**Some ideas developed from Samuel Halpern’s “Brace for Collision”. ^{1}**

Edward Wilding, a senior naval architect from Harland & Wolff shipbuilders, gave evidence to the British inquiry into the loss of the *Titanic* concerning the design and construction of the ship. On the 19^{th} day of the inquiry, 7th June 1912, the following exchange took place: ^{2}

20266.

QPerhaps I ought to put this general question to you. The contact with this iceberg was the contact of a body weighing 50,000 tons moving at the rate of 22 knots an hour? -AYes.20267.

QI gather to resist such a contact as that you could not build any plates strong enough, as plates? -AIt depends, of course, on the severity of the contact. This contact seems to have been a particularly light one.20268.

QLight? -AYes, light, because we have heard the evidence that lots of people scarcely felt it.20269.

QYou mean it did not strike a fair blow? -AIf she struck it a fair blow I think we should have heard a great deal more about the severity of it, and probably the ship would have come into harbour if she had struck it a fair blow, instead of going to the bottom.20270.

QYou think that? -AI am quite sure of it.20271.

(I am rather interested about that. Do you mean to say that if this ship had driven on to the iceberg stem on she would have been saved? -The Commissioner.)AI am quite sure she would, my Lord. I am afraid she would have killed every firemen down in the firemen’s quarters, but I feel sure the ship would have come in.…..

20275.

QDo you think that if the helm had not been starboarded there would have been a chance of the ship being saved? -AI believe the ship would have been saved, and I am strengthened in that belief by the case which your Lordship will remember where one large North Atlantic steamer, some 34 years ago, did go stem on into an iceberg and did come into port, and she was going fast?

The Commissioner:I am old enough to remember that case, but I am afraid my memory is not good enough.Mr. Laing:The “Arizona” - I remember it.The Witness:The “Arizona,” my Lord.20276.

QYou said it would have killed all the firemen? -AI am afraid she would have crumpled up in stopping herself. The momentum of the ship would have crushed in the bows for 80 or perhaps 100 feet.20277.

QYou mean the firemen in their quarters? -AYes, down below. We know two watches were down there.20278.

QDo you mean at the boilers? -AOh, no, they would scarcely have felt the shock.

The Commissioner:Any person, fireman or anybody else, who happened to be in that 100 feet, would probably never have been seen again?20279.

QThe third class passengers are there too, I think, some of them? -AI do not think there are any third-class passengers forward of the second bulkhead, and I believe she would have stopped before the second bulkhead was damaged. It is entirely crew there, and almost entirely firemen - firemen, trimmers, and greasers.20280.

QYour opinion is that the ship would have suffered that crushing-in in the first two compartments, but that the shock would not have shattered or loosened the rivets in any other part of the ship? -ANot sufficiently. As it would take a considerable length, 80 or 100 feet to bring up, it is not a shock, it is a pressure that lasts three or four seconds, five seconds perhaps, and whilst it is a big pressure it is not in the nature of a sharp blow.20281.

(It would, I suppose, have shot everybody in the ship out of their berths? -The Commissioner.)AI very much doubt it, my Lord.20282.

QAt 22^{1}/_{2}knots an hour, and being pulled up quite suddenly? -ANot quite suddenly, my Lord. 100 feet will pull up a motor car going 22 miles an hour without shooting you out of the front.20283.

QWhat you mean is that the ship would have telescoped herself? -AYes, up against the iceberg.20284.

QAnd stopped when she telescoped enough? -AYes, that is what happened in the “Arizona.”

This appears to have come as a surprise to the inquiry. The two questions at the start of this excerpt suggest that the lawyers were expecting answers along the lines that it would be difficult to design a ship to withstand a collision at high speed. The subsequent ones suggest surprise at Wilding’s view that the contact with the iceberg was actually a very light one and the ship was capable of surviving a much heavier one had it been head on.

It seems unlikely that the inquiry would have posed questions about a hypothetical head-on collision had Wilding not introduced the subject, and therefore there was no reason why he should have prepared his replies in anticipation of this, yet his subsequent confident replies giving estimates of distances and timing suggest that he may have put considerable effort into doing so. It is *possible* that he was merely making rough estimates by scaling up from known previous cases, such as that of the *Arizona* mentioned in his testimony, though I personally doubt it. Wilding was a naval architect who had been deeply involved in the design of the *Titanic* and had ready access to all the specifications so I believe that, if only out of professional pride, he would have made calculations based on real structural strength data.

