MultiPhysicsVault/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_012.md

Loads. From the second integral in Eq. 3.4-6, with the substitution of u = u^{T} for axial displacement and F_{x} = q/A for axial body force, the potential of the load is

\begin{array}{r l} \Omega & = - \int_ {0} ^ {L} \left(f _ {x} A\right) d s = - \int_ {0} ^ {L} u ^ {T} q d s \\ & \quad f _ {x} = \frac {f - c e}{\Delta t} = \frac {q}{A} = \frac {c x}{A} \end{array} \tag {3.9-4}

From Eqs. 3.8-3 and 3.9-4,

\Omega = - \{\mathbf {d} \} ^ {T} \left\{\mathbf {r} _ {e} \right\}, \quad \text { where } \quad \left\{\mathbf {r} _ {e} \right\} = \int_ {0} ^ {L} \left\lfloor \mathbf {N} \right] ^ {T} q d s \tag {3.9-5}

Vector \{\mathbf{r}_e\} is called a consistent load vector. It tells how a distributed load should be allocated to nodes in a way that is consistent with the displacement field assumed. Further explanation appears in Section 4.3.

For the present illustration we will take q = cx, which is the linearly varying distributed axial load used in Fig. 3.5-1. In elements 1, 2, and 3, respectively, q = cs, q = c(x_{2} + s) , and q = c(x_{3} + s) . For convenience we now give all elements the same length, L = L_{T}/3 . Thus, for elements 1, 2, and 3,

\left\{\mathbf {r} _ {e} \right\} _ {1} = \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 1 \\ 2 \end{array} \right\} \quad \left\{\mathbf {r} _ {e} \right\} _ {2} = \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 4 \\ 5 \end{array} \right\} \quad \left\{\mathbf {r} _ {e} \right\} _ {3} = \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 7 \\ 8 \end{array} \right\} \tag {3.9-6}

Global (Structural) Equations. The total potential of the three-element structure is the sum of the three element contributions:

\Pi_ {p} = U + \Omega = U _ {1} + U _ {2} + U _ {3} + \Omega_ {1} + \Omega_ {2} + \Omega_ {3} \tag {3.9-7}

Let element matrices be expanded to “structure size” as explained in Section 2.5, so that nodal d.o.f. vector \{d\} of each element is replaced by the “global” vector \{D\} = \left[u_{1} \quad u_{2} \quad u_{3} \quad u_{4}\right]^{T} , which contains all d.o.f. of the structure. Thus, from Eqs. 3.9-3 and 3.9-5, with AE taken as constant over the entire length L_{T} = 3L ,

\begin{array}{l} \Pi_ {p} = \frac {1}{2} \{\mathbf {D} \} ^ {T} \left(\frac {A E}{L} \left[ \begin{array}{r r r r} 1 & - 1 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] + \frac {A E}{L} \left[ \begin{array}{r r r r} 0 & 0 & 0 & 0 \\ 0 & 1 & - 1 & 0 \\ 0 & - 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \right. \\ \left. + \frac {A E}{L} \left[ \begin{array}{c c c c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & - 1 \\ 0 & 0 & - 1 & 1 \end{array} \right]\right) \{\mathbf {D} \} = \{\mathbf {D} \} ^ {T} \left(\frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 1 \\ 2 \\ 0 \\ 0 \end{array} \right\} + \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 0 \\ 4 \\ 5 \\ 0 \end{array} \right\} + \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 0 \\ 0 \\ 7 \\ 8 \end{array} \right\} + \left\{ \begin{array}{l} R _ {1} \\ 0 \\ 0 \\ 0 \end{array} \right\}\right) \tag {3.9-8} \\ \end{array}

in which R_{1} represents the support reaction at x = 0. Four equilibrium equations [K]\{D\} = \{R\} are provided by the stationary condition \{\partial\Pi_{p}/\partial D\} = \{0\} . Differentiation rules given in Appendix A make this process easy. The resulting global equations [K]\{D\} = \{R\} are

\frac {A E}{L} \left[ \begin{array}{r r r r} 1 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 1 \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \\ u _ {4} \end{array} \right\} = \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 1 \\ 6 \\ 1 2 \\ 8 \end{array} \right\} + \left\{ \begin{array}{l} R _ {1} \\ 0 \\ 0 \\ 0 \end{array} \right\} \tag {3.9-9}

The essential boundary condition u_{1} = 0 is imposed by striking out the first equation and the first column of the matrix, as explained in Section 2.3. Solution of the remaining three equations yields nodal d.o.f. u_{2}, u_{3} , and u_{4} . The complete solution vector is

\{\mathbf {D} \} = \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \\ u _ {4} \end{array} \right\} = \frac {c L ^ {3}}{3 A E} \left\{ \begin{array}{l} 0 \\ 1 3 \\ 2 3 \\ 2 7 \end{array} \right\} \tag {3.9-10}

These are the axial displacements at x = 0 , x = L , x = 2L , and x = 3L .

