Finite Element Method - Semi - analytical finite element processes - use of orthogonal functions and finite strip methods _09

Finite Element Method - Semi - analytical finite element processes - use of orthogonal functions and finite strip methods _09 This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.

Semi-analytical finite element -

functions and ’finite strip’ methods

Standard finite element methods have been shown to be capable, in principle, of dealing with any two- or three- (or even four-)* dimensional situations Nevertheless, the cost of solutions increases greatly with each dimension added and indeed, on occasion, overtaxes the available computer capability It is therefore always desirable

to search for alternatives that may reduce computational effort One such class of processes of quite wide applicability will be illustrated in this chapter

In many physical problems the situation is such that the geometry and material properties do not vary along one coordinate direction However, the ‘load’ terms may still exhibit a variation in that direction, preventing the use of such simplifying assumptions as those that, for instance, permitted a two-dimensional plane strain

or axisymmetric analysis to be substituted for a full three-dimensional treatment

In such cases it is possible still to consider a ‘substitute’ problem, not involving the particular coordinate (along which the geometry and properties do not vary), and

to synthesize the true answer from a series of such simplified solutions

The method to be described is of quite general use and, obviously, is not limited to structural situations It will be convenient, however, to use the nomenclature of structural mechanics and to use potential energy minimization as an example

We shall confine our attention to problems of minimizing a quadratic functional such as described in Chapters 2-6 of Volume 1 The interpretation of the process involved as the application of partial discretization in Chapter 3 of Volume 1 followed (or preceded) by the use of a Fourier series expansion should be noted

Let (x, y , z ) be the coordinates describing the domain (in this context these do not necessarily have to be the Cartesian coordinates) The last one of these, z , is the coordinate along which the geometry and material properties do not change and which is limited to lie between two values

O < z < a

The boundary values are thus specified at z = 0 and z = a

See finite elements in the time domain in Chapter 18 of Volume 1

We shall assume that the shape functions defining the variation of displacement u

can be written in a product form as

for body force, with similar form for concentrated loads and boundary tractions (see

Chapter 2 of Volume 1) Indeed, initial strains and stresses, if present, would be

expanded again in the above form

Applying the standard processes of Chapter 2 of Volume 1 to the determination of

the element contribution to the equation minimizing the potential energy, and limit-

ing our attention to the contribution of body forces b only, we can write

Without going into details, it is obvious that the matrix given by Eq (9.4) will

contain the following integrals as products of various submatrices:

Introduction 291

These integrals arise from products of the derivatives contained in the definition of B'

and, owing to the well-known orthogonality property, give

when 1 = 1,2, and m = 1,2, The first integral I I is only zero when 1 and m are

both even or odd numbers The term involving I , , however, vanishes in many

applications because of the structure of B' This means that the matrix K' becomes

a diagonal one and that the assembled final equations of the system have the form

and the large system of equations splits into L separate problems:

Further, from Eqs (9.5) and (9.2) we observe that owing to the orthogonality

property of the integrals given by Eqs (9.6), the typical load term becomes simply

f f = 11 1 (Nf)=b' dx dy dz

V

(9.1 1)

This means that the force term of the lth harmonic only affects the lth system of

Eq (9.9) and contributes nothing to the other equations This extremely important

property is of considerable practical significance for, if the expansion of the loading

factors involves only one term, only one set of equations need be solved The solution

of this will tend to the exact one with increasing subdivision in the xy domain only

Thus, what was originally a three-dimensional problem now has been reduced to a

two-dimensional one with consequent reduction of computational effort

The preceding derivation was illustrated on a three-dimensional, elastic situation