Many people reading Wilding’s testimony remain sceptical of his findings. Most have a similar view to the Wreck Commissioner, Lord Mersey, and find it intuitively difficult to accept that such a collision would not necessarily involve a large and destructive ‘shock’ force. Others simply question the reliability of Wilding’s conclusions, suspecting that there most be a large range of uncertainty in the underlying assumptions leading to great imprecision in the calculations.

Halpern’s paper “Brace for Collision” ^{1} reproduced as an appendix in his book *Prelude to an Allision*^{3} has done much to address the first of these issues. He has developed a method for working back from Wilding’s crush distance estimate to the forces and decelerations involved and calculates that the event would have taken place over 3.74 seconds during which the deceleration would rise to a maximum of 0.39 g. The time is within Wilding’s estimate of 3-4 seconds. The deceleration is too high to support his suggestion that those in the boiler rooms would “scarcely have felt the shock” (Q20278) but it agrees with Q20280, “whilst it is a big pressure it is not in the nature of a sharp blow”. Halpern agrees with Wilding that the impact would not have caused significant structural damage aft of the immediate damage area and concludes that “Although very seriously damaged the ship would very likely have remained afloat”.

In this paper I propose to address the second concern about Wilding’s scenario; the dependence of the reliability of the results on the accuracy of the starting assumptions, or in other words the error tolerance. We can do this by using Halpern’s method to generate strength and deceleration values for a wide range of damage distances and then plot these with the axes reversed to illustrate how the damage and deceleration would vary according to the assumed strength as our starting point. This will enable us to examine the range of input values over which Wilding’s overall conclusions remain valid, and how things might break down at the extremes of that range.

I have also taken further Halpern’s scaling of the damage length to a range of speeds, keeping everything else constant, which leads to some interesting speculation on the implications for the conditions under which bulkhead “A” could be regarded as a fully effective collision bulkhead.

#### Analytic section – damage length and deceleration over a range of strength of the ship.

I am assuming that Wilding’s starting point was an estimate of some aspect of the ‘strength’ of the ship, from which he then calculated the likely distance and forces involved in bringing the ship to a standstill. We do not know his starting point but Halpern has shown that we can work the problem in reverse to obtain a ‘strength’ figure from the predicted length of crushing and use that to calculate the force, deceleration and time course involved. I have briefly summarised how he derived the mathematical equations in an appendix but for a full and much clearer explanation, I recommend reading his original paper on the subject. ^{1}

The main equation he derived is:

(1) K = ½ (n+1) Mv^{2}/X^{(n+1)}

M is the ship’s mass, v is the speed, X is the length crushed and n is a factor relating to how the cross-sectional area of the ship increases with distance from the bow. In the original paper, Halpern describes K simply as a constant relating the force to the crush length without stating anything about the units, though clearly, it is in some general way related to the ‘strength’ of the ship. In fact, if we examine the derivation of the equation in a little more detail we can show that K would come out in units of force/area. More specifically, with M in metric tonnes x in metres and v in metres/second (as used by Halpern), it would be in kilonewtons/square metre.

These are exactly the sort of units normally used to describe the compressive strength of construction materials or components, so we can see that K is analogous to a composite compressive strength for the whole ship girder.

Equation 1 can be rearranged to express damage length as a function of K, M and v, thus:

(2) X = [(n+1) K Mv^{2}/2]^{1/(n+1)}

This enables us to explore the effect of varying the compressive strength estimate (K) keeping the original value of 22.5 knots (11.6 m/s) for v, or to calculate the damage length for a range of speeds using the original value of K based on Wilding’s figures.

Using Wilding’s estimate of 80 ft (24.4 m), a speed of 22.5 knots (11.6 m/s) and a mass of 49075 tonnes at the time of the collision we get a value for K of 52607 kilonewtons/m^{2}. Now we aren’t really interested in the *absolute* value of K, and the large numbers would make it very inconvenient to compare a range of different K values.

I therefore calculated the damage length and K values over a wide range, keeping the speed at the original value, and then divided all the K values by 52607. The resulting figures are a measure of the *relative* strength of the hull compared with that corresponding to Wilding’s estimate. By plotting the damage length against the relative strength, and assuming that Wilding based his length and force figures on a strength estimate, we can examine how much the length estimate would change over a large range of assumed error in his strength estimate.

