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FIGS 58

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Originally appearing in Volume V24, Page 953 of the 1911 Encyclopedia Britannica.
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FIGS 58 .—Cargo See also:Vessel of 12,000 Tons on FIG. 59.—Cargo Vessel of '12,000 Tens See also:Wave See also:Crest. in Wave'. Trough. See also:section was approximately in accordance with the linear See also:law assumed in the theory of bending.' (c) The modulus of See also:elasticity E was obtained by equating the sum of the moments about the neutral See also:axis of the stresses deduced from the observed strains to the bending moment. (d) The value of E was also deduced from the deflections by means the See also:case, the comparison is made between two See also:ships of similar type. The relation between stress and See also:strain has therefore to be investigated, which involves the experimental determination of the modulus of elasticity of the structure. The assumptions on which the theory of bending is based are : (a) At any transverse section the material lying on the neutral See also:surface,-which passes through the C.G. of the effective sectional material, is neither extended nor compressed. (b) The material is homogeneous; and the layers comprised between adjacent surfaces parallel to the neutral surface See also:act inde- pendently. (This is probably more nearly the case in a See also:ship than In a See also:beam of solid section:) (c) The material situated at a distance y from the neutral surface is compressed (or extended) longitudinally by an amount 2 of its See also:original length; where See also:lip is the curvature of the neutral surface if originally straight, or the alteration of curvature if originally curved. (d) The stress is proportional to the strain and equal to Py, E being See also:Young's modulus for the material. It follows that the resultant See also:longitudinal force across a section is zero, and the moment of the See also:internal forces about the neutral axis (i.e. about the trace of the neutral surface in the section) is PI, which is equal and opposite to the See also:external bending moment M. (e) Taking axes—Ox longitudinal, Oz See also:vertical, since p is large, I may be replaced .by dx =,, and Edx2=M or Ez=If M dx dx, giving the deflection z at any point.

The validity of the theory as applied to a ship was tested and confirmed in 1903 at See also:

Portsmouth Dockyard when experiments were made-on H.M.S." See also:Wolf " by See also:Professor J. H. Biles for the Committeeof the See also:formula Ez=ffMdxdx; and its value under a sagging moment is in agreement with that found by (c). Under a hogging moment the mean value obtained from the deflection is less than that from the strain, showing that the curvature obtained from the deflections is greater than that to which the structure is actually See also:bent. The table at the See also:top of the following See also:page shows the values obtained for E, the modulus of elasticity. By observing the deflections of two vessels when loaded with See also:ballast, the following values for E were obtained by T. C. Read and G. Stanbury (Traits. Inst. See also:Nail. See also:Arch., 1894), and are given for purposes of comparison,: See also:Principal Dimensions Load in Deflection Value of E of Vessel.

Tons. in Inches. deduced. 347' X45'- 6" X27'-2" 5000 2 31 11,000 300'X41'-6"X21'-2" 1800 62 9,000 After the experiments the " Wolf " was sent to See also:

sea in rough See also:weather with the See also:object of comparing the stresses then observed with those calculated under the See also:standard conditions on trough or crest. The strains at various portions of the structure were again measured with Stromeyer's indicators, and the stresses deduced from the values for E found from the See also:dock experiments. The maximum stresses were as follows: See also:Condition. Stress—Tons per Square See also:Inch. See also:Keel. See also:Deck. Maximum observed stresses when hogging . . 2.9 C. 2.0 T. Maximum observed stresses when sagging . 5.4 T.

2'5 C. Calculated stress (sagging) when in a wave hollow of height 7'1 T. 5.3 C. -26th length . . . . C. = Compressive. T. = Tensile. It appears from these experiments that (at least in ships of similar See also:

character to H.M.S. " Wolf ") the stresses corresponding to any particular external conditions closely agree with those calculated from the usual theory of bending; on the other See also:hand the waves encountered during the sea trials were such that the maximum stress then obtained was considerably less than that in the condition assumed for the standard calculations. Finally, the material of the ship was subjected 'in dock to a tensile stress of nearly 9 tons and a compressive stress of nearly 7 tons per sq. in. without See also:distress.

