김경종
31 KiB
commonly known as the direct stiffness method for obtaining the total structure stiffness matrix. Along with the development of the high-speed digital computer in the early 1950s, the work of Turner et al. [8] prompted further development of finite element stiffness equations expressed in matrix notation. The phrase finite element was introduced by Clough [9] in 1960 when both triangular and rectangular elements were used for plane stress analysis.
A flat, rectangular-plate bending-element stiffness matrix was developed by Melosh [10] in 1961. This was followed by development of the curved-shell bendingelement stiffness matrix for axisymmetric shells and pressure vessels by Grafton and Strome [11] in 1963.
Extension of the finite element method to three-dimensional problems with the development of a tetrahedral stiffness matrix was done by Martin [12] in 1961, by Gallagher et al. [13] in 1962, and by Melosh [14] in 1963. Additional three-dimensional elements were studied by Argyris [15] in 1964. The special case of axisymmetric solids was considered by Clough and Rashid [16] and Wilson [17] in 1965.
Most of the finite element work up to the early 1960s dealt with small strains and small displacements, elastic material behavior, and static loadings. However, large deflection and thermal analysis were considered by Turner et al. [18] in 1960 and material nonlinearities by Gallagher et al. [13] in 1962, whereas buckling problems were initially treated by Gallagher and Padlog [19] in 1963. Zienkiewicz et al. [20] extended the method to visco-elasticity problems in 1968.
In 1965 Archer [21] considered dynamic analysis in the development of the consistent-mass matrix, which is applicable to analysis of distributed-mass systems such as bars and beams in structural analysis.
With Melosh’s [14] realization in 1963 that the finite element method could be set up in terms of a variational formulation, it began to be used to solve nonstructural applications. Field problems, such as determination of the torsion of a shaft, fluid flow, and heat conduction, were solved by Zienkiewicz and Cheung [22] in 1965, Martin [23] in 1968, and Wilson and Nickel [24] in 1966.
Further extension of the method was made possible by the adaptation of weighted residual methods, first by Szabo and Lee [25] in 1969 to derive the previously known elasticity equations used in structural analysis and then by Zienkiewicz and Parekh [26] in 1970 for transient field problems. It was then recognized that when direct formulations and variational formulations are difficult or not possible to use, the method of weighted residuals may at times be appropriate. For example, in 1977 Lyness et al. [27] applied the method of weighted residuals to the determination of magnetic field.
In 1976 Belytschko [28, 29] considered problems associated with large-displacement nonlinear dynamic behavior, and improved numerical techniques for solving the resulting systems of equations. For more on these topics, consult the text by Belytschko, Liu, and Moran [58].
A relatively new field of application of the finite element method is that of bioengineering [30, 31]. This field is still troubled by such difficulties as nonlinear materials, geometric nonlinearities, and other complexities still being discovered.
From the early 1950s to the present, enormous advances have been made in the application of the finite element method to solve complicated engineering problems. Engineers, applied mathematicians, and other scientists will undoubtedly continue to
develop new applications. For an extensive bibliography on the finite element method, consult the work of Kardestuncer [32], Clough [33], or Noor [57].
1.2 Introduction to Matrix Notation
Matrix methods are a necessary tool used in the finite element method for purposes of simplifying the formulation of the element stiffness equations, for purposes of longhand solutions of various problems, and, most important, for use in programming the methods for high-speed electronic digital computers. Hence matrix notation represents a simple and easy-to-use notation for writing and solving sets of simultaneous algebraic equations.
Appendix A discusses the significant matrix concepts used throughout the text. We will present here only a brief summary of the notation used in this text.
