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NAVY DEPARTMENT
DAVID TAYLOR MODEL BASIN

WASHINGTON, D. C.

HYDRODYNAMIC FORCES ON AN ANCHOR CABLE

by

L. Landweber

November 1 9^7

Report R-317

I

â€˘ 03

no. 3/7

HYDRODYNAMIC FORCES ON AN ANCHOR CABLE

ABSTRACT

The holding power required of an anchor for a ship anchored in shoal
water with a length of anchor line at least five times the water depth, i.e.,
with a scope of five, can normally be assumed to be equal to the estimated
drag of the ship. Anchoring in deep water necessitates a relatively shorter
anchor line, which results in a considerable hydrodynamic force on the anchor
cable and a tension in the cable much greater than the drag of the ship.

Curves have been computed from which the magnitude and direction of
the tensions in the anchor cable can be determined when the drag of the ship,
the velocity of the current, the depth of the water, and the type and length
of the anchor cable are known. Formulas are given for ship drag, current
parameter, breaking strength of wire-rope and chain cables, safe working loads
on cables, and holding power of an anchor. An illustrative example applies
these calculations to the determination of diameter and length of a wire-rope
anchor cable and of size of anchor required in a given problem.

INTRODUCTION

In March 1 9^5 the Bureau of Ships (1)* requested the David Taylor
Model Basin to calculate the forces acting on the anchor cable of a ship. An
anchor selected for a ship must have sufficient holding power to take the
greatest loads normally encountered in service. Ordinarily, however, it is
not practicable to carry anchors which will hold a ship under any storm con-
ditions and on any types of bottom. Likewise the length of anchor cable
carried is limited by practical considerations. The holding power of models
of several types of anchors in different types of sea bottom has been investi-
gated experimentally (2) (3). Also, tests of the holding power of full-scale
anchors of new designs have been conducted by the Norfolk Naval Shipyard from
time to time, as requested by the Bureau of Ships.

The holding power required of an anchor for a given ship is usually
computed for the standard conditions of a 70-knot wind and a 4-knot tide in
the same direction. It is desirable where practicable to anchor with a length
of anchor line at least five times the water depth, i.e., with a scope of five
or greater. With such a scope, under the standard conditions of wind and cur-
rent, an anchor cable would make a small angle with the horizontal at both the
anchor and the ship, resulting in a tension in the cable at the anchor very

* Numbers in parentheses indicate references on page 12 of this report.

nearly equal to the drag of the ship. Consequently the required holding pow-
er of an anchor in depths of water where a proper scope can be used may be
assumed to be equal to the estimated drag of the ship.

When anchoring in deeper water, however, it is not practicable to
pay out a length of line which is five times the depth of the water. Under
this condition, the hydrodynamic force on the anchor cable cannot be neglect-
ed, and the tension in the cable may be much greater than the drag of the
ship. An anchor with a holding power greater than the estimated drag on the
ship would then be necessary to prevent dragging of the anchor. Since this
condition would also impose greater tension in the cable, the cable size
should be selected accordingly.

To meet tne various conditions of anchoring, curves have been com-
puted from which the magnitude and direction of the tensions in the anchor
cable at the anchor and at the ship can be determined when the drag of the
ship, the velocity of the current, the depth of the water, and the type and
length of the anchor line are known. The application of these curves is il-
lustrated by a numerical example .

CHARTS OP FORCES ON AN ANCHOR LINE

Figure 1 shows diagrammatically and serves to define the geometrical
quantities and the forces in the anchor line which are employed in the sub-
sequent figures and discussion.

The laws of force on a cable in a stream are well known (4). On
each element of the cable the hydrodynamic forces consist of a component
normal to the cable, whose magnitude diminishes as the square of the sine of
the inclination of the element with the horizontal, and a tangential compon-
ent whose magnitude is small compared with the normal component, except when
the inclination of the cable is very small.

