MultiPhysicsVault/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_028.md

supporting effect of the roadbed or soil must be modeled. Elastic support can be represented by a foundation stiffness matrix, [k_{f}] for a foundation element and [K_{f}] = \Sigma [k_{f}] for the entire foundation structure. If [K_{s}] is the stiffness matrix of the supported structure, then [K_{s}] + [K_{f}] is the net stiffness matrix of the supported structure on its elastic foundation.

In what follows we presume that [K_{f}] is to operate on interface d.o.f. only—that is, on d.o.f. of only the nodes shared by the supported structure and its foundation [8.20,8.21]. Physically, the jth column of [K_{f}] represents forces (and perhaps moments as well) that must be applied to nodes on the surface of the foundation to cause the jth foundation d.o.f. to have unit value while all other d.o.f. on the interface are zero.

For an elastic solid foundation model, [K_{f}] is a full matrix, as loads must be applied to all interface d.o.f. when only one of these d.o.f. is activated. An approximate foundation model, simple and inexpensive yet often adequate, is the Winkler foundation model.

A Winkler foundation, Fig. 8.10-1a, deflects only where load is applied. Adjacent foundation material is utterly unaffected. A Winkler foundation of modulus \beta applies a vertical pressure \beta w when deflected vertically an amount w. In this regard the foundation acts exactly like a liquid of density \beta . However, we assume that, unlike the pressure of buoyancy, foundation pressure can act either upward or downward. If instead part of a structure lifts off the foundation, the problem is nonlinear, as then contact forces and contact geometry are both unknown at the outset.

We define a foundation element as the area on the foundation surface that makes contact with an element (or element face) of the supported structure. Thus a rectangular plate element in a supported paving slab would define a rectangular foundation element of identical shape and size. To determine [k_{f}] for a Winkler foundation element, we can use the following strain energy argument. Let dA be an increment of the area A of the foundation element. Then deflection w normal to A produces a force increment dF = \beta w \, dA . By analogy with a linear spring, whose strain energy is F\Delta/2 when deflected an amount \Delta , the strain energy increment in the foundation is dU = dF(w/2) = \beta w^{2} \, dA/2 . If w is governed by d.o.f. \{d\} of the aforementioned plate element, then w = \lfloor N \rfloor\{d\} , where \lfloor N \rfloor is the lateral-displacement shape function matrix of the plate element. Hence

U = \frac {1}{2} \int \beta w ^ {2} d A = \frac {1}{2} \int w ^ {T} \beta w d A = \frac {1}{2} \left\{\mathbf {d} \right\} ^ {T} \left[ \mathbf {k} _ {f} \right] \left\{\mathbf {d} \right\} \tag {8.10-1}

in which

[ \mathbf {k} _ {f} ] = \int \beta [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d A \tag {8.10-2}

Figure 8.10-1. Deflections of elastic foundations. Uniform pressure p is applied directly to the foundation; no structure is interposed. (a) Winkler foundation model. (b) Elastic solid foundation model.

is a foundation stiffness matrix that operates on the same d.o.f. as the plate element in contact with the foundation.

If the supported element were a beam rather than a plate, then [N] in Eq. 8.10-1 would contain the cubic shape functions of a beam and dA = b \, dx , where b is the width of the beam. However, cubic functions are not exact because the beam element is not loaded only by forces and moments at its end nodes; it is also loaded by distributed foundation pressure. Use of the exact deflected shape [8.22] leads to stiffness coefficients in the combined matrix [\mathbf{k}_{\text{beam}}] + [\mathbf{k}_f] . These coefficients involve lengthy expressions, but they are much more accurate in a coarse mesh than coefficients based on a cubic polynomial.

If the supported element is neither a plate nor a beam, but (say) an eight-node hexahedron, then \{\mathbf{d}\} in Eq. 8.10-1 would not contain nodal rotation d.o.f. It would contain only the w d.o.f. of the four corner nodes of the quadrilateral contact area, and the N_{i} would be those of Eq. 6.3-2. Indeed, one could ignore nodal rotation d.o.f. even if the supported element is a plate. Then [\mathbf{k}_f] becomes more sparse and does not resist nodal rotations. Ultimately one can imagine for the supported structure element only a rigid-body lateral translation w , and divide the foundation resisting force \beta A w equally among element nodes in contact with the foundation. Thus, for a supported element that has n contacting nodes, [\mathbf{k}_f] becomes a diagonal matrix whose n nonzero coefficients are k_{fii} = \beta A / n . By this "lumping" procedure the foundation is reduced to a set of linear springs at the contacting nodes.

