Plating that gives way slows the water that slams it, but only on a flat bottom

Water climbs a falling wedge, and the load comes from the climbing found that the slamming force on a V-bottom is not the momentum of the water the hull strikes but the rate at which it recruits water: the free surface rises up the bottom ahead of the body, the wetted width grows faster than the geometry alone suggests, by Wagner’s factor of $\pi/2$, and the pressure is a narrow ridge riding at the spray root. A hull bottom feels the slam it has time to answer let one panel of the bottom’s plating, a strip between two stiffeners, bend under that ridge: its period is of the order of the time the ridge takes to cross it, so it overshoots its static stress by up to half.

That calculation was one-way. The plating bent, and its motion carried the water’s added mass, but the wetted width was the rigid wedge’s: the water climbed the bottom as though the plating had not moved. It ended on the feedback it had left out — a panel that bends inward as the ridge approaches changes the shape the water is climbing, and so the speed at which the ridge arrives.

The feedback slows the water, and the relief is set by the plating’s deflection against the bottom’s own rise across the panel. On a nearly flat bottom, where the bottom rises only a few centimetres across a panel, a deflection of a few millimetres is enough to hold the water back by a tenth of the panel’s width and take a seventh off the peak stress. On a deep V the same deflection changes nothing.

Wagner’s condition, with the plating in it #

Wagner’s theory treats the wetted part of the bottom as a flat plate pushing on the water and finds where its edge is by a geometrical condition: the free surface, lifted by the potential flow under the plate, meets the body where the body is. For a body whose height above its keel is $f(y)$, sunk a depth $h$ into the water, the wetted half-width $c$ satisfies

$$\int_0^{\pi/2} f(c\sin\theta)\,\mathrm{d}\theta = \frac{\pi}{2}\,h .$$

For a wedge, $f = y\tan\beta$, and this gives the rigid answer, $c = (\pi/2)\,h/\tan\beta$. For a bottom whose plating has deflected inward by $w(y)$, the body’s height is $y\tan\beta + w$: the water has further to climb over the panel. Differentiated in time the condition gives the speed at which the wetted edge moves,

$$\dot c = \frac{(\pi/2)V - \int_0^{\pi/2} \dot w(c\sin\theta)\,\mathrm{d}\theta}{\int_0^{\pi/2}\sin\theta\,\big(\tan\beta + w'(c\sin\theta)\big)\,\mathrm{d}\theta},$$

and both corrections slow it while the plating is being pushed in: plating moving away from the water subtracts from the rate at which the body sinks into it, and its inward deflection adds to the slope the water must climb at the panel’s leading side. When the plating springs back the signs reverse.

The model keeps everything else the earlier essay had. The strip is 0.3 metres of 10 millimetre aluminium, simply supported between stiffeners 0.2 and 0.5 metres from the keel of a wedge of half-beam one metre, in five bending modes; Wagner’s pressure is integrated through its spray-root singularity exactly; the water’s added mass for each mode is switched on as the strip wets. Only the wetted width, now a state of its own marched with the plate, is new; the two integrals over $\theta$ in its equation are taken by forty-point Gauss–Legendre quadrature, which resolves the modes’ shapes across the panel at every wetted width. With the feedback switched off, the march reproduces the earlier panel’s peak stress to $10^{-14}$, and a plate eight times thicker, which barely bends, gives the same answer coupled or not to five parts in ten thousand.

The relief, by deadrise #

The first figure is the result, and its comparison is chosen carefully. A panel’s stress depends on the impact speed, so comparing deadrises at one speed compares panels at very different stresses. Instead each deadrise is struck at the speed that loads the one-way panel to 200 megapascals — near the yield of marine aluminium, the largest load at which a linear elastic panel is a fair model — and the coupled panel is compared with it there: 2.27 metres a second at 4° deadrise, 3.94 at 10°, 5.8 at 20°.

At 4° the coupled panel peaks at 0.847 of the one-way panel’s stress. At 6°, 0.894; at 8°, 0.934; at 10°, 0.972; at 15°, 1.00; at 20°, 0.983. On a nearly flat bottom the feedback takes a seventh off the peak; on a bottom of ordinary deadrise it is a few per cent; on a deep V it is nothing.

