11.3.1 Derivation of Interpolation Function for Simplex Elements with Scalar Quantities at Nodes

Derivation of interpolation functions for a particular geometry of the elements begins with assumption of a form of the function that describes the variation of the primary quantity in the elements. The simplest function that one may use for this purpose is a linear polynomial function. We will derive such an interpolation function for a relatively simple triangular plate element with three nodes, as illustrated in Figure 11.5 The interpolation function [N(x,y)] that we will derive will relate the primary unknown quantity Φ(x,y) in the element to the same unknown quantity at the associated nodes with set coordinates ϕ₁ at (x₁,y₁), ϕ₂ at (x₂,y₂), and ϕ₃ at (x₃,y₃).

We assume the function that represents the unknown quantity in the element to be a linear polynomial function that has a mathematical form:

11.2

in which the matrix , and the coefficients α₁, α₂ and α₃ are constants that can be determined by substituting the specified coordinates of the nodal values ϕ₁, ϕ₂, and ϕ₃, to give

or in a matrix form:

11.3

where the matrix

and the inverse of this matrix is obtained with the result

11.4

in which the determinant |A| can be evaluated to

where A = the plane area of the triangular plate element

Substituting Equations 11.3 and 11.4 into Equation 11.2, the function Φ(x,y) in the element can be evaluated by the three nodal quantities ϕ₁, ϕ₂, ϕ₃, or {ϕ} to be

11.5

The interpolation function, as defined in Equation 11.1 takes the form

11.6

where the matrix and the matrix [h] is given in Equation 11.4.

11.3.2 Derivation of Interpolation Function for Simplex Elements with Vector Quantities at Nodes

Primary unknown quantities in triangular plate elements with three fixed nodes in a discretized finite-element model may contain only one component as shown in Figure 11.5. However, in general, they contain two components such as Φ_x(x,y) along the x-coordinate and for the other component as Φ_y(x,y) along the y-coordinate, as illustrated in Figure 11.6. Consequently, there are two corresponding components of the same primary quantity at each of the three associated nodes with ϕ_1x(x₁,y₁) along the x-direction and ϕ_1y(x₁,y₁) along the y-coordinate at node 1 situated at (x₁,y₁), and so on.

According to Figure 11.6, the primary unknowns in the element can be represented by two linear polynomial functions as expressed in Equation 11.7:

11.7

where α₁, α₂, α₃, …, α₆ are constants to be determined by the nodal values at the specific coordinates of the nodes.

The unknown quantities in the element in Equation 11.7 may be related to those at the three associated nodes as in Equation 11.8:

11.8

in which N₁(x,y), N₂(x,y) and N₃(x,y) are components of the interpolation function [N(x,y)] associated with the unknown nodal quantities ϕ₁, ϕ₂, and ϕ₃ at nodes 1, 2, and 3 respectively.

We realize that each of the three nodes of the element in Figure 11.6 consists of two components:

By following a similar procedure to that in deriving the interpolation function in the proceeding case, we may derive the three components of the interpolation function, N₁(x,y), N₂(x,y), and N₃(x,y) in Equation 11.8, as in the following expressions (Segerlind, 1976):

11.9a

11.9b

11.9c

where

is the area of the triangular element in Figure 11.6.

Step 8: Solve for Secondary Unknown Quantities

As was mentioned early in Step 2 of finite-element analysis, the selection of primary quantities for the analysis may vary from case to case. For instance, element displacements are denoted as where U_x and U_y are the two components of the unknown displacement U in the element along the x- and y-coordinate, respectively. These unknown primary quantities in the elements of the discretized model may be related to the quantities at the associated nodes, designated by {ϕ} using the adopted interpolation functions. These unknown nodal quantities are later computed by the overall stiffness equations of the discretized model in Equation 11.18. In reality, however, there is the need to determine other quantities required in the analysis. For instance, in stress analysis of machine structures such as the case illustrated in Figure 11.4, the “stress” and the “maximum stresses” in the structures were required to determine whether the structures would be strong enough to withstand the applied forces F at nodes 7, 14, and 21. We thus need to determine the stress distributions and the magnitude and locations of maximum stress in the structure from the finite-element analysis. The “stresses” and the related “strains” are referred to as the required “secondary unknown quantities” in finite-element analysis.