A matrix is a rectangular array of quantities arranged in rows and columns that is often used as an aid in expressing and solving a system of algebraic equations. As examples of matrices that will be described in subsequent chapters, the force components ( F _ { 1 x } , F _ { 1 y } , F _ { 1 z } , F _ { 2 x } , F _ { 2 y } , F _ { 2 z } , \ldots , F _ { n x } , F _ { n y } , F _ { n z } ) acting at the various nodes or points ( 1 , 2 , \ldots , n ) on a structure and the corresponding set of nodal displacements ( d _ { 1 x } , d _ { 1 y } , d _ { 1 z } , d _ { 2 x } , d _ { 2 y } , d _ { 2 z } , \dotsc , d _ { n x } , d _ { n y } , d _ { n z } ) can both be expressed as matrices:
\{F \} = \underline {{{F}}} = \left\{ \begin{array}{l} F _ {1 x} \\ F _ {1 y} \\ F _ {1 z} \\ F _ {2 x} \\ F _ {2 y} \\ F _ {2 z} \\ \vdots \\ F _ {n x} \\ F _ {n y} \\ F _ {n z} \end{array} \right\} \quad \{d \} = \underline {{{d}}} = \left\{ \begin{array}{l} d _ {1 x} \\ d _ {1 y} \\ d _ {1 z} \\ d _ {2 x} \\ d _ {2 y} \\ d _ {2 z} \\ \vdots \\ d _ {n x} \\ d _ {n y} \\ d _ {n z} \end{array} \right\} \tag {1.2.1}
The subscripts to the right of F and d identify the node and the direction of force or displacement, respectively. For instance, F _ { 1 x } denotes the force at node 1 applied in the x direction. The matrices in Eqs. (1.2.1) are called column matrices and have a size of n \times 1 . The brace notation f g will be used throughout the text to denote a column matrix. The whole set of force or displacement values in the column matrix is simply represented by \{ F \} or \{ d \} . A more compact notation used throughout this text to represent any rectangular array is the underlining of the variable; that is, F and \underline { d } denote general matrices (possibly column matrices or rectangular matrices— the type will become clear in the context of the discussion associated with the variable).
The more general case of a known rectangular matrix will be indicated by use of the bracket notation ½ . For instance, the element and global structure stiffness
matrices ½k and [ K ] , respectively, developed throughout the text for various element types (such as those in Figure 1–1 on page 10), are represented by square matrices given as
[ k ] = \underline {{k}} = \left[ \begin{array}{c c c c} k _ {1 1} & k _ {1 2} & \dots & k _ {1 n} \\ k _ {2 1} & k _ {2 2} & \dots & k _ {2 n} \\ \vdots & \vdots & & \vdots \\ k _ {n 1} & k _ {n 2} & \dots & k _ {n n} \end{array} \right] \tag {1.2.2}
and [ K ] = \underline { { K } } = \left[ \begin{array} { c c c c } { K _ { 1 1 } } & { K _ { 1 2 } } & { \ldots } & { K _ { 1 n } } \\ { K _ { 2 1 } } & { K _ { 2 2 } } & { \ldots } & { K _ { 2 n } } \\ { \vdots } & { \vdots } & & { \vdots } \\ { K _ { n 1 } } & { K _ { n 2 } } & { \ldots } & { K _ { n n } } \end{array} \right] ð1:2:3Þ
where, in structural theory, the elements k _ { i j } and K _ { i j } are often referred to as stiffness influence coefficients.
You will learn that the global nodal forces \underline { { F } } and the global nodal displacements \underline { d } are related through use of the global stiffness matrix K by
\underline {{{F}}} = \underline {{{K}}} \underline {{{d}}} \tag {1.2.4}
Equation (1.2.4) is called the global stiffness equation and represents a set of simultaneous equations. It is the basic equation formulated in the stiffness or displacement method of analysis. Using the compact notation of underlining the variables, as in Eq. (1.2.4), should not cause you any difficulties in determining which matrices are column or rectangular matrices.