The results are shown in Figure 1. The values on the x-axis represent the real strength of the hull relative to the value calculated for Wilding’s 80 ft damage estimate.

The crush distance (green line) is the value of X as given by Equation 2. This represents the length of hull that would be crushed in absorbing the energy of impact but the real damage length would be more. This is because the crushed and telescoped material builds up behind the point of impact and is pushed on, to crush the hull further behind, so the actual limit of crush damage moves back further than the length of hull lost. Halpern has suggested, based on research on other ship collisions, that the true damage length is likely to be about ⅓ greater than the actual loss of hull length. I have used the same figure to calculate my total damage length (red line).

Wilding made no specific mention of this point in his testimony. His length estimates are variable. He mostly talks about “80-100 ft” but we don’t know whether he is thinking of an 80 ft loss of length plus an allowance for further damage aft of that or merely expressing a range of uncertainty in his calculations. The reason his 80 ft lower bound is more commonly cited and used as the baseline in my calculations, is that he stated in answer to Question 20279 “I believe she would have stopped before the second bulkhead was damaged”, which implies a maximum of 80 feet of damage.

I have marked the positions of the first 4 bulkheads in Figure 1 and it is clear, according to current thinking, that the total damage length at an 80 ft crush length (relative strength 1) would have gone well beyond bulkhead “B” so there *would* have been casualties among third class passengers and deck crew in addition to the firemen and trimmers mentioned by Wilding. However, flooding would still be confined to the first three compartments so his confidence that the ship would still have floated is entirely justified.

The point at which damage would reach the next bulkhead, “C”, occurs at a true strength ⅔ of his estimate or, looking at the inverse calculation, where he had overestimated the true value by 50%; (⅔ plus 50% of ⅔ equals 1). The casualty figure would probably be no higher than the previous scenario. More third-class passengers might be killed by crushing as the third compartment is compressed further but in either case, the compartment would be widely breached and would flood rapidly with no possibility of escape, so the majority of the occupants who escaped crushing would be drowned in either scenario. Again, the ship would still have floated but there might now be problems with command, control and communication as the foremast would have been brought down, possibly damaging the bridge and wheelhouse and certainly bringing down the Marconi aerial system.

In fact, if the ship had been only half as strong as his estimate - meaning he had overestimated by 100% of the true value – the ship should *still* have continued to float! It would now be crushed almost up to the bridge, third-class casualties would be very extensive and even some first-class cabins on C, D and E decks might be involved. However, only the first four compartments would be opened up so if the watertight doors on F deck were closed quickly the ship could float (just).

Now let us look at the possibilities if Wilding had *underestimated* the true strength of the hull. We can see from the graph that if the relative strength was more 1.25 the total damage would stop short of bulkhead “B”. That point represents Wilding having underestimated by 20% of the true strength. It is at this point that his prediction that casualties would be limited entirely to the firemen, trimmers and greasers becomes true.

Beyond that strength, the total damage length remains confined to the second compartment, so the potential casualty count remains the same until we reach a strength almost 4 times the estimate. At that point, the damage would fall short of bulkhead “A” so the only crushed compartment would be the unoccupied forepeak and there would not be any direct, crush casualties at all. The 4 times increased strength would have to apply to the *whole* ship, not just the first compartment. (If only the first compartment was strengthened the effect would be that it would remain relatively intact and get pushed bodily aft while crushing the weaker compartments *behind* it.)

Such a massively increased strength for the whole ship would appear to be impracticable but, in any case, we shall see that consideration of the forces involved imposes serious limits long before that point.

To explore the dynamics more fully we need to look at both the forces and the times involved over a range of possible hull strengths. Calculating the maximum force or deceleration at the various strengths and crush distances is straightforward, but plotting the time course is not. It involves a differential equation that does not have an explicit algebraic solution and can only be solved by a rather tedious successive numerical approximation. Halpern has done it for the 80 ft case and, to save a great deal of effort, I have scaled his graph approximately to other lengths and relative strengths (see appendix).

The results are shown in Figure 2 as graphs of deceleration against time. The black curve shows Halpern’s original results calculated for the Wilding 80 ft case, relative strength 1. My curves for other relative strengths are scaled to the correct maximum deceleration for the strength. The *time *scale is approximate, but the general shape of the curves is close enough to illustrate the points I want to discuss here.

