soil mechanics -I (chapter five ).pdf

ONE- DIMENSIONAL CONSOLIDATION
SETTLEMENT OF FINE GRAINED SOILS
CHAPTER five
AMBO UNIVERSITY
Soil Mechanics- I
Lecture Note

1. Introduction
 Load applied to soil from super structure through foundation
develops stress in the soil, cause compression of soil (volume
change).
 The contact pressure is not uniform, non- uniform compression is
occur, that causes tilting, example is leaning tower of Pisa, Italy.
 Under load, all soils will settle, causing settlement of structures
founded on or within them, related to displacement structures
undergo.
 If the settlement is not kept to a tolerable limit, the desired use of
the structure may be impaired and the design life of the structure
may be reduced.
 Structures may settle :
 uniformly or
 Non- uniformly : is called differential settlement
and is often the crucial design consideration.
 The total settlement usually consists of three parts
 immediate or elastic compression
 primary consolidation
 secondary compression

2. Basic Concepts
 Assumption on consolidation settlement development are:
A homogeneous, saturated soil.
The soil particles and the water to be incompressible.
Vertical flow of water.
The validity of Darcy’s law.
Small strains.
 conduct a simple experiment to establish the basic concepts of the
1D- consolidation settlement of fine-grained soils.
 Let us take a thin, soft, saturated sample of clay and place it
between porous stones in a rigid, cylindrical container whose
inside wall is frictionless (figure below. Next slide)
 The porous stones facilitate drainage of the pore water from the
top and bottom faces of the soil.
 The top half of the soil will drain through the top porous stone
and the bottom half will drain through the bottom porous stone.

2. Basic Concepts
 A platen on the top of the top porous stone transmits applied loads to
the soil.
 Expelled water is transported by plastic tubes to a burette.
 A valve is used to control the flow of the expelled water into the
burette.
Fig. Experimental setup for illustrating basic concepts on
consolidation.

 Three pore water transducers are mounted in the side wall of the
cylinder to measure the excess pore water pressure near the porous
stone at the top (A), at a distance of one-quarter the height (B), and
at the mid-height of the soil (C).
 A displacement gauge with its stem on the platen keeps track of the
vertical settlement of the soil.
 We will assume that the pore water and the soil particles are
incompressible and the initial pore water pressure is zero.
 The volume of excess pore water that drains from the soil is then a
measure of the volume change of the soil resulting from the
applied loads.
 Since the side wall of the container is rigid, no radial displacement
can occur.
 The volume of excess pore water that drains from the soil is then a
measure of the volume change of the soil resulting from the
applied loads.
 Since the side wall of the container is rigid, no radial displacement
can occur.
2. Basic Concepts

 The lateral and the circumferential strains are then equal to zero
(𝜀𝑟= 𝜀𝜃 = 0) and the volumetric strain (𝜀𝑝 = 𝜀𝑧 + 𝜀𝑟 + 𝜀𝜃) is
equal to the vertical strain 𝜀𝑧 =
∆𝑧
𝐻𝑜
where ∆𝑧 is the change in
height or thickness and 𝐻𝑜 is the initial height or thickness of the
soil.
1. Instantaneous Load
2. Basic Concepts
fig. Instantaneous or initial excess pore water pressure when a vertical
load is applied

apply a load P to the soil through the load platen and keep the valve
closed.
no excess pore water can drain from the soil, the change in volume
of the soil is zero(∆𝑉 = 0)
 The pore water carries the total head i.e.∆𝑢𝑜 = ∆𝜎𝑧 =
𝑃
𝐴
or more
appropriately the change in mean total stress, ∆𝑃 =
∆𝜎𝑧+2∆𝜎𝑟
3
Where ∆𝑢𝑜 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑟𝑒 𝑤𝑎𝑡𝑒𝑟 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
∆𝜎𝑧 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠
∆𝜎𝑟 = 𝑐ℎ𝑎𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠
 For our thin soil layer, we will assume that the initial excess pore
water pressure will be distributed uniformly with depth so that at
every point in the soil layer, the initial excess pore water pressure
is equal to the applied stress. For example if the
∆𝜎𝑧 = 100𝑘𝑃𝑎, 𝑡ℎ𝑒𝑛 ∆𝑢𝑜 = 100𝑘𝑃𝑎
2. Basic Concepts

2. Consolidation Under a Constant Load: Primary Consolidation
 open the valve and allow the initial excess pore water to drain.
2. Basic Concepts
Excess pore water pressure distribution and settlement
during consolidation.

