Chapter briefs for GATE Naval Architecture.

What the paper asks

Numerical answer questions dominate here: compute an area or volume by Simpson's rules, find TPC or MCT at a given draught, work out sinkage and trim after loading a weight, or read a value off the curves of form. Conceptual questions test what each form coefficient physically means and how it moves with hull shape.

The concepts, in order

Displacement and buoyancy. A floating body displaces its own weight of water. Volume of displacement ∇ is fixed by the hull and the draught; displacement Δ is that volume times the water density, so moving from sea to river water changes Δ but not the shape of the hull — the ship sinks deeper until ∇ has grown enough to restore the balance.

Form coefficients. C_B tells you how box-like the underwater body is, C_M how full the midship section is, C_P how the volume is distributed along the length, and C_W how full the waterplane is. C_P is the one that matters for resistance, because it describes longitudinal fullness — the driver of wave making.

Waterplane properties. The waterplane area gives TPC. Its second moment about the centreline gives BM_T, and about a transverse axis through F gives BM_L, which in turn gives MCT. The centre of flotation F is the centroid of the waterplane and is the point about which the ship trims — not amidships.

Numerical integration. Areas, volumes and moments come from Simpson's rules applied to a table of ordinates. First rule needs an even number of equal intervals; second rule (the 3/8 rule) needs a multiple of three. Half-ordinates and half-intervals near the ends are where careless marks go.

Weight added or shifted. A small weight added at the centre of flotation sinks the ship bodily by w/TPC with no trim. Added anywhere else, resolve it into that bodily sinkage plus a trimming moment w·d about F, then split the resulting trim between forward and after draughts in proportion to the distance of F from each perpendicular.

Where marks are lost

• Trimming about amidships. The ship trims about F. When F is well aft of midships, splitting trim equally forward and aft gives a wrong draught by a decimetre or more.
• Density substituted in the wrong place. ∇ is geometry, Δ is force. In a dock-water problem it is ∇ that changes, driven by ρ.
• Simpson's rule with the wrong number of intervals. Five ordinates is four intervals — fine for the first rule. Six ordinates is five intervals, and neither rule applies without splitting the range.
• Half-breadths treated as full breadths. Offset tables are usually half-breadths; the waterplane area is twice the integral.

What the paper asks

Compute GM from KM and KG; apply a free surface correction; find the list from a transverse weight shift; read GZ off a KN curve; integrate a GZ curve for dynamical stability; check a loading condition against the IS Code criteria; and the damage-stability basics — lost buoyancy, permeability, floodable length.

The concepts, in order

Initial stability. KM = KB + BM is pure geometry: it depends on the hull and the draught, not on the loading. KG is the loading. GM = KM − KG is therefore the one number the ship's staff can actually change, and it sets the slope of the GZ curve at the origin.

The righting lever at large angles. Small-angle theory (GZ ≈ GM·sin θ) fails once the deck edge nears the water. Beyond that, GZ comes from cross curves: GZ = KN − KG·sin θ. KN is tabulated for an assumed KG of zero, so the correction is always subtracted.

Free surface. A slack tank does not move the centre of gravity, but it behaves as though it did — the virtual rise in G is i·ρ_liquid/Δ, where i is the second moment of the free liquid surface about its own centreline. Because i goes as b³, breadth is everything: one undivided tank is far worse than two half-breadth tanks, and how much liquid is in the tank barely matters.

List versus loll. A list comes from an off-centre weight with positive GM, and it grows with the moment. A loll comes from negative GM at upright, and the ship settles at the angle where BM has grown enough to restore GZ to zero. The corrective actions are opposite: for a loll you lower G first and never move weight to the high side.

Damaged stability. Use the lost-buoyancy method: the flooded space stops contributing buoyancy, so the ship sinks and trims until the intact volume replaces it. Permeability is the fraction of the space that can actually flood — about 0.95 for an empty hold, 0.85 for machinery, 0.60 for stores.

Where marks are lost

• Degrees left in a GZ-curve integration. Dynamical stability is Δ × the area under GZ in metre-radians. Simpson's rule on a curve plotted in degrees needs the interval converted — multiply by π/180.
• Free surface applied to KM instead of KG. It is a virtual rise of G; the effective GM is KM − (KG + FSC).
• Forgetting that KN already excludes KG. Subtract KG·sin θ once, not twice.
• Treating a loll as a list. Loading against a loll on the high side can capsize the ship — an examiner favourite as a conceptual question.