Inspection of the Solution. Axial displacements are plotted in Fig. 3.9-1a. Upon comparing Eq. 3.9-10 with the exact solution, Eq. 3.5-8, we find that nodal d.o.f. u_{1} , u_{2} , and u_{3} are exact. This happens only because the present problem is of a special mathematical type [3.3]. In most problems, nodal d.o.f. \{D\} are not exact. Between nodes, the linear finite element field cannot match the exact cubic field. For example, at the midpoint x = 3L_{T}/2 , for which s = L/2 in element 2, Eq. 3.8-3 yields

u = \left\lfloor \frac {L - (L / 2)}{L} \quad \frac {L / 2}{L} \right\rfloor \left\{ \begin{array}{l} u _ {2} \\ u _ {3} \end{array} \right\} = \frac {u _ {2} + u _ {3}}{2} = \frac {6 c L ^ {3}}{A E} \tag {3.9-11}

The exact result at x = 3L_{T} / 2 , from Eq. 3.5-8, is u = 6.1875cL^{3} / AE .

The state of stress is uniaxial. Therefore, in a typical element, using Eq. 3.9-1,

text_image

y u₁=0 u₂ q=cx u₃ u₄ 1 2 3 4 x 3@L=Lₜ

(a)

line

| x | σ_x (dashed line) | σ_x = c/(2A) (9L² - x²) | | ---- | ----------------- | ------------------------ | | 0 | ~1.0 | ~1.0 | | L | ~0.8 | ~0.8 | | 2L | ~0.6 | ~0.6 | | 3L | ~0.0 | ~0.0 |

(b)
Figure 3.9-1. Axial displacements and axial stresses in a bar under linearly varying axial load q = cx, modeled by three elements of equal length. The dashed line represents the exact stress field.

\sigma_ {x} = E \epsilon_ {x} = E [ \mathbf {B} ] \{\mathbf {d} \} = \frac {E}{L} [ - 1 1 ] \{\mathbf {d} \} \tag {3.9-12}

Applying Eq. 3.9-12 to the three elements in turn, we obtain the stresses plotted in Fig. 3.9-1b. As is typical in finite element problems, stresses change abruptly at nodes and are less accurate than displacements. A bar element based on a linear axial displacement field will always approximate the exact stress curve in stairstep fashion. Nevertheless, the exact curve can be approached arbitrarily closely by using more and more elements.

That stresses are most accurate near element centers is a consequence of the mean value theorem for derivatives. As applied to Fig. 3.5-1b, the theorem says that slopes of the two curves agree near the middle of the interval even though slopes disagree at x = 0 and at x = L_{T} . The same argument applies to Fig. 3.9-1a: slopes of the piecewise linear curve match the exact u_{,x} from Eq. 3.5-8 somewhere near element centers but not at nodes.

3.10 FINITE ELEMENT FORMULATIONS DERIVED FROM A FUNCTIONAL

In Section 3.9, a finite element solution based on a two-d.o.f. bar element is derived from the potential energy functional \Pi_{p} . In the present section we consider another example, this time in two-dimensional heat conduction, but without requiring that the element have a particular shape or a particular number of nodes. Similar formulations for stress analysis are considered in Chapter 4.

Let temperature T within a plane element be interpolated from n nodal temperatures \{T_{e}\} ,

T = \left\lfloor \mathrm{N} \right] \left\{\mathrm{T} _ {e} \right\} \tag {3.10-1}

where each of the n shape functions in [N] is a function of x and y. If elements have corner nodes only, then n = 3 for a triangle, n = 4 for a rectangle, and so on.

