The Standard Desrete System

7/28/2019 The Standard Desrete System

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Some preliminaries: th e standarddiscrete system

1.1 IntroductionThe limitations of the human mind are such that it cannot grasp the behaviour of itscomplex surroundings and creations in one operation. Thus the process of sub-dividing all systems into their individual components or elements, whose behaviouris readily understood, and then rebuilding the original system from such componentsto study its behaviour is a natural way in which the engineer, the scientist, or even theeconomist proceeds.

In many situations an adequate model is obtained using a finite number of well-defined components. We shall term such problems discrete. In others the subdivisionis continued indefinitely and the problem can only be defined using the mathematicalfiction of an infinitesimal. This leads to differential equations or equivalent statementswhich imply an infinite number of elements. We shall term such systemscontinuous.

With the advent of digital computers, discrete problems can generally be solvedreadily even if the number of elements is very large. As the capacity of all computersis finite, continuous problems can only be solved exactly by mathematical manipula-tion. Here, the available mathematical techniques usually limit the possibilities tooversimplified situations.

To overcome the intractability of realistic types of continuum problems, variousmethods of discretization have from time to time been proposed both by engineersand mathematicians. All involve an approximation which, hopefully, approachesin the limit the true continuum solution as the number of discrete variablesincreases.

The discretization of continuous problems has been approached differently bymathematicians and engineers. Mathematicians have developed general techniquesapplicable directly to differential equations governing the problem, such as finite dif-ference approximations,,2 various weighted residual proced~res,~.~r approximatetechniques for determining the stationarity of properly defined functionals. Theengineer, on the other hand, often approaches the problem more intuitively by creat-ing an analogy between real discrete elements and finite portions of a continuumdomain. For instance, in the field of solid mechanics McHenry, Hrenikoff,6Newmark7, and indeed Southwel19 n the 1940s,showed that reasonably good solu-tions to an elastic continuum problem can be obtained by replacing small portions


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2 Some preliminaries: the standard d iscrete systemof the continuum by an arrangement of simple elastic bars. Later, in the same context,Argyris and Turner et showed that a more direct, but no less intuitive, substitu-tion of properties can be made much more effectively by considering that smallportions or elements in a continuum behave in a simplified manner.

It is from the engineering direct analogy view that the term finite element wasborn. Clough appears to be the first to use this term, which implies in it a directuse of a standard methodology applicable to discrete systems. Both conceptually andfrom the computational viewpoint, this is of the utmost importance. The firstallows an improved understanding to be obtained; the second offers a unifiedapproach to the variety of problems and the development of standard computationalprocedures.Since the early 1960s much progress has been made, and today the purely mathe-matical and analogy approaches are fully reconciled. It is the object of this text topresent a view of the finite element method asa general discretizationprocedureof con-tinuum problems posed by mathematically dejined statements.In the analysis of problems of a discrete nature, a standard methodology has beendeveloped over the years. The civil engineer, dealing with structures, first calculatesforce-displacement relationships for each element of the structure and then proceedsto assemble the whole by following a well-defined procedure of establishing localequilibrium at each node or connecting point of the structure. The resulting equa-tions can be solved for the unknown displacements. Similarly, the electrical orhydraulic engineer, dealing with a network of electrical components (resistors, capa-citances, etc.) or hydraulic conduits, first establishes a relationship between currents(flows) and potentials for individual elements and then proceeds to assemble thesystem by ensuring continuity of flows.All such analyses follow a standard pattern which is universally adaptable to dis-crete systems. It is thus possible to define astandard discrete system, and this chapterwill be primarily concerned with establishing the processes applicable to such systems.Much of what is presented here will be known to engineers, but some reiteration atthis stage is advisable. As the treatment of elastic solid structures has been themost developed area of activity this will be introduced first, followed by examplesfrom other fields, before attempting a complete generalization.

