Compressibility and Settlement

Prepared by

Dr. Md. Rokonuzzaman

Professor
Department of Civil Engineering, KUET
 Compressibility and Settlement

Reference: Geotechnical Engineering-principles and practice (Chapter 11)-Donald P. Coduto

Introduction
• In this topic, we will learn the mechanical and behavioral issues: non-linearity, time dependency, and elasto-
plastic response to loading-unloading, and the effect of stress history.

Consolidation vs Compaction
• The process of consolidation is often confused with the process of compaction.
• The difference between consolidation and compaction can be appreciated using three-phase diagrams as
shown below:

• Compaction increases the density of an unsaturated soil by reducing the volume of air in the voids.
• Consolidation is a stress-strain-time process of increasing the density of a saturated fine-grained soil by
draining some of the water out of the voids. Consolidation is generally related to fine-grained soils such as
silts and clays. Saturated clays consolidate at a much slower rate due to their low permeability. This process
may continue for months, year or decades.

The Necessity of Consolidation Theory

• Consolidation theory is required for the prediction of both the magnitude and the rate of consolidation
settlements to ensure the serviceability of structures founded on a compressible soil layer.
• Differential settlements that can lead to structural failures due to tilting should be avoided. Otherwise, you’ll
need extreme measures to save your structure!

1-D Consolidation Process

The Spring Analogy Field Case
The spring is
analogous to the soil
skeleton. The outlet
opening size is
analogous to the
permeability of the
soil.

Initial condition Initial condition

Spring force=P, Water Force=0 Effective stress at P= σv0ʹ= σv0-u, PWP at P (hydrostatic)=u=hγw

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 2

 Compressibility and Settlement

t=0 t=0
Spring force=P, Water Force=ΔP Effective stress at P= σv0ʹ, PWP at P (NOT hydrostatic)=hγw + Δσv
Intermediate time

t >0 t >0
Spring force>P, Water Force<ΔP Effective stress at P >σv0ʹ, PWP at P (NOT hydrostatic)<hγw +Δσv

t =α (or 100% Consolidation) t =α (or 100% Consolidation)

Spring force=P+ ΔP, Water Force=0 Effective stress at P =σv0ʹ+Δσv, PWP at P (Hydrostatic)=hγw
Consolidation Process in Cutting vs Filing
(a) Cutting (b) Filling

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 3

 Compressibility and Settlement

FSL

In excavation problem, the soil may fail after long time It may not be possible to construct an embankment
when the negavive excess PWP (due to unloading) will on soft clay entirely in a single operation, because
dissipate to zero and the effective stress will be the loads that would be applied to the surface of the
minimum. soft clay would cause the clay to fail. This problem
can be solved by constructing the embankment in a
number of stages. The soft clay must be allowed to
consolidate between stages so that there is a gain of
undrained shear strength (Ladd gives for soft clay,
Δsu=0.25Δσvʹ using CU Tests). (eg. Embankment
of the approach road of Padma Bridge)
Practical Cases of One-dimensional Consolidation
• Since water can flow out of a saturated soil sample in any direction, the process of consolidation is
essentially three-dimensional.
• However, in most field situations, water will not be able to flow out of the soil by flowing horizontally
because of the vast expanse of the soil in horizontal direction. Therefore, the direction of flow of water is
primarily vertical or one-dimensional. As a result, the soil layer undergoes one-dimensional or 1-D
consolidation settlement in the vertical direction.
• Following practical cases can be considered as 1D problem (FOX, 1995)
(a) Width of the loaded area at least four times the thickness of the compressible layer (eg. large area fill).
(b) Depth to the top of the compressible layer is at least twice the width of the loaded area.
(c) Compressible layer lies between stiffer layers whose presence tends to reduce the horizontal strain.

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 4

 Compressibility and Settlement

B>4H B B

Clay H Sand z>2B Rock

Sand Clay
Clay
Rock
Fig: Case a Fig: Case b Fig: Case c
Example 1: Consolidation Process in Field (11.2/Coduto)
Compute the stresses and pore water pressure in soil at point A
before the fill, immediately after the fill and sufficiently long time 1.0m Proposed fill, γ=20 kN/m3
with 100% consolidation after the fill.
Time and Loading Total Effective Pore water 1m
Stages stress stress pressure 2.0m 3 A
Clay, γ=20 kN/m
(kPa) (kPa) (kPa)
Before fill 20 10 10
Immediately after fill 40 10 30
Long time after fill 40 30 10

Sample quality: For soft to medium silts and clays, rotary wash boring or hollow-stem auger methods with
heavy drilling mud to reduce stress relief at the base of the boreholes must be used. It also generally requires
sampling with thin-walled samplers with low area ratios; at a minimum, normally 3 inch Shelby tubes should
be used and various types of fixed piston samplers are even more desirable.

