Soil Mechanics

Preface

Atque neque, uti docui, solido cum corpore mundi naturast, quoniam admixtumst in rebus inane …

Titus Lucretius Caro De Rerum Natura

According to the engineering nomenclature, soil mechanics is concerned with the behavior of clastic rocks, or soils, under different loading conditions: external loading, such as that transmitted by the foundations of any structure, or generated by the seepage of water, and also by its own weight as a consequence of geometric changes, induced for instance by excavation or tunneling.

Knowledge of soil mechanical behavior is, in fact, an essential element for the prediction of the displacements and internal actions of a structure founded on or interacting with it. Soil Mechanics is, therefore, the fundamental subject of geotechnical engineering, the branch of civil engineering concerned with soil and with the interacting soil structure, dealing with the design and the construction of civil and industrial structures and environment defense works against geological hazards.

Aristotle said "Φαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητική": any prediction is based either on a rational calculation or on intuitive perception. Although the latter has been for a long time the starting-point of any construction and still plays a relevant role in design, it is the former that allows the definition of the structure’s dimensions and safety assessment. In fact, it allows rational prediction of the structure’s behavior in the different construction phases and during its life.

This calculation must be based on a mathematical model of the structure and the soil. This should schematize the geometry of the problem, the mechanical behavior both of materials and structures, as well as the loading. The definition of an overall mathematical model of the structure and the soil is a very complex problem that is beyond the scope of this book. In the following, only the bases upon which a mathematical model of soil behavior can be formulated will be outlined.

Though limited in scope, soil modeling is rather complex and requires different levels of abstract thinking. First, it is necessary to pass from the physical nature of soil, composed of a discrete and innumerable number of solid mineral particles and voids, into which fluids such as air, water or mineral oils can seep, to its representation as a continuum. In fact, this allows a much more feasible mathematical formulation. In order to achieve this goal, it is necessary to assume the soil to be a special medium obtained by overlapping two continua: a solid continuum, modeling the skeleton composed of the mineral particles, the solid skeleton, and a fluid continuum, modeling the fluid, or the fluid mixture, seeping through the voids.

The most relevant aspect lies in the fact that both these continua completely occupy the same region of space. They interact by parting the stress state in a way that directly derives from the conditions of conservation of energy and mass, and that is a function of how the behavior of the solid continuum under loading and the fluid seepage in the soil are independently modeled. Hence, it is necessary to mathematically formulate models for the description of the mechanical behavior of the solid skeleton (stress-strain relationship) and a conceptually equivalent law ruling the motion of fluid with respect to the solid skeleton.

Once the model is defined, in order to mathematically reproduce with the best approximation possible the experimental results obtained by elementary tests, the parameters describing the soil (or the different soil layers) behavior have to be specified for the case under examination.

Finally, a further step in modeling is necessary to transform the system of differential equations and boundary conditions ruling any soil mechanics problem in the light of continuum mechanics into a system of algebraic equations that can be solved by means of a computer.

This book will be developed in logical sequence according to what has been previously outlined.

Chapter 1 presents some elementary concepts necessary to pass from the discrete nature of soil to its continuum representation. Differential and boundary equations for a generic soil mechanics problem will then be presented in Chapter 2. Special cases will be analyzed, such as stationary seepage conditions (Chapter 3), rapid loading conditions (undrained conditions), and transient seepage conditions (Chapter 4). In this last case, under constant loading, the stress state is transferred from the water to the solid skeleton, inducing soil deformations and structure assessments over time (consolidation). For the sake of simplicity, in this case soil will be assumed to be characterized by an incrementally linear behavior.

Nevertheless, the mechanical behavior of the solid skeleton is much more complex. In fact, it is nonlinear, irreversible, and highly influenced by the average pressure to which it is subjected. These aspects will be detailed in Chapter 5, which is dedicated to the study of the response of elementary soil samples in laboratory tests. In Chapter 6, mathematical models of increasing complexity describing the behavior outlined in the previous chapter will be formulated. Finally, in Chapter 7, methods of discretizing the continuum and integration procedures will be mentioned. A few examples, referring to archetypes of geotechnical problems (foundations, sheet piles, slopes), will illustrate the results that can be obtained in this way.

This book is not intended to be exhaustive on all the geotechnical issues or to give practical suggestions. For these purposes several good and topical books already exist and there is no reason to write another. On the contrary, the goal of this work is to tackle the fundamental aspects of a very complex subject at a deeper level than current works. These aspects can have remarkable consequences on the choices that the engineer has to make in order to build the geotechnical model of the soil that is appropriate for the particular case under examination (geometry of the problem, type of model to describe soil behavior, parameters to be assumed, type of numerical solution) and thus, as a consequence, on the design.

