A hull bottom feels the slam it has time to answer

A light hull slams softly, and a round one does not let a V-bottomed hull slow down as it hit the water. The force that slams it also decelerates it, and the deceleration depends on the hull’s mass against the mass of water it has set moving. For a wedge the slamming force stops growing and peaks at a definite moment, when the water it has recruited weighs a fifth of the hull, so a light hull feels a smaller slam than a heavy one of the same shape. Every number there treated the hull as rigid and gave the force on the whole bottom.

A bottom is not rigid, and a naval architect does not design it for the whole force. The plating between two stiffeners is a strip a few tenths of a metre wide, and it is that strip that dents, cracks or fatigues. The strip has a natural period of a few milliseconds. The slam, at the speeds a fast boat meets, sweeps across it in a few milliseconds too. Two time scales of the same order mean the answer is neither the static one nor the impulsive one, and this essay works out which way it falls, for a panel on the same 10-degree wedge.

The answers: the panel overshoots the stress the same pressure would give statically, by up to about half; the overshoot depends on one ratio, the wetted panel’s period over the time the water takes to cross it; a design rule that applies the peak pressure uniformly asks for half as much plating again as the panel needs; and a light hull’s softer slam is gentler on its plating twice over, because its pressure is lower and the water crosses its panels more slowly.

A ridge that crosses the panel #

Water climbs a falling wedge gave Wagner’s solution for the wetted width: as a wedge enters, the water rises up its sides, and the wetted half-width grows π/2 times faster than the geometric one. The pressure on the wetted part, from Wagner’s flat-plate potential, is the pressure that depends on the past — the rate of change of the potential — and it has two parts. One is proportional to the hull’s deceleration and spread across the wetted width like an ellipse. The other is proportional to the hull’s speed times the speed at which the wetted width spreads, and it is concentrated at the edge, at the root of the spray sheet the water throws up the sides, where Wagner’s outer solution is infinite and his inner solution caps it at half the water’s density times the spreading speed squared.

The first figure is that pressure across the panel at four instants of a constant-speed entry at 5 metres a second. It is a narrow ridge riding on a broad floor. The ridge is 0.99 megapascals high, roughly ten atmospheres, and it moves outward at 44.5 metres a second, nine times the hull’s own speed, because a 10-degree wedge is nearly flat and a small descent wets a lot of width. It crosses the 0.3-metre panel in 6.7 milliseconds. Behind it the floor is far lower: by the time the ridge is near the panel’s far edge, the pressure at its keel-side edge has fallen to a quarter of a megapascal.

The singularity is not a numerical inconvenience to be smoothed but part of the physics. The total force on the wetted width is finite, and integrating the outer solution across it — through the infinity at the edge, by a change of variable that makes the integrand smooth — gives exactly the rate of change of the added mass’s momentum, the force the free-entry essay computed, to a part in 10¹⁶. So the panel can be loaded with the outer pressure as it stands, and the cap matters only to anyone who asks what the pressure is at one point.

A panel with the water’s mass on it #

The plating is modelled as a strip of 10-millimetre aluminium spanning 0.3 metres between two stiffeners, 0.2 and 0.5 metres out from the keel of a hull of half-beam one metre, simply supported at each, and described by its first five bending modes. Each mode is a spring and a mass. The spring is the plate’s bending stiffness; the mass is the plate’s own, plus — once the water reaches it — the added mass of the water it must push when it bends. The mass a body has to borrow applies to a panel as much as to a hull. A sinusoidal ripple on a plate of wavelength twice the panel’s span carries a layer of water nearly a third of the span deep, which for this panel is more than three times the plate’s own mass.

So the strip’s natural frequency falls from 272 hertz in air to 128 once it is fully wetted, and its period rises to 7.8 milliseconds. The water crosses it in 6.7. In this calculation the added mass switches on in proportion to the share of the panel already wetted. The panel’s motion does not feed back into the pressure’s form or the wetted width — a one-way coupling — which is the simplest model that keeps the term that matters most and is named as a limit below.

