Foundation Design: Theory and Practice

Chapter 1

Introduction

1.1 Foundations, Soils and Superstructures

Foundations are essential to transfer the loads coming from the superstructures such as buildings, bridges, dams, highways, walls, tunnels, towers and for that matter every engineering structure. Generally that part of the structure above the foundation and extending above the ground level is referred to as the superstructure. The foundations in turn are supported by soil medium below. Thus, soil is also the foundation for the structure and bears the entire load coming from above. Hence, the structural foundation and the soil together are also referred to as the substructure. The substructure is generally below the superstructure and refers to that part of the system that is below ground level. Thus, the structural foundation interfaces the superstructure and the soil below as shown in Figures 1.1 and 1.2. The soil supporting the entire structure above is also referred to as subsoil and/or subgrade. For a satisfactory performance of the superstructure, a proper foundation is essential.

Figure 1.1 Building with spread foundations.

Figure 1.2 Superstructure with pile foundations.

The manmade superstructures or facilities/utilities are expected to become very intricate and complex depending on creativity, architecture and infinite scope in modern times. However, the soil medium is mother earth which is a natural element and very little can be manipulated to achieve the desirable engineering properties to carry the large loads transmitted by the superstructure through the interfacing structural foundation (which is usually referred to as the foundation). Further, almost all problems involving soils are statically indeterminate (Lambe and Whitman, 1998) and soils have a very complex behavior, as follows:

1. Natural soil media are usually not linear and do not have a unique constitutive (stress–strain) relationship.

2. Soil is generally nonhomogeneous, anisotropic and location dependent.

3. Soil behavior is influenced by environment, pressure, time and several other parameters.

4. Because the soil is below ground, its prototype behavior cannot be seen in its entirety and has to be estimated on the basis of small samples taken from random locations (as per provisions and guidelines).

5. Most soils are very sensitive to disturbances due to sampling. Accordingly, their predicted behavior as per laboratory samples could be very much different from the in situ soil.

Thus, foundation design becomes a challenging task to provide a safe interface between the manmade superstructure and the natural soil media whose characteristics have limited scope for manipulation. Hence, the above factors make every foundation or soil problem very unique which may not have an exact solution.

The generally insufficient and conflicting soil data, selection of proper design parameters for design, the anticipated mode for design, the perception of a proper solution and so on require a high degree of intuition – that is, engineering judgment. Thus, foundation engineering is a complex blend of soil mechanics as a science and its practice through foundation engineering as an art. This may be also referred to as geotechnique or geotechnical engineering.

1.2 Classification of Foundations

Foundations are classified as shallow and deep foundations based on the depth at which the load is transmitted to the underlying and/or surrounding soil by the foundation as follows.

1.2.1 Shallow Foundation

A typical shallow foundation is shown in Figure 1.3(a). If Df/B ≤ 1, the foundations are called shallow foundations, where Df = depth of foundation below ground level, and B = width of foundation (least dimension). Common types of shallow foundations are continuous wall footing, spread footing, combined footing, strap footing, grillage foundation, raft or mat foundation and so on. These are shown in Figure.

Figure 1.3 Shallow and deep foundations.

All design and analysis considerations of shallow foundations are discussed in Chapters 4, 5, 6, 7, 8 and 12. The shallow foundations are thus used to spread the load/pressure coming from the column or superstructure (which is several times the safe bearing pressure of supporting soil) horizontally, so that it is transmitted at a level that the soil can safely support. These are used when the natural soil at the site has a reasonable safe bearing capacity, acceptable compressibility and the column loads are not very high.

1.2.2 Deep Foundations

A typical deep foundation is shown in Figure 1.3(b). If Df/B ≥ 1, the foundations are called deep foundations such as piles, drilled piers/caissons, well foundations, large diameter piers, pile raft systems. The details of analysis and design of such foundations are discussed in Chapters 9 and 10.

Deep foundations are similar to shallow foundations except that the load coming from columns or superstructure is transferred to the soil vertically. These are used when column loads are very large, the top soils are weak and the soils with a good strength and compressibility characteristics are at a reasonable depth below ground level. Further, earth retaining structures are also classified under deep foundations.