The Oedometer Test: Procedure (ASTM D 2435, Incremental Load Consolidation Test)

Floating Type Consolidometer (less wall friction) Fixed Type Consolidometer

• The oedometer test is used to investigate the 1-D consolidation behavior of fine-grained soils.
• An undisturbed soil sample 20 mm in height and 75 mm in diameter is confined in a steel confining ring and
immersed in a water bath. Two-way drainage is permitted through porous disks at the top and bottom as
shown in the figure above.
• It is subjected to a compressive stress by applying a vertical load.
• After a few seconds, the pore water begins to flow out of the
voids, which results in a decrease in pore water pressure and void
ratio of the soil sample and an increase in effective stress. As a
result, the soil sample settles. The vertical compression of the soil
sample is recorded using highly accurate dial gauges.
• Several increments of vertical stress are applied in an oedometer
test usually by doubling the previous increment (LIR=1). For
example, after the completion of consolidation for the first
increment under a vertical stress of 50 kPa, another 50 kPa of

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 5

 Compressibility and Settlement

vertical stress is applied so that the vertical stress during the second increment is 100 kPa. For the third
increment, another 100 kPa of vertical stress is applied so that the vertical stress during the third increment is
200 kPa.

Δh1 Δh2
50 kPa

Settlement (Δh)
100 kPa
200 kPa

Time (day)
The Oedometer Test-Data Analysis and Plotting
𝑉𝑠 𝑀 𝑀
(a) ℎ𝑠 = = 𝐴𝜌𝑠 = 𝐴𝐺 𝜌𝑠
𝐴 𝑠 𝑠 𝑤
e0Vs hv0 (b) ℎ𝑣0 = ℎ0 − ℎ𝑠
hv eVs 𝑉 ℎ 𝐴 ℎ
(c) 𝑒0 = 𝑉𝑣0 = ℎ𝑣0𝐴 = ℎ𝑣0
𝑠 𝑠 𝑠
h ∆ℎ
Vs hs Vs (d) 𝜀𝑣 = 𝜀𝑥 + 𝜀𝑦 + 𝜀𝑧 = 𝜀𝑧 = (as no lateral strain)
ℎ0
𝑉𝑜 −𝑉 𝑉𝑣0 +𝑉𝑠 −(𝑉𝑣 +𝑉𝑠 )
(e) 𝜀𝑧 = 𝜀𝑣 = = =
𝑉0 𝑉𝑣0 +𝑉𝑠
𝒆𝟎 𝑉𝑠 +𝑉𝑠 −(𝒆𝑉𝑠 +𝑉𝑠 ) 𝒆𝟎 −𝒆 ∆𝒉
= 𝟏+𝒆 =
𝒆𝟎 𝑉𝑠 +𝑉𝑠 𝟎 𝒉𝟎
∆ℎ
(f) 𝑒 = 𝑒0 − 𝜀𝑧 (1 + 𝑒0 ) = 𝑒0 − ℎ (1 + 𝑒0 )
0
• Since the void ratio and vertical strain of the soil sample at different stages of an oedometer test can be
estimated using the above equation from the settlement of the soil, it is customary to plot the results in terms
of vertical effective stress and void ratio or vertical strain as shown in the figure below. The figure shows that
the soil is strain hardening material; that is, the instantaneous modulus increases as the stress increases. Since
the stress-strain relationship is highly non-linear and elasto-plastic, more common ways to represent the
results in semi-logarithmic scale.

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 6

 Compressibility and Settlement

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 7

 Compressibility and Settlement

Sample Laboratory Data Sheet

A. Sample Data D. Computation:

Diameter, d (cm) = Initial mass of water, Mw0 (gm)=Mrsw+Md-Mdrs=
Initial height, h0 (cm) = Mass of dry soil, Ms (gm)=Mdrs-Mr-Md =
Area, A0 (cm2)= Initial water content, w0 (%)=Mw0*100/Ms=
Volume, V0 (cc)= Height of soil solids, hs (cm)=Ms/GsA0=
Specific gravity, Gs= Initial height of void, hv0 (cm)=h0-hs=
B. Data before Test Initial void ratio, eo=hv0/hs=
Mass of ring, Mr (gm) = Initial degree of saturation, Sr0 (%)=woGs/e0=
Mass of ring + wet soil, Mrsw (gm) = Final mass of water, Mwf (gm) = Mdrsw-Mdrs=
C. Data after Test Final water content, wf (%)=Mwf*100/Ms=
Mass of dish, Md (gm)= Final height of void, hvf (cm) = Mwf/A0=
Mass of dish+ring+wet soil, Mdrsw (gm) = Final void ratio, ef=hvf/hs=
Mass of dish+ring+dry soil, Mdrs (gm)= Final degree of saturation, Srf (%)=wfGs/ef=

Coefficient of
After load increment From t graph
consolidation, Cv
Effective stress,  = P*98/A0

Drainage path, hdr= havg/2

Vertical strain, z =-h/h0

Avg. height, havg=h0-D50

(1 + e0 )
Applied load, P

(1div. = 0.0025 mm)