What the paper asks

A model test extrapolation is close to guaranteed — given model resistance at a model speed, find the ship's effective power. Around it sit questions on the components of resistance, the ITTC-1957 line, form factor, the humps and hollows of the wave-making curve, and roughness and appendage allowances.

The concepts, in order

Froude's hypothesis. Total resistance = frictional + residuary. Friction is estimated from an equivalent flat plate of the same wetted surface and length; whatever remains is residuary, and residuary resistance coefficients are equal between model and ship at the same Froude number. This is the only reason a 5 m model can predict a 200 m ship.

Why you cannot scale both. Equal Froude number requires V_m = V_s/√λ. Equal Reynolds number would require V_m = V_s·λ. Both cannot hold in the same water, so the test satisfies Froude and friction is computed analytically for each of model and ship at its own Reynolds number.

Wave making. The bow and stern wave systems interfere. When their crests coincide, resistance shows a hump; between humps, a hollow. The last and largest hump sits near F_n ≈ 0.5, which is why a displacement hull becomes expensive to drive beyond about 1.34√L in knots and feet.

Form factor. The modern (1978 ITTC) treatment writes C_T = (1+k)C_F + C_W, acknowledging that a three-dimensional hull has more viscous resistance than a flat plate. k is found from a low-speed test where wave making is negligible.

Allowances. A correlation allowance C_A covers hull roughness and the model-ship correlation; air resistance and appendages are added separately. All of these go on the ship side, never the model side.

Where marks are lost

• Using one C_F for both. Model and ship have different Reynolds numbers, so each gets its own C_F from the ITTC line.
• Scaling wetted surface by λ instead of λ². Areas go as λ², volumes and displacement as λ³.
• Fresh water in the tank, sea water at sea. Both ρ and ν differ; check which the question intends.
• Reporting brake power when the question asks effective power. P_E = R_T·V is the towrope figure and includes no propulsive losses.

What the paper asks

Read K_T and K_Q off an open-water diagram at a given J and work through to delivered power; compute wake fraction and thrust deduction from given values; distinguish apparent from real slip; assess cavitation with a Burrill-type criterion; and describe propeller geometry — pitch ratio, blade area ratio, skew, rake.

The concepts, in order

Speed of advance. The hull drags a layer of water along with it, so the propeller works in a flow slower than the ship: V_a = V(1 − w). Every open-water coefficient uses V_a. Substituting ship speed into J is the single most common error in this topic.

Thrust deduction. A working propeller lowers the pressure at the stern and increases the hull's resistance, so the thrust required exceeds the towed resistance: R = T(1 − t). It is a hull effect, not a propeller loss.

The efficiency chain. Hull efficiency η_H = (1−t)/(1−w) can exceed unity, which surprises people — it reflects the propeller recovering energy from the wake. Open-water efficiency η_O comes from the K_T–K_Q–J diagram. The relative rotative efficiency η_R corrects for the non-uniform flow behind the hull. Their product is the quasi-propulsive coefficient η_D.

Slip. Apparent slip uses ship speed and is what a bridge log reports; real slip uses speed of advance and is the physically meaningful figure. Apparent slip can even go negative in a following current.

Cavitation. When local pressure on the back of the blade falls to the vapour pressure, cavities form and collapse — thrust breakdown, erosion, noise. The defence is blade area: raise the developed area ratio to reduce mean thrust loading, and check against the Burrill diagram at the cavitation number for the shaft depth.

Where marks are lost

• J computed with V instead of V_a. Wake fraction exists precisely so this substitution is wrong.
• Confusing which power is asked. P_E → P_D → P_S → P_B, each one larger than the last.
• n in rpm. K_T and K_Q want revolutions per second.
• Blade rate confused with shaft rate when a question links propulsion to hull vibration — multiply by the number of blades.

What the paper asks

Section modulus of a simplified midship section; bending stress at deck and keel; shear flow in a closed section; still-water and wave bending moment; Euler and plate buckling; the standard beam and column results; and conceptual questions on panting, pounding, racking and the reason for continuous longitudinal material.