A functional for plane heat conduction is given in Eq. 3.7-5. With T = T^T , T_{,x}^2 = T_{,x}^T T_{,x} , and T_{,y}^2 = T_{,y}^T T_{,y} , Eq. 3.7-5 has the form

\Pi = \iint \frac {1}{2} \left(T _ {, x} ^ {T} T _ {, x} + T _ {, y} ^ {T} T _ {, y}\right) k d x d y - \iint T ^ {T} Q d x d y + \iint T ^ {T} \dot {T} \rho c d x d y \tag {3.10-2}

As in Eq. 3.9-2, the transposition symbol is placed on the scalars to make it easier to differentiate subsequent matrix expressions. Differentiation of Eq. 3.10-1 yields

T _ {, x} = \left\lfloor \mathrm{N} _ {, x} \right\rfloor \left\{\mathrm{T} _ {e} \right\} \quad T _ {, y} = \left\lfloor \mathrm{N} _ {, y} \right\rfloor \left\{\mathrm{T} _ {e} \right\} \quad \dot {T} = \left\lfloor \mathrm{N} \right\rfloor \left\{\dot {\mathrm{T}} _ {e} \right\} \tag {3.10-3}

in which, for example,

\left[ \mathrm{N} _ {, x} \right] = \left[ N _ {1, x} \quad N _ {2, x} \quad \dots \quad N _ {n, x} \right] \tag {3.10-4}

The expression for \dot{T} in Eqs. 3.10-3 indicates that nodal temperatures are regarded as functions of time, but the N_{i} are independent of time. Thus T and \dot{T} are interpolated from nodal values \{\mathbf{T}_{e}\} and \{\dot{\mathbf{T}}_{e}\} by means of the same shape function matrix [\mathbf{N}] .

Substitution of Eqs. 3.10-3 into 3.10-2 yields, for a single element,

\Pi = \frac {1}{2} \{\mathbf {T} _ {e} \} ^ {T} [ \mathbf {k} ] \{\mathbf {T} _ {e} \} + \{\mathbf {T} _ {e} \} ^ {T} [ \mathbf {c} ] \{\dot {\mathbf {T}} _ {e} \} - \{\mathbf {T} _ {e} \} ^ {T} \{\mathbf {r} _ {Q} \} \tag {3.10-5}

in which we have defined terms as follows:

[ \mathbf {k} ] = \int \int (\lfloor \mathbf {N}, _ {x} \rfloor^ {T} \lfloor \mathbf {N}, _ {x} \rfloor + \lfloor \mathbf {N}, _ {y} \rfloor^ {T} \lfloor \mathbf {N}, _ {y} \rfloor) k d x d y \tag {3.10-6a}

[ \mathbf {c} ] = \int \int [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] \rho c d x d y \tag {3.10-6b}

\left\{\mathbf {r} _ {Q} \right\} = \iint [ \mathbf {N} ] ^ {T} Q d x d y \tag {3.10-6c}

Finite element equations are obtained by making \Pi stationary with respect to variations of nodal temperature:

\left\{\frac {\partial \Pi}{\partial \mathbf {T} _ {e}} \right\} = \{\mathbf {0} \} \quad \text { yields } \quad [ \mathbf {k} ] \{\mathbf {T} _ {e} \} + [ \mathbf {c} ] \{\dot {\mathbf {T}} _ {e} \} = \{\mathbf {r} _ {Q} \} \tag {3.10-7}

Upon assembly of elements, \{T_{e}\} is replaced by the global vector \{T\} , which contains all nodal temperatures of the structure. The global equations are therefore

(\sum [ \mathbf {k} ]) \{\mathbf {T} \} + (\sum [ \mathbf {c} ]) \{\dot {\mathbf {T}} \} = \sum \{\mathbf {r} _ {Q} \} \tag {3.10-8}

in which summation signs imply the usual assembly process of summing overlapping terms of element matrices. (Equation 3.10-8 is again obtained if assembly is indicated earlier, i.e., by summing element contributions from Eq. 3.10-5 to a global II.)

global II.) For the sake of having notation like that used in structural mechanics, we can write [k] of Eq. 3.10-6a in the form

[ \mathbf {k} ] = \int \int [ \mathbf {B} ] ^ {T} k [ \mathbf {B} ] t d x d y, \quad \text { where } \quad [ \mathbf {B} ] = \left[ \begin{array}{l} \mathbf {N} _ {, x} \\ \mathbf {N} _ {, y} \end{array} \right] \tag {3.10-9}

Thickness t -is taken as unity in the preceding development.

