# S0300-A6-MAN-010 CHAPTER 2 STABILITY

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1 CHAPTER 2 STABILITY 2-1 INTRODUCTION This chapter discusses the stability of intact ships and how basic stability calculations are made. Definitions of the state of equilibrium and the quality of stability as they apply to ships are given in the following paragraphs Equilibrium. A ship floating at rest, with or without list and trim, is in static equilibrium; that is, the forces of gravity and buoyancy are balanced. They are equal and acting in opposite directions and are in a vertical line with each other Stability. Stability is the measure of a ship s ability to return to its original position when it is disturbed by a force and the force is removed. A ship may have any one of three different kinds of stability, but only one at a time Positive Stability. If the ship tends to return to its original position after being disturbed by an external force, it is stable or has positive stability Negative Stability. If the ship tends to continue in the direction of the disturbing force after the force is removed, it is unstable or has negative stability Neutral Stability. A third state, neutral stability, exists when a ship settles in the orientation it is placed in by the disturbing force. Neutral stability seldom occurs to floating ships, but is of concern in raising sunken ships because a ship rising through the surface passes through a neutral condition. While the ship is neutrally stable, even a very small disturbing force may cause it to capsize. 2-2 TRANSVERSE STABILITY Transverse stability is the measure of a ship s ability to return to an upright position after being disturbed by a force that rotates it around a longitudinal axis. The following paragraphs define the elements of transverse stability and provide a method to calculate the transverse stability characteristics of a vessel Height of the Center of Gravity (KG). One of the primary concerns in transverse stability is the height of the center of gravity above the keel. This distance is measured in feet. In most ships, the center of gravity lies between a point six-tenths of the depth above the keel and the main deck. The position of the center of gravity depends upon the position of weights in the ship and changes whenever weight is added, removed or shifted. To calculate the height of the center of gravity, the following steps are necessary: 2-1

2 a. Classify all the weights in the ship. b. Determine the height of each weight above the keel. c. Multiply each weight by the height above the keel to determine the moment of the weight. d. Total the weights and the moments of weight. e. Divide the total of the moments of weight by the total weight to determine the height of the center of gravity. EXAMPLE 2-1 CALCULATION OF THE HEIGHT OF THE CENTER OF GRAVITY (KG) A ship has the following weights on board: Material Weight (w) Height above the keel (kg) Ship s structure 2, Machinery Stores Fuel Cargo What is the height of the center of gravity? A tabular format is convenient for this type of calculation. Weight Height above keel Moment of Weight w kg w x kg 2, , , , , ,200 Sums 3,950 55,

3 Height of the center of gravity: KG = KG = sum of the moments of weight total weight 55, , 950 KG = feet Height of the Center of Buoyancy (KB). The height of the center of buoyancy above the keel or baseline is another important distance in stability. This distance is measured in feet. Because the center of buoyancy is the geometric center of the underwater body of the ship, the height of the center of buoyancy depends upon the shape of the ship. In flat-bottomed full ships, such as carriers and tankers, the center of buoyancy is lower than in finer lined ships, such as destroyers or frigates. Calculation of the location of the center of buoyancy for ship shapes is a lengthy and tedious process. The height of the center of buoyancy is contained in the curves of form. When curves of form are not available, estimates sufficient for salvage work may be made as follows. The height of the center of buoyancy is half the draft for a rectangular barge. In a ship s form, the center of buoyancy lies between 0.53 and 0.58 of the draft. A reasonable first approximation of the height of the center of buoyancy that is sufficiently accurate for salvage work is 0.55 times the mean draft Transverse Metacentric Radius (BM). The transverse metacentric radius is the distance between the center of buoyancy and the metacenter. Transverse metacentric radius is measured in feet. It is defined as the moment of inertia around the longitudinal axis of the waterplane at which the ship is floating divided by the displacement volume. BM = I V If the shape of the waterplane is known, the moment of inertia of the waterplane can be defined exactly. For salvage work, a reasonably accurate approximation may be made by: where: I = C IT x L x B 3 C IT = The transverse inertia coefficient and is equal to C WP 2 /11.7 L = Length between perpendiculars B = Beam 2-3

4 The transverse inertia coefficient may also be obtained from Figure 2-1 by entering along the horizontal scale with the waterplane coefficient, reading up to the curve, then across to the vertical scale. Figure 2-1. Transverse Inertia Coefficient. 2-4