The name scalar element is given to a linear or torsional spring that connects a node to a support. A scalar element can resist only a deformation along its axis or a twist about its axis.

8.11 MEDIA OF INFINITE EXTENT

Many physical problems deal with an unbounded medium. Examples include a wing moving through air, diffraction of water around an island, and a load supported by the ground (Fig. 8.11-1a). In all these problems a finite element model must be terminated somewhere short of infinity. Simple truncation at a rigid boundary, Fig. 8.11-1b, is usually adequate in static problems. However, it is

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P r r Infinite extent Symmetric

(a)

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P Rigid boundary

(b)

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P ← a ← b Infinite elements (shaded)

(c)
Figure 8.11-1. (a) Load P on plane or axially symmetric body of infinite extent below the x axis. (b) Large mesh of conventional elements. (c) Smaller mesh, bounded by infinite elements.

unclear where the rigid boundary should be placed, and the analysis may be expensive because many elements are used. In dynamic problems a rigid boundary reflects a wave, regardless of the size of the mesh; therefore, the model misrepresents reality.

Various methods for numerical analysis of unbounded field problems, both static and dynamic, have been devised [8.23] . In what follows we summarize a particular kind of “infinite element” for static analysis that is simple and effective [8.23-8.25] . In Fig. 8.11-1c, infinite elements permit satisfactory results to be obtained from fewer elements than would otherwise be required.

Infinite Elements. In stress analysis, infinite elements are analogous to an elastic foundation in that they provide correct or approximately correct support conditions for a region of interest that is modeled by a mesh of standard elements. Stresses in the infinite elements are usually not of interest and may not be accurate.

In formulating an infinite element, one makes use of two sets of shape functions. These are the standard shape functions [N] and either one of the following: (1) “decay” shape functions [N_{d}] , which approach zero as a coordinate approaches infinity, or (2) “growth” shape functions [M], which grow without limit as a coordinate approaches infinity. In the first method [N] is applied to geometry and [N_{d}] to the field variable, so that the element remains of finite size while the field variable decays. In the second method [N] is applied to the field variable and [M] to geometry, so that the element grows to infinite size. The second method yields what are called “mapped” infinite elements. They are easy to implement and are described as follows.

In order to illustrate concepts and introduce procedures, we consider a one-dimensional element—namely, element 1–2–3 in Fig. 8.11-2 [8.25]. Distance a between nodes 1 and 2 may be considered a characteristic length of the element. Point 0, a distance a to the left of node 1, is not a node; it is a “pole” whose significance is discussed subsequently. Geometry of the element is interpolated according to

x = M _ {1} x _ {1} + M _ {2} x _ {2}, \quad \text { where } \quad \begin{array}{l} M _ {1} = - \frac {2 \xi}{1 - \xi} \\ M _ {2} = \frac {1 + \xi}{1 - \xi} \end{array} \tag {8.11-1}

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x → r ← x₀ → ← a → ← a → 0 • 1 2 ∥ 3 ← x₁ → ← x₂ → ← x₃ = x ∥

(a)

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ξ = -1 ξ = 0 ξ = +1 1 2 3 M₂ = (1 + ξ)/1 - ξ 1 0 M₁ = -2ξ/1 - ξ

(b)
Figure 8.11-2. (a) One-dimensional infinite element in physical space. (b) The same element in natural-coordinate space.

which yields x = x_{1} at \xi = -1 and x = x_{2} at \xi = 0 . As for x_{3} , from Eq. 8.11-1,

x _ {3} = \lim _ {\xi \rightarrow 1} \frac {- 2 \xi x _ {1} + (1 + \xi) x _ {2}}{1 - \xi} = \infty \tag {8.11-2}

Accordingly, the mapping of Eq. 8.11-1 automatically places node 3 at infinity, and node 3 need not be explicitly present in Eq. 8.11-1. A field variable \phi can be interpolated by standard shape functions. For the present three-node line element, from Eq. 6.2-2, the field interpolation \phi = \lfloor N \rfloor \{\phi_e\} is the usual quadratic

\phi = \left\lfloor - \frac {\xi + \xi^ {2}}{2} \quad (1 - \xi^ {2}) \quad \frac {\xi + \xi^ {2}}{2} \right\rfloor \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \\ \phi_ {3} \end{array} \right\} \tag {8.11-3}

