Naval Architecture and Ocean Engineering II: Manoeuvring, Seakeeping, Ship Structures and Ocean Engineering

Manoeuvring is the horizontal-plane motion of a ship: path keeping (holding a straight course against disturbances) and path changing (turning, zig-zagging, stopping). For small deviations from a straight course at speed u, the sway velocity v and yaw rate r obey the linearised equations m(v̇ + ur) = Y_v v + Y_v̇ v̇ + Y_r r + Y_ṙ ṙ + Y_δ δ and I_z ṙ = N_v v + N_v̇ v̇ + N_r r + N_ṙ ṙ + N_δ δ, whose coefficients Y_v, N_r, … are the hydrodynamic derivatives — the rates of change of side force Y and yaw moment N with each motion variable and with rudder angle δ. With the rudder held fixed, the ship is directionally (straight-line) stable if a disturbance dies out and it settles on a new straight path; with the centre of gravity at the origin the condition is the stability index C = Y_v N_r − N_v (Y_r − mu) > 0. An unstable ship can still be steered, but only by continuous rudder action.

Eliminating v gives Nomoto’s first-order model T ṙ + r = K δ: the gain K is the steady yaw rate per unit rudder angle (turning ability) and the time constant T measures how quickly the ship answers the helm (a small T is quick; a negative T marks an unstable ship). The definitive manoeuvres, run on sea trials and in free-running model tests, measure these qualities: the turning circle at full rudder gives the advance, transfer, tactical diameter and steady turning diameter (IMO criteria ask for an advance under 4.5 L and a tactical diameter under 5 L); the zig-zag (Kempf) manoeuvre, 10°/10° or 20°/20°, gives the overshoot angles and the time to check a swing; the spiral (Dieudonné) and reverse spiral (Bech) tests plot steady yaw rate against rudder angle and reveal instability as a hysteresis loop about zero rudder; the pull-out test shows whether the yaw rate decays to zero after the rudder is centred; and the crash-stop gives the head reach and track reach from full ahead to full astern.

Rudder hydrodynamics. A rudder is a low-aspect-ratio foil: its lift rises almost linearly with angle until it stalls, and the normal force acts at a centre of pressure that moves aft with angle, so the stock torque depends on how much area lies forward of the stock — a balanced rudder, with part of its area forward, needs a smaller steering gear. Types include the spade rudder, the semi-balanced horn rudder, rudders with a sole piece, and high-lift flap and fish-tail rudders. Helm is normally limited to 35°, and the steering gear must move the rudder from 35° on one side to 30° on the other within 28 s. The propeller slipstream multiplies the flow speed over the rudder and makes it far more effective ahead than when the engine is stopped or astern; the hull, skegs and the stern shape straighten or shadow the inflow; and a rudder close behind a propeller is also exposed to its cavitation and vibration.

Regular waves are described by linear (Airy) theory: in deep water the celerity is c = gT/(2π) and the length L = gT²/(2π), so a 10 s wave travels at 9.81 × 10/6.283 = 15.6 m/s and is 156 m long. The trochoidal wave, the profile traced by a point inside a rolling circle, has sharper crests and flatter troughs than the sine wave and was the classical standard wave for longitudinal strength, with a height of L/20. A real sea is irregular: it is modelled as a superposition of many regular components of random phase, whose energy distribution over frequency is the wave spectrum S(ω). Its moments m_n = ∫ωⁿS(ω)dω give the statistics: the significant wave height H_s = 4√m₀ (the mean of the highest third of the waves) and the mean zero-crossing period 2π√(m₀/m₂). Standard spectra are the Pierson–Moskowitz spectrum of a fully developed sea and the more peaked JONSWAP spectrum of a fetch-limited sea.