While dealing with longitudinal bending, some of the refinements suggested for calculating stresses among waves may be cited, although the additional labour involved in their application has prevented their introduction in See also:

general practice. Since the See also:distribution of pressure in the See also:water of a wave See also:system differs from that in still water, the buoyancy of a vessel or the resultant vertical thrust of the water is then not equal to the See also:weight of the water displaced; and the position of the ship when in See also:equilibrium and the stresses upon it are changed in consequence. By assuming the pressure at any point of the water to be in accordance with the trochoidal theory of wave See also:motion, and undisturbed by the intrusion of the ship, the equilibrium position can be obtained and the modified stresses evaluated. This See also:process was first applied to ships by Mr W. E. See also:Smith (Trans. I.N.A., 1883), who obtained the arithmetical sum of the sagging and hogging moments on vessels placed in the trough and on the crest of a wave, thereby eliminating the effect of the distribution of weight; and compared it with the sum of the moments as ordinarily obtained. The correction for the ships considered involved a reduction of the bending, moment to about f,of the value calculated in the. See also:ordinary manner, and in a See also:torpedo-See also:boat destroyer a reduction of about 100A has been obtained. This reduction increases as the See also:draught and fullness of the ships are increased, and the bending moment on a square-bilged ship deeply immersed is almost uninfluenced by wave motion, since the reduction in orbital motion at considerable depths below the surface ensures the bottom of a fairly deep ship being in comparatively undisturbed water. In the foregoing the vessel is assumed to occupy at every instant a See also:horizontal position on the wave with the correct displacement; a ship proceeding perpendicularly to the crests of a wave system will, however, undergo heaving and pitching oscillations which See also:lead to a further modification in the bending moment obtained (see See also:paper by T. C. Read, Trans.

LN.A., 1890). Considering first the effect of pitching only, imagine the ship at her proper displacement (See also:

allowance being made for the altered buoyancy of the wave system as before), but momentarily out of her correct See also:trim; the longitudinal restoring couple, due to the wedges of See also:immersion and emersion, is balanced by the moment of the reversed See also:mass-accelerations of the component parts. If the ship is longitudinally symmetrical about her midship section, one See also:half of the moment of the restoring forces and one half of the moment of the reversed mass-accelerations about amidships are due to the forward end, and one half to the after end. These moments are therefore equal and opposite for each half of the ship and have no See also:influence oa the midship bending moment: Itappears, therefore, that in the See also:majority of ships whose departure from longitudinal symmetry is slight, pitching has little effect on the amount of the maximum longitudinal bending moment; nevertheless it considerably increases the bending moments near the ends. The effect of heaving is investi- gated by obtaining the positions of equilibrium of the C.G. of the ship when on wave crest and in wave trough; intermediate positions of equilibrium are assumed to be recorded in See also:order that the given by y= a. See also:sin sr't where" TI is the are Tl apparent semi-See also:period of the wave. On taking into See also:account the mass of the ship, assumed originally stationary; the height of the C.G. above its mean position becomes a X TI2 Ts f sinyiri -TT sinZ. 7rt i - where T=ir'vgp=period of See also:dip in still water.; W is the displacement, and p the tons per See also:foot immersion; theresistance to vertical motion being neglected. When T and Tl are nearly equal, allowance has to be made for the resistance by using a process. of graphic integration. On applying the correction to two vessels; and comparing the bending moments in their positions of the wave, given by the formula,. with those in the equilibrium position, the effect on.. the maximum hogging moment was found small; but the sagging moment of a moderately See also:fine vessel was increased by over 20%, and that of a full vessel by about i0 Allowance has also been made for the effect of the superposed heaving, pitching and See also:rolling oscillations undergone by a ship moving obliquely across the crests of a wave system (see papers by See also:Captain See also:Kriloff , Trans. I.N.A., 1896 and 1898). The maximum calculated stress on vessels inclined to considerable angles of See also:heel has been found in some instances to be slightly greater than that for the upright condition; and the stress on the material towards the ends is then usually more nearly equal to that amidships: In addition to the See also:direct stresses on keel, bottom, and upper See also:works resulting from longitudinal bending, shearing stresses are experienced which in some cases are of appreciable magnitude. The in- tensity of shear stress in the See also:side plating is equal to 2t' . where F is the shearing force over the transverse section, At the moment about the neutral axis of the sectional See also:area above or below a horizontal See also:line through the point considered, and t~ the thickness of side plating.

