FEA Based Level 3 Assessment of Deformed Tanks with Fluid Induced Loads
1 s2.0-002980189090003 o-main
1. Ocean Engng, Vol. 17, No 3, pp. 201-233, 1990. 0029-8018/90 $3.00 + .00
Printed in Great Britain. Pergamon Press plc
b
CDC
CLb
Cp
Cn
f (A)
m'
m"
PLANING AND IMPACTING PLATE FORCES AT LARGE
TRIM ANGLES
PETER R. PAYNE
Payne Associates, 108 Market Court, Stevensville, MD 21666, U.S.A.
Abstract--Added mass theory has been shown to give excellent agreement with experimental
measurements on planing surfaces at normal planing angles [e.g. Payne, P.R. (1982, Ocean
Engng 9, 515-545; 1988, Design of High-Speed Boats, Volume 1: Planing. Fishergate. Inc.,
Annapolis, Maryland)] and to agree exactly with more complex conformal transformations
where such a comparison is possible. But at large trim angles, it predicts non-transient pressures
that are greater than the free-stream dynamic pressure and so cannot be correct. In this paper,
I suggest that the reason is because, unlike a body or a wing in an infinite fluid, a planing plate
only has fluid on one side--the "high pressure" side. So the fluid in contact with the plate
travels more slowly as the plate trim angle (and therefore static pressure) increases. This results
in lower added mass forces than Munk, M. (1924) The Aerodynamic Forces on Airship Hulls
(NACA TR-184) and Jones, R. T. (1946) Properties of Low-Aspect-Ratio Pointed Wings at
Speeds Below and Above the Speed of Sound (NACA TR-835) originally calculated for wings
and other bodies in an infinite fluid.
For simplicity of presentation, I have initially considered the example of a triangular (vertex
forward) planing plate. This makes the integration of elemental force very simple and so the
various points are made without much trouble. But the penalty is that there seem to be no
experimental data for such a configur.ation; at least none that I have been able to discover. But
at least the equations obtained in the limits of zero and infinite aspect ratio, small trim angles
(~) and "r = 90° all agree with established concepts and the variation of normal force with trim
angle looks like what we would expect from our knowledge of how delta wings behave in air.
I then employed the new equation to calculate the force on a rectangular planing surface at
a trim angle -r, having a constant horizontal velocity u,, and a vertical impact velocity of k.
This happens to have been explored experimentally by Smiley, R. F. [(1951) An Experimental
Study of Water Pressure-Distributions During Landings and Planing of a Heavily Loaded
Rectangular Flat Plate Model (NACA TM 2453)] up to trim angles of "r = 45°, and so a
comparison between theory and experiment is possible. The results of this comparison are
encouraging, as is also a comparison with the large trim angle planing plate measurements of
Shuford, C. L. [(1958) A Theoretical and Experimental Study of Planing Surfaces Including
Effects of Cross Section and Plan Form (NACA Report)].
As two practical applications, I first employed the new equations to calculate the "'design
pressures" needed to size the plating of a transom bow on a high-speed "Wavestrider" hull.
The resulting pressures were significantly different to those obtained using semi-empirical design
rules in the literature. Then I used the theory to critically review data obtained from tank tests
of a SES bow section during water impact to identify how the "real world" of resilient deck
plating diverged from the "model world" of extreme structural rigidity.
NOMENCLATURE
beam
the cross-flow force coefficient, which in this analysis turns out to be equal to unity
R cos T/~pu~bz
np/qo (= Cn) the average pressure coefficient
R/qo S, the normal force coefficient on wetted area
an aspect ratio correction which fairs slender wing theory into two-dimensional theory
added mass per unit length
added mass (per unit length) at the maximum span or trailing edge
201
2. 2( 2 P.R. P,x~qf:
P
p~
qo
R
S
Svv
U
blo
V
X
Y
Y
Z
O~
"7
z~p
p
'T
4,
static pressure
ambient static pressure
lp u~,, the dynamic pressure associated with the horizontal velocity ,,,
the dynamic pressure associated with the absolute velocity (u7~, v ~7:,)at the momenl
of impact
force normal to the body axis
body plan area viewed normal to its major axis. In planing, the nominally "wetted"
area made by the intersection of the undisturbed water surface and the plate
actually wetted area of a planing surface (S~ > S because of splash-up)
velocity parallel to a body's axis (Fig. 1)
free-stream velocity or plate horizontal velocity at impact
velocity normal to a body's axis (Fig. 1)
distance along the body axis (Fig. 1)
body half width
maximum body half width, or span
draft
vertical velocity dz/dl
vertical acceleration dez/dt2
angle through which a stream tube of air is deflected by a finite aspect ratio wing
sin "r + ~/u~, cos 1", an "'equivalent planing angle"
p-p-,
vertex half angle of a triangular wing or planing plate
mass density of the fluid
trim angle (Fig. 1)
an unknown constant which is found to be equal to C~r.