Total deformation,

Cv= 0.197* hdr2/t50

Dial reading, DR
Loading criteria

h=DR*0.0025
Void ratio,

D100
∆h

D50
h0

D0

t50
e = e0 −

Kg kPa div. mm unit mm/mm mm mm mm min mm mm mm2/min

10
20
Loading

40
80
160
320
160
Unloading

80
20
10

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 8

 Compressibility and Settlement

The Oedometer Test –Coefficients

• The slope of the compression curve, when the results are plotted
arithmetically, is called the coefficient of compressibility:
𝑑𝑒 ∆𝑒
𝑎𝑣 = − ′ = − ′
𝑑𝜎𝑣 ∆𝜎𝑣

• The coefficient of volume compressibility is defined as the ratio of

volumetric strain over change in effective stress:
𝑒0 − 𝑒1 𝑒0 − 𝑒2
𝑑𝜀𝑣 ∆𝜀𝑣 ( 1 + 𝑒0 ) − ( 1 + 𝑒0 ) 𝑒1 − 𝑒2
𝑚𝑣 = = = = −
𝑑𝜎𝑣′ ∆𝜎𝑣′ ∆𝜎𝑣′ (1 + 𝑒0 )∆𝜎𝑣′
𝑎𝑣 1
= =
1 + 𝑒0 𝐷
• The units for mv are the inverse of pressure, i.e. m2/kN and its value
depends on the stress range over which it is calculated. The inverse of
volume compressibility is constrained modulus (D).

• The e-σʹv curve becomes almost linear if σʹv is plotted on a log scale
as shown in the figure on the right. The slopes of the loading curve
(compression index, cc) and unloading or recompression curve
(swelling or recompression index, cr) and is dimensionless:
𝑑𝑒 𝑒1 − 𝑒2
𝑐𝑐 𝑜𝑟 𝑐𝑟 = − =
𝑑𝑙𝑜𝑔𝜎𝑣 𝑙𝑜𝑔𝜎2′ − 𝑙𝑜𝑔𝜎1′
′

• The negative sign is used because the void ratio decreases when the
effective stress is increased.
• Empirical formula given for estimating cc can be used for
preliminary design and checking the validity of test results [e.g.
Terzaghi and Peck: 𝑐𝑐 = 0.009(𝐿𝐿 − 10)]. cr appreciably smaller than
compression index (one-fifth to one-tenth).
• Soil classification based on compressibility
cc/(1+e0) Compressibility
0~0.05 Very slightly
0.05~0.10 Slightly
0.10~0.20 Moderate
0.20~0.35 Highly
>0.35 Very highly
• If you are interested to compute settlement of sands and gravel using the classical method, see Table
11.8/Coduto to estimate cc. Compacted fill can be assumed as overconsolidated soil.

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 9

 Compressibility and Settlement

The Oedometer Test-Preconsolidation Stress or Stress History

• An oedometer test of an undisturbed sample of an OC soil shows an e-
log(σʹv) curve as shown in the figure on the right.
• The slope of the e-log(σʹv) curve (reconsolidation part) is fairly flat until a
vertical effective stress, known as preconsolidaion stress or maximum
historical stress, (σʹc) is reached. Beyond this point, the slope of the e-log(σʹv)
curve (virgin part) becomes steeper, i.e. the soil becomes more compressible
while the soil structure breaks. The greater part of the deformation is due to
slippage of the soil particles as the soil skeleton rearranges itself to
accommodate higher loads. This component of deformation is irrecoverable
or plastic. On an unload/reload line, changes in stress can be accommodated
without the need for a rearrangement of the soil skeleton. Deformation is
primarily due to distortion of the soil particles. It is recovered on unloading
and may, in this sense, be described as elastic.
• The pre-consolidation pressure is like a yield stress-so to speak “memory of
soil”-which is analogous to yield stress in metal. This fact can be appreciated
by rotating the curve by 90° in anti-clockwise direction.

Plastic Elastic

• Over Consolidation Ratio (OCR) is defined as the ratio of preconsoldiation pressure to present vertical
effective stress (σʹc/σʹv0).
• A soil that has never experienced a vertical effective stress that was greater than its present vertical effective
stress is called a Normally Consolidated (NC) soil. The OCR for an NC soil is equal to 1. Most NC soils have
fairly low shear strength and high compressibility.
• A soil that has experienced a vertical effective stress that was greater than its present vertical effective stress
is called an Over Consolidated (OC) soil. The OCR for an OC soil is greater than 1. Most OC soils have fairly
high shear strength and less compressible.
• If OCR less than 1, the soil is underconsolidated. It can occur in soils that have only recently been deposited
or in soil with recent lowering of water table, and are still consolidating under its self-weight, equilibrium has
not yet been reached. The pore pressure, if measured, will be in excess of hydrostatic pressure.
• Soil classification based on over consolidation:
σʹc- σʹv0 (kPa)=Overconsolidation margin (σʹm) OCR Classification
0 1 Normally consolidated
0~100 1~2 Lightly overconsolidated
100~400 2~8 Moderately consolidated
>400 >8 Heavily consolidated
• Reasons for overconsolidation at the field: preloading by previously overlying strata, desiccation,
groundwater lowering, glacial overriding or an engineered preload.

