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.

Introduction

Beams and bars alone do not a vehicular structure make. Vehicular structures are typically beam frames and grids enveloped in a thin skin. See, for example, the illustrations in the front of this textbook, or the illustration in Endnote (2) of this chapter, which discusses the function of the thin skin, the stringers, and the frames. Characteristically, the skin thickness is much smaller than the distances between the beams that support the skin. In order to extend the finite element method to analytical models of vehicular structures, it is now necessary to begin to consider the analysis of thin skins in combination with the beam grids and frames that provide support for the skin. For the sake of simplicity, the discussion of thin skins in this introductory chapter is restricted to those that have a midsurface that parallels a single plane.

There are two distinct load cases for a planar, thin skin. In the first load case the edge traction vectors and the internal stress vectors parallel the plane of the skin, and are constant across the small thickness dimension of the skin. Furthermore, in this plane stress case, the skin is thick enough, or the lateral beam supports are sufficiently close to each other, that the skin does not buckle. Thus the deflection vectors also parallel the unloaded skin midsurface. The purpose of this chapter is to introduce a finite element-based deflection formulation to this plane stress problem.

Introduction

The theory of elasticity problems of Chapters 7 and 8 are restricted to plane stress problems. In most of those illustrative problems the elastic body has a simple geometry that is either a circular disk or a straight beam with a uniform, rectangular cross-section. In each of those example problems the loadings and material properties are also chosen to be mathematically simple. While there are many theory of elasticity solutions much more complex than those illustrated in Chapters 7 and 8, it may generally be said that almost all theory of elasticity solutions involve relatively simple geometries, simple material descriptions, and simple loadings. For example, there are no theory of elasticity solutions for straight beams with any of the efficient cross-sectional geometries usually used in engineering, such as I- or H-shaped cross-sections. Thus it is quite rare that a theory of elasticity solution is, in any precise sense, directly applicable to an everyday engineering problem. Nevertheless, the theory of elasticity problem is the problem to be solved in one manner or another.

One possible approach to actual engineering problems involving the common components (often called “elements”) of vehicular structures such as bars or beams, and plates or shells, is to seek what is called a numerical solution to all four sets of equations that comprise the theory of elasticity rather than seek an analytical solution. The solutions of Chapters 7 and 8 exemplify analytical solutions, that is, solutions expressed in terms of mathematically smooth functions.

Introduction

This chapter introduces just one of the many applications of the Principle of Complementary Virtual Work (PCVW). There are many small variations on this one application. The unit load method, the dummy load method, the virtual load method, the Maxwell–Mohr method, (Ref. [16]) the Mueller-Breslau method, (Ref. [16]) and the complementary virtual work method are all names given to what is essentially this same procedure. Since, in the case of linearly elastic materials, these same methods of analysis can also be derived from the PVW, (Ref. [16]) this same procedure is sometimes even called the method of virtual work. For the time being, the general form of this basic PCVW analysis procedure is called the unit/dummy/virtual load method. Later, the slight distinctions between the unit load method, the dummy load method, and the virtual load method are described, and the unit load method is chosen for full development.

To add to the confusion of names, there is an equally popular method based upon the Principle of the Minimum Value of the Total (Complementary) Energy that is only stylistically different from the unit/dummy/virtual load method. This method is called Castigliano's second theorem. About half of the engineering students in the United States are first taught the unit/dummy/virtual load method, while the other half are first taught Castigliano's second theorem. There is very little advantage to one of these methods with respect to the other.

Integration of the Strains to Obtain Displacements

There are two aspects to the following discussion of strains and displacements. The first aspect is an outline of the process that is the general integration of the six strains to obtain the three displacements. The second aspect is the redirection of the series of equations developed during the process of obtaining the displacements towards the second goal, which is the partial differential equations that relate the strains to each other. The equations that relate the strains are called the compatibility equations. In this textbook, the compatibility equations are of more immediate concern than the process of integrating the strains to obtain the displacements. As is proved in Endnote (1) of Chapter 3 there are six second order compatibility equations that occur in two sets or three equations of similar form. The second of the two sets of three compatibility equations is rederived here because the form of those compatibility equations is less obvious than that of the first set.

Throughout the process begun below for obtaining the displacements from the strains, it is of course presumed that the strains are known functions of the cartesian coordinates, and if necessary, time as well. The process begins with the first order partial differential equations that are the linear form of the strain–displacement equations.