By ShipCalculators.com · Published 25 April 2026 · last updated 6 June 2026

Hull Strength and Longitudinal Bending

Contents

The hull girder is the primary structural spine of every ship. It carries the longitudinal bending moment that results from the mismatch between the longitudinal distributions of weight and buoyancy, with an additional dynamic component from wave-induced buoyancy variation. When those bending moments exceed the capacity of the hull cross-section, the consequences are catastrophic: the Erika broke in two off Brittany in December 1999, the Prestige broke off Galicia in November 2002, and the MOL Comfort parted at midships in the Indian Ocean in June 2013. Each of those events killed crew or caused major environmental damage and forced revisions to the governing rules.

The governing framework is the IACS Common Structural Rules for Bulk Carriers and Oil Tankers (CSR BC&OT), in force since 2006 for bulk carriers above 90 m and oil tankers above 150 m, harmonised into a single document (CSR-H) in 2015 and updated through the 2024 consolidated edition. Behind the CSR sit two Unified Requirements that apply across all IACS member society ships: UR S11 for the longitudinal strength standard (wave bending moment values, section modulus requirement, and permissible stress) and UR S7 for the absolute floor on midship section modulus. The loading instrument that monitors SWBM limits in real time draws its approved envelope from those same design calculations.

This article covers the physics of hull-girder bending, the SWBM and VWBM calculation framework, the section modulus and stress checks, shear force distribution, the incremental-iterative ultimate strength method (Smith method), fatigue, and the practical implications for bulk carriers and tankers operating under Class and flag-state requirements.

The hull girder as a box beam

A ship’s hull acts as a hollow box beam spanning the full length of the vessel. The box cross-section at any transverse frame consists of: the bottom plating and the bottom longitudinal stiffeners (flanges in hogging, compressed in sagging); the deck plating and deck longitudinals (flanges in sagging, compressed in hogging); the side shell plating and side longitudinals (web, carrying shear); and for double-bottom ships, the inner bottom plating and its stiffeners.

The hull beam is not prismatic: the cross-section changes from almost full amidships to sharply reduced sections at the bow and stern. However, for the design bending moment check, the critical section is almost always the midship region, typically defined as the 0.4L amidships zone extending from 0.3L to 0.7L from the aft perpendicular. IACS UR S11 and CSR BC&OT both focus the mandatory section modulus requirement on this zone.

The structural response of the hull girder to a vertical bending moment $M$ can be described by the Euler-Bernoulli beam bending equation. At any point in the cross-section at a distance $y$ from the neutral axis, the bending stress is:

\sigma = \frac{M \cdot y}{I_N}

where $I_N$ is the second moment of area of the hull cross-section about the horizontal neutral axis. The maximum bending stress occurs at the extreme fibres: the deck (distance $y_D$ above the neutral axis) and the keel (distance $y_K$ below). The section moduli for the two extreme fibres are:

Z_D = \frac{I_N}{y_D} \quad \text{(deck section modulus)}

Z_K = \frac{I_N}{y_K} \quad \text{(keel section modulus)}

and the maximum bending stresses are:

\sigma_D = \frac{M}{Z_D} \qquad \sigma_K = \frac{M}{Z_K}

For a typical merchant ship with a neutral axis at 40 to 50% of the depth from the keel, $Z_D$ and $Z_K$ are of similar magnitude. However, the sign of the bending stress reverses between hogging and sagging, so the deck is in tension under hogging and in compression under sagging. Compressive stress in the deck under sagging is the condition that drives buckling.

You can compute $Z_D$ and $Z_K$ for any proposed midship cross-section using the deck section modulus calculator. The hull girder stress calculator converts bending moment and section modulus to the deck and keel stresses.

Hogging and sagging: sign conventions and physical origin

A ship hogging bends with midships raised above the bow-stern line. This occurs when the bow and stern are supported on wave crests while midships is over a wave trough, or when the weight distribution is concentrated amidships (heavy cargo amidships, ballast at ends). The bending moment is positive in the conventional IACS sign convention: positive $M$ produces tensile stress in the keel and compressive stress in the deck.

A ship sagging bends with midships lower than the bow-stern line. This occurs when midships is on a wave crest and the ends are in troughs, or when heavy cargo loads are concentrated at the ends (the alternate-hold loading case for bulk carriers). Sagging produces compressive stress in the keel and tensile stress in the deck.

The IACS sign convention for bending moments in CSR BC&OT and UR S11 defines:

Hogging bending moment as positive ( $M_{SW,H} > 0$ , $M_{WH} > 0$ )
Sagging bending moment as negative ( $M_{SW,S} < 0$ , $M_{WS} < 0$ )

The design check uses the algebraic combination $M_T = M_{SW} + M_W$ for both the maximum hogging and maximum sagging cases separately.