The secondary unknown quantities can be determined from the primary quantities computed in finite-element analysis by using appropriate laws of physics. For instance, the strain components in the elements may be determined by the element displacement using the “displacement–strain relations” derived in the theory of linear elasticity, and the “element strains” may be related to the element “stress components” by use of Hooke's law. Similar formulations are available for heat transfer analysis, in which Fourier's law of heat conduction and Bernoulli's law for fluid dynamics analysis are frequently used to determine the secondary quantities from the primary unknown quantities obtained from finite-element analyses.

11.5.1 Stresses

Two physical responses occur in a homogeneous solid with isotropic material properties when it is subjected to a set of applied forces {P} as illustrated in Figure 11.14a. In these responses (1) the solid deforms into a new shape, and (2) the solid develops internal resistance to the applied forces while it deforms into new shapes. The induced internal resistance by the solid eventually reaches a state of equilibrium with the applied forces, and the solid ceases further deformation under this new equilibrium condition.

The intensity of the induced resistance to the applied force in the solid is termed the “stress” at any given location. These stresses may be oriented in different directions in the space defined by the x-, y-, and z-coordinates in Figure 11.14a. The magnitude, location, and orientation of the induced stresses can be denoted by the special notation system described next.

Let us consider an infinitesimally small cubic element located at (x₁, y₁, z₁) with its edges parallel to the three coordinate axes as shown in Figure 11.14b. We envisage that there are six faces of this infinitesimally small cubic element, and there exist stress components with some normal to the faces of the cube, and some others on the faces. A commonly accepted way to denote these stress components is to attach two subscripts to the symbol σ, which denotes the stress magnitude, with the first subscript indicating the coordinate normal to the specific face of the cube, and the second subscript indicating the direction of the stress component. We can thus account for the nine stress components on each of the six faces of the small cubic element inside the deformed solid with the following matrix expression:

11.22a

where σ_xx, σ_yy, and σ_zz are the stress components on the faces perpendicular to the respective x-, y-, and z-coordinates, and oriented along the same respective coordinates. We will call these stress components the normal stresses in the deformed solid. The other six stress components with different subscripts—σ_xy, σ_xz, and so on—are called shearing stress components; for example, σ_xy acts on the cube face that is normal to the x-coordinate and points in the y-direction, and σ_xz is another shearing stress on the same cube face but pointing in the z-direction.

We may account a total of nine stress components on the small cubic element as included in Equation 11.22a. This number is reduced to six in Equation 11.22b via the relationships , and because of the new equilibrium condition of the solid in the deformed condition.

We thus have the six independent stress components in deformed solid in three-dimensional stress analyses:

11.22b

We may also express these stress components in the form of a column matrix in our subsequent derivations:

11.23

in which σ₁ = σ_xx, σ₂ = σ_yy, σ₃ = σ_zz , σ₄ = σ_xy, σ₅ = σ_yz, σ₆ = σ_xz

One will note that the stress components with identical subscripts are the normal stresses, whereas those with different subscripts are the shearing stresses.

11.5.2 Displacements

The change of shape of the solid due to externally applied forces is illustrated in Figure 11.15, in which the change of the positions of the points inside and on the boundary of the solid is represented by the movement of point P₁ at position (x,y,z) before the loading to point P₂ at a new location (x + u, y + v, z + w) after the deformation. The net movement of point P₁ to P₂ is represented by a vector U with components u, v, and w along the coordinates x, y, and z, respectively. The displacement vector of a point located at r(x,y,z) can thus be expressed as

11.24

where the bold-face r is the position vector representing the coordinates as defined in Chapter 3.

