An extremum principle is basically a mathematical concept that relies on some physical law. In mechanics, extremum principles such as the principle of minimum total potential energy and minimum total complementary energy form an important base of knowledge that has provided the means for obtaining approximate solutions to a variety of problems in engineering. This is particularly the case with the theory of elasticity. The celebrated principles of least work attributed to Alberto Castigliano, are also in the realm of extremum principles that have been used extensively in the solution of problems in classical structural mechanics dealing with elastic materials. In general, extremum principles and for that matter variational principles start with the basic premise that the solution to a problem can be represented as a class of functions that would satisfy some but not all of the equations governing the exact solution. It is then shown that a certain functional expression, usually composed of scalar quantities such as the total potential energy, strain energy, energy dissipation rate, etc., that have physical interpretations associated with them and are defined through the use of this class of functions, will yield an extremum (i.e. either a maximum or a minimum) for that function. Moreover, the extremum will satisfy the remaining equations required for the complete solution. For example, the principle of minimum total potential energy states that of all the kinematically admissible displacement fields in an elastic body, which also satisfy the governing constitutive equations, only those that satisfy the equations of equilibrium will give rise to a total potential energy that has a stationary value or an extremum.

Introduction

The simplicity of the collapse load theorems masks some of the more complex aspects of engineering applications involving plasticity. Solutions for fully three-dimensional elastic–plastic response will generally be difficult if not impossible to obtain in closed form. There is, however, one more class of problem for which relatively simple solutions are possible. This is the class of two-dimensional problems concerning plane plastic flow for which regions of the material are in a failure condition. The failure regions need not cover the entire body, but within the failing zone we must be assured that the yield condition is satisfied everywhere.

For these plane problems there will be three unknown components of stress: for example, σxx, σyy,σxy, where the (x, y)-plane is taken to be the plane of the problem. Within the failing region the three stresses are related by three equations: two equations of equilibrium plus the equation of the yield surface. Only first-order derivatives are involved. While this in itself does not appear overly complex, it will become apparent that considerably more simplification is possible by invoking a new coordinate system. Introducing coordinates that coincide with the potential failure surfaces, we cause the system of equations to become extremely simple. In our two-dimensional problem, the potential failure surfaces are seen simply as lines and they have come to be called slip lines.

When a continuum region consists of either a rigid strain hardening or an elastic strain hardening material, the strains and displacements of the region for a given history of loading can be determined. If, on the other hand, the continuum region is made of either a rigid perfectly plastic or an elastic perfectly plastic material the situation is quite different. A qualitative picture of the behaviour of such a material was discussed at the very start of this volume. For example, at sufficiently small loads the region can either remain rigid or experience small elastic deformations. As the loading increases parts of the continuum region can become plastic, but the region as a whole can withstand collapse due to the restraining effect of elastic regions. As the loads increase, larger regions of the continuum can experience plastic yield and eventually the continuum region can undergo ‘indefinite’ plastic deformations leading to what we term ‘collapse’ of the region. Two interesting examples that illustrate the definition of a ‘collapse’ state in the context of limit analysis are given by Drucker et al. (1952). In this process we implicitly assume that the deformations experienced by the continuum region are small enough so that changes in the geometry of the region may be neglected and that all the deformations take place in a quasi-static fashion so that any dynamic effects can be ignored.

Introduction

This final chapter presents a collection of ideas related to work hardening, as well as some thoughts on modern descriptions of the mechanical response of soils. Recall from Chapter 3 that work hardening materials, in contrast to perfectly plastic materials, may change their response during yielding. These changes are accomplished by altering, in some fashion, the shape or size of the yield surface as plastic flow occurs. Initially the concept of work hardening was introduced to give a better representation of the stress–strain response of metals. The ideas involved are straightforward, although there is a price to pay in terms of increased levels of complexity. We have avoided the topic until now, not because it is unimportant, but because it plays such an important role in the modern theories of soil plasticity that are also considered in this chapter.

Geotechnical engineers have found the general concept of work hardening extremely useful whenever there is a need for response calculations that are more detailed than is possible with perfect plasticity. In particular, the closed yield surfaces described in Chapter 3 would be nearly useless without work hardening. With a closed yield surface there is a possibility of plastic response under increasing isotropic stress, but one cannot arbitrarily limit the amount of stress increase by establishing a fixed yield surface at some arbitrary stress level. We must be able to increase the isotropic stress, to move the stress point out along the space diagonal, without limitation.

