Abstract: A somewhat unusual type of drawbridge was quite widely used in New South Wales in the first quarter of this century. In this design, the moving span is balanced by a counterweight running on a track that is, to a fair approximation, a cardiodal arc. The designer was Henry Harvey Dare who had a long and distinguished career in the NSW Public Works Department.
Figure 1 is a redrawing of a diagram in an 18th-Century book of military engineering. The author of that work was Bernard Forest de B\'elidor, a mathematician and engineer. The picture concerns the design of a drawbridge mechanism, which achieves the twin aims of making the raising and lowering of the movable span as easy as possible and also the putting as much as the mechanism as possible within the fortress (i.e. to the left of BE), rather than having it protrude and thus be vulnerable to attack. This same general idea is also seen (rather crudely) implemented in a 9th-Century structure in Corsica [23]. Such bridges are now referred to as ``B\'elidor bascules''. (``Bascule'' is the technical term applying to drawbridges.) \midspace{9.5cm} \caption{ Fig. 1. The design of the curved-track bascule bridge (after de B\'elidor). }
The simplest type of bascule bridge is the so-called trunnion (or fixed pivot) type, in which the movable span is hinged at one end. (For details of other types, see Waddell [22, vol. 1, p.701], Hovey [14] or Hool and Kinne [13].) A natural design would in this instance use a cable to raise and lower the span, with a counterweight at the other end to balance the weight of the span. But then, ``if the counterweight remains constant on a vertical axis, it rapidly overcomes the weight of the opening span which eventually slams up against the tower [supporting the cable, and then there is also] the problem of forcing the bridge to close against the excessive counter-balance'' [10].
de B\'elidor's solution (one among many possible) to this difficulty was to run a counterweight down a curved track. See again Figure 1. His book La Science des Ing\'enieurs [4], in which this design was mooted, was obviously influential. It was first published in 1729, reprinted frequently, and ran to at least five editions, including one with notes by Navier (issued in 1813, after its author's death and yet again reprinted in 1830). Clearly it was much in demand (see e.g. [12]).
It invoked the design criterion that the movable parts (span plus counterweight) remain in equilibrium at all stages of operation. A mathematical analysis to this effect (by Navier) is added to the later editions as a footnote. When such bridges came to be built, this criterion was (in most instances) employed. Hovey has it thus: ``If the bridge is balanced by the counterweight in all positions, no work is done during the movement of the span.'' In other words, potential energy is conserved. Cundy [6] writes that the object would be ``to choose the curve in which the counterweight moves so that the bridge [is] in equilibrium in every position''.
Figure 2 shows an idealised version of the situation in which the following assumptions are made.
\item{1.}{The cable is supposed to engage the span at its far end.} \item{2.}{The tower has the same height as the length of the span.} \item{3.}{The span is modelled as planar.} \item{4.}{The wheel on top of the tower has negligible radius.} \item{5.}{The wheel is directly above the hinge.} \item{6.}{The initial configuration has the counterweight hanging vertically at the top of the tower.} \item{7.}{Friction is neglected.}
This is (in essence) the case discussed by Cundy [6] who then, by means of an anlysis involving the resolution of forces, determined that the track is to be a cardioidal arc. (Cundy and indeed many others avoided the use of Assumption 4 above by implicitly assuming an approximate equality between the radius of the wheel and that of the counterweight. Figure 3 shows that, in the design stage of an actual bridge, Assumption 4 was adopted.) Use of Hovey's principle, however, allows the rather simpler derivation given here. For simplicity and consistency, Cundy's notation will be followed.
Let 2a be the length of the span and 2\alpha the angle it makes with the tower. Let the distance between the counterweight and the top of the tower be r and let the cable make an angle of \theta with the tower. Let the weight of the span be W (= Mg) and that of the counterweight be w (= mg). Then becuase of the initial conditions, we have the relation M = m\surd 2.