11.5.3 Strains

As we have defined stress in a deformed solid to be the intensity of internal resistance by the solid induced by applied forces, it is appropriate to define the induced strain to be the rate of change of displacement of the solid. Like stresses, strain components can be classified as normal strain components and shearing strain components. However, the reader should bear in mind that normal strain components result in changes in dimensions (and thus the size) of the solid, whereas the shearing strain components cause change only of the shape of the solid, which change is given by the angles of the sides of the cubic faces of the cubic element, which is 90° in the undeformed cubic element before the deformation and the angles between the same faces after the deformation, with a unit of “radians.” Strain components are usually expressed as column matrices corresponding to the six independent stress components, as show in Equation 11.25:

11.25

with ϵ₁ = ϵ_xx, ϵ₂ = ϵ_yy, = ϵ_zz, ϵ₄ = ϵ_xy, ϵ₅ = ϵ_yz, ϵ₆ = ϵ_xz.

11.5.4 Fundamental Relationships

The three essential physical quantities of stress, strain, and displacement in stress analysis of solid structures are mutually interrelated by mathematical expressions derived from the theory of elasticity. We will present only those formulations that are relevant to the subsequent derivations of equations and formulae required for finite-element analysis.

11.5.4.1 Strain–Displacement Relations

As illustrated in Figure 11.15, the displacements {U(r)} from point P₁ at (x,y,z) to point P₂ at (x + u, y + v, z + w) with for the net movement of the point along the x-coordinate, for the net movement of the same point along the y-coordinate, and for the net movement of the point along the z-coordinate. As we have defined the strain components that account for the rate of change of the displacements along the coordinates that define the deformed solid, and the strain components that exist in the same cubic element as stresses in Figure 11.14b with , , , , , and , we may relate the various strain components to the related displacement components as shown in the following equations:

11.26a

11.26b

11.26c

11.26d

11.26e

11.26f

The relationships in Equations 11.26a to 11.26f may be expressed in the following matrix form:

11.27

We may further condense the relationship in Equation 11.27 into the following form for the subsequent derivation of element equations:

11.28

where matrix [D] takes the form shown in Equation 11.29 and the element displacement matrix {U} is given in Equation 11.24.

11.29

11.5.4.2 Stress–Strain Relations

In Figure 11.14 we represented the six independent stress components existing at any point (represented by infinitesimally small cubic elements in the figure) located at (x,y,z). We realize that there are equal numbers of corresponding strain components associate with these stress components. The relationship between these stress and strain components can be expressed by the generalized Hooke's law, as given, fore example, in Hsu (1986). Equation 11.30 shows the matrix form of such a generalized Hooke's law:

11.30

or in a compact form:

11.31

in which E is the Young's modulus and ν is the Poisson's ratio of the material. The matrix [C], called the elasticity matrix, has the form:

11.32

11.5.4.3 Strain Energy in Deformed Elastic Solids

The above formulations are derived from a postulate that in elastic solids, the deformations, stresses, and strains induced by external forces applied to the solid in the equilibrium state will disappear altogether after removal of the applied forces. This postulate implies that there must be a mechanism that restores the solid to its original state after removal of the applied loads. This mechanism is referred to as the “strain energy.” This energy is induced in the solid during the deformation process, and it is “stored” in the deformed solid at the equilibrium state of the solid under external loading. It will be released to restore the deformed solid to its original state upon removal of the applied actions. Since this energy is created by the deformation of the solid expressed by the strains induced by the externally applied forces in the form of the induced stresses, we may mathematically express this energy in the following form:

11.33

or in a compact version:

11.34

11.5.5 Finite-element Formulation

We will follow the formulations described in Step 5 of Section 11.3 in deriving the element and overall stiffness equations for stress analysis of elastic solid continua. Figure 11.16 illustrates how a cam-shaft assembly may be discretized by a finite number of tetrahedral elements interconnected at nodes. These elements are typically used for discretization of solids with complicated geometry in three-dimensional finite-element analyses. Shapes of these elements are illustrated in Figure 11.2.

It was mentioned earlier that hexahedral elements are commonly used in the finite-element modeling of solids of complicated geometry in three dimensions, but since each hexahedral element can be subdivided into four or five tetrahedral elements in actual computations, we will present the subsequent finite-element formulations in terms of typical tetrahedral elements. As illustrated in Figure 11.17, tetrahedral elements have shapes that are similar to pyramids with four apexes (or nodes) for each element. These nodes may be designated by nodes i, j, k, and m as indicated in Figure 11.17.

