Chapter 9 is devoted to one of the most important new developments in soil research in the last 50 years: the linkage of geomorphological models with soils on hillslopes. Soils on hillslopes are dynamic entities that are constantly in motion and that exhibit aspects of complex systems, with feedbacks that stabilize them and help ensure their presence on all but the more extreme conditions on Earth. The processes (and models) to describe hillslope mass balance (erosion and production) are introduced. The importance of hillslope erosion and soil production to biogeochemistry is considered. Finally, soils on depositional landforms, which receive the material from the hillslopes, are also considered.

In this chapter we consider the finite element formulation for bending of slender bodies under a tensile or compressive axial force. In order to capture the effect of axial force we look at the force and moment equilibrium in the deformed configuration, but still assuming small translational displacement and rotation of the cross-section. In the finite element formulation, it is shown that the effect of axial force on bending manifests as an effective bending stiffness. It will be shown that the finite element formulation of a slender body under compressive axial force results in a matrix equation for eigenvalue analysis from which we can determine the static buckling load and the buckling mode. Subsequently, we consider the finite element formulation for vibration analysis of slender bodies to investigate the effect of axial force on the natural frequencies and modes. Finally, we introduce the finite element formulation of slender bodies subjected to a compressive follower force in which the direction of the applied force is always parallel to the body axis in the deformed configuration.

Chapter 3 is devoted to the biology of soil biogeochemistry. This is a rapidly evolving field. The chapter begins with our present understanding of the tree of life, and how little of it we have been able to detect. The second section examines the role of biology, its enzymatic impacts on the nature of chemistry over geological time, and the impacts it has had via oxidation-reduction pathways. The section considers how minerals and compounds that are now common on Earth are present largely (or only) because of biological processes. The next section examines what is presently known about the geography of soil microbiology, and what that means for the spatial diversity of metabolic capabilities. The chapter considers the challenges and emerging opportunities of explicitly embedding microbial parameters into biogeochemical models. Finally, the role and impact of vascular plants on nutrient cycling, distributions, and weathering are introduced.

Chapter 6 is an introduction to the importance and role of time in soil biogeochemistry. Time is one of the five major factors of soil formation, but many students are unfamiliar with the concept and terms of geological time, and with the concept and understanding of soil age. New developments in geomorphology and geochemistry in the last two decades have further added important insights into the concept and determination of age on hillslopes, which is also introduced here.

Chapter 10 focuses on what might be the most important reason for studying soil biogeochemistry: its importance for and interactions with us. First, the sheer magnitude of human disturbance of the soil mantle on Earth is introduced, especially the effect of farming. The impact of farming on soil C is examined in detail, and the effects of irrigated agriculture on soil geochemistry are used as an example of further comparisons that can be made by the reader. The uncertain and potential impact of our warming climate system on soil C, and potential feedbacks that might be ensuing, are considered. The recently renewed interest in accelerated soil erosion on managed landscapes is examined in light of new methods to measure the rates of erosion and also our emerging concepts and data on the true rate of soil production and regeneration.

Under certain conditions, a finite element may lose its ability to deform and become excessively stiff. This phenomenon is called "element locking." In this chapter we will consider three forms of locking, including transverse shear locking, membrane locking, and incompressibility locking. Approaches for alleviating or avoiding locking will also be described.

Truss structures are built up from individual slender-body members connected at common joints. The members are connected through hinge joints which are free to rotate and thus cannot transmit moment. Individual members carry only axial tensile or compressive force. In this chapter, the truss is comprised of uniaxial elements introduced in the previous chapter. However, in order to construct the global stiffness matrix of a truss structure in 3D space, it is necessary to construct the element stiffness matrices with 3 DOF at each node, corresponding to three displacement components in the Cartesian coordinate system. After developing the finite element formulation for 3D truss structures, the effects of thermal expansion and uniaxial members subject to torsional deformation are treated.

In this chapter, we consider time-dependent problems of discrete systems with N DOF. We will show how the finite element formulation is used to construct the element mass matrices, which are assembled into the global mass matrix. We will consider free vibration, to determine natural frequencies and natural modes of a finite element model through eigenvalue analysis, and numerical methods for integrating the equation of motion in time which can be used to determine dynamic response under applied loads and given initial conditions. The Lagrange equation will be shown to demonstrate how it can be applied to construct equations of motion. Once again we will consider slender bodies undergoing uniaxial vibration, torsional vibration, and bending vibration. A formal derivation of the Lagrange equation will be considered in a later chapter.