Remarks. The foregoing derivation has significance that goes beyond the problem of plane heat conduction. The derivation shows that a finite element formulation of a physical problem is available from only two basic ingredients—namely, a functional that describes the physical problem and a shape function matrix [N] that describes the element. From these we obtain definitions of element properties (e.g., Eqs. 3.10-6) and algebraic equations of the structure (e.g., Eqs. 3.10-8).

To obtain numerical results one must next attend to specifics by choosing the element shape, number of d.o.f., distribution of d.o.f. over the element, and the shape function matrix [N]. These choices have great influence on the efficiency of calculation and the accuracy of results.

3.11 INTERPOLATION

To interpolate is to approximate the value of a function between known values by operating on the known values with a formula different from the function itself. This is done in each of the three spans of length L in Fig. 3.9-1a, where a linear operator [N] is applied to known values u_{1}, u_{2}, u_{3} , and u_{4} that happen to lie on a cubic curve. In a finite element context, the “known values” are d.o.f. to be found by solving algebraic equations, and they are usually approximate rather than exact. Operator [N] , the shape function matrix, serves as a basis from which a finite element can be formulated.

One can regard interpolation as the basic motivation of the finite element method, in that a sufficiently small portion of even a complicated field can be modeled well enough by a simpler interpolating field. A linear interpolating field, as used in each element of Fig. 3.9-1a, may be adequate if many elements are used. Elements based on a quadratic field or a cubic field would provide a better fit of the actual field: fewer elements would be needed, but each element would be more complicated. The limiting case of a single element with many d.o.f. yields the classical Rayleigh–Ritz method.

Should one use many simple elements or a few complicated elements? There is no easy answer. A good analyst is familiar with how various elements behave in various circumstances. In solving a transient or nonlinear problem, for example, many analysts prefer simpler (and therefore cheaper) elements because of the need to seek low cost in every computational step.

Degree of Continuity. For future use, we introduce the following symbolism to define the degree of continuity of a function or a field. A field is said to have C^m continuity if derivatives of the field through order m are continuous. Thus \phi = \phi(x) is C^0 continuous if \phi is continuous but \phi_{,x} is not. An example of C^0 continuity appears in Fig. 3.11-1a. Another example of C^0 continuity is axial displacement u in Fig. 3.9-1. Figure 3.11-1b shows an example of C^1 continuity: both \phi and \phi_{,x} are continuous but \phi_{,xx} is discontinuous at x = x_c . In general, it is necessary that derivatives of \phi of order m be used as nodal d.o.f. if the field \phi produced by a mesh of finite elements is to be C^m continuous.

Element Node Identification. Heretofore we have labeled element nodes with letters and structure nodes with numbers. We will subsequently encounter ele-

line

x	φ
x_c	φ_x

(a)

text_image

φ or φₓ φ φₓ x xₑ

(b)
Figure 3.11-1. Function \phi = \phi(x) is (a) C^0 continuous and (b) C^1 continuous.

ments that have many nodes, for which letters would be awkward as node labels. Therefore, we will henceforth use numbers as element node labels. At this stage in our study, the reader should be able to distinguish between an element and a structure without the artifice of separate labeling systems.

3.12 SHAPE FUNCTIONS FOR C^0

ELEMENTS

A C^0 element provides interelement continuity of the field quantity \phi but not interelement continuity of all first derivatives of \phi . Thus, in a mesh of C^0 elements, \phi_{xx}, \phi_{yy} , and/or \phi_{,z} exhibit a jump as one passes from one element into another.

Interpolation formulas lead to shape functions [N], from which finite elements can be formulated. In this section and the next we consider shape functions for some simple elements. These and other elements are considered in more detail in subsequent chapters.

A field \phi is interpolated over an element from n element nodal values \{\phi_e\} = [\phi_1 \quad \phi_2 \ldots \phi_n]^T according to the formula

\phi = \lfloor \mathrm{N} \rfloor \{\phi_ {e} \} \quad \text { that   is, } \quad \phi = \sum_ {i = 1} ^ {n} N _ {i} \phi_ {i} \tag {3.12-1}

where the N_{i} are functions of the coordinates. A shape function N_{i} defines the distribution of \phi within the element when the i th nodal d.o.f. \phi_{i} nas unit value and all other nodal \phi 's are zero.