Having worked in the field of soil mechanics for many years, I know that there is some confusion concerning the fundamental principles which this subject is based on. Frequently, even people working in the geotechnical engineering field do not completely understand the formulae that they use, especially the computer methods, whose bases they do not have knowledge of. The dialog between the several actors involved in a geotechnical project (civil and environmental engineers, geologists, architects) risks becoming a dialog between deaf people, in which not even the specific role of each of them is clear.

As any good geotechnical engineer knows, a safe structure has to be based on solid foundations. The book is therefore intended to give, to those who will have the patience to read it, the bases necessary to understand the fundamentals of soil mechanics. It is my firm belief that only through the thorough understanding of such fundamentals can appropriate geotechnical characterization and soil modeling be carried out. Though the main point of this book is undoubtedly theoretical, its final goal is very practical: to give adequate means for a correct framing of geotechnical design.

In writing this book, I was privileged to collaborate with some young colleagues: Claudio di Prisco, Roberta Matiotti, Silvia Imposimato, Riccardo Castellanza, Francesco Calvetti, Cristina Jommi, Rocco Lagioia, Claudio Tamagnini, Stefano Utili, Giuseppe Buscarnera, Matteo Oryem Ciantia, Giuseppe Dattola and Federico Pisanò. They helped me to clarify the text (in addition to taking care of the graphics). To them and to all those who have been so kind as to highlight mistakes and omissions or simply been willing to discuss the non-traditional approach followed in this book, my most sincere thanks.

This book is dedicated to my Maddalena, Tommaso and Tobia, who patiently bore the consequences of its writing.

Chapter 1

Introduction: Basic Concepts

1.1. Soils and rocks

The term soil is used in civil engineering to describe a material composed of a natural accumulation of mineral particles, whose sizes range between specified limits, according to a conventional classification system.

Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering), rocks being the primary element that constitutes the Earth’s crust. Soil particles can be completely uncemented or weakly cemented, depending on the degree of alteration of the parent rock. On the other hand, soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles, so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis).

Since the processes of weathering and diagenesis are gradual, the distinction between soil and rock is to a certain extent arbitrary. To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds, such that the stress levels typical of civil engineering applications can easily exceed their strength. On the other hand, rock is defined as a material with strong chemical bonds. The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself.

Several geological materials (e.g. tuff, clay stone, marble, limestone, etc.) have an intermediate behavior. These materials behave as rocks if subjected to relatively low stresses, and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles.

Soil grains are mainly composed of silica minerals (e.g. silicon dioxide and other silica-based minerals), which are more resistant to chemical-physical attack by weathering than other minerals. Quartz (SiO2) is almost insoluble in water, is relatively acid proof, and is a very stable mineral. It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands, followed by feldspars.

Feldspars are chemically altered by water, oxygen and carbon dioxide. The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin. Similarly, phillosillicates, existing in large quantities in igneous rocks, delaminate along their basal plane, due to their mineralogical foil structure, and form illite and smectite. Kaolin, illite and smectite are the primary minerals appearing in clay; they are characterised by plate-like particles with length and width in the order of a micron.

Soil particles are also composed of calcite and gypsum, as well as of minerals of volcanic origin (pyroclasts). Particles formed by these types of minerals are usually weaker than those formed by silica minerals; therefore, they have a greater influence on the strain behavior of these materials.

The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history.

For example, on the one hand, rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation. On the other hand, sand grains that remain in their original location, where weathering of the parent rock took place, are angular and have an irregular shape.

The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways. If clay particles align in the same direction (face-to-face orientation, Figure 1.1a) it is referred to as dispersed structure, while a structure similar to a card house (edge-toface or edge-to-edge orientation) is referred to as flocculated structure (Figure 1.1b) and is much more unstable than the former. With the change in the deposit chemical conditions, the structure can pass from dispersed to flocculated and vice versa.

Figure 1.1. a) Dispersed clay structure; b) flocculated clay structure (cardhouse)

1.2. Engineering properties of soils

As seen in the previous section, several types of minerals compose a soil, its solid skeleton, and its fabric are influenced by its geological history and by the chemical environment. However, for the majority of engineering aims, different types of soil can be initially classified according to the size of the constituent particles. The classification of the different types of soils is somewhat arbitrary. Examples of classifications adopted by British Standards (BS), Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 1.1.