 The total volume of soil at time 𝑡1 decreases by the amount of
excess pore water that drains from it as indicated by the change in
volume of water in the burette.(fig. above)
 At the top and bottom of the soil sample, the excess pore water
pressure is zero because these are drainage boundaries.
 The decrease of initial excess pore pressure at the middle of the soil
(position C) is the slowest because a water particle must travel from
the middle of the soil to either the top or bottom boundary to exit the
system.
 Most of the settlement of soil ∆𝑧 with time 𝑡, occurs shortly after
the valve was opened and not linear.
 Before the valve opened, an initial hydraulic head(
∆𝑢𝑜
𝛾𝑤
) was created
by the applied vertical stress.
 When the valve opened, the initial excess pore water forced out of
the soil by this initial hydraulic head and with time, the initial
hydraulic decreases and, consequently, smaller amount of excess
pore water are forced out.
2. Basic Concepts

 We call the initial settlement response, soon after the valve was
opened, the early time response or primary consolidation.
 Primary consolidation is the change in volume of the soil caused by
the expulsion of water from the voids and the transfer of load from
the excess pore water pressure to the soil solid particles
3. Secondary Compression
 Primary consolidation ends with ∆𝑢 = 0
 The later time settlement response is called secondary compression
or creep.
 Secondary compression is the change in volume of fine-grained soils
caused by the adjustment of the soil fabric (internal structure) after
primary consolidation has been completed.
 The rate of settlement from secondary compression is very slow
compared with primary consolidation.
 In reality, the distinction is not clear because secondary compression
occurs as part of the primary consolidation phase especially in soft
clays.
2. Basic Concepts

4. Drainage Path
 It is the longest vertical path taken by water to exit the soil.
 If we allowed the soil to drain on the top and bottom faces (double
drainage) ,the length of the drainage path, 𝐻𝑑𝑟,is
𝑯𝒅𝒓 =
𝑯𝒂𝒗
𝟐
=
𝑯𝒐 + 𝑯𝒇
𝟒
where 𝐻𝑎𝑣 is the average height and 𝐻0 and 𝐻𝑓 are the initial and final
heights, respectively, under the current loading.
 If drainage were permitted from only one face of the soil, then
𝐻𝑑𝑟 = 𝐻𝑎𝑣
5. Rate of Consolidation
 The rate of consolidation for a homogeneous soil depends on the
soil’s permeability, the thickness, and the length of the drainage
path.
2. Basic Concepts

6. Effective Stress Changes
According to the principle of the effective stress:
∆𝝈𝒛
′
= ∆𝝈 − ∆𝒖
 Increases in vertical stresses lead to soil settlement caused
by changes to the soil fabric.
 As time increases, the initial excess water pressure
continues to dissipate and the soil continues to settle.
 After some time the initial excess pore water pressure in
the middle of the soil reduces to approximately zero and
the rate of decrease of the volume of the soil becomes very
small, and from the principle of effective stress all of the
applied vertical stress is transferred to the soil; i.e.
∆𝝈𝒛
′
= ∆𝝈𝒛
2. Basic Concepts

7. Void Ratio and Settlement Changes Under a Constant Load
 The initial volume (specific volume) of a soil is 𝑽 = 𝟏 + 𝒆𝒐 where
𝒆𝒐 =initial void ratio.
 Any change in volume of the soil ∆𝑽 is equal to the change in
void ratio ∆𝒆 .
 Volumetric strain 𝜀𝑝=
∆𝑉
𝑉
=
∆𝑒
1+𝑒𝑜
 For one-dimensional consolidation, 𝜺𝒛 = 𝜺𝒑
 A relationship between settlement and the change in the void
ratio can be written as 𝜺𝒛 =
∆𝒛
𝑯𝒐
=
∆𝒆
𝟏+𝒆𝒐
, ∆𝒛 = 𝑯𝒐 ∗
∆𝒆
𝟏+𝒆𝒐
 denote 𝝆𝒑𝒄 for primary consolidation settlement rather than
∆𝒛
𝝆𝒑𝒄= 𝑯𝒐 ∗
∆𝒆
𝟏 + 𝒆𝒐
2. Basic Concepts