5 EXAMPLE 2-2 CALCULATION OF THE TRANSVERSE METACENTRIC RADIUS An FFG-7 Class ship is 408 feet long with a beam of 44 feet and draws 14.5 feet. Her block coefficient is and her waterplane coefficient is What is her transverse metacentric radius (BM)? a. Determine the transverse inertia coefficient. C IT = C WP C IT = C IT = b. Calculate the moment of inertia of the waterplane. l = C IT x L x B 3 l = x 408 x (44) 3 l = 1,689,096 feet 4 c. Calculate the displacement volume. V = C B x L x B x T V = x 408 x 44 x 14.5 V = 126,768 feet 3 d. Divide the moment of inertia by displacement volume to determine the transverse metacentric radius. BM = BM = l -- V 1, 689, , 768 BM = feet The value of the metacentric radius derived from the curves of form is 13.4 feet. The calculated value is sufficiently accurate for salvage work. 2-5

6 NOTE The expression for transverse inertia coefficient is derived from the analysis of numerous ships and is a reasonable approximation for use in salvage. For a vessel or a barge with a rectangular waterplane (C WP = 1.0), an exact calculation is: I = ( L x B 3 ) Height of the Metacenter (KM). The height of the metacenter is the distance between the keel and the metacenter. The height of the metacenter is measured in feet. It is the sum of the height of the center of buoyancy and the metacentric radius, that is: KM = KB + BM For an upright ship, the metacenter lies on the same vertical line as the center of buoyancy and the center of gravity. If the curves of form are available, KM can be determined directly from them Metacentric Height (GM). The metacentric height, measured in feet, is the distance between the center of gravity and the metacenter and is the principal indicator of initial stability. A ship whose metacenter lies above the center of gravity has a positive metacentric height and is stable; conversely, a ship with the metacenter below the center of gravity has negative metacentric height and is unstable. With the distances KB, BM and KG known, GM can be calculated: and GM = KB + BM - KG GM = KM - KG Righting Arm (GZ). In an upright ship in equilibrium, the forces of gravity and buoyancy act equally in opposite directions along the vertical centerline. As the center of buoyancy shifts when the ship heels, these two opposing forces act along parallel lines. The forces and the distance between them establish the couple which tends to return a stable ship to the upright position. The righting arm is the distance between the lines of action of the weight acting through the center of gravity and the force of buoyancy acting through the center of buoyancy at any angle of inclination. Righting arms are measured in feet. Figure 2-2 shows the righting arm for an inclined stable ship. 2-6

7 Figure 2-2. Righting Arm. The length of the righting arm varies with the angle of inclination. The ratio of the righting arm to the metacentric height, GZ/GM, is equal to the sine of the angle for any small angle. If the sine of the angle is represented by Sin θ, the equation can be written as: or Sin θ = GZ GM GZ = GM x Sin θ NOTE The sine is one of three natural functions of angles that are used in basic naval architectural and salvage calculations. The others are the tangent and the cosine. A table of these natural functions is given in Appendix D. 2-7

8 This equation provides a convenient means of calculating righting arm for small angles of inclination. At angles of heel greater than about ten to fifteen degrees, the metacenter moves away from the centerline and the relationship between the metacentric height and righting arm is no longer exact. The righting arm at large angles of heel can be determined from the statical stability curve described in Paragraph Righting Moment (RM). The righting moment is the couple of the weight and buoyancy of an inclined ship. This moment acts to return the ship to an upright position. Righting moment is measured in foot-tons. Because forces creating the couple (the weight and buoyancy of the ship) are equal, opposite and equal to the displacement, the righting moment is the product of the displacement of the ship and the righting arm or: RM = W x GZ The size of the righting moment at any displacement and angle of inclination is a measure of the ship s ability to return to an upright position. As the righting moment at any displacement is directly proportional to the righting arm, the righting arm may be used as an indicator of stability Cross Curves of Stability. The cross curves of stability are a set of curves, each for a different angle of inclination, that show righting arm changes with displacement. A set of cross curves of stability for the FFG-7 Class ship are shown in Figure 2-3. Note that a particular height of the center of gravity has been assumed in computing the curves. The importance of this assumption is explained in Paragraph To use the cross curves, enter on the horizontal scale with the displacement of the ship, read up to the curve representing the angle of interest and read across to the vertical scale to determine the value of the righting arm. For example, to obtain the righting arm at 3,200 tons displacement at an angle of 30 degrees, enter the curves of Figure 2-3 along the horizontal scale with 3,200 tons, then: a. Read up to the intersection with the 30-degree curve. b. Read across to the vertical scale where it can be seen that the righting arm (GZ) is 1.67 feet. The principal use of the cross curves of stability is in constructing the curve of statical stability The Curve of Statical Stability. The curve of statical stability (or simply the stability curve) shows righting arm changes as the ship inclines at a particular displacement. Righting arm, in feet, is plotted on the vertical scale while the angle of inclination, in degrees, is plotted on the horizontal scale. The curve of statical stability, when plotted and corrected as described below, will provide the following information: 2-8

