Some preliminaries: the standard discrete system 1.1 Introduction The limitations of the human mind are such that it cannot grasp the behaviour of its complex surroundings and creations in one operation. Thus the process of subdividing all systems into their individual components or ‘elements’, whose behaviour is readily understood, and then rebuilding the original system from such components to study its behaviour is a natural way in which the engineer, the scientist, or even the economist proceeds. In many situations an adequate model is obtained using a finite number of welldefined components. We shall term such problems discrete. In others the subdivision is continued indefinitely and the problem can only be defined using the mathematical fiction of an infinitesimal. This leads to differential equations or equivalent statements which imply an infinite number of elements. We shall term such systems continuous. With the advent of digital computers, discrete problems can generally be solved readily even if the number of elements is very large. As the capacity of all computers is finite, continuous problems can only be solved exactly by mathematical manipulation. Here, the available mathematical techniques usually limit the possibilities to oversimplified situations. To overcome the intractability of realistic types of continuum problems, various methods of discretization have from time to time been proposed both by engineers and mathematicians. All involve an approximation which, hopefully, approaches in the limit the true continuum solution as the number of discrete variables increases. The discretization of continuous problems has been approached differently by mathematicians and engineers. Mathematicians have developed general techniques applicable directly to differential equations governing the problem, such as finite difference approximations,’,2 various weighted residual p r o c e d ~ r e s , or ~ . ~approximate techniques for determining the stationarity of properly defined ‘functionals’. The engineer, on the other hand, often approaches the problem more intuitively by creating an analogy between real discrete elements and finite portions of a continuum domain. For instance, in the field of solid mechanics McHenry,’ Hrenikoff,6 Newmark7, and indeed Southwel19 in the 1940s, showed that reasonably good solutions to an elastic continuum problem can be obtained by replacing small portions
2
Some preliminaries: the standard discrete system
of 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, substitution of properties can be made much more effectively by considering that small portions or ‘elements’ in a continuum behave in a simplified manner. It is from the engineering ‘direct analogy’ view that the term ‘finite element’ was born. Clough” appears to be the first to use this term, which implies in it a direct use of a standard methodology applicable to discrete systems. Both conceptually and from the computational viewpoint, this is of the utmost importance. The first allows an improved understanding to be obtained; the second offers a unified approach to the variety of problems and the development of standard computational procedures. Since the early 1960s much progress has been made, and today the purely mathematical and ‘analogy’ approaches are fully reconciled. It is the object of this text to present a view of the finite element method as a general discretizationprocedure of continuum problems posed by mathematically dejined statements. In the analysis of problems of a discrete nature, a standard methodology has been developed over the years. The civil engineer, dealing with structures, first calculates force-displacement relationships for each element of the structure and then proceeds to assemble the whole by following a well-defined procedure of establishing local equilibrium at each ‘node’ or connecting point of the structure. The resulting equations can be solved for the unknown displacements. Similarly, the electrical or hydraulic engineer, dealing with a network of electrical components (resistors, capacitances, etc.) or hydraulic conduits, first establishes a relationship between currents (flows) and potentials for individual elements and then proceeds to assemble the system by ensuring continuity of flows. All such analyses follow a standard pattern which is universally adaptable to discrete systems. It is thus possible to define a standard discrete system, and this chapter will 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 at this stage is advisable. As the treatment of elastic solid structures has been the most developed