INTRODUCTION--THE BASIC PROBLEM
To CALCULATEthe normal force on a slender body, Munk (1924) resolved the free-
stream velocity into a component v, at right-angles to the body's axis and u, parallel
to the axis. The plane normal to the body is a form of Trefftz plane, but through the
body instead of behind it, as in the original work of Trefftz (1921). Munk (1924) then
assumed that the cross-flow in any given Trefftz plane was two-dimensional (not
influenced by the cross-flow at other longitudinal positions) so that the elemental force
dR could be expressed as
d ,
dR = dt (m v) dx (1)
where m' is the two-dimensional added mass for the cross-flow in plane x. (m' = ~rp
y2 for an elliptical cylinder, for example, including the limiting case of a fiat lamina.)
0 -~- "
uosJn l"
O~V = U o SJl'l "L"
FIG. |. Munk's (1924) velocity components.
3. Planing and impactingplate forces 203
Performing the differentiation in Equation (1)
dR dv dm'
-- = m' --. (2)
dx dt +v dt
Munk (1924) was concerned only with steady-state forces, so he ignored the transient
dv/dt term. And for the remaining term, he wrote
dR dm' dm' dx dm' dx
dx-V~=V~-dt =u°sin~ dx dt' (3)
On the assumption that the axial flow velocity was u = Uo cos "r, he wrote
dx
dt - Uo cos "r (4)
and thus obtained
dR 2dm' 1 2dm' .
- Uo ~ sin ~ cos r = ~ Uo ~- sin 2"r (5)
which can be integrated directly to give
1 2 ,
R = ~ Uo mx sin 2~ (6)
where m" is the value of the added mass (per unit x) at the maximum span (Jones,
1946).
So far, we have covered familiar ground; and for small values of "r, Equation (6) is
known to give excellent agreement with experiment (Payne, 1988).
The average loading Ap = /5 - t% on the body will be
R 1 2 ,Uomx .
Ap S 2 S sm2"r (7)
where S is the area of the body projected normal to the velocity vector v. To put values
into this, assume that the body is a fiat plate wing, of span 2Y, so that
m" = "troY2 (8)
and
1 'trY2
Ap = ~ pu~ ~- sin 2-r.
If, for example, the planform is triangular (y = x tan 0),
y2 X 2tan 20
- - tan 0
S X 2tan 0
and
A p = qo rr tan 0 sin 2-r
for a fully immersed triangular plate.
(9)
(lO)
4. 204 P.R. PAVNL
r X -,
Scheme 1.
For a planing plate, the added mass approaches half the value given by Equation (8)
as the angle "r~ 0, but has an additional "cavity term" (Payne, 1981) for finite angles.
Let us ignore this additional term for the moment and write
~rtan 0
Ap = q,, ---f sin 2"r for a triangular planing plate (11)
then for -r = 45°, sin 2 = 1, and
0 = 0° 10° 20° 30° 40°
Ap_ 0 0.277 0.572 0.907 1.318.
qo
It was the anomaly of Ap > qo that presumably led to the idea that added mass
theory is only valid for slender surfaces and bodies at small trim angles.
The effect of axial velocity
The axial velocity u will actually diminish in regions of high static pressure and
increase in low pressure regions. A body of revolution has both high pressure (upwind)
and low pressure in its lee, so the average axial velocity is not much different, perhaps,
to Uo cos -r. But the planing plate only has fluid in its "high pressure" side.
At the leading edge (of a triangular planing plate, for example) the axial velocity
will clearly be u = Uo cos -~. But as a normal force develops when moving aft, the local
average static pressure will increase to
dR 1 1 1
A/~ = dx 2y = 2 pu~,cos: -r - 2
by Bernoulli. So, from Equation (3)
uv dm' 1 1
___ = 2 ~ IX2
2y dx 2 9u°c°s~'r- 2 9
and
vu dm'
U 2 "}- . . . . . . U2 COS2T = 0
yp dx
pU 2 (12)
(12a)
5. Planing and impacting plate forces 205
SO
and
dx v dm' A(v dm'l 2
u = It = 2yp dx + ~12y9 dx ] + uI c°s2 "r. (13)
dx
Substituting for dt and v in Equation (3) gives
dR 2dm' ./ 1 dm' 1 dm' .
- Uo~ sin "r ~cos 2 'r + sin 2 "r sin "r.
2py dx 2py dx
For the triangular plate, where conventional slender plate thoery gives
m r ,/1-= ~ p tan 2 0 x2
dm t
dx - ~rp tan 2 0 x = 7rpy tan 0
(14)
1 dm' -:r
tan 0.
2py dx 2
Therefore,
iR 2 ' 'IT
= Uomxsin • cos2T + tan e sin2T - ~ tan 0 sin -~
/1 2,. )i 7r 'rr
= ~ Uomx sin 2"r 1 + ~(tan 0)2tan2 "r- ~ tan 0 tan ~" (15)
-- (Munk's result) × (Bernoulli correction).
The Bernoulli correction is plotted in Fig. 2 for various vertex angles 0. Applying
this to Equation (11)
qo 2tan0sin2"r 1+ tan0 tan2~-~tan0tan'r. (16)
Figure 3 compares this modified theory with Munk's, for slender triangular plates,
and shows that it predicts substantially reduced pressures above about T = 20-40°. For
a trim angle of 45°, Fig. 4 shows that the average pressure coefficient (or force
coefficient) reaches a maximum of 1/2 at the limit of 0 = 90°, instead of the obviously
incorrect value of infinity given by the Munk theory.