This stress is usually greatest at or near a See also:

quarter of the length from either end and at the height of the neutral axis, since here F and Af respectively attain their maximum values. In some cases the thickness of 'plating and arrangement of 'riveting have to be specially considered in relation to these shearing stresses. The stresses due to transverse bending are not, in general, capable of definite determination; as, however, they are frequently severe when the ship is in dry dock, and may also attain consider-able magnitude during 'heavy rolling, a means of See also:corn- Trans'',, paring the transverse strength of vessels is of some 'See also:verse See also:interest. A transverse bulkhead forms a region of almost bending. See also:infinite transverse stiffness, and it is therefore difficult in ships internally subdivided by Numerous bulkheads, to' determine how far the stresses at intermediate sections are influenced by the neighbpuring bulkheads. In many vessels carrying See also:cargo, 3r however, in which trans-verse bulkheads are widely spaced, a section midway along a hold may be so far removed from all bulkheads as to be uninfluenced by' their See also:local support; and the following ' method has been proposed for comparing the transverse strengths of such ships: A See also:frame and a See also:strip 0 of plating one 'frame space in width are re- garded as a stiff inextensible See also:bar subjected to the known external forces and to the unknown' tension, shearing force, and bending moment, at any fixed point. Let OP (fig. 6o) be a portion of the framing under See also:consideration; O being the keel, and Ox, Oy, horizontal and vertical axes. On consideration of the forces on the arc OQ, which are in Sagging. Hogging. Draught Maximum Maximum E by Dena- Maximum Maximum Eby Deflec- of Water. Compressive Tensile Eby Strain non over the Compressive Tensile E by strain See also:lion over the Stress. Stress: See also:Indicator. whole length.

Stress. Stress. Indicator. whole length. Feet. 6 I.7 2'3 12,100 11,800 I.0 '9 .. .. 5 2'9 3'7 12,100 12,000 2.7 2.6 16,000 I1,8o0 4 41 5.4 11,400 11,400 4.2 4.0 15,100 10,800 3 5.2 6.6 11,400 11,500 5.3 5.0 13,000 10,400 2 6o 7'7 I0,800 I1,100 6'1 5'8 .12,700 9,600 1 6.5 8.4 10,700 10,600 6'6 6.4 12,790 9,900 Dry 6-7 86 10,200 10,300 6.8 6 6 11,80o 9,800 See also:

Note.—The maximum stresses above are approximate, and are variation of E with the stress in the material may be seen. Ton employed. Tons per square-inch See also:units equilibrium, the tension T, shearing force F, and bending moment M, at Q can be algebraically expressed in terms of its coordinates (x, y,), the water or other external pressures on OQ, and the values of T, F and M at 0 (To, Fo, Me,). Neglecting the effects of T and F on the See also:element QR, it follows from the equations of bending that M=EI( de do ds) where 4 and 0' are the respective inclinations of the element OR to Ox before and after the strain caused by bending, and ds is the length of QR. Due to the effect of M on QR, the bar at the point P (xi, yl) is rotated through an See also:angle d¢'—d¢ and moved through distances (yI—y) (d4'—d¢) and (xi—x) (4'—4) in directions parallel to Ox and Oy respectively. On integrating along OP the See also:total See also:movement of P due to the bending of all such elements as QR in OP is obtained; when P is moved See also:round the See also:complete section so as to return to 0, where the total movement is zero, it follows, on subtraction and reduction, that See also:IM ds=o, f —FM f hIyds=o; the integrations being taken completely round the section.