\midspace{8cm} \caption{ Fig. 2. The idealised bridge in a general configuration }
Conservation of potential energy now gives
2a - r\cos\theta + a\surd 2 \cos 2\alpha = 2a. \tag1
Furthermore the length of the cable is constant, being 2\surd 2a, so that
we have also:
4a\sin\alpha + r = 2\surd2a. \tag2
Elimination of \alpha between these equations yields the desired result:
r = 4\surd 2 a(1 - \cos\theta). \tag3
This derivation was given in this form by Deakin~[8]. More general cases
may also be analysed by similar means. The equation
r^2 + 2(A\cos\theta - \ell)r = B, \tag4
where A,B are constants and \ell is the contstant length of the cable,
covers many much more general situations. This equation applies even
when Assumptions 1,2,3,6 above are relaxed. It may be derived by adjoining
to the above principles (conservation of potential energy and constancy of
cable-length) the use of the cosine rule in the triangle formed by the
tower, the bascule and the cable. The details are left as an exercise for
the reader. Analyses equivalent to this are given by Hool and Kinne~[13] and
by Navier in a footnote to the later editions of de B\'elidor's book~[4].
Neither Navier nor de B\'elidor made use of Assumption 6.
Almost certainly, Cundy was the first to identify correctly the ideal curve as a cardiod. de B\'elidor referred to the track as a sinusoide , which Hovey states to be more nearly a cycloid. (The designers of bridges of this type seem to have, except in one case, calculated the curve by one means or another, but not to have named it. The exception is Morison~[1] who employed an elliptical arc.) That de B\'elidor did not recognise the link to the cardioid is ironic for he had studied under Phillipe de la Hire who first explored the properties of that curve! See~[12,21].
The design reached the United States and several such bridges were built along these lines in the years 1890-1910. Hovey [14, pp.33-34] lists five. (Apart from the Corsican bridge noted above, these are the only ones of which he seems to have been aware. He includes B\'elidor bascules among the ``Types rarely used''. The fifth, moreover, is rather different in that stiff struts rather than cables connect the movable span and the counterweights. The reader may care to analyse this case as an exercise.)
Hovey was himself associated with the two earliest of these bridges and because he was clearly aware of the history of the design, it follows that he was also one of the major influences in bringing that design to the US. His list, however, is far from complete (see below) and indeed may well not include all the B\'elidor bascules in the United States.
Morison's Chicago bridge (designed in fact by Hovey) used an elliptical track, as has been mentioned, even though his bridge comes closest of the five to Cundy's idealisation. The track was so set up that the vertex of the ellipse was placed at the top of the track, the tangent being vertical; calculations then determined the major and the minor axes. These were determined by exact calculation either of its position for three configurations - fully lowered, fully raised and intermediate - or else of position and orientation for the two extremes. (The description is not clear as to which of these approache was in fact employed.) This design thus comes closest to that put forward in a discussion by Kitchen~[15], the one that prompted Cundy's article~[6].
The curve-track design was brought to Australia by Henry Harvey Dare (25.8.1867 - 20.8.1949) and was based on American models~[7]. Dare had a brilliant and distinguished career. He graduated from The University of Sydney in 1888 and took a master's degree from the same institution in 1894, winning a University Medal on each occasion. He joined the NSW Department of Public Works in 1888 and held a number of posts first within and later with the Water Conservation and Irrigation Commission, serving a total of just over 47 years before his retirement in 1935. He is probably best remembered for work on the Burrinjuck Dam, the Murrumbidgee Irrigation Area and the Sydney Water Supply. For further details see [2; 5, p.62]. Dare was a slightly senior contemporary and lifelong friend of the (now better remembered) J.J.C. Bradfield whose biography [20] contains obiter dicta much material on the older man.
All told, Dare designed and built eight such bridges in NSW over the years 1901-1925. Evidently, Hovey~[14] was unaware of the Australian bridges as he lists none of them although all were in place when he wrote.
Dare explicitly refers to the B\'elidor design as ``American'' and he would, before the turn of the century, have had easy access to descriptions of three of the US bascules. Of the available descriptions, some would have been available to him through the State Library of NSW and others via the Institution of Engineers. For fuller details, see~[9].
Dare's method of calculating the shape of the track [7, Figure 8;10, Figure 15] was to consider four positions of the span (down, fully raised and two intermediate). The first of these gives the initial conditions, which differ slightly from Cundy's. The fully raised position also differs, being somewhat (about 20^o) short of the vertical. For each of the four positions, the position of the counterweight and the orientation of the track were calculated, essentially by the same techniques as employed by Cundy. The track was then set up as a smooth curve comprising three circular arcs interpolated between these nodal points and directions.