The cavity and cross-flow complications
So far, we have treated the problem in a very simple way. In fact, there are additional
normal force components due to the added mass of the cavity and to the cross-flow,
which gives additional normal force components, so that we then have
dR dm'
dx - uv ~ + Cocqo2y sin2 T (17)
6. 206 P.R. PAv~
0.8
oo ,",4,
0
0.2
0 20° 40° 60 ° 80°
TRIM ANGLE ~-
F=G. 2. The Bernoulli correction to Munk's (1924) equation, as a function of trim angle, for triangular planing
plates; see Equation (15).
CR = ZX5
qo
0.6
MUNK'S (1924) THEORY (~ fen e sin 2r )
PRESENT THEORY, EQ. (16)
LINEAR APPROXIMATION
0.4
0.2
S
0 = 15°
"
O 20 ° 40 ° 60 ° IO°
TRIM ANGLE "t"
Fl6. 3. Average added mass pressure component as a function of trim angle for triangular planing plates.
7. Planing and impacting plate forces 207
MUNK(1924)
c.= ~ -~ ton eq.
2.5
2.0
1.5
/I.O /
/ PRESENTTHEORY
EQUATION 06)
O 20* 40* 60* 80*
PLATE VERTEX HALF ANGLE O
FIG. 4. Average added mass pressure at • = 45°, as a function of plate vertex angle.
where Coc is the cross-flow coefficient, normally assumed to be the value determined
by Bobyleff (1881) = 0.88 for a flat plate with a free-streamline separated wake in an
infinite fluid. Also we have, for the added mass
where
m' ~_ 2 pyZ (1 + sin "Of(A)
is an empirical "aspect ratio" correction obtained by Payne (1981).
The local static pressure will now be governed by
dR 1 1 2 1 2 - uv din'
AI~ - ~ ~y - d~22pUo - 2 pu 2y dx + Coc qo sin2 I" (18)
where the value of + was cos2 "r before, but is now initially unknown. Re-arranging
Equation (18) gives the quadratic relationship
v dm'
u2 + -- ~- u + UZo(Coc sin2 "r - +) = 0. (19)
PY
8. 2118 P.R. PAYNE
If we use +qo = qo cos2 T as the upstream dynamic head, Equation (19) would give
u = 0 at ~ = tan ' (1/X/Q)c-) or "r = 46.38 ° for Co~. = 0.88. This is obviously unrealistic.
If + = 1, u > 0 when "r = 90°. The only value which gives u --~ 0 as -r --~ 90° is 6 --
CDC. But then, in the limit "r --~ O, u/uo --~ X/G~.. So the only value of Co<, which
gives sensible results at both angular limits is CDC = 1.0 (Fig. 5).* In what follows,
we shall retain Co<, instead of replacing it with unity in order to preserve generality.
The solution to Equation (19) is, therefore
vdm' ~/(2~y d~') 2u = - - + + sin 2 ~') (20)u,2CDc (1 -
2py dx
Making the substitution in Equation (17)
-dx = 2yCDcq°sin2"r+vd~ ' 2py + bloCD(.2 (1- sin2.r)- 2pyV . (21)
For the triangular planing plate,
m' "rr ( 2 sin[ t
=~ptane0x 2 l+~tan0/f(A )
dm' ( 2 sin "r
~lx = rrp tan 2 0x 1 + 3 ian O)f(z) (22)
U
Uo C0$ T
0.8~-
0.6
0.4
0.2
I
i
i
I
20* 40o
TRIM ANGLE T °
60* 80*
U
FlG. 5. The ratio -- for a flat plate. CDC = 1. The circles are cos "r, so u =u,, cos2 "r in this example.
/'/o COS I"
• dm'
Equation (19) gives u = u,, cos ,, if ~ = 0, so the difference is due to the added mass term in this example.
* Oddly enough, Bollay (1937, 1939) concluded CDC = 2.0 for a wing (and hence CDc- = 1.0 for a planing
surface) on entirely different theoretical grounds.
9. Planing and impactingplate forces 209
and
v dm' ~r [ 2 sin'r
2py dx -uosin'r~tan0 11 + ~)f(A).
Since the square brackets of Equation (21) do not contain any terms in x, direct
integration gives
rr (2sin'r)
R = CocSqosin2"r + U2osin'r~pY 2 1 + 3 tan-0 f(A) ×
+ CDc(1 - sin2,r) ~..
~2Oy / 2fly t.ld(,
and
2 sin ,r
CR = Coc sin2"r+ 7r sin r tan 0 1 + ~ tanOn0) f(A) x
~/~fpy + Coc (1 - sin2r) 2py " (23)
Equation (23) is plotted in Fig. 6 for various vertex angles. This looks very reasonable
and like what we see with delta wings, except for the latter's dip in normal force
between r -~ 30-60 ° because of the bursting of the air vortex structure above the wing
and the concomitant loss of upper surface suction. See, for example, Fig. 7 on p. 18-5
of Hoerner and Borst (1975).
