In Chapter 7, we considered the kinetic theory of a granular gas composed of smooth inelastic spheres. However, the particles in most granular materials one normally encounters are rough, and it is therefore desirable to extend the theory to rough spheres. It is clear that particle roughness will result in the transmission of a tangential impulse during collision. We shall see that the tangential impulse is partly determined by the angular velocities of the colliding pair. As a consequence, the hydrodynamic balance of the angular momentum field, which was implicitly satisfied for smooth particles, must be enforced here.

The dimensional analysis of §7.1 does not depend on whether the particles are smooth or rough. Hence the scaling of the stress with the shear rate, particle size, and density for a granular gas composed of rough particles will be the same as that for smooth particles. However, we shall see that the form and symmetry of the stress tensor is different and that additional hydrodynamic variables are required to describe the flow of a rough granular gas.

Pidduck (1922) was the first to develop a kinetic theory for a gas of rough spherical molecules. He considered “perfectly rough” spheres, which conserve the total kinetic energy of a particle pair during collision. He determined the equilibrium distribution function and applied the Chapman–Enskog analysis to determine the first correction from equilibrium for a dilute gas; his analysis is presented (with a few corrections) in Chapman and Cowling (1964, chap. 11). McCoy et al. (1966) extended the analysis to dense gases, and Lun (1991) further extended it to a dense gas of nearly elastic, nearly perfectly rough spheres.

A bunker is a combination of a bin and a hopper (Fig. 1.5). The abrupt change in geometry at the bin–hopper transition results in a rich variety of flow patterns, and oscillatory wall stresses. These features pose formidable difficulties for modeling bunker flow. At present, there are no satisfactory models which can capture all the observed features.

Some of the experimental observations are summarized below. This is followed by a discussion of models for the bin section, the transition region, and the hopper section.

EXPERIMENTAL OBSERVATIONS

Flow Regimes

The patterns observed by Nguyen et al. (1980) for the flow of sand through a bunker are shown in Fig. 4.1. For θw < 40°, mass flow (type A) occurs regardless of the value of H/W (Fig. 4.2). Here W and H are the half-width of the bunker and the height of the free surface of the material relative to the exit slot, respectively. (For small values of H, the free surface is not flat. In this case, H is the elevation of the point of intersection of the free surface with the bunker wall.) For θw > 70°, funnel flow of type B occurs when H/W exceeds a critical value (≍3) and funnel flow of type C occurs for a smaller value of H/W. As indicated by the hatched regions in Fig. 4.2, the boundaries between various regimes are not sharply defined. For θw = 60°, there is an interesting transition from mass flow to funnel flow, and back again to mass flow as H/W decreases from high values.

The flow of granular materials such as sand, snow, coal, and catalyst particles is a common occurrence in natural and industrial settings. Unfortunately, the mechanics of these materials is not well understood. Experiments reveal complex and, at times, unexpected behavior, whereas existing theories are often tentative and do not represent the entire range of observed behavior. Nevertheless, significant advances have been made in the understanding of the mechanics of granular flows, and the time is ripe for an account of experimental observations and theoretical models pertaining to flow in relatively simple geometries.

The importance of understanding granular flows need not be overstated – a large fraction of the materials handled and processed in the chemical, metallurgical, pharmaceutical, and food-processing industries are granular in nature. The flow and transportation of these materials are often critical operations in these processes. In most cases, the design of processes and equipment is based largely on experience and empirical rules. An appreciation of the underlying principles may be helpful in developing better design and operating procedures.

Some of the early investigations of granular flow were motivated by the need to understand the deformation of soils subjected to external loads, such as large structures. The deformation rates in these processes are usually very small. Theoretical models for these slow flows have increased in sophistication and complexity over the years, borrowing concepts from metal plasticity and soil mechanics. A contrasting picture of granular flow has emerged over the last three decades. This is believed to be applicable to rapid flows, where the deformation rates are large.

Introduction

A thin plate is a structural element, just as a long beam is a structural element. The thin plate also is characterized by its special geometry. While a long beam has two dimensions very much shorter than the length dimension, a thin plate, as is pictured in Fig. 1.11, has one dimension that is very much less than the other two dimensions. The thin plate's least dimension is, of course, called the plate thickness. The thickness is usually a constant in most vehicular structures, but the possibility of a gradually changing thickness can be incorporated into thin plate bending theory (Ref. [16]). The difference between a thin plate and a membrane is that a thin plate can resist both stretching and bending, while a membrane can only resist lateral or inplane loads by stretching.