The primary unknown quantity in typical finite-element analysis in stress analysis is the displacements of the element U(r) with r being the position vector in a discretized model representing the coordinates of the solution point (usually located at the center of gravity) in the element. The element displacement vector is expressed as U(x,y,z) in the finite-element formulation.

The displacement in a deformed solid is a vector quantity with components along the three coordinates that define the space in which the element is situated. Mathematically, we may represent the element displacement as

for the element, in which the components U_x, U_y, and U_z along the coordinates x, y, and z are often denoted by U(x,y,z), V(x,y,z), and W(x,y,z), respectively.

The discretization process in the finite-element analysis requires a relationship between the displacements in the element and those at the associated nodes according to Equation 11.1 or as expressed in Equation 11.35 for the current situation:

11.35

for the element displacement in a discretized finite-element model with {u} being the displacement components of the associated nodes at fixed coordinates in the discretized model. N(x,y,z) is the interpolation function for the tetrahedral element in Figure 11.17. The displacement components at each of the four nodes of the element in Figure 11.17 consist of three components as in the following:

These nodal displacements are point values in the form of real numbers. The total number of displacement components of the four nodes of a tetrahedral element in Figure 11.17 is thus equal to 12, as expressed in the following matrix form:

11.36

where the subscripts identify the nodes: for example, the subscripts i, j, k, and m denote nodes i, j, k, and m, respectively. The notations u, v, and w in Equation 11.36 represent the displacement components along the x-, y-, and z-directions, respectively.

Referring to the tetrahedral element in Figure 11.17, Equation 11.35 can be expanded to give

or, in an compact form:

11.37

where N_i, N_j, N_k, and N_m are the components of the interpolation function associated with node i, j, k, and m, respectively, in Figure 11.17.

Substituting Equation 11.35 into Equation 11.28 results in the relationship between the element strains and nodal displacements:

11.38

where the matrix [B] has the form:

11.39

The above expression indicates that the matrix [B(x,y,z)] may be obtained from the derivatives of [N(x,y,z)] with respect to the coordinates x-, y-, and z-, respectively. The matrix [D] is given in Equation 11.29.

The element stresses and the nodal displacement components are related by substituting Equation 11.38 into Equation 11.31 to give Equation 11.40:

11.40

And finally, the strain energy in the element is obtained by substituting Equations 11.38 and 11.40 into Equation 11.34, resulting in the following equation for the strain energy of the deformed element:

11.41

The strain energy U({u}) in Equation 11.41 is the energy stored in the deformed solid (in the elements in this case). It is induced by the deformation of the elements in a discretized finite-element models by the applied forces at the nodes of the elements.

Strain energy in the deformed elements constitutes an important part of the potential energy of discretized solid continua such as shown in Figure 11.1b. The potential energy in a deformed solid is the difference between the induced strain energy in the deformed solid and the work done on the solid by applied external forces. For a continuum such as shown in Figure 11.1a, this difference should be zero; or conversely, the induced strain energy in the deformed solid should be equal to the work done on the solid by the applied forces. In the discretized solid model consisting of individual elements such as that illustrated in Figure 11.1b, however, the equality of the induced strain energy in elements U{u} in Equation 11.41 and the work performed on the same element by applied forces on the surface in the form of surface tractions {t} and body forces {f} may likely not hold. To ensure that the discretized solid is stable and in mechanical equilibrium, such a difference must be kept to a minimum.

The formulation of the potential energy (Π) in a deformed discretized model such as shown in Figure 11.1b can thus be expressed mathematically in the form

where the work produced by applied forces in the deformed element by the body forces {f} and surface tractions {t} can be expressed as

in which {U(x,y,z)} are the displacement components of the element, v and s are the respective volume and surface of the element, and {f} and {t} are the respective body force components and surface tractions applied to the element.

The potential energy in the element can thus be expressed in terms of nodal displacement with the interpolation function [N(x,y,z)] in the following expression:

11.42

By applying the principle of variational calculus, the potential energy function in Equation 11.42 has either a maximum or minimum value by solving the following equation:

with nodal displacement components {u} to be “point values” which are independent to the coordinates in differentiation. This variation will thus lead to the following equality:

11.43

One may also demonstrate the second-order derivative of the potential energy function Π({u}) to be positive:

We may thus ensure that the solution of the induced nodal displacements {u} by applied loads as obtained from Equation 11.43 will result a minimum value of the potential function in Equation 11.42.

