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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– -~oÂ£S

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 [2]

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 [3] and [4], 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 [7]. 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 [8]

1 - 0.017400

where O is in degrees. In using Equation [8] 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 [2], 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 ,

Â£ ,

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 [8], 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

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– -~oÂ£S

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 [2]

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 [3] and [4], 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 [7]. 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 [8]

1 - 0.017400

where O is in degrees. In using Equation [8] 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 [2], 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 ,

Â£ ,

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 [8], 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

1