There are two goals for this chapter. The first and lesser goal is to develop the equations of classical thin plate bending theory. More than an introductory discussion of the extensive body of classical thin plate bending solutions is outside the purposes of this textbook. The second and more important goal is to develop the equations necessary to describe a simple plate bending finite element comparable to the previously developed beam bending finite element. The greater importance assigned to the second, rather limited, goal not only suggests the present relative importance of the two topics but also suggests that certain simplifications like constant plate thickness are appropriate for the thin plate theory to be presented here.

Introduction

This chapter examines the standard engineering approaches to beam twisting used with those types of beam cross-sections that are commonly used in vehicular structures. These analytical approaches are far simpler than the theory of elasticity approach of Chapter 12, which requires the satisfaction of a second order partial differential equation and, in general, the satisfaction of a simple boundary condition on a difficult boundary. The basis of these approaches is the division of engineering beam cross-sections into two categories. The first category is that of thin-walled “open cross-sections” and the second category is that of thin walled “closed cross-sections.” A closed cross-section encloses one or more voids. A thin-walled pipe, or better yet, a thin-walled box beam built up from two oppositely facing channel cross-sections (i.e.,] [or []) connected by top and bottom plates so as to produce a rectangular or roughly rectangular interior void, are examples of closed cross-sections. On the other hand, an open cross-section does not have any interior voids. A single channel beam, or an H or I beam are examples of open cross-section beams.

Thin open section beams are generally more efficient than equally strong closed section beams when bending moments and shearing forces are the only significant loadings. That is, in those circumstances, thin open section beams generally weigh less than thin closed section beams. Open cross-section beams also have the advantage that they are easier to connect to other structural components, and inspect for damage.

Introduction

Chapter 5 pointed out that the mechanical and thermal response of engineering structural materials is quite complex. Nevertheless, if a uniaxial stress value lies within the bounds of the compressive and tensile elastic/proportional limits, that complex behavior becomes relatively simple. Within those two limits, or as an approximation, slight extensions of those limits to the limits of the compressive and tensile yield stresses, there is very nearly a straight-line relation between stress and strain. Moreover, that straight-line relation is very nearly the same for both loading and unloading. That is, permanent set (plastic deformation) is negligible, and the relation between stress and strain is single-valued. In other words, the stress–strain relation no longer depends upon the previous load history. Under these circumstances materials are called Hookean, or linearly elastic. In summary, on the basis of extensive experimental evidence, engineering structural materials within the yield stress limits can be and are described by a mathematical model wherein the plot of stress versus strain is exactly a straight line, and this same straight line serves the dual purpose of being a loading line and an unloading line. Thus in this chapter the focus shifts from the complicated behavior of actual engineering materials to the justifications and implications of the relatively simple linearly elastic material model.

There are three important reasons to justify restricting almost all further studies within this text to stresses whose values are limited to being within the yield stress bounds of nearly linearly elastic behavior.

Introduction

Chapter 7 illustrated the direct approach to solving selected problems in structural mechanics. In that chapter solutions for structural displacements and stresses were obtained through the separate use of the four sets of equations that constitute the theory of elasticity. There were no major difficulties in solving the example problems of that chapter because the original three-dimensional problems were reduced to one-dimensional problems (i.e., problems involving only one independent spatial variable) by means of plane stress and symmetry concepts. More challenging problems are the theory of elasticity problems that involve two independent spatial variables, such as plane stress problems in general. The purpose of this chapter is to present two of the simplest of such problems and their solutions in order for the reader to obtain some familiarity with their characteristics. The selected example and exercise problem solutions of this chapter are sufficiently valuable to be referenced in succeeding chapters as proof of the accuracy of the approximations adopted in those chapters.

While the followinge theory of elasticity solutions provide valuable practice in all the essentials of structural engineering analysis, it is also important to know that from the point of view of everyday structural engineering practice, theory of elasticity solutions are rarely, if ever, referenced. The advent of modern digital computers and the development of modern numerical methods, particularly the finite element method that is explained in Part V of this textbook, have relegated theory of elasticity solutions to the role of mere curiosities.