\midspace{19cm}\caption{ Fig. 3: A page from Dare's notebooks for the Darlington Point Bridge }
I thought to see how close this composite curve lay to the cardioidal
arc
determined by Cundy. Let y be the angle between the track and the
horizontal. Then it may readily be shown that in the case of the
cardioid \theta = (\pi - 2\psi)/3.
We also have
\rho = (16\surd2)(a/3)\sin(\theta/2), \tag5
where \rho is the radius of curvature.
For each of the three circular arcs in Dare's [7] Figure 8 (reproduced as Figure 4 of this paper), the centre point was determined and \psi measured there. The value of \sin(\theta/2) was then calculated for each of these readings. This gave values of 0.162, 0.301 and 0.396 respectively. Dare's corresponding values of \rho were respectively 17.42, 44.08 and 60.25 (in feet). (I retain the old units, as these are the ones Dare used.) \midspace{10cm} \caption{ Fig. 4: Dare's design for the Telegraph Point Bridge. }
A least-squares best fit \rho =b\sin(\theta/2) was then embarked upon to give a value of 146 for b. This corresponds to a value of 20.9 (feet) for a. The height of the tower was 35.26 (feet), giving a value for a of 17.6 (again in feet), while the length of the span was 47 (feet), giving a value for a of 23.5 (feet). The agreement between the theoretical (54.75 feet) and actual (about 55 feet) values for the total length of cable was very good.
Where the agreement was not so good was in the equation M = m\surd2. The actual ratio involved was not \surd 2 but 1.03. The discrepancy is explained principally by the fact that the cable engaged the span some distance from the end, about one-third of the way in.
However, if an unconstrained regression line \rho = b\sin(\theta/2) + k is fitted to the above data, the fit is vastly improved. The sum of the squares of the residuals is reduced from 44.73 to 0.9793, a factor of 45.7. This shows that the deviations from Cundy's idealisation are significant, and the cardioidal fit only fair.
These calculations apply to Dare's first such bridge, at Telegraph Point. A somewhat different, though related, analysis was applied to his slightly later Darlington Point bridge whose track was designed as a composite of five circular arcs~[9]. (For reasons that will appear below, this bridge lends itself best to measurement by amateurs like myself without specialist apparatus.) In this instance Equation (3) was employed directly and the same conclusions apply as were reached above. A cardioidal fit estimates some 34 metres of cable which is consistent with the remnant that is now to be seen (some 22-23 metres). A fit of Equation (4) however yields values of about 33 metres for \ell, 39 metres for A and 177 m^2 for B. As A is not very close to \ell and B not particularly small compared to \ell^2, the reasons for the relative failure of the cardioidal fit are clear.
Sadly, time has not been kind to Dare's B\'elidor bascules. Three still stand (at Coraki, at Maclean and at Carrathool Crossing on the Murrum-bidgee - see Figures 5, 6), though their mechanisms are long since locked. A railway bridge on Sydney's Botany Line remains in use although its towers have gone.
The Telegraph Point structure was bypassed and subsequently demolished= in 1974, when the Pacific Highway was rerouted (see [3]). Another bridge at Kyalite has likewise been removed. (This latter bridge is the subject of a rather splendid photograph in the mathematical literature [19].)
A similar fate almost befell the Darlington Point bridge but in this case the tower and track were reassembled (in a nearby caravan park) as a historic monument by local volunteers in co-operation with engineering students from UNSW. The pieces lay in an untidy heap for some ten years before this. The story of their ultimate reassembly is best told by Fraser~[11], who oversaw the work.
In a personal communication, Bruce Aird, now of Marmong Point, recalls operating the mechanism of the today long-demolished bridge at Swansea at the entrance to Lake Macquarie. On his account the bascule mechanism was still in action in the early 1950s, at a time when the other bridges were probably all long out of routine raising.
For more detail on these bridges and their history, see [9]. There is also information in two standard works of reference [17, 18]. The Telegraph Point bridge provided the cover for an award-winning project [16]; the Darlington Point bridge is to be the subject of a forthcoming book by Mona Finley of the Darlington Point Advancement and Historical Association Inc.