A ship moving at speed V meets the waves at the encounter frequency ω_e = ω − (ω²V/g) cos μ, μ being the heading (180° in head seas, where ω_e = ω + ω²V/g, and 0° in following seas). A 0.6 rad/s wave met head-on at 10 m/s is encountered at 0.6 + 0.36 × 10/9.81 = 0.967 rad/s. The ship has six degrees of freedom — surge, sway and heave (translations) and roll, pitch and yaw (rotations). For a ship symmetric about its centre plane, heave and pitch are coupled with each other in the vertical plane, and sway, roll and yaw with each other in the horizontal; the two groups are uncoupled in linear theory. Each motion obeys (m + a)ẍ + bẋ + cx = F cos ω_e t, with a the added mass, b the wave-making damping and c the restoring stiffness: ρgA_W for heave, ΔgGM for roll and ΔgGM_L for pitch. The response per unit wave amplitude is the response amplitude operator (RAO), and the motion spectrum is RAO² × wave spectrum.

The undamped natural periods follow: roll T_φ = 2πk/√(gGM), k being the roll radius of gyration including added inertia — k = 8 m and GM = 1.0 m give 2π × 8/√9.81 = 16.05 s, and a stiff ship with a large GM rolls quickly and violently; heave T_z = 2π√[(m + a)/(ρgA_W)], which for a box of draught T without added mass is 2π√(T/g) = 4.01 s at T = 4 m. Resonance between ω_e and a natural frequency gives large motions. The dynamic effects of motion in a seaway are slamming (bottom or bow-flare impact when the forefoot emerges and re-enters), deck wetness and green water, propeller racing when the stern lifts, added resistance and involuntary speed loss, high vertical accelerations that cause motion sickness and limit crew work, cargo shift, and the loads that make the hull whip. Stabilisers reduce roll, the motion with the least natural damping: passive bilge keels (the cheapest and most common), passive anti-roll tanks (U-tube or free-surface flume tanks tuned to the roll period), and active fin stabilisers, active tanks, gyroscopic stabilisers and rudder-roll stabilisation. Pitch and heave are too powerful to stabilise; they are reduced by hull form and by changing course and speed.

Methods. Strip theory divides a slender ship into two-dimensional transverse strips, computes each section’s added mass, damping and wave exciting force as if it were an infinite cylinder, and integrates along the length; it is fast and good for heave and pitch at moderate speed, weaker at low encounter frequency, high speed and for roll, where viscous damping matters. Boundary-element (panel) methods solve the three-dimensional potential problem on the wetted surface with Green-function or Rankine sources; finite-element methods discretise the fluid domain or couple the fluid to an elastic hull (hydroelasticity); and viscous CFD adds what potential theory leaves out. Experiments in a seakeeping basin use regular and irregular waves from wave makers. Among high-performance vehicles, the SWATH hull, with a small waterplane on deep struts, has low heave and pitch stiffness and hence long natural periods above most wave periods, and so moves little; the catamaran has great transverse stability but a quick, jerky roll; planing craft suffer severe vertical accelerations from slamming; and hydrofoil craft fly clear of the waves on submerged foils.

Shipbuilding materials. Most hulls are welded mild (normal-strength) steel of 235 MPa minimum yield, supplied in grades A, B, D and E of increasing notch toughness for use where low temperature or thick plate makes brittle fracture a risk, and higher-tensile steels (AH32, AH36, AH40, of 315, 355 and 390 MPa yield) in the highly stressed deck and bottom of large ships. Aluminium alloys save weight in superstructures and fast craft; fibre-reinforced plastics serve small craft; stainless and clad steels line chemical tanks. Joining: the hull is built of prefabricated blocks joined by welding — manual metal-arc, flux-cored and gas metal-arc for fitting and positional work, submerged-arc for long flat seams, electro-gas for vertical butts — as butt welds in plating and fillet welds between stiffeners and plate. Riveting survives in repair and some crack-arrest details; bolting is used where parts must be removable.