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 10

 Compressibility and Settlement

Sampling

Erosion
Figure: Consolidation stress history of soil in field and experiment
Logσʹ

Example:

Let us evaluate the stresses, past and present at point A situated at a depth of z:
-the maximum effective vertical stress applied in the past: 𝜎𝑝′ = 𝛾 ′ (𝑧 + 𝑧1 ),
-the present effective vertical stress:𝜎𝑣′ = 𝛾 ′ 𝑧.
𝜎𝑝′
The over-consolidation ration, OCR, defined as 𝑂𝐶𝑅 = = 1 + 𝑧1 /𝑧 is a hyperbolic function of z. Clearly,
𝜎𝑣′
as the depth increases, the OCR decreases towards its ultimate value of 1 corresponding to a normally
consolidated state.

Determination of Pre-consolidation Pressure from 1D Consolidation Test

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 11

 Compressibility and Settlement

• Pre-consolidation pressure (σʹp) for an OC soil

can be estimated from the e-logσʹv curve using
Casagrande’s procedure with the following
steps:
1. Fit data with smooth curve
2.Determine the point of maximum curvature
(point A) on the consolidation the curve (by eye).
3.Draw a horizontal line and a tangent at point A
and bisect the angle between these two lines.
4.Extend the straight-line portion of the virgin
curve upto where it meets the bisector line. The
intersection point (point B) of the bisector and the
extended straight line gives the preconsolidation
pressure (σʹp).
• Factors that affect the preconsolidation pressure
are: sample disturbance, load increment ratio
(LIR) and duration of load increment.
• If the variation of preconsolidation pressure with
depth is unknown, e.g., only one consolidation
test was conducted in the soil profile, actual
settlements could be higher or lower than the
computed value based on a single value of
preconsolidation pressure.
𝜎𝑝′
• From index properties: 𝑝 = 1011.11−1.62𝑃𝐼 (Stas and Kulhaway, 1984).
𝑎
• The Schmertmann procedure (1953) was designed to minimize the effects of disturbance, represented by
the shaded area in the figure due to in situ sampling and laboratory handling of the soil. He devised the
procedure as follows:
1. Apply increment loading to the sample, including an unload-
reload cycle, until point A corresponding to a void ratio of
0.42e0 is reached.
2. Draw the line BC within the unload-reload loop.
′
3. Plot point D with the co-ordinates (e0- 𝑙𝑜𝑔𝜎𝑣0 ), from which
the line DE is drawn parallel to BC
4. Select a point on DE (point P, for instance) thus fixing the
presumed value of 𝜎𝑝′ .
5. Plot the (presumed) virgin consolidation line PA.
6. Measure the difference in void ratio ∆𝑒𝑝 between the selected
point and the actual experimental graph.
7. Plot the point of co-ordinates (∆𝑒𝑝 , 𝜎𝑝′ ) in the space
(∆𝑒 , 𝑙𝑜𝑔 𝜎𝑣′ ).
8. Repeat the procedure from step 4. The estimated 𝜎𝑝′ will result
in the most symmetrical (∆𝑒 , 𝑙𝑜𝑔 𝜎𝑣′ ) graph.
Note: For stiff clays, the Schmertmann method requires the
application of very high vertical pressures (eg. 20000 kPa) in
order to achieve a void ratio of 0.42e0.

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 12

 Compressibility and Settlement

Prediction of Field Compaction Curve

•Schmertmann found out that the oedometer compression line meets the in-situ compression line at a void
ratio of approximately 0.42 times the initial in-situ void ratio e0.

Figure: Illustration of the Schmertmann (1955): (a) NC soil; (b) OC soil.

1. Perform the Casagrande (1936) construction or use Schmertmann procedure (1953) and evaluate the
pre-consolidation pressure σʹp
2. Calculate the initial void ratio eo. Draw a horizontal line from eo, parallel to the log effective stress axis,
to the existing vertical overburden pressure σʹvo. This establishes control point 1, illustrated by triangle 1
in Fig. b.
3. From control point 1, draw a line parallel to the rebound-reload curve to the preconsolidation pressure
σ'p. This will establish control point 2, as shown by triangle 2 in Fig. b.
4. From a point on the void ratio axis equal to 0.42eo, draw a horizontal line, and where the line meets the
extension of the laboratory virgin compression curve L, establish a third control point, as shown by
triangle 3. The coefficient of eo is not a “magic number,” but is a result of many observations on
different clays.
5. Connect control points 1 and 2, and 2 and 3 by straight lines. The slope of the line F joining control
points 2 and 3 defines the compression index cc for the field virgin compression curve. The slope of the
line joining control points 1 and 2 of course represents the recompression index cr.

Pre-consolidation Pressure – Key Points

• The pre-consolidation pressure for an overconsolidated soil should not be exceeded in construction, if
possible.
• Consolidation settlements will small if the effective vertical stress in the soil layer remains below its
preconsolidation pressure.
• If effective vertical stress in the soil layer exceeds its preconsolidation pressure, the consolidation
settlements will be large due to further yielding of the soil layer.
• Normally consolidated and underconsolidated soils should be considered unsuitable for direct support of
spread footings due to the magnitude of potential settlement, the time required for settlement, for low shear
strength concerns, or any combination of these design considerations. Preloading or vertical drains may be
considered to mitigate these concerns.