Condition	Deck stress	Keel stress	Critical structural check
Maximum hogging	Compressive	Tensile	Deck buckling; keel yield
Maximum sagging	Tensile	Compressive	Deck yield; keel buckling
Zero crossing	Near zero	Near zero	Shear maximum (quarter-length)

Still-water bending moment: derivation and Class approval

The still-water bending moment ( $M_{SW}$ ) at any transverse section $x$ along the ship length is derived from the longitudinal distribution of downward loads (weights) and upward loads (buoyancy). Define $w(x)$ as the total downward force per unit length at position $x$ (lightship steel, cargo, ballast, fuel, stores), and $b(x)$ as the upward buoyancy force per unit length from the displaced water. The net load intensity is $q(x) = w(x) - b(x)$ .

The shear force at position $x$ from the aft perpendicular is:

V(x) = \int_0^x q(\xi) \, d\xi

The bending moment at $x$ is the integral of the shear force:

M_{SW}(x) = \int_0^x V(\xi) \, d\xi

For the entire ship in equilibrium, total weight equals total displacement, which means $V(L) = 0$ and $M_{SW}(L) = 0$ . The SWBM distribution reaches a maximum somewhere in the 0.3L to 0.7L region; for conventional cargo ships with uniform loading, the maximum is close to midships. The still-water bending moment calculator (IACS UR S11) applies the UR S11 standard SWBM limits to a given loading condition.

The Class society approves a SWBM envelope for each loading condition at newbuild design. The envelope defines permissible maximum hogging and sagging SWBM values at every frame station, derived from the design wave bending moment envelope with a reserve for uncertainty in at-sea wave loading. Under SOLAS II-1 Reg.5-1 and IACS UR S1, ships above 65 m length (and all ships above 100 m) must carry an approved loading instrument that computes $M_{SW}(x)$ for any proposed loading condition and checks it against the approved envelope. The loading instrument is tightly linked to the marine stability booklet and loading computer; the same software platform typically handles both stability and hull-girder checks.

The master is legally required to ensure the vessel doesn’t exceed the approved SWBM limits at any point during the loading sequence, during port stays, and at sea. Exceeding the approved limit even temporarily during loading operations (for example, when part of the cargo is loaded before ballast is discharged) is a Class non-conformity.

Wave-induced vertical bending moment: IACS UR S11

IACS UR S11 (Revision 7, 2022) provides the standard rule values of the vertical wave bending moment that any ship on unrestricted ocean service must be designed to withstand. The rule VWBM values represent the expected maximum in a 25-year operational life with a probability of exceedance of $10^{-8}$ per cycle, derived from long-term wave statistics for the North Atlantic as the standard reference sea state.

The hogging VWBM at midships is:

M_{WH} = +0.19 \, C_W \, L^2 \, B \, C_B \quad [\text{kN m}]

The sagging VWBM at midships is:

M_{WS} = -0.11 \, C_W \, L^2 \, B \, (C_B + 0.7) \quad [\text{kN m}]

where:

$L$ = ship length between perpendiculars (m), with $L$ taken between 90 m and 350 m
$B$ = ship moulded breadth (m)
$C_B$ = block coefficient at summer load line draught, not less than 0.6
$C_W$ = wave coefficient:

C_W = \begin{cases} 10.75 - \left(\dfrac{300 - L}{100}\right)^{1.5} & 90 \leq L \leq 300 \\ 10.75 & 300 < L \leq 350 \end{cases}

For a Capesize bulk carrier with $L = 280$ m, $B = 45$ m, $C_B = 0.83$ :

$C_W = 10.75 - (20/100)^{1.5} = 10.75 - 0.089 = 10.661$
$M_{WH} = +0.19 \times 10.661 \times 280^2 \times 45 \times 0.83 \approx +7,030{,}000$ kN m (approximately 7.0 GN m)
$M_{WS} = -0.11 \times 10.661 \times 280^2 \times 45 \times 1.53 \approx -7,170{,}000$ kN m (approximately 7.2 GN m)

The sagging VWBM is characteristically larger in absolute value than the hogging VWBM for full-form ships with $C_B > 0.75$ . This asymmetry arises because full-form ships lose more buoyancy amidships when the trough is amidships (the hogging wave case) than a slender ship would, but they also gain relatively more buoyancy at the bow and stern in the same wave, producing a larger net bending effect. The IACS wave bending moment calculator applies the UR S11 formulae directly.

The distribution of VWBM along the ship length follows a standard shape function $C_M(x/L)$ defined in UR S11, reaching 1.0 at midships and tapering to zero at the forward and aft perpendiculars.