The existenceof a unified treatment of standard discrete problems leadsus to thefirst definition of the finite element process as a method of approximation to con-tinuum problems such that(a) the continuumisdivided into a finite number of parts (elements), the behaviour of(b) the solution of the complete system as an assembly of its elements follows pre-

Itwill be found that most classical mathematical approximation procedures as wellas the various direct approximations used in engineering fall into this category. It isthus difficult to determine the origins of the finite element method and the precisemoment of its invention.Table 1.1 shows the process of evolution which led to the present-day concepts offinite element analysis. Chapter 3 will give, in more detail, the mathematical basiswhich emerged from these classical ideas. 1-20

which is specified by a finite number of parameters, andcisely the same rules as those applicable to standard discrete problems.


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Some p relimi naries: th e s tandard dis crete system

1.2 The structural element and the structural system

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Atypical element (1)Fig. 1.1 A typical structure built up from interconnectedelementsTo introduce the reader to the general concept of discrete systems we shall firstconsider a structural engineering example of linear elasticity.

Figure 1.1 represents a two-dimensional structure assembled from individualcomponents and interconnected at the nodes numbered 1 to 6. The joints at thenodes, in this case, are pinned so that moments cannot be transmitted.Asa starting point it will be assumed that by separate calculation, or for that matter

from the results of an experiment, the characteristics of each element are preciselyknown. Thus, if a typical element labelled (1) and associated with nodes 1, 2, 3 isexamined, the forces acting at the nodes are uniquely defined by the displacementsof these nodes, the distributed loading acting on the element (p), and its initialstrain. The last may be due to temperature, shrinkage, or simply an initial lack offit. The forces and the corresponding displacements are defined by appropriate com-ponents (U , V and u, v) in a common coordinate system.

Listing the forces acting on all the nodes (three in the case illustrated) of the element(1) as a matrixt we have

tA limited knowledge of matrix algebra will be assumed throughout this book. This is necessary forreasonable conciseness and formsa convenient book-keepingform. For readers not familiar with the subjecta brief appendix (Appendix A) is included in which sufficient principles of matrix algebra are given to followthe development intelligently. Matrices (and vectors) will be distinguished by bold print throughout.


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The struc tural element and th e structural system 5and for the corresponding nodal displacements

Assuming linear elastic behaviour of the element, the characteristic relationship willalways be of the form

q1=K'a' +f j +froin whichf j represents the nodal forces required to balance any distributed loads actingon the element and fro the nodal forces required to balance any initial strains such asmay be caused by temperature change if the nodes are not subject to any displacement.The first of the terms represents the forces induced by displacement of the nodes.

Similarly, a preliminary analysis or experiment will permit a unique definition ofstresses or internal reactions at any specified point or points of the element intermsof the nodal displacements. Defining such stresses by a matrix c1a relationshipof the form

0 = ~ ac rois obtained in which the two term gives the stresses due to the initial strains when nonodal displacement occurs.

The matrix K e is known as the element stiffness matrix and the matrix Q' as theelement stress matrix for an element (e).Relationships in Eqs (1.3) and (1.4) have been illustrated by an example of an ele-ment with three nodes and with the interconnection points capable of transmittingonly two components of force. Clearly, the same arguments and definitions willapply generally. An element (2) of the hypothetical structure will possess only twopoints of interconnection; others may have quite a large number of such points. Simi-larly, if the joints were considered as rigid, three components of generalized force andof generalized displacement would have to be considered, the last of these correspond-ing to a moment and a rotation respectively. For a rigidly jointed, three-dimensionalstructure the number of individual nodal components would be six. Quite generally,therefore,

(1.4)1 1

with eachq; andaipossessing the same number of components or degrees o freedom.These quantities are conjugate to each other.