~"t"
CR I"01~7"~ ~ C D C
o.,I o I/ I
0"6~ ~T/
0.4
I// //I ./ I
20° 40~ 60* 80°
TRIMANGLEINDEGREES
FIG.6. Normalforce coefficienton a triangularplaningplate, accordingto Equation (23).
10. 21(i P. R, P.',',~t:
Note that in the limit T ~ 0
10CR tan 0 A
---> = (24)
7r 0~ l + tan 2 0 /16 + A 2
which is correct in the limits A ~ 0 and A --> ~. Also for infinite aspect ratio (no
cross-flow or cavity), Equation (16) becomes
[,/ I2 ]CI~ = :2 sin 2-r 1 + tan T - 2 tan "r . (25)
As shown by Fig. 7, this agrees with small angle theory (CR = ~r'r). Remarkably,
instead of going off to some infinity as "r -~ 90°, the solution "stalls" like a real wing,
at a believable C~.,,,,.,and "r....... and then the force coefficients drop off to zero at -r =
90°.
Rectangular flat plate with vertical velocity
The added mass at a location x behind the still water surface intersection is
approximately
m ('2)2( )= ~ p 1 + ~/~ sin "r f(A) (26)
7r-(
r
o. il
0.3 CL~~
02. ~
0.1
O 20° 40° 60° 80°
TRIM ANGLE IN DEGREES
Fro. 7. Predictions of the theory for the added massnormal and lift coefficients at infinite aspectratio. (No
cavity or cross-flow terms.)
11. Planing and impacting plate forces 211
where
SO
,/ 2
f(A)= l/ 1+
dm t 'IT
dx - 6 pb sin "rf(A)
v = Uosin r + 2 cos r = `/Uo
where `/ = sin -r + -- cos r
Uo
v dm' rr .
2py ~- - 6 sin r `/Uo.
(say)
(27)
(28)
(29)
(30)
using the splash-up equation of Savitsky and Neidinger (1954).
Because of the vertical velocity ~r, the variation of added mass with time is
~-= 1 +0.3~ ~>1
=l.6-0.3eb (b <1)
where
Once again, the square bracket in Equation (21) does not contain terms in x, so that
direct integration gives
R=CDcSwqo`/a+u2o`/2 9 1 + 3b sin~ f(A)
`/sin T + CDC (1 -- `/2) _ 6 `/sin "r (31)
_ R Sw ~b ( 42.)
CR qoS- s Coc,/2 + `/~ e 1 + ~ ~ sin -r f(A)
× ~ `/sin "r + CDC(1 -- `/2) _ 6 `/sin "r (32)
which for planing (2 = 0) simplifies to
)CR = CDC sin2"r+ ~ ~ sin "r 1 + ~ ~ sin -r f(A)
x sin 2T + CDC (1 -- sin2~) - ~ sin2"r (33)
13. Planing and impacting plate forces 213
and
ARI - sin "r dt 2 p
[[ 2}1+ ~ ~sm'r+ l+~sinv
(38)
A comparison with Smiley's experiments
Smiley (1951) dropped a heavy, one-foot wide plate into the water when travelling
at a speed Uo. He read acceleration data from accelerometers whose accuracy was only
--+ 0.2 g, linear to about 100 Hz. He integrated this to obtain velocity and again to
obtain draft, the latter also being measured independently with a resistance bridge
device. As Table 1 shows, the accuracy of the accelerometer was not particularly good
by 1987 standards. His overall accuracy estimates were as follows:*
Horizontal velocity:
Initial values for landings (ft./sec) . . . . . .
Time histories for planing runs (ft./sec)
Initial vertical velocity (fl./sec) . . . . . . . . .
Draft (ft.) . . . . . . . . . . . . . .
Model weight (lb.) . . . . . . . . . .
Vertical acceleration (g) . . . . . . . . . .
Time (sec) . . . . . . . . . . . . . .
+ 0.5
-+1
-+ 0.2
-+ 0.03
-+2
--- 0.2
-+ 0.005.
In Fig. 8, we have compared Equations (32) and (38) with Smiley's data (by integrating
the equation of motion) indicating the accuracy limits in the usual way. It seems clear
that predicted acceleration velocity and immersion fall well within the experimental
range, particularly as the initial impact velocity can vary by - 0.2 feet/sec.
Figures 9-13 contain additional comparisons between theory and those experiments
in which more than two or three data points were obtained. An analysis of this
comparison shows that theory is always within Smiley's data accuracy tolerance and
that the errors are randomly distributed with respect to both impact speed and
immersion depth Froude number Uo/X/~. So on this evidence, the theory seems to be
correct.
The average pressures associated with these calculations are given in Fig. 14 and as
can be seen, pressure not only exceeds the dynamic head ½pU2obut also exceeds that
absolute value ½p(U2o+ Zo) by a significant margin. These pressures are based on the
actually wetted area.
* Smiley did not give a tolerance on trim angle setting, which is not as easy to control as one might think.