A general solution for the shape and tension of a cable in a ptream,
when the weight of the cable or the tangential force component or both are
taken into account, cannot be expressed in simple analytical terms. It is
usually necessary to express these solutions in terms of new functions, de-
fined as integrals, for which tables must be computed. However, the use of
these tables is as simple as the use of tables of the trigonometric functions,
and by their use numerical solutions of cable problems can be obtained readily,
The application of such tables to the solution of cable problems in which the
weight of the cable can be neglected has been illustrated by numerous examples
in a recent Taylor Model Basin report (5).

New charts have now been constructed in which the weight of the
cable is not neglected; see Figures 2, 3, 4, and 5- The tangential component

Woterline

Ship

Figure 1 - Dimensions and Forces on an Anchor Cable

Here D is the horizontal component of tension in the anchor line at the ship (drag of
the ship), in pounds,
L is the vertical component of tension in the anchor line at the ship, in pounds,
T = VL'l + D't, the tension in the anchor cable at the ship, in pounds,
T = T - WY, the tension in the anchor cable at the anchor, in pounds,
W is the weight per unit length of the anchor cable in water, in pounds per foot,

Y is the depth of the water, in feet,

^0 is the angle of the anchor line with the horizontal at the anchor, in degrees,
M = R/w, the current parameter,

R is the drag per unit length of the anchor cable, 0.34- V 2 d for wire rope and 0.20
V^h for chain,

V is the current speed, in knots,

d is the diameter of the wire rope, in inches,

h is the outside width of a link of chain, in inches, and

S is the length of the anchor line, in feet.

of the hydrodynamic force on the line is neglected in these computations, but
the resulting error is partly offset by the assumption that the velocity of
the current is the same at all depths; actually the current velocity is known
to approach zero rapidly near the bottom.

Figures 2, 3, 4, and 5 are polar diagrams of the vertical and hori-
zontal components of the cable tension at the waterline, expressed as dimen-
sionless quantities in terms of the weight WY in water of a length of anchor
line equal to the depth. Plots of this type are called polar diagrams be-
cause the polar coordinates, i.e., the radius from the origin and the angle
of the radius with the D/WY axis, give the magnitude of the tension and the
inclination of the anchor cable with the horizontal at the ship.

In anchoring a ship it is considered desirable to pay out enough
line to permit the cable to be horizontal at the anchor. The reason for this
is that the holding power of an anchor falls off linearly with the angle of
inclination of the cable at the anchor; the holding power is reduced by one-
half for an angle of 30 degrees with the horizontal, as shown in Figure 13
of Reference (3). Consequently, it is desirable to have a method for esti-
mating the length of line needed to ensure that the cable will be horizontal
at the anchor. Figure 2 was devised for this purpose.

H = I0

7

5

'0.2-

o5

^<D.*

â– -~oÂŁS

s^\

^^

<<i .0*

Figure 2 - Forces in an Anchor Cable with Anchor Line
Horizontal at Anchor [<j> a = 0)

These curves also apply when part of the anchor line is lying on the bottom.
In this case the value for S in Y/S is the length of cable above the bottom.

15

10

M = 10-
7-
5-
3-
2-
1-

o.i-

b

6

y =5.0

5

4

fc

4>o

25
D
WY

Figure J - Forces in an Anchor Cable for a Scope of 5-Â°

3-

\0 ,

^

8

D

_5 ,

zS

i

%.

s

â– f^Z.

"o>

Figure 4 - Forces in an Anchor Cable for a Scope of 3.0

Expressed in functional form, Figure 2 gives a graphical presenta-
tion of the functions

L _ F { D

WY'")

and

S ~ F \WY '^)

when the angle of the anchor line at the anchor, <f> , is zero. Here n is a
current parameter, which is defined in Figure 1 and discussed in a following
section. When the drag of the ship, D, the weight per unit length of the
line in water, W, the depth of water, Y, and the value of fi are given, then,
from Figure 2, the values of Y/S and L/WYcan be read, and hence the length
of cable, S, and the corresponding downward load L at the ship can be deter-
mined such that O = 0.