The stiffness matrices of the element will clearly always be square and of the form


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6 Some preliminaries: the standard disc rete system

Y

t

Fig. 1.2 A pin-ended bar.

in whichKZ.,etc., are submatrices which are again square and of the sizeE x 1,where1is the number of force components to be considered at each node.As an example, the reader can consider a pin-ended bar of uniform sectionA andmodulus E in a two-dimensional problem shown in Fig. 1.2.The bar is subject to auniform lateral loadp and a uniform thermal expansion strain

Eo =aTwherea is the coefficient of linear expansion and T is the temperature change.calculated asIf the ends of the bar are defined by the coordinatesxi , i andx yn ts length can be

and its inclination from the horizontal as1 Y n -Yi/3 =tan- ~x n - xi

Only two components of force and displacement have to be considered at thenodes.The nodal forces due to the lateral load are clearly

and represent the appropriate components of simple reactions, pL /2. Similarly, torestrain the thermal expansion an axial force ( E a T A ) s needed, which gives the


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The structu ral element and th e s truc tural system 7

cos2p sinpcosp I -cos2p -sin p cosp -sinp cos,O sin2p 1 -sinpcosp -sin2p-sinpcosp I cos2p- sinp cosp

-sin pcosp -sin 2p I sinpcosp sin2p -

components

{k)

-cos pf' 0 ={ -{-sinp](EaTA)osp

sinpFinally, the element displacements

The components of the general equation (1.3) have thus been established for theelementary case discussed. It is again quite simple to find the stresses at any sectionof the element in the form of relation (1.4). For instance, if attention is focused onthe mid-section C of the bar the average stress determined from the axial tensionto the element can be shown to be

bL' M c=- -cos p, -sin p,cosp, sinP]ae- EaT

where all the bending effects of the lateral loadp have been ignored.For more complex elements more sophisticated procedures of analysis are requiredbut the results are of the same form. The engineer will readily recognize that the so-

called 'slope-deflection' relations used in analysis of rigid frames are only a specialcase of the general relations.It may perhaps be remarked, in passing, that the complete stiffness matrix obtainedfor the simple element in tension turns out to be symmetric (as indeed was the casewith some submatrices). This is bynomeans fortuitous but follows from the principleof energy conservation and from its corollary, the well-known Maxwell-Bettireciprocal theorem.


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8 Some prelimin aries: the standard d iscrete systemThe element properties were assumed to follow a simple linear relationship. In

principle, similar relationships could be established for non-linear materials, butdiscussion of such problems will be held over at this stage.

The calculation of the stiffness coefficients of the bar which we have given here willbe found in many textbooks. Perhaps it is worthwhile mentioning here that the firstuse of bar assemblies for large structures was made as early as 1935 when Southwellproposed his classical relaxation method.22

1.3 Assembly and analysis of a structureConsider again the hypothetical structure of Fig. 1.1. To obtain a complete solutionthe two conditions of(a) displacement compatibility and(b) equilibriumhave to be satisfied throughout.

Any system of nodal displacements a:

a = {}an

listed now for the whole structure in which all the elements participate, automaticallysatisfies the first condition.As the conditions of overall equilibrium have already been satisfiedwithin an ele-ment, all that is necessary is to establish equilibrium conditions at the nodes of thestructure. The resulting equations will contain the displacements as unknowns, andonce these have been solved the structural problem is determined. The internalforces in elements, or the stresses, can easily be found by using the characteristicsestablished a priori for each element by Eq. (1.4).

Consider the structure to be loaded by external forces r:

r =

applied at the nodes in addition to the distributed loads applied to the individualelements. Again, any one of the forcesr i must have the same number of componentsas that of the element reactions considered. In the example in question

as the joints were assumed pinned, but at this stage the general case of an arbitrarynumber of components will be assumed.If now the equilibrium conditions of a typical node, i , are to be established, eachcomponent of r i has, in turn, to be equated to the sum of the component forcescontributed by the elements meeting at the node. Thus, considering all the force


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The boundary conditions 9components we have

(1.10)e= 1

in which q! is the force contributed to node i by element 1, q by element 2, etc.Clearly, only the elements which include point i will contribute non-zero forces,but for tidiness all the elements are included in the summation.