9 Range of inclination through which the ship is stable (Range of Stability) Righting arm at any inclination Righting moment at any inclination Angle at which maximum righting arm and maximum righting moment occur Metacentric height. Figure 2-3. Cross Curves of Stability Plotting the Curve of Statical Stability. Figure 2-4 is the curve of statical stability taken from the cross curves shown in Figure 2-3 for a displacement of 3,200 tons. The curve was constructed by: a. Entering the cross curves along the 3,200-ton displacement line. b. Reading up to the angle of inclination and across to determine the righting arm for that angle of inclination and displacement. c. Repeating the last step for each angle plotted in the cross curves. d. Plotting the values obtained and drawing the curve. 2-9

10 Figure 2-4. Statical Stability Curve Height of Center of Gravity Correction. The cross curves of stability are calculated for a particular height of the center of gravity. When the center of gravity has a different height, the metacentric height and the stability curve change. If the actual center of gravity lies above the assumed center of gravity, the metacentric height is decreased and the ship is less stable; conversely, if the actual center of gravity is below the assumed center of gravity, the metacentric height is increased and the ship is more stable. The correction at any angle of inclination is the product of the difference between the actual and assumed heights of the center of gravity and the sine of the angle of inclination or: correction = GG 1 x sin θ where: GG 1 is the difference between the actual and assumed heights of the center of gravity. Thus, if the center of gravity is two feet above the assumed center of gravity, the correction can be calculated as: 2-10

11 Angle θ Sine of the angle sin θ Height difference GG 1 Correction GG 1 sin θ The corrections are plotted to the same scale as the curve of statical stability as shown in Figure 2-5. The corrected curve of statical stability is drawn by plotting the difference between the two curves as shown in Figure 2-6. If the actual height of the center of gravity is less than the assumed height, the calculation is done in the same manner, however, the correction curve is plotted below the horizontal axis as shown in Figure 2-7. The new statical stability curve is again the difference between the two curves as shown in Figure Off-Center Weight Correction. When there is off-center weight and the center of gravity is no longer on the centerline, a list results and there is deterioration in the stability characteristics, including a reduction in righting arm toward the side to which the ship is listing. The reduction in righting arm is equal to the product of the distance between the new center of gravity and the centerline times the cosine of the angle of inclination or: where: correction = GG 1 x cos θ GG 1 is the distance between the centerline and the new position of the center of gravity. Thus, if the center of gravity is 0.5 feet from the centerline, the correction can be calculated as: Angle θ Cosine of the angle cos θ Horizontal distance GG 1 Correction GG 1 x cos θ

12 Figure 2-5. Correction to Statical Stability Curve, CG is 2 Feet Above Assumed Point. Figure 2-6. Corrected Statical Stability Curve. 2-12

13 Figure 2-7. Correction to Statical Stability Curve, CG is 2 Feet Below Assumed Point. Figure 2-8. Corrected Statical Stability Curve. 2-13

