FIGS 58.—Cargo Vessel of 12,000 Tons on FIG. 59.—Cargo Vessel of '12,000 Tens
Wave Crest. in Wave'. Trough.
section was approximately in accordance with the linear law assumed in the theory of bending.'
(c) The modulus of elasticity E was obtained by equating the sum of the moments about the neutral 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 case, the comparison is made between two ships of similar type. The relation between stress and 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 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 act inde
pendently. (This is probably more nearly the case in a ship than In a 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
original length; where 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 Young's modulus for the material. It follows that the resultant longitudinal force across a section is zero, and the moment of the 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 external bending moment M.
(e) Taking axes—Ox longitudinal, Oz 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 Portsmouth Dockyard when experiments were madeon H.M.S." Wolf " by Professor J. H. Biles for the Committeeof the 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 bent.
The table at the top of the following page shows the values obtained for E, the modulus of elasticity.
By observing the deflections of two vessels when loaded with ballast, the following values for E were obtained by T. C. Read and G. Stanbury (Traits. Inst. Nail. Arch., 1894), and are given for purposes of comparison,:
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 sea in rough weather with the object of comparing the stresses then observed with those calculated under the 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 dock experiments. The maximum stresses were as follows:
Condition. Stress—Tons per Square Inch.
Keel. 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 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 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 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 general practice.
Since the distribution of pressure in the water of a wave 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 weight of the water displaced; and the position of the ship when in 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 motion, and undisturbed by the intrusion of the ship, the equilibrium position can be obtained and the modified stresses evaluated. This process was first applied to ships by Mr W. E. 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. ordinary manner, and in a torpedoboat destroyer a reduction of about 100A has been obtained. This reduction increases as the draught and fullness of the ships are increased, and the bending moment on a squarebilged 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 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 lead to a further modification in the bending moment obtained (see paper by T. C. Read, Trans. LN.A., 1890). Considering first the effect of pitching only, imagine the ship at her proper displacement (allowance being made for the altered buoyancy of the wave system as before), but momentarily out of her correct trim; the longitudinal restoring couple, due to the wedges of immersion and emersion, is balanced by the moment of the reversed massaccelerations of the component parts. If the ship is longitudinally symmetrical about her midship section, one half of the moment of the restoring forces and one half of the moment of the reversed massaccelerations 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 influence oa the midship bending moment: Itappears, therefore, that in the 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 order that the given by y= a. sin sr't where" TI is the
are
Tl
apparent semiperiod of the wave. On
taking into 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 dip in still water.;
W is the displacement, and p the tons per 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 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 rolling oscillations undergone by a ship moving obliquely across the crests of a wave system (see papers by Captain Kriloff , Trans. I.N.A., 1896 and 1898).
The maximum calculated stress on vessels inclined to considerable angles of 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 direct stresses on keel, bottom, and upper 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 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 area above or below a horizontal line through the point considered, and t~ the thickness of side plating. This stress is usually greatest at or near a 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 considerable magnitude during 'heavy rolling, a means of corn Trans'',, paring the transverse strength of vessels is of some 'verse interest. A transverse bulkhead forms a region of almost bending. 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 cargo, 3r however, in which transverse bulkheads are widely spaced, a section midway along a hold may be so far removed from all bulkheads as to be uninfluenced by' their local support; and the following ' method has been proposed for comparing the transverse strengths of such ships:
A frame and a strip 0 of plating one 'frame space in width are re
garded as a stiff inextensible 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 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 lion over the
Stress. Stress: 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 67 86 10,200 10,300 6.8 6 6 11,80o 9,800
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 squareinch 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 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 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 total movement of P due to the bending of all such elements as QR in OP is obtained; when P is moved round the complete section so as to return to 0, where the total movement is zero, it follows, on subtraction and reduction, that
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 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 part
of the structure, local stresses are experienced including those caused
by waterpressure; forces on sails, masts and rigging; reactions
of moving parts of machinery; heavy blows from the
Local sea on side, deck and upper works; anchor, cable and stresses.
mooring gear, and blast from gunfire. 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 waterpressure are of small amount where the curvature is appreciable, since the plating, by compression, directly resists any tendency towards change of curvature; in a deep flatbottomed ship, on the other hand, resistance to waterpressure is chiefly due to the bending of the plating, the slight extension having little influence. The plating is supported at the transverse and longitudinal frames, and, to some extent, at the edges. The close spacing of transverse frames usually adopted in 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 special provision is frequently made to prevent the side plating near the bow from panting due to the great and rapid fluctuation of waterpressure when pitching.
The material of the structure is arranged so that the distribution of stress over any localized section of material is maintained as uniform Utrilorm as possible in order that the ratio of maximum to mean
m or stress may not be unduly large. For this reason abrupt
stress. discontinuities and sudden changes of section are avoided,
and " 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 hatchways has been known to take 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 gunfire 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 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 plane and of two frequencies, one equal to, and the other twice, the speed of revolution, the latter being due to the secondary action introduced by the connecting rod. In twinscrew 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 crank effort tends to cause torsional oscillation of the same frequency, particularly in single or twocylinder 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 blades. Such resistances occur when (I) the blades pass too close to the hull; (2) when the screw breaks the surface of the water; and (3) when the supply of water to the propeller is imperfect, due either to " cavitation " or to the screening effect of 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 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 model." The instrument used for measuring and recording vibrations consists of a weight suspended, and held laterally in position, by springs, so as to have a long period of oscillation; pens or pencils attached to the weight record the vibrations upon revolving cylinders fixed to the vessel and fitted with time records. The formula (of the same 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 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 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. 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 density p
are known, and on a linear 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 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 WIND. <(ZK`K>S~*
M.M s
,g 1:•55.23.4
°'4 .M 404e
,.
12.32
12.42
End of Article: FIGS 58 

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