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 13

 Compressibility and Settlement

Settlement Analysis
• The settlement is defined as the compression of a soil layer due to the loading applied-for example, by
structure or manmade fill at or near its top surface. Heaving is the opposite which may be due to excavation.
• The total settlement of a soil layer consists of three parts:
𝑆𝑡 = 𝑆𝑖 + 𝑆𝑐 + 𝑆𝑠
– Distortional compression (Si): no change in water content and volume, due to distortion or change is shape.
– Primary consolidation settlement (Sc): water content changes, due to expulsion of water (volume change)
and change in effective stress whatever the soil type.
– Secondary compression (Ss): no change of water content, due to particle reorientation, creep, decomposition
of organic content with constant effective stress.

• The soil settlement can often be categorized in time frame: short term (immediate) settlement and long term
(delayed) settlement. The following table illustrates the relationship among soil type, sources of settlement
and their time dependence:

Time Soil Type

Frame Clays and Silts Sands and Gravels
Process Magnitude Process Magnitude
Short Term Distortion Negligible to small Distortion Negligible to small
-- -- Consolidation Small to moderate
Long Term Consolidation Moderate to large Secondary compression Negligible to small
Secondary compression Small to large -- --
•Sands and gravels undergo also consolidation but at a much faster rate due to their high permeability. For
these soils, the immediate settlement cannot be distinguished from consolidation settlement. Most of the
compression occurs during construction period. However, because they occur so fast, even relatively small
total settlements may be detrimental to the structure which is particular sensitive to rapid settlement.
•The induced stress (Figure given below) and settlement of a flexible foundation (eg. steel tank foundation) is
greater under the center than it is at the same depth under the edge. But the settlement is uniform for the rigid
foundation (which is perfectively simulated in 1D consolidation test). But mat is between perfectly rigid and
flexible footing having some differential settlement between the edge and center. You may consider the
rigidity factor of the footing to save money when the extensive sub-soil investigation is conducted. The total
corrected settlement of the foundation is obtained applying the correction factor to the settlement computed
under the center of the presumed flexible footing as: 𝑆𝑐 = 𝑟𝑆𝑡(1−𝐷) .

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 14

 Compressibility and Settlement

Figure: Stress bulb for square footing Figure: Influence of foundation rigidity on settlement.
Table: r-values for computation of total settlement at the center of a shallow foundation and methodology for
computing differential settlement.
Rigidity r for maximum settlement at center Method for differential settlement
Perfectively flexible (eg. steel 1.0 Compute Δσz below edge and use
tank) r=1.0
Intermediate (eg. Mat foudnations) 0.85-1.00 (typically about 0.9) Winkler/FEM Method
Perfect Rigid (eg. RCC spread 0.85 Compute differential settlement
footings) between footings. For strip footing
use Winkler Method.

Primary Consolidation Settlement–Classical Terzaghi’s 1-D Method

•For the calculation of consolidation settlement using the 1-D
method, the value of either the coefficient of volume
compressibility (mv) or the Compressibility Indices (cc
and/or cr) are required.
•Let’s consider a layer of saturated soil of thickness Hi
subjected to an increase in total vertical stress of Δσ as
shown in the figure on the right.
•At the completion of consolidation, the soil layer will experience an increase in effective vertical stress of
Δσ’ and as a result, its void ratio will reduce from e0 (initial void ratio) to e1 (final void ratio).
• Therefore, the volumetric or vertical strain in the soil layer can be written in terms of void ratio as:
𝑉 −𝑉 𝑒0 −𝑒1
𝜀𝑧 = 𝜀𝑣 = 𝑜𝑉 = 1+𝑒 .
0 0
Since the soil layer has undergone 1-D consolidation, the lateral strains are zero and therefore, the volumetric
strain is equal to the vertical strain (the change in thickness of the soil layer over its initial thickness).
• Therefore, the settlement of the soil layer is given by:
𝐻𝑖 𝐻𝑖 𝜎′1 𝐻𝑖 𝐻𝑖
∆𝜎′𝑖
𝑆𝑐𝑖 = ∫ 𝜀𝑧𝑖 𝑑𝑧 = ∫ ∫ 𝑚𝑣𝑖 𝑑𝜎′𝑖 𝑑𝑧 = ∫ 𝑚𝑣𝑖 ∆𝜎′𝑖 𝑑𝑧 = ∫ 𝑑𝑧
0 0 𝜎′0 0 0 𝐷

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 15

 Compressibility and Settlement

If mv or D and Δσʹ are constant with depth (logical for thin layer), then 𝑆𝑐 = ∑ 𝑆𝑐𝑖 = ∑ 𝑚𝑣 ∆𝜎′𝑖 𝐻𝑖 . In order to
take the variation of mv and/or Δσʹ with depth, a graphical procedure shown in the figure below can be used to
determine Sc. The area in (c) gives the settlement sc and can be estimated using Simpson’s rule.

