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.

Part I. True or False?

(Answers at the end of this section)

1. The principal stress axes are always the same as the principal strain axes in an isotropic material.

2. The principal stress axes are always the same as the principal strain axes in an orthotropic material.

3. The equations of compability derived in this text, such as εxx,yy + εyy,xx = γxy,xy, are valid for large strains as well as small strains. (Recall that small strains are linear expressions involving the derivatives of displacements, while large strains further include quadratic expressions involving derivatives of the displacements.)

4. The coordinate rotation equation for stresses, that is, [σ*]= [c][σ][c]t, is valid for any material, not just an isotropic material.

5. The equations of equilibrium apply to a material undergoing plastic deformations as well as a material undergoing purely elastic deformations.

6. The line paralleling the linear portion of the stress–strain curve that defines an offset yield stress originates at a strain value of 0.01 in/in.

7. Not counting the coefficient of thermal expansion, there are three independent material constants that require specification for the linearly elastic, isotropic material model.

8. Young's modulus is the same as the modulus of rigidity.

9. The Cauchy equations are a set of algebraic equations that relate the tractions to the internal stresses, but do not include the body forces.

10. Both the constitutive and strain–displacement equations for an isotropic material are different from those for an orthotropic material.

11. The Cauchy equations can be viewed as an application of Newton's second law; that is, they are based upon a summation of forces.

12. The material stiffness matrix [E] is the matrix transpose of the material compliance matrix [S].

13. […]

Introduction

At this point, all that is available for the purpose of accomplishing a general structural analysis are the three equilibrium equations, Eqs. (1.6). These equations pertain to the stress state in a general structural body subjected to a general mechanical loading (including dynamic loads) and a temperature change. These three equations are insufficient to deduce the six stresses that define the stress state. Since the reader's ambition and good sense require nothing less than a complete set of equations that are applicable to a structure of any shape or material, as well as any loading, it is necessary to look beyond equilibrium considerations in order to describe fully the response of a general structural body. The two other physical phenomena that need to be investigated in order to obtain additional equations are the geometry of the deformations of the general structure, and the response of materials to mechanical and thermal loadings. Descriptions of the deformations of loaded structures are the focus of this chapter and Chapter 4. Chapters 5 and 6 discuss the response of structural materials to loadings and temperature changes.

Displacements

The general concept of a displacement is simply that of a movement; a change in position that has been completed or is in progress. The change in position involves both a direction and a distance. Thus a displacement is defined as a vector quantity. For engineering purposes, the displacement concept must be susceptible to precise description and measurement.

Introduction

In Parts I and III, equilibrium equations are established via Newton's laws by summing the forces and moments acting on isolated (free) bodies. Equilibrium equations assure continuity of forces and moments. Continuity of displacements is established by requiring either that (i) the orthogonal displacement or deflection components are described by differentiable functions in the case of a displacement formulation, or (ii) the strain compatibility equations are satisfied in the case of a stress formulation. Since the stresses acting upon fixed surfaces are vector quantities, and the displacement and deflection components are also vector quantities, the methods of Parts I, II, and III are often called “vector methods.” This chapter introduces an alternate approach to vector methods. This alternate approach, called “energy methods,” is often analytically superior to vector methods, particularly for complex problems. Energy methods have the same bases as vector methods. That is, just like vector methods, energy methods can be based upon Newton's laws and the same geometric description of structural deformations. However, energy methods involve scalar quantities such as work and potential energy. The shift from vector quantities to work and potential energy results in four important advantages.

The first advantage of energy methods is that the work or energy statements permit the choice of any coordinate system without a change in form for the basic equations.

Aircraft and Other Vehicular Structures

The structure of a flight or marine vehicle usually has a dual function: (i) it transmits and resists the fluid and other forces that are applied to the vehicle; and (ii) it acts as a cover that provides an aerodynamic or hydrodynamic shape that protects the contents of the vehicle from the environment. This combination of roles is fortunate since, from the standpoint of structural weight, the most efficient location for the structural material is at the outer surface of the vehicle. Thus the structures of flight and marine vehicles, and some land vehicles, are essentially thin shells. If these shells are not reinforced by stiffening members, they are referred to as monocoque. When the cross-sectional dimensions of the shell are large, the wall of a monocoque structure must be relatively thick to resist bending, compressive, and torsional loads without buckling. In such cases a much more weight-efficient type of construction is one which contains stiffening members that permit a much thinner covering shell. Stiffening members may also be required to diffuse concentrated loads into the skin. (Most aircraft have “strong points” where the structure can be put on jacks or lifted by a crane.) Construction of this type is called semimonocoque. Typical segments of semimonocoque flight structures are sketched in Fig. III.