 The void ratio at any time under load P is
𝑒 = 𝑒𝑜 − ∆𝑒 = 𝑒𝑜 −
∆𝑧
𝐻
1 + 𝑒𝑜 = 𝑒𝑜 1 −
∆𝑧
𝐻
−
∆𝑧
𝐻
 For a saturated soil,𝑒𝑜 = 𝑤𝐺𝑠 then, 𝑒 = 𝑤𝐺𝑠 1 −
∆𝑧
𝐻
−
∆𝑧
𝐻
2. Basic Concepts
8. Effects of Vertical Stresses on Primary Consolidation

 By applying additional loads to the soil and for each load increment
we can calculate the final void ratio from 𝑒 = 𝑒𝑜 1 −
∆𝑧
𝐻
−
∆𝑧
𝐻
and
plot the results as shown by segment AB in figure (a)
 The segment AB is called the virgin consolidation line or normal
consolidation line (NCL) & plotted from soil data test.
 At point where soil reaches equilibrium under the previous loading
step to the vertical effective stress, 𝜎′𝑧𝑐,let us unload the soil
incrementally.
 When an increment of load is removed, the soil will start to swell
by absorbing water from the burette.
 The void ratio increases but the increase is much less than the
decrease in void ratio from the same magnitude of loading that was
previously applied.
 Now let us reload the soil after unloading it to 𝜎′𝑧𝑐 .
 The reloading path CD is convex compared with the concave
unloading path, BC.
2. Basic Concepts

 We will represent the unloading-reloading path by an average
slope BC and refer to it as the recompression line or the
unloading-reloading line (URL).
 The path BC represents the elastic response while the path AB
represents the elastoplastic response of the soil.
 Loads that cause the soil to follow path BC will produce elastic
settlement.
 Loads that cause the soil to follow path AB will produce
settlements that have both elastic and plastic (permanent)
components.
 Unloading and reloading the soil at any subsequent vertical
effective stress would result in a soil’s response similar to paths
BCDE.
2. Basic Concepts

9. Primary Consolidation Parameters
 The primary consolidation settlement of the soil (settlement that
occurs along path AB in Figure can be expressed through the
slopes of the curves.
 Two slopes for primary consolidation are coefficient of
compression or compression index,
𝐂𝐜 = −
𝐞𝟐−𝐞𝟏
𝐥𝐨𝐠
𝝈𝒛𝟐
′
𝝈𝒛𝟏
′
=
∆𝒆
𝐥𝐨𝐠
𝝈𝒛𝟐
′
𝝈𝒛𝟏
′
and
modulus of volume compressibility,
𝐦𝐯 = − 𝜺𝐳 𝟐− 𝜺𝒛 𝟏
𝝈′
𝒛 𝟐− 𝝈𝒛
′
𝟏
= ∆𝜺𝒛
𝝈′
′
𝟏
2. Basic Concepts

 Similarly, the slope BC in Figure the recompression
index, 𝐶𝑟, which we can express as:
𝐂𝒓 = −
𝐞𝟐−𝐞𝟏
𝐥𝐨𝐠
𝝈𝒛𝟐
′
𝝈𝒛𝟏
′
=
∆𝒆𝒓
𝐥𝐨𝐠
𝝈𝒛𝟐
′
𝝈𝒛𝟏
′
 The slope BC in Figure the modulus of volume
recompressibility, 𝒎𝒗𝒓and is expressed as
𝒎𝒗𝒓 = − 𝜺𝐳 𝟐− 𝜺𝒛 𝟏
𝝈′
′
𝟏
=
∆𝜺𝒛𝒓
𝝈′
′
𝟏
From Hooke’s law:
𝐸𝑐
′
=
∆𝜎𝑧
′
∆𝜀𝑧
𝑡ℎ𝑒𝑛 , 𝑚𝑣𝑟 =
1
𝐸𝐶
′
2. Basic Concepts