I have expressed the deceleration in units of g, the normal acceleration of an object falling under gravity, or 9.81 m/s^{2}. Since this is the basis of the definition of the SI unit of force, the newton, it can equally be considered as a force of 9.81 newtons/kg. It is a convenient way of expressing how the deceleration “feels” in comparison with the weight of any object to which it applies.

In the lower strength cases, the decelerations are lower, the rise in force is spread over a longer time and the slope is gentler than in the ‘Wilding’ case. This clearly adds no new problems to the effects of the collision so casualties and ship damage remain as in the previous discussion of Wilding’s figure being an overestimate of the true strength. It doesn’t alter the limitations.

If the relative strength is higher than 1 (i.e. Wilding has *underestimated*) the deceleration or force increases rapidly and the time shortens, and in particular the initial rise of the deceleration with time increases sharply. The rate of increase of acceleration or deceleration with time is mathematically known as either the first derivative of acceleration or the second derivative of velocity but is often known in mechanical engineering as the “jerk”. It is a good name because that is exactly what it feels like.

In human terms, the jerk rate is often more of a problem than the overall acceleration. Pretty much anyone can tolerate an acceleration of one g – equivalent to being pressed upon by their own weight, or about the maximum g force typically experienced in driving ordinary cars – if they are in a secure position, prepared for it and it doesn’t happen faster than they can react. If not they will fall over, adding a new source of potential injury. This is less of a problem for inanimate objects because they have no reaction delay and simply respond in direct proportion to the decelerating force, regardless of the rate of increase.

The purple curve in Figure 2, for a relative hull strength of 4 times the Wilding estimate, represents the first point at which damage could be limited to the first compartment thus avoiding any direct crush casualties. The maximum deceleration is not much more than 1 g which, in itself, is well within human tolerance but 90% of the increase occurs in the first second and it has already reached the same level as the ’80 ft’ scenario in little more than a tenth of a second. The jerk is very high and would undoubtedly lead to numerous injuries by falling throughout the whole ship.

This is the scenario that many people would imagine in such an impact, though they also tend to imagine numerous injuries caused by loose objects being violently thrown forwards. At 1 g those objects would experience the same acceleration as they would if they were simply dropped and falling under gravity. They could cause injuries, certainly, but their motion would be much less violent than one might intuitively expect.

It might therefore appear that deliberately designing the hull for such a scenario might be a good idea with a view to minimising casualties. The problem is that structural limitations would give rise to problems long before the human tolerance limit. For any rigid connection between components the breaking point depends mainly on the force applied and much less upon the rate of increase, so if we know the force required for such a breaking point we can plot it as a horizontal line on the graph, in the same way we plotted bulkhead locations on the distance vs strength graph.

I have plotted one such line as the “boiler bed shear limit”. To derive that I started from a later part of Wilding’s testimony, on day 20 of the British inquiry ^{2}, when he was answering questions about events during the sinking. The relevant excerpt is as follows:

20915.

QSome witnesses spoke of noises, and some of them suggested that the noises might have been caused by the machinery falling forward when she got tipped up considerably. Do you think there is anything in that? -AThe boilers might have moved; I do not think the machinery did.

(The Commissioner:)It was thought the boilers had got loose from their seats.20916.

QYes. Is that a reasonable theory? -AWhen the ship was about 35 degrees by the head.20917.

QThat might have happened? -AWhen the bow was down so that her stern was up, so that the slope fore and aft of the ship was about 35 degrees.

Wilding has twice made a fairly precise statement about the angle of trim at which he would expect the boilers to detach from their beds. The relevant force would be shear, or sliding, force between the boilers and their bedplates. He presumably started from a known or estimated value for the shear force required to break the boilers free and then calculated the angle at which that force would be generated by the weight of the boilers. We can work the same problem in reverse. Either way the problem involves the principle of resolving a force into two components at right angles to each other, using the ‘triangle of forces’ principle, and a little elementary trigonometry.

This is illustrated in Figure 3, which is intended to depict a boiler resting on the tank top, both tilted to an angle of 35 degrees. The weight of the boiler acts directly downwards with a force mg, the boiler mass times gravitational acceleration. This can be resolved into components perpendicular and parallel to the bed as shown, with values (cos 35^{o}) mg and (sin 35^{o}) mg respectively. The parallel component is the shear force tending to pull the boiler from its bed.