Foundations can be classified in terms of the materials used for their construction and/or fabrication. Usually reinforced concrete (RCC) is used for the construction of foundations. Plain concrete, stone and brick pieces are also used for wall footings when the loads transmitted to the soil are relatively small. Engineers also use other materials such as steel beams and sections (such as in grillage foundations and pile foundations), wood as piles (for temporary structures), steel sheets (for temporary retaining structures and cofferdams) and other composite materials.

Sometimes, these are also encased in concrete depending on the load and strength requirements (Bowles, 1996; Tomlinson, 2001).

1.3 Selection of Type of Foundation

While engineering judgment and cost play a very important role in selecting a proper foundation for design, the guidelines given in Table 1.1 can be helpful (please see also Chapters 4, 5, 6, 7, 8, 9, 10, 11, 12).

Table 1.1 Foundation types.

1.4 General Guidelines for Design

Following broad guidelines may be useful for foundation design and construction, depending on site.

1. Footings should be constructed at an adequate depth below ground level to avoid passive failure of the adjacent soil by heaving.

2. The footing depth should be preferably below the zone of seasonal volume changes due to freezing, thawing, frost action, ground water and so on.

3. Adequate precautions have to be taken to cater for expansive soils causing swelling pressure (upward pressure on the footing).

4. The stability of the footing has to be ensured against overturning, sliding, uplift (floatation), tension at the contact surface (base of the footing), excessive settlement and bearing capacity of soil.

5. The foundation needs to be protected against corrosion and other harmful materials that may be present in the soil at site.

6. The design should have enough flexibility to take care of modifications of the superstructure at a later stage or unanticipated site conditions.

1.5 Modeling, Parameters, Analysis and Design Criteria

All practical problems need to be reduced to physical models and behavior represented by corresponding analytical equations. The physical parameters of the system form the inputs in the mathematical equations for computing the responses. The models used should be simple enough that the physical parameters needed for computations are accurately and reliably determined using inexpensive test procedures. For example, in a foundation–soil system, the foundation can be modeled as rigid, while the soil may be assumed to be elastic. The physical parameters needed in such a model are the elasticity parameters of the soil, that is. Young's modulus of elasticity, E, and Poisson's ratio, v, of the soil. Naturally E and have to be accurately determined for the soil under consideration as they will be needed for the computation of the responses of the system. Thus modeling, evaluation of parameters and analysis are closely linked and the solutions obtained are highly dependent on all these aspects.

The responses thus obtained have to be judged using appropriate design criteria specified either by codes or evolved from practice and/or experience.

The design process necessarily has two vital components, namely the methods of analysis and experimental data which have to be integrated with them to yield accurate results. However, both the methods and data depend entirely on the mechanism chosen for mathematical idealization of the system components. At this juncture, engineering judgment and experience is very useful. It may be noted that optimum accuracy in analysis and design can be achieved only by properly matching the data and analytical methods used. It is also obvious that any improvement in the data alone or any sophistication in the analytical methods alone can even reduce the accuracy of the results/predictions (Lambe, 1973).

1.6 Soil Maps

Most countries have prepared maps of soil deposits, based on the geological and geotechnical data available. These are very useful for a quick assessment of the project and its requirements. A map of soil deposits in India is given in Figure 1.4 (Ramiah and Chickanagappa, 1981).

Figure 1.4 Soils of India.

Adapted from B.K. Ramiah and L.S. Chickanagappa, Soil Mechanics and Foundation Engineering, p. 3 (Figure 1.1), Oxford and IBH Publishing Co., New Delhi, India. © 1981.

Chapter 2

Engineering Properties of Soil

2.1 Introduction

The physical and engineering properties of soil are necessary for foundation design, as all loads are ultimately supported by the soil media and occasionally by rock medium if present at the site. For engineering applications, soils include all earth materials, organic and inorganic, present in the zone overlying the rock crust of the planet earth.

This chapter presents the engineering properties of soils relevant to foundation design, such as simple soil properties, strength and compressibility characteristics and so on. The laboratory and field tests necessary to evaluate the parameters are also discussed briefly.

