For the engineering types on the board, could someone provide me with a concise and understandable explanation of the Bonjean Curve Method as it applies to Edward Wilding's calculation of 12 square feet of aggregate hull openings? Wilding mentions the Bonjean Curve method but does not provide any further details or clarification. --Nate R.

Nate-- Bonjean curves are a representation of the area of the transverse sections of a ship at successive waterlines. They are normally plotted as a fair cure on a profile of the ship. "Waterline" in this case does not mean the physical load waterline, but rather a horizontal plane running fore-and-aft. Think of them as the layers of frosting in a multi-layer wedding cake. When combined with other lines called "sections" and "buttocks," a 3-dimensional curved ship can be represented on a flat sheet of paper. The three--waterlines, buttocks, and sections--are combined in one drawing called the "lines drawing." When plotted on a profile of the vessek, each Bonjean curve begins on the base of the station it represents. It rises and curves forward on the profile, indicating the transverse area of the hull at that location on the hull. Bonjean curves can also be plotted on graph paper. Bonjean curves are used in calculating the volume of displacement and the center of buoyancy at any waterline, or angle of trim. Most often they are used in stability calculations, determining the capacity of the ship, or in launching calculations. --David G. Brown

Bonjean curves form part of the set of hydrostatic curves provided to the vessel by the builder. The underwater volume of a vessel and her longitudinal centre of buoyancy can be derived from her Bonjean curves. Customarily, a ship is divided into a number of equidistant stations, usually ten (note that these stations are not necessarily coincident with her watertight bulkheads). Bonjean curves are curves of immersed cross-sectional area plotted against draught and are drawn on the profile of the ship at each station. Thus waterlines at any angle to the vessel's baseline can be superimposed upon the profile. It then becomes possible to read off the immersed areas by drawing lines parallel with the baseline from the intercept of the waterline with the section to the Bonjean curve for that section. The use of Bonjean curves thus enables the volume to be found for waterlines that are not parallel to the baseline. Since the resultant is a volume of displacement your Mr Wilding must then have gone one to adduce some COEFFICIENT OF PERMEABILITY for each station to arrive at a volume of inundation. From the summation of such inundations at two representative waterlines (presumably as per witness observation) and the elapsed time between them, it is possible to compute a rate of inundation and thereby arrive at a notional summation of the areas of all the breaches in the hull, in this case, 12 square feet. As to the validity of his computation, how your Mr Wilding dealt with the effect of the changing head of water as the vessel settled I would not know. I seem to recall that the eponymous Monsieur Bonjean was a French naval officer. Noel

Noel -- this discussion is so entertaining that perhaps we could take our act on the road. I'm sure we have lost most of the ET contributors--or at least put them to sleep. The significance of this is that Wilding used a standard method technique of naval architects as the basis of his calculations. The same curves are still used for similar purposes today. You mention the problem of the coeffficient of permiability. In effect, a "fudge factor." This factor is an attempt to approximate how the internal components of the ship restrict the flow of water. Other variables include the duration over which the flooding occurred and its location. On both of these, it appears that Wilding was misled by the data provided by Lord Mersey. In specific, Wilding was told that boiler room #6 flooded perhaps 20 minutes earlier than it did. And, he was told that hold #2 and the forepeak tank were not flooded. What is curious is that Wilding's initial calculations--using Bonjean curves--showed that Titanic should not have foundered. So, Wilding dutifully changed the flooding pattern until it matched the results desired. It was this second set of calculations that became the basis for his famous "12 square feet" estimate of the damage. -- David G. Brown

Daved, Noel, while I'm no mathmatician, I'm not having a problem following what you two are saying. Keep it up guys...I'm learning something.

In this case, it might be inferred that the Bonjean Curves were deficient. It was said for many years that "according to the Principles of Aerodynamics, the bumblebee cannot fly." Then vortex lift was discovered, and the Principles were modified to "save the phenomenon," as the ancient Greeks would say.