It is assumed in the foregoing that rigid connexions are made at discontinuities, such as deck edges, in order to prevent any alteration in the angle due to strain. The values of M Mx My can 'T, I— can be calculated at varying points and expressed in terms of To, Fo, Mo; by using a method of approximate See also:

quadrature, To, Fo, Mo are found by solving, the 3 equations obtained, and M is deduced giving the corresponding stress at any point. In applying this method to the determination of the stresses caused by rolling, the centrifugal forces on each element are included in the external forces when estimating M. This method of estimating the transverse strength of ships is due to Dr Bruhn, who in Trans. I.N.A., 190I, 1904 and 1905, gives illustrations of its application. In addition to the stresses due to longitudinal and transverse bending, which are distributed over the whole or a considerable See also:part of the structure, local stresses are experienced including those caused by water-pressure; forces on sails, masts and See also:rigging; reactions of moving parts of machinery; heavy blows from the Local sea on side, deck and upper works; See also:anchor, See also:cable and stresses. mooring See also:gear, and blast from See also:gun-See also:fire. General methods are usually inapplicable to such cases; the support provided is determined by experience and by the particular requirements. The stresses in bottom plating due to water-pressure are of small amount where the curvature is appreciable, since the plating, by See also:compression, directly resists any tendency towards See also:change of curvature; in a deep See also:flat-bottomed ship, on the other hand, resistance to water-pressure is chiefly due to the bending of the plating, the slight See also:extension having little influence. The plating is supported at the transverse and longitudinal frames, and, to some extent, at the edges. The See also:close spacing of transverse frames usually adopted in See also:merchant ships reduces the stress to a small amount; but in large warships, whose frame spacing varies from 3 to 4 ft., it is probable that the flat plating near the keel amidships is subjected to considerable stress, although, as experience shows, not beyond the limits of safety. In fine ships See also:special See also:provision is frequently made to prevent the side plating near the See also:bow from panting due to the See also:great and rapid fluctuation of water-pressure when pitching.

The material of the structure is arranged so that the distribution of stress over any localized section of material is maintained as See also:

uniform Utrilorm- as possible in order that the ratio of maximum to mean m or stress may not be unduly large. For this See also:reason abrupt stress. discontinuities and sudden changes of section are avoided, and " See also:compensation " is introduced where large openings are cut in plating. The corners of hatchways in ships whose upper decks are subjected to considerable tension are frequently rounded, since failure of the material near the square corners of such See also:hatch-ways has been known to take See also:place, pointing to the existence of abnormal stress intensities, which are also, evident from theoretical considerations. Similarly, local stiffening required for the support of a heavy weight or for resisting the blast of gun-fire is reduced in sectional area at the ends, or continued for a length greater than absolutely necessary, to ensure an even distribution of stress. Among the stresses to which a ship is subjected are those caused by its mode of propulsion. The stresses due to the reactions of vmraiion. the moving parts of the machinery are, in general, of small amount, but owing to their periodic character vibrations are induced in the structure which are frequently of sufficient magnitude to cause considerable inconvenience and even damage. It is known that when a periodic force of frequency n is applied to a structure capable of vibrating naturally with frequency p, the See also:amplitude of the forced vibrations assumed by the structure is inversely proportional to (P —nz)2 +K2, where K is a coefficient depending on the resistance t0 vibration. If the period of the force synchronizes or nearly synchronizes with the natural period of the structure, the amplitude is considerable, but otherwise it is of relatively small amount. If, therefore, the natural period of vibration has been found for a ship, the causes of vibration at various speeds. can be readily traced, since marked vibration is usually attributable to a synchronizing source. Vibration in a steamship is due to various causes, the principal of which are: I. The reciprocating parts of the engines, if unbalanced, cause vibrating forces and couples in a vertical See also:plane and of two frequencies, one equal to, and the other twice, the See also:speed of revolution, the latter being due to the secondary See also:action introduced by the connecting See also:rod. In twin-See also:screw ships torsional oscillations in transverse planes may also result when the engines are working in opposite phase.