When the depth of water is very great, it may be impracticable to
pay out the length of line required to obtain a zero angle at the anchor. In
any case, however, it is desirable to keep the inclination of the cable and
its tension at the anchor as small as possible. Consequently, for preliminary
design purposes, it is desirable to have a method for determining the tension
and angle at the anchor for various assumed types and lengths of anchor cable.

&

â– \J

<bs

1 -h i^

r ^r

A.<

Ir"/

s^\$>

^ r

/ A

H

â€˘f - 2.0

^s%

if

i '

w

Figure 5 - Forces in an Anchor Cable for a Scope of 2.0

Figures 3, 4, and 5 were prepared for this purpose. The components of the
tension at the ship are also included in these figures to ascertain whether
the safe working load of the cable is exceeded.

Figure 3 is a graphical presentation of the functions

WY = G \WY "")

for S/Y = 5. Figures 4 and 5 present the same function for S/Y = 3 and 2
respectively. Hence, when D, W, Y, 11, and S are given, L and 4> can be
determined.

The tension at the ship, T, is determined from the length of the
radius vector T/WY to the point D/WY, L/WY. The tension at the anchor, T ,
is then given by

T = T - WY [la]

WY WY 1 [1b ]

In using the charts to solve anchor problems it will be necessary
frequently to estimate ship drags, to determine the weight per foot and the
safe working load of anchor cable of given types, and to compute the current
parameter fi and the required holding power. To facilitate the use of the
charts, simple approximate expressions for these quantities have been derived
and are assembled in the following section.

ESTIMATE OF SHIP DRAG

The drag of a ship due to current can be calculated from the approx-
imate formula

D c = 0.12 V 2 VU

where D c is the drag due to current, in pounds,

I is the length of the ship at the waterline, in feet,
A is the displacement of the ship, in tons, and
7 is the current speed in knots.

The drag due to wind can be calculated from an approximate formula
proposed by Captain E.F. Eggert, USN (Retired),

D a = 0.0022 B 2 V 2

where D a Is the drag due to wind, in pounds,

B is the beam of the ship, in feet, and
V a is the wind speed in knots.

The drag of the ship will be greatest when V a and V are in the same direction.
In this case the drags will be additive and the total drag D is

D = D a + D c = 0.0022 B 2 V 2 + 0.12V 2 VTZ 

EVALUATION OF CURRENT PARAMETER

The parameter n is defined as

"-#

where R is the drag per unit length of the line when the cable is normal to
the stream and W is the weight per unit length of the line in water .

The following recommended values of R are based on tests made at
the Taylor Model Basin:

R = 0.34V 2 d pound per foot, for wire rope [3a]

R = Q.2QV 2 h pound per foot, for chain [3b]

where d is the diameter of the cable in inches and h is the outside width of
a link of chain in inches.

If anchor cables of different sizes were geometrically similar, W
would be proportional to the square of the width of a link. The data in engi-
neering handbooks and manufacturers' catalogues, corrected to give weight in
water, show a small variation. Approximately, however,

W = 1 .40d 2 pound per foot, for wire rope [4a]

W= 0.64/i 2 pound per foot, for chain [4b]

From Equations  and , the expressions for n can be written as

V = ^^2 V z = 0.24 \ for wire rope [5a]

and

\$\$fe V* = 0.31 \ for chain [5b;

BREAKING STRENGTH AND SAFE WORKING LOAD OP CHAIN AND WIRE ROPE

Theoretically, the breaking strength of geometrically similar cables
should vary as the square of the width of a link. However, data in handbooks
and catalogues show a small variation with size. The following approximate
values for the breaking strength T B are recommended:

T B = 70,000d 2 pounds, for plow-steel wire rope

T B = 4,000/i 2 pounds, for forged stud-link anchor chain

Then, from Equations [4a] and [4b]

