void ratio correlations

CORRELATIONS OF VOID RATIO

Presented By: Muhammad Ali Rehman

Soil

Void Ratio

INTRODUCTION

Soil

To a Geotechnical Engineer,

Soil is considered to be a

three-phase material

composed of, solid (mineral

particle), water & air.

Void Ratio

Total volume of a soil sample canbe expressed as:

V = Vs + Vv

V = Vs + Vw + Va

The ratio of volume of voids to the volume of solids is known as void ratio.

e = Vv/Vs

Void Ratio

Value of void ratio depends on:

the volumetric changes of the soil.

the consistency

packing of soil.

Void ratio characterizes the compactness of the soil.

Void ratio of loose soil is higher than that of dense soil.

Can be determined only from undisturbed samples.

Typical Void ratio value for

Dense gravel: 0.3

Loose sand: 0.6

Clays: 0.5 < to <1.0.

Correlations

Porosity

Void ratio is usually used in parallel with soil

porosity.

Porosity is the ratio of volume of voids to the total

volume.

e = 𝑛

1−𝑛

Or

n = 𝑒

1+𝑒

Unit Weight

The relation of Dry Unit Weight with void ratio:

𝛾𝑑 = 𝐺𝑠𝛾𝑤

1+𝑒

e = 𝐺𝑠𝛾𝑤

𝛾𝑑− 1

Moisture content, Dry Density

S.e = 𝐺𝑠. 𝑤𝑛

𝑤𝑛 = 𝑆.𝑒

𝐺𝑠

e = 𝐺𝑠. 𝑤𝑛 (if S=1)

𝜌𝑑 = 𝐺𝑠.𝜌𝑤

1+𝑒(𝜌𝑤 = 1000 kg/𝑚3)

emax & emin

The maximum and minimum void ratios for granular soils

depend on several factors, such as:

Grain size

Grain shape

Fine contents, Fc (that is, fraction smaller than 0.075 mm)

emax is the void ratio of soil in loosest state

emin is the void ratio of soil in densest state

emax & emin

The amount of non-

plastic fines present in

a given granular soil

has a great influence

on emax and emin.

Influence of fines on void ratio

of Nevada Sand, Lade et al.

(1998)

emax & emin

Miura et al. (1997) determined the maximum and

minimum void ratios of a larger number of clean

sand samples.

emax ≈ 1.62emin

emax & emin

Cubrinovski and Ishihara (2002) and Patra et al.

(2010).

emax & emin

Cubrinovski & Ishihara (2002) studied the variation

of emax and emin for very large number of soils.

Clean Sand (Fc = 0 to 5%)

emax = 0.072 + 1.53emin

Sands with fines (5 < Fc ≤ 15%)

emax = 0.25 + 1.37emin

Sands with fines (15 < Fc ≤ 30%)

emax = 0.44 + 1.21emin

Silty soils (30 < Fc ≤ 70%)

emax = 0.44 + 1.32emin

emax & emin With Mean Grain Size

Plot of emax - emin versus the mean grain size (D50):

Cubrinovski and Ishihara (2002)

Relative Density

Relative density is commonly used to indicate the in situ denseness or looseness of granular soil.

Dr = 𝑒𝑚𝑎𝑥 − 𝑒

𝑒𝑚𝑎𝑥 −𝑒𝑚𝑖𝑛

Dr = relative density (usually in percentage)

e = in situ void ratio

emax = void ratio of soil in loosest state

emin = void ratio of soil in densest state

Shear Modulus

The small-strain shear modulus of soils, Gmax, is an important parameter for many geotechnical design applications, including site characterization, settlement analyses, seismic hazard analyses, and site response analysis and soil-structure interaction.

Hardin (1978) suggested that Gmax for clays depends on the in situ (or applied) stress (σ'), void ratio (e), and OCR.

The effects of OCR are, to a large extent, taken into account by the effect of void ratio and could be neglected, (Leroueil and Hight, 2003).

Shear Modulus

Hardin (1978) and

Hight & Leroueil (2003):

Void Ratio, e

Hydraulic Conductivity

One of the most important and useful parameter in the study of percolation process in porous media, consolidation & settlement of soils and foundation, water regime in stratified deposits, and other geotechnical problem.