Clearly, the arguments could equally well be applied for reduction of two-

dimensional problems to one-dimensional ones, etc., and the arguments are not

restricted to problems of elasticity Any physical problem governed by a minimization

of a quadratic functional (Chapter 3 of Volume 1) or by linear differential equations is

amenable to the same treatment

A word of warning should be added regarding the boundary conditions imposed on

u For a complete decoupling to be possible these must be satisfied separately by each

and every term of the expansion given by Eq (9.1) Insertion of a zero displacement in

the final reduced problem implies in fact a zero displacement fixed throughout all

terms in the z direction by definition Care must be taken not to treat the final

matrix therefore as a simple reduced problem Indeed, this is one of the limitations

of the process described

When the loading is complex and many Fourier components need to be considered the advantages of the approach outlined here reduce and the full solution sometimes becomes more efficient

Other permutations of the basic definitions of the type given by Eq (9.1) are

obviously possible For instance, two independent sets of parameters ae may be specified with each of the trigonometric terms Indeed, on occasion use of other orthogonal functions may be possible The appropriate functions are often related

to a reduction of the differential equation directly using separation of variables.'

As trigonometric functions will arise frequently it is convenient to remind the

reader of the following integrals:

1; sinylzcosylzdz = 0 when I = 0,1,

(9.12)

2 cos y l z d z = - when I = 1,2,

In this, N j are simply the (scalar) shape functions appropriate to the elements used in

the xy plane and again 7/ = h / a If, as shown in Fig 9.1, simple triangles are used

then the shape functions are given by Eqs (4.7) and (4.8) in Chapter 4 of Volume 1,

but any of the more elaborate elements described in Chapter 8 of Volume 1 would

be equally suitable (with or without the transformations given in Chapter 9 of

Volume 1) The displacement expansion ensures zero u and z1 displacements at the

ends and the zero t , traction condition can be imposed in a standard manner

As the problem is fully three-dimensional, the appropriate expression for strain

involving all six components needs to be considered This expression is given in

Eq (1.15) of Chapter 1 On substitution of the shape function given by Eq (9.13)

for a typical term of the B matrix we have

It is convenient to separate the above as

In all of the above it is assumed that the parameters are listed in the usual order:

ai I = [uf vf w;IT (9.16) and that the axes are as shown in Fig 9.1

The stiffness matrix can be computed in the usual manner, noting that

Ve

On substitution of Eq (9.15), multiplying out, and noting the value of the integrals

from Eq (9.12), this reduces to

(9.18) when I = 1,2, The integration is now simply carried out over the element area.*

* It should be noted that now, even for a single triangle, the integration is not trivial as some linear terms

from N , will remain in B

The contributions from distributed loads, initial stresses, etc., are found as the loading terms To match the displacement expansions distributed body forces may

be expanded in the Fourier series

sinylz 0 ; ] { b x ( x l ~ l z ) }

0 0 COSYIZ bZ(X,Y,Z) Similarly, concentrated line loads can be expressed directly as nodal forces

sinylz 0 f X ( X I Y 7 z)

fi =c [ 0 sinylz ; ] { & ( x l y l z ) } d z (9.20)

in which f f are intensities per unit length

Fig 9.2 A thick box bridge prism of straight or curved platform

Thin membrane box structures 295

The boundary conditions used here have been of a type ensuring simply supported

conditions for the prism Other conditions can be inserted by suitable expansions

The method of analysis outlined here can be applied to a range of practical

problems - one of these being a popular type of box girder, concrete bridge,

illustrated in Fig 9.2 Here a particularly convenient type of element is the distorted,

serendipity or lagrangian quadratic or cubic of Chapters 8 and 9 of Volume 1.2 Finally, it should be mentioned that some restrictions placed on the general shapes

defined by Eqs (9.1) or (9.13) can be removed by doubling the number of parameters

and writing expansions in the form of two sums:

(9.21)

Parameters aA' and aB' are independent and for every component of displacement two

values have to be found and two equations formed

u = C N ( x , y ) cosyIzaA' + C N ( x , y ) sinyIza BI

I = 1 I = 1

An alternative to the above process is to write the expansion as

u = C [ W Y ) exP(i7/z)lae

and to observe that both N and a are then complex quantities

of the above expression with Eq (9.21) will be observed, noting that

Complex algebra is available in standard programming languages and the identity

exp i0 = cos 0 + i sin 0

9.3 Thin membrane box structures

In the previous section a three-dimensional problem was reduced to that of two

dimensions Here we shall see how a somewhat similar problem can be reduced to

one-dimensional elements (Fig 9.3)