Substituting the forces contributing to node i from the definition (1.3) and notingthat nodal variables ai are common (thus omitting the superscript e), we have

(1.11)where

f e=f; +fZ0The summation again only concerns the elements which contribute to node i. If all

such equations are assembled we have simplyK a =r - f (1.12)

in which the submatrices arem

e =lmfi =xf:(1.13)

e= 1

with summations including all elements. This simple rule for assembly is veryconvenient because as soon as a coefficient for a particular element is found it canbe put immediately into the appropriate location specified in the computer. Thisgeneral assembly process can be found to be the common and fundamental feature ofalljinite element calculations and should be well understood by the reader.

If different types of structural elements are used and are to be coupled it must beremembered that the rules of matrix summation permit this to be done only ifthese are of identical size. The individual submatrices to be added have therefore tobe built up of the same number of individual components of force or displacement.Thus, for example, if a member capable of transmitting moments to a node is to becoupled at that node to one which in fact is hinged, it is necessary to complete thestiffness matrix of the latter by insertion of appropriate (zero) coefficients in therotation or moment positions.

1.4 The boundary conditionsThe system of equations resulting from Eq. (1.12) can be solved once theprescribed support displacements have been substituted. In the example of Fig. 1.1,where both components of displacement of nodes 1 and 6 are zero, this will mean


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10 Some preliminaries: he standard discrete systemthe substitution of

al =a6={:}which is equivalent to reducing the number of equilibrium equations (in this instance12) by deleting the first and last pairs and thus reducing the total number of unknowndisplacement components to eight. It is, nevertheless, always convenient to assemblethe equation according to relation (1.12)so as to include all the nodes.

Clearly, without substitution of a minimum number of prescribed displacements toprevent rigid body movements of the structure, it is impossible to solve this system,because the displacements cannot be uniquely determined by the forces in such asituation. This physically obvious fact will be interpreted mathematically as thematrix K being singular, i.e., not possessing an inverse. The prescription of appropri-ate displacements after the assembly stage will permit a unique solution to beobtained by deleting appropriate rows and columns of the various matrices.If all the equations of a system are assembled, their form is

K l l al+K 12a2+. . .=r l - lK zlal+K Z2a2+. . .=r2- 2 (1.14)etc.

and it will be noted that if any displacement, such asal =al , s prescribed then theexternal force r l cannot be simultaneously specified and remains unknown. Thefirst equation could then bedeleted and substitution of known values of al made inthe remaining equations. This process is computationally cumbersome and thesame objective is served by adding a large number, aI, to the coefficient K l l andreplacing the right-hand side, r l - l , by ala. f a is very much larger than otherstiffness coefficients this alteration effectively replaces the first equation by the equa-tion

aal =aal (1.15)that is, the required prescribed condition, but the whole system remains symmetricand minimal changes are necessary in the computation sequence.A similar procedurewill apply to any other prescribed displacement. The above artifice was introduced byPayne and Irons.23An alternative procedure avoiding the assembly of equationscorresponding to nodes with prescribed boundary values will be presented inChapter 20.When all the boundary conditions are inserted the equations of the system can besolved for the unknown displacements and stresses, and the internal forces in each ele-ment obtained.

1.5 Electrical and f luid networksIdentical principles of deriving element characteristics and of assembly will be foundin many non-structural fields. Consider, for instance, the assembly of electricalresistances shown in Fig. 1.3.


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Electrical and fluid networks 11

Fig. 1.3 A network of electrical resistances.If a typical resistance element, j,is isolated from the system we can write, byOhm'slaw,the relation between the currentsentering the element at the ends and the end voltages as