10. Effects of Loading History
 If a soil were to be consolidated to stresses below its past maximum
vertical effective stress, then settlement would be small because the
soil fabric was permanently changed by a higher stress in the past.
 However, if the soil were to be consolidated beyond its past
maximum effective stress, settlement would be larger for stresses
because the soil fabric undergo further change from a current loading
 The practical significance of this soil behavior is that if the loading
imposed on the soil by a structure is such that the vertical effective
stress in the soil does not exceed its past maximum vertical effective
stress, the settlement of the of the structure would be small,
otherwise significant permanent settlement would occur.
 The preconsolidation stress defines the limit of elastic behavior.
For stresses lower than the preconsolidation stress, the soil will
follow the URL and soil behave like an elastic material.
For stresses greater than the preconsolidation stress the soil
would behave like an elastoplastic material.
2. Basic Concepts

11. Overconsolidation Ratio
 A soil whose current vertical effective stress or overburden effective
stress 𝝈𝒛𝒐is less than its past maximum vertical effective stress or
preconsolidation stress 𝝈𝒛𝒄is called as an over consolidated soil.
 An over consolidated soil will follow a void ratio versus vertical
effective stress path similar to CDE during loading. (refer figure)
 The degree of overconsolidation called, overconsolidation ratio, OCR,
is defined as 𝑶𝑪𝑹 =
𝝈𝒛𝒄
𝝈𝒛𝒐
 If OCR = 1, the soil is normally consolidated soil.
 Normally consolidated soils follow paths similar to ABE (refer figure)
12.Possible and Impossible Consolidation Soil States
 The normal consolidation line delineates possible from impossible soil
states.
 Unloading of a soil or reloading it cannot bring it to soil states right of
the NCL, impossible soil states.
 Possible soil states only occur on or to the left of the normal
consolidation line.
2. Basic Concepts

3. Calculation of Primary Consolidation Settlement
1. Effects of Unloading/Reloading of a Sample Taken from Field
 Consider a soil sample taken from the field at a depth z & assume the
groundwater level is at the ground surface.
 The current vertical effective stress or overburden effective stress at z
is:𝝈𝒛
′ = 𝜸𝒔𝒂𝒕 − 𝜸𝒘 ∗ 𝐳 = 𝜸′𝒛
a. Soil sample at a depth z below
ground surface.
b. Expected one-dimensional response.

 The current void ratio can be found from 𝛾𝑠𝑎𝑡
 On a plot of e versus log 𝜎′ the current vertical effective stress
represented as in figure above
 To obtain a sample, make a borehole and remove the soil above it.
 The act of removing a soil reduces the total stress to zero; that is, the
soil is fully unloaded.
 From the principle of effective stress 𝝈𝒛= −∆𝒖𝒐
 Since 𝝈 can’t be -ve ,i.e., soil can’t sustain tension – the pore water
pressure must be -ve.
 As the pore water pressure dissipates with time, volume changes
occur.
 If soil sample reload, the reloading path followed depends on the
OCR.
 If OCR=1 ,the path followed during reloading would be BCD
(Figure b) and the average slope ABC is Cr.

 Once σzois exceeded, the soil follows the normal consolidation line,
CD, of slope Cc.
 If OCR>1, the reloading path followed BEF because the soil is
reloaded beyond its preconsolidation stress before it behaves like a
normally consolidated line.
 The average slope of ABE is Cr and the slope of EF is Cc.
 The point E, marks the preconsolidation stress.
2. Primary Consolidation Settlement of Normally
Consolidated Fine-grained Soils
 If a building constructed on a site consisting of a normally
consolidated soil, the increase in vertical stress due to the building at
depth z, where soil sample taken is ∆𝜎𝑧
 The final vertical stress is 𝝈′𝒇𝒊𝒏 = 𝝈𝒛𝒐
′
+ ∆𝝈𝒛