T B

yy = 50,000 feet for plow-steel wire rope [6a]

and

T

w = 6 250 feet for forged stud-link anchor chain [6b]

iased on a factor of safety of approximately 3> the values of the ratio of
tne safe working load T s to W may be taken as

T

Tjf = 16,000 feet for plow-steel wire rope [7a]

and

T

yf = 2000 feet for forged stud-link anchor chain [7b]

When 7 is given, the ratio T s /WYcan be computed from one of the
Equations . A circle of radius T s /WYon any of the polar diagrams deter-
mines the region within which the anchor-line tensions are under the safe
working load but outside of which they are unsafe.

REQUIRED HOLDING POWER

The holding power of an anchor falls off rapidly as the angle of
the anchor line with the horizontal at the anchor is increased from zero. The
variation of the holding power with this angle has been investigated (2), and
the results are reproduced in Figure 13 of Reference (3). Expressed analy-
tically, the ratio of the holding power H^ at a finite angle O to the hold-
ing power H at O = is given by the formula

-]f- = 1 - 0.017400

H = ^*o 

1 - 0.017400

where O is in degrees. In using Equation  to design an anchor, if^ is
set equal to the maximum value of T that is anticipated.

APPLICATION OF CHARTS

When the drag of the ship, D, the weight per unit length of cable
in water, W, the depth of the water, Y, the speed of the current, V, the type
and size of the anchor cable have been given, and the value of n has been
computed, the charts can be applied directly. The length of line and the
tension corresponding to a zero angle of the cable with the horizontal at the
anchor can then be determined from Figure 2. If this indicates that the
length of cable for O = Â° ls impracticably long, then Figures 3, 4, and 5
can be applied to find the tensions in the line and the angle of the cable
at the anchor for lengths of line corresponding to S/Y = 5. 3, and 2. The
points spotted as solutions on the charts should be examined to ascertain
whether they lie within the circle of safe working loads.

In an anchor problem where a suitable size of anchor cable must be
determined it will be necessary to repeat the foregoing procedure for various

10

assumed sizes of cable. The optimum solution is the size of cable which will
be loaded up to the safe working load under the given conditions.

EXAMPLE

Consider a ship whose displacement is 2000 tons, whose length is 360
feet, and whose beam is 40 feet, anchored in water 300 fathoms deep. Assume
a 4-knot tide and a 70-knot wind, both in the same direction. It is desired
to find the diameter and length of a wire-rope anchor cable and the size of
anchor required. From Equation , the drag of the ship is

D = 0.0022 x (40) 2 x 4900 + 0.12 x 16 V360 x 2000
= 17,200 + 1630 = 18,830 pounds

The forces on the anchor will be examined for cable diameters of 1 ,
1.5, 2, and 2.5 inches. For convenience in the subsequent calculation, the
values of W, WY, n, and D/WY for these diameters and for Y = 1800 feet have
been computed from Equations [4a] and [5a] and assembled in Table 1. From
Equations [6a] and [7a] the values of T B /WYand T a /WY are found to be 27.8
and 8.89 respectively for all diameters.

TABLE 1
Values of W, WY, n, and ^

d

inches

W

pounds
per foot

WY
pounds

V

D
WY

1.0

1.5

2.0

2.5

1 .40

3.15
5.60

8.75

2,520

5,670

10,080

15,740

3.84
2.56
1.92
1.54

7.47
3.32
1.87
1 .20

Figure 6 illustrates the procedure for d = 1 inch and S/Y = 3, i.e.,
for the chart of Figure 4. The point corresponding to n = 3.84 and D/WY=
7.47 is spotted on the chart. The values of L/WY '= 5.4 and <j> = 7-0 degrees
are then read by interpolation. The length of the radius vector to the point
is readily obtained by swinging an arc with center at the origin back to the
D/WY axis and reading the value at the intersection of the arc with this axis.
This gives T/WY= 9.25 and then, from Equation [la], T = 20,800 pounds. The
circle of safe working loads with radius T,/WY= 8.89 is also shown in Figure
6. The factor of safety, T B /T, is computed as the ratio of T B /WY to T/WY
where

11

t2

T s

WY = 8 - 9 "

J

\0 .