14. 214 P. R. P.x'~Ni
TABI,E 1. SMII_EY'S I)RAFI READIN(iS
Run
Run No. 1
Run No. 2
Run No. 3
t (scc)
0.007
0.(117
0.029
0.071
O.0O8
0.017
O.O27
0.054
0.072
0.086
0.091
0. 138
(}.0l(I
0.02 l
0.030
0.056
0.073
0.085
0.089
0.123
Error
I).33
0.06
0.0
-0.067
{).33
{).33
0.125
0.20
0.12
0.(155
0.1t)
0.19
0.0
0.17
0.11
0.13
0.11
0.10
0.095
(I.042
= 60, w = 1,176 lb.
Error = bridge circuit rcading-accclcrometer inlegjation
bridgc circuit reading
IC
¢.D
z 0.8
z
_o
~ 0.6J
~_ 0.4
z_
~ 02
~A
SMILEY'S[ 0 IMMERSION
DATA ~ [] VELOCITY
(1951) k • ACCELERATION
Y
=
//
4~
//
/
>
:/
z
o
z
43
o
2~
uJ
0 .05 O. I 0.15 0.2
TIME iN SECONDS
Fro, 8. Accuracy analysis of Smiley's run 14, using his formal precision data: 1 = 15°, w = 1,176 lb.
4.1 ft./sec, u,, = 39.3 -+ 0.5 ft./sec.
18. 2h~ P.R. PAV~.
A comparison with Shuford's planing plate measurements
Shuford (1958) tested flat planing plates up to trim angles of 34° and his results are
compared with both the classical Munk calculation and the theory of this paper in Figs
15-19. It is unfortunate that no data are available for trim angles larger than 34° because
the two theoretical approaches are only just beginning to diverge significantly at this
angle, thanks to the dominant effect of cross-flow.
It is also unfortunate that Shuford did not measure the draft of his models, so that
we have to infer it by using the splash-up equation of Savitsky and Neidinger (1954).
With that caveat, the present theory appears to give a low estimate of the force below
T = 30° and a slight over-estimate at 34° for l,Jb > 2.0. Figure 19 puts this in the
perspective of the total angular range from 0-90 °.
BRASS 0 WITH WINDSCREEN
BRASS [] NO WINDSCREEN
BRASS <~ NO WINDSCREEN-NO SPRAYSCREEN SHUFORD
PLASTIC ~, (1958)
• WEINSTEIN 8 KAPRYAN
MUNK (1924)
CLB THEORY
0.7
o.o S/
.ej0.5
0.4 /
0.5
0.2
OA
0 2 4 6 8 I0
WETTED LENGTH / BEAM
Fro. 15. Comparison between theory and experiment. Flat plate at -r = 12°, C, = 18.2, C/~( = l.
21. Planingand impactingplateforces 221
A practical application
The equations presented replicate Smiley's data to within the accuracy of his
measurements, and so may be employed to calculate forces on impacting surfaces in
other situations. It so happens that two of my recent designs, Figs 20-23, have bows
which to a first approximation may be regarded as rectangular flat plates at trim angles
of 45°! While there is an obvious urgent need to determine realistic "design pressures"
for the plating of these bows, I have not been able to find anything at all helpful in
the literature concerned with "design pressures" to help me with this.
To simplify the calculation, I considered only vertical motion (no pitch) with the
plate impacting on a calm water surface with a vertical velocity of 10 ft./sec. Taking
the weight associated with the bow to be one-fifth the total displacement, or 10,000 lb.
acting on the bow, Fig. 24 shows the submerged motion of the bow at various speeds,
and Fig. 25 the peak average pressure (normal force divided by area actually wetted,
including the splash-up area) as a function of speed and bow plate angle. Notice that
impact pressure varies as Vn, where 1 < n < 2.
Figure 26 gives the variation of average pressure with bow immersion. Damping
contributes a substantial hysteresis loop.
In Figs 27 and 28, we can see the effect of varying the weight carried by the plate.
Comparing Fig. 27 with Fig. 26, we see that peak acceleration occurs much later in the
impact than maximum average pressure; an observation first made, I believe, by Heller
and Jasper (1961), "... the maximum effective pressure for the entire boat does not
occur when accelerations are greatest". Not surprisingly, the greater the load, the larger
the plate's penetration into the water. But in Fig. 28, we see that the pressures are not
much affected by the load carried, only the vertical acceleration. And then most
remarkable of all, the peak vertical acceleration is not much affected by trim angle.
A comparison with some SES bow panel impact tests
During the 3K SES program, a full-size bow panel was fabricated with the same
scantlings as those intended for the prototype vehicle.
The model weighed 740 lb. (after run No. 20) and had a beam of (about) 44 in.
Gersten (1975) describes the test set-up; Band (1977) gives some of the results for
~o = 18.76ft./sec
:?o(= Uo) = 70 ft./sec
for • = 0.75°, 2°, 5° and 10°. Here we have considered 2°, 5° and 10° because of the
obvious air entrainment and water smoothness problems at the lowest angle.
Considering the problems of "ringing" and local dynamic deflection locally, the
agreement between the theory (of this paper) assuming that the flow separated cleanly
at the knuckle and acceleration measurements (Figs 29-30) is probably acceptable.