Introduction

In Chapter 1 the choice of the origin and orientation of the analysis Cartesian coordinate system was made arbitrarily. For a body of general shape there simply is no geometric feature with which to align the coordinate system for the advantage of the analyst. Even if the structure has a special geometry that clearly suggests an advantage for a particular origin or orientation of a Cartesian coordinate system, it is still necessary to consider the following question. What effect would there be upon the analysis results (e.g., the calculated values of the stresses) if another Cartesian coordinate system were selected that is different from the first in its origin and orientation? In general, the answer is that the set of stresses associated with the second Cartesian coordinate system are different from those that are associated with the first Cartesian coordinate system. For example, consider the first bar of Fig. 1.1 (a). If the first Cartesian coordinate system is such that the x axis runs along the length of the bar, then with the end forces applied as uniform tractions, the first set of stresses would be σxx = N/A with the other five stresses equal to zero. Similarly, if the second Cartesian coordinate system were such that the y axis ran the length of the bar, the second set of stresses would be σyy = N/A with the other five stresses equal to zero.

Preface

There is only one chapter in Part IV so as to emphasize the important role work and energy principles play in modern structural analysis. Every chapter past this one depends upon the material of this chapter. The content of this chapter admittedly will be challenging to all those who have not had any previous experience with this material. For the student, the concept of work is not likely to have had a particularly prominent place in prior studies. The concept of potential energy is likely to have made a previous appearance only in relation to the gravitational potential. The calculus of variations, a key aspect of this chapter, is likely to be entirely new to the student (be sure to read Endnote (1)). Obviously in this chapter there are abundant opportunities for intellectual growth. In order to take advantage of those opportunities, the Green–Gauss theorem must be mastered. The next section provides a review of that theorem.

The Green–Gauss Theorem

Consider a closed area A on an x plane such as shown in Fig. IV.1(a). Let this area be such that any line paralleling the z axis cuts ∂A, the piecewise smooth boundary of A, only twice. Let the various boundary segments that comprise the total boundary be divided into a lower and upper curve which meet at the points y = a and y = b, where those points are the minimum and maximum y values on the boundary. In the case illustrated in Fig. IV.1, the “point” y=a is actually a line segment.

Introduction

To a close approximation, when a beam bends and extends, each planar cross-section translates in each of the three Cartesian coordinate directions, and rotates about the y and z axes. The one motion that is excluded from those approximations is the twisting of the beam cross-section about the x axis, called φx. The primary reason for separating this one motion from the other five is that its inclusion substantially complicates the treatment of finite beam bending deflections. However, for small deflections, there is no interaction between the twisting motion and the other five motions. Thus by limiting the discussion of this chapter and Chapter 13 to the situation where the beam bending deflections, if present, are small, the following discussion of twisting deflections and torsional loadings can proceed without taking any notice at all of the extensional and bending deflections caused by axial and shearing forces and bending moments.

Recall that a bar is a beam that is loaded only in extension or torsion. Since only twisting deflections are to be discussed in this chapter and Chapter 13, here the terms beam and bar can be, and are, used interchangeably. Consider a bar with a noncircular cross-section. When the bar is twisted, the bar cross-section of arbitrary shape does not remain plane after twisting. On the contrary, the cross-section warps out of its original plane in apparently complicated ways.

This text has a single purpose. That purpose is to provide clear instruction in the fundamental concepts of the theory of structural analysis as applied to vehicular structures such as aircraft, automobiles, ships, and spacecraft. To this end, the text offers explanations and applications of the fundamental concepts of structural analysis and indications of how those concepts are employed in everyday engineering practice. The text endeavors to foster in the reader the habit of asking questions until the reader is thoroughly clear on all important details within the scope of this text.

Three strategies are followed to achieve clarity with regard to the basic concepts of structural analysis. The first strategy is to be thoroughly logical within the scope of the presented material. No “assumptions” with regard to method of analysis are made anywhere in this textbook. All approximations are accompanied by a full explanation of their validity. The second strategy is to be repetitious and redundant. Repetition is an important learning tool, and redundancy dispels misunderstandings. The third strategy to obtain the goal of clarity is to limit the number of topics covered in detail in this text to only those that are essential to an introduction to modern structural analysis.