One Dimension. Linear interpolation in one dimension is depicted in Fig. 3.12-1a. The interpolated function \phi = \phi(x) is to have value \phi_{1} at x = x_{1} and

N _ {1} = \frac {L - x}{L}

N _ {2} = \frac {x}{L}

\phi = \left\lfloor N \right\rfloor \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \end{array} \right\}

N _ {1} = \frac {(x _ {2} - x) (x _ {3} - x)}{(x _ {2} - x _ {1}) (x _ {3} - x _ {1})}

N _ {2} = \frac {(x _ {1} - x) (x _ {3} - x)}{(x _ {1} - x _ {2}) (x _ {3} - x _ {2})}

N _ {3} = \frac {(x _ {1} - x) (x _ {2} - x)}{(x _ {1} - x _ {3}) (x _ {2} - x _ {3})}

\varphi = [ N ] \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \\ \phi_ {3} \end{array} \right\}

Figure 3.12-1. (a) Linear interpolation and shape functions. (b) Quadratic interpolation and shape functions.

value \phi_{2} at x = x_{2} . Two data points define a linear polynomial \phi = a_{1} + a_{2}x . The procedure for obtaining shape functions N_{i} from this polynomial is discussed in Section 3.8. In the notation of Fig. 3.12-1a, with x_{1} = 0 and x_{2} = L , the result is

[ \mathrm{N} ] = \left\lfloor \frac {L - x}{L} \quad \frac {x}{L} \right\rfloor \tag {3.12-2}

Quadratic interpolation in one dimension is depicted in Fig. 3.12-1b. Here there are three data points. They define a parabola \phi = a_{1} + a_{2}x + a_{3}x^{2} , which must display the values \phi = \phi_{1} , \phi = \phi_{2} , and \phi = \phi_{3} at x = x_{1} , x = x_{2} , and x = x_{3} , respectively. The x_{i} values need not be uniformly spaced. Quadratic shape functions can be determined by the procedure used for linear shape functions in Eqs. 3.8-6 to 3.8-8, but now there are three a_{i} and the algebra is more tedious. The three shape functions are most easily obtained from Lagrange's interpolation formula, which is discussed subsequently. The result is

[ \mathrm{N} ] = \left\lfloor \frac {(x _ {2} - x) (x _ {3} - x)}{(x _ {2} - x _ {1}) (x _ {3} - x _ {1})} \frac {(x _ {1} - x) (x _ {3} - x)}{(x _ {1} - x _ {2}) (x _ {3} - x _ {2})} \frac {(x _ {1} - x) (x _ {2} - x)}{(x _ {1} - x _ {3}) (x _ {2} - x _ {3})} \right\rfloor \tag {3.12-3}

In Eqs. 3.12-2 and 3.12-3 we note the following characteristics, which are true of all C^0 polynomial shape functions (in one dimension).

1. All shape functions N_{i} , and function \phi itself, are polynomials of the same degree.

2. For any shape function N_{i}, N_{i} = 1 when x = x_{i} and N_{i} = 0 when x = x_{j} where i \neq j .

3. C^0 shape functions sum to unity. This is not obvious in Eq. 3.12-3 but can be shown as follows. If \phi_i = 1 at all n data points, then \phi = 1 everywhere in the interpolated function \phi . Equation 3.12-1 becomes

1 = \sum_ {i = 1} ^ {n} N _ {i} \tag {3.12-4}

For C^{1} elements, in which derivatives of \phi are also used as nodal d.o.f., Eq. 3.12-4 is valid if the N_{i} are those associated with translational d.o.f. only.

Lagrange's Interpolation Formula. A function \phi = \phi(x) , of degree n - 1 and defined by n values \phi_i at corresponding abscissae x_i , has the form

\phi = \sum_ {i = 1} ^ {n} N _ {i} \phi_ {i} \quad \text { or } \quad \phi = N _ {1} \phi_ {1} + N _ {2} \phi_ {2} + \dots + N _ {n} \phi_ {n} \tag {3.12-5}

in which shape functions N_{i} have been devised by Lagrange as follows:

N _ {1} = \frac {(x _ {2} - x) (x _ {3} - x) (x _ {4} - x) \cdot \cdot \cdot (x _ {n} - x)}{(x _ {2} - x _ {1}) (x _ {3} - x _ {1}) (x _ {4} - x _ {1}) \cdot \cdot \cdot (x _ {n} - x _ {1})}