The right-hand panel orders the same points by the deflection over the panel’s rise across its width, $w/(\ell\tan\beta)$: 0.19 at 4°, 0.06 at 15°. The deflections themselves hardly differ, between 4.0 and 4.8 millimetres, because every panel is at the same stress; what differs is the height the bottom climbs across the panel, five centimetres on a 10° bottom and two at 4°. A deflection of four millimetres against two centimetres of climb holds the water back; against five it barely does. The plating’s give matters in proportion to how flat the bottom it belongs to already is.

The water climbs more slowly #

The second figure shows the mechanism at 4°. The rigid wedge’s wetted width grows steadily, its edge running across the bottom at fifty metres a second — a flat bottom wets almost explosively, since the water has only a few degrees to climb. Over the panel the coupled width falls behind: at its largest the lag is 13 per cent of the panel’s width, and at the panel’s middle the water is climbing at 38.6 metres a second against the rigid wedge’s 50.9.

That is where the relief comes from. Wagner’s pressure has two parts, one from the hull’s deceleration and one, the spray-root ridge, proportional to the speed of the wetted edge, and it is the second that loads the panel. A wetting edge a quarter slower is a ridge a quarter lower and a little broader, and the panel meets it while it is itself already moving away. The relief is not the panel absorbing energy, which an elastic panel returns; it is the panel changing the geometry the water sees, so that less load arrives.

Only where the plating has stopped being linear #

The third figure holds the deadrise at the earlier essay’s 10° and raises the impact speed. At 2 metres a second the feedback changes nothing; at 4, the largest speed at which the one-way panel stays below 250 megapascals, it relieves 3.5 per cent; at 5, the earlier essay’s case, 12 per cent; at 7 and 8, about 18 per cent. The relief grows with the impact, because the deflection does.

But the right-hand panel measures the deflection against the plating’s thickness, and it says where the growth happens. At 4 metres a second the plating deflects half its own thickness; at 5, three quarters; at 7, one and a half. A plate that deflects more than about half its thickness is no longer a linear bending plate: it carries its load partly in tension, as a membrane, and stiffens. And at 5 metres a second the one-way stress is 361 megapascals, past the yield of any marine aluminium. On a 10° bottom, then, the feedback becomes large only where the plating has left the theory that computes it, both in its geometry and in its material. Within that theory it is a few per cent — which is the earlier essay’s one-way model’s justification on an ordinary hull, and its limitation on a flat one.

Later and lower #

The fourth figure follows the stress through the slam. At the dial’s first setting, 6° and 2.9 metres a second, the one-way panel’s stress climbs as the ridge crosses it and peaks at 200 megapascals; the coupled panel’s climbs a little later and peaks at 179, as the water it holds back arrives late and lower. After the ridge has passed both ring at the panel’s wetted frequency, at amplitudes that differ by about the same proportion. The dial shows the difference growing at 4° and vanishing at 10° and 20°.

A free hull gains more #

A real hull is not driven at constant speed; it decelerates as it recruits water, and a light hull slams softly found that a hull of a fraction of its own added mass loads its bottom far less. The fifth figure frees the hull at 6° deadrise. The coupled panel’s peak is 0.858 of the one-way panel’s for a hull of three tenths of its full width’s added mass, 0.818 for one of equal mass and 0.856 for three times it, against 0.894 at constant speed. A decelerating hull’s wetting edge slows as the slam goes on, so the panel spends longer under a slower ridge and its own deflection has more time to hold the water back.

A feedback with a sign #

The coupling is a loop: the load bends the plating, the bending changes the wetted width, the wetted width sets the load. Whether a loop of that kind relieves or amplifies depends on the sign of the plating’s response at the moment it matters, which is why the deadrise figure’s points are not all below one. While the ridge is approaching, the plating is being pushed inward and moving away from the water, and both corrections in Wagner’s condition slow the wetting: relief. Once the ridge has passed and the plating rings, it springs back towards the water, and the same corrections speed the wetting of whatever part of the panel the water has not yet reached. At 15° the two very nearly cancel, and the peak is unchanged; on a flatter bottom the approach dominates.

That makes the feedback’s structure the one the lag that makes flutter possible found in a wing: a fluid load that depends on the structure’s displacement and velocity, with a phase set by how the structure’s period compares with the time the load takes to act. Here the time is the wetting’s, and the earlier essay showed that the panel’s period against that time is the ratio that decides its overshoot. The feedback acts through the same ratio, and its sign can change with it; at the speeds and deadrises drawn here it relieves.