 The increase in vertical stress will cause the soil to settle following
the NCL and the primary consolidation settlement is:
𝜌𝑝𝑐 = 𝐻𝑜 ∗
∆𝑒
1+𝑒
=
𝐻𝑜
1+𝑒
𝐶𝑐 log
𝜎𝑓𝑖𝑛
′
𝜎𝑧𝑜
′ ; Where 𝑂𝐶𝑅 = 1,
∆𝑒 = 𝐶𝑐 ∗ log
𝜎𝑓𝑖𝑛
′
𝜎𝑧𝑜
′
3. Primary Consolidation Settlement of Overconsolidated Fine-
grained Soils
 If the soil is Overconsolidated, we have to consider two cases
depending on the magnitude of ∆𝜎𝑧
 Case-1 :In this case, consolidation occurs along the URL.
𝜌𝑝𝑐 = 𝐻𝑜 ∗
∆𝑒
1 + 𝑒𝑜
=
𝐻𝑜
1 + 𝑒𝑜
𝐶𝑟 log
𝜎𝑓𝑖𝑛
′
𝜎𝑧𝑜
; 𝜎𝑓𝑖𝑛
′
< 𝜎𝑧𝑐
′
 Case-2 :In this case, we have to consider two equations, one
along the URL and the other along the NCL
𝜌𝑝𝑐 =
𝐻𝑜
1 + 𝑒
𝐂𝐫 log
𝝈𝒛𝒄
′
𝝈𝒛𝒐
′ + 𝑪𝒄 log
𝝈𝒇𝒊𝒏
′
𝝈𝒛𝒄
′ ; 𝛔𝒇𝒊𝒏
′
> 𝝈𝒛𝒄
′

𝝆𝒑𝒄 =
𝑯𝒐
𝟏+𝒆
(𝒄𝒓 𝐥𝐨𝐠 𝑶𝑪𝑹 + 𝐂𝐜 𝐥𝐨𝐠
𝝈𝒇𝒊𝒏
′
𝝈𝒛𝒄
′ ); 𝝈𝒇𝒊𝒏
′
> 𝝈𝒛𝒄
′
4. Procedure to Calculate Primary Consolidation Settlement
 The procedure to calculate primary consolidation settlement is as follows:
1. Calculate the current vertical effective stress (𝝈𝒛𝒐
′ ) and the current void ratio
(𝒆𝒐) at the center of the soil layer for which settlement is required.

2. Calculate the applied vertical stress increase (∆𝜎𝑧) at the center of
the soil layer using the appropriate method.
3. Calculate the final vertical effective stress 𝜎𝑓𝑖𝑛
′
= 𝜎𝑧𝑜
′ + ∆𝜎𝑧
4. Calculate the primary consolidation settlement.
 If the soil is normally consolidated (𝑶𝑪𝑹 = 𝟏 ), the primary
consolidation settlement is 𝝆𝒑𝒄 =
𝑯𝒐
𝟏+𝒆
𝑪𝒄 𝒍𝒐𝒈
𝝈𝒇𝒊𝒏
′
𝝈𝒛𝒐
′ ;
 If the soil is Overconsolidated and 𝝈𝒇𝒊𝒏
′
< 𝝈𝒛𝒄
′ the primary
𝑯𝒐
𝟏+𝒆
𝑪𝒓 𝒍𝒐𝒈
𝝈𝒇𝒊𝒏
′
𝝈𝒛𝒐
′ ;
 If the soil is Overconsolidated and 𝝈𝒇𝒊𝒏
′
> 𝝈𝒛𝒄
′ the primary
𝑯𝒐
𝟏+𝒆
(𝒄𝒓 𝐥𝐨𝐠 𝑶𝑪𝑹 + 𝐂𝐜 𝐥𝐨𝐠
𝝈𝒇𝒊𝒏
′
𝝈𝒛𝒄
′ );
 The primary consolidation settlement be calculated using mv
𝜌𝑝𝑐 = 𝐻𝑜𝑚𝑣∆𝜎𝑧𝜎
 However, unlike Cc, which is constant, mv varies with stress
levels.
 Compute an average value of mv over the stress ranges𝜎𝑧𝑜
′ 𝑡𝑜 𝜎𝑓𝑖𝑛
′

