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.

Introduction

The use of a separate chapter for truss problems is prompted by the convenience of using the simpler bar element, as opposed to the beam element, as a means of illustrating the following topics that are less vital to an overall understanding of the basics of the FEM: (i) bars (or beams, or whatever) that are neither horizontally nor vertically oriented so that the element DOF are rotated relative to the global DOF; (ii) the equivalent loads that arise from enforced boundary deflections; and (iii) the equivalent loads that arise as a result of temperature changes or other initial strains. For the sake of simplicity, the developments of this chapter will be kept within the confines of planar trusses. The only slightly more complicated geometry of three-dimensional structures, which is available in all major commercial programs, is not necessary to the development of the reader's understanding of the concepts to be explored here. However, the bar element in three dimensions is adequately introduced in Exercise 18.10(b).

Bar elements do not differ much from beam elements. A distinction is usually made in a commercial FEM program between a beam and a bar in order to use the simpler bar element formulation wherever possible. It can be expected that any such program offers a beam element that includes the axial deformation properties of a bar. That is, there is always available a beam element with six DOF at each beam end, for a total of 12 DOF.

Introduction

In the course of a career, an engineer can expect to be called upon to deal with the behavior of a great many different types of materials from which an object with a structural purpose is to be, or has been, fashioned. Metals, plastics, woods, and man-made composites of all kinds are plentiful in structural engineering practice. The paragraphs below provide a brief overview of these materials, starting with the metals.

The material that is used most extensively in modern engineering practice is steel. In comparison with other metals and composites, steels are cheap, and have the advantages of being stiff, strong, and hard. Thus steels are often the material of choice for vehicular applications such as automobile frames and aircraft landing gear. The superior high-temperature properties of certain alloy steels often results in their choice as a material for the structure of fuel-burning engines such as rockets. The principal drawback of steel for vehicular use is its relatively high weight density of approximately 0.28–0.29 lb/in3. There is presently a great variety of types of steel and steel alloys (Refs. [10–14]). There is also a great variety of designations of steels. Lists of specification equivalents can be found, for example, in Ref. [13]. The American Iron and Steel Institute (AISI) and the Society of Automotive Engineers (SAE) have agreed upon an AISI–SAE four-digit numbering system for steels that is widely, but not exclusively, used. It is outlined on p. 143 of Ref. [10].

Introduction

The finite element method introduced in this chapter is the routine choice for the analysis of structures in government and industry, large companies and small. The finite element method is especially useful in the aerospace sciences and all related fields. For engineering applications, the finite element method was introduced to the engineering community by aerospace structural dynamicists in Ref. [49]. It is not the only useful, or necessary-to-know, method (one other method requires discussion), but it dominates because it is the only method suitable for large, complicated structures such as airplanes, helicopters, ships, and land vehicles, as well as the small redundant structures discussed here. It is particularly broad in scope. As a mathematical concept the finite element method is applicable to a wide variety of problems including such diverse problems as those of fluid dynamics, heat transfer, electromagnetic fields, and electrical circuits. As a mathematical concept, it is an adaptation of the Rayleigh–Ritz method. In this adaptation, instead of the requirement for choosing approximation functions for each new analysis, a finite element analysis can be computer programed because the same functions are repeatedly used for the same type of structural element. Instead of using the complicated functions that are sometimes necessary to meet boundary condition requirements of complicated structures, this adaptation uses simple functions, almost exclusively polynomial terms for quick computer processing.

Introduction

Routine analyses of structures whose mathematical models contain more than two or three structural elements (e.g., beams, plates, springs) are accomplished today using one or another of the commercially available, large or small capacity, standard structural analysis digital computer software packages. The larger commercially available structural analysis programs have become so inclusive, that few types of analyses are beyond their reach. In addition to producing what is called a numerical analysis of the structure's mathematical model, such software packages, perhaps in combination with auxiliary programs, are also commonly used in creating a structure's mathematical model, and in simplifying the interpretation of the analysis results. The basis for the large majority, if not all, of these general use software packages is a numerical analysis technique called the finite element method, which is introduced in Chapter 17.

The essence of any numerical analysis is the replacement of a collection of partial or ordinary differential equations (or sometimes integral equations) by a much larger set of simultaneous, algebraic equations. The set of algebraic equations involve an equal number of algebraic variables that, in one sense or another, are used to approximate the continuous, unknown functions of the differential equations being replaced by the algebaric equations. In almost every case, the differential equations are those obtained from strength of materials. If the differential equations are nonlinear, so too are the algebraic equations.

Introduction

While the finite element method, with occasional assistance from the unit load method or some such application of the principle of complementary virtual work, is a completely satisfactory approach to all linear structural analysis tasks, there are still many topics that need to be touched upon in this introductory textbook in order to provide a reasonably complete overview of aerospace structural analysis. The first of these topics is the mechanics of two-dimensional structural elements other than merely the equations for the out-of-plane stretching of thin membranes and the finite elements that can be used to determine the in-plane deflections and stresses of thin skins. Chapter 22 considers the strength of materials approach to thin plate bending and buckling. Again, the objective is to provide a sufficient understanding of the mechanics and the mathematics of thin plate bending so that a simple, but representative, thin plate finite element can be introduced. The next step in complexity after thin plates is thin shells. A treatment of thin shells of general geometry is complicated. Hence the analysis of thin shells is left entirely to textbooks that are specific to that topic.

Chapter 23 returns to the essential topic of elastic stability by discussing a simple finite element for beam buckling. Just as any finite element analysis is much superior to a differential equation–based analysis in its ability to address geometric and material complexities, a beam buckling finite element greatly expands the range of beam and beam frame buckling problems that can be so analyzed.

Portions of this chapter are taken from Introduction to Chemical Engineering Analysis by Russell and Denn (1972) and are used with permission.

In Chapter 2, a constitutive equation for reaction rate was introduced, and the experimental means of verifying it was discussed for some simple systems. The use of the verified reaction-rate expression in some introductory design problems was illustrated in Chapter 2. Chapter 3 expanded on the analysis of reactors presented in Chapter 2 by dealing with heat exchangers and showing how the analysis is carried out for systems with two control volumes. A constitutive rate expression for heat transfer was presented, and experiments to verify it were discussed.

This chapter considers the analysis of mass contactors, devices in which there are at least two phases and in which some species are transferred between the phases. The analysis will produce a set of equations for two control volumes just as it did for heat exchangers. The rate expression for mass transfer is similar to that for heat transfer; both have a term to account for the area between the two control volumes. In heat exchangers this area is determined by the geometry of the exchanger and is readily obtained. In a mass contactor this area is determined by multiphase fluid mechanics, and its estimation requires more effort. In mass contactors in which transfer occurs across a membrane the nominal area determination is readily done just as for heat exchangers, but the actual area for transfer may be less well defined.