5, σθ = 0


Stresses due to point load

can be written as:

⇒Expression for σz is independent of elastic modulus(E) and Poisson’s ratio (ν).


Influence factors for vertical stress due to point load

As r/z ↑ Ip ↓ σz ∞


Vertical stress distribution along a vertical line

As z increases, r/z ↓ for a constant value of rAs z2 is involved in the denominator of theexpression for σz, first it increases withdepth and attains a maximum value andthen decreases with further increase indepth.

Q

θ = 39°13′50′′

σz, max =

zr 0.0888Q


Stress Isobar (or Pressure bulb) Stress contour or a line which connects all points

below the ground surface at which the verticalpressure is the same.

Pressure at points inside the bulb are greater thanthat at a point on the surface of the bulb; andpressures at points outside the bulb are smaller thanthat value.

Any number of stress isobars can be drawn for anyapplied load.

A system of isobars indicates the decrease in stressintensity from the inner to outer ones.

Isobars are Leminscate curves


Procedure for plotting Isobars

For example, σz = 0.1Q per unit area (10% Isobar)

= 0.1Z2

Z IP r/z r σz0.5 0.025 1.501 0.75 0.1Q

1 0.1 0.9332 0.832 0.1Q

1.5 0.255 0.593 0.890 0.1Q

2 0.40 0.271 0.542 0.1Q

2.185 0.4775 0 0 0.1Q


A single concentrated load of 1000 kN acts at theground surface. Construct an isobar for σz = 40 kN/m2

by making use of the Boussinesq equation.

Example Problem

Now for Q = 1000 kN, σz, = 40 kN/m2 , we obtain thevalues of r1 r2, r3 etc. for different depths z1,, z2, z3 etc.The values so obtained are:


Example Problem


Westergard stress distribution under a point load

Boussinesq assumed that the soil is elastic, isotropicand homogeneous; However, the soil is neitherisotropic nor homogeneous. The most common typeof soils that are met in nature are the water depositedsedimentary soils

The soils of this type can be assumed as laterallyreinforced by numerous, closely spaced, horizontalsheets of negligible thickness but of infinite rigidity,which prevent the mass as a whole from undergoinglateral movement of soil grains (For this case ν or µ = 0)


Westergard stress distribution under a point load

(For this case ν or µ = 0)

For r/z = 0; Iw = 1/π = 0.3183


Values of IB or Iw for use inthe Boussinesq orWestergaard equation

The value of Iw=0.3183 at r/z = 0 (Which is 33% less than Boussinesq IP)

Geotechnical engineersprefer to useBoussinesq's solution asthis gives conservativeresults.


Vertical stress due to a line loade.g, long brick wall or railroad track

x

z

y ∞

-∞

Series of point loads (~qdy)(From Boussinesq’s solution)


Vertical stress due to a line load

At x/z = 0 σz = 0.6366q/Z

can be used to estimate the lateralpressure on an earth-retaining structuredue to a line load on the surface of thebackfill.


Vertical stress due to a line load

However, the structure will tend tointerfere with the lateral strain due to theload q and to obtain the lateral pressureon a relatively rigid structure a secondload q must be imagined at an equaldistance on the other side of thestructure.Then, the lateral pressure is given by

The total thrust on the structure is given by:


Vertical stress due to a Strip load

A strip load is the loadtransmitted by a structureof finite width and infinitelylength of the soil surface.

qs (uniformly applied stress)

qs

39 - NPTELStress Isobar (or Pressure bulb) Stress contour or a line which connects all points below...

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