Table 1.1. Classification of different types of soils. Sizes are in mm

Note that in the proposed classifications there is no direct reference to the grain chemical composition, to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration). This type of classification has two advantages. Firstly, it is a quantitative classification, and hence it is almost free from the subjectivity of the operator. Secondly, it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior. The range of possible particle sizes is enormous. Soil particle sizes range from sub microscopic clay particles, discernible only by a scanning electron microscope, to rounded sand grains with a diameter a thousand times larger, to cobbles with a diameter a hundred times larger.

On a single particle, both body forces (weight) and surface forces (electrostatic forces) have effect. The former, are proportional to the volume of the particle, while the latter are proportional to the external surface. An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface. An indicator of the relative role played by the two types of forces is the specific surface, Ss, defined as the ratio of the area of the surface of the particle to the mass of the particle, ρV:

[1.1]

where ρ is the density and V is the volume of the particle.

In the case of a rounded particle of silica sand, the specific surface is inversely proportional to the diameter of the grain, dg.

[1.2]

Quartz density is equal to 2.65 g/cm³; hence, for a rounded particle of diameter 1 mm the specific surface will be 0.00226 m²/g. A clay particle, of plate-like shape, has instead a specific surface equal to:

[1.3]

where s is the thickness of the particle. The particle thickness largely depends on the type of clay. For kaolin it can be of the order of a tenth of a micron, while it can be of the order of only 10 Å (10-3 μm) for the smallest particles, this is typical of montmorillonite.

For kaolin the specific surface is of the order of 10 m²/g (more than 3,000 times the value of the sand considered). For montmorillonite the specific surface is of the order of 1,000 m²/g. Electrostatic forces are then negligible in sand, however, they become relevant when dealing with clay. In the presence of water, clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes. This layer is referred to as adsorbed water. The water forming this layer has very different mechanical properties in comparison with those of free water: for instance, adsorbed water is capable of transferring shear stresses. In practice, adsorbed water can be considered, as a first approximation, as an integral part of the mineral clay particle. Unless otherwise specified, the mineral particle is assumed to be coated by a layer of adsorbed water. From the mechanical point of view, interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains.

Moving away from the surface of the particle, the attractive force decreases and progressively water starts behaving as free water, which can be gradually removed from a sample of soil; for example, by applying compression stresses.

From an engineering point of view, the most relevant aspect related to the particle size distribution is the ability of water or other fluids, such as oil, to seep through the soil pores.

Figure 1.2. A soil element as an aggregation of particles

A soil element can be visualized as an aggregation of solid particles, weakly cemented or uncemented, the void space between the particles containing one or more fluids, principally air and/or water (Figure 1.2). A fluid can seep through a soil more or less easily depending on the width of the flow channel section. The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius, which is of the same order of magnitude of the soil particle size. The size of a clay particle is approximately a thousand times smaller than the one of a sand grain. Thereafter, water discharge velocity in a clay layer must be a million times lower than the one in a sand layer, all other conditions being equal. As will be observed in the following, this difference has relevant practical consequences.

A load applied on a sample of soil provokes the rearrangement of its structure. Since soil grains are principally composed of extremely resistant and rigid minerals, the deformability of a soil element is mainly associated with a change in the configuration among grains, which is related to a change in the volume occupied by voids. In fact, grain deformability is negligible; with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads. Water is also considered, under the stress levels typical of civil engineering applications, to be an incompressible fluid. If soil is fully saturated by water, a change in volume can take place only if water is free to drain throughout the soil. If soil is coarsely grained, drainage is instantaneous and the particles are free to change their configuration while loads are applied. On the contrary, in fine grained soils, water flow is subjected to a higher resistance. The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (e.g. the construction of a building, a road embankment or an excavation). In the initial phases of the load, referred to as short term, the possible configurations are only the ones that maintain the total volume constant, which means that the soil has an internal kinematic constraint. With time, water gradually drains through the soil and at long term also fine grained soils can freely change their configuration without any internal constraints.

The first and main difference between coarsely and fine grained soils is then apparent. Fine grained soils change their configuration after a change in load, even though initially without a change in volume. Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process. The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles. Therefore, the strain process continues also after the stabilization of the load.

It is worth noting, however, that in relatively coarsely grained soils, such as fine sands, there can be kinematic constraints preventing changes in volume after rapid variations in load, as in the case of earthquakes. Moreover, a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles.