Physical models of rivers have existed at least since 1875, when Louis Jerome Fargue built a model of the Garonne River at Bordeaux. Physical models are usually built to test various river engineering structures and to carry out experiments under controlled laboratory conditions as opposed to costly field programs. The main purposes of physical models include: (1) a small-scale laboratory duplication of a flow phenomenon observed in a river; (2) the examination of the performance of various hydraulic structures or alternative countermeasures to be considered in the final design; and (3) investigation of the model performance under different hydraulic and sediment conditions.

This chapter first describes hydraulic similitude in Section 10.1 in terms of geometric, kinematic, and dynamic similitude. Hydraulic models can be classified into two categories: (1) rigid-bed models, discussed in Section 10.2 and (2) mobile-bed models, discussed in Section 10.3. The analysis leads to the definition of model-scale ratios, and several examples and case studies are presented.

Hydraulic similitude

The prototype conditions, denoted by the subscript p, refer to the full-scale field conditions for which a hydraulic model, subscript m, is to be built in the laboratory. Model scales, subscript r, refer to the ratio of prototype to model conditions. For instance, the gravitational acceleration in the prototype is gp, the gravitational acceleration in the model is gm, and the scale ratio for gravitational acceleration is defined as gr = gp/gm.

River-stabilization structures are designed to protect the riverbanks and prevent lateral migration of alluvial channels through bank erosion. River-stabilization methods can be classified according to two different approaches: (1) strengthening the banks and (2) reducing hydrodynamic forces. This chapter first examines the bank stability of alluvial streams in Section 8.1. Bank-protection methods through strengthening the banks with riprap are discussed in Section 8.2, and other bank-strengthening methods are covered in Section 8.3. Flow-control structures offer an alternative approach by reducing the hydrodynamic forces applied against the riverbanks. The flow-control structures covered in Section 8.4 aim at gaining control over the flow depth and the direction and magnitude of flow velocity near the river banks. Some engineering considerations are discussed in Section 8.5.

Riverbank stability

Bank stability is examined in this section. First, the processes are reviewed in Subsection 8.1.1, followed by conceptual solutions for slope reduction in Subsection 8.1.2 and subsurface drainage in Subsection 8.1.3.

Bank-erosion processes

Processes of bank erosion are directly linked to the lateral migration of alluvial channels. Bank erosion is the result of flowing water that applies active forces met by the passive forces of the bank material to resist motion. As discussed in Chap. 6, the hydrodynamic forces in river bends induce secondary flow where the free-surface streamlines are deflected toward the outer bank and the near-bed streamlines are deflected toward the inner bank.

Deviations from equilibrium conditions will trigger a dynamic response from the alluvial river system to restore the balance between inflowing and outflowing water and sediment discharges. Section 7.1 of this chapter deals with the dynamics of stream response to changes in water and sediment discharges. Sections 7.2 and 7.3 describe the dynamic response of alluvial systems to degradation and aggradation, respectively, particularly the effects on hydraulic geometry and channel morphology. Section 7.4 focuses on river confluences and branches. Finally, Section 7.5 provides guidelines on river databases, data sources, and field surveys.

River dynamics

Conceptually, the fluvial system of the watershed sketched in Fig. 7.1 can be divided into three main zones: (1) an erosional zone of runoff production and sediment source; (2) a transport zone of water and sediment conveyance; and (3) a depositional zone of runoff delivery and sedimentation. The second zone is characterized by near-equilibrium conditions between the inflow and the outflow of water and sediment. The bed elevation in this equilibrium zone is fairly constant and the hydraulic geometry is described in Chap. 6 and this section. The upper zone is characterized by net erosion of bed material and channel degradation. The dynamic response of degrading fluvial systems is discussed in Section 7.2. The lower zone is characterized by net sedimentation and channel aggradation. The dynamic response of aggrading fluvial systems is discussed in Section 7.3.

This chapter presents a discussion of various types of river engineering structures. The aim of this chapter is to familiarize the reader with solutions to a wide spectrum of river engineering problems. This chapter covers an overview of the following types of river engineering structures: (1) flood control in Section 9.1; (2) river closure and local jet scour in Section 9.2; (3) canal headworks in Section 9.3; (4) bridge crossings in Section 9.4; (5) navigation and waterways in Section 9.5; and (6) dredging in Section 9.6.