However, for more detailed discussion, one may refer to classical and recent books on Soil Mechanics, Geotechnical Engineering, and Foundation Engineering, such as Terzaghi (1943), Taylor (1964), Terzaghi and Peck (1967), Ramiah and Chickanagappa (1981), Shamsher Prakash and Sharma (1990), Cemica (1994), Coduto (2001), Tomlinson (2001), Das (2002, 2007)), Reese, Isenhower and Wang (2005), Budhu (2006), Salgado (2007). In the case of foundations on rock, the relevant properties of rock have to be studied, as discussed in standard rock mechanics books, such as Goodman (1989), Brady and Brown (2006), Jaeger and Cook (2007).

2.2 Basic Soil Relations

Soil is formed by the weathering of parent rock as a continuous geological process. It may be identified broadly as residual and/or transported. Residual soils are formed due to weathering of parent rock at its present location. Usually such soils consist of angular grains of different sizes. Residual soils are considered good for supporting a foundation. Transported soils are those that are formed at one location and are transported to their present location by nature, that is, wind, water, ice or gravity. They are of poor quality and are fine grained with low strength and high compressibility.

Thus, soils consist of irregular shaped particles of different sizes and shapes, that is, solids. In addition, there are voids between these particles (pores), which may be filled partly or fully by air and water. Thus, the soil mass can be symbolically represented as a three phase material, as shown in Figure 2.1. The various parameters shown in the figure are defined as follows

Figure 2.1 Representation of soil as a three-phase material.

V, W = total volume and weight of soil mass respectively

Vs, Ws = volume and weight of soil solids respectively

Vw, Ww = volume and weight of water respectively

Vg = volume of gas.

(2.1) equation

The basic parameters used in geotechnical engineering studies are void ratio, e, porosity, n, water content, w and degree of saturation, S. These are defined as follows

(2.2) equation

Besides these parameters, the unit weights of the soil mass and its variations with changes in water content are important and they can be expressed as follows

(2.3)

where , which varies between 2.65 and 2.85 for the majority of soils.

It can be shown from Equation (2.2) that, for a soil mass

(2.4) equation

Hence

(2.5)

When layers of soils are submerged due to ground water present at site, then the soil mass in saturated and is subjected to buoyancy. Accordingly, we can define

(2.6)

All these soil properties are routinely determined by standard laboratory tests and also by field tests (Lambe, 1951; Taylor, 1964).

2.2.1 Grain Size Distribution

Grain size distribution (GSD) is also a basic soil property which affects its engineering properties considerably and is used in most soil classification systems. Mechanical sieve analysis is used to determine the grain size distribution of coarse grained soils such as sands. For fine grained soils, hydrometer analysis is used for determining the distribution of grain size (Lambe, 1951; Taylor, 1964) as grain sizes less than 0.074 mm (sieve size No. 200 BS and US) are the smallest sizes that are visible to the naked eye and can be mechanically sieved. Typical sieve sizes used for sieve analysis of coarse grained soils are given in Table 2.1.

Table 2.1 Sieve sizes.

Also typical grain size distribution curves are shown in Figure 2.2. If the curve is smooth and is spread evenly with almost constant slope as shown in curve 1, it is called a well graded soil. If the slope of the curve is wavy as shown in curve 2, it is called poorly graded. If the curve has very steep slope with most of the soil particles being of almost same size as shown in curve 3, it is called uniformly graded soil. A commonly accepted method to express the general features of the GSD curve is due to Hazen (Taylor, 1964) which uses the grain sizes D10 and D60 (respectively, diameter finer than 10 and 60%) to define the uniformity coefficient, cu as

Figure 2.2 Typical grain size distribution curves.

(2.7) equation

where D10 = effective size which is used in several engineering applications such as in permeability studies. For example, Hazen's formula (Taylor, 1964) for the coefficient of permeability, k in filter sands is

(2.8)

GSD curves are used in almost all soil classification systems, as shown in Figures 2.3 and 2.4. A typical classification system of soils using grain sizes of particles is given in Table 2.2 (Das, 2007), besides the ones shown in Figures 2.3 and 2.4.

Figure 2.3 Classifications based on grain size (in mm).

Figure 2.4 United States Bureau of Soils triangular classification chart.

Table 2.2 General classification of soils.