The concepts, in order

The hull as a beam. Weight and buoyancy do not match at every station, so the difference is a load curve. Integrate once for shear force, again for bending moment. Shear peaks near the quarter lengths, bending moment near amidships — which is why the midship section is the one that gets calculated.

Hogging and sagging. Hogging puts the deck in tension and the bottom in compression; sagging reverses it. Still-water condition depends on the loading; the wave contribution is conventionally taken with a trochoidal wave of ship length and height L/20 (or the classification-society formula).

Section modulus. Only continuous longitudinal material counts. Find the neutral axis by taking moments of area, get I about it, then Z = I/y separately for deck and keel — the two differ whenever the neutral axis is not at half depth, and the smaller Z governs.

Shear flow. q = VQ/I, distributed around the section; in a plated hull the side shell carries most of it. Shear stress is τ = q/t, so thin side plating is where shear buckling starts.

Buckling. Slender columns fail by Euler buckling at π²EI/L_e²; stocky ones squash. Plate panels between stiffeners buckle at a critical stress that goes as (t/b)², which is why stiffener spacing matters more than plate thickness for a given weight.

Where marks are lost

• Using the same Z for deck and keel. Compute y to each, and check both.
• Including non-continuous material — superstructure that stops short, or transverse members — in the hull girder section.
• Effective length in Euler's formula. Fixed–fixed is L/2, pinned–pinned is L, fixed–free is 2L. The factor is squared, so a slip here is a factor of four.
• Sign of the bending moment. Sagging and hogging swap which fibre is in tension, and questions on deck stress depend on it.

What the paper asks

Natural frequency of a spring–mass or a shaft in torsion; damping ratio and logarithmic decrement; the magnification factor near resonance; blade-rate excitation from the propeller; hull girder vibration modes; and fatigue through S–N curves and Miner's rule.

The concepts, in order

Free vibration. ω_n = √(k/m) for translation, √(k_t/J) for torsion. Everything else in the topic is a modification of this: adding damping lowers the observed frequency slightly, adding entrained water lowers a hull's frequency substantially through added mass.

Damping. The ratio ζ = c/c_c classifies the response — under-damped and oscillatory below 1, critically damped at 1. From a decay trace, the logarithmic decrement between successive peaks gives ζ directly.

Forced vibration and resonance. The amplitude ratio peaks near a frequency ratio of one, and the peak height is set entirely by damping. Away from resonance, stiffness controls the response below ω_n and mass controls it above.

Ship excitation. The propeller excites the hull at blade rate — shaft rev/s multiplied by the number of blades — and at its harmonics. Main engine unbalanced forces and moments excite at engine order. Design practice is to keep the hull's two-node vertical frequency clear of blade rate at service revolutions.

Fatigue. Cracks start at details, not at mid-panel: hatch corners, bracket toes, weld toes. The S–N curve gives cycles to failure at a stress range, and Miner's rule sums the fractional damage across the load spectrum. A stress concentration factor multiplies the nominal stress before you enter the curve.

Where marks are lost

• Blade rate taken as shaft rate. A four-bladed propeller at 120 rpm excites at 8 Hz, not 2 Hz.
• ω and f interchanged. f = ω/2π; questions mix rad/s and Hz deliberately.
• Ignoring added mass in a hull frequency question — a hull in water is far heavier dynamically than its own weight.
• Miner's rule with stress amplitude instead of stress range. S–N curves are plotted against range.

What the paper asks

Pump head and power, NPSH and cavitation, affinity laws; friction loss in a pipe run; heat exchanger duty by LMTD; refrigeration COP; the air-standard cycles and specific fuel consumption; and electrical basics — three-phase power, power factor, and the difference between kW and kVA.

The concepts, in order

Pumps. Total head is the sum of static lift, pressure difference, velocity head and friction. Hydraulic power is ρgQH; divide by efficiency for shaft power. Centrifugal pumps follow the affinity laws with speed, so a 10 % speed rise costs about a third more power.

NPSH. Available NPSH is what the system offers at the impeller eye — atmospheric or tank pressure, minus vapour pressure, minus static lift and suction friction. It must exceed the pump's required NPSH or the pump cavitates. Hot liquids are the hard case because vapour pressure rises steeply with temperature.

Piping. Darcy–Weisbach for friction, with the friction factor from the Moody chart or, in laminar flow, simply 64/R_e. Minor losses from bends and valves are handled as equivalent lengths or as K·V²/2g.