Failure to reach the "r = 2° peak acceleration of 59 g is possibly due to air entrainment,
but it is more probably due to a soft accelerometer mount (< 2,000 Hz) or an indication
that rigid body response cannot be expected of such a model when the excitation force
peaks in 1 msec. Note that in all the cases, the integral of the experimental data (with
respect to time) is roughly the same as for the calculated curve. Thus, the calculations
must be giving realistic velocities and displacements.
23. FIG. 21. The 24-foot Wavestrider in action on the Chesapeake Bay.
UmYEI'rRID£R
!
FroG.22. Side and plan elevations of the 60-foot Wavestrider.
223
24. FI(,. 23 The 60-foot Wavestrider. Enterprise.
o~ I [
I
O6
0.5
0.1
•0.5 0.1 0.15 0.2
TIME IN SECONDS
0.4
z
~0.3
0.2
TS
0.25
FIG. 24. Immersion-time history for a vertical impact at 10 ft./sec, v = 45°, A = l(J,000 lb., beam = 20 ft.
Figures on the curves give speed in knots.
224
26. 226 P.R. PA'JNI-
0.5
0.4
0.3
F-
W
W
U-
Z
~ o.z
Q
0.1
MAXIMUM IMMERSION
--IMMERSION AT PEAK ACCELERATION
O 5,000 IO,OOO 15,OO0 20,000
WEIGHT tN POUNDS
r=45 o
T =15°
FIG. 27. Immersion for peak acceleration and maximum immersion as a function of load on a flat plate
impacting vertically at 10 ft./sec. V = 50 knots, beam = 20 ft.
In Fig. 31, we compare the measured stagnation line velocity with calculations. At r
= 2°, the measurements fall substantially below the C,,/f = 1 curve, and the same trend
is clear for T = 5° and 10°. Assuming no errors in data reduction (the original data and
analysis have not been located), this probably means that the model is "whipping"
(pitching-up) during the impact.
The comparison between measured and calculated pressures is given in Figs 32-34.
Once again, the original data are not available, and E.G.U. Band has remarked of the
average pressures that they were '%.. some kind of average calculated by Rohr".
Particularly for 1 --- 5°, they imply twice the acceleration measured and calculated in
Fig. 29.
The maximum pressures were calculated from
p =½ova2
where Vs is the speed of the stagnation relative to stationary water axes. This speed
can be calculated as follows:
Uot
Scheme 2.
27. Planing and impactingplate forces 227
70
60
ca
z
_~5o
r~
U.I
_1
o° 40 . 1 PRESSURE,,~ pE,.~.~,~_~.~K I T = 45 °
~ 30
z
-- pE/~K ~ : 15°
z_
,,, /
~.) I( -
li,l F= 15°
n.. .T=45o(3_
0 5,000 I0,000 15,000 20,000
WEIGHT IN POUNDS
FIG. 28. Peak average pressure and peak acceleration as a function of load on a flat plate impacting vertically
at 10 ft./sec. V = 50 knots, beam = 20 ft.
In time t, the stagnation line has moved a horizontal distance:
z ew
S= Uot + - - -
tan r t
and therefore
ds z ~w
V__,Sn=dt=uo+ tan-r t "
So, if Vs were the absolute speed of the stagnation line
Cem.x-- ~
2pUO
Band (1977) obtains a slightly larger value for the absolute velocity Vs because the
stagnation line is on the plate, above the water, at a height
,)sin, 1/
29. Planing and impacting plate forces 229
RUN 40,T=2"
5,4o c~
II
i
b
20 i
I0.
_B
I i
1J.002 .0~ .006 .008
ELAPSED TIME IN SECONDS
.OI
Fro. 30. Comparison between theory and measured acceleration for "r = 2°.
0
,u I00
_---J
800
k-
600
o 400
_J
z 20C
o_
QT'-2° - - T FROM SAVITSKY ETAL
Or = 5" O
/~T =10" . . . . . ~ = 1.0
[]
r-
C []
O
[] []
~-. T-_IO o
T= 2 °
'~ r = 5°
io 15 zo~
DISTANCE FROM TRAILING EDQE IN INCHES
FIG. 31. Comparison between calculated and measured stagnation line velocity relative to the trailing edge.
30. 230 P. R, PA'¢NE
z
g
z
z_
0.
.5ck !
/ i o apM~
[] APAvG
-- J~,WFROM SAVITSKY
I . . . . . . . . . __~. I. ET AL
300 ............. ~ ................. "-- ~- : ,.0
2OC ~ j - ~ .................. O~. . . . . . . .
APMAx
i 0 =
15C ~ .....................
I
",. ]
10c
5 I0 15 20
WETTED LENGTH IN INCHES
FIG. 32. Maximum and average pressures for "r = 10°.
Reference to Fig. 31 shows that the measured stagnation line velocity was, for
T = 2 °
at ~w = 2 4.8 8.0 in.
measured Vs = 760 730 400 ft./sec
sothat ½9V~ = 3930 3627 1089 lb./sq, in.