Typically, \phi_{3} is set to a constant value (usually zero) as a boundary condition. Formulation of the element stiffness matrix, Eq. 6.2-6, proceeds in standard fashion except that mapping functions M_{1} and M_{2} of Eq. 8.11-1 are used to form the Jacobian J . Specifically, in Eq. 6.2-6 we require the strain-displacement matrix [B] and the Jacobian J , which for the infinite line element are

\left\lfloor \mathbf {B} \right\rfloor = \frac {1}{J} \left\lfloor \frac {d}{d \xi} \mathbf {N} \right\rfloor \quad \text { and } \quad J = M _ {1, \xi} x _ {1} + M _ {2, \xi} x _ {2} \tag {8.11-4}

where J = dx / d\xi is obtained from Eq. 8.11-1 and \lfloor \mathbf{N}\rfloor is given by Eq. 8.11-3.

To show how the foregoing infinite element represents field quantity \phi , we first solve Eq. 8.11-1 for \xi . With x = x_{0} + r and other dimensions shown in Fig. 8.11-2a,

\xi = \frac {x - x _ {2}}{x - 2 x _ {1} + x _ {2}} = 1 - \frac {2 a}{r} \tag {8.11-5}

Substitution of Eq. 8.11-5 into Eq. 8.11-3 yields

\phi = \phi_ {3} + \left(- \phi_ {1} + 4 \phi_ {2} - 3 \phi_ {3}\right) \frac {a}{r} + \left(2 \phi_ {1} - 4 \phi_ {2} + 2 \phi_ {3}\right) \frac {a ^ {2}}{r ^ {2}} \tag {8.11-6}

We see that as r approaches infinity, \phi approaches \phi_3 (which may be set to zero as a boundary condition). The constant value \phi = c prevails if \phi_1 = \phi_2 = \phi_3 = c , but linear variations of \phi with r are not represented. In general, the two parenthetic expressions in Eq. 8.11-6 do not vanish, so \phi becomes infinite at point 0 because r = 0 at point 0. Point 0 is therefore a pole or singular point about which field quantity \phi decays. This suggests that in a problem such as that of Fig. 8.11-1c, in which there is indeed a singularity at r = 0 , one should use a = b .

It is not necessary that the mapping and the field interpolation rely on identical sets of nodes. For example, we can use the three-node mapping of Fig. 8.11-2 and Eq. 8.11-1, but replace Eq. 8.11-3 by a linear field interpolation between nodes 1 and 3,

\phi = \left\lfloor \frac {1 - \xi}{2} \quad \frac {1 + \xi}{2} \right\rfloor \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {3} \end{array} \right\} \tag {8.11-7}

This is perhaps the simplest possible infinite element.

Equation 8.11-7 offers the following physical interpretation. Let \phi be axial displacement u and let node 3 be fixed. Then, from Eqs. 8.11-1 and 8.11-7, axial strain is

\epsilon_ {x} = \frac {1}{J} \left\lfloor \frac {d}{d \xi} \mathbf {N} \right\rfloor \left\{ \begin{array}{l} u _ {1} \\ 0 \end{array} \right\} = \frac {(1 - \xi) ^ {2}}{2 a} \left(- \frac {1}{2}\right) u _ {1} = - \frac {u _ {1}}{2 a} \frac {(1 - \xi) ^ {2}}{2} \tag {8.11-8}

We see that for an imagined element of physical length 2a between nodes 1 and 3, axial strain decays parabolically from \epsilon_{x} = -u_{1}/a at end \xi = -1 to \epsilon_{x} = 0 at end \xi = +1 , rather than being the constant value \epsilon_{x} = -u_{1}/2a throughout as would be the case for a standard two-node element of length 2a.

For analysis of plane and axially symmetric bodies, one needs infinite elements that are mathematically two-dimensional. Such an element is shown in Fig. 8.11-3. It extends to infinity in the \xi direction and is directly analogous to the element of Fig. 8.11-2. If the field variable \phi is set to zero at element nodes 5 and 6, one need not use N_{5} and N_{6} in element formulation, and d.o.f. \phi_{5} and \phi_{6} need not appear in \{\mathbf{D}\} . However, nodal d.o.f. \phi_{i} on outer edges of infinite elements may be left unspecified, as d.o.f. to be determined, if unrestrained outer boundaries do not imply the possibility of rigid-body motion. An axially symmetric plane problem, in which only axially symmetric deformations are allowed, is a case in point.