• For a normally consolidated (NC) soil (Case I), settlement of a

thin layer can be calculated using the Compression Index (cc) as:
𝑒0 − 𝑒1 𝑐𝑐 𝑙𝑜𝑔(𝜎1′ /𝜎0′ )
𝑆𝑐𝑖 = 𝜀𝑧 𝐻𝑖 = 𝐻 = 𝐻𝑖
1 + 𝑒0 𝑖 1 + 𝑒0

•For an overconsolidated (OC) soil (Case II), if the

preconsolidation pressure (σʹc) is not exceeded, the settlement
can be calculated using the Expansion Index (ce) as:
𝑒0 − 𝑒1 𝑐𝑒 𝑙𝑜𝑔(𝜎1′ /𝜎0′ )
𝑆𝑐𝑖 = 𝜀𝑧 𝐻𝑖 = 𝐻 = 𝐻𝑖
1 + 𝑒0 𝑖 1 + 𝑒0

•If, however, the preconsolidation pressure is exceeded (Case

III), both cc and ce values will be required for the calculation of
settlement:
𝐻𝑖
𝑆𝑐𝑖 = [𝑐𝑒 𝑙𝑜𝑔(𝜎𝑐′ /𝜎0′ ) + 𝑐𝑐 𝑙𝑜𝑔(𝜎1′ /𝜎𝑐′ )]
1 + 𝑒0

The settlement of the entire layer is

𝑛

𝑆𝑐(1−𝐷) = ∑ 𝑆𝑐𝑖
𝑖=1
n being the number of and it is related to the type of pressure
distribution generated by the loaded area. For a circular or a
rectangular uniformly loaded area, the vertical induced stress
decays at a depth of about 3B, whereas for a long strip, the
vertical induced stress decays at a depth of 8B. To account for
the decreasing stress with increased depth below a footing and
variations in soil compressibility with depth, the compressible
layer should be divided into vertical increments for manual
computation of consolidation settlement of shallow foundations
Layer Approximate Layer Thickness
Number Square Continuous Footing
Footing
1 B/2 B
Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 16
 Compressibility and Settlement

2 B 2B
3 2B 4B
Notes: For rectangular footings, use intermediate thickness
between those given for square and strip. If the GWT is
shallow, use thicker layers (1.5 times thickness shown in the
table).

Note: for simplicity, for all sublayers of a soil stratum initial void ratio is assumed same, though it depends on
current effective overburden stress.

1D Settlement Calculation – An Example

A 3.0 m deep compacted fill is to be placed over the soil
profile as shown in the right. A consolidation test on a 3.0m Proposed fill, γ=20 kN/m3, E=23000 kPa
sample from point A produced the following results: cc=0.5,
cr=0.05, e0=1.0 and σ'c=60 kPa. This sample is representative 5.0m Sand
3
γ=18 kN/m , E=15000 kPa
of the entire soft clay stratum. After the settlement due to the
fill is completed, a 20 m diameter, 10 m tall cylindrical steel
3.0m
water tank is to be built. The bottom of the tank will be at the
1.0m in the fill, and it will have an empty mass of 3000 kN. 6.0m Clay, γ=15 kN/m
3 A
Ultimately, the water inside will be 9.0m deep. Compute the
ultimate consolidation settlement of the clay layer: Eu=15000 kPa
(a) Due to the weight of fill.
Dense Sand and Gravel
(b) Beneath the center and edge of the tank due to the weight
of the tank and its contents.
(c) Due to the weight of the fill and weight of the tank as well Note: It will be solved in the class
as its contents if the tank were built immediately after the fill.

Solution:
′ (𝑎𝑡
Check the consolidation state of clay layer: 𝜎𝑧0 𝐴) = ∑ 𝛾𝐻 − 𝑢ℎ = 18 ∗ 55 + 15 ∗ 3 − 10 ∗ 8 = 55 𝑘𝑃𝑎,
𝜎𝑐′ (𝑎𝑡 𝐴) ≈ 𝜎𝑧0
′ (𝑎𝑡
𝐴) (Difference is less than 8%), Clay layer is NC.
Part (a) Settlement due to filling
Stress at mid-point of layer (kPa) Compressibility Ultimate settlement (mm)
′
Case I: Case II: CaseIII:
𝜎𝑧𝑓 𝑐𝑟
Thickness

′ 𝜎′
cc/(1+e0)

cr/(1+e0)

𝑐𝑟
Layer No.

