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.

Introduction

There are only two chapters in this optional part of the textbook. Chapters 7 and 8 simply provide examples in different circumstances of putting together the four basic sets of equations for a stress formulation (the equilibrium equations, compatibility equations, the strain–stress material equations, and the stress boundary conditions) or a displacement formulation (the equilibrium equations, the stress–strain constitutive equations, the strain–displacement equations, and the displacement or stress boundary conditions).

This section is labeled optional for two reasons. The first and foremost reason is that the type of problems that are solved in Part II, particularly those of Chapter 8, in which the geometry of the structural element involves straight line (planar) boundaries, are more easily, indeed routinely, solved by the finite element method, which is presently the dominant approximate method of structural analysis. Only the very simple circular geometries of Chapter 7 offer a rare exception to the rule, which is to use the finite element method and forget the exact solutions offered in these two chapters and elsewhere. The second reason that this part of the textbook is optional is because it is expected that most, if not all, undergraduate curriculums cannot provide the time to consider this material, which, as mentioned, is far from the center of present engineering practice. Thus, the question arises as to why this material should be included at all.

True or False?

1. The Principles of Virtual Work and Complementary Virtual Work are valid only for elastic materials.

2. The virtual work and complementary virtual work of a entire structural system are simply the sum of the virtual work and complementary virtual work, respectively, of each part of the structural system.

3. Typical virtual work and complementary virtual work statements are δW = u δF, and δW* = F δu, respectively.

4. The PCVW for small strains is entirely equivalent to the linear strain–displacement equations, and therefore is entirely equivalent to the linear compatibility equations.

5. One possible form of complementary virtual work is a real moment moving through a virtual rotation.

6. The PCVW requires that the virtual forces and moments form a system of forces and moments that are in equilibrium with the actual forces and moments.

7. There are only stylistic differences between the virtual load method, the unit load method, and applications of Castigliano's second theorem.

8. The unit load method is a special application of the PCVW wherein a unit load system consisting of a unit load and its equilibrating reactions are the set of virtual loads.

9. For both statically determinate and indeterminate structural systems, any unit load system must be distributed as if the unit load system were another actual load system where, generally, any actual load system satisfies more than just equilibrium conditions. Note that all “load systems” include support reactions.

10. There need be no connection between any unit load system and the applied load system, and each unit load system must be linearly independent of all other unit load systems used in that analysis of the indeterminate structure.

11. […]

Introduction

As was previously noted, all complementary virtual work (CVW) methods of analysis, such as the unit load method (ULM), are force or flexibility types of analyses, and thus are a special case of a stress formulation. The internal CVW terms are always written in terms of force-type quantities such as actual and virtual bending moments, or actual and virtual twisting moments, or actual and virtual axial forces, and so forth. Flexibility is the inverse of stiffness, and these force-type quantities of the internal CVW expressions are always multiplied by the inverse of a corresponding structural element stiffness coefficient such as 1/(EI), or 1/(GJ), or 1/(EA). All force-type analyses always divide loaded structures and their supports into two categories. With respect to force-type analyses, a structure is either statically determinate or statically indeterminate, and the analysis procedure depends upon to which category the structure belongs. When a beam structure is statically determinate, all external support reactions and all internal stress resultants can be calculated by use of equations of force and moment equilibrium. Then the calculation of the internal stresses is simply a matter of algebra. See, for example, for beam extension and bending, Eq. (9.8). The only quantities not immediately derivable from the stress resultants are the deflections. Chapter 20 explains how the ULM can be used to calculate deflections when the structure is statically determinate, or when the indeterminate stress resultants have already been determined by means such as those discussed in this chapter.

Introduction

In Chapter 3 the concept of strains was developed for a single Cartesian coordinate system. The same two questions that were raised with regard to stresses referenced to a single Cartesian coordinate system can be raised with respect to strains: (i) do the strains in one Cartesian coordinate system uniquely determine the strains in another Cartesian coordinate system that is translated and rotated with respect to the first coordinate system; and (ii) how are the maximum strains determined? After the experience gained with stresses, it may be tempting to assume that the answer to the first question is in the affirmative. However, unlike stresses which involve force components, and therefore can be summed using Newton's second law, strains are only a matter of geometry. Therefore, at this point, since they do not share a common basis, no parallel can be drawn between stresses and strains with regard to the translation and rotation of Cartesian coordinate axes. The question has to be investigated anew.

Strains in Other Cartesian Coordinate Systems

The translation of Cartesian coordinate axes has no effect on longitudinal strains simply because longitudinal strains are measures of the differences between displacements at neighboring material points in a structural body. The description of the magnitudes of relative displacements in any particular coordinate axis direction is wholly unaffected by a translation of the origin of that coordinate axis.

Preface

In Chapters 16–2, the focus shifts from solving differential equations to employing the Principle of Virtual Work, or the Principle of Complementary Virtual Work, or, for those who prefer them, the corresponding energy principles. The goal remains the same: to solve larger and more complicated structural analysis problems. The shift in focus is more stylistic than fundamental. As the last chapter's endnotes demonstrate, the beam differential equations follow from either the Principle of Virtual Work, in the case of bending and extension, or the Principle of Complementary Virtual Work, in the case of twisting. Although not demonstrated here, the reverse path from differential equations to a work or energy principle is also possible when the differential equation meets certain requirements as described by Ref. [3], p. 158. Hence it is essentially a matter of convenience whether a differential equation or a work principle is the starting point of an analysis. If the structure contains more than a couple of structural elements, it is usually, if not always, the work or energy principles that are most convenient. Indeed, one particular application of the Principle of Virtual Work – the finite element method – coupled with modern digital computers, permits the routine analysis of structures with many thousands of structural elements. The finite element method is a numerical method that is unperturbed by geometric or material complexity, and it allows the analyst to minutely model (and thus analyze) one part of a structure while getting by with a crude model of other parts of the structure.

In an attempt to improve the first edition, more topics, figures, examples, and exercises have been added. The author hopes all the old errors have been removed and few new errors have been introduced. The primary change has been a greater emphasis on preparing the student for a broad understanding of the finite element method of analysis. In the author's experience, various finite element method software packages are almost, if not totally, the only means of structural analysis used today in the aerospace industry and in the associated federal and state government agencies. The three chapters dealing with the finite element method of analysis, Chapters 16, 17, and 18, are hopefully just the right amount of exposure suitable for undergraduates.

The style of presentation has remained the same. Clarity rather than brevity has been the consistent goal. Hence, there is a purposeful use of extra words and sentences in order to try to assist the reader who is new to the material. This strategy of being wordy admittedly makes this textbook less useful as a reference for the instructor who already is quite familiar with the chosen material. Perhaps this wordiness will allow that instructor the luxury of being brief in his or her lectures, knowing that this textbook is available as a backup to those lectures.