\midspace{18cm} \caption{ Fig. 5: The B\'elidor bascule at Carrathool Crossing }
\midspace{12cm} \caption{ Fig. 6. Plaque at Carrathool Crossing. The date 1903 should be 1901. }
It is a pleasure to record with great thanks my deep indebtedness to Don Fraser and to Colin O'Connor for much technical and historical information, and to Bruce Aird, Mona Finley and Kay Kneale (Telegraph Point) for further historical background and to all of these for reference material and indeed for copies of some of the references used in this article. I also thank Aidan Sudbury who tendered valuable statistical advice.
\item{[2]} {Anon., Obituary Notice - Henry Harvey Dare. Journal and Proceedings, Roy. Soc. NSW , 84(1) (1950), p. xxv.}
\item{[3]} {Anon., New Bridge and Deviation at Telegraph Point. Main Roads (Journal of the Dept. of Main Roads, NSW) 39(4) (June 1974), pp. 98-101.}
\item{[4]} {de B\'elidor, B. F., La Science des Ing\'enieurs dans la Conduite des Travaux de Fortification et d'Architecture Civile. Paris: Jombert, 1729. Subsequent editions include those with notes by C.-L.-M.-H. Navier (Paris: Didot, 1813 and 1830).}
\item{[5]} {Coltheart, L. and Fraser, D., Landmarks in Public Works. Sydney: Hale and Ironmonger, 1987.}
\item{[6]} {Cundy, M., The Bascule bridge - an unexpected cardioid. Math. Gaz. 74 (1990) pp. 124-127.}
\item{[7]} {Dare, H. H., Recent Road-Bridge Practice in New South Wales. Min. Proc. Inst. C. Eng. 155 (1904) pp. 382-400 and Plates 2-4. (Reprinted in edited form in [16].)}
\item{[8]} {Deakin, M. A. B., More on Bascule Bridges. Math. Gaz. 79 (1995) 107-108.}
\item{[9]} {Deakin, M. A. B. and Fraser, D. J., Cardioid-Tracked Bascule Bridges in NSW. Trans. Inst. Eng. Aust. (Multidisciplinary Eng.). GE18(2) (1994/5) pp. 163-171.}
\item{[10]} {Fraser, D. J., Moveable Span Bridges in New South Wales prior to 1915. Trans. Inst. Eng. Aust. (Multidisciplinary Eng.) GE9(2) (1985) pp. 71-81.}
\item{[11]} {Fraser, D. J., Darlington Point Bridge Reconstruction. In Proc. 5th Nat. Conf. on Engineering Heritage, Institution of Engineers, Aust. (1990) pp. 81-87.}
\item{[12]} {Gillespie, C. C., Article on B\'elidor. In Dictionary of Scientific Biography v.1, pp. 581-582.}
\item{[13]} {Hool, G. A. and Kinne, W. S. (Eds.) Movable and Long-Span Steel Bridges. (2nd Edition revised by R. R. Zipprodt and H. E. Langley) New York: McGraw-Hill, 1943.}
\item{[14]} {Hovey, O. E., Movable Bridges. Vol. 1. New York: Wiley, 1926.}
\item{[15]} {Kitchen, A., Building a Bridge between GCSE and `A'-Level. Mathematics in School. 18(3) (May 1989), pp. 2-7.}
\item{[16]} {Kneale, K. (Ed.), Ripples on the River. Telegraph Point Primary School's entry in Hastings Council's 1993 Cultural Contribution Award.}
\item{[17]} {O'Connor, C., Register of Australian Historic Bridges. Barton, ACT: Institution of Engineers, Australia and Australian Heritage Commission, 1983.}
\item{[18]} {O'Connor, C., Spanning two Centuries: Historic Bridges of Australia. St. Lucia: University of Queensland Press, 1985.}
\item{[19]} {Peregrine, D. H., A Drawbridge in Balance. Math. Gaz. 76 (1992), pp. 280-281.}
\item{[20]} {Raxworthy, R., The Unreasonable Man: The Life and Works of J. J. C. Bradfield. Sydney: Hale and Ironmonger, 1989.}
\item{[21]} {Taton, R., Article on Phillipe de la Hire [the elder], Dictionary of Scientific Biography. 7, pp. 576-579.}
\item{[22]} {Waddell, J. A. L., Bridge Engineering. New York: Wiley, 1916.}
\item{[23]}
{Williams, M. O.,
The Coasts of Corsica.
National Geographic. 44(3) (Sept. 1923), pp. 221-312; see esp. p.307.}
Department of Mathematics
Monash University
Clayton, Vic., 3168