N _ {2} = \frac {(x _ {1} - x) (x _ {3} - x) (x _ {4} - x) \cdot \cdot \cdot (x _ {n} - x)}{(x _ {1} - x _ {2}) (x _ {3} - x _ {2}) (x _ {4} - x _ {2}) \cdot \cdot \cdot (x _ {n} - x _ {2})} \tag {3.12-6}

N _ {n} = \frac {(x _ {1} - x) (x _ {2} - x) (x _ {3} - x) \cdot \cdot \cdot (x _ {n - 1} - x)}{(x _ {1} - x _ {n}) (x _ {2} - x _ {n}) (x _ {3} - x _ {n}) \cdot \cdot \cdot (x _ {n - 1} - x _ {n})}

Note that the N_{i} have characteristics 1 and 2 just cited. Characteristic 3 is present but is not obvious. Note also that the N_{i} in Eqs. 3.12-2 and 3.12-3 are special cases of Eq. 3.12-6, for which n = 2 and n = 3 , respectively.

If there is a “true curve” for which the interpolated curve \phi = \Sigma N_{i}\phi_{i} is but an approximation, the two curves are coincident only at the n values of x_{i} that provide the \phi_{i} used for interpolation. Moreover, the interpolated curve yields only exact ordinates \phi_{i} , not exact slopes \phi_{,xi} as well. An example appears in Fig. 3.12-2.

Two Dimensions. Imagine that a dependent variable \phi = \phi(x, y) is to be interpolated from four nodal \phi_{l} at corners of a rectangle (Fig. 3.12-3). Here \phi has the form

\phi = a _ {1} + a _ {2} x + a _ {3} y + a _ {4} x y \tag {3.12-7}

Shape functions are products of the N_{i} of Lagrange's formula. We argue as follows.

In Fig. 3.12-3, one can linearly interpolate \phi along the left edge between nodal values \phi_1 and \phi_4 , and along the right edge between nodal values \phi_2 and \phi_3 . Thus, in Eqs. 3.12-6, y replaces x and n = 2 . Calling the edge values \phi_{14} and \phi_{23} , we have

line

| x | φ | | ---- | ---- | | x₁ | 1 | | x₂ | 2 | | x₃ | 3 | | x₄ | 4 |

Figure 3.12-2. Possible discrepancies between a “true curve” (solid line) and the fit produced by Lagrange’s formula (dashed line).

text_image

y a a 4 3 b x 1 2

Figure 3.12-3. Four-node "bilinear" element.

\phi_ {1 4} = \frac {b - y}{2 b} \phi_ {1} + \frac {b + y}{2 b} \phi_ {4} \quad \text { and } \quad \phi_ {2 3} = \frac {b - y}{2 b} \phi_ {2} + \frac {b + y}{2 b} \phi_ {3} \tag {3.12-8}

Next we linearly interpolate in the x direction between \phi_{14} and \phi_{23} :

\phi = \frac {a - x}{2 a} \phi_ {1 4} + \frac {a + x}{2 a} \phi_ {2 3} \tag {3.12-9}

Substitution of Eq. 3.12-8 into Eq. 3.12-9 yields \phi = \Sigma N_{i}\phi_{i} , where

N _ {1} = \frac {(a - x) (b - y)}{4 a b} \quad N _ {2} = \frac {(a + x) (b - y)}{4 a b} \tag {3.12-10}

N _ {3} = \frac {(a + x) (b + y)}{4 a b} \quad N _ {4} = \frac {(a - x) (b + y)}{4 a b}

One can easily check that each N_{i} = 1 at the coordinates of node i , is zero at other nodes, and that N_{1} + N_{2} + N_{3} + N_{4} = 1 .

The element associated with Eqs. 3.12-10 is called “bilinear,” as each of its shape functions is a product of two linear polynomials. Similarly, a nine-node element (nodes at corners, midsides, and the center) is called “biquadratic,” a 16-node element in which four of the nodes are internal is called “bicubic,” and so on. Shape functions for all these elements are products of one-dimensional Lagrange interpolation shape functions [3.4]. These elements, and analogous elements in three dimensions, are called Lagrange elements.