Computer programming of mapped infinite elements is straightforward. In terms of Figs. 6.5-1 and 6.5-2, the essential change is alteration of the loop on statement 30 in Fig. 6.5-1: mapping functions [M] must be used to generate the Jacobian matrix, its inverse, and its determinant. Throughout the remainder of the subroutine one uses shape functions [N] and shape function derivatives (appropriate to the number of element nodes used for the field variable) in the manner already programmed.

Boundary Element Method (BEM). The BEM is an alternative to the finite element method (FEM). BEM can be applied to bounded or unbounded domains, but seems best suited to the latter. Like FEM, BEM uses nodes and elements, but only on the boundary. Thus, as compared with FEM, dimensionality is reduced by one; for example, a solid analyzed by BEM uses a two-dimensional mesh that covers only its surface. BEM and FEM can be coupled, so that BEM might replace infinite elements as the supporting medium for a structure modeled by FEM. BEM accurately models response in the domain bounded by its mesh (unlike infinite

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Decay origin a 1 2 3 4 η ξ 5 6

Mapping Functions

M _ {1} = \frac {- 2 \xi}{1 - \xi} \frac {1 - \eta}{2}

M _ {2} = \frac {- 2 \xi}{1 - \xi} \frac {1 + \eta}{2}

M _ {3} = \frac {1 + \xi}{1 - \xi} \frac {1 - \eta}{2}

M _ {4} = \frac {1 + \xi}{1 - \xi} \frac {1 + \eta}{2}

Shape Functions

N _ {1} = \frac {1}{4} (- \xi + \xi^ {2}) (1 - \eta)

N _ {2} = \frac {1}{4} (- \xi + \xi^ {2}) (1 + \eta)

N _ {3} = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta)

N _ {4} = \frac {1}{2} (1 - \xi^ {2}) (1 + \eta)

N _ {5} = \frac {1}{4} (\xi + \xi^ {2}) (1 - \eta)

N _ {6} = \frac {1}{4} (\xi + \xi^ {2}) (1 + \eta)

Figure 8.11-3. A two-dimensional infinite element. Several additional elements are described in [8.25].

elements, which provide support but do not offer internal accuracy). However, the computational expense of BEM increases quickly if the response at several interior locations is needed. Although [K] of FEM is usually large, sparse, and symmetric, the analogous matrix of BEM is small, full, and unsymmetric. With an increase in the ratio of surface to volume, BEM becomes a less attractive alternative to FEM, because a mesh must be supplied for each boundary (each surface, hole, joint plane, or other discontinuity).

The theory of BEM is not easily explained. The mathematics required is more advanced than that needed for FEM. The interested reader will find several texts, conference proceedings, journal articles, and surveys [8.26,8.27].

8.12 FINITE ELEMENTS AND FINITE DIFFERENCES

Both the finite element method and the older finite difference method discretize a continuum, and both generate simultaneous algebraic equations to be solved for nodal d.o.f. Otherwise, the methods are superficially different. Finite difference stencils overlap one another and sometimes have nodes outside the structure boundary. Finite elements do not overlap and have no nodes outside the structure boundary. Finite differences are usually explained as a way to solve differential equations; finite elements are usually explained as a way to minimize a functional.

But a finite difference model can be derived from a functional [8.28]. For example, if \Pi_p is the functional and \{\mathbf{D}\} are nodal d.o.f., we can write finite difference expressions for the derivatives in \Pi_p and generate algebraic equations from the stationary condition \{\partial \Pi_p / \partial \mathbf{D}\} = \{\mathbf{0}\} . This procedure is called the finite difference energy method. It produces a symmetric coefficient matrix if the finite element method produces a symmetric coefficient matrix for the same physical problem.

Thus the finite difference and finite element methods differ only in the choice of d.o.f. and in the location of nodes. Indeed, we can say that finite elements are a device for generating finite difference equations. Sometimes the two methods produce identical equations.

Both methods have about the same accuracy. Computer cost is often less when finite differences are used. Inevitably, cost comparisons depend on the type of problem and program organization as well as on the analysis method.

The finite difference energy method is well suited to shells of revolution [8.28] . It is also suited to “pure” continua, where there is only one medium, such as a homogeneous solid or fluid. It is not well suited to a structure with a complicated boundary shape or to a structure that must be modeled by a mixture of materials or a mixture of forms, such as a vehicle that combines bar, beam, plate, and shell components. For such a problem the finite element method has no rival.