Heat exchangers. Q = U·A·LMTD. Counterflow gives a larger LMTD than parallel flow for the same terminal temperatures, so it needs less area for the same duty. Where an outlet temperature is unknown, ε–NTU is faster than iterating on LMTD.

Refrigeration and cycles. COP is useful effect over work in, so a refrigerator's COP is Q_e/W and a heat pump's is one greater. For prime movers, know the air-standard efficiencies of Otto, Diesel and dual cycles and how compression ratio and cut-off ratio move them.

Where marks are lost

• Suction lift sign in NPSH. A pump above the liquid subtracts static head; a flooded suction adds it.
• Affinity laws applied to a change in impeller diameter as if it were speed. The exponents differ.
• Parallel-flow LMTD used for a counterflow exchanger. Check which stream enters at which end.
• kW and kVA treated as interchangeable. kW = √3·V·I·cos φ for three phase.

What the paper asks

Split a displacement into lightweight and deadweight; estimate steel, outfit and machinery weights; work through freeboard and tonnage concepts; identify stages of the design spiral; and production questions on block construction, welding processes, distortion and non-destructive testing.

The concepts, in order

The design spiral. Requirements → preliminary dimensions → form and coefficients → powering → weight and capacity → stability → structure → cost, and around again. It is iterative because every choice invalidates an earlier estimate; the first pass is deliberately crude.

Weight groups. Δ = LWT + DWT. Lightweight is steel, outfit and machinery — what the yard delivers. Deadweight is cargo, fuel, fresh water, stores, crew and ballast — what the owner puts aboard. Steel weight is commonly estimated from the cubic number L·B·D, corrected for block coefficient.

Freeboard and tonnage. Freeboard is a safety requirement expressed as a minimum, computed from the load line rules by ship type and length with corrections for depth, superstructure and sheer. Tonnage is volumetric: gross tonnage measures enclosed volume, not weight, and drives port dues and manning rules.

Production. Modern hulls are built as blocks, outfitted before erection because work is far cheaper on the shop floor than up on the berth. Accuracy control and welding sequence matter because distortion is cumulative and expensive to fair out afterwards.

Welding and inspection. Heat input governs the microstructure and the residual stress. Radiography and ultrasonics find internal defects, dye penetrant and magnetic particle find surface-breaking ones — a distinction worth remembering, since it is a common single-mark question.

Where marks are lost

• Deadweight confused with cargo weight. Deadweight includes bunkers, water and stores.
• Gross tonnage treated as a mass. It is a dimensionless volumetric measure.
• Freeboard treated as a design output rather than a rule-based minimum.
• NDT methods mismatched to defect type — penetrant will never find a subsurface inclusion.

What the paper asks

Wave celerity, length and group velocity in deep and shallow water; significant wave height and zero-crossing period from spectral moments; encounter frequency; response amplitude operators and the idea of a transfer function; Morison's equation for a slender member; and an overview of offshore platform types and mooring.

The concepts, in order

Linear wave theory. The dispersion relation ω² = gk·tanh(kd) collapses to ω² = gk in deep water (d > λ/2) and to c = √(gd) in shallow water (d < λ/20). Almost every wave question is a matter of deciding which regime applies before touching a formula.

Group velocity. Energy travels at the group velocity, which is half the phase speed in deep water and equal to it in shallow water. Questions about how fast a wave system's energy reaches a coast rest on this.

Sea states as spectra. A real sea is a superposition of components described by an energy spectrum — Pierson–Moskowitz for a fully developed sea, JONSWAP for a fetch-limited one. Significant wave height comes from the zeroth spectral moment, and periods from ratios of the higher moments.

Response and RAO. The ship is treated as a linear filter: response amplitude divided by wave amplitude, plotted against frequency, is the RAO. Multiply the RAO squared by the wave spectrum to get the response spectrum. Roll shows a sharp peak because roll damping is small; heave and pitch are heavily damped and follow the wave at long periods.

Encounter frequency. The frequency the ship feels depends on its own speed and heading. In head seas it rises, in following seas it falls and can pass through zero — the condition where the ship is overtaken by its own wave system, and where broaching risk lives.