area of activity this will be introduced first, followed by examples from other fields, before attempting a complete generalization. The existence of a unified treatment of ‘standard discrete problems’ leads us to the first definition of the finite element process as a method of approximation to continuum problems such that (a) the continuum is divided into a finite number of parts (elements), the behaviour of which is specified by a finite number of parameters, and (b) the solution of the complete system as an assembly of its elements follows precisely the same rules as those applicable to standard discrete problems. It will be found that most classical mathematical approximation procedures as well as the various direct approximations used in engineering fall into this category. It is thus difficult to determine the origins of the finite element method and the precise moment of its invention. Table 1.1 shows the process of evolution which led to the present-day concepts of finite element analysis. Chapter 3 will give, in more detail, the mathematical basis which emerged from these classical ideas. 1-20
r-
1
4 Some preliminaries: the standard discrete system
1.2 The structural element and the structural system
4
v3
Atypical element (1)
Fig. 1.1 A typical structure built up from interconnected elements
To introduce the reader to the general concept of discrete systems we shall first consider a structural engineering example of linear elasticity. Figure 1.1 represents a two-dimensional structure assembled from individual components and interconnected at the nodes numbered 1 to 6. The joints at the nodes, in this case, are pinned so that moments cannot be transmitted. As a 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 precisely known. Thus, if a typical element labelled (1) and associated with nodes 1, 2, 3 is examined, the forces acting at the nodes are uniquely defined by the displacements of these nodes, the distributed loading acting on the element ( p ) , and its initial strain. The last may be due to temperature, shrinkage, or simply an initial ‘lack of fit’. The forces and the corresponding displacements are defined by appropriate components ( 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
t A limited knowledge of matrix algebra will be assumed throughout this book. This is necessary for reasonable conciseness and forms a convenient book-keeping form. For readers not familiar with the subject a brief appendix (Appendix A) is included in which sufficient principles of matrix algebra are given to follow the development intelligently. Matrices (and vectors) will be distinguished by bold print throughout.
The structural element and the structural system 5
and for the corresponding nodal displacements
Assuming linear elastic behaviour of the element, the characteristic relationship will always be of the form q1 = K'a'
+ f j +fro
in which f j represents the nodal forces required to balance any distributed loads acting on the element and fro the nodal forces required to balance any initial strains such as may 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 of stresses or internal reactions at any specified point or points of the element in terms of the nodal displacements. Defining such stresses by a matrix c1a relationship of the form 1
0
=
1 1
~ +cro a
(1.4)
is obtained in which the two term gives the stresses due to the initial strains when no nodal displacement occurs. The matrix Ke is known as the element stiffness matrix and the matrix Q' as the element stress matrix for an element (e). Relationships in Eqs (1.3) and (1.4) have been illustrated by an example of an element with three nodes and with the interconnection points capable of transmitting only two components of force. Clearly, the same arguments and definitions will apply generally. An element (2) of the hypothetical structure will possess only two points of interconnection; others may have quite a large number of such points. Similarly, if the joints were considered as rigid, three components of generalized force and of generalized displacement would have to be considered, the last of these corresponding to a moment and a rotation respectively. For a rigidly jointed, three-dimensional structure the number of individual nodal components would be six. Quite generally, therefore,
with each q; and ai possessing the same number of components or degrees of freedom. These quantities are conjugate to each other. The stiffness matrices of the element will clearly always be square and of the form
6 Some preliminaries: the standard discrete system
t Y
Fig. 1.2 A pin-ended bar.