14 As is done with the height corrections, the off-center weight corrections are plotted to the same scale as the curve of statical stability. The corrected curve of statical stability is drawn by plotting the difference between the two curves as shown in Figures 2-9 and The angle at which the corrected curve of statical stability crosses the horizontal axis is the angle of list caused by the off-center weight Range of Stability. The range of stability is the number of degrees through which the ship is stable or the number of degrees through which the ship can heel without capsizing. The range of stability may be measured directly from the statical stability curve and its limit is the intersection of the curve and the horizontal axis. For instance: In Figure 2-4, the uncorrected stability curve, the range of stability is from 0 degrees to more than 90 degrees. In Figure 2-6, the stability curve corrected for height of the center of gravity, the range of stability is from 0 degrees to 77 degrees. In Figure 2-10, the stability curve corrected for off-center weight, the range of stability is 20 degrees to 75 degrees. Figure 2-9. Correction to Statical Stability Curve for Transverse Shift of G. 2-14

15 Figure Corrected Statical Stability Curve for Transverse Shift in G Righting Arm and Righting Moment. The righting arm at any inclination may be read directly from the curve. Because each stability curve applies only to a specific displacement, the righting moment can be obtained directly for any angle by multiplying the righting arm by the displacement. In Figure 2-6, the maximum righting arm is 1.1 feet, the maximum righting moment is 3,520 foot-tons and the angle where the maximums occur is 51 degrees. Similarly, in Figure 2-10 the maximum righting arm is 0.83 feet, the maximum righting moment is 2,656 foot-tons and the angle where the maximums occur is 49 degrees Metacentric Height. The metacentric height may be obtained directly from the curve of statical stability by: a. Erecting a perpendicular to the horizontal axis at 57.3 degrees (one radian) b. Drawing the tangent to the statical stability curve at the origin. The intersection of the two lines indicates the metacentric height. In Figure 2-11, the metacentric height of the ship with stability curve 1 is 3.47 feet and that of the ship with stability curve 2 is 1.47 feet. 2-3 LONGITUDINAL STABILITY Longitudinal stability is the measure of a ship s ability to return to its original position after being disturbed by a force that rotates it around a transverse axis. Longitudinal stability is important to refloating operations because changes in the longitudinal stability of a stranded or sunken ship will not be apparent since the ship does not respond in the same manner as a ship afloat. The changes must be calculated to ensure salvors have an accurate assessment of the actual longitudinal stability situation. 2-15

16 Figure Metacentric Height from Statical Stability Curves Longitudinal Position of Center of Gravity (LCG). The longitudinal position of the center of gravity is as important to longitudinal stability as the height of the center of gravity is to transverse stability. Its position is determined solely by the distribution of weight along the length of the ship. The longitudinal position of the center of gravity is measured in feet from the midships section or the forward perpendicular. It is determined in a manner similar to determining the height of the center of gravity above the keel in that the sum of the moments of the weights about either the forward perpendicular or the midships section is divided by the total weight to obtain the desired position. The following steps are necessary: a. Classify all the weights in the ship. b. Determine the longitudinal distance of each weight from the reference. c. Multiply each weight by the longitudinal distance from the reference to determine the moment of the weight. d. Total the weights and the moments of weight. e. Divide the total of the moments of weight by the total weight to determine how far the longitudinal position of the center of gravity lies from the reference. 2-16

17 EXAMPLE 2-3 CALCULATION OF THE LONGITUDINAL CENTER OF GRAVITY A ship has the following weights on board: Material Weight (w) Distance from the FP (Icg) (Long Tons) (Feet) Ship s structure 2, Machinery Stores Fuel Cargo What is the longitudinal position of the center of gravity? A tabular format is convenient for this type of calculation. Weight (w) Distance from FP (Icg) Moment of Weight (w x Icg) (Long Tons) (Feet) (Foot Tons) 2, , , , , ,000 Sums 3, ,400 Distance of LCG from FP = sum of the moments of weight total weight LCG = 856, , 950 LCG = feet (or 217 feet) abaft the FP Longitudinal Position of the Center of Buoyancy (LCB). The longitudinal position of the center of buoyancy is measured in feet from either the forward perpendicular or the midships section. For a ship in equilibrium, the longitudinal position of the center of buoyancy is in the same vertical line as the longitudinal position of the center of gravity. At any given time, there is only one point that is the center of buoyancy; the height and longitudinal position are two coordinates of that single point. Determination of the longitudinal position of the center of buoyancy is lengthy and tedious, requiring calculation of the underwater volume of the ship and its distribution. The longitudinal position of the center of buoyancy may be obtained from the curves of 2-17