To obtain a clearer understanding of elements K _ { i j } in Eq. (1.2.3), we use Eq. (1.2.1) and write out the expanded form of Eq. (1.2.4) as
\left\{ \begin{array}{c} F _ {1 x} \\ F _ {1 y} \\ \vdots \\ F _ {n z} \end{array} \right\} = \left[ \begin{array}{c c c c} K _ {1 1} & K _ {1 2} & \dots & K _ {1 n} \\ K _ {2 1} & K _ {2 2} & \dots & K _ {2 n} \\ \vdots & & & \\ K _ {n 1} & K _ {n 2} & \dots & K _ {n n} \end{array} \right] \left\{ \begin{array}{c} d _ {1 x} \\ d _ {1 y} \\ \vdots \\ d _ {n z} \end{array} \right\} \tag {1.2.5}
Now assume a structure to be forced into a displaced configuration defined by d _ { 1 x } = 1 , d _ { 1 y } = d _ { 1 z } = \cdot \cdot \cdot d _ { n z } = 0 . Then from Eq. (1.2.5), we have
F _ {1 x} = K _ {1 1} \quad F _ {1 y} = K _ {2 1}, \dots , F _ {n z} = K _ {n 1} \tag {1.2.6}
Equations (1.2.6) contain all elements in the first column of K. In addition, they show that these elements, K _ { 1 1 } , K _ { 2 1 } , \ldots , K _ { n 1 } , are the values of the full set of nodal forces required to maintain the imposed displacement state. In a similar manner, the second column in K represents the values of forces required to maintain the displaced state d _ { \mathrm { l } y } = 1 and all other nodal displacement components equal to zero. We should now have a better understanding of the meaning of stiffness influence coefficients.
Subsequent chapters will discuss the element stiffness matrices k for various element types, such as bars, beams, and plane stress. They will also cover the procedure for obtaining the global stiffness matrices K for various structures and for solving Eq. (1.2.4) for the unknown displacements in matrix d.
Using matrix concepts and operations will become routine with practice; they will be valuable tools for solving small problems longhand. And matrix methods are crucial to the use of the digital computers necessary for solving complicated problems with their associated large number of simultaneous equations.
1.3 Role of the Computer
As we have said, until the early 1950s, matrix methods and the associated finite element method were not readily adaptable for solving complicated problems. Even though the finite element method was being used to describe complicated structures, the resulting large number of algebraic equations associated with the finite element method of structural analysis made the method extremely difficult and impractical to use. However, with the advent of the computer, the solution of thousands of equations in a matter of minutes became possible.
The first modern-day commercial computer appears to have been the Univac, IBM 701 which was developed in the 1950s. This computer was built based on vacuum-tube technology. Along with the UNIVAC came the punch-card technology whereby programs and data were created on punch cards. In the 1960s, transistorbased technology replaced the vacuum-tube technology due to the transistor’s reduced cost, weight, and power consumption and its higher reliability. From 1969 to the late 1970s, integrated circuit-based technology was being developed, which greatly enhanced the processing speed of computers, thus making it possible to solve larger finite element problems with increased degrees of freedom. From the late 1970s into the 1980s, large-scale integration as well as workstations that introduced a windows-type graphical interface appeared along with the computer mouse. The first computer mouse received a patent on November 17, 1970. Personal computers had now become mass-market desktop computers. These developments came during the age of networked computing, which brought the Internet and the World Wide Web. In the 1990s the Windows operating system was released, making IBM and IBMcompatible PCs more user friendly by integrating a graphical user interface into the software.
The development of the computer resulted in the writing of computational programs. Numerous special-purpose and general-purpose programs have been written to handle various complicated structural (and nonstructural) problems. Programs such as [46–56] illustrate the elegance of the finite element method and reinforce understanding of it.
In fact, finite element computer programs now can be solved on single-processor machines, such as a single desktop or laptop personal computer (PC) or on a cluster of computer nodes. The powerful memories of the PC and the advances in solver programs have made it possible to solve problems with over a million unknowns.
To use the computer, the analyst, having defined the finite element model, inputs the information into the computer. This information may include the position of the element nodal coordinates, the manner in which elements are connected, the material properties of the elements, the applied loads, boundary conditions, or constraints, and the kind of analysis to be performed. The computer then uses this information to generate and solve the equations necessary to carry out the analysis.
1.4 General Steps of the Finite Element Method
This section presents the general steps included in a finite element method formulation and solution to an engineering problem. We will use these steps as our guide in developing solutions for structural and nonstructural problems in subsequent chapters.
For simplicity’s sake, for the presentation of the steps to follow, we will consider only the structural problem. The nonstructural heat-transfer and fluid mechanics problems and their analogies to the structural problem are considered in Chapters 13 and 14.