But the measured Apmax = 250 500 500 lb./sq, in. if one draws a line through the
data points in Fig. 34. Possibly, the pressure transducer diameter was too large to
record the peak pressure ½pV~ or possibly could not respond to the nsec duration of
its presence.
This comparison shows that it is just as important to have a good theory to check
an experiment with, as is the converse. In the present case, the theory checks well with
31. Planing and impactingplate forces 231
z
_z
900
800
700
600
400
0 ApMAX
J~ FROM SAVITSKY
x, El" AL
,ew
i-- E = t.O
-~"'-~. O!
0
0
!_1
~,<~
%.
0 ~PMAx
" O
iO0 ~ "" ~, ~, E3
~- ~ .~-vG
0 5 I0 15 ZO
WETTED LENGTH IN INCHES
Flo. 33. Maximumand averagepressures for • = 5 °.
Smiley's experiments with an essentially rigid model and then alerts us to problems and
inconsistencies in similar tests of a flexible model.
CONCLUSIONS
The Munk/Jones approach to calculating the added mass normal force on a body has
been modified to account for the fluid being on only one side (the bottom) of a planing
plate and the fact that it must be moving significantly slower than freestream when the
plate is developing high static pressures at large trim angles. The equations resulting
from this modification are well-behaved at the unlikely limits of infinite aspect ratio
and 90° trim angle, as well as for more practically reasonable values.
The difference between the new equations and classical theory is not significant for
trim angles less than about 20°. Even for "r= 45°, the total calculated force on a planing
plate is reduced by only 20% if the length-to-beam ratio is large. The effect of the
modification on added mass force is large, but the total force is dominated by the cross-
32. _3_ P.R. PAv~,l
6,000
• 5,00C
0
4~00( - - - -
z
z_ 5,00C - - --
a. 2,00¢
1,000
. . . . . . . [- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-------L. F
o i !o o
o o i
5 I0 15 20
WETTED LENaTH IN INCHES
FIG. 34. Maximum and average pressures for -r : 2°.
flow force component at these large trim angles. At smaller length-to-beam ratios,
where the cross-flow component of lift is less important, the new theory predicts a
markedly smaller lift than the Munk formulation.
The theory explains the impacting plate drop tests conducted by Smiley to within his
stated experimental error limits. It is less satisfactory in explaining Shuford's planing
plate data at high length/beam ratios. This discrepancy may possibly be due to the fact
that his nominal wetted length has to be inferred, using a relationship of unknown
validity.
Applying the equations developed to a practical "real boat" problem, we find that
few of the semi-empirical rules--rules admittedly meant to apply to conventional boat
lines--for estimating "design pressure" seem to be correct for this rather unusual
problem. The pressures developed are not related to the weight associated with the
surface nor do they vary as V2.
Finally, we have used theory to critically review data obtained from tests with a "real
world" SES bow section, built like the real ship, and so necessarily somewhat resilient.
The comparison seems to indicate that the accelerometer is ringing on its mount and
too softly mounted to follow the maximum acceleration at "r = 2°. The model seems to
be "whipping" (bow up) during the impact, so that the stagnation line velocities are
lower than they would be for a rigid model. Also, a combination of inadequate
frequency response and/or too large a diameter results in the pressure gauges reading
peak pressures (at "r = 2°) which are an order of magnitude less than we would expect
for a rigid model, and almost an order of magnitude less than they should be, based
on the observed stagnation line velocity. It would have been difficult to detect these
problems without the theoretical comparison.
33. Planing and impacting plate forces 233
Acknowledgements--I am greatly indebted to John D. Pierson and E.G.U. ("Bill") Band for many helpful
suggestions and some significant corrections during the course of this investigation.
REFERENCES
BAND, E.G.U. 1977. Contributing to design criteria report slam and wave-induced analysis report: comparison
of impact/planing methods. In StructuralDesign CriteriaReport. Rohr Marine, Inc., Chula Vista, California
(CDRL No. E02V).
BOBVLEFF,D. 1881. Journal of the Russian Physio-ChemicalSociety, xiii. In Lamb's Hydrodynamics, 1945,
6th Edition, p. 104, Dover Publications.
BOLLAY, W. 1937. A theory for rectangular wings of small aspect ratio. Presented at the Fluid Mechanics
Session, Fifth Annual Meeting. I.Ae.S. (January).
BOLLAV, W. 1939. A contribution to the theory of planing surfaces. Proceedings of the Fifth International
Congress for Applied Mechanics, pp. 474-477. John Wiley, New York.
DANAHY,P.J. 1968. Adequate strength for small high-speed vessels. Mar. Tcchnol. (January), 63-71.
GERSXEN, A. 1975. Impact experiments with a full scale wet deck panel of a two thousand ton SES. Naval
Ship Research and Development Center, Bethesda, Maryland. SPD-P-534-01 (August).
HELLER, S.R. and JASPER, H. 1961. On the structural design of planing craft. Trans. R. lnstn, nay. Archit.
103, 49-65.
HOERNER, S.F. and BORST, H.V. 1975. Fluid Dynamic Lift: Practical Information on Aerodynamic and
Hydrodynamic Lift. Liselotta A. Hoerner, Brick Town, New Jersey.