Introduction

The use of Eq. (9.8), the strength of materials solution for the bending and extensional axial stress σxx(x, y, z) in a long, straight beam, requires a knowledge of the internal axial force N(x) and the internal bending moments My(x) and Mz(x). The difficulty is that a free body diagram (FBD) of a beam under study is often insufficient by itself to determine the axial force or bending moments at any point along the beam x-axis. As previously mentioned, whenever the equilibrium equations alone are insufficient to determine the internal stress resultants, the structure is called indeterminate. Most beams, or beam elements, that are parts of aerospace structures are indeterminate because these beams are mostly elements of beam grids and frames with, as much as possible, rigid connections. The grids and frames are often covered by thin sheeting, referred to as the vehicle skin. Thus there are many internal unknown reations. One purpose of this type of construction is to make the structure, and hence individual beams, as stiff as possible within the constraint of least weight. Increased stiffness has many advantages in a beam or a structure. For example, the stiffer the beam or structure, the higher the load required to buckle the beam or structure, and the lesser the chance of aeroelastic instabilities such as those discussed in Chapter 9. The stiffer the beam or structure, the higher its natural frequencies and the lesser the chance that gusts or control motions will stress the structure.

Introduction

The Euler beam buckling type of elastic instability, where an entire beam axis moves laterally, is discussed in Sections 11.6 and 11.7. Plate buckling and local flange and web type beam buckling are discussed in Section 22.8. These previous elastic instability discussions centered upon the solution of appropriate differential equations. In the first part of this chapter the focus is upon the use of the finite element method to calculate elastic buckling loads. The use of the finite element method makes practical the elastic buckling analysis of structures as opposed to the buckling analysis of one or two isolated structural elements. These FEM beam buckling solutions have the same limits of applicability as the Euler beam buckling solutions.

The second part of this chapter examines certain instabilities of structures that arise from fluid–structure interactions. These instabilities have much in common with the purely elastic instabilities. Whereas the object of an elastic instability analysis is to discover the critical magnitude of a particular type of load that will cause a sudden and sizable lateral deflection of the elastic structure, the object of the aeroelastic analysis is to discover the critical value of the airspeed (or Reynolds number or Mach number) beyond which the airloads cause an ever increasing deflection or vibration amplitude. A representative static instability (divergence) and a representative dynamic instability (wing flutter) are studied in some detail. Again the FEM is useful for describing the linear elastic properties of the structure being studied.

An Overview of Part I

Vehicular weight, particularly that of aircraft and spacecraft, has a strong effect on the performance or economics of all such vehicles. Thus it is well worth spending many engineering man-hours on their design and analysis so as to make those vehicles as light-weight as possible. To make those many engineering hours of analysis as effective as possible, it is important that all the different types of loads that the vehicle will bear be well estimated, and then the structural response to those loads be carefully calculated. To carefully calculate the response of structures to estimated or measured loadings, it is important to use structural analysis techniques to which considerable confidence can be assigned. High degrees of confidence are achieved through experience and through thorough understanding of any approximations that are incorporated within the derivations of the selected structural analysis techniques. Thus it would seem that, in general, the fewer and the smaller the approximations, the more useful the structural analysis technique. This surmise is only partially true. As will be seen as the material of this textbook unfolds, the use of structural analysis techniques that contain essentially no approximations for many circumstances can be much too expensive and time consuming. Hence a compromise between cost and accuracy is necessary for good engineering practice. To understand how that compromise is found, this introduction to aerospace structures begins with the fundamentals of structural mechanics where the approximations are few in number and small in impact.

Introduction

Chapters 1–6 developed (i) the general equilibrium equations from a free body diagram (FBD) of a differential rectangular parallelepiped taken from a structural body of any shape and material; (ii) the strain–displacement equations and the equivalent compatibility equations from the geometry of the deformations of the same parallelepiped; and finally (iii) the constitutive equations for the isotropic and orthotropic linearly elastic material models. Again, these three sets of equations, which apply over the interior of the structural body, are called the field or domain equations. The Cauchy equations, which relate the tractions and the stresses at the boundary, were also discussed at length, and mention was made of prescribed displacement equations that specify displacements at the boundary. Again, the Cauchy and the prescribed displacement equations are collectively called the boundary condition equations. Together, the three sets of field equations and the boundary condition equations form the four sets of equations that are the basis for what is called the theory of elasticity. This chapter demonstrates what can be done with these four sets of equations.