in which KZ., etc., are submatrices which are again square and of the size E x 1, where 1 is the number of force components to be considered at each node. As an example, the reader can consider a pin-ended bar of uniform section A and modulus E in a two-dimensional problem shown in Fig. 1.2. The bar is subject to a uniform lateral load p and a uniform thermal expansion strain Eo
= aT
where a is the coefficient of linear expansion and T is the temperature change. If the ends of the bar are defined by the coordinates x i ,y iand x, yn its length can be calculated as
and its inclination from the horizontal as /3 = tan- 1 Y n - Yi ~
xn - xi
Only two components of force and displacement have to be considered at the nodes. The nodal forces due to the lateral load are clearly
and represent the appropriate components of simple reactions, p L / 2 . Similarly, to restrain the thermal expansion an axial force ( E a T A ) is needed, which gives the
The structural element and the structural system 7
components
f'€0 =
{ -{
-cos
p
-sinp](EaTA) cos p sin p
Finally, the element displacements
-
cos2p sin p cos ,O
sinpcosp sin2p -sinpcosp -sin 2p
-sin pcos p
I 1 I I I
-cos2p -sinpcosp cos2p sinpcosp
-sin p cos p -sin2p
-
sin p cos p sin2p -
{ k)
The components of the general equation (1.3) have thus been established for the elementary case discussed. It is again quite simple to find the stresses at any section of the element in the form of relation (1.4). For instance, if attention is focused on the mid-section C of the bar the average stress determined from the axial tension to the element can be shown to be b c' M c = - [-cos
L
p, -sin p, cos p, sin P]ae - E a T
where all the bending effects of the lateral load p have been ignored. For more complex elements more sophisticated procedures of analysis are required but the results are of the same form. The engineer will readily recognize that the socalled 'slope-deflection' relations used in analysis of rigid frames are only a special case of the general relations. It may perhaps be remarked, in passing, that the complete stiffness matrix obtained for the simple element in tension turns out to be symmetric (as indeed was the case with some submatrices). This is by no means fortuitous but follows from the principle of energy conservation and from its corollary, the well-known Maxwell-Betti reciprocal theorem.
8 Some preliminaries: the standard discrete system
The element properties were assumed to follow a simple linear relationship. In principle, similar relationships could be established for non-linear materials, but discussion of such problems will be held over at this stage. The calculation of the stiffness coefficients of the bar which we have given here will be found in many textbooks. Perhaps it is worthwhile mentioning here that the first use of bar assemblies for large structures was made as early as 1935 when Southwell proposed his classical relaxation method.22
1.3 Assembly and analysis of a structure Consider again the hypothetical structure of Fig. 1.1. To obtain a complete solution the two conditions of (a) displacement compatibility and (b) equilibrium have to be satisfied throughout. Any system of nodal displacements a:
a=
{:}
an
listed now for the whole structure in which all the elements participate, automatically satisfies the first condition. As the conditions of overall equilibrium have already been satisfied within an element, all that is necessary is to establish equilibrium conditions at the nodes of the structure. The resulting equations will contain the displacements as unknowns, and once these have been solved the structural problem is determined. The internal forces in elements, or the stresses, can easily be found by using the characteristics established 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 individual elements. Again, any one of the forces ri must have the same number of components as 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 arbitrary number of components will be assumed. If now the equilibrium conditions of a typical node, i, are to be established, each component of ri has, in turn, to be equated to the sum of the component forces contributed by the elements meeting at the node. Thus, considering all the force
The boundary conditions 9
components 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 noting that nodal variables ai are common (thus omitting the superscript e), we have (1.11) where
f e = f;
+ fZ0
The summation again only concerns the elements which contribute to node i. If all such equations are assembled we have simply
Ka=r-f
(1.12)
in which the submatrices are m
e=l
m
(1.13)
fi = xf: e= 1
with summations including all elements. This simple rule for assembly is very convenient because as soon as a coefficient for a particular element is found it can be put immediately into the appropriate ‘location’ specified in the computer. This general assembly process can be found to be the common and fundamental feature of alljinite 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 be remembered that the rules of matrix summation permit this to be done only if these are of identical size. The individual submatrices to be added have therefore to be 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 be coupled at that node to one which in fact is hinged, it is necessary to complete the stiffness matrix of the latter by insertion of appropriate (zero) coefficients in the rotation or moment positions.