Equation 11.43 may be expressed in the following form, called the “element equations”:

11.44

in which

11.45

with the matrix [B(x,y,z)] as in Equation 11.39 and the matrix[C] available from Equation 11.32. The matrix {p} has the form:

11.46

and the matrix [N(x,y,z)] is the interpolation function defined in Equation 11.1

In a strict sense, a deformed solid is in equilibrium only when the potential energy Π in Equation 11.42 in all elements in the discretized model is minimum. Consequently, we postulate that the potential energy of the entire discretized solid can be obtained by the summation of potential energies in all individual elements in the discretized solid. This postulate legitimizes our using the following equation for the entire discretized structure:

11.47

where

11.48

with

as in Equation 11.45; M is total number of elements in the discretized model, and {p} represents the total nodal forces as expressed in Equation 11.46.

The assembly of the element stiffness matrices for the overall stiffness matrix in Equation 11.47 entails following the rule stipulated in Step 6 in Section 11.3 and illustrated in Example 11.3.

11.5.6 Finite-element Formulation for One-dimensional Solid Structures

We will derive the element equation of a bar element such as shown in Figure 11.7 in Example 11.1 to illustrate the general procedure for developing finite-element formulations.

We will begin by referring to the bar element illustrated in Figure 11.7, made of a material with Young's modulus E. We locate the bar element in a plane defined by the x–y coordinate system as shown in Figure 11.18.

The interpolation function [N(x)] that relates the displacement in the bar element to its two nodes may be derived from a simple polynomial function as we did in Example 11.1. This function has the form shown in Equation (11.10):

11.10

The matrix [B(x)] that relates the element strain and the nodal displacement may be obtained using Equation 11.39:

11.49

Since the bar element is subjected to two applied forces with –F₁ at node 1 and +F₂ at node 2 and both these forces act along the longitudinal direction, that is, along the x-coordinate of the bar, the only induced stress component in the bar is σ_xx along the x-coordinate. Likewise, the only induced strain component is ϵ_xx along the same direction. The relationship between the induced stress and strain obeys Hooke's law with where E is the Young's modulus of the bar material. We thus obtain the elasticity matrix [C] = E.

By substituting the above expression for [N(x)] from Equation 9.10, [B(x)] in Equation 11.49, and with [C] = E and , where A is the cross-sectional area of the bar element and into Equations 11.42 and (11.43), we obtain the element stiffness matrix from Equation 11.45:

11.50

The nodal forces matrix in this case is

From Equation 11.44 the element equation of the bar element thus has the form

11.51

11.6.1 Common Features in General-purpose Finite-element Codes

Finite-element programs that can be used to solve a large class of engineering problems go under the generic name of “general-purpose programs.” Common desirable features of such a program are described below.