1i.e1re

J f = - ( V I - v, )J ; =-(v, - VI )

or in matrix form {;i)=f[-: - ; I{ ;}which in our standard form is simply

J e=K eVe (1.16)This form clearly corresponds to the stiffness relationship (1.3); indeed if an exter-

nal current were supplied along the length of the element the element 'force' termscould also be found.To assemble the whole network the continuity of the potential ( V )at the nodes isassumed and a current balance imposed there. If P,now stands for the external inputof current at node i we must have, with complete analogy to Eq. (1.1l),n nip,=c K b V , (1.17)

where the second summation is over all 'elements', and once again for all the nodesP =K V (1.18)/ = I e=l

in whichI 71

KIJ =C K ' ,e= 1


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12 Some prelim inaries: the s tandard discrete systemMatrix notation in the above has been dropped since the quantities such as voltageand current, and hence also the coefficients of the stiffness matrix, are scalars.If the resistances were replaced by fluid-carrying pipes in which a laminar regimepertained, an identical formulation would once again result, with V standing forthe hydraulic head and J for the flow.

For pipe networks that are usually encountered, however, the linear laws are ingeneral not valid. Typically the flow-head relationship is of a formJ i =C( Vi - 6) (1.19)

where the indexy lies between0.5and 0.7. Even now it would still be possibletowriterelationships in the form (1.16) noting, however, that the matrices K are no longerarrays of constants but are known functions of V. The final equations can onceagain be assembled but their form will be non-linear and in general iterative techniquesof solution will be needed.Finally it is perhaps of interest to mention the more general form of an electricalnetwork subject to an alternating current. It is customary to write the relationshipsbetween the current and voltage in complex form with the resistance being replacedby complex impedance. Once again the standard forms of (1.16)-(1.18) will beobtained but with each quantity divided into real and imaginary parts.Identical solution procedures can be used if the equality of the real and imaginaryquantities is considered at each stage. Indeed with modern digital computers it ispossible to use standard programming practice, making use of facilities availablefor dealing with complex numbers. Reference to some problems of this class will be

made in the chapter dealing with vibration problems in Chapter 17.1.6 The general pattern

An example will be considered to consolidate the concepts discussed in this chapter.This is shown in Fig. 1.4(a) where five discrete elements are interconnected. Thesemay be of structural, electrical, or any other linear type. In the solution:Thefirststep is the determination of element properties from the geometric materialand loading data. For each element the stiffness matrix as well as the correspond-ing nodal loads are found in the form of Eq. (1.3). Each element has its own iden-tifying number and specified nodal connection. For example:

element 1 connection 1 3 42 1 4 23 2 54 3 6 7 45 4 7 8 5Assuming that properties are found in global coordinates we can enter each stiff-ness or force component in its position of the global matrix as shown in Fig.1.4(b), Each shaded square represents a single coefficient or a submatrix of typeK i j if more than one quantity is being considered at the nodes. Here the separatecontribution of each element is shown and the reader can verify the position of


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The general pattern 13

Fig. 1.4 The general patternthe coefficients. Note that the various types of elementsconsidered here present nodifficulty in specification. (All forces, including nodal ones, are here associatedwith elements for simplicity.)The second step is the assembly of the final equations of the type given by Eq. (1.12).This is accomplished according to the rule of Eq. (1.13) by simple addition of allnumbers in the appropriate space of the global matrix. The result is shown inFig. 1.4(c) where the non-zero coefficients are indicated by shading.

As the matrices are symmetric only the half above the diagonal shown needs, infact, to be found.

All the non-zero coefficients are confined within a band or projile which can becalculated a priori for the nodal connections. Thus in computer programs onlythe storage of the elements within the upper half of the profile is necessary, asshown in Fig. 1.4(c).The third step is the insertion of prescribed boundary conditions into the finalassembled matrix, as discussed in Sec. 1.3. This is followed by the final step.The inal step solves the resulting equation system. Here many different methodscan be employed, some of which will be discussed in Chapter 20. The general


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14 Some preliminaries: the standard disc rete systemsubject of equation solving, though extremely important, is in general beyond thescope of this book.The final step discussed above can be followed by substitution to obtain stresses,

All operations involved in structural or other network analysis are thus of anWe can now define the standard discrete system as one in which such conditions

currents, or other desired output quantities.extremely simple and repetitive kind.prevail.