Wilding would have started from an estimated value for the shearing force and divided it by the weight to obtain the sine of the angle and thus ascertained the angle. We can do the opposite and calculate the shearing force from the angle.

The sine of 35^{o} is just over 0.57. We can omit the mass of the boiler to express the force purely in terms of deceleration and thus conclude that, if Wilding was correct, the boilers would shear from their beds at a deceleration of 0.57 g. When we plot this limit on Figure 2 we see that it would be reached at the end of the collision if the relative strength is 1.5 or more, and as the strength increases it would be exceeded sooner.

Wilding implied that the “machinery” would not move at this point but we don’t know what kinds of machinery he was thinking of. If he meant the main engines he was probably correct, but there are many other items of machinery that I suspect would not have been more securely held down than the boilers. The shifting of the boilers themselves would have caused secondary effects through fracture of high-pressure steam pipes and there could have been many more cascades of secondary damage from the movement of other machinery. I think we can reasonably regard the boiler bed limit as a point where major damage would commence throughout the ship and, even though only two compartments would have been directly opened by the impact, the survival of the ship becomes questionable beyond that limit.

A relative strength of 1.5 is equivalent to Wilding having underestimated by 33%. Therefore there is quite a small range of hull strength, representing an underestimate of between 20% and 33%, in which damage would not reach bulkhead “B” but before major damage would begin throughout the rest of the ship.

In the 1997 film “*Titanic*” Thomas Andrews utters the line “*I'm sorry that I didn't build you a stronger ship, young Rose”*. If the collision had been head-on perhaps that would not have been such a good idea!

Returning briefly to Figure 2, notice that in all cases the deceleration drops abruptly to zero at the end of the collision. Does this sudden *decrease* in force cause a “jerk” effect? It depends on what kind of object it acts on. You could think of it as a “pseudo-jerk”.

A decrease in force cannot impart any additional energy so the effect on objects in rigid contact is simply to remove any stress that they were under. Freely moving objects, such as items of furniture sliding along the floor, would simply slow down and stop. The problem arises with objects (or people) that have adjusted themselves into a new equilibrium with the deceleration force. Thus a hanging object that had swung forward would now be out of equilibrium and would suddenly swing back. In doing so it might overshoot and fall off its supports so, for example, some utensils and crockery hanging from hooks in the galley might fall off. The biggest problem would be for people who had either braced themselves or adjusted their balance to avoid falling over in the earlier stages. They would now be instantly out of equilibrium and might well fall in a sternward direction before they had time to react. Thus there would probably be another wave of injuries right at the moment the collision *ended*.

#### Speculative section – length of damage at various speeds

Let us now consider the effects of a head-on collision in relation to the speed of the ship. To do this we can use Equation 2, keeping the same value for the strength, K, but varying v to calculate the expected damage length. In Figure 4 I have plotted the results for a speed range from 5 to 24 knots. I have calculated the total damage length, as before, by increasing the crush distance by a third. Thus at a speed of 22.5 knots, we obtain 107 ft rather than Wilding’s optimistic 80 ft.

The maximum decelerations involved at these speeds range from 0.39 g at 22.5 knots down to 0.16 g at 5 knots, so no new concerns arise about the forces and I have not presented them here.

The first, minor, point of interest is that the damage distance reaches the “B” bulkhead limit at a speed of 20 knots. Even a very modest reduction in speed would have made it possible to limit casualties to the second compartment as Wilding predicted and it is possible that *Titanic’s* speed was already decreasing as a result of the engine orders given by Murdoch. All that means, though, is that perhaps Wilding’s prediction was correct. There would still have been a very substantial immediate death toll and it makes no difference to discussing Murdoch’s avoidance action.

It is of more interest to consider the speed at which damage would reach “A” bulkhead, as that is the point below which the ship could safely collide head-on without either suffering critical damage or sustaining significant human casualties. We find that this speed is about 11.5 knots which is half the designed maximum speed in normal service. Is this merely coincidence?

Bulkhead “A”, the forwardmost transverse watertight bulkhead, fulfils the function of what would later become more generally known as the “collision bulkhead”. Without going into the full details this is a bulkhead situated not less than 5% of the ship’s length aft of the sternpost, or 10 metres, whichever is less, and is constructed to more stringent standards than the other bulkheads. For example, it has no watertight doors and only a very limited number of essential structures may pass through it.