The general names given to various soils in the above table and figures convey additional information about their engineering behavior. For example, clays are cohesive with plasticity. The cohesion of the clay is represented by c. Similarly, sands and gravel are nonplastic with only frictional properties represented by angle of internal friction, ϕ. Silts have low plasticity and have cohesion and very low friction. These soils can be identified by simple tests like the dispersion test, shaking test and rolling test (Taylor, 1964).

2.2.2 Plasticity and the Atterberg's Limits

Plasticity (mainly in clays or cohesive soils) is a predominant feature of fine grained soils such as clays or cohesive soils. It is defined as the ability of the material or soil to undergo deformation/distortion/change of shape without rupture or crack. Water content affects the physical properties of clays. Atterberg (Taylor, 1964) proposed a series of tests for determining these effects which are known as Atterberg Limits (also referred to as Consistency Limits). A lot of useful empirical formulae have been developed over the years to correlate these limits to strength, compressibility and other important engineering properties of the soil. These are simple tests and are routinely conducted in the laboratories and throw lot of information on the soil for soil mechanics and foundation engineering applications. The Atterberg limits are shown in Figure 2.5.

Figure 2.5 Representation of Atterberg limits.

These are briefly explained below depending on their physical state as functions of water content. If a lot of water is added to a clayey soil, it may start flowing and behave like a semiliquid state. The limit at which the soil behaves like a semiliquid is called the liquid limit (LL). This is determined in the laboratory by Casagrande's LL device and is defined as the water content at which a groove closure of 12.7 mm occurs at 25 standard blows.

If the soil is dried gradually, it behaves in a plastic, semi solid or solid state. The limit between plastic and semi solid states is called the plastic limit (PL), as shown in Figure 2.5. It is determined in the laboratory as the moisture content at which the soil shows visible cracks/crumbles when rolled into a thread 3.18 mm in diameter.

The water content limit at which the soil changes from a semi solid to solid state is called the shrinkage limit (SL). It is also easily determined in the laboratory as the water content at which the soil does not undergo any further volume change with loss of moisture (Figure 2.5). The liquid and plastic limits of few well studied clays and silts are given in Table 2.3.

Table 2.3 Liquid and plastic limits of clay minerals and clayey soils.

The following indices are also useful in analyzing the behavior of soils.

(2.9)

where

w = natural water content of the soil

wp = water content at plastic limit

wL = water content at liquid limit.

If LI ≥ 1, it may indicate the possibility of liquefaction, that is, a loss of soil strength after a few cycles of loading and unloading resulting in liquid like behavior.

2.3 Soil Classification

Based on the Atterberg's limits and Grain size distribution, soils are classified by several agencies in most countries, like the AASHTO and Unified systems. The focus in classification is on the purpose for which the soil is used. The most popular classification is due to Casagrande and is referred to as the Unified classification. It is presented as a plasticity chart shown in Figure 2.6; Table 2.4 shows the procedure for assigning symbols for various soils.

Figure 2.6 Plasticity chart.

Table 2.4 Unified soil classification system (based on material passing 75-mm sieve).

2.4 Permeability

Since soil is porous, water can flow through the pores, which is also referred to as seepage. The ease with which water flows through the soils is represented by the coefficient of permeability of soils, k. The velocity follows Darcy's Law as

(2.10a) equation

where

v = superficial velocity (assuming water is flowing through the entire cross section including pores and soil particles)

i = hydraulic gradient =

h = loss of head between any two cross sections of flow

L = straight distance between the cross sections.

k can be determined in the laboratory using constant head permeameter and/or variable head permeameter (Taylor, 1964). k can also be determined in the field (in situ) by pumping tests.

2.4.1 Quick Sand Condition and Critical Hydraulic Gradient

As the hydraulic gradient increases, the seepage force acting on the soil particles gradually increases and starts pulling the particles out in the direction of flow. This phenomenon is called the quick sand condition where the soil particles appear to be boiling. This happens when the buoyant weight or submerged weight of the soil equals the seepage force when the flow is opposite to the direction of gravity. This gradient is called critical hydraulic gradient, ic and can be obtained as

Hence

(2.10b) equation

This value generally ranges from 0.8 to 1.3 and it may be taken as 1.0 for average conditions in the absence of data.