It also leaves the earlier essay’s design conclusion standing in a refined form. That essay found that a rule designing a panel for the peak pressure as a static load asks for half as much plating again as the dynamic panel needs. On a flat bottom the feedback widens that margin further, since the dynamic panel’s peak is lower again; on an ordinary bottom it does not change it.

Where flat bottoms slam #

Bottoms struck nearly flat are not rare. A planing boat’s bottom loses its deadrise towards the transom, where it often slams; a catamaran’s wet deck, the flat underside between its hulls, is struck almost exactly flat when a wave rises between them; and a ship’s flat bottom slams forward when the bow emerges in a head sea. In each, the water climbing the body climbs a surface whose own rise across a panel is a few centimetres or less, and the plating’s give is a large fraction of it. Those are the cases in which hydroelastic slamming has been studied most, and in which a rigid calculation is most conservative.

The pressure that does the damage is a pressure that depends on the past: it is the rate of change of the water’s momentum, and the water’s momentum is the mass the body has to borrow times its speed. The feedback is a change in how fast that mass is borrowed. A panel that slows the borrowing lowers the force, exactly as a hull that slows down does in the free-entry calculation, and for the same reason.

The convention: deadrise, deflection and the design stress #

Deadrise is the bottom’s angle above the horizontal at the keel. Deflection is the plating’s displacement into the hull, positive inward, and the panel’s rise is the height the bottom climbs across the panel’s width, $\ell\tan\beta$. Stresses are the largest bending stress anywhere on the strip’s surface, from the plate’s curvature. “One-way” is the earlier essay’s model, the wetted width the rigid wedge’s; “coupled” feeds the plating’s deflection and its speed back into Wagner’s condition. The comparison speed at each deadrise is the one at which the one-way panel’s peak stress is 200 megapascals.

What the calculation leaves out #

The pressure’s own change of form. The pressure keeps the form Wagner’s flat-plate potential gives, with the coupled wetted width; the plating’s own motion enters only as the added mass of each mode. A full hydroelastic solution would solve the potential problem with the deflected, moving bottom as its boundary.

Large deflection and yield. The plate is linear and elastic; the third figure shows where that ends.

The stiffeners and the hull girder. The strip is simply supported on stiffeners that do not move; real stiffeners bend under the slam too, more slowly, and the hull girder whips. A stiffener’s deflection lowers the bottom over two panels at once, which on a flat bottom would feed back into the wetting in the same way and over a longer length, and its period is long enough that the feedback would act through most of the slam rather than only as the ridge passes.

Air. A flat bottom traps air as it strikes, and the cushion delays and spreads the slam on just the bottoms where the feedback here is largest; the cushion that is there after all is the same physics in another setting.

Three dimensions. The hull is a two-dimensional wedge, struck flat; a real bottom panel is bounded by frames as well as stiffeners and is wetted from one corner.

Who worked it out #

Wagner’s theory of the expanding wetted plate is from 1932. Hydroelastic slamming — the bending of plating changing its own load — was studied by Faltinsen and co-workers from the late 1990s, who identified the panel’s natural period against the wetting time as the controlling ratio and showed that hydroelastic effects matter most at small deadrise. The calculation here adds one feedback, the plating’s deflection in Wagner’s condition, to the earlier essay’s panel and measures its effect against the deflection over the panel’s rise; the numbers are this model’s.

How it was checked #

With the feedback switched off the new march, which carries the wetted width as a state, reproduces the rigid wedge’s width and the earlier essay’s peak stress to $10^{-14}$. A plate 80 millimetres thick, which deflects a few hundredths of a millimetre, gives the same peak coupled or not to five parts in ten thousand, and its wetted width never lags the rigid one by more than two ten-thousandths of the panel. The Wagner pressure’s integral through its singularity was checked against the free-entry force in the earlier essay.

Still open: the cushion a flat bottom traps #

The feedback is largest exactly where a real slam is least like Wagner’s: on a nearly flat bottom, which strikes a layer of air before it strikes the water. The air is squeezed out from under the bottom at high speed, its pressure rises before contact, and it lifts the water surface into a dome that meets the bottom earlier and more gently. The next calculation would add an air layer of stated initial thickness under the flat bottom, compute its pressure from the squeezed flow and the water’s response to it, and ask whether the cushion, which acts before the wetting begins, leaves anything for the plating’s feedback to relieve — or whether on the flattest bottoms the air, not the plating, is what keeps the slam below its rigid value.