Previous chapters examined the topics of equilibrium, compatibility, strain–displacement relations, and stress–strain relations. When these elements are combined, we can form up different complete sets of governing differential and algebraic equations. In order to solve those sets of equations we must also specify the conditions that arise from having known loads or geometric constraints. These are called the boundary conditions for the problem.In this chapter we will examine some of the choices we have for formulating complete sets of the governing equations and how those governing equations can be combined with appropriate boundary conditions to solve for the stresses and deformations. We will also discuss the principle of Saint-Venant, which gives us some flexibility in how we specify the boundary conditions. Finally, we will also show how structural analysis problems can be expressed in terms of algebraic matrix–vector equations, which are the counterparts of the governing differential/algebraic equations. A classical deformable body problem, Navier's table problem, will be used as an example of these purely algebraic methods.

The engineering beam bending theory summarized in Chapter 1 assumed that the beam cross-sectional area has a plane of symmetry and that bending moments were acting along a single axis. In this chapter we want to remove those restrictions and to examine the multiaxis bending of beams with nonsymmetrical cross-sections. This will lead to a generalization of the flexure formula for the normal stress in the beam.In Chapter 1 we also obtained an expression for the shear stresses induced in symmetrical beams. It is difficult to obtain similar analytical shear-stress forms for beams with general unsymmetrical cross-sections. However, we will show that when the cross-section is thin one can obtain explicit expressions for the shear stresses. Analysis of the bending of thin beams will demonstrate that the shear force in the beam must pass through a specific point, called the shear center, if the beam is to bend without twisting. A new cross-sectional area property, called the principal sectorial area function, will be shown to play a key role in locating the shear center.

Most complex deformable body problems commonly found in practice can only be solved with numerical methods. The stiffness-based finite element method is today the numerical method of choice for analyzing deformable bodies as well as many other engineering problems. In this chapter we will describe both stiffness-based and force-based finite elements. It will be shown that while the stiffness-based finite element method is based on solving equilibrium equations for the displacements, the force-based method relies on appropriately combining equilibrium and compatibility to solve directly for the forces (or stresses). We will examine both stiffness-based and force-based finite elements for axial load and bending problems. Simple examples will be used that allow us to work through the application of these finite element methods in detail and to compare results with analytical solutions. Finally, we will outline an important alternative to a finite element approach - the boundary element method – for solving stress problems numerically.

Stresses describe the local distributions of forces within a deformable body and strains describe the local deformations. In this chapter we want to describe the relations between stresses and strains as these are the key relationships that allow us to connect the loads applied to a body to its changes in shape. We will only consider linear elastic materials in this book where the stresses are proportional to the small strains present. Both isotropic and anisotropic linear elastic materials will be discussed. How the elastic constants appearing in the general stress–strain relations for an anisotropic material change with choice of orientation of the coordinate system being used will be given explicitly. The use of strain gages and stress–strain relations to determine the state of stress on the surface of a body will be discussed.

The torsion of a solid or hollow bar with a circular cross-section is one of the important problems considered in elementary strength of material texts. In this chapter we consider the torsion of bars having more general cross-sections, where the axial warping deformations produced requires that one develop a much more complex solution procedure. We will first consider the idealized case of uniform torsion where the bar is completely free to warp. Solutions of uniform torsion problems are obtained using both a warping function and a Prandtl stress function approach. The case of nonuniform torsion, where the rate of twist varies along the length of the bar and the warping of the bar is restrained, is also considered using a warping function approach. Uniform and nonuniform torsion problems for general cross-sections typically require a numerical solution. However, in the case of thin members, one can obtain more direct solutions for both open and closed cross-sections. It will be shown that the sectorial area function plays a major role in the solution of both uniform and nonuniform torsion problems for thin cross-sections.

The chapter begins by defining mixing and then discusses how ocean mixing is studied by a combination of direct observations, process studies, and studies integrated with modeling. The role of mixing in the meridional overturning circulation is examined in detail, including current suggestions termed ‘upside-down’ mixing. The chapter concludes with energy and scalar budgets that determine average mixing levels throughout the ocean.

This chapter discusses the interactions generating and dissipating internal waves, the primary mechanism mixing the stratified ocean. Generation is primarily by wind stress at the surface, bottom stress at the seafloor, and tidal flows over irregular bottoms. Energy is transferred from large to small scales by resonant interactions, primary triads, that lead to ultimate breaking and mixing. Testing of expressions for predicting dissipation rates resulting from wave–wave interactions is examined in detail. The mechanisms of breaking affect the efficiency of mixing and therefore the buoyancy fluxes it produces. Shear instabilities and advective overturning are the two primary mechanisms, but the form of instability remains open to question. Finally, the saturated range separating linear internal waves from turbulence is examined.

After explaining the mechanism producing double diffusion, its representation analytically is developed and applied to linear stability analysis to determine conditions for double diffusion to occur. Laboratory observations of salt fingers are summarized, as well as the existence of thermohaline staircases produced by fingers in the thermocline. A similar development for the diffusive regime includes layering in Lake Kivu and in polar regions. Lastly, the role of double diffusion in thermohaline intrusions is explored.

This chapter explores how mixing is measured, focusing on what can be measured and how accurately. It begins with probes measuring dissipation scales, i.e. thermistors and airfoils, but also includes other devices not used routinely, such as pitot tubes. Finescale sensors are also examined, with an emphasis on absolute accuracy and salinity spiking. Then the vehicles carrying the probes are described, i.e. profilers, tows, AUVs, and submersibles. Moorings and fixed platforms are also briefly examined. Finally, remote sensing, principally using backscatter from high-frequency acoustics, is reviewed, along with tracer releases for measuring net diffusivity.

Dimensional analysis for stratified turbulence defines the variables describing turbulence in the ocean, which are then used with the Reynolds decomposition to develop equations of motion and define key parameters, such as turbulent kinetic energy and the viscous dissipation rate. These are used in developing the Kolmogorov energy cascade and the accompanying cascade of scalar variance. Considering turbulent evolution and decay leads to consideration of strongly stratified turbulence and pancake eddies. Turbulent intermittence and statistics include expressions for estimating confidence limits. The turbulent equations are then used to develop expressions for estimating turbulent eddy coefficients from what can be measured. The chapter ends with a consideration of mixing efficiency.

The last chapter combines themes from earlier chapters to summarize what has been learned about mixing in the stratified ocean. The nature of finestructure is explored first, as it contains signatures of the processes producing the mixing, as well as modulating the development of turbulent patches. After exploring how patches are identified, their characteristics are discussed. Evidence for the other major mixing process, double diffusion, is presented, principally as signatures in horizontal tows. The chapter concludes with summaries of mixing in three important regions differing from the open ocean pycnocline: the Southern Ocean, the Arctic, and ocean ridges.