1.4 The boundary conditions The system of equations resulting from Eq. (1.12) can be solved once the prescribed 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
10 Some preliminaries: the standard discrete system
the substitution of
al = a6 =
{ :}
which is equivalent to reducing the number of equilibrium equations (in this instance 12) by deleting the first and last pairs and thus reducing the total number of unknown displacement components to eight. It is, nevertheless, always convenient to assemble the equation according to relation (1.12) so as to include all the nodes. Clearly, without substitution of a minimum number of prescribed displacements to prevent 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 a situation. This physically obvious fact will be interpreted mathematically as the matrix K being singular, i.e., not possessing an inverse. The prescription of appropriate displacements after the assembly stage will permit a unique solution to be obtained by deleting appropriate rows and columns of the various matrices. If all the equations of a system are assembled, their form is
+ + Kzlal + KZ2a2+ . . . = r2 - f2
Kllal K12a2 . . . = rl - fl (1.14)
etc. and it will be noted that if any displacement, such as al = a l , is prescribed then the external ‘force’ rl cannot be simultaneously specified and remains unknown. The first equation could then be deleted and substitution of known values of al made in the remaining equations. This process is computationally cumbersome and the same objective is served by adding a large number, aI, to the coefficient K l l and replacing the right-hand side, rl - f l , by ala. If a is very much larger than other stiffness coefficients this alteration effectively replaces the first equation by the equation
aal = aal
(1.15)
that is, the required prescribed condition, but the whole system remains symmetric and minimal changes are necessary in the computation sequence. A similar procedure will apply to any other prescribed displacement. The above artifice was introduced by Payne and Irons.23 An alternative procedure avoiding the assembly of equations corresponding to nodes with prescribed boundary values will be presented in Chapter 20. When all the boundary conditions are inserted the equations of the system can be solved for the unknown displacements and stresses, and the internal forces in each element obtained.
1.5 Electrical and fluid networks Identical principles of deriving element characteristics and of assembly will be found in many non-structural fields. Consider, for instance, the assembly of electrical resistances shown in Fig. 1.3.
Electrical and fluid networks
Fig. 1.3 A network of electrical resistances.
If a typical resistanceelement, ij, is isolated from the system we can write, by Ohm's law, the relation between the currents entering the element at the ends and the end voltages as 1 Jf = - ( V I -
v,)
i.e
J;
1 =-(v, re
- VI)
or in matrix form
{;i)=f[-:
-;I{
;}
which in our standard form is simply Je = KeVe
(1.16)
This form clearly corresponds to the stiffness relationship (1.3); indeed if an external current were supplied along the length of the element the element 'force' terms could also be found. To assemble the whole network the continuity of the potential ( V ) at the nodes is assumed and a current balance imposed there. If P, now stands for the external input of current at node i we must have, with complete analogy to Eq. (1.1 l),
=c n
p,
ni
CKbV,
/=I
(1.17)
e=l
where the second summation is over all 'elements', and once again for all the nodes P=KV (1.18) in which I71
KIJ =
CK', e= 1
11
12 Some preliminaries: the standard discrete system
Matrix notation in the above has been dropped since the quantities such as voltage and 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 regime pertained, an identical formulation would once again result, with V standing for the hydraulic head and J for the flow. For pipe networks that are usually encountered, however, the linear laws are in general not valid. Typically the flow-head relationship is of a form
Ji = C ( Vi -
6)’
(1.19)
where the index y lies between 0.5 and 0.7. Even now it would still be possible to write relationships in the form (1.16) noting, however, that the matrices K‘ are no longer arrays of constants but are known functions of V. The final equations can once again be assembled but their form will be non-linear and in general iterative techniques of solution will be needed. Finally it is perhaps of interest to mention the more general form of an electrical network subject to an alternating current. It is customary to write the relationships between the current and voltage in complex form with the resistance being replaced by complex impedance. Once again the standard forms of (1.16)-(1.18) will be obtained but with each quantity divided into real and imaginary parts. Identical solution procedures can be used if the equality of the real and imaginary quantities is considered at each stage. Indeed with modern digital computers it is possible to use standard programming practice, making use of facilities available for 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. These may be of structural, electrical, or any other linear type. In the solution:
Thefirst step is the determination of element properties from the geometric material and loading data. For each element the ‘stiffness matrix’ as well as the corresponding ‘nodal loads’ are found in the form of Eq. (1.3). Each element has its own identifying number and specified nodal connection. For example: element
1 2 3 4 5
connection
1 1 2 3 4
3 4 5 6 7
4 2
7 8
4 5
Assuming that properties are found in global coordinates we can enter each ‘stiffness’ 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 type K i j if more than one quantity is being considered at the nodes. Here the separate contribution of each element is shown and the reader can verify the position of
The general pattern
Fig. 1.4 The general pattern
the coefficients. Note that the various types of ‘elements’considered here present no difficulty in specification. (All ‘forces’, including nodal ones, are here associated with 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 all numbers in the appropriate space of the global matrix. The result is shown in Fig. 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, in fact, to be found. All the non-zero coefficients are confined within a band or projile which can be calculated a priori for the nodal connections. Thus in computer programs only the storage of the elements within the upper half of the profile is necessary, as shown in Fig. 1.4(c). The third step is the insertion of prescribed boundary conditions into the final assembled matrix, as discussed in Sec. 1.3. This is followed by the final step. The jinal step solves the resulting equation system. Here many different methods can be employed, some of which will be discussed in Chapter 20. The general
13
14 Some preliminaries: the standard discrete system
subject of equation solving, though extremely important, is in general beyond the scope of this book. The final step discussed above can be followed by substitution to obtain stresses, currents, or other desired output quantities. All operations involved in structural or other network analysis are thus of an extremely simple and repetitive kind. We can now define the standard discrete system as one in which such conditions prevail.
1.7 The standard discrete system In the standard discrete system, whether it is structural or of any other kind, we find that: 1. A set of discrete parameters, say ai, can be identified which describes simultaneously the behaviour of each element, e, and of the whole system. We shall call these the system parameters. 2. For each element a set of quantities qf can be computed in terms of the system parameters ai. The general function relationship can be non-linear
qf = sT(a)
(1.20)
but in many cases a linear form exists giving qf = Kflal Kf2az+ . . . + f f
+
(1.21)
3. The system equations are obtained by a simple addition
... m
ri = C q S
(1.22)
e= 1
where ri are system quantities (often prescribed as zero). In the linear case this results in a system of equations Ka+f=r
(1.23)
such that m
m
e= 1
e= 1
from which the solution for the system variables a can be found after imposing necessary boundary conditions. The reader will observe that this definition includes the structural, hydraulic, and electrical examples already discussed. However, it is broader. In general neither linearity nor symmetry of matrices need exist - although in many problems this will arise naturally. Further, the narrowness of interconnections existing in usual elements is not essential. While much further detail could be discussed (we refer the reader to specific books for more exhaustive studies in the structural context24p26),we feel that the general expose given here should suffice for further study of this book.
Transformation of coordinates 15
Only one further matter relating to the change of discrete parameters need be mentioned here. The process of so-called transformation of coordinates is vital in many contexts and must be fully understood.
1.8 Transformation of coordinates It is often convenient to establish the characteristics of an individual element in a coordinate system which is different from that in which the external forces and displacements of the assembled structure or system will be measured. A different coordinate 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 force components of Eq. (1.3) to any other coordinate system. Clearly, it is necessary to do so before an assembly of the structure can be attempted. Let the local coordinate system in which the element properties have been evaluated be denoted by a prime suffix and the common coordinate system necessary for assembly have no embellishment. The displacement components can be transformed by a suitable matrix of direction cosines L as a’ = La
(1.25)
As the corresponding force components must perform the same amount of work in either systemt qTa = q I TaI
(1.26)
On inserting (1.25) we have qTa = q I TLa or q = LTq’
(1.27)
The set of transformations given by (1.25) and (1.27) is called contravariant. To transform ‘stiffnesses’ which may be available in local coordinates to global ones note that if we write q’ = K’a’
( 1.28)
then by (1.27), (1.28), and (1.25) q = LTK’La
or in global coordinates
K =L~K’L
(1.29)
The reader can verify the usefulness of the above transformations by reworking the sample example of the pin-ended bar, first establishing its stiffness in its length coordinates. t With (
)T
standing for the transpose of the matrix.