It overshoots, by up to half #

The second figure follows the largest bending stress anywhere on the panel through the slam, and beside it, dashed, the static stress the same instantaneous pressure would give a panel with no mass at all. At constant speed the static stress peaks at 287 megapascals, when the ridge is about three quarters of the way across. The panel’s own response peaks at 361, a quarter higher and a little later, at 10.3 milliseconds. It has been loaded in less than its own period, and it overshoots, as a spring loaded suddenly does. The dashed line at 215 megapascals is a typical proof stress for marine aluminium plate: the constant-speed panel is well past it.

The third figure shows that the overshoot is not a property of this particular hull and panel. It plots the dynamic peak over the static peak against a single ratio, the wetted panel’s period over the time the wetted width takes to cross it, for entries at constant speeds from half a metre a second to fourteen, under a 10-degree and a 20-degree wedge.

The two deadrises fall on one curve. When the water crosses slowly, the ratio is small and the panel follows its static stress exactly; at a ratio of one it overshoots by about a tenth; the overshoot is largest, 1.49 times the static stress, at a ratio of 2.1, when the ridge crosses in about half the panel’s period; and beyond that it begins to fall again, as the load starts to look like an impulse the panel answers only after it has passed. The curve is the slam’s version of the dynamic load factor every structural engineer knows for a suddenly applied load, and its peak below two reflects the moving ridge: the panel is never loaded everywhere at once.

The far end of the curve, where the slam is an impulse and the stress grows only as the first power of the entry speed rather than its square, is the regime Faltinsen called hydroelastic. This calculation stops before it, because at high speeds the chine wets before the panel has finished answering, and the flow round the chine, which Wagner’s solution does not contain, decides the rest.

A light hull is gentler on its plating twice over #

The free-entry essay found that a light hull slams with less force. The fourth figure asks what that does to the panel. It plots the panel’s peak stress against the hull’s mass in units of the added mass of its full width, with a dial over the entry speed.

At 5 metres a second, a hull as heavy as the added mass of its full width brings the panel’s peak down from 361 megapascals to 245, and one of three tenths of it to 132, under the proof stress. Two things are at work. The hull is slower when the water reaches the panel, so every pressure is lower: by the time the wetted width is at the panel’s middle the light hull has lost nearly a third of its speed. And the slower hull spreads its wetted width more slowly, 31.6 metres a second across the panel’s middle against 44.5, so the water takes longer to cross and the panel is closer to following its static stress. The overshoot is barely changed in this case — 1.25 against 1.26 — because the panel sits near the keel and is crossed before the hull has slowed much; a panel further out, crossed later in the slam, would gain more from the second effect.

The dial shows how strongly all of it depends on the speed. At 3 metres a second the constant-speed panel peaks at 110 megapascals, and at 7 the light hull’s panel passes the proof stress too. Every stress grows close to the square of the entry speed while the water crosses slowly, and the overshoot grows with it as the ratio of periods rises.

So the free-entry essay’s question — whether a light hull’s softer slam is also a gentler one for its structure — has a clear answer. It is, and by more than the force alone would say, because its slam is slower as well as smaller.

Designing for the peak pressure #

There is a simple way to size plating against slamming that does not need any of this: find the largest pressure the water can exert, apply it uniformly over the panel, and make the panel strong enough to carry it statically. It is conservative by construction, and the dashed upper curve in the fourth figure shows how conservative. At constant speed it gives 670 megapascals against the panel’s 361; for the light hull, 338 against 132.

The fifth figure turns that into plate thickness. To keep the constant-speed panel under 215 megapascals takes 12.2 millimetres; the peak-pressure rule asks for 17.6. For the light hull, 8.2 against 12.5. In both cases the rule asks for about half as much plating again as the panel needs, and plating is most of a planing hull’s bottom weight, which is the weight that sets how hard it slams. A rule that is conservative about stress is therefore not conservative about the slam: it adds mass, and on a hull light enough for its own inertia to soften the slam, mass is the thing that made it softer.

The opposite shortcut errs the other way. Wagner’s own pressure, applied statically, gives 287 megapascals for the constant-speed panel, a fifth below the panel’s actual response. A design that used the outer solution with no allowance for the panel’s dynamics would under-size it. The rules naval architects actually use for planing craft sit between the two: a design pressure well below the peak, averaged over the panel, with factors calibrated against hulls that cracked and hulls that did not — which is a measured answer to exactly the question this essay computes.