There are six unknown stresses, six unknown strains, and three unknown displacements throughout the domain (interior) of the structural body of interest. There are three equilibrium equations, six stress–strain equations, and six strain–displacement equations. Thus there are a total of 15 unknown stresses, strains, and displacements, and 15 independent equations relating those quantities.

Introduction

This chapter introduces three topics that expand the usefulness of the Bernoulli–Euler beam bending and extension equations developed in the previous chapter. The first topic is elastic beam end supports. The use of elastic end supports begins the process, developed further in Part V, of modeling beams that are parts (elements) of larger elastic structures. The second topic is partial span distributed loads, and concentrated loads acting at points other than the beam ends. Then, both as another form of loading, and as a prelude to the third topic, combined lateral and axial loading cases are also examined. The third topic is beam buckling. This chapter provides only a brief introduction to beam buckling theory. However, some of the complexities of the topic are mentioned without being explored mathematically. The mathematical differences between the one standard type of buckling analysis introduced here and all the other beam analyses of this chapter and Chapter 10 are underscored. Additional aspects of beam and plate buckling theory are provided in Part VI.

Before proceeding to these three topics, it is worthwhile mentioning again a limitation on the scope of the beam bending theory developed in Chapter 10 which is retained in this chapter. That limitation is that the bending deflections are small. Thus it is possible to confine the axial and bending interactions to the bending equations, and to deal with the bending and twisting deflections separately, without regard for any interaction between them.

Part III. True or False?

(Answers at the end of this section)

1. A basic difference between the theory of elasticity and strength of materials (“applied elasticity”) is that strength of materials solutions are based upon an approximation of either the stress field or the displacement field, while the theory of elasticity uses neither approximation.

2. Even for a nonhomogeneous beam, in Bernoulli–Euler straight beam bending theory both the displacements and the strains vary linearly in both centroidal coordinate directions over a compact beam cross-section.

3. Even for a nonhomogeneous beam, in Bernoulli–Euler beam bending theory the stresses vary linearly in both centroidal coordinate directions over the compact beam cross-section.

4. The number of Prandtl stress function BCs equals the number of internal boundaries plus one BC for the external boundary.

5. The membrane analogy for uniform torsion is based upon the extension of the Bernoulli–Euler beam bending approximations to membrane bending theory.

6. The membrane analogy for uniform torsion is useful for visualizing the torsional shearing stress distribution for both compact singly connected and compact multiply connected, beam cross-sections.

7. The equations that are useful for the analysis of a uniformly twisted, multicell, closed bar cross-section, with a sufficiently stiffened cross-sectional shape, are those deflection equations that say that the twist per unit length of each individual cell is the same, and those equilibrium equations that say that the resisting torque for each cell is the same.

8. For uniform torsion, the maximum shearing stress may occur at a fillet, but always occurs at an outer boundary point of the open, that is, singly connected, cross-section.

9. […]

The Concept of Stress

Structural engineers are concerned with the effects that forces produce on structures. That forces produce results such as deformations or structural collapse is the usual structural engineering cause-to-effect point of view. Even though this viewpoint is not the only possible or even useful viewpoint, it is the one adopted implicitly in Parts I, II, and III of this text as a temporary convenience until it becomes necessary to adopt a more general viewpoint. In other words, the usual engineering viewpoint is that the forces are an input, the structure is the system, and the effects of the forces acting on the structure (deformations, cracking, etc.) are the output. If a structural effect in turn influences the forces acting on the structure, then a feedback loop involving the forces and the structural effect exists. An example of structural feedback is first encountered in Part III of this text in the form of a beam buckling problem.

The theory that is developed in the next four chapters is valid for any type of force or combination of forces (within certain limits), and any type of structure. The task of classifying types of forces and structures can wait until it becomes necessary. What is necessary now is to begin to discuss the types of effects that forces produce on structures. One effect that forces can produce is structural failure. Structural failure is defined simply as occurring whenever a structure no longer can serve its intended use.