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.

As in the case of viscous fluids, very good approximations to exact results for viscoelastic fluids can be obtained from purely irrotational studies of stability. Here we consider RT instability (§21.1) and capillary instability (§21.2) of an Oldroyd B fluid. Viscoelastic effects enter into the irrotational analysis of RT instability through the normal stress at the free surface. For capillary instability, the short waves are stabilized by surface tension, and an irrotational viscoelastic pressure must be added to achieve excellent agreements with the exact solution. The extra pressure gives the same result as the dissipation method as is true in viscous fluids where VPF works for short waves and VCVPF and DM give the same results for capillary instability.

Rayleigh–Taylor instability of viscoelastic drops at high Weber numbers

Movies of the breakup of viscous and viscoelastic drops in the high-speed airstream behind a shock wave in a shock tube have been reported by Joseph, Belanger, and Beavers (1999; hereafter JBB). They performed a RT stability analysis for the initial breakup of a drop of Newtonian liquid and found that the most unstable RT wave fits nearly perfectly with waves measured on enhanced images of drops from the movies, but the effects of viscosity cannot be neglected. Snapshots from these movies are displayed in figures 21.1 to 21.4 and 21.9; data for the experiments are shown in table 21.1.

We study the force on a 2D cylinder near a wall in two potential flows: the flow that is due to the circulation 2πκ about the cylinder and the uniform streaming flow with velocity V past the cylinder. The pressure is computed with Bernoulli's equation, and the viscous normal stress is calculated with VPF; the shear stress is ignored. The forces on the cylinder are computed by integration of the normal stress over the surface of the cylinder. In both of the two cases, the force perpendicular to the wall (lift) is due to only the pressure and the force parallel to the wall (drag) is due to only the viscous normal stress. Our results show that the drag on a cylinder near a wall is larger than on a cylinder in an unbounded domain. In the flow induced by circulation or in the streaming flow, the lift force is always pushing the cylinder toward the wall. However, when the two flows are combined, the lift force can be pushing the cylinder away from the wall or toward the wall.

The flow that is due to the circulation about the cylinder

Figure 10.1 shows a cylinder with radius a near the wall x = 0.

When a vessel containing liquid is made to vibrate vertically with constant frequency and amplitude, a pattern of standing waves on the gas–liquid surface can appear. For some combinations of frequency and amplitude, waves appear; for other combinations the free surface remains flat. These waves were first studied in the experiments of Faraday (1831), who noticed that the frequency of the liquid vibrations was only half that of the vessel. Nowadays, this would be described as a symmetry-breaking vibration of a type that characterized the motion of a simple pendulum subjected to a vertical oscillation of its purpose.

The first mathematical study of Faraday waves are due to Rayleigh (1883a, 1883b) but the first definitive study is due to Benjamin and Ursell (1954; hereafter BU) who remark that “The present work has been made possible by the development of the theory of Mathieu functions.”

Faraday's problem is a rich source of problems in pattern formation, bifurcation, chaos, and other topics within the framework of fluid mechanics applications in the modern theory of dynamical system. Under the excitation of different parameters governing the Faraday system, different patterns, stripes, squares, hexagons, and time-dependent states can be observed. These features have spawned a large recent literature on Faraday waves. The experiments of Ciliberto and Gollub (1985) and Simonelli and Gollub (1989) on chaos, symmetry, and mode interactions are often cited.