Fig 9.3 A 'membrane' box with one-dimensional elements

A box-type structure is made up of thin shell components capable of sustaining

stresses only in its own plane Now, just as in the previous case, three displacements have to be considered at every point and indeed similar variation can be prescribed for

these However, a typical element i j is 'one-dimensional' in the sense that integrations have to be carried out only along the line i j and only stresses in that direction be

considered Indeed, it will be found that the situation and solution are similar to that of a pin-jointed framework

Consider now a rectangular plate simply supported at the ends and in which all strain energy is contained in flexure Only one displacement, w, is needed to specify fully the state of strain (see Chapter 4)

For consistency of notation with Chapter 4, the direction in which geometry and

material properties do not change is now taken as y (see Fig 9.4) To preserve

slope continuity the functions need to include now a 'rotation' parameter Oi

Use of simple beam functions (cubic Hermitian interpolations) is easy and for a typical element i j we can write (with 7' = h / a )

shell problem [Chapter 7, Eq (7.9)J

Fig 9.4 The 'strip' method in slabs

Axisymmetric solids with non-symmetrical load 297

Table 9.1 Square plate, uniform load q; three sides simply supported one clamped

Multiplier q a 4 i 0 4a2 4a2

Using all definitions of Chapter 4 the strains (curvatures) are found and the B

matrices determined; now with C, continuity satisfied in a trivial manner, the problem

of a two-dimensional kind has here been reduced to that of one dimension

This application has been developed by Cheung and others,3p17 named by him the

‘finite strip’ method, and used to solve many rectangular plate problems, box girders,

shells, and various folded plates

It is illuminating to quote an example from the above papers here This refers to a

square, uniformly loaded plate with three sides simply supported and one clamped

Ten strips or elements in the x direction were used in the solution, and Table 9.1

gives the results corresponding to the first three harmonics

Not only is an accurate solution of each I term a simple one involving only some nineteen unknowns but the importance of higher terms in the series is seen to decrease rapidly

Extension of the process to box structures in which both membrane and bending

eflects are present is almost obvious when this example is considered together with

the one of the previous section

In another paper Cheung6 shows how functions other than trigonometric ones can

be used to advantage, although only partial decoupling then occurs (see Sec 9.7 below)

In the examples just quoted a thin plate theory using the single displacement

variable w and enforcing CI compatibility in the x direction was employed

Obviously, any of the independently interpolated slope and displacement elements

of Chapter 5 could be used here, again employing either reduced integration or

mixed methods Parabolic-type elements with reduced integration are employed in

references 13 and 14, and linear interpolation with a single integration point is

shown to be effective in reference 15

Other applications for plate and box type structures abound and additional

information is given in the text of reference 17

9.5 Axisymmetric solids with non-symmetrical load

One of the most natural and indeed earliest applications of Fourier expansion occurs

in axisymmetric bodies subject to non-axisymmetric loads Now, not only the radial

Fig 9.5 An axisymmetric solid; coordinate displacement components in an axisymmetric body

(u) and axial (w) displacement (as in Chapter 5 of Volume 1) will have to be

considered but also a tangential component (v) associated with the tangential angular direction 0 (Fig 9.5) It is in this direction that the geometric and material properties

do not vary and hence here that the elimination will be applied

To simplify matters we shall consider first components of load which are symmetric about the 0 = 0 axis and later include those which are antisymmetric Describing now only the nodal loads (with similar expansion holding for body forces, boundary conditions, initial strains, etc.) we specify forces per unit of circumference as

the direction of T has to change for 13 > T

The displacement components are described again in terms of the two-dimensional

(r, z ) shape functions appropriate to the element subdivision, and, observing

symmetry, we write, as in Eq (9.13),

cosyle o

u l = { %) = F N i [ 0 siny10 : ] { j,} (9.25)

Axisymmetric solids with non-symmetrical load 299