′ 𝜎𝑧𝑓
𝜎𝑧0 ′
𝑐𝑐
𝐻𝑙𝑜𝑔 ( ′ ) 𝐻 𝐻𝑙𝑜𝑔 ( ′𝑐 ) +
∆𝜎𝑖𝑛𝑑. = 𝜎𝑧0 1+𝑒0 𝜎𝑧0 1 + 𝑒0
1+𝑒0 𝜎𝑧0
H, (m)

= ∑ 𝛾𝐻 − 𝑢ℎ + ∆𝜎𝑖𝑛𝑑. ′
𝜎𝑧𝑓 𝑐𝑐 𝜎′
𝐻𝑙𝑜𝑔 ( ′𝑐 )
𝑙𝑜𝑔 ( ′ ) 1+𝑒 0 𝜎𝑧𝑓
𝜎𝑧0
18*5+15*1- 3*20 105
1 2 105 0.250 -- 0.25*2*𝑙𝑜𝑔 ( )=184 -- --
10*(5+1)=45 =60 45
18*5+15*3- 115
2 2 60 115 0.250 -- 0.25*2*𝑙𝑜𝑔 ( )=160 -- --
10*(5+3)=55 55
18*5+15*5- 125
3 2 60 125 0.250 -- 0.25*2*𝑙𝑜𝑔 ( )=142 -- --
10*(5+5)=65 65

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 17

 Compressibility and Settlement

Sum 486 mm
Note: If 𝜎𝑐′ is slightly greater than 𝜎𝑧0
′
(perhaps less than 20% greater), then it may not be clear if the soil is truly OC. In such cases
it is acceptable to consider NC soil (Case I) or OC (Case III). The settlements of the fill (due to self-weight) and sand layer are
ignored. Sublayer thickness has negligible effect on the result due to uniform induced stress distribution with depth.
𝜋
𝑆𝑒𝑙𝑓 𝑤𝑒𝑖𝑔ℎ𝑡+𝑊𝑎𝑡𝑒𝑟 𝑠𝑡𝑜𝑟𝑎𝑔𝑒 3000+ (202 )∗9.0∗10
Part (b) Net pressure due to water tank, ∆𝑞 = 𝐵𝑎𝑠𝑒 𝑎𝑟𝑒𝑎
-𝛾𝐷= 4
𝜋
(202 )
− 20 = 80 𝑘𝑃𝑎
4
Ultimate settlement below

Compressibility,
At mid-point of the layer
center of tank(mm)
Thickness
Layer No

H, (m)

∆𝜎𝑖𝑛𝑑𝑢𝑐𝑒𝑑

cc/(1+e0)
′
′ 1.5 ′ 𝑐𝑐 𝜎𝑧𝑓
𝜎𝑧0 z 1 𝜎𝑧𝑓 𝐻𝑙𝑜𝑔 ( ′ )
= ∆𝑞 [1 − { } ] 1 + 𝑒0 𝜎𝑧0
1 + (𝐵/2𝑧)2

1 2 105 8.0 60.5 165.5 0.250 98.8

2 2 115 10.0 51.7 166.7 0.250 80.6
3 2 125 12.0 43.7 168.7 0.250 65.1
Sum 244 mm
Note: Here, for simplicity, Polous and Davis formula is used to compute induced stress but Bousinessq method can be used for more
accurate analysis for the settlement at the edge and center. Sublayer thickness may affect the result due to decreasing stress.

For the edge, you have to use Bousinessq method. The settlement will be less than the settlement below the
center.

Part (c) Do yourself (Hints: Induced stress will be summation of fill and tank pressure at a time)

Home practices: All examples and problems of Coduto and Holts and Kovacs 2nd Edn.

Assignment: Problems/Coduto: 11.12, 11.25, 11.28, and 11.31

Effect of Soil Type and Foundation Size on Consolidation Settlement

The majority of cases in the field, the size of the loaded area is too small compared with the thickness of the
saturated clay layer and more importantly when clay is overconsolidated (i.e. small pore pressure coefficient,
A values) the consolidation settlement calculated from Terzaghi’s equations is corrected to make allowance
for the effect of foundation size. Skempton and Bjerrum (1957) suggested a correction factor for
overconsolidation and three-dimensional loading for consolidation settlement: 𝑆𝑐 = 𝜇𝑆𝑐(1−𝐷) .

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 18

 Compressibility and Settlement

𝐻
𝑆𝑐 ∫0 𝑚𝑣 ∆𝑢 𝑑𝑧
𝜇= = 𝐻
𝑆𝑐(1−𝐷) ∫0 𝑚𝑣 ∆𝜎′𝑣 𝑑𝑧
Using Skempton’s pwp coefficient (𝐴𝑓 ),
dissipated pwp for saturated soil considering
3D effect is ∆𝑢 = ∆𝜎ℎ′ + 𝐴𝑓 (∆𝜎𝑣′ − ∆𝜎ℎ′ )
Figure: Correction factor for foundation size (Skempton and Bjerrum, 1957)
Immediate Settlement for Shallow Foundation
•In theory (Terzaghi 1D method), the method used to predict the settlement of spread footings in clays and
silts also could be used for sands and gravels with some correction. However, to use these methods we would
need to evaluate cc and cr in these soils, which would be very difficult or impossible because of difficulties in
obtaining undisturbed samples. In this case, Schmertmann’s Method (1978) is widely used for granular soils
to compute the average settlement:
Iε H
Se = C1 C2 C3 (q − σ′zD ) ∑ Es
where, q=bearing pressure, σ′zD = effective vertical stress at depth D from GL (if heave settlement is computed
separately, set it to zero), Es = modulus of elasticity of soil layers= 10 N60Pa for clean sands, 5 N60Pa for clayey
and silty sands (Kulhawy and Mayne, 1990); For compacted (97% of Modified Proctor) structural granular
fill, you may assume, N60=32. For mat foundation, it is better to increase its value progressively.
σ′
C1 = depth factor=1 − 0.5 (q−σzD′ )
zD
t
C2 = secodnary creep factor = 1 + 0.2log (0.1); t=time since application of load (yr) ≥0.1, if no time is
given use 50 years.
0.03L
C3 = shape factor = 1 for suqre and circular, = 1.03 − ≥ 0.73for rectangular; B=width of
B
foundation and L=length of foundation.