River flood control

This section discusses river engineering solutions to flood control. The methods include flow regulation with reservoirs in Subsection 9.1.1, floodways in Subsection 9.1.2, channel conveyance in Subsection 9.1.3, and levees in Subsection 9.1.4.

Reservoirs

The most direct method of flood control is the storage of surface runoff. Flood control with reservoirs redistributes floodwaters and attenuates floodwaves. A reservoir fills up when the inflow exceeds the outflow, and water storage acts as a buffer to decrease the peak discharges and increase the low discharges. An increase in low discharges is beneficial to hydropower, navigation, and irrigation. It is best to keep the reservoir as full as possible in order to maintain sufficient storage capacity to increase the low discharge reserve in periods of drought. For flood control, however, the reservoir should be kept as low as possible to store unexpected large floods and decrease flood peaks.

Numerous river engineering problems can be conveniently investigated by means of mathematical models. Mathematical models must properly describe the physical processes and provide a numerical solution to a system of differential equations that are solved together with suitable boundary conditions and empirical relationships that describe resistance to flow and turbulence.

The differential equations describing river mechanics problems are usually simplified forms of the equations of conservation of mass and momentum, leading to a set of partial differential equations involving two independent variables (time and space or two spatial variables). Examples that use the finite-difference method are presented in this chapter. The finite-element method also provides useful solutions to river engineering problems but is beyond the scope of this chapter.

The algorithms to be used in the finite-difference method depend on the type of differential equation to be solved. Table 11.1 provides a simple classification of river engineering problems. The information propagates at a celerity c in hyperbolic equations, and the celerity is effectively infinite in parabolic equations.

Once a river engineering problem has been defined and a mathematical model chosen, field data need to be gathered to describe initial and boundary conditions, geometrical similitude, material properties, and design conditions. Additional data are also required for calibration and verification. The governing equations can be simplified to preserve the main features of the physical problem; the time and the space increments are determined at this stage. A schematization can be made of the design conditions to be investigated.

As a natural science, the variability of river processes must be examined through the measurement of physical parameters. This chapter first describes dimensions and units (Section 2.1), physical properties of water (Section 2.2), and sediment (Section 2.3). The equations governing the motion of water and sediment from upland areas to oceans include kinematics of flow (Section 2.4), the equation of continuity (Section 2.5), the equation of motion (Section 2.6), and the concept of hydraulic and energy grade lines (Section 2.7). In this chapter and in the rest of the book, a solid diamond (♦) denotes equations and problems of particular significance. Problems with a double diamond (♦♦) are considered most important.

Dimensions and units

Physical properties are usually expressed in terms of the following fundamental dimensions: mass (M), length (L), time (T), and temperature (T°). Throughout the text, the unit of mass is preferred to the corresponding unit of force. The fundamental dimensions are measurable parameters that can be quantified in fundamental units.

In the SI system of units, the fundamental units of mass, length, time, and temperature are the kilogram (kg), the meter (m), the second (s), and degrees Kelvin (°K). Alternatively, the Celsius scale (°C) is commonly preferred because it refers to the freezing point of water at 0°C and the boiling point at 100°C.

A Newton (N) is defined as the force required for accelerating 1 kg at 1 m/s2.

Steady flow refers to flow conditions that do not change with time. Steady flows can be either uniform when the conditions do not change with space or nonuniform when flow conditions change with space. Steady flow in rivers (Section 4.1) includes description of at-a-station hydraulic geometry, followed by a description of steady-uniform flow and resistance to flow. Steady-nonuniform flows (Section 4.2) include an analysis of the momentum equations followed by rapidly varied flow and gradually varied flow. Sediment transport in rivers (Section 4.3) includes a simple description for sediment transport in steadyuniform flow followed with calculations of aggradation and degradation in river reaches.

Steady river flow

Drainage networks have been studied by geomorphologists and topologists. In general, topologists search mathematical ranking and order among subwatersheds without specific reference to physical entities. The results of several years of experimental studies from the Rainfall Erosion Facility at Colorado State University by Schumm et al. (1987) attempted to relate basin morphology and sediment yield. Although considerable sediment-yield variability is assumed to result from climatic fluctuations and land-use changes, the experiments show that sediment yield is highly variable under steady rainfall conditions. The complex response of channel network evolution seems to be characterized by an exponential decrease in sediment yield as the channel network develops.