Where marks are lost

• Deep-water formulas used in shallow water. Check d/λ before choosing.
• Group velocity assumed equal to phase velocity. In deep water it is half.
• H_s read as an amplitude. It is a height — 4√m₀, roughly twice the RMS amplitude times two.
• Heading sign in the encounter frequency. μ = 180° is head seas; getting the cosine's sign wrong inverts the result.

What the paper asks

Eigenvalues and rank; gradient, divergence and curl with the integral theorems; first and second order ODEs; Laplace transforms; complex variables through Cauchy–Riemann and residues; probability distributions and Bayes; and numerical methods — root finding, integration, and marching an ODE forward.

The concepts, in order

Linear algebra. Solve |A − λI| = 0 for eigenvalues, then (A − λI)x = 0 for each eigenvector. Two checks catch most arithmetic slips instantly: the eigenvalues sum to the trace and multiply to the determinant. Rank determines whether a system has no solution, one, or infinitely many.

Vector calculus. Gradient of a scalar gives the direction of steepest increase; divergence measures source strength; curl measures rotation. Green's, Stokes' and the divergence theorems each convert an integral over a boundary into one over the interior — recognise which dimension you are moving between and the choice is automatic.

Differential equations. First order linear: integrating factor e^∫P dx. Second order with constant coefficients: complementary function from the auxiliary equation, plus a particular integral matched to the forcing term. This is also the mathematics behind the vibration topic, so the two reinforce each other.

Probability. Know when each distribution applies — binomial for a fixed number of independent trials, Poisson for rare events in an interval, normal for sums of many small effects. Bayes' theorem questions are almost always about reversing a conditional that has been stated the other way round.

Numerical methods. Newton–Raphson converges quadratically but needs a sound starting point; the trapezoidal rule under- or over-estimates depending on curvature; Simpson's rule is exact for cubics and needs an even number of intervals — the same rule that runs through the hydrostatics topic.

Where marks are lost

• Rounding a numerical answer too early. NAT questions have a tolerance band; carry full precision until the final line.
• Calculator in degrees for a calculus question. Derivatives of trigonometric functions assume radians.
• Simpson's rule with an odd number of intervals. Split the range or use the 3/8 rule.
• Confusing P(A|B) with P(B|A). Write the tree before applying Bayes.

What the paper asks

Verbal ability — grammar, word usage, inference from a short passage — and quantitative aptitude: ratios and percentages, time–speed–distance, work and time, averages and mixtures, permutations and combinations, elementary probability, mensuration, number series, and data interpretation from a chart or table. Analytical and spatial reasoning appear as well.

The concepts, in order

Ratio thinking. Most quantitative aptitude questions are ratio questions wearing different clothes. Percentage change, mixtures, speed, and work rates all reduce to setting up a proportion and being careful about what the base quantity is.

Work and rates. Convert everything to rate per unit time and add the rates, never the times. Two workers taking 6 and 12 days together take 4, because 1/6 + 1/12 = 1/4.

Speed and relative speed. Add speeds when closing, subtract when moving apart, and be careful that average speed over equal distances is the harmonic mean, not the arithmetic one.

Counting. Decide whether order matters before choosing a permutation or a combination, then check for restrictions — repeated letters, forbidden adjacencies, at-least-one conditions — which usually resolve fastest through the complement.

Verbal. For inference questions, the correct option is the one that cannot be false given the passage. Anything requiring outside knowledge or an extra assumption is wrong, however reasonable it sounds.

Where marks are lost

• Time spent, not marks lost. These are the cheapest marks on the paper and should be taken early, not left for the last twenty minutes.
• Negative marking on guessed MCQs. Aptitude MCQs carry the same penalty as subject questions; eliminate rather than guess blindly.
• Averaging speeds arithmetically over equal distances instead of using the harmonic mean.
• Choosing the "most reasonable" verbal option rather than the one the passage actually supports.

What it covers

The processes a yard actually uses and where each one lives; joints, edge preparation and reading a weld symbol; heat input, the heat affected zone and the metallurgy behind preheat and hydrogen cracking; defects with their causes and cures; distortion and how yards keep it out; the WPS–PQR–welder qualification chain; NDT and acceptance; and a quick reference to every IACS requirement that touches welding.

Who it is for

GATE candidates meeting welding inside ship production, and anyone facing a shipyard, design office or QC interview — the last section is twenty-six questions in three bands, each with a model answer.