Example
1. The soil profile at a site for a proposed office building consists of a
layer of fine sand 10.4 m thick above a layer of soft, normally
consolidated clay 2 m thick. Below the soft clay is a deposit of coarse
sand. The groundwater table was observed at 3 m below ground
level. The void ratio of the sand is 0.76 and the water content of the
clay is 43%. The building will impose a vertical stress increase of
140 kPa at the middle of the clay layer. Estimate the primary
consolidation settlement of the clay.
Assume the soil above the water table to be saturated, 𝐶𝑐 = 0.3 and
𝐺𝑆 = 2.7

example
2. Assume the same soil stratigraphy as in example 1. But now the
clay is Overconsolidated with an 𝑂𝐶𝑅 = 2.5, 𝑤 = 38%, 𝑎𝑛𝑑 𝐶𝑟 =
0.05. All other soil values given in Example 1 remain unchanged.
Determine the primary consolidation settlement of the clay.
3. Assume the same soil stratigraphy and soil parameters as in
Example 2 except that the clay has an overconsolidation ratio of 1.5.
Determine the primary consolidation settlement of the clay.

4. Terzaghi's One-dimensional Consolidation Theory
1. Derivation of Governing Equation
 Assumptions to derive Terzaghi’s 1D consolidation Equation:
1. The soil is saturated, isotropic and homogeneous
2. Darcy’s law is valid.
3. Flow only occurs vertically.
4. The strains are small.
 The basic concepts:
1. The change in volume of soil(∆𝑉) is equal to the change of pore
water expelled (∆𝑉𝑊)which is equal to the change in the volume
of the voids(∆𝑉
𝑣)
2. At any depth, the change in vertical effective stress is equal to the
change in excess pore water pressure at that depth.i.e. 𝜕𝜎𝑧 = 𝜕𝑢

 For soil element in figure the
inflow of water is= 𝑣𝑑𝐴
and the outflow over the elemental
thickness 𝑑𝑧 is 𝑣 +
𝜕𝑣
𝜕𝑧
𝑑𝑧 𝑑𝐴
 the flow rate is the product of the
velocity and the cross-sectional
area normal to its (velocity)
direction.
The change in flow is then
𝜕𝑣
𝜕𝑧
𝑑𝑧 𝑑𝐴
The rate of change in volume of water expelled, which is equal to the
rate of change of volume of the soil, must equal the change in flow
𝜕𝑉
𝜕𝑡
=
𝜕𝑣
𝜕𝑧
𝑑𝑧 𝑑𝐴 … … … … … .
𝜕𝑢
𝜕𝑡
= 𝐶𝑣
𝜕2𝑢
𝜕𝑧
 This equation describes the spatial variation of excess pore water
pressure (∆𝑢) with time (t) and depth (z)

2. Solution of Governing Consolidation Equation Using Fourier
Series
 The solution of any differential equation requires a knowledge of
the boundary conditions.
 By specification of the initial distribution of excess pore water
pressures at the boundaries, we can obtain solutions for the spatial
variation of excess pore water pressure with time and depth.
 Various distributions of pore water pressures within a soil layer are
possible.
 For Example
 Uniform distribution of initial excess pore water pressure with depth
 Triangular distribution of initial excess pore water pressure with depth
 The boundary conditions for a uniform distribution of initial excess pore
water pressure in which double drainage occurs are