2.5 Over Consolidation Ratio

A soil whose present overburden pressure is the largest pressure ever experienced by this soil is referred to as normally consolidated soil. If otherwise, it is called an over consolidated soil. The ratio of the past effective pressure, , to the present overburden pressure, , is called the over consolidation ratio (OCR), that is

(2.11) equation

For normally consolidation soils, OCR = 1. For over consolidation soils, OCR > 1.

If OCR < 1, it has no significance.

If OCR > 1–3, the soils are lightly over consolidated.

If OCR > 3–8 or more, the soils are heavily over consolidated.

OCR has a very significant effect in the behavior of clayey soils though its effect is marginal in sandy soils. OCR can be determined by the consolidation test (oedometer test) in the laboratory as described in Section 2.9.

2.6 Relative Density

The degree of compaction in granular soils in the field can be determined by the relative density, Dr, expressed as a percentage as

(2.12) equation

where

emax = void ratio of the soil in the loosest state

emin = void ratio of the soil in the densest state

e = in situ void ratio.

The various void ratios can be determined in the laboratory using standard methods. The relative density can also be expressed in terms of dry unit weights as

(2.13) equation

where

γd = in situ dry density of soil

γd(max) = dry unit weight in the densest state (corresponding emin)

γd(min) = dry unit weight in the loosest state (corresponding emax).

The denseness of the soil is correlated to the relative density, Dr, as given in the Table 2.5.

Table 2.5 Denseness of soils.

Figure 2.7 Intergranular or effective stress.

2.7 Terzaghi's Effective Stress Principle

If a soil mass shown in Figure 2.7 is subjected to a total stress, σ, then from equilibrium we can express

(2.14) equation

where

As = contact area between solid grains

A = total area of cross section of the soil mass

u = pore water pressure

σ′ = vertical component of stress of the contact (over the unit cross sectional area)

= vertical effective stress.

Usually a is negligible in comparison to 1 and hence Equation 2.14 can be expressed as

(2.15) equation

where

σ = total stress at any point in the soil mass

σ′ = effective stress (stress between the solid to solid contact)

u = pore water pressure.

This is called the effective stress principle formulated by Terzaghi (1943) and is one of the important concepts in soil mechanics and foundation engineering. It can be readily recognized that stresses and hence strains and displacements (settlements) occur only due to changes in effective stresses.

2.8 Compaction of Soils

A soil mass can be made denser by compacting with some mechanical energy (static or dynamic) and its unit weight generally increases. The dry unit weight increases with the gradual increase of water content and subsequent compaction. This is because the additional water acts as a lubricant and helps in rearranging the soil particles into a denser state of packing. The dry unit weight increases with the water content up to a maximum or limiting value beyond which it decreases with increase in water content, as shown in Figure 2.8.

Figure 2.8 Standard and modified Proctor compaction curves for a fine grained soil.

The moisture content at which the soil reaches its maximum dry density is called the optimum moisture content (OMC).

The OMC and maximum dry density of soils can be determined by standard laboratory tests such as the standard Proctor Test (using a 2.5 kg rammer and a drop of 305 mm) and the modified Proctor Test (using a 4.54 kg rammer and a drop of 457 mm; Taylor, 1964; Das, 2002).

Typical curves from these compaction tests are shown in Figure 2.8. These results are used for specifying the methods of field compaction. Usually the field compaction is required to achieve a relative compaction (RC) of 90% or more of the max dry density obtained in laboratory using either the standard or modified Proctor test (or other tests specified by local codes), that is

(2.16) equation

where

Dr = relative density defined in Equation (2.13)

Another empirical relationship between RC and Dr is given by Lee and Singh (Das, 2007) as

(2.17) equation

The field compaction of soils is done by rollers such as sheep foot rollers, vibratory rollers, pneumatic rubber tired rollers, smooth wheel rollers.