Many popular accounts state that the requirement for a collision bulkhead was introduced by the 1914 Convention for the Safety of Life at Sea (SOLAS) ^{4 }in response to the *Titanic* disaster. This is not so. The term “collision bulkhead” appeared in the report of the Bulkhead Committee of the Board of Trade in 1891. Those same rules, with minor amendments in 1905 and 1907, were those with which the *Titanic’s* design complied. Edward Harland, the founder of Harland & Wolff, was a member of the committee that drafted them. The relevant passages of the rules are quoted extensively in discussions between the Commissioner and the Attorney-General on day 21 of the British inquiry. ^{5} Much of the text is identical to the SOLAS definitions of 1914 and even much later. The subsequent modifications are mainly further definitions of what may be carried ahead of the collision bulkhead, clarifying for example that it may not include crew accommodation spaces. While the British Board of Trade regulations in 1912 may have been very badly outdated in terms of lifeboat provision it appears that they were well ahead of their time with regard to watertight subdivision.

The concept of the collision bulkhead is clearly to provide a second line of defence against water ingress in the event of a bow collision but at no point are there any rules or recommendations regarding the conditions, such as speed, at which such protection should be effective. It seems to me quite natural, though, that a naval architect might make a few calculations regarding this and possibly use them as a basis for a design ‘target’. Is it possible that the Harland & Wolff team were consciously aiming for a collision bulkhead fully effective at up to half speed, which would cover most of the scenarios in congested waters, such as harbour approaches, where collisions would be considered most likely? If so it might explain the “coincidence” of the 11.5 knot calculation with half the design maximum speed.

That may be over-speculation, but even if they had done the calculation with no particular target in mind it would mean that Wilding already had a calculated example at hand which he could scale up to obtain the 80-100 foot figure he volunteered at the inquiry. He might, in fact, have done no more than made a rough calculation that the damage at full speed would be a bit more than double that at half speed. Perhaps this is why he seemed to have ready answers to questions that might never have been asked had he not introduced the idea.

Regardless of the actual value of the bulkhead “A” speed calculation it is clear that at *some* speed damage would be restricted to the first compartment. The regulations did not require that such a ‘safe collision speed’ should be calculated with regard to the collision bulkhead and it is only speculation on my part that it was probably known to the H&W design team. If it *had* been required then no doubt it would have been published in the ship’s performance data. How might such knowledge have influenced the handling of the ship that night?

I very much doubt that Captain Smith would have reduced to half speed. The idea that one should do so on the grounds that it would make head-on ice impacts ‘safe’ for the passengers would have seemed ludicrous to him or any other ship’s master. It would, after all, still cause expensive damage to the ship so they would still prefer to avoid the collision and they nearly all believed that ice could be sighted far enough away to enable them to do this safely, even at full speed. Shipping lines certainly wouldn’t have encouraged it either. After all, ice was common in the North Atlantic and it would be an economic disaster to slow all ships to a crawl every time there were a few ice warnings. It is possible that a few lines, or a few more cautious captains, might have defined certain high-risk scenarios in which it was considered advisable to slow to ‘safe collision’ speed, but it would be a very uncommon course of action.

But suppose that conditions on the night of April 14^{th} 1912 had been considered to fit one of those high-risk scenarios and *Titanic* had indeed reduced speed to 11.5 knots. Previous discussions of First Officer Murdoch’s actions have been based on the full-speed scenario, in which he would have been very unlikely to opt for a head-on collision knowing that it would lead to a large number of certain deaths. At this lower speed, he might now know that the ship should be able to strike ice head-on without causing significant casualties and that he could mitigate the damage further by ordering engines to stop and take the additional precaution of closing the watertight doors. Of course, he would still prefer to take avoiding action if possible and would still have believed that ice would be seen at a sufficient distance to do so. What, then, should he do if an iceberg was seen dead ahead at a much shorter distance than he expected?

The intuitive answer is that he should still attempt to avoid the collision and would have a much better chance of success at this lower speed because the iceberg would be spotted ‘sooner’, giving him more time to make the turn. It is true that he would have more time, but the iceberg would only be seen ‘sooner’ in relation to the projected time of impact. It could not have been seen at a greater *distance*, so he would still have to achieve the turn within the same distance albeit over a longer time. This is not necessarily as achievable as it may appear.