 The boundary conditions for a uniform distribution of initial excess
pore water pressure in which double drainage occurs are
when 𝑡 = 0, ∆𝑢 = ∆𝑢𝑜 = ∆𝑢𝑧
At top boundary 𝑍 = 0, ∆𝑢 = 0.
At bottom boundary, 𝑧 = 2𝐻𝑑𝑟, ∆𝑢 = 0,
𝑤ℎ𝑒𝑟𝑒 𝐻𝑑𝑟 𝑖𝑠 𝑑𝑟𝑎𝑖𝑛𝑎𝑔𝑒 𝑝𝑎𝑡ℎ 𝑙𝑒𝑛𝑔𝑡ℎ.
 A solution for the governing consolidation equation, which satisfies
these boundary conditions, is obtained using the Fourier series,
∆𝑢 𝑧, 𝑡 =
2∆𝑢𝑜
𝑀
𝑠𝑖𝑛
𝑀𝑧
𝐻𝑑𝑟
∞
𝑚=0 𝑒𝑥𝑝 −𝑀2𝑇𝑣
where 𝑀 =
𝜋
2
2𝑚 + 1 and 𝑚 is a positive integer with values from
0 𝑡𝑜 ∞ 𝑎𝑛𝑑
𝑇𝑣=
𝐶𝑣𝑡
𝐻𝑑𝑟
2 where 𝑇𝑣 is known as the time factor; it is a dimensionless
term.

3. Degree of consolidation or consolidation ratio (𝑼𝒛) - gives us
the amount of consolidation completed at a particular time and
depth.
This parameter can be expressed mathematically as:
𝑼𝒛 = 𝟏 −
∆𝒖𝒛
∆𝒖𝟎
= 𝟏 −
𝟐∆𝒖𝒐
𝑴
𝒔𝒊𝒏
𝑴𝒛
𝑯𝒅𝒓
∞
𝒎=𝟎
𝒆𝒙𝒑 −𝑴𝟐𝑻𝒗
An isochrones illustrating
the theoretical excess
pore water pressure
distribution with depth.

 𝑈𝑧 = 0 everywhere at the beginning of the consolidation(∆𝑢𝑧 =
∆𝑢𝑜) but increases to unity as the initial excess pore water pressure
dissipates.
 A geotechnical engineer is often concerned with the average degree
of consolidation(U) of a whole layer at a particular time rather than
the consolidation at a particular depth(𝑈𝑧).
• The shaded area in Figure represents the amount of consolidation of
a soil layer at any given time.
4. The average degree of consolidation(U)
 U expressed mathematically from the solution of the one
dimensional consolidation equation as:
𝑈 = 1 −
2
𝑀2
exp −𝑀2𝑇𝑣
𝑚
𝑚=𝑜
 Consider the figure below shows the variation of the average degree
of consolidation with time factor 𝑇𝑣 for a uniform and triangular
distribution of excess pore water pressure.

Relationship between time factor and average degree of consolidation

• A convenient set of equations for double drainage, found by curve
fitting from above figure is
𝑻𝒗 =
𝝅
𝟒
𝑼
𝟏𝟎𝟎
𝟐
𝐟𝐨𝐫 𝐔 < 𝟔𝟎%
𝑻𝒗 = 𝟏. 𝟕𝟖𝟏 − 𝟎. 𝟗𝟑𝟑 𝐥𝐨𝐠 𝟏𝟎𝟎 − 𝑼 , 𝑼 ≥ 𝟔𝟎%
𝑇𝑣 = 0.197 corresponding to 50% and 𝑇𝑣 = 0.848 corresponding to
90% consolidation are often used in interpreting consolidation test
results.
Example

5. Secondary Compression Settlement
consolidation settlement consists of two parts.
The first part is primary consolidation, which occurs at early
times
The second part is secondary compression, or creep, which
takes place under a constant vertical effective stress.
 Primary consolidation is assumed to end at the intersection of the
projection of the two straight parts of the curve as shown in Figure
Fig. Secondary
compression

 The secondary compression index is
𝑪𝜶 = −
𝒆𝒕 − 𝒆𝒑
𝒍𝒐𝒈
𝒕
𝒕𝒑
=
∆𝒆
𝒍𝒐𝒈
𝒕
𝒕𝒑
; 𝐭 > 𝒕𝒑
 Overconsolidated soils do not creep significantly but creep settlements
in normally consolidated soils can be very significant.
 The secondary consolidation settlement is 𝜌𝑠𝑐 =
𝐻𝑜
1+𝑒𝑝
𝐶𝛼 log
𝑡
𝑡𝑝
5. Secondary Compression Settlement