1.7 The standard discrete systemIn thestandard discrete system, whether it is structural or of any other kind, we findthat:1. A set of discrete parameters, say ai, can be identified which describes simulta-

neously the behaviour of each element, e, and of the whole system. We shall callthese thesystem parameters.2. For each element a set of quantities qf can be computed in terms of the systemparameters ai .The general function relationship can be non-linear

qf =sT(a) (1.20)but in many cases a linear form exists giving

qf =K flal +K f2az+. . .+ff3. Thesystem equations are obtained by a simple additionm...

r i =C q S

(1.21)

(1.22)e= 1

where ri are system quantities (often prescribed as zero).In the linear case this results in a system of equations

K a +f =r (1.23)such that

m m

e= 1 e= 1from which the solution for the system variables a can be found after imposingnecessary boundary conditions.The reader will observe that this definition includes the structural, hydraulic, and

electrical examples already discussed. However, it is broader. In general neitherlinearity nor symmetry of matrices need exist - although in many problems thiswill arise naturally. Further, the narrowness of interconnections existing in usualelements is not essential.While much further detail could be discussed (we refer the reader to specific booksfor more exhaustive studies in the structural context24p26),we feel that the generalexpose given here should suffice for further study of this book.


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Transform ation of coord inates 15Only one further matter relating to the change of discrete parameters need be

mentioned here. The process of so-called transformation of coordinates is vital inmany contexts and must be fully understood.

1.8 Transformationof coordinatesIt is often convenient to establish the characteristics of an individual element in acoordinate system which is different from that in which the external forces anddisplacements of the assembled structure or system will be measured. A differentcoordinate system may, in fact, be used for every element, to ease the computation.It is a simple matter to transform the coordinates of the displacement and forcecomponents of Eq. (1.3) to any other coordinate system. Clearly, it is necessary todo sobefore an assembly of the structure can be attempted.

Let the local coordinate system in which the element properties have been evalu-ated be denoted by a prime suffix and the common coordinate system necessary forassembly have no embellishment. The displacement components can be transformedby a suitable matrix of direction cosinesL as

a =L a (1.25)As the corresponding force components must perform the same amount of work in

either systemt(1.26)I T Iq a = q a

On inserting (1.25) we haveT I Tq a = q L a

orq=LTq (1.27)

The set of transformations given by (1.25) and (1.27) iscalledcontravariant.To transform stiffnesses which may be available in local coordinates to globalq =K a (1.28)

ones note that if we write

then by (1.27), (1.28), and (1.25)q=LTKLa

or in global coordinatesK =L ~K L (1.29)

The reader can verify the usefulness of the above transformations by reworkingthe sample example of the pin-ended bar, first establishing its stiffness in its lengthcoordinates.

tWith ( )T standing for the transposeof the matrix.


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16 Some prelim inaries: the standard discrete systemIn many complex problems an external constraint of some kind may be imagined,enforcing the requirement (1.25) with the number of degrees of freedom of aandabeing quite different. Even in such instances the relations (1.26) and (1.27) continueto be valid.An alternative and more general argument can be applied to many other situations

of discrete analysis. We wish to replace a set of parameters a in which the systemequations have been written by another one related to it by a transformationmatrix T asa=Tb (1.30)

In the linear case the system equations are of the formK a=r -f (1.31)

and on the substitution we haveK Tb=r -f (1.32)

The new system can be premultiplied simply by TT ,yielding(TTKT)b=TTr- TTf (1.33)

which will preserve the symmetry of equations if the matrix K is symmetric. However,occasionally the matrix T is not square and expression (1.30) represents in fact anapproximation in which a larger number of parameters a is constrained. Clearly thesystemof equations (1.32) gives more equations than are necessary for a solutionof the reduced set of parameters b, and the final expression (1.33) presents a reducedsystem which in some sense approximates the original one.We have thus introduced the basic idea of approximation, which will be the subjectof subsequent chapters where infinite setsof quantities are reduced to finite sets.A historical development of the subject of finite element methods has been pre-sented by the a~thor .~ ,~~