Rivers follow the low points along the watershed topographic profiles. With very few exceptions in arid areas, the lowest point of awatershed is located at the river outlet.

This chapter covers the characteristic of river basins (also called watersheds) that affect surface runoff and sediment yield. The main watershed characteristics are illustrated in Section 3.1, followed by rainfall precipitation (Section 3.2), interception and infiltration (Section 3.3), excess rainfall (Section 3.4), and surface runoff (Section 3.5). Soil eroded from upland areas is usually the source of most sediments that are transported by rivers to reservoirs and estuaries. Methods are presented to calculate upland erosion (Section 3.6) and to estimate sediment yield from watersheds (Section 3.7).

River-basin characteristics

The hydrologic cycle describes processes that contribute to the source and the yield of water and sediment from upland areas to the fluvial system. Figure 3.1 depicts a portion of a watershed during a precipitation event. Shown in this figure are the processes of condensation, precipitation, interception, evaporation, transpiration, infiltration, subsurface flow, exfiltration, deep percolation, groundwater flow, surface flow, surface-detention storage, channel precipitation, evaporation, and streamflow. All of these processes play a role in hydrology; however, precipitation, infiltration, overland flow, and streamflow are most important in surface runoff, upland erosion, and river mechanics.

River basins or watersheds define areas of the Earth's surface where rainwater drains into a particular stream. The terms basin and catchment are synonymously used in the literature. Watershed characteristics can often be described in geographical terms, including physiography and topography, geology and pedology, and climatology and forestry.

Rivers have fascinated humanity for centuries. Most prosperous cities around theworld have been founded along rivers. Today, river engineers are still designing structures to draw benefits from the fluvial system for developing societies. It is clear that river engineering is not based solely on a simple understanding of local hydrodynamic forces, but also on an encompassing knowledge of the watershed that supplies water and sediment to dynamic river systems. Expertise in river mechanics combines knowledge of watershed climatology, geomorphology, and hydrology, with a deep understanding of hydrodynamic forces governing the motion of water and sediment in complex river systems. State-of-the-art teaching of river mechanics clearly requires study material that emphasizes both theoretical concepts and practical engineering technology. Ideally, scientists should develop new concepts that may be applicable to engineering design, and practitioners should understand why certain structures work and why others fail.

This textbook has been prepared for engineers and scientists seeking broadbased technical expertise in river mechanics. It has been specifically designed for graduate students, for scholars actively pursuing scientific research, and for practitioners keeping up with developments in river mechanics. The prerequisites simply include basic knowledge of undergraduate fluid mechanics and partial differential equations. The textbook Erosion and Sedimentation, by the same author and Cambridge University Press, serves as prerequisite material for the graduate course on river mechanics taught at Colorado State University.

In a strict sense, a channel is stable when all particles along the wetted perimeter are not moving. This implies that, without transport of bed material, a crosssectional geometry cannot change with time. The geometry of stable channels, also termed nonalluvial channels, depends on rock outcrops and artificial riprap and does not depend on sediment transport. Many rivers, however, flow in their own deposits, called alluvium, to form alluvial channels. Equilibrium of alluvial channels implies a balance between incoming and outgoing water discharge and sediment load. Poised and graded streams are synonyms describing equilibrium conditions. Whenever a balance is obtained between incoming and outgoing sediment discharges, the cross-sectional geometry may locally change as long as the deposition volume with a river reach is equal to the erosion volume. For instance, river bends may reach equilibrium condition between the rate of erosion on the outside bank and the rate of sedimentation on the point bar. In a broad sense, the cross-sectional geometry of a meandering channel is in equilibrium. However, lateral migration of the bend implies that the plan form geometry of the stream is not stable.

This chapter deals with stable and equilibrium river conditions. Section 6.1 details particle stability in a strict sense, and the concept of stable channel is used in Section 6.2 to define the ideal at-a-station cross-sectional geometry of a straight channel. Empirical regime relationships in Section 6.3 are followed by flow conditions in river bends in Section 6.4.