Another important difference between fine and coarsely grained soils is represented by capillary rise. Let Ts be the surface tension of water, α the angle between the tangent to the meniscus and the wall of the capillary tube, γw the unit weight of water and d the diameter of the tube (see Figure 1.3). Equilibrium in the vertical direction implies that the capillary height, hc, of the liquid column is:

[1.4]

Surface tension of water in standard conditions is equal to 0.075 N/m, therefore, in a capillary of 1 mm diameter the rise is of the order of 2 cm. If the diameter of the tube is instead 1 μm the capillary rise is 20 meters. In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry. Conversely, fine grained soils are saturated up to several tens of meters over the ground-water table.

Figure 1.3. Rise in a capillary tube

1.3. Soils as an aggregation of particles

As a first approximation, the structural arrangement of an elementary volume of soil can be schematized as in Figure 1.2. Solid particles occupy only a portion of the space relative to an element of soil. The remaining portion, called volume of voids, is occupied by a fluid, usually air and/or water.

The ratio of the volume occupied by voids, Vv, to the volume occupied by solids, Vs, is called void ratio, e:

[1.5]

Alternatively, porosity, n, is defined as the ratio of the volume of voids to the total volume, V, of a soil element:

[1.6]

It is clear that the higher the porosity, the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads. On the other hand, a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration. Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior.

In order to define the range of soil porosities, an ideal material composed of rigid spheres of equal radius is considered. A simple cubic structure, in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below, is characterized by a porosity equal to 0.476. This configuration is highly unstable. A small external perturbation is sufficient to reduce its porosity. Conversely, in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral, in contact among them), porosity is much lower and equal to 0.259. In this case, the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case. It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated, an external perturbation will cause a decrease in volume in the case of loose sand, while it will cause an increase in volume in the case of dense sand (dilatancy).

The proposed model is only an example. Firstly, a soil is composed of particles of different sizes and non-rounded shapes. Moreover, smaller particles have a greater possibility of occupying a minor total volume, the volume of solids being equal. For instance, a sphere of radius R occupies a cube of radius 2R with a porosity of 0.476. On the other hand, eight spheres of radius R/2, and hence occupying the same volume of the spheres just considered, would fill the same cube only if disposed in the most unstable configuration previously described. However, the eight small spheres can dispose in several ways, for example in the tetrahedral configuration that is characterized by a much lower porosity. A sample of sand composed of several particle sizes will be characterized, in general, by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size). To take into account the effect of grading in sands, it is more appropriate to refer, other than to porosity, to the relative density (density index), which is traditionally defined as:

[1.7]

where emax and emin are two void ratios, conventionally determined (refer to ASTM D2216-66), which define the loosest and the densest state for a criterion of sand.

Moreover, particles are not rigid. Calcareous and volcanic sands are composed of fragile grains that can crush under loading. As stated before on the effect of particle size, porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves.

Finally, when particles develop cohesive bonds, configurations with very high void ratios are possible. For example, loess deposits and cohesive silts deposited by wind, can be characterized by porosities higher than 60% (e > 1.5). These configurations are stable only for tensional levels that are lower than the bond strength. For higher tensional levels, the bonds break and the soil assumes a much more compact configuration. The collapse of these kinds of soils usually causes big problems from an engineering viewpoint.

1.4. Interaction with pore water

Inter-granular voids can be partially or totally filled with water. The degree of saturation, Sr, is defined as the percent ratio of the volume occupied by water to the volume of voids:

[1.8]

A soil is referred to as saturated when Sr = 100% and as dry when Sr = 0%. In general, soil is not in its limit conditions. A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater. However, full saturation is not reached even under the groundwater table, due to small air bubbles trapped within the soil voids.

The soil water content, w, is defined as the ratio of the mass of water within the sample, Ww, to the mass of the solid part, Ws, this being the dry weight of the considered sample,

[1.9]

Let γw be the unit weight of water and γs the unit weight of the material composing the grains. The water content is then linked to the degree of saturation and to the void ratio by the relationship:

[1.10]

where

[1.11]

The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 2.5 to 2.9. The Gs value of quartz is 2.65, of calcite is 2.71, with 2.7 the typical average value of clayey minerals.

It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content. Let γ be the total unit weight of a certain sample of soil:

[1.12]

In particular, for a dry sample, the dry unit weight, γd, is equal to:

[1.13]

while for a saturated sample the total unit weight is equal to:

[1.14]

Finally, a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced. The buoyant unit weight of a soil is hence equal to:

[1.15]

Notice that this result is equally achieved by considering the soil sample as composed of a unique material, or by considering the solid part, namely the