8.13 REANALYSIS METHODS

Imagine that an initial solution has been obtained. Then the structure is altered: by changing member sizes, changing materials, or otherwise altering the finite

element mesh. Loads on the structure are not changed. ^{2} Symbolically, we have

\text { Initial   system: } \quad [ \mathbf {K} ] \{\mathbf {D} \} = \{\mathbf {R} \} \tag {8.13-1}

\text { Altered   system: } \quad [ \mathbf {K} ^ {*} ] \{\mathbf {D} ^ {*} \} = \{\mathbf {R} \} \tag {8.13-2}

where

[ \mathbf {K} ^ {*} ] = [ \mathbf {K} ] + [ \Delta \mathbf {K} ] \quad \text { and } \quad \{\mathbf {D} ^ {*} \} = \{\mathbf {D} \} + \{\Delta \mathbf {D} \} \tag {8.13-3}

D.o.f. \{D^{*}\} are desired. The obvious approach is complete re-solution: that is, solve Eq. 8.13-2. Alternatives, called reanalysis methods, intend to obtain \{D^{*}\} with less computational effort than complete re-solution, by using information available from the previously obtained solution of Eq. 8.13-1. In vibration analysis, the analogous problem is to obtain modified frequencies without redoing the eigenvalue extraction.

Many methods of reanalysis have been proposed [8.29]. They have been categorized as follows [8.30].

1. Direct methods require a finite and predictable number of steps. They produce the exact \{D^{*}\} and work best when only a small portion of the structure is altered. If [K] has semibandwidth b, and less than roughly b rows of [K] are altered, then a direct method of reanalysis may be more efficient than complete re-solution.
2. Iterative methods converge from \{D\} toward \{D^{*}\} at a rate that is case-dependent. Iterative methods work best when alterations are small. Large differences between [K] and [K^{*}] make the iterations converge slowly or even diverge.
3. Approximate methods are usually based on a truncated series expansion or on a reduced set of structural equations. They are best suited to problems where exact results are not needed, for example, in intermediate stages of design or optimization.

To do reanalysis, one must choose among the many methods, and revise and enlarge the computer program. The option of complete re-solution is easier and may also be more efficient in many problems.

However, the option of substructuring should be noted. Substructuring (Section 8.14) is done for various reasons. One of its benefits is that the effect of alterations in a single substructure is efficiently computed. In this regard, substructuring is a direct method of reanalysis.

8.14 SUBSTRUCTURING

Mathematically, a substructure is a partially solved portion of the complete set of structural equations. Physically, a substructure is one of two or more parts

into which a structure or a finite element mesh is divided. Multilevel substructuring is possible (Fig. 8.14-1). Substructuring has other names in other contexts: blocking or dissection when used by numerical analysts, and diakoptics or tearing when used by electrical engineers. We will describe the procedure in structural terms [8.31,8.32].

Procedure. In brief, a substructure is a “superelement,” that is, a single element with many nodes on its boundary and many interior d.o.f. The name “macroelement” is also appropriate. The process is that of condensation and recovery, as described in Sections 8.1 and 8.2. Indeed, elements in Fig. 8.1-1 are substructures having few d.o.f. After the division of a structure into substructures has been selected, static analysis proceeds as follows.

1. Evaluate [k] and {r} for each substructure, where [k] and {r} pertain to all d.o.f. of the substructure. Eliminate internal d.o.f. by condensation; that is, apply Eq. 8.1-3. The condensed [k] and {r} pertain to only the boundary d.o.f. {d} of the substructure, which may be called “attachment” d.o.f.
2. Assemble substructures by connecting attachment nodes (i.e., nodes shared by substructures). Thus generate structural equations [K_{m}]\{D_{m}\} = \{R_{m}\} , in which \{D_{m}\} contains the attachment d.o.f. of all substructures. (Attachment nodes on mating boundaries of adjacent substructures must match in physical placement and in orientation of their d.o.f.) Solve for \{D_{m}\} .
3. For each substructure, extract from \{D_{m}\} the attachment d.o.f. \{d_{r}\} of that substructure. Use Eq. 8.1-2 to compute interior d.o.f. \{d_{c}\} . Now all d.o.f. of the substructure are known. Hence, stress calculation proceeds in the usual way.