18 form. In salvage, the longitudinal position of the center of buoyancy is important primarily in making strength calculations when buoyancy must be distributed to place the longitudinal positions of the centers of gravity and buoyancy in the same vertical line Longitudinal Center of Flotation (LCF). The longitudinal center of flotation is the point about which the ship trims. It is the geometric center of the waterline plane. The longitudinal position of the center of flotation is measured in feet from either the midships section or the forward perpendicular. In ships of normal form, it may lie either forward or aft of the midships section. In fine-lined ships, the longitudinal center of flotation is usually slightly abaft the midships section. The longitudinal position of the center of flotation is required to calculate final drafts when trim changes. It can be calculated if the exact shape of the waterplane is known or it may be obtained from the curves of form. If the position cannot be obtained, it can be assumed to be amidships Longitudinal Metacenter (M L ). The longitudinal metacenter is an imaginary point of importance in longitudinal stability. Like the transverse metacenter, it is located where the force of buoyancy s lines of action running through the longitudinal center of buoyancy intersect the vertical line through the center of buoyancy of the untrimmed ship. While the center of gravity and the center of buoyancy are points whose heights above the keel and longitudinal position are two coordinates of the same point, the transverse metacenter and the longitudinal metacenter are separate points, each with its own set of coordinates Longitudinal Metacentric Radius (BM L ). The longitudinal metacentric radius is the distance between the center of buoyancy and the longitudinal metacenter. The longitudinal metacentric radius is measured in feet. It is defined as the moment of inertia of the waterline waterplane about a transverse axis divided by the volume of displacement. BM L = I L V --- If the shape of the waterplane is known, the moment of inertia of the waterplane can be defined exactly. For salvage work, a reasonably accurate approximation may be made by: where: I L = C IL x B x L 3 C IL = The longitudinal inertia coefficient and is equal to (0.143 x C WP ) B = Beam L = Length between perpendiculars 2-18

19 NOTE The longitudinal inertia coefficient may be obtained directly from Figure 2-12 by entering along the horizontal scale with the waterplane coefficient, reading up to the curve and then reading across to the vertical scale. Figure Longitudinal Inertia Coefficient. 2-19

20 EXAMPLE 2-4 CALCULATION OF THE LONGITUDINAL METACENTRIC RADIUS An FFG-7 Class ship is 408 feet long with a beam of 44 feet and draws 14.5 feet. Her block coefficient is and her waterplane coefficient is What is her longitudinal metacentric radius (BM L )? a. Determine the longitudinal inertia coefficient: C IL = (0.143 x C WP ) C IL = (0.143 x ) C IL = b. Calculate the moment of inertia of the waterplane: I L = CIL x B x L 3 I L = x 44 x (408) 3 I L = 125,212,356 feet 4 c. Calculate the displacement volume: V = C B x L x B x T V = x 408 x 44 x 14.5 V = 126,768 feet 3 d. Divide moment of inertia by displacement volume to get longitudinal metacentric radius: BM L = BM L = BM L = l --- L V 125, 212, , feet (or 988 feet) 2-20

21 2-3.6 Height of the Longitudinal Metacenter (KM L ). The height of the longitudinal metacenter is the distance between the metacenter and the keel and is measured in feet. The height of the longitudinal metacenter is the sum of the height of the center of buoyancy and the longitudinal metacentric radius or: KM L = KB + BM L where: KM L = The longitudinal height of the metacenter KB = The height of the center of buoyancy BM L = Longitudinal metacentric radius Longitudinal Metacentric Height (GML). The longitudinal metacentric height is the distance between the height of the center of gravity and the longitudinal metacenter measured in feet. The longitudinal metacentric height is the difference between the longitudinal height of metacenter and the height of the center of gravity or: GM L = KM L - KG where: GM L = Longitudinal metacentric height KM L = Height of the longitudinal metacenter KG = Height of the center of gravity Trim. The quantities necessary for determining trim and how they are used are explained in the following paragraphs Trimming Moment. A trimming moment is a moment exerted by a weight anywhere in the ship acting about the center of flotation. A trimming moment causes the ship to trim or tip, around the center of flotation. There is no special symbol for trimming moment, though M is convenient to use. Trimming moment is measured in foot-tons. For instance, a weight of 25 tons placed 100 feet forward of the center of flotation would have a trimming moment of 2,500 foottons by the bow. Paragraph provides an example for longitudinal effects of weight additions and removals Moment to Change Trim One Inch (MT1). The trimming moment required to cause a change in trim of one inch (1") is known as the Moment to Change Trim One Inch. The MT1 for 2-21

