John Wallis and the Numerical Analysis of Structures
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Abstract
In 1695, in his Opera Mathematica, John Wallis (Savilian Professor of Geometry at the University of Oxford) published an analysis of the forces in a reciprocal grillage structure. The structure itself was an extended version of a famous design often attributed to Serlio, but first sketched much earlier by Villard de Honnecourt. The extended structure was also sketched by Leonardo da Vinci. Wallis’s analysis is remarkable for the fact that he systematically (and correctly) solved a set of 25 simultaneous equations to obtain the required forces. The main steps in the analysis closely parallel the essential stages in modern Finite Element Analysis, and Wallis’s calculations can be seen as a key step in the development of structural analysis techniques.
Keywords
Reciprocal structures Numerical analysis Structural mechanics John Wallis Christopher WrenPreamble: a Classical Structural Problem
Many accidents like unto this may fall to a workman’s hand, which is, that a man should lay a ceiling of a house in a place which is fifteen foote long, and as many foote broad, & the rafters should be but fourteen foote long, and no more wood to be had… [Serlio 1611: fol. 12r (reprinted in Serlio, 1982)].
This simple form was also sketched by Leonardo da Vinci, in the Codex Madrid I (ca. 1490–96, conserved in the Biblioteca Nacional de España, Madrid, Spain).^{2} In the Codex Atlanticus (ca. 1480–1518, conserved in the Biblioteca Ambrosiana, Milan, Italy), Leonardo da Vinci also drew a much more complex structure in which the fourbeam pattern was repeated many times to form a structure that can in principle be extended indefinitely.^{3} At least in principle, he had solved the problem of how to span an opening of arbitrary size with beams of fixed length much smaller than the span.
John Wallis
It is an intriguing question whether Wallis had any knowledge of Leonardo da Vinci’s drawing, and in turn whether Leonardo da Vinci was aware of Villard de Honnecourt’s. It is likely that neither of the authors was aware of the earlier work, but that is not our concern here. Wallis set himself the problems of (a) presenting a solution which allowed a space to be spanned by beams all of which were much shorter than the span and (b) calculating all the forces in the structure, assuming that these arise from the weight of the beams themselves. He was therefore addressing exactly the same type of problem that faces structural engineers today. There are two important omissions from his analysis: firstly he considered only the forces and not the deflections of the structure, which would now be regarded as of equal importance. Secondly, he considered only the forces due to the weight of the beams (in modern terms the dead load), neglecting the additional forces that would arise from the floor loading (live load).
I did first, saith the Doctor, contrive and delineate it, in the year 1644, at Queen’scollege in Cambridge. When afterwards I was made Professor of geometry at Oxford, about the Year 1650, I caused it to be framed of small Pieces of Timber, prepared by a Joiner, and put together by myself (Wren 1750: 338).
He is known to have lectured on its construction and the analysis of the forces in the floor in 1652 and 1653. He had a second model made, and presented to the King, in 1660, and again lectured on the topic in 1669 and published his analysis in 1670. The most accessible description is, however, found in his collected works, Opera Mathematica (1695). What was novel about his contribution is not just his description of the structure, and some alternative designs that also exploit reciprocity, but his mathematical analysis of the forces in the structure. It is possible that his analysis represents the first recorded example of a comprehensive structural analysis of a nontrivial structure. Furthermore, the elements of his analysis have many useful parallels with modern, computerbased, numerical analysis of structures.
Reciprocal Structures
First of all we clarify the meaning of a “reciprocal” structure.^{4} By reciprocal we mean that all the members in the structure depend on each other for mutual support. No substructure is stable until the entire structure is complete, and furthermore the removal of a single structural element renders every part of the structure mobile with respect to every other part. This feature is clear in the fourbeam structure in Fig. 2.
Reciprocal structures are a subclass of “statically determinate” structures, i.e., those for which all the forces may be determined by the equations of statics alone. Large, complex statically determinate structures are relatively rare—larger structures are much more often “statically indeterminate”, i.e., they include redundant members, and determination of the forces in them requires knowledge of material properties and the use of compatibility conditions as well as the equations of statics. It was the special feature of statical determinacy that made Wallis’s grillage amenable to analysis in the seventeenth century.
Secondly, we need to debunk a myth. Wallis’s grillage structure is often, but erroneously, referred to as his design for the ceiling of the Sheldonian Theatre in Oxford. There is, however, no known evidence to support this contention. It seems to have arisen because a reference to Wallis appears in Parentalia (1750) by Christopher Wren (Junior), immediately following a discussion of the designs by Wren (Senior) for the roof trusses for the Sheldonian.^{5} The Sheldonian was begun in 1664, but as discussed above, Wallis was already considering his structure in 1650. Wren’s solution for the Sheldonian was to support the ceiling from trusses, whilst Wallis’s analysis is of a flat grillage.
Wallis’s Analysis