JONES, R.T. 1946. Properties of low-aspect-ratio pointed wings at speeds below and above the speed of
sound. Report NACA TR-835.
MUNK, M. 1924. The aerodynamic forces on airship hulls. Report NACA TR-184.
PAYNE, P.R. 1981. The virtual mass of a rectangular flat plate of finite aspect ratio. Ocean Engng 8, 541-545.
PAYNE, P.R. 1982. Contributions to the virtual mass theory of hydrodynamic planing. Ocean Engng 9,
515-545.
PAYNE, P.R. 1988. Design of High-Speed Boats, Volume 1: Planing. Fishergate, Inc., Annapolis, Maryland.
SAVZTSKY,D. and NEIDINGER,J.W. 1954. Wetted area and center of pressure of planing surfaces at very low
speed coefficients. Sherman M. Fairchild Publications Fund Paper No. FF-11. Institute of the Aeronautical
Sciences, New York.
SHUFORO, C.L. 1958. A theoretical and experimental study of planing surfaces including effects of cross
section and plan form. NACA Report 1355.
SMILEY, R.F. 1951. An experimental study of water pressure-distributions during landings and planing of a
heavily loaded rectangular flat plate model. Report NACA TM 2453 (September).
TREFFTZ, E. 1921. Prandtlsche traglfachen- und propeller-theorien. Z. angew, math. Mech. 1, 206.
![Ocean Engng, Vol. 17, No 3, pp. 201-233, 1990. 0029-8018/90 $3.00 + .00
Printed in Great Britain. Pergamon Press plc
b
CDC
CLb
Cp
Cn
f (A)
m'
m"
PLANING AND IMPACTING PLATE FORCES AT LARGE
TRIM ANGLES
PETER R. PAYNE
Payne Associates, 108 Market Court, Stevensville, MD 21666, U.S.A.
Abstract--Added mass theory has been shown to give excellent agreement with experimental
measurements on planing surfaces at normal planing angles [e.g. Payne, P.R. (1982, Ocean
Engng 9, 515-545; 1988, Design of High-Speed Boats, Volume 1: Planing. Fishergate. Inc.,
Annapolis, Maryland)] and to agree exactly with more complex conformal transformations
where such a comparison is possible. But at large trim angles, it predicts non-transient pressures
that are greater than the free-stream dynamic pressure and so cannot be correct. In this paper,
I suggest that the reason is because, unlike a body or a wing in an infinite fluid, a planing plate
only has fluid on one side--the "high pressure" side. So the fluid in contact with the plate
travels more slowly as the plate trim angle (and therefore static pressure) increases. This results
in lower added mass forces than Munk, M. (1924) The Aerodynamic Forces on Airship Hulls
(NACA TR-184) and Jones, R. T. (1946) Properties of Low-Aspect-Ratio Pointed Wings at
Speeds Below and Above the Speed of Sound (NACA TR-835) originally calculated for wings
and other bodies in an infinite fluid.
For simplicity of presentation, I have initially considered the example of a triangular (vertex
forward) planing plate. This makes the integration of elemental force very simple and so the
various points are made without much trouble. But the penalty is that there seem to be no
experimental data for such a configur.ation; at least none that I have been able to discover. But
at least the equations obtained in the limits of zero and infinite aspect ratio, small trim angles
(~) and "r = 90° all agree with established concepts and the variation of normal force with trim
angle looks like what we would expect from our knowledge of how delta wings behave in air.
I then employed the new equation to calculate the force on a rectangular planing surface at
a trim angle -r, having a constant horizontal velocity u,, and a vertical impact velocity of k.
This happens to have been explored experimentally by Smiley, R. F. [(1951) An Experimental
Study of Water Pressure-Distributions During Landings and Planing of a Heavily Loaded
Rectangular Flat Plate Model (NACA TM 2453)] up to trim angles of "r = 45°, and so a
comparison between theory and experiment is possible. The results of this comparison are
encouraging, as is also a comparison with the large trim angle planing plate measurements of
Shuford, C. L. [(1958) A Theoretical and Experimental Study of Planing Surfaces Including
Effects of Cross Section and Plan Form (NACA Report)].
As two practical applications, I first employed the new equations to calculate the "'design
pressures" needed to size the plating of a transom bow on a high-speed "Wavestrider" hull.
The resulting pressures were significantly different to those obtained using semi-empirical design
rules in the literature. Then I used the theory to critically review data obtained from tank tests
of a SES bow section during water impact to identify how the "real world" of resilient deck
plating diverged from the "model world" of extreme structural rigidity.
NOMENCLATURE
beam
the cross-flow force coefficient, which in this analysis turns out to be equal to unity
R cos T/~pu~bz
np/qo (= Cn) the average pressure coefficient
R/qo S, the normal force coefficient on wetted area
an aspect ratio correction which fairs slender wing theory into two-dimensional theory
added mass per unit length
added mass (per unit length) at the maximum span or trailing edge
201](https://image.slidesharecdn.com/1-s2-140707081052-phpapp01/85/1-s2-0-002980189090003-o-main-1-320.jpg)