22 any ship depends upon the hull form and may be either obtained from the curves of form or calculated by: MT1 = ( GM L W) ( 12 L) where: GM L = Longitudinal metacentric height W = Displacement L = Length between perpendiculars As the longitudinal metacentric radius, BML, is easily obtained and is not very different from the longitudinal metacentric height, GML, it is often used to determine an approximate MT1 by: MT1 = ( BM L W) ( 12 L) Additional methods of calculating the approximate moment to change trim one inch can be found in Appendix C. EXAMPLE 2-5 CALCULATION OF MOMENT TO TRIM ONE INCH What is the approximate moment to change trim one inch for the FFG-7 Class ship described in Example 2-4? To calculate the approximate moment to trim one inch: MT1 = ( BM L W) ( 12 L) From the calculation in Example 2-4: BM L = 988 feet L = 408 feet V = 126,768 cubic feet 2-22

23 Displacement can be calculated from displacement volume: 126, 768 W = W = 3,622 tons then ( 988 3, 622) MT1 = ( ) MT1 = foot-tons MT1 from the curves of form is 760 foot-tons. The value based on calculation is within 10 percent of the actual MT1 and is a reasonable approximation for salvage work. NOTE The Greek letter delta (À) is used in conjunction with symbols to mean the change in the quantity that the symbol represents Trim Calculations. In salvage, trim calculations are usually made to determine the changes in draft fore and aft resulting from a change in trimming moment. The calculation has three parts: a. Determination of the trimming moment b. Determination of the total change of trim by dividing the trimming moment by the moment to change trim one inch or: δ trim = trimming moment MT1 c. Determination of the new drafts. When a ship trims, one end gains draft while the other end loses draft. For instance, a ship that trims by the bow gains draft at the bow and loses draft at the stern. The total trim is the sum of the amount gained at one end and lost at the other. As a ship trims about the center of flotation, the amount of change at the bow is proportional to the ratio of the product of the change of trim multiplied by the distance between the forward perpendicular and the center of flotation and the length of the ship or: δ T f = δ trim x (FP to LCF) L 2-23

24 The new draft forward will be the old draft forward with the change added or subtracted as appropriate: New T f = Original T f + δ T f Likewise, the change in trim aft is equal to the ratio of the product of the change of trim multiplied by the distance between the after perpendicular and the center of flotation and the length of the ship or: δ T a = δ trim x (AP to LCF) L The new draft aft will be the old draft aft with the change added or subtracted as appropriate: New T a = Original T a + δ T a EXAMPLE 2-6 CALCULATION OF TRIM The FFG-7 Class ship described in Example 2-5 is floating at a draft both forward and aft of 14.5 feet. The center of flotation is twenty-five feet abaft the midship section. A trimming moment of 25,000 foot-tons by the bow is introduced. What are the new drafts? The first step, the determination of trimming moments, is not necessary because that information is already available; i.e., M = 25,000 foot-tons. To obtain the total change in trim, divide the trimming moment by the moment to change trim one inch: M δ trim = MT1 Moment to trim one inch is 760 foot-tons from the curves of form; therefore: δ trim = 25, δ trim = 32.9"" 2-24

25 To obtain the new draft forward, determine the distance from the FP to the LCF: FP to LCF = L FP to LCF = FP to LCF = 229 then δ T f = δ trim x (FP to LCF) L δ T f = δ T f = 18.5, or New T f = Old T f + δ T f New T f New T f = = To obtain the new draft aft, determine the distance from the AP to the LCF: AP to LCF = L AP to LCF = AP to LCF = 179 then δ T a = δ trim x (AP to LCF) L δ T a = δ T a = 14.5, or New T a = Old T a + δ T a New T a New T a = = A method to check the accuracy of the draft calculations is to add the change in 2-25

26 draft forward and the change in draft aft and compare the result with the total change in trim. These quantities should be equal. In the previous example the total change in trim was 32.9 inches. δ T f = 18.5 δ T a = Since the sum of the draft changes is equal to the total change in trim, the calculation was performed correctly. 2-26

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