6. One-dimensional Consolidation Laboratory Test
1. Oedometer Test- It is the 1D consolidation test, is used to find
𝐶𝑐, 𝐶𝑟, 𝐶𝛼, 𝐶𝑣, 𝑚𝑣 𝑎𝑛𝑑 𝜎𝑧𝑐
′
Read the procedures in laboratory
2. Determination of the Coefficient of Consolidation
 There are two popular methods to calculate 𝐶𝑣
a. Root time method: proposed by Taylor
𝑪𝒗 =
𝟎. 𝟖𝟒𝟖𝑯𝒅𝒓
𝟐
𝒕𝟗𝟎
b. Log time method: proposed by Casagrande's and Fadum.
Read the procedure how we can determine 𝐶𝑣 using
these methods
𝑪𝒗 =
𝟎. 𝟏𝟗𝟕𝑯𝒅𝒓
𝟐
𝒕𝟓𝟎

3. Determination of Void Ratio at the End of a Loading Step.
To calculate the void ratio for each loading step as follows:
1. Calculate the final void ratio, 𝑒𝑓𝑖𝑛 = 𝑤𝐺𝑠
2. Calculate the total consolidation settlement of the soil sample
during the test, ∆𝑧 𝑓𝑖𝑛 = 𝑑𝑓𝑖𝑛𝑎𝑙 − di where 𝑑𝑓𝑖𝑛 is the final
displacement gage reading and 𝑑𝑖 is the displacement gage
reading at the start of the test.
3. Back-calculate the initial void ratio, 𝒆𝒐 =
𝒆𝒇𝒊𝒏+
∆𝒛𝒇𝒊𝒏
𝑯𝒐
𝟏−
∆𝒛𝒇𝒊𝒏
𝑯𝒐
4. Calculate e for each loading step

4. Determination of the Past Maximum Vertical Effective Stress
 From e versus log 𝜎𝑧𝑐
′ curve determine the preconsolidation stress
using a method proposed by Casagrande's.
 The procedure, with reference to Figs. below, is as follows:
1. Identify the point of maximum curvature, point D, on the initial
part of the curve.
2. Draw a horizontal line through D
3. Draw a tangent to the curve at D.
4. Bisect the angle formed by the tangent and the horizontal line at D
5. Extend backward the straight portion of the curve (the normal
consolidation line), BA, to intersect the bisector line at F.
6. The abscissa of F is the past maximum vertical effective
stress= 𝜎𝑧𝑐
′

Fig. Determination of the past maximum vertical effective stress using
(a) Casagrande’s method.(b) simplified method

5. Determination of Compression and Recompression Indices
6. Determination of the Secondary Compression Index

7. Evaluation of Total Soil Settlement
• In general, the total settlement 𝜌𝑡𝑜𝑡𝑎𝑙 total of a foundation can be
given as:𝜌𝑡𝑜𝑡𝑎𝑙 = 𝜌𝑒 + 𝜌𝑝𝑐 + 𝜌𝑠𝑐
where 𝜌𝑒 = 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑜𝑟 𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 𝑠𝑒𝑡𝑡𝑙𝑒𝑚𝑒𝑛𝑡
𝜌𝑝𝑐 = primary consolidation settlement
ρ𝑠𝑐 = secondary consolidation settlement
 The immediate settlement is sometimes referred to as the elastic
settlement.
 In granular soils this is the predominant part of the settlement,
whereas in saturated inorganic silts and clays the primary
consolidation settlement predominates.
 The secondary compression settlement forms the major part of
the total settlement in highly organic soils and peats.

8. Relationship Between Laboratory and Field Consolidation
 The time factor (TV) provides a useful expression to estimate the
settlement in the field from the results of a laboratory consolidation
test.
 If two layers of the same clay have the same degree of consolidation,
then their time factors and coefficients of consolidation are the same.
Hence,𝑇𝑣 =
𝐶𝑣𝑡 𝑙𝑎𝑏
𝐻𝑑𝑟
2
𝑙𝑎𝑏
=
𝐶𝑣𝑡 𝑓𝑖𝑒𝑙𝑑
𝐻𝑑𝑟
2
𝑓𝑖𝑒𝑙𝑑

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