Typically, for the structural stress-analysis problem, the engineer seeks to determine displacements and stresses throughout the structure, which is in equilibrium and is subjected to applied loads. For many structures, it is difficult to determine the distribution of deformation using conventional methods, and thus the finite element method is necessarily used.
There are two general direct approaches traditionally associated with the finite element method as applied to structural mechanics problems. One approach, called the force, or flexibility, method, uses internal forces as the unknowns of the problem. To obtain the governing equations, first the equilibrium equations are used. Then necessary additional equations are found by introducing compatibility equations. The result is a set of algebraic equations for determining the redundant or unknown forces.
The second approach, called the displacement, or stiffness, method, assumes the displacements of the nodes as the unknowns of the problem. For instance, compatibility conditions requiring that elements connected at a common node, along a common edge, or on a common surface before loading remain connected at that node, edge, or surface after deformation takes place are initially satisfied. Then the governing equations are expressed in terms of nodal displacements using the equations of equilibrium and an applicable law relating forces to displacements.
These two direct approaches result in different unknowns (forces or displacements) in the analysis and different matrices associated with their formulations (flexibilities or stiffnesses). It has been shown [34] that, for computational purposes, the displacement (or stiffness) method is more desirable because its formulation is simpler for most structural analysis problems. Furthermore, a vast majority of general-purpose finite element programs have incorporated the displacement formulation for solving structural problems. Consequently, only the displacement method will be used throughout this text.
Another general method that can be used to develop the governing equations for both structural and nonstructural problems is the variational method. The variational method includes a number of principles. One of these principles, used extensively
throughout this text because it is relatively easy to comprehend and is often introduced in basic mechanics courses, is the theorem of minimum potential energy that applies to materials behaving in a linear-elastic manner. This theorem is explained and used in various sections of the text, such as Section 2.6 for the spring element, Section 3.10 for the bar element, Section 4.7 for the beam element, Section 6.2 for the constant-strain triangle plane stress and plane strain element, Section 9.1 for the axisymmetric element, Section 11.2 for the three-dimensional solid tetrahedral element, and Section 12.2 for the plate bending element. A functional analogous to that used in the theorem of minimum potential energy is then employed to develop the finite element equations for the nonstructural problem of heat transfer presented in Chapter 13.
Another variational principle often used to derive the governing equations is the principle of virtual work. This principle applies more generally to materials that behave in a linear-elastic fashion, as well as those that behave in a nonlinear fashion. The principle of virtual work is described in Appendix E for those choosing to use it for developing the general governing finite element equations that can be applied specifically to bars, beams, and two- and three-dimensional solids in either static or dynamic systems.
The finite element method involves modeling the structure using small interconnected elements called finite elements. A displacement function is associated with each finite element. Every interconnected element is linked, directly or indirectly, to every other element through common (or shared) interfaces, including nodes and/or boundary lines and/or surfaces. By using known stress/strain properties for the material making up the structure, one can determine the behavior of a given node in terms of the properties of every other element in the structure. The total set of equations describing the behavior of each node results in a series of algebraic equations best expressed in matrix notation.
We now present the steps, along with explanations necessary at this time, used in the finite element method formulation and solution of a structural problem. The purpose of setting forth these general steps now is to expose you to the procedure generally followed in a finite element formulation of a problem. You will easily understand these steps when we illustrate them specifically for springs, bars, trusses, beams, plane frames, plane stress, axisymmetric stress, three-dimensional stress, plate bending, heat transfer, and fluid flow in subsequent chapters. We suggest that you review this section periodically as we develop the specific element equations.
Keep in mind that the analyst must make decisions regarding dividing the structure or continuum into finite elements and selecting the element type or types to be used in the analysis (step 1), the kinds of loads to be applied, and the types of boundary conditions or supports to be applied. The other steps, 2–7, are carried out automatically by a computer program.