Presentation of a clear drawing of the structure, in which the joints are labelled;

Exploitation of a fourfold symmetry to simplify the subsequent analysis:

An analysis based on the systematic application of a simple principle (moment equilibrium of each beam) to all the members in the structure, resulting in 25 simultaneous linear equations:

A systematic elimination between the equations to solve for the 25 variables:

From the result obtained, extraction of important structural conclusions about loads on salient members in the structure.
Of course, many earlier authors had provided diagrams of structures, and labelling of diagrams with letters for reference in the accompanying text was already commonplace, but Wallis’s labelling of his frame goes a step further, and is more akin to the systematic numbering of nodes in a modern structural analysis. Leaving aside the supports (which Wallis does not label) there are a hundred joints in the structure, but Wallis cleverly exploits a fourfold rotational symmetry to identify just twentyfive independent quantities: the downward forces exerted from the end of each beam on its neighbour. These he identifies with the symbols A…Z and & (he omits J, and uses T for the weight of the longer beams).
Many analysts would choose to write the moment equation about one end of the beam together with the vertical equilibrium equation, rather than the moment equations about both ends of the beam. Wallis’s choice of the latter perhaps reveals his very systematic mathematical approach.
It is at this point that he was in greatest danger of making a mistake. Taking the longer beams to be of unit length, the shorter beams would be of length 2/3 and weight 2T/3. If he had adopted exactly the same procedure as for the longer beams, one would expect the first of his short beam equations to read \( \frac{2}{3}\;W\; = \;\frac{1}{3}\;\left( {\frac{2}{3}T} \right)\; + \;\frac{1}{3}\;V \), but Wallis multiplies by 3/2 and writes instead \( W\; = \;\frac{1}{3}\;T\; + \;\frac{1}{2}\;V \). He has thus combined two operations in a single stage of working; one cannot help guessing that he must have worked through an intermediate stage that is omitted in the published version.
His solution, which was expressed in fractions, was checked by Halliwell (2001) using Mathematica^{®}, and confirmed as correct. His correct solution of 25 simultaneous linear equations is a remarkable feat in itself, as anyone who has solved simultaneous equations by hand will attest. It is also remarkable because of the underlying “reciprocal” nature of the structure. Mathematically, this results in a set of equations in which solutions for individual variables cannot be picked off one at a time. It is not possible to solve for any variable until all the equations have been combined, and then an elaborate backsubstitution exercise is necessary to solve for the rest.
Having obtained his solution, Wallis then picked out the most heavily loaded beam, which is in fact the one closest to the centre (of course there are four such beams because of symmetry). One should contrast this with Leonardo da Vinci’s assumption that the most heavily loaded beams are at the periphery of the structure, an assumption based on an incorrect intuition rather than on an analysis. Sadly, however, Wallis’s analysis breaks down at this stage. In modern terms we would identify either the maximum shear force or (more likely) the maximum bending moment in any beam as being the critical value for design. It can easily be shown that the maximum shear force in the central beam is 9.08T, and that the maximum bending moment is 2.97TL (if the weight of each beam is assumed to be concentrated at its centroid) or 3.03TL (if the weight is more correctly assumed to be uniformly distributed along each beam). However, Wallis simply calculates a force of \( 13\frac{167771}{453556}\;T \), which is equal to the supporting force at the more heavily loaded end (A) plus half the supporting force at the other end (B). Quite why Wallis felt that this quantity was critical is not clear; he refers back to a previous section in his book, but the relevance of that section is not readily apparent. Perhaps, however, we can forgive Wallis the fact that he fell at this final hurdle.
Parallels with Modern Analysis
It is worth recapitulating each of the stages of Wallis’s analysis, and comparing these to a modern structural analysis using the Finite Element method. Until relatively recent software developments, introducing sophisticated graphical interfaces which hide many of these stages from the user, each of these steps would be immediately recognisable to a modern structural analyst.
For those not familiar with Finite Element method we provide a very brief description here. Finite Element Analysis (FEA) has become the standard method for the analysis of most structural and mechanical problems. It involves breaking a structure down into small “elements” each of a simple shape, for instance a cuboid. The mathematical equations defining the stresses and strains within each element are approximated in a way that requires just the forces and displacements at certain “nodes” (typically the corners of the elements) to be known. The connections between the elements are established by “assembling” the equations into a single large system of linear equations, expressed in matrix form. The method was first applied to problems in which the structure was entirely elastic, with the advantage that all the necessary equations are linear. It has since been extended to cope with nonlinear material properties and a host of other complicating factors. Indeed it is the versatility of FEA that is one of its most attractive features. There are other numerical techniques that are used for specialised applications, but none is as versatile and adaptable to different materials, different geometries and different loading conditions. At its heart FEA depends on some simple steps: the breaking down of the problem into small repeating components; the mathematical description of the mechanics of each of these standard components; the assembly of the resulting equations into a complete system; the solution of the equations and the presentation of useful results.