Along with Mike I am still reading, although I must say I had to read Dave's and Noel's post twice. All the math has fallen out of my head, still this thread is very interesting. Especially that Wilding had to recalculate his figures, and the issue of internal fittings impeding flooding. Brian

Tom -- The Bonjean curves were accurate enough. They are not a theoretical construct, but a realistic method of describing vessels. Wilding undoubtedly had a full set of Bonjean curves for Titanic produced as part of the ship's construction. They were needed to launch the vessel without damage. As to percolation...well, that's where the black arts come into play. There are various "fudge factors" which are used for various types of ships and/or cargoes. These approximate reality, but are not reality--sort of like computer projections of tomorrow's weather. --David G. Brown

"As to percolation...well, that's where the black arts come into play. There are various "fudge factors" which are used for various types of ships and/or cargoes. These approximate reality, but are not reality--sort of like computer projections of tomorrow's weather." Quite. There are promulgated in the textbooks various parameters of permeability depending upon the use to which a compartment is put. Considering the infinitely variable conditions of lading of such as a tweendeck or lower hold containing general cargo, once you get past the design stage 'permeability' must border on wild guesswork. Especially when you consider that the vessel's stations are somewhat arbitrary and don't necessarily conveniently encompass whole compartments. Mr Wilding must have been in possession of a deal of detailed information to arrive at his '12 square feet' - which I would take with a large pinch of salt. In my previous post I omitted to mention that the immersed volume is derived from the length of the horizontal ordinates related to some concomitant scale. I also omitted to add that "the above does not purport to be a definitive or complete exposition of the procedure"! Noel

On seeing that the ET Discussion Board is veritably a-buzz with this gripping topic I feel compelled, at risk of some considerable re-iteration, to re-address it in more measured terms than hitherto, the better to answer Nathan Robison's original query.... Bonjean curves form part of the set of hydrostatic curves put aboard by the builder. The underwater volume of a vessel and her longitudinal centre of buoyancy can be derived from inter alia her Bonjean curves whenever she floats on an uneven keel. At the design stage the drafted profile of a ship is customarily divided into several (usually ten) equidistant sections, referred to as 'stations'. For each station a Bonjean curve is got out and inscribed on the profile. Those stations at the extremities where curvature is severe can be further divided into sub-stations. Note that a ship's stations do not necessarily coincide with her w/t compartments. The Bonjean Curve of Section is a fair curve generated by wrangling the 'five, eight, minus one' rule on a tabulated work sheet in accordance with the formulae handed down to us by M.Bonjean. The curve initially allows the determination of the median immersed cross-sectional area of each station to any oblique waterline superimposed upon the profile. This is done via a system of rectangular Cartesian co-ordinates whereby the y-axis represents the vertical distance between baseline and maindeck and the x-axis returns the corresponding immersed area, the curve springing from the axial origin. The abscissa thus generated (that is, the rectangular or horizontal distance between the intercept of the waterline with the curve) represents the immersed area, discernible in practice by dropping an ordinate to the graduated x-axis. Having thus determined the immersed cross-sectional area to the given oblique waterline, the prismatic projection of that area throughout the length of the station will render the corresponding immersed volume and therefore the volume of displacement to that waterline. The findings can be adjusted for angles of heel by taking a mean of port and starboard draughts. Having thus determined the volume, the mass of displacement can be determined from the prevailing specific gravity (i.e., sea- or fresh water). As to Mr Wilding's aspirations to aggregate the total area of the breaches in Titanic's hull, since the Bonjean curves can provide the volume of displacement for each station it would then be necessary for him to adduce some coefficient of permeability for each station to arrive at a volume of inundation. From the aggregation of such inundations at two representative waterlines and the elapsed time between them, it would be possible to compute a rate of inundation and thereby arrive at a notional aggregation of the areas of all the breaches in the hull. In the present case this seems to have been 12 square feet. However, while of value at the design stage and thereafter in managing the operational conditions of lading of the ship, I would not have thought Bonjean curves of much practical value in determining the damaged condition of a ship. Better recourse would be had to her 'floodable length' (a known quantity, being that flooded length which takes her down to her margin line and no further). That and the soundings, which themselves would directly return the rate of inundation. That is probably what Thomas Andrews succinctly did in the present case. Rather than redundantly computing to two successive waterlines, on being notified of the carpenter's soundings he could simply work back to the time of the collision to determine the rates of inundation for the affected w/t compartments (never mind theoretical 'stations'). From the outset he would have know that the Titanic's floodable length had been exceeded. The most urgent requirement therefore was to estimate the time left for her to remain afloat and tenable. From the determined rate of inundation and the capacity of the pumps to meet it he was able to give Captain Smith such a time. All this without reference to my textbooks! – albeit I can see them out of the corner of my eye fulminating on the shelf and threatening due retribution after I've sent off this post. Reverting to Mr Wilding's recourse to Bonjean curves in his attempt to aggregate the area of the breaches through which the sea entered, I see him as overly reliant upon dubious evidence as to waterlines, elapsed times and the permeability of the bilged compartments. There are various parameters promulgated in the textbooks for permeability depending upon the use to which a compartment is put but these can be very approximate in actual service. Remember also that 'stations' survive from the design stage and do not necessarily conveniently coincide with compartmentation, let alone w/t subdivision. Wilding probably took recourse to Bonjean curves because the soundings book did not survive to help him, even assuming it was filled in (the damage control soundings probably arrived at the bridge on such as the back of a fag packet in the time-honoured manner). I note Edward Wilding was a qualified naval architect on the H&W design team and it must therefore be assumed he knew precisely what he was talking about when he put forward his computation. I've been unable to trace the context but presumably he was being asked to supply an answer to a specific question and went about it will nilly. I cannot see however that such a retrospective calculation had any material bearing on the casualty or the prevention of future such casualties. Note that the above does not purport to be a definitive or complete exposition of the concept or the procedures. Ever anxious to prise apart the synapses with more essential information with which to face down the vicissitudes of life, I am trying to track down a biography or obituary of the originator of Bonjean curves (we speak of little else where I come from). The name, as far as I can ascertain, originates in Provence and should be accorded the French pronunciation. Short of my asking the RINA or its French equivalent, can anybody help? I could continue with this peroration but others are advising me to end it now, their reasoning being that the arrival of the Horlicks trolley is imminent. Noel