References1. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, 1946.2. D.N. de G. Allen. Relaxation Methods. McGraw-Hill, 1955.3. S.H. Crandall. Engineering Analysis. McGraw-Hill, 1956.4. B.A. Finlayson. The Method of Weighted Residuals and Variational Principles. Academic5. D. McHenry. A lattice analogy for the solution of plane stress problems. J . Znst. Civ. Eng. ,6. A. Hrenikoff. Solution of problems in elasticity by the framework method. J . Appl. M ech.,7. N.M. Newmark. Numerical methods of analysis in bars, plates and elastic bodies, in8. J .H. Argyris.Energy Theorems and Structural Analysis. Butterworth, 1960 (reprinted from9. M.J. Turner, R.W. Clough, H.C. Martin, and L.J . Topp. Stiffness and deflection analysis

Press, 1972.21, 59-82, 1943.AS, 169-75, 1941.Numerical Methods in Analysis in Engineering (ed. L.E. Grinter), Macmillan, 1949.Aircraft Eng., 1954-5).of complex structures. J . Aero. Sci., 23, 805-23, 1956.


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1710. R.W. Clough. The finite element in plane stress analysis. Proc. 2nd ASC E Conf.onElectro-11. Lord Rayleigh (J .W. Strutt). On the theory of resonance. Trans. Roy. SOC.London),A161,12. W. Ritz. Uber eine neue M ethode zur L osung gewissen Variations - Probleme der math-13. R. Courant. Variational methods for the solution of problems of equilibrium and vibra-14. W. Prager and J .L . Synge. Approximation in elasticity based on the concept of function15. L .F. Richardson. The approximate arithmetical solution by finite differences of physical16. H. L iebman. Die angenaherte Ermittlung: harmonischen, functionen und konformer17. R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, 1962.18. C .F . Gauss, See Carl Friedrich Gauss Werks. Vol. V II, Gottingen, 1871.19. B.G. Galerkin. Series solution of some problems of elastic equilibrium of rods and plates20. C.B. Biezeno and J .J . K och. Over een Nieuwe Methode ter Berekening van V lokke Platen.21. O.C. Zienkiewicz and Y.K. Cheung. The finite element method for analysis of elastic22. R.V . Southwell. Stress calculation in frame works by the method of systematic relaxation23. N.A. Payne and B.M . Irons, Private communication, 1963.24. R .K . L ivesley. Matrix Methods in Structural Analysis. 2nd ed., Pergamon Press, 1975.25. J .S. Przemieniecki. Theory of Matrix Structural Analysis. McGraw-Hill, 1968.26. H.C. M artin. Introduction to Matrix Methods of Structural Analysis. McGraw-Hill, 1966.27. O.C. Zienkiewicz. Origins, milestones and directions of the finite element method. Arch.Comp. Methods Eng., 2, 1-48, 1995.28. O.C. Zienkiewicz. Origins, milestones and directions of the finite element method - Apersonal view. Handbook of Numerical Analysis, I V , 3-65. Editors P.C. Ciarlet and J .L .L ions, North-Holland, 1996.

nic Computation. Pittsburgh, Pa., Sept. 1960.77-118, 1870.ematischen Physik. J . Reine Angew. Math., 135, 1-61, 1909.tion. Bull. Am. Math. SOC., 9, 1-23, 1943.space. Q. J . Appl. Math., 5 , 241-69, 1947.problems. Trans. Roy. Soc. (London),A210, 307-57, 1910.Abbildung. Sitzber. Math. Physik K1. Buyer Akad. Wiss. Miinchen. 3, 65-75, 1918.

(Russian). Vestn. Inzh. Tech., 19, 897-908, 1915.I g . Grav.,38,25-36, 1923.isotropic and orthotropic slabs. Proc. Inst. Civ. Eng., 28,471-488, 1964.of constraints, Part I & 11. Proc. Roy. SOC. ondon (A),151, 56-95, 1935.

The Standard Desrete System

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