The Hull Girder as a Beam

Treat the whole ship as one long, hollow box girder floating freely in the water, and the entire machinery of beam bending — shear force and bending moment — applies to it directly.

Marine EngineeringShip Structures & MaterialsFree preview

⏱️ About 16 min

The Hull Girder as a Beam — illustration

Illustrative image (AI-generated).

A 200-metre ship looks nothing like a lab specimen — but to a structural engineer it is just a very long, very hollow beam, floating freely in the water and bending under the mismatch between its weight and its buoyancy.

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The big idea: The whole ship behaves as a hull girder — a free-free beam — and the global loads that decide its structural survival are the longitudinal shear force and bending moment, whose sign convention distinguishes hogging from sagging.

🎯 By the end, you'll be able to

Treat the whole ship as a single longitudinal hull girder (a free-free box beam)
Define shear force and bending moment along the hull and explain why they arise from the weight/buoyancy mismatch
State the hogging versus sagging sign convention and which puts the deck in tension versus compression
Estimate an idealised maximum bending moment by modelling the hull as a beam under a uniformly distributed load

📎 Helpful to know first

The Whole Ship as One Beam

A ship's deck plating, bottom plating, side shell, longitudinal bulkheads, and internal structure are not independent parts — they act together as a single long hollow tube called the hull girder (or box girder). When you stand back far enough, every ship is just a very large beam floating in the water, and the beam-bending theory you met in the Solid Mechanics course applies to it almost unchanged.

The one feature that makes a ship unlike a bridge or a roof beam is its support. A ship is a free-free beam: it is not pinned or held up at its ends. Instead it is supported everywhere at once by the distributed upward pressure of the water (buoyancy), and loaded everywhere at once by its distributed downward weight. At any cross-section along the length the net load per unit length is the local weight minus the local buoyancy. Where weight exceeds buoyancy the section is pushed down; where buoyancy exceeds weight it is pushed up. It is precisely this varying net load that bends the hull girder.

From that net load the two internal actions follow directly, just as in any beam: the shear force at a section is the running integral of the net load up to that point, and the bending moment is the running integral of the shear force. The bending moment is the quantity that ultimately stresses the hull girder, and it is what the rest of this module is about.

\[ V(x) = \int \bigl(\,w(x) - b(x)\,\bigr)\,dx, \qquad M(x) = \int V(x)\,dx, \qquad M_{\max} \approx \frac{wL^{2}}{8} \]

w(x) and b(x) are the distributed weight and buoyancy per unit length; the wL²/8 form is the classic simply-supported-beam estimate used only as a quick idealisation, not the true hull-girder moment.

Hogging Versus Sagging

The sign of the bending moment tells you which way the hull is bending, and naval architects use two everyday words for the two cases:

Hogging — the hull bends so that the middle rises relative to the ends, like a hog's arched back. The deck is in tension and the bottom (keel) is in compression. Hogging occurs when buoyancy is concentrated amidships (a wave crest amidships) or when weight is concentrated toward the ends.
Sagging — the hull bends so that the middle drops relative to the ends. The deck is in compression and the bottom is in tension. Sagging occurs when weight is concentrated amidships or when a wave trough sits amidships.

The distinction matters because steel behaves differently in tension and compression: tension governs fatigue cracking at details, while compression governs the buckling of slender deck plating. A hull girder must be designed to survive both hogging and sagging, because the same ship will experience each at different moments as waves pass underneath and as cargo distribution changes.

The dashed line is the undeflected hull; the coloured curve is the hogging (deck tension) or sagging (deck compression) deflection. A real deflection is far smaller than drawn here.

✨ Why the sign convention drives design

Because hogging puts the deck in tension and sagging puts it in compression, the two cases stress the hull in different ways and are checked separately. Compression can buckle slender deck plating between stiffeners; tension grows fatigue cracks at welded details. A hull girder that comfortably survives one case may still be vulnerable to the other — which is why both hogging and sagging moments are computed for every loading and wave condition.

📝 Worked example: As a quick idealisation, model a hull as a beam of length L = 100 m carrying a uniformly distributed load w = 200 kN/m. Estimate the maximum bending moment using the classic M = wL²/8 form.

M = wL²/8 = 200 × 100²/8 = 200 × 10,000/8
= 2,000,000/8 = 250,000 kN·m
Converting to MN·m: 250,000 kN·m ÷ 1000 = 250 MN·m

✓ M = wL²/8 = 250,000 kN·m ≈ 250 MN·m (an idealisation; the real distribution comes from the weight/buoyancy curves)

✏️ Practice: Using the same idealisation, model a hull as a beam of length L = 120 m carrying w = 150 kN/m. Find the estimated maximum bending moment (in MN·m).

MN·m

Solution

M = wL²/8 = 150 × 120²/8 = 150 × 14,400/8
= 2,160,000/8 = 270,000 kN·m
= 270,000 ÷ 1000 = 270 MN·m

Check your understanding

1. In a hogging condition, the deck plating of the hull girder is in:

Hogging arches the hull so the midships rises; the deck is stretched and so carries tension, while the bottom (keel) carries compression.

2. In the idealised M = wL²/8 estimate, doubling the loaded length L while keeping w the same increases the estimated maximum bending moment by a factor of:

M ∝ L², so doubling L multiplies the estimated moment by 2² = 4 — one reason longer ships face much larger hull-girder bending moments.

✅ Key takeaways

The whole ship is a hull girder — a free-free beam supported by distributed buoyancy and loaded by distributed weight; the net load bends it
Shear force is the running integral of net load, and bending moment is the running integral of shear force — the quantity that stresses the hull
Hogging (midships rises, deck in tension) and sagging (midships drops, deck in compression) are the two cases every hull girder must survive

➡️ A still ship already bends — but the loads that really size the hull girder come from waves. Next we see how a single wave reshapes the buoyancy distribution and pushes the hull between hogging and sagging.

Want to test yourself on this? Try the Marine Engineering Aptitude test →