Noel -- Wilding's first iteration showed the floodable length had not been exceeded. That was the problem. The data given to him showed the ship was not in a sinking condition. Wilding new his his calculations that were at fault, so he turned to "massaging" the data. In other words, he introduced what is called "expectation error." He made his conclusion first and then forced the proof to follow. The strange twist to all of this is that his first calculation was undoubtedly much closer to reality. Givin Wilding's skill at his profession, I believe he recognized the truth. -- David G. Brown

This can only be a brief response to various representations of Wilding's 12 sq. ft. I find it extremely interesting that NOBODY has ever questioned the margin of error for this supposedly precise calculation. Just what is Wilding's margin for error? It is unbelievably large, + or - 25%. That is, the aggregate hull openings could have totaled as little as 9 sq. ft. Or they could have totaled 15 sq. ft. 12 sq. ft. is simply the average. Because it was the average, Wilding deemed 12 sq. ft. the most probable. Note...this information is from Wilding's 1915 deposition for the limitation of liability hearings. There is more on his calculations, but I don't have the deposition in from of me at the moment. I would be happy to post Wilding's explanation of how he arrived at 12 sq. ft. as stated in the limitation of liability deposition if anyone is interested. --Nate R.

A contemporaneous post by Tom Pappas under Collision/Sinking Theories gives a RINA retrospective (1996) aggregation as 12.6 square feet as well as adducing that had pumping capacity been intelligently re-directed the vessel would have survived. This infers that, had the damage control computations been processed by modern software instead of hyper-stressed human agency in the form of Thomas Andrews, the outcome might indeed have been thus different. In the light of this and the present discussion it would indeed be interesting if Nathan could post Wilding's 1915 deposition. I assume it gives the parts of the sum - which we could compare with the 1996 re-working. ERRATUM: In my post of Sunday 23 March 12.06 am: the fifth paragraph should have read: "The abscissa thus generated (that is, the rectangular or horizontal distance between the intercept of the waterline with the y-axis and the curve) represents the immersed area, discernible in practice by dropping an ordinate to the graduated x-axis." My apologies for that. Noel