2.9 Consolidation and Compressibility

When a fine grained soil or cohesive soil is subjected to loads or stresses, some or all the additional load or stress is supported by the pore water present in the soil mass initially. This excess pore pressure creates hydraulic gradients in the pore water and the water flows out (due to the soil permeability) and simultaneously transfers the load or stress to the soil particles gradually. This amounts to the gradual transfer of pore water pressure to the intergranular stress or effective stress, until the entire load or total stress becomes effective stress (as per Equation 2.15). This simultaneously produces compression/settlement of the soil mass (as only effective stresses produce settlements). This gradual process involves simultaneously a slow escape of water, a gradual load transfer and a gradual compression of the soil mass and is called consolidation. The compressibility and consolidation characteristics of the soil are determined in the laboratory using a consolidometer/oedometer, as shown in Figure 2.9.

Figure 2.9 Schematic diagram of oedometer/consolidometer.

The saturated soil sample (usually 64 mm diameter and 25 mm thick) is placed inside the metal ring with porous stones at top and bottom to facilitate escape of water, as shown in the Figure 2.9.

A load is applied on the specimen which becomes the total vertical stress, σ. Compression or settlement readings are taken at 15 s, 1 min, 4 min, 16 min and so on, in time ratios of four, up to 24 h or until no further settlement is noticeable, signifying the consolidation is practically complete under the present load. Then the load on the specimen is doubled and the test is repeated for several cycles to include the range of design stresses anticipated in the field. The results of these tests can be plotted as a graph of void ratio at the end of consolidation (corresponding to each applied load) versus corresponding vertical effective stress, as shown in Figure 2.10. While the total effective stress can be directly calculated by dividing the applied load by the area of cross section of the specimen, the change in void ratio (being directly proportional to the change in thickness of the sample) can be obtained as

(2.18) equation

where

e = change in void ratio

e = void ratio (initial)

= change in thickness of the sample

H = initial thickness of the sample.

Figure 2.10 Compressibility curves for a clayey soil.

Figure 2.10(a) shows the semi log plot of e versus log σ′. Figure 2.10(b) shows the e versus σ′ curve.

After completing the test up to the desired pressure, the specimen can be gradually unloaded resulting in some recovery of the compression recorded, that is, increase in thickness as shown in these figures.

2.9.1 Compressibility Characteristics and Settlement of Soils

Following compressibility characteristics can be determined from Figure 2.10:

1. Compression index, Cc

The slope of the straight line portion of the e log σ′ graph (loading part) shown in Figure 2.10(a) is called the compression index, Cc.

Accordingly

(2.19)

There are several correlations of compression index with the other soil parameters (Bowles, 1996). The most popular one is due to Terzaghi and Peck (1967) and is expressed as

(2.20) equation

where LL is the liquid limit of the soil.

2. Swelling index or recompression index, Cs

This is the slope of the unloading portion of the e log σ′ graph, (Figure 2.10(a)), that is

(2.21) equation

In most cases

(2.22) equation

3. The coefficient of compressibility, av, and the coefficient of volume decrease, mv.

av is the slope of the e − σ′ graph which is idealized as a straight line between the ranges of σ′ needed for computations, as shown in Figure 2.10(b).

Accordingly

(2.23) equation

Also, coefficient of volume decrease

(2.24) equation

Change in the thickness or settlement of the soil sample or layer (ΔH) of total thickness (H) is due to primary consolidation, Sc.

From Equations 2.18, 2.19, 2.23 and 2.24, we can write

(2.25)

where e and are the initial void ratio at effective stress and Δ is the change in effective stress = .

Similarly the increase in thickness during swelling can be calculated using the swelling index or coefficient of swelling.

4. Preconsolidation pressure,

i. Locate O on the e log σ′ curve where the curve has maximum curvature, that is, smallest radius of curvature.

ii. Draw the line OA horizontally.

iii. Draw the line OB tangentially to the e log σ′ curve.

iv. Draw the line OC bisecting the angle AOB.

v. Extend the straight line portion of the e log σ′ curve backward to intersect line OC at D. The pressure corresponding to point D on the e log σ′ curve is the preconsolidation pressure .

This may also be called the over consolidation pressure, . This is the maximum past effective pressure to which the soil specimen is subjected to, as mentioned in Section 2.5. It can be determined from Figure 2.10(a), as shown there. The preconsolidation pressure can be determined using Casagrande's method (Taylor, 1964) as follows.