Clearly this is a finite element process in which elements have many internal d.o.f. and are given the name “substructures.” It differs from a standard finite element process in that one does not form a single stiffness matrix that operates on all

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A 6 A 2' 1 2 3' 4 5 3 2 1 3 4 5 (a) (b)

Figure 8.14-1. (a) An aircraft divided into substructures 1, 2, 2', and so on. (b) Division of substructure 2' into further substructures.

d.o.f. of the structure. Instead, there are several stiffness matrices, one for each substructure, and there is information about how to connect them so as to form [K_{m}] .

Remarks. The names “masters” and “slaves” are sometimes used for retained and condensed d.o.f., respectively. In static analysis, all attachment d.o.f. \{D_{m}\} are masters and all d.o.f. interior to the substructures are slaves. In dynamic analysis, most interior d.o.f. are slaves but some may be masters. (Condensation in dynamics is discussed in Section 13.7.) Thus, in dynamic analysis, master d.o.f. \{D_{m}\} may not consist entirely of attachment d.o.f. shared by substructures. In static analysis no approximation is introduced by substructuring. In dynamic analysis some loss of accuracy is produced by substructuring.

In both static and dynamic analysis one desires that boundaries between substructures be small, so that the ratio of masters to slaves is small and [K_{m}] is kept to more manageable size. Accordingly, some structures are more amenable to substructuring than others; for example, a long cylinder can be more effectively substructured than a sphere.

No two substructures need be alike. However, there is special advantage if a structure contains many repetitions of the same form, particularly if the ratio of masters to slaves is small. Consider Fig. 8.14-2a. After the first analysis step has produced the condensed substructure [k] that operates on d.o.f. along boundaries AB and CD, this [k] need only be replicated with different node numberings to form the structure matrix [K_{m}] = \Sigma [k] . (Indeed, substructures within ABCD can be identified: the condensed [k] of the shaded substructure in Fig. 8.14-2b can be reflected about vertical and horizontal centerlines of ABCD, after which condensation of d.o.f. along these centerlines produces the condensed [k] of ABCD.) Thus substructuring is computationally efficient, as internal d.o.f. are processed only once, in forming the condensed [k] of the typical substructure. Without substructuring, the analysis would take longer, even though the total number of d.o.f. is unchanged.

Other advantages of substructuring include the following. There is a managerial advantage in breaking a large problem into smaller and more tractable parts. Different substructures can be studied simultaneously by different design groups. The work of one group can be almost independent of the others if the interaction between substructures is small. Design changes or analysis of nonlinearities, if

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A D B C Typical substructure

(a)

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A H D G E J F L I B K C

(b)
Figure 8.14-2. (a) Typical repeating substructure ABCD in an I beam with holes in its web. (b) A possible substructure of ABCD is shown shaded.

confined to a single substructure, leave matrices of all other substructures unchanged. The results of substructure analyses can be checked separately and revised if necessary before substructures are combined to form the complete structure.

Disadvantages of substructuring include the following. Substructuring replaces one long computer run by several shorter runs. Although this can be an advantage, it is a disadvantage if turnaround is slow. The computer program is more complicated because of increased file handling and data transfers, the need for efficient data structures, and user conveniences. Thus bookkeeping and overhead expense increase. If only one analysis is to be performed and there are no repeating substructures, it is cheaper to analyze the structure entire than to use substructuring. (In practice, design changes are expected, so it is unlikely that there will be but a single analysis.)

In vibration analysis it is possible to compute modes and frequencies of a structure from modes of its component substructures (see Section 13.8).

8.15 STRUCTURAL SYMMETRY

Figure 8.15-1a represents a thin square plate under lateral load. Imagine that the plate is homogeneous, isotropic, uniformly loaded, and has all four edges simply supported. Axes x, y, s, and t are all axes of symmetry. Accordingly, in static analysis, one need not analyze the entire plate: analysis of a single quadrant or a single octant—for example, one of those shown shaded in Fig. 8.15-1a—tells all that there is to know. As compared with analysis of the entire plate, data preparation time and computational expense are reduced.

How can symmetry be recognized? To be symmetric, a structure must have symmetry of shape, material properties, and support conditions. Symmetry can be classed as reflective with respect to an axis or to a plane, or rotational with respect to an axis. A symmetric structure is one for which one or more reflections and/or rotations brings the structure to a configuration indistinguishable from the

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t a y a s D C x a A B

(a)

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q = q/2 Original loading Symmetric component

• q/2 Antisymmetric component L/2 L/2

(b)
Figure 8.15-1. (a) A laterally loaded square plate with simply supported edges. (b) A uniform beam whose loading is broken into symmetric and antisymmetric components.