The rate at which a ship changes its heading in a turn is determined by the sideways force on the rudder which is, in turn, determined by the speed of water flow over the rudder and therefore related to the forward motion of the ship. At a lower forward speed the rate at which the heading changes will also be lower. The two are not necessarily directly proportional and many other factors related to the hull geometry affect the rate of turn but, in many ships, the initial radius of turn and the diameter of the turning circle are much the same over a wide range of speeds.

Another factor of possible relevance is the arrangement of the turbine engine and centre propeller. At full speed, the centre propeller would be providing additional water flow over the rudder and augmenting the turning force, but the turbine normally only operated at “half ahead” or above on the engine orders telegraphs. This is not the same as half-speed. To run at 11.5 knots would actually require a somewhat lower rate of revolutions at which, normally, the turbine would not be running. Sustained running at half speed was not a normal procedure so we cannot be certain whether the turbine would have been engaged, but if the centre propeller was not turning the rudder effectiveness would be determined by the ship’s speed alone, giving further support for the suggestion that the turning radius might be no less than at full speed.

It is possible that the longer and slower turn might actually take the ship on a very similar track to the full-speed turn and a grazing or side-on impact with the iceberg would still be a distinct possibility

Of course, the kinetic energy of the ship at half speed would be only a quarter of that at full speed but that might be of little consequence. In another paper ^{6} Halpern calculated that energy dissipated in the full-speed event was just 1.5% of the initial kinetic energy, so the 25% still available at half speed would be more than sufficient to cause equally serious damage. It depends very much on the angle and position of the impact, and in that respect, there was a large element of ‘bad luck’ in the nature of the grazing collision.

It might be that Murdoch would recognise these uncertainties and realise that, from a strictly logical point of view, a head-on collision offered the surest way to avoid any deaths. I suspect, though, that he would still have attempted an avoidance manoeuvre because it offered the possibility of avoiding both casualties and damage to the ship, and the likelihood of *serious* damage would have appeared very small.

### Conclusions

- Wilding’s prediction of casualties being limited to firemen, trimmers and greasers is over-optimistic because the total length of damage from an 80 ft crush would extend beyond bulkhead “B”. The prediction that the ship would remain afloat remains valid
- If Wilding had overestimated the compressive strength of the hull by up to 50% or underestimated by up to 20% those modified predictions would remain the same.
- Even if he had overestimated by up to 100% the prediction that the ship could float would
*still*remain valid though the mortality, mainly among third class passengers, would become much higher and there would be risks to the ship’s command, control and communication systems. - If he had underestimated by over 20% the damage would not extend beyond bulkhead “B”, as he had hoped, but if his overestimate was more than 33% extensive damage would commence throughout the rest of the ship.
- A modest speed reduction to 20 knots would have made Wilding’s casualty prediction a realistic outcome.
- At half the design maximum speed the damage would be limited to entirely forward of bulkhead “A”. This may have been a calculated design figure.
- Even if such a “safe collision speed” had been known to the captain and officers it is unlikely that it would have altered decisions on the ship’s speed or the avoidance manoeuvre carried out.

### Acknowledgement

I would like to thank Samuel Halpern for discussing my ideas for further development of his mathematical analysis and for suggesting that I should write these up as a paper for *Encylopedia Titanica*. He also kindly reviewed the draft paper and suggested some valuable improvements, but any remaining errors are entirely my own responsibility.

### Appendix – summary of methodology

The essential principle underlying the calculations is the law of conservation of energy. The energy dissipated in bringing the ship to a standstill must equal the initial kinetic energy of the ship, so

(1) E_{dissipated} = E_{kinetic} = ½ Mv^{2}

The energy expended by a constant force would simply be equal to the force times the distance through which it acts, but since the force will vary with distance (as the cross-section of the ship changes) we must write this as the integral of force with respect to distance.

(2) E_{dissipated} = ∫ F dx

To solve this we need a function relating F to x, describing how the crushing force varies with distance from the bow. Standard textbooks on naval architecture such as Rawson & Tupper’s “Basic Ship Theory” ^{7} reveal that the compressive strength of a whole ship girder can be regarded as closely proportional to the area of the cross-section and that the areas of the profiles in curved bow sections, for most conventional bow designs, increase in proportion to a factor between the square root and cube root of the distance from the bow. We can therefore write:

(3) F = K x^{n}

Where K is some coefficient of proportionality and n lies between 0.33 and 0.5.