I am happy to post the relevant information on Wilding's deposition. However, be advised that the post will be highly critical of the RINA article that so many people have put so much faith in. I would implore everyone who has the Bedford/Hackett article to read it again with a degree of caution. Wilding's deposition: Q (Claimants attorney Betts) Will you tell us how you made that calculation? A (Wilding) Yes. It was known that certain spaces were filled up in a certain number of minutes, approximately. You couldn't say to a stop watch, but in about 40 minutes certain compartments were filled up to a certain level. The capacity of those compartments was known, and therefore the amount of water which got in in the 40 minutes was known. The approximate depth of the position of the damage was known and it was possible to calculate the rate of inflow per square foot of opening. I have the total inflow and I can divide that by the number of minutes and get the total inflow per minute. I can get the inflow per square foot per minute, and by dividing the one by the other I can get the square feet. Q. What was the area? A. My memory is 12 square feet; 4 feet by 3. Q. (Mr. Robinson) In making that calculation as to the flooding plan, what factor of permeability did you allow? A. (Wilding) A factor in the coal bunkers of 50% and in the cargo holds of 75%; in the mail and baggage rooms of 83 1/3%; in the engine and boiler spaces 80%; in the passenger accomodation 95%. Q. Was due consideration given to the loss of water plane? A. Certainly; the calculations were made by what is known as the Bonjean Curve Method, the direct method; he, Bonjean, was a French mathematician of about 1790. They retain Wilding's permeability figures as given in the BOT inquiry, but also state "...and more importantly he does not mention what values were used for engine and boiler rooms." To enable us to proceed we have decided to use the SOLAS value of 0.85 for engine and boiler rooms and also for the bunkers, since this is not too different from Wilding's figure if we assume that the bunkers were about 1/3 full at the time of the collision." (Bedford/Hackett, p. 172). Note that Bedford/Hackett claimed Wilding did not mention the engine and boiler room permeability at the BOT. But he DID mention it at the liability hearings. Bedford/Hackett's value should result in a slightly different computation than Wilding's 12 square feet. Indeed, the initial Bedford/Hackett value, using Wilding's figures and the SOLAS number was 9.53 sq. ft. of hull openings. The formula for flow through a submerged orifice is found on p. 182. The formula is too complex for me to try to post. "To achieve an answer of 12 sq. ft. we must reduce the coefficient [of discharge] to 0.485 [Wilding's coefficient of discharge was 0.61] which seems not unreasonable for the type of opening we suspect was caused by the collision." Thus, there are now two instances of Bedford/Hackett altering Wilding's figures to produce an answer that agrees with the famous 12 sq. ft. But recall my previous post where I stated that Wilding's margin for error for the calculation was 25%. Note also that the original Bedford/Hackett calculation was 9.53 sq. ft., near the higher end of a 25% margin of error. But the RINA article alters Wilding's formula values to obtain an answer that equals the famous 12 sq. ft. There is no justification given for why Bedford and Hackett made those alterations aside from that it "seems not unreasonable for the type of opening we suspect was caused by the collision." I simply cannot accept the scientific methodology presented in the RINA article as empirically valid. Perhaps it is time that Titanic scholars consider the possibility that the aggregate hull openings were less than 12 square feet. Bedford and Hackett calculated 9.53 sq. ft. and this falls within the margin of error that Wilding indicated for his calculation. --Nate R.

Thank you Nathan for the abstract from Wilding's deposition. I am content to leave the anachronistic contretemps vis-Ã -vis Wilding and Messrs Bedford & Hackett at the RINA. to those now better placed to contest it. As for the curvaceous M.Bonjean, at least now I have a date! Q. Was due consideration given to the loss of water plane? A. Certainly; the calculations were made by what is known as the Bonjean Curve Method, the direct method; he, Bonjean, was a French mathematician of about 1790. Noel

M. Bonjean M. Bonjean was a French Naval Architekt about 1790. He is quoted in Pollard et Dudebout: . THEORIE DU NAVIRE. Tomes I . 1892. by Pollard et Dudebout. on page145. Derived from Johows Hilfsbuch fuer den Schiffbau (1910) page 305. Cheers Michael