Framing systems. In transverse framing, closely spaced transverse frames, floors and deck beams support the plating; it suits short ships and local loads and leaves the hold clear. In longitudinal framing (the Isherwood system), closely spaced longitudinal stiffeners run fore and aft, supported by widely spaced deep web frames and transverses; the longitudinals add to the hull-girder section modulus and stiffen the plating against buckling, so it is used in long ships dominated by longitudinal bending — tankers and the deck and bottom of bulk carriers. Combined framing puts longitudinals in the deck and bottom and transverse frames in the sides. The bottom is usually a double bottom — outer shell, inner bottom (tank top), centre and side girders and solid or bracket floors — which carries cargo, forms tanks and survives grounding; the side has shell plating, frames or longitudinals, web frames and stringers; the deck has plating on beams or longitudinals and girders, with hatch coamings; bulkheads — the collision bulkhead forward, transverse watertight bulkheads and longitudinal bulkheads, plane-stiffened or corrugated — subdivide the hull and support decks. The end structures carry panting and slamming forward (panting beams and stringers, breasthooks, the bulbous bow) and the stern frame, rudder horn and shaft bossings aft. Members meet through structural connections — brackets, knees, collar plates where longitudinals pass through webs — which are where cracks begin if they are badly shaped.

The superstructure (full width, erected on the strength deck) shares in hull-girder bending if it is long enough, while a short deckhouse does not and must be isolated by expansion joints or tapered in to avoid cracking at its ends. Hatch covers — steel pontoon, folding or rolling covers on coamings — must be weathertight and carry cargo and green-sea loads, and the openings they close cut the deck, so their corners are rounded and thickened. Machinery foundations (seatings) are deep girders tied into the double bottom, stiff enough to keep the engine aligned and to move structural natural frequencies away from engine and propeller excitation. Cargo-handling systems — cranes, derricks, king posts and their pedestals — impose concentrated loads that are carried down by local strengthening under the deck.

Loads on a ship in a seaway are the static still-water loads from the distributions of weight and buoyancy; wave-induced vertical and horizontal bending, shear and torsion; dynamic loads — slamming, whipping and springing; local pressures of sea, cargo and ballast and green water; and thermal, launching and docking loads. For longitudinal strength the ship is a free–free beam: the load per unit length q(x) = b(x) − w(x) (buoyancy minus weight) integrates to the shear force F(x) = ∫q dx, and that to the bending moment M(x) = ∫F dx, both zero at the ends. The bending moment is greatest near amidships, the shear force near the quarter lengths. With a wave crest amidships and troughs at the ends the ship hogs — the deck is in tension and the bottom in compression; with a trough amidships it sags, and the deck is in compression. A box barge 100 m long whose hull weight is balanced by uniform buoyancy, carrying 1000 t at amidships, has an extra uniform buoyancy of 10 t/m and a sagging moment P L/8 = 9810 kN × 100/8 = 122.6 MN·m.

The hull girder’s bending stress is the beam formula σ = M y/I = M/Z, with I the second moment of all longitudinally continuous material about the neutral axis and Z = I/y the section modulus at the deck or the bottom. With M = 1.2 × 10⁹ N·m, I = 400 m⁴ and the deck 10 m above the neutral axis, the deck stress is 1.2 × 10⁹ × 10/400 = 30 MPa, and the bottom, 8 m below, 24 MPa. Transverse strength resists racking and the water pressure on the frame rings; torsion matters in open ships such as container ships, whose wide hatches leave little closed cell, so warping stresses concentrate at hatch corners and a torsion box is built under the coaming. Stiffened-plate analysis treats the plate between stiffeners: under in-plane compression a long plate simply supported on its edges buckles at σ_cr = kπ²E/[12(1 − ν²)](t/b)² with k = 4 — a 10 mm plate between stiffeners 800 mm apart buckles elastically at 113 MPa — and a value above about half the yield is reduced by the Johnson–Ostenfeld correction σ_y(1 − σ_y/4σ_E). Local strength covers panels, stiffeners and girders under lateral pressure, and the ultimate strength of the hull girder is the bending moment at which progressive buckling and yielding of the compressed flange collapse the section.