We will return later to selecting an appropriate value for n, but let us first consider the units and meaning of K. We can re-arrange equation (3) to obtain:

(4) K = F/x^{n}

The numerator is in units of force, but what are the units of x^{n} ? As x represents length it is tempting to think that the units should be length^{n}. However from the discussion of how we obtained equation (3) it is apparent that x^{n} is an expression describing a change in cross-sectional *area* per unit length. So although the arithmetic value is a power of length the result should *functionally* be expressed in units of area.

We can then see from (4) that K is in base units of force per unit area; the same units we would use to express the compressive strength of any material. K can therefore be considered simply as an aggregate compressive strength for the whole ship girder and could presumably be calculated from structural data by a naval architect.

Returning to the main theme, we can now combine equations (2) and (3) to obtain:

(5) E_{dissipated} = ∫ K x^{n} dx

Solving this between the limits x=0 to x=X, where X is the maximum length of damage, we obtain:

(6) E_{dissipated} = K X^{(n+1)}/(n+1)

And equating this to the kinetic energy we have:

(7) K X^{(n+1) }/(n+1) = ½ Mv^{2}

Which we rearrange to solve for K.

(8) K = ½ (n+1) Mv^{2}/X^{(n+1)}

(This can of course, be rearranged to solve for an unknown X if K is known, as discussed further below, and would then resemble the calculation which I suspect was done by Wilding.)

In metric units, using tonnes for mass and metres for length, the calculated value of K is in units of kilonewtons/sq.m.

Having obtained a value for K we can then calculate the maximum force at the end of the crushing event as F = KX^{n }and the corresponding deceleration from Newton’s second law, F = Ma.

It remains to choose an appropriate value for n, between the limits of 0.33 and 0.5. Halpern chose the value of 0.4 as the midpoint. (It is the mid-point of the denominators if you express the decimal values as fractions). One might question whether this is appropriate. It is not obvious on inspection of equation (8) whether small errors in the choice of n might greatly alter the value of K.

This is easily checked by a simple sensitivity analysis, calculating the differences that would be seen if n was at either of the limits of the range. For n = 0.5 the forces increase by 7% and at n = 0.33 they decrease by 5%. Although errors of this size might be important in a real-life, safety-critical scenario they are trivial in the hypothetical cases discussed in the paper.

We can therefore say that the results are relatively insensitive to the exact choice of n. Any value within the accepted limits would suffice.

To calculate the maximum crush distance at any lower speed, for the same compressive strength, we begin by rearranging equation (8) to give X as a function of M, K and v, thus:

(9) X = [(n+1) K Mv^{2}/2]^{1/(n+1)}

The maximum force and deceleration are both proportional to X^{n} so we can calculate the new F_{max} by scaling from the original values at 22.5 knots as follows:

(10) F_{max}(new speed) / F_{max}(22.5 knots) = [X(new speed) / X(22.5 knots]^{n}

The deceleration, either absolute or as a g force, scales by the same factor.

The approximation for the time scale in Figure 2 is simply to scale the time in proportion to the inverse square root of the force. This is based on the assumption that F ∝ a and thus proportional to 1/t^{2}. This would only be true if the acceleration increased *linearly* with time which it clearly does not, but for the purpose of illustrating that the initial slope of the curve becomes much steeper as the force increases and time decreases it is more than adequate, A hand-drawn sketch would have sufficed but rescaling the original curve to the new (correct) force end-point and very approximate time end point in PowerPoint was both quicker and more elegant!

### References

- Halpern, S., “Brace for Collision”, online publication, www.titanicology.com/Titanica/BraceForCollision.pdf
- The Official Transcript of the British Inquiry into the sinking of the RMS Titanic May 2-July 3, 1912, www.titanicinquiry.org/downloads/BritishInquiry.pdf
- Halpern, Samuel (2022)
*Prelude to an Allision*, independently published, ISBN 9798843039981 - Text of the Convention for the Safety of Life at Sea (SOLAS) 1914, archive.org/details/textofconvention00inte/page/n5/mode/2up
- British Wreck Commissioner's Inquiry, Day 21, (bulkhead discussion), www.titanicinquiry.org/BOTInq/BOTInq21Howell02.php
- Halpern, S., “The Energy of Collision”, online publication, www.titanicology.com/Titanica/EnergyOfCollision.pdf
- Rawson, K.J., Tupper, E.C., (2001)
*Basic Ship Theory 5*, Butterworth-Heineman, ISBN 0 7506 5398 1^{th}Ed.