16 Some preliminaries: the standard discrete system
In 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 a and a’ being quite different. Even in such instances the relations (1.26) and (1.27) continue to 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 system equations have been written by another one related to it by a transformation matrix T as
a = Tb
(1.30)
In the linear case the system equations are of the form
Ka=r-f
(1.31)
KTb=r-f
(1.32)
and on the substitution we have
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 an approximation in which a larger number of parameters a is constrained. Clearly the system of equations (1.32) gives more equations than are necessary for a solution of the reduced set of parameters b, and the final expression (1.33) presents a reduced system which in some sense approximates the original one. We have thus introduced the basic idea of approximation, which will be the subject of subsequent chapters where infinite sets of quantities are reduced to finite sets. A historical development of the subject of finite element methods has been presented by the a ~ t h o r . ~ ’ , ~ ~
References 1. 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. Academic Press, 1972. 5. D. McHenry. A lattice analogy for the solution of plane stress problems. J. Znst. Civ. Eng., 21, 59-82, 1943. 6. A. Hrenikoff. Solution of problems in elasticity by the framework method. J. Appl. Mech., AS, 169-75, 1941. 7. N.M. Newmark. Numerical methods of analysis in bars, plates and elastic bodies, in Numerical Methods in Analysis in Engineering (ed. L.E. Grinter), Macmillan, 1949. 8. J.H. Argyris. Energy Theorems and Structural Analysis. Butterworth, 1960 (reprinted from Aircraft Eng., 1954-5). 9. M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp. Stiffness and deflection analysis of complex structures. J. Aero. Sci., 23, 805-23, 1956.
17 10. R.W. Clough. The finite element in plane stress analysis. Proc. 2nd ASCE Conf. on Electronic Computation. Pittsburgh, Pa., Sept. 1960. 11. Lord Rayleigh (J.W. Strutt). On the theory of resonance. Trans. Roy. SOC.(London),A161, 77-118, 1870. 12. W. Ritz. Uber eine neue Methode zur Losung gewissen Variations - Probleme der mathematischen Physik. J . Reine Angew. Math., 135, 1-61, 1909. 13. R. Courant. Variational methods for the solution of problems of equilibrium and vibration. Bull. Am. Math. SOC.,49, 1-23, 1943. 14. W. Prager and J.L. Synge. Approximation in elasticity based on the concept of function space. Q. J . Appl. Math., 5, 241-69, 1947. 15. L.F. Richardson. The approximate arithmetical solution by finite differences of physical problems. Trans. Roy. Soc. (London), A210, 307-57, 1910. 16. H. Liebman. Die angenaherte Ermittlung: harmonischen, functionen und konformer Abbildung. Sitzber. Math. Physik K1. Buyer Akad. Wiss. Miinchen. 3, 65-75, 1918. 17. R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, 1962. 18. C.F. Gauss, See Carl Friedrich Gauss Werks. Vol. VII, Gottingen, 1871. 19. B.G. Galerkin. Series solution of some problems of elastic equilibrium of rods and plates’ (Russian). Vestn. Inzh. Tech., 19, 897-908, 1915. 20. C.B. Biezeno and J.J. Koch. Over een Nieuwe Methode ter Berekening van Vlokke Platen. I g . Grav., 38, 25-36, 1923. 21. O.C. Zienkiewicz and Y.K. Cheung. The finite element method for analysis of elastic isotropic and orthotropic slabs. Proc. Inst. Civ. Eng., 28, 471-488, 1964. 22. R.V. Southwell. Stress calculation in frame works by the method of systematic relaxation of constraints, Part I & 11. Proc. Roy. SOC.London ( A ) , 151, 56-95, 1935. 23. N.A. Payne and B.M. Irons, Private communication, 1963. 24. R.K. Livesley. 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. Martin. 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 - A personal view. Handbook of Numerical Analysis, IV, 3-65. Editors P.C. Ciarlet and J.L. Lions, North-Holland, 1996.