What was checked #

The pressure integral through the spray root was checked against the closed form for the entry force at three states, with and without deceleration, to a part in 10¹⁵. A very slow entry onto a thick panel was checked to give a dynamic stress equal to the static one, to two parts in ten thousand, which tests the modal dynamics against its own quasi-static limit. The impulse delivered to the panel was checked to double when the entry speed doubles, as it must at constant speed, since every pressure scales as the square of the speed and every time as its inverse. The two frequencies, 272 hertz dry and 128 wetted, follow from the plate’s stiffness and the water’s added mass. The calculation refuses an entry at no speed and a panel beyond the chine.

The checks above, like the force of getting going and slow enough to be steady before them, test the model’s internal consistency. Nothing here was checked against a drop test, and the absolute stresses should be read with that in mind.

What the calculation leaves out #

The coupling is one-way. The panel’s deflection changes the shape the water meets: a panel bending inward lowers the local deadrise and speeds the wetting. That feedback is the essence of full hydroelastic slamming and would raise the pressure on a flexible panel. It is largest for the thin, soft panels of the overshoot’s far end, and this calculation stays short of them. The lag that makes flutter possible showed what a two-way coupling between a structure and the flow round it can do when the flow’s answer arrives late: it can feed the structure’s motion rather than damp it. A slammed panel is loaded once rather than continually, so it cannot flutter, but the same reasoning says the feedback’s sign is not settled by intuition and has to be computed.

The panel is a strip. A real panel is a plate supported on four sides, between stiffeners along the hull and frames across it, and its stresses are two-dimensional; a strip gives the stress across the short span, which is usually the largest.

The plate is linear and elastic. Past its proof stress, aluminium yields, a panel takes a permanent set, and membrane stretching starts to carry load that bending alone would not. The calculation’s stresses above 215 megapascals mean the plate would dent, not that it would carry 361.

The chine ends the calculation. After the water reaches the chine the flow separates from its edge and Wagner’s solution stops applying. For the panel here, near the keel, the peak comes well before.

The lesson for the slam #

The pressure on a slamming hull is often quoted as its peak, because the peak is the number a pressure gauge records and the number that frightens. The panel does not feel the peak. It feels the ridge’s passage, integrated over the time it has to answer, and how much that matters is a single ratio of two times. The momentum with no value found that an impulsive pressure is best read through what it does over its duration rather than at an instant; the slammed panel is the structural version of the same reading.

It also changes what a measurement of slamming means. A pressure transducer a few millimetres across, set flush in a hull bottom, records the ridge as it passes and reports a spike close to Wagner’s inner peak; a strain gauge on the plating beside it records the panel’s answer, which is smoother, later and, measured against the spike, far smaller than a static calculation from the spike would predict. Both readings are right. They are answers to different questions, and the one a designer needs is the second, which is a property of the panel and the slam together rather than of the water alone.

Who worked it out #

Herbert Wagner’s water-entry solution, with its spray-root singularity and the π/2 width, is of 1932, worked out for seaplane floats. The added mass of a vibrating plate in water was measured and computed through the 1950s for ship hull vibration. Odd Faltinsen showed in the late 1990s that slamming on flexible panels at small deadrise is governed by the panel’s natural period against the wetting time, and that in the hydroelastic regime the maximum strain grows with the entry speed itself rather than its square; the classification societies’ rules for planing craft encode design pressures calibrated against service experience.

Still open: a panel that shapes its own slam #

Every pressure here is the rigid wedge’s. A panel that bends inward as the ridge approaches presents the water with a locally flatter bottom, which spreads the wetted width faster and raises the pressure on itself. The next calculation would couple the panel’s deflection into Wagner’s wetted-width condition, so that the wetting and the bending are solved together, and ask how far into the hydroelastic regime the overshoot curve continues before the feedback reverses it — whether there is a thickness below which a softer panel is stressed more, not less, because it invites a harder slam.