Kozney-Carman relation, (Kozney 1927 and Carman 1937):

k: hydraulic conductivity (m/s)

e: void ratio,

Ss: specific surface area (𝑚2/𝑔)

CF: shape factor { ≈ 0.2 (Taylor, 1948)}

γw = unit weight of water (N/𝑚3)

ρm = density of soil (kg/𝑚3)

μ = Viscosity of fluid (N.s/𝑚2)

Hydraulic Conductivity

Carrier (2003) has modified the Kozney-Carman

relation into:

k = 𝟏. 𝟗𝟗 × 𝟏𝟎𝟒 𝟏

𝑺𝑺

𝟐×

𝒆𝟑

𝟏+𝒆

Carrier (2003) further suggested that:

Hydraulic Conductivity

Taylor (1948) and Lambe & Whitman (1969)

proposed:

Ck: permeability change index i.e. the

slope of e versus log(k) plot

k0: hydraulic conductivity for

reference void ration e0

Hydraulic Conductivity

Samarasinghe et al. (1982) proposed an equation:

C: constant with same unit as k,

n: constant that depends on type

of soil and varies from 3.2 to 14.2

Hydraulic Conductivity

Variation in hydraulic conductivity with void ratio:

Mesri & Olson (1971)

Void Ratio-Pressure Plot

Typical plot of void ratio against effective pressure

(semi logarithmic scale)

e0: initial void ratio of specimen

e1: void ratio after consolidation caused

by pressure increment σ’1

e2: void ratio at the end of consolidation

caused by next increment of

pressure σ’2

Pre-consolidation Pressure

Cassagrande (1936) proposed a simple graphical method to determine the pre-consolidation pressure from laboratory e-logσ’ plot.

Draw a horizontal line ab.

Draw the line ac tangent at a.

Draw the line ad, which is thebisector of the angle bac.

Project the straight-line portiongh of the e-log σ’ plot back tointersect line ad at f.

The abscissa of point f is thepre-consolidation pressure (σ’c).

Pre-consolidation Pressure

Nagaraj & Murty (1985):

e0: in situ void ratio

eL: void ratio at liquid limit = 𝐿𝐿 (%)

100. 𝐺𝑠

Gs: specific gravity of soil

σ’0: in situ effective overburden pressure (kN/𝑚2)

Cc : Compression Index

Rendon-Herrero (1983) gave the relationship for the compression index in the form:

𝐶𝑐 = 0.141𝐺𝑠2 1+𝑒0

𝐺𝑠

2.38

𝐶𝑐 = 1.15(𝑒0- 0.27) by Nishida (1956) – all clays

𝐶𝑐 = 0.156𝑒0 + 0.0107) by Hough (1957) – All clays

𝐶𝑐 = 0.30(𝑒0- 0.27) by Hough (1957) – inorganic cohesive soils

𝐶𝑐 = 0.30(𝑒0- 0.27) by Hough (1957) – low plasticity soils

𝐶𝑐 = 0.208𝑒0 + 0.0083 by Hough (1957) – Chicago Clays

Variation of Void Ratio with Shearing

Displacement

At large shear displacement, the void ratios of loose and dense sands become practically the same, and this is termed the critical void ratio.

Variation of tan𝜙’ with Void Ratio

Acar, Durgunoglu, and Tumay (1982)

Figure shows the results of direct shear tests

conducted with a quartz sand and concrete,

wood, and steel as foundation

materials, with σ’ =100 (kN/𝑚2).

References

Principles of Geotechnical Engineering, 7th Ed, B.M. Das

An Introduction to Geotechnical Engineering, Robert D. Holtz, and William D. Kovacs

Soil Mechanics in Engineering Practice, 3rd Ed, Karl V. Terzaghi, Ralph B. Peck, and Gholamreza Mesri

Correlations between Shear Wave Velocity and Geotechnical Parameters in Norwegian Clays, J. S. L’Heureux and M. Long

S. M. Rezwan Hossain, MD. Abdul Qaiyum Talukder, Shariful Islam, MD. Rafiue Islam, “Significance of Silt Content and Void Ratio on the Hydraulic Conductivity of Sand-Silt Mixtures”, International Journal of Advanced Structures and Geotechnical Engineering ISSN 2319-5347, Vol. 02, No. 04, October 2013