2. The rotating parts of the engines cause vertical and horizontal oscillations of frequency equal to the speed of revolution. 3. The variation in the See also:

crank effort tends to cause torsional oscillation of the same frequency, particularly in single or two-See also:cylinder engines. 4. Vibrations, principally at the stern, may result from an unbalanced screw; these are similar to those caused by the rotating parts of the machinery. 5. A screw propeller which experiences uneven resistance during its revolution is the cause of vibrations, whose frequency is the product of the revolution and the number of See also:blades. Such resistances occur when (I) the blades pass too close to the See also:hull; (2) when the screw breaks the surface of the water; and (3) when the See also:supply of water to the propeller is imperfect, due either to " cavitation " or to the screening effect of See also:shaft and propeller supports. The natural vibration of a ship's structure (irrespective of local vibrations) is analogous to that of an unsupported rod of suitable dimensions, the principal difference being that the vibrations in the rod are undamped and those in the ship are damped rapidly • through the communication of the motion at the hull surface to the surrounding water. A thin._uniform rod vibrating laterally has a minimum frequency (per See also:minute) equal to 1210 ,E', in this mode of vibration there are two nodes situated at a distance .224 L from either end. Vibrations of a higher order having three, four or more nodes are also possible, the fre- quencies increasing approximately in the ratio I : 2.8 : 5.4, &c. he complex variation of the weight, inertia and modulus in a ship prevent a corresponding result being obtained by direct mathematical investigation; recourse is therefore made either to direct experiments on ships, or to a " dynamic See also:model." The See also:instrument used for measuring and recording vibrations consists of a weight suspended, and held laterally in position, by springs, so as to have a See also:long period of oscillation; pens or pencils attached to the weight See also:record the vibrations upon revolving cylinders fixed to the vessel and fitted with See also:time records.

The formula (of the same See also:

form as that for a rod) N=c WL, where N is the frequency per minute, was used by Dr Schlick for the vibration of ships; the value of c found by him for vertical vibrations varied from 1600 in very fine vessels to 1300 in those having moderately full lines. The nodes were found to be at about a third of the length from the See also:stem and about a quarter of the length from the after perpendicular. The frequency with three nodes was slightly more than twice that of the See also:primary vibrations. Horizontal and torsional vibrations were also observed; their minimum frequency is, however, generally considerably more than that of the vertical vibrations, and they are therefore generally of much smaller amplitude. (See papers in Trans. Inst. 1Vay. Archs. from 1884 to 1901, by Dr O. Schlick, and in 1895 by Mr A. See also:Mallock.) The dynamic model," suggested by Mr Mallock, forms a convenient means of approximately investigating the positions of the nodes and the frequencies of vibration of a ship. The formula given above suggests that by making a model of material whose modulus E and See also:density p are known, and on a linear See also:scale of n, then if N„ N. refer to ship and model, N. Er , P„, Nm n\JEm p, This relation is unaffected if the lateral distribution of material is changed in the model, provided that Im and the weight of the model per foot run are unaltered at each point in the length; the model is therefore made solid and of rectangular or other convenient section, so that I I Im=n • I, and Wm=n, ' Pp .

Wo; the weight being also similarly distributed in a longitudinal direction to that in the ship. The model is supported at points, whose positions are obtained by trial, giving the highest frequency for the mode of vibration considered; these points are the nodes corresponding to the See also:

free vibrations when the model is unsupported, and the influence of the supports is thus eliminated. On comparison with the results obtained in a ship, the reliability of such model experiments has 12.34 12.39 T14vNQERER, SURVEY OYTRACKTRAVERSED BYSHIP UNDER THE ACTION OF 31°OF HELM CiORRESPONDIN3 TO THE INITIAL SPEED Of IO.S KNOTS., 41 12.35 2.33 M M A 12.32 /is 401R!CTION Oft See also:WIND. -<(ZK`K>S~* M.M s ,g 1:•55.23.4 °'4 .M 404e ,.

End of Article: FIGS 58

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