Step 1 Discretize and Select the Element Types
Step 1 involves dividing the body into an equivalent system of finite elements with associated nodes and choosing the most appropriate element type to model most closely the actual physical behavior. The total number of elements used and their
variation in size and type within a given body are primarily matters of engineering judgment. The elements must be made small enough to give usable results and yet large enough to reduce computational effort. Small elements (and possibly higherorder elements) are generally desirable where the results are changing rapidly, such as where changes in geometry occur; large elements can be used where results are relatively constant. We will have more to say about discretization guidelines in later chapters, particularly in Chapter 7, where the concept becomes quite significant. The discretized body or mesh is often created with mesh-generation programs or preprocessor programs available to the user.
The choice of elements used in a finite element analysis depends on the physical makeup of the body under actual loading conditions and on how close to the actual behavior the analyst wants the results to be. Judgment concerning the appropriateness of one-, two-, or three-dimensional idealizations is necessary. Moreover, the choice of the most appropriate element for a particular problem is one of the major tasks that must be carried out by the designer/analyst. Elements that are commonly employed in practice—most of which are considered in this text—are shown in Figure 1–1.
The primary line elements [Figure 1–1(a)] consist of bar (or truss) and beam elements. They have a cross-sectional area but are usually represented by line segments. In general, the cross-sectional area within the element can vary, but throughout this text it will be considered to be constant. These elements are often used to model trusses and frame structures (see Figure 1–2 on page 16, for instance). The simplest line element (called a linear element) has two nodes, one at each end, although higher-order elements having three nodes [Figure 1–1(a)] or more (called quadratic, cubic, etc. elements) also exist. Chapter 10 includes discussion of higher-order line elements. The line elements are the simplest of elements to consider and will be discussed in Chapters 2 through 5 to illustrate many of the basic concepts of the finite element method.
The basic two-dimensional (or plane) elements [Figure 1–1(b)] are loaded by forces in their own plane (plane stress or plane strain conditions). They are triangular or quadrilateral elements. The simplest two-dimensional elements have corner nodes only (linear elements) with straight sides or boundaries (Chapter 6), although there are also higher-order elements, typically with midside nodes [Figure 1–1(b)] (called quadratic elements) and curved sides (Chapters 8 and 10). The elements can have variable thicknesses throughout or be constant. They are often used to model a wide range of engineering problems (see Figures 1–3 and 1–4 on pages 17 and 18).
The most common three-dimensional elements [Figure 1–1(c)] are tetrahedral and hexahedral (or brick) elements; they are used when it becomes necessary to perform a three-dimensional stress analysis. The basic three-dimensional elements (Chapter 11) have corner nodes only and straight sides, whereas higher-order elements with midedge nodes (and possible midface nodes) have curved surfaces for their sides [Figure 1–1(c)].
The axisymmetric element [Figure 1–1(d)] is developed by rotating a triangle or quadrilateral about a fixed axis located in the plane of the element through 360 . This element (described in Chapter 9) can be used when the geometry and loading of the problem are axisymmetric.
text_image
y 1 2 x
text_image
y 1 2 3 x
(a) Simple two-noded line element (typically used to represent a bar or beam element) and the higher-order line element
text_image
y 3 1 2 x
Triangulars
text_image
1 2 3 4
natural_image
Simple geometric shape composed of connected dots forming a quadrilateral (no text or symbols)
Quadrilaterals
(b) Simple two-dimensional elements with corner nodes (typically used to represent plane stress/ strain) and higher-order two-dimensional elements with intermediate nodes along the sides
text_image
y x z 1 2 3 4
Tetrahedrals
text_image
8 7 4 6 3 5 1 2
Regular hexahedral
natural_image
Simple 3D wireframe cube diagram with solid and dashed lines indicating visible and hidden edges (no text or labels)
Irregular hexahedral
(c) Simple three-dimensional elements (typically used to represent three-dimensional stress state) and higher-order three-dimensional elements with intermediate nodes along edges
text_image
z 3 1 2 θ r Triangular ring
text_image
4 3 1 2
Quadrilateral ring
(d) Simple axisymmetric triangular and quadrilateral elements used for axisymmetric problems
Figure 1–1 Various types of simple lowest-order finite elements with corner nodes only and higher-order elements with intermediate nodes
Step 2 Select a Displacement Function
Step 2 involves choosing a displacement function within each element. The function is defined within the element using the nodal values of the element. Linear, quadratic, and cubic polynomials are frequently used functions because they are simple to work with in finite element formulation. However, trigonometric series can also be used. For a two-dimensional element, the displacement function is a function of the coordinates in its plane (say, the x- y plane). The functions are expressed in terms of the nodal unknowns (in the two-dimensional problem, in terms of an x and a y component). The same general displacement function can be used repeatedly for each element. Hence the finite element method is one in which a continuous quantity, such as the displacement throughout the body, is approximated by a discrete model composed of a set of piecewise-continuous functions defined within each finite domain or finite element.