1. 1. The program should offer extensive capabilities of solving engineering problems ranging from static and dynamic elastic to elastoplastic stress analysis of solid structures made either of traditional materials or of unusual materials such as composites, biomedical materials, and so on, with applied loads from nontraditional sources such as piezoelectric and electrostatic forces as well as molecular forces in the stress analysis of structures at micrometer and nanometer scales.
2. 2. It must have a large element library with many more element types to choose from than the simplex elements offered in Figure 11.2. There are other types of elements that can make the finite-element analysis more efficient in terms of setting discretization models and that produce more accurate analytical results. Figure 11.21 shows some of these elements available in the element library of the ANSYS code (Lee, 2015). One will observe from Figure 11.21 that the first of the three listed types of elements is for beam structures using bar elements and the second and third types are for modeling plates of planar geometry. Thin-shell elements are available for structures such as pipes and pressure vessels. One may also note that the last six types of elements have additional nodes at the middle of the edges of the elements. These are the elements normally called the “serendipity” elements, with many such elements having additional nodes on selected locations along the edges of the elements (Bathe and Wilson, 1976; Hsu, 1986). The functions that represent the variation of the primary unknowns in these elements are higher-order polynomial functions, such as the Lagrange polynomials. There are many advantages of using these elements and high-order polynomial interpolation functions, with one obvious advantage of better fitting the element shapes to solids of curved geometry. The higher order of the polynomial functions used in the computations can yield more accurate results from the analyses. The additional nodes in the elements may result in the use of fewer elements in the discretized model for the finite-element analysis. There are many other types of elements that are often offered by general-purpose finite-element codes, such as “spring elements,” “contact elements,” and “interface elements” with unusually large aspect ratio (defined as the ratio of the longest to shortest edges of the element). Other special elements may include the “birth and death elements,” such as are found in the element library of the ANSYS code. These elements are used for cases simulating the “growth” of solid substances involving welding, or the “disappearance” of solid substances such as in etching of structures in microfabrication process, or the growth and propagation of cracks in solid structures in fracture mechanics analyses.
3. 3. The package must have user-friendly pre-processor with sufficient versatility. A good pre-processor should include, in addition to extensive material database, easy-to-use automatic mesh generation for users to establish the required discretized model with user-selected density and configurations of finite-element meshes, such as illustrated in Figures 11.3 and 11.13b.
4. 4. It must have a comprehensive and easy-to-use post-processor that can display the results of the analysis with visual outputs, either in static graphical form or in animated graphical output for continuous action if so desired. Users should be offered ample options in choosing the desired form for outputs of their analyses.
5. 5. It is desirable to offer cost-effective ways to use the code in terms of minimal effort in learning the use of the code, minimal human input, and high computational efficiency. Most commercial finite-element codes adopt highly efficient solution methods.
6. 6. Effective interfacing of computer-aided design (CAD) with the finite-element analysis (FEA) is also a desirable feature of general-purpose FE code. It allows the user to perform finite-element analysis using the imported solid model geometry from a CAD package. Figure 11.22a shows a solid model of a coupler from a computer-aided design package to be automatically translated to the ANSYS code for finite-element stress analysis; and Figure 11.22b shows the variation of deformation of the structure under applied load. There are also codes that offer interfacing of CAD/FE with computer-aided-manufacturing (CAM), with the solid models of structures being downloaded to a CAM package ready for mechanical analysis as well as coding for computer-numerically controlled (CNC) machine tools after satisfactory finite-element analysis.

11.6.2 Simulation using general-purpose finite-element codes

A relatively new application of general-purpose finite-element code is to simulate critical engineering processes, in particular those involved in new product development. Such simulations may result in significant cost saving for the development process, and also reduce the time to market for these products. While the process of product development may vary from case to case, Figure 11.23 illustrates the general procedure with various stages in such a development.

A recent article on “What does product development really cost” by K. Douglass [http://www.pivotint.com] reports some resounding outcomes, such as that a 12-month in reduction in time-to-market (TTM) would result in 92% increase in “internal rate of return” (IRR) on the product development investment. A 63% increase in IRR was achieved with a 9-month reduction in TTM. Further observation indicates that other than the costs of personnel and management, and the market research activities, the most costly activities in new product development are the design, production, and testing of the two prototypes, as indicated in Figure 11.23. Not only are activities relating to the prototyping costly, they also prolong time-to-market.

General-purpose finite-element code such as ANSYS has developed such capability that it can simulate many of the activities involved in product development that are shown in Figure 11.23. It can produce high-quality simulation based on expected functions of products of complex geometry subjected to all conceivable loading conditions. It also provides graphic simulations of what “real” physical prototypes can achieve under all conceivable conditions and beyond, which results not only in substantial savings in costs of building the real prototypes but also in substantial time saving in production and testing of the product using the virtual prototypes.

Figure 11.24 illustrates a case of simulation by the ANSYS code of various vibrational modes of the disk-and-pad assembly of an automobile braking system. The design objective of this system was to identify the onset of instability in braking using a modal analysis. The ANSYS solution from simulating the performance of a new design of a braking system such as shown in Figure 11.24 enabled engineers to achieve a first-time virtual design of a cost-effective, quiet braking system for multiple platforms in far less time than would have been possible with conventional methods using physical prototyping and field testing.