We now pursue the parallels between Wallis’s analysis and the key steps in modern FEA.
First of all Wallis recognised that he could use symmetry to reduce the variables he had to solve for from a hundred to just twentyfive. Structural engineers typically use very similar tricks today, although we note that rotational symmetries can be rather more difficult to deal with than mirror symmetries. When using FEA, analysts will often divide a symmetric problem along the plane of symmetry, solving for the variables in just half the problem, with resulting savings in computing time. Modern analysts also often reduce threedimensional problems to simplified twodimensional sections.
Next, Wallis labelled the joints between the beams, using 25 letters for these. This is just like the numbering system for nodes in a finite element analysis; in the early years of finite element analysis this was a laborious task carried out by hand by the analyst. These days it is hidden from the user behind sophisticated meshgeneration software, but the need for a numbering system nevertheless underpins the method.
Wallis’s identification of a typical beam (Fig. 4) and his systematic writing down of the equations for all the beams in the same way has perhaps the most striking parallel with finite element analysis. It is standard practice (see, for instance, any textbook on Finite Element Analysis) to derive the equations governing standard elements, later combined into a complete analysis. Because Wallis was examining a statically determinate problem, he only needed the equilibrium equations. In modern structural analyses these are combined with the constitutive model and the compatibility equations to determine an element stiffness matrix, but Wallis’s analysis does not contain this aspect. At the time he was writing the underpinning concepts required to develop a satisfactory theory for the bending of beams were beginning to emerge, but it was not Claude Navier published his Leçons (1826) that the full and correct solution to the elastic bending problem was given.
Wallis combined his equations by hand, carefully substituting them into each other to eliminate variables one at a time. The modern equivalent is the “assembly” of the finite element equations and the solution process using matrix methods. Much research has of course been devoted to the efficient solution of the equations, and no doubt this problem preoccupied Wallis too. The efficiency of a modern analysis can be greatly improved by numbering of the nodes in such a way as to minimise the “bandwidth” of the stiffness matrix. In a similar way, Wallis did not simply combine his equations in some random order, but clearly picked his way through them carefully, minimising his effort. We may speculate that he tried various strategies to obtain his solution, only publishing the most satisfactory one.
His analysis revealed the solutions for the variables in a somewhat inconvenient order, but Wallis then reordered the solution to present them in a usable form by the reader—a simple example of the “postprocessing” that is done today to present the engineer with results in a way that is easily assimilated, usually these days in some graphical form.
As mentioned above, it is unfortunate that Wallis’s final step involves a calculation for which we can see no modern justification. Nevertheless, his purpose was clear: to identify the element with the greatest load and subject that element alone to more detailed analysis, exactly as an analyst would do today.
Of course Wallis’s analysis does not contain every element of a modern analysis—there is no elasticity, no matrix methods and no analysis of stresses as opposed to forces. However, it contains sufficient of the elements of a modern analysis that it may be regarded as the first numerical analysis of a structure of any significant complexity, and he should be acknowledged as a pioneer of structural analysis.
Footnotes
 1.
As an aside, however, it is important to note that Serlio’s design cannot actually be constructed as the beams could not in fact be interwoven in the way he depicts.
 2.
The Leonardo da Vinci reference to the Codex Madrid is Folio 49 verso (visible on page 101 of the online resource http://leonardo.bne.es/index.html).
 3.
The reference to the Codex Atlanticus is Folio 899 verso (visible at http://www.leonardodigitale.com/Sfoglia/Menu.swf?direct=1&codex=Atlantico&folio=0001r).
 4.
For a more detailed discussion of reciprocity see (Pugnale and Sassone 2014).
 5.
The trusses devised by Wren for the Sheldonian are interesting in themselves although unrelated to Wallis’s work, in that they are variations of kingpost trusses whose long bottom chords were composed of shorter pieces of timber joined together by means of bolts and scarf joints (“… lockages being so quite different from any beforementioned”); see (Wren 1750: 335–339).
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