Step 3 Define the Strain= Displacement and Stress=Strain Relationships
Strain/displacement and stress/strain relationships are necessary for deriving the equations for each finite element. In the case of one-dimensional deformation, say, in the x direction, we have strain ex related to displacement u by
\varepsilon_ {x} = \frac {d u}{d x} \tag {1.4.1}
for small strains. In addition, the stresses must be related to the strains through the stress/strain law—generally called the constitutive law. The ability to define the material behavior accurately is most important in obtaining acceptable results. The simplest of stress/strain laws, Hooke’s law, which is often used in stress analysis, is given by
\sigma_ {x} = E \varepsilon_ {x} \tag {1.4.2}
where sx ¼ stress in the x direction and E ¼ modulus of elasticity.
Step 4 Derive the Element Stiffness Matrix and Equations
Initially, the development of element stiffness matrices and element equations was based on the concept of stiffness influence coefficients, which presupposes a background in structural analysis. We now present alternative methods used in this text that do not require this special background.
Direct Equilibrium Method
According to this method, the stiffness matrix and element equations relating nodal forces to nodal displacements are obtained using force equilibrium conditions for a basic element, along with force/deformation relationships. Because this method is most easily adaptable to line or one-dimensional elements, Chapters 2, 3, and 4 illustrate this method for spring, bar, and beam elements, respectively.
Work or Energy Methods
To develop the stiffness matrix and equations for two- and three-dimensional elements, it is much easier to apply a work or energy method [35]. The principle of virtual work (using virtual displacements), the principle of minimum potential energy, and Castigliano’s theorem are methods frequently used for the purpose of derivation of element equations.
The principle of virtual work outlined in Appendix E is applicable for any material behavior, whereas the principle of minimum potential energy and Castigliano’s theorem are applicable only to elastic materials. Furthermore, the principle of virtual work can be used even when a potential function does not exist. However, all three principles yield identical element equations for linear-elastic materials; thus which method to use for this kind of material in structural analysis is largely a matter of convenience and personal preference. We will present the principle of minimum potential energy—probably the best known of the three energy methods mentioned here—in detail in Chapters 2 and 3, where it will be used to derive the spring and bar element equations. We will further generalize the principle and apply it to the beam element in Chapter 4 and to the plane stress/strain element in Chapter 6. Thereafter, the principle is routinely referred to as the basis for deriving all other stress-analysis stiffness matrices and element equations given in Chapters 8, 9, 11, and 12.
For the purpose of extending the finite element method outside the structural stress analysis field, a functional1 (a function of another function or a function that takes functions as its argument) analogous to the one to be used with the principle of minimum potential energy is quite useful in deriving the element stiffness matrix and equations (see Chapters 13 and 14 on heat transfer and fluid flow, respectively). For instance, letting p denote the functional and f ( x , y ) denote a function f of two variables x and y, we then have \pi = \pi ( f ( x , y ) ) Þ, where p is a function of the function f . A more general form of a functional depending on two independent variables u ( x , y ) and v ( x , y ) , where independent variables are x and y in Cartesian coordinates, is given by:
\pi = \iint F (x, y, u, v, u _ {x}, u _ {y}, v _ {x}, v _ {y}, u _ {x x}, \dots , v _ {y y}) d x d y \tag {1.4.3}
Methods of Weighted Residuals
The methods of weighted residuals are useful for developing the element equations; particularly popular is Galerkin’s method. These methods yield the same results as the energy methods wherever the energy methods are applicable. They are especially useful when a functional such as potential energy is not readily available. The weighted residual methods allow the finite element method to be applied directly to any differential equation.