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 19

 Compressibility and Settlement

Iε = strain influence factor. The exact value of Iε at any

depth may be computed using following equations:
Square and circular foundations:
For zf= 0 to B/2: Iεs = 0.1 + (zf /B)(2Iεp − 0.2)
For zf= B/2 to 2B: Iεs = 0.667Iεp (2 − zf /B)
Continuous foundations:
For zf= 0 to B: Iεc = 0.2 + (zf /B)(Iεp − 0.2)
For zf= B to 4B: Iεc = 0.333Iεp (4 − zf /B)
Rectangular foundations (1 <L/B<10):
L
Iεr = Iεs + 0.111(Iεc − Iεs ) ( − 1)
B
where
q−σ′zD
Iεp = peak strain influence factor = 0.5 + 0.1√ ;
σ′zp
σ′zp =initial vertical effective stress at the depth of peak
strain influence factor (for square and circular foundations,
it is at a depth of D+B/2 below GL; for continuous
footings (L/B≥10), it is at D+B)

•For clay soil, Skempton and Bjerrum (1957) can be used:

B
Se = (q − σ′zD ) I1 I2
Eu
where Eu = undrained modulus of elasticity of soil=300su. Alternatively it can be found from triaxial
compression test (CU). Skempton (1951) points out that when the factor of safety is 3.0 the maximum shear
stress induced in the soil is not greater than 65 per cent of the ultimate shear strength, so that a value of E can
be obtained directly from the triaxial test results by simply determining the strain corresponding to 65 per cent
of the maximum deviator stress.

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 20

 Compressibility and Settlement

Figure: Influence factors I1 and I2

Homework Problem: Computation of Total Settlement
A concrete rectangular water tank having a base of 10m by
30m and height of 10 m is to be built. The bottom of the tank 3.0m Proposed fill, γ=20 kN/m3, E=23000 kPa
will be at the 1.0m in the fill, and it will have an empty mass
of 3000 kN. The water inside will be 9.0m deep. Assume that 5.0m Sand
3
γ=18 kN/m , E=15000 kPa
𝜇 = 0.55 and the tank will be constructed long after the
placement of the filling.
(a) Compute short term settlement due to the construction of
the tank and its content. 6.0m NC Clay, γ=15 kN/m
3

(b) Compute the long term settlement due to the construction

of the tank and its content. Eu=15000 kPa, cc=0.5, e0=1.0
(c) Compute the total settlement due to the construction of
Very dense sand
the tank and its content.
Note: (a) will be solved in the class.

Secondary Settlement
• It appears to be due to particle reorientation,
creep, decomposition of organic content
• Highly plastic clays, organic soils, and sanitary
landfills have significant secondary compression.
• If secondary compression is estimated to
exceed serviceability limitations, either deep
foundations or ground improvement should be
considered to mitigate the effects of secondary
compression. Experience indicates preloading
and surcharging may not be effective in
eliminating secondary compression.
• The secondary compression index can be
defined as (see right figure) in lab. deformation
curve

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 21

 Compressibility and Settlement

∆𝑒
𝐶𝛼 = − 𝑡
𝑙𝑜𝑔 𝑡2
1
• The secondary settlement of a thin layer can be
derived as
H
𝐶𝛼 𝑡
Ss = ∫ εz dz = 𝐻𝑙𝑜𝑔
0 (1 + 𝑒𝑝 ) 𝑡𝑝
where 𝑒𝑝 is the void ratio at the end of primary consolidation (can use ep≈e0 without introducing much error),
𝑡𝑝 (≈ 𝑡95 ) is time required to complete primary consolidation in the field for the thin layer.

• Practice Example 11.10/Coduto

Sources of Errors in Settlement Prediction:

• Differences between soil profile used in analysis and the real soil profile especially proper identification of
crusts.
• Differences between the engineering properties of the soil samples and the average properties of the strata
they represent (i.e. are they truly representative?)
• Sample disturbances.
• Errors introduced due to testing technique in the laboratory.
• Error in assessing preconsolidation pressure.
• The assumption that consolidation in the field is one dimensional.
• Differences between Terzaghi’s theory of consolidation and the real behavior of soil in the field.

Dr. M. Rokonuzzaman, Department of Civil Engineering, KUET 22