2.9.2 Time Rate of Consolidation

The one-dimensional consolidation equation (Terzaghi, 1943) for the laboratory soil sample shown in Figure 2.9 is

(2.26) equation

where

Cv = coefficient of consolidation =

u = pore water pressure

z = vertical coordinate of the soil sample

t = time parameter

k = coefficient of permeability

mv = coefficient of volume decrease (Equation 2.24)

γw = unit weight of water.

The above equation was solved by Terzaghi (1943), and the following curve fitting methods were developed for determining Cv, which is useful for calculating time rate of settlements. These are:

1. Square root of time ( ) fitting method (Taylor, 1964)

2. Logarithm of time (log t) fitting method – Casagrande's method (Taylor, 1964).

From the exhaustive solution of Equation 2.26 given by Terzaghi, the most important ones used for settlement calculations are given in Figure 2.11. These are in terms of value of average degree of consolidation, U (%) versus nondimensional time factors, T where

Figure 2.11 Consolidation curves as per Terzaghi's theory.

(2.27) equation

(2.28) equation

where

u = pore pressure at time t

ui = initial pore pressure at t = 0

H = total thickness of the soil layer or sample

z = vertical coordinate

Cv = coefficient of consolidation.

Noting that the solutions shown in Figure 2.11 are close to each other, only curve 1 (case 1) is used for most of the calculations.

Using these results shown in Figure 2.11, Cv is determined using curve fitting methods developed by Taylor (1964) and Casagrande (Taylor, 1964). Taylor's method is called the square root of time (√t) fitting method and uses 90% consolidation results from experiments and theory (Figure 2.11) for comparison; that is, he compares t90 from experiments and T90 = 0.848 from theory (Figure 2.11).

Casagrander's method is called the logarithm of time (log t) fitting method and uses 50% consolidation results from experiments and theory (Figure 2.11) for comparison; that is, he uses t50 from experiments and T50 = 0.197 from theory (Figure 2.11).

The solution details and several examples are given in all standard books in Geotechnical Engineering (Taylor, 1964; Bowles, 1996; Das, 2007).

2.10 Shear Strength of Soils

Engineering materials may generally fail due to tension, compression, shear or a combination of these factors. However, soils and rocks fail essentially due to shear. The corresponding shear stress beyond which the soil fails is called the shear strength of the soil and is expressed by Coulomb's equation, that is

(2.29)

where

s = shear strength of the soil

c, c′ = cohesion of the soil

ϕ, ϕ′ = angle of internal friction of the soil

σ′ = effective stress = (as in Equation 2.15)

u = pore water pressure

= .

Generally shear strength parameters depending on the total stresses, that is, c and ϕ are used to check the stability of the supporting soil at the end of construction stage, while c′ and ϕ′ (shear strength parameters of the soil with reference to effective stress) are used for analyzing long term stability. Hence, most of the following details are presented in terms of c and ϕ, though they equally apply for c′ and ϕ′.

The cohesion c of the soil is independent of the normal stress. However, the frictional component between the grains (i.e., ) depends on the normal stress, σ. The shear strength of soils given by Equation 2.29 is shown in Figure 2.12 for different soils, such as (a) cohesive soils, (b) cohensionless soils (sands and gravels) and (c) purely cohesive soils (clays) or sometimes for end of construction analysis with (Taylor, 1964; Terzaghi and Peck, 1967; Lambe and Whitman, 1969).

Figure 2.12 Shear strength of soils.

The shear strength parameters of the soils can be determined in the laboratory by:

1. Direct shear test for sandy soils

2. Vane shear test for clayey soils

3. Triaxial shear test for general soils and loading conditions

4. Unconfined compression test for clayey soils.

Some important aspects of these tests are briefly described below while more details can be obtained from Lambe (1951), Taylor (1964), Das (2002) and other books.

2.10.1 Direct Shear Test

This test is mainly done on frictional soils/coarse grained soils/sandy soils using a shear box, as shown in Figure 2.13 (a). The sandy soil is to be tested in the shear box, which is split into two halves (Figure 2.13(a)). A normal load N is applied and then a shear force, Q is applied in steps until the specimen fails along the horizontal plane dividing the two halves of the split shear box. A plot of normal stress, σ, versus shear stress, s, is drawn as shown in Figure 2.13(b), where

Figure 2.13 Direct shear test for sands.