Additional Dependent Variables. The element of Fig. 3.12-3 can be used to solve problems of plane stress and plane strain. For such problems there are two dependent field variables—namely, u = u(x, y) and v = v(x, y) . The four-node element then has eight d.o.f. Displacements u and v are each interpolated from four nodal values, that is,

u = \sum_ {i = 1} ^ {4} N _ {i} u _ {i} \quad \text { and } \quad v = \sum_ {i = 1} ^ {4} N _ {i} v _ {i} \tag {3.12-11}

in which the N_{i} are defined by Eqs. 3.12-10.

3.13 SHAPE FUNCTIONS FOR C^1 ELEMENTS

A C^{1} element provides interelement continuity of the field quantity \phi and its first derivatives at nodes, but not interelement continuity of all second derivatives of \phi . An example appears in the analysis of a thin plate in bending, where the field quantity is displacement w in the z direction, where w = w(x, y) , and x and y are coordinates in the plane of the plate. Typical thin-plate elements use w, w_{,x} , and w_{,y} as nodal d.o.f. Second derivatives w_{,xx} , w_{,yy} , and w_{,xy} are not all continuous across interelement boundaries. (Even the first derivative w_{,n} , where n is a direction normal to an edge, is typically discontinuous except at nodes.)

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φ 1 θ₁ 2 θ₂ φ₁ 0 L x

Figure 3.13-1. A curve interpolated between two points at which ordinates \phi_{1} and \phi_{2} and inclinations \theta_{1} and \theta_{2} are known.

One Dimension. Fitting a curve to both ordinate and slope information at data points is known as Hermitian interpolation. The simplest and most common Hermitian interpolation is between two points at which both ordinate and slope are known (Fig. 3.13-1). We will assume that slope \phi_{,x} is small, so that rotation \theta is practically the same as \phi_{,x} . Four data items define a cubic curve,

\phi = a _ {1} + a _ {2} x + a _ {3} x ^ {2} + a _ {4} x ^ {3} \quad \text { or } \quad \phi = \lfloor \mathbf {X} \rfloor \{\mathbf {a} \} \tag {3.13-1a}

where

[ \mathbf {X} ] = \left[ \begin{array}{l l l l} 1 & x & x ^ {2} & x ^ {3} \end{array} \right] \quad \text { and } \quad \{\mathbf {a} \} = \left[ \begin{array}{l l l l} a _ {1} & a _ {2} & a _ {3} & a _ {4} \end{array} \right] ^ {T} \tag {3.13-1b}

To express the a_{i} in terms of ordinates and slopes at x = 0 and at x = L , we make the substitutions

\phi = \phi_ {1} \quad \text { and } \quad \phi_ {, x} = \theta_ {1} \quad \text { at } x = 0 \tag {3.13-2}

\phi = \phi_ {2} \quad \text { and } \quad \phi_ {, x} = \theta_ {2} \quad \text { at } x = L

Thus Eq. 3.13-1 yields

\left\{ \begin{array}{l} \phi_ {1} \\ \theta_ {1} \\ \phi_ {2} \\ \theta_ {2} \end{array} \right\} = \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & L & L ^ {2} & L ^ {3} \\ 0 & 1 & 2 L & 3 L ^ {2} \end{array} \right] \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \\ a _ {3} \\ a _ {4} \end{array} \right\} \quad \text { or } \quad \{\mathbf {d} \} = [ \mathbf {A} ] \{\mathbf {a} \} \tag {3.13-3}

Therefore \{\mathbf{a}\} = [\mathbf{A}]^{-1}\{\mathbf{d}\} , and Eq. 3.13-1 becomes

\phi = \lfloor \mathbf {N} \rfloor \{\mathbf {d} \}, \quad \text { where } \quad \lfloor \mathbf {N} \rfloor = \lfloor \mathbf {X} \rfloor [ \mathbf {A} ] ^ {- 1} \tag {3.13-4}

Shape function matrix [N] is 1 by 4. The four N_{i} are shown in Fig. 3.13-2. As expected, three of the N_{i} and three of the dN_{i}/dx are zero at end x = 0, whereas the remaining N_{i} and the remaining dN_{i}/dx have unit value. The same is true at end x = L. This behavior is required if Eqs. 3.13-2 are to be satisfied by the interpolation \phi = \Sigma N_{i}d_{i} . These shape functions may be used to generate the stiffness matrix of a beam element (Section 4.2).

Two Dimensions. Hermitian interpolation of a function w = w(x, y) can be used to generate elements for the analysis of thin plates in bending. A great many