Reliability analysis treats the capacity R and the load effect S as random variables and the margin g = R − S as the limit state; the reliability index β = μ_g/σ_g (mean over standard deviation of the margin) measures safety, and the probability of failure is Φ(−β) — a margin of mean 50 MPa and standard deviation 20 MPa has β = 2.5. Structural vibration of the hull girder, in its two-node vertical mode and higher modes, and of local panels is excited by the propeller at blade frequency (blades × rev/s) and by the main engine; springing is resonant vibration excited by waves, and whipping the transient vibration after a slam. Fatigue accumulates under the millions of wave-load cycles of a ship’s life and starts at stress concentrations — hatch corners, bracket toes, the connections of longitudinals to webs — and damage from blocks at different stress ranges is summed by Miner’s rule D = Σnᵢ/Nᵢ, failure being predicted at D = 1. Fracture: a crack grows until the stress intensity K = Yσ√(πa) reaches the toughness; brittle fracture at low temperature broke welded ships of the 1940s in two, which is why tough grades D and E and crack-arrester strakes are specified at the sheer strake, bilge and deck.

Physical properties of sea water. Its salinity averages about 35 grams of salt per kilogram; its density, about 1020–1030 kg/m³, rises with salinity and pressure and falls with temperature; it freezes near −1.9 °C; and sound travels through it at about 1500 m/s. A thermocline separates the warm, mixed surface layer from the cold deep water. Tides are the long waves raised by the gravitational pull of the moon and the sun on the rotating earth: semi-diurnal tides give two highs and two lows a day (period about 12 h 25 min), diurnal tides one, and mixed tides unequal pairs; spring tides of large range come when sun and moon are in line at new and full moon, neap tides at the quarters. Tides decide the depth available in a port, the timing of docking, and tidal currents. Wind waves grow with wind speed, fetch and duration until the sea is fully developed; once they leave the generating area they become long-crested swell; approaching the shore they shoal, refract and break. Waves set the design loads of every marine structure, taken as an extreme sea of long return period.

Offshore structures. Fixed platforms: the steel jacket, a welded tubular space frame piled into the seabed and carrying the topsides above the waves, and the concrete gravity platform, held in place by its own weight and often storing oil in its base caissons. Floating platforms: the semi-submersible, a deck on columns standing on submerged pontoons, whose small waterplane gives long natural periods and small motions; the jack-up, a mobile barge that lowers its legs to the seabed and jacks its hull clear of the waves, used in shallow water; the tension-leg platform (TLP), whose excess buoyancy is held down by vertical tendons in tension, so that heave, roll and pitch are nearly eliminated while surge, sway and yaw remain compliant; and the FPSO (floating production, storage and offloading), a ship-shaped hull, usually turret-moored so that it weathervanes into the weather, that processes and stores oil for shuttle tankers. Mooring holds a floating unit by catenary chains or wires, whose restoring force comes from lifting their weight off the seabed, or by taut synthetic-rope legs, whose restoring force is elastic, in a spread or single-point pattern; dynamic positioning keeps station with computer-controlled thrusters fed by satellite, acoustic and taut-wire position references, in classes of increasing redundancy.

The in-line wave force on a slender member of such a structure is given by Morison’s equation, f = ½ρC_D D u|u| + ρC_M(πD²/4) u̇ per unit length. A 2 m diameter leg in sea water, at an instant when u = 1.5 m/s and u̇ = 0.8 m/s², with C_D = 0.7 and C_M = 2.0, carries a drag force of ½ × 1025 × 0.7 × 2 × 2.25 = 1614 N/m and an inertia force of 1025 × 2.0 × π × 0.8 = 5152 N/m, 6767 N/m in all. Inertia dominates for large members in small waves, drag for thin members in large waves; members wider than about a fifth of the wavelength diffract the waves and need diffraction theory instead.