8 ,

ÂŁ ,

n=>o

*0

x

vj.

,^ -

^5

^

>^

\

'

^V

Tj?''

^

^

-â€” *

y = 3.0

T^S

"6"*

\\

T
WY "

9.25

\\

\

\

\
i

i

Figure 6 - Illustration of Application of Figure 4 for an
Anchor Cable of 1-Inch Diameter and a Scope of 3.0

Hence

WY

50,000
1800

T B
T

27.8
~ 9.25

= 27.

= 3.01

The holding power required, computed from Equation , is

H =

20,800

1 â€” 0.0174 x 7 = 23 > 700 pounds

The procedure for using the curves in Figure 2 is the same as that for the
curves in Figures 3, k, and 5 except that <j> = is given initially and the
value of S/Y is read by interpolation.

These values and the results of similar calculations for other
lengths and diameters for this example are summarized in Table 2.

It will be assumed that it is impracticable to use more than 360O
feet of cable, or an S/Y greater than 2.0. For an S/Y of 2, Table 2 shows
that the factor of safety increases almost linearly with diameter of cable.
The holding power, however, reaches a minimum of about 30,700 pounds with a
cable of 2-inch diameter. If a safety factor of 3-1 is assumed to be ade-
quate, interpolation from Table 2 indicates that the ship could be anchored
with 3600 feet of 17/1 6-inch wire rope with an anchor having a holding power

12

TABLE 2

Holding Power and Safety Factors for Various Lengths and
Diameters of Wire Rope for Conditions of Example

00

degrees

S

T

W7

JJL
WY

To

pounds

Safety
Factor

Holding

Power
Required

pounds

d

=1.0 Inch

7

21

3.8

3

2

5.10

5-4

6.55

9.05
9.25
9.94

20,200
20,800
22,500

3.08
3.01

2.80

20,200
23,700
35,500

d

= 1 .5 Inch

13.3

2.75
2

3.80
4.i6

5.04
5-30

22,900
24,400

5.52

5.24

22,900
31,700

d =

=2.0 Inches

7.5

2.27
2

3.10
3.15

3.62
3.65

26,400
26,700

7.68
7.61

26,400
30,700

d =

=2.5 Inches

0.5

2

2J0

2.96

30,900

9.40

31,200

of 35,000 pounds. This would require a lightweight (LWT) anchor of about
1750 pounds if the experimental value for the holding power of a lightweight
anchor as recommended by the Bureau of Ships, 20 pounds per pound of anchor
weight, is assumed. A safety factor of 5 would require a cable of 1.5-inch
diameter with a 1580-pound anchor.

REFERENCES

(1) Telephone conversation of 23 March 1945 between Comdr. P.W.
Snyder, USN, BuShips, Hull Design (440), and Comdr. E.A. Wright, USN, David
Taylor Model Basin.

(2) "Investigation of Anchor Characteristics by Means of Models,"
by W.E. Howard and R.K. James, Student Thesis, Massachusetts Institute of
Technology, 1933-

(3) "Determining Anchor Holding Power from Model Tests," by W.H.
Leahy and J.M. Farrin, Jr., Transactions of the Society of Naval Architects
and Marine Engineers, Vol. 43, 1935-

13

(4) "On the Resistance of a Heavy Flexible Cable for Towing a Sur-
face Float behind a Ship," by J.G. Thews and L. Landweber, EMB Report 4l8,
March 1936.

(5) "The Shape and Tension of a Light, Flexible Cable in a Uniform
Current," by L. Landweber and M.H. Protter, TMB Report 533, October 1944.

PRNC- 8639- 11-17-47- 60

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