(2.30) equation

where A is the area of the failure plane, that is, the cross sectional area of the shear box. From the graph, it can be noted that

(2.31) equation

For sandy soils, varies from 20° to 45° increasing with relative density Dr.

2.10.2 Vane Shear Test

This test can be done both in the laboratory as well as in the field and is applicable more for cohesive soils. The vane consists of four thin plates welded to a torque rod as shown in Figure 2.14. A torque is then gradually applied at the top of the torque rod (as shown in the sketch) and the cylindrical surface of soil of height h and diameter d resists the applied torque until the soil fails. Then, the shear strength (undrained, since practically no drainage occurs during the test) can be computed by this expression

Figure 2.14 Sketch of vane shear test equipment.

(2.32) equation

where

s = undrained shear strength

= cohesion, c (since for cohesive soils)

T = torque at failure

d = diameter of the shear vane

h = height of the shear vane

β = a factor depending on the slope of the zone of resistance/shear strength at the periphery of cylindrical surface

= 1/2 for triangular mobilization

= 2/3 for uniform mobilization

= 3/5 for parabolic mobilization.

Uniform mobilization factor of 2/3 is commonly used and accordingly s (= c) is calculated as

(2.33) equation

The commonly used laboratory vane size has and . The field vane is generally bigger and there are several sizes prescribed by standards such as ASTM (Das, 2002).

2.10.3 Triaxial Shear Test

This is a very comprehensive test that can be conducted on any general soil with cohesion, c, and friction, ϕ, components. The test set up is shown in Figure 2.15.

Figure 2.15 Sketch of triaxial test equipment.

In this test, a cylindrical soil specimen of standard size (around 36 mm diameter, 76 mm long; usually the length diameter ratio is 2.0 to 2.5) confined by a rubber membrane is placed in a lucite chamber. Then an all round confining pressure, σ3, is applied to the specimen using either water (mostly) or glycerin as the chamber fluid. This is also called hydrostatic stress, all round pressure or cell pressure, σ3. Then a vertical stress, Δσ1, is applied in the vertical direction until failure. This is also called the deviator stress. Thus the normal stress in the vertical direction at failure becomes .

If drainage is allowed in the test, it is called a drained test. Otherwise, it is called an undrained test where pore pressures are developed due to the applied deviator stress. The soil specimen is usually tested after complete saturation but also can be tested at any desired water content. Out of the several customized triaxial tests, following three main types of tests are commonly conducted in the triaxial equipment (Lambe, 1951; Lambe and Whitman, 1969).

1. Unconsolidated undrained test (UU test)

2. Consolidated undrained test (CU test)

3. Consolidated drained test (CD test).

Figure 2.16 Mohr's circles and failure envelopes for different triaxial tests.

The test results are analyzed using Mohr's circle, knowing the major and minor principal stresses, that is, σ3 (minor principal stress) and (major principal stress), as shown in Figure 2.16. Usually three or four samples are tested at different cell pressures, and a common envelope is drawn tangential to the circumferences of these Mohr's circles obtained for each sample using minor and major principal stresses at failure. This is called the failure envelope or Mohr–Coulomb failure envelope. These details are shown in Figure 2.16 for the above types of tests. Noting that the failure envelope represents the shear strength of the soils, as shown in Figure 2.12, the cohesion, c (or c′), and the angle of internal friction, ϕ (or ϕ′), can be determined from these figures, as marked therein.

Thus, the triaxial test is very comprehensive and versatile with lots of flexibility to customize the test to simulate the design requirement. The literature available on this test is very exhaustive (Lambe, 1951; Lambe and Whitman, 1969).

2.10.4 Unconfined Compression Test

The unconfined compression test (also called the UCC test) is more relevant to cohesive soils. This is a special case of the unconsolidated undrained trixial test with no cell pressure (that is, , as shown in Figure 2.17 (a)), Hence, it is called unconfined compression test since the vertical compression stress, Δσ1, is applied until failure. The corresponding Mohr's circle and failure envelope are shown in Figure 2.17(b). It may be noted that only one Mohr's circle can be drawn with for the same soil