Required section modulus: UR S11 and UR S7

UR S11 permissible-stress approach

IACS UR S11 requires that the hull girder section modulus at midships not be less than the value derived from the combined total design bending moment and the permissible bending stress. The permissible stress for ordinary mild steel (yield 235 MPa) is set at 175 MPa (a utilization factor of $175/235 = 0.745$ ). For higher-tensile steels the permissible stress is adjusted by $k^{0.5}$ where $k = 235/\sigma_y$ .

The required section modulus from UR S11 is:

Z_{\text{req}} \geq \frac{M_{SW} + M_W}{\sigma_{\text{perm}}}

where $M_{SW}$ is the maximum approved still-water moment and $M_W$ is the rule wave moment (hogging and sagging checked separately). The required section modulus calculator applies this check.

UR S7 absolute floor

IACS UR S7 (Revision 6, 2022) sets a minimum section modulus floor that must not be violated regardless of the loading condition analysis. For ships with $90 \leq L \leq 500$ m:

Z_{\min} = C_1 \, L^2 B (C_B + 0.7) \quad [\text{cm}^3]

where $C_1$ is a coefficient that depends on ship length (ranging from 0.070 for $L = 90$ m to 0.075 for $L \geq 300$ m). This floor prevents a designer from arguing that a lightly loaded condition justifies a structurally lighter midship section. In practice, for large bulk carriers and VLCCs, $Z_{\text{req}}$ from UR S11 governs over $Z_{\min}$ from UR S7.

CSR BC&OT gross vs net scantlings

The CSR BC&OT system works with gross scantlings (the as-built plate thickness including a corrosion margin) and net scantlings (the structural plate thickness after full corrosion wastage, used for the strength check). The corrosion margins per CSR range from 1.0 mm to 3.5 mm depending on structural member type and exposure: inner bottom of ballast tanks is at the higher end; shielded internal structure of holds is at the lower end. The section modulus $Z$ used in the strength check is computed from the net scantling cross-section.

Shear force and shear stress

Shear force distribution

The shear force $V(x)$ in the hull girder is the integral of the net load intensity $q(x)$ from the aft end to section $x$ . For a typical merchant ship with cargo concentrated amidships, the shear force distribution shows:

Zero at the aft perpendicular
A maximum (positive, cutting the midship section from below) near the quarter-length aft
Zero crossing near midships (where the bending moment is maximum)
A maximum (negative, cutting from above) near the quarter-length forward
Zero at the forward perpendicular

The maximum shear force is therefore not at midships but near the quarter-length stations. The quarter-length shear force calculator provides this check.

Shear flow and shear stress

Shear in a thin-walled open or closed section is distributed as shear flow $q_s$ . In the hull girder, the shear flow in the side shell is:

q_s(s) = -\frac{V}{I_N} \int_0^s y \, t(s) \, ds

where $s$ is the arc length along the shell, $t(s)$ is the shell thickness, and the integral is the first moment of area of the section between the shear reference point and $s$ . The maximum shear stress in the shell plating is:

\tau_{\max} = \frac{q_{s,\max}}{t_{\min}}

For typical merchant ships, the maximum shear stress occurs in the side shell at or near the neutral axis. CSR BC&OT Chapter 4 sets permissible shear stress limits; the side shell and longitudinal bulkhead scantlings in the 0.25L to 0.75L zone are driven by combined bending and shear. The shear flow calculator handles multi-cell closed sections.

Hull-girder ultimate strength: the Smith method

Why ultimate strength is a separate check

The yield-based section modulus check assumes that once the extreme-fibre stress reaches the permissible value, the section has reached its design capacity. But real hull girder behavior is more complex: as bending moment increases, yielded elements lose capacity, buckled panels carry reduced post-buckling load, and the neutral axis shifts. The section as a whole can carry a moment substantially higher than the first-yield moment before it collapses. Conversely, premature buckling of the compression flange (deck in sagging) can reduce the ultimate moment below the first-yield moment for sections with slender plate panels.

IACS CSR BC&OT Part 1 Chapter 5 Section 5 mandates the incremental-iterative method (commonly called the Smith method, after C.S. Smith’s 1977 paper in the Transactions of the Royal Institution of Naval Architects) for computing the hull-girder ultimate bending moment $M_U$ .

The incremental-iterative procedure

The cross-section is divided into structural units: each plate-stiffener combination (a stiffener and its associated plating) forms one unit. For each unit, a load-shortening curve (LSC) is derived that describes the average stress as a function of average compressive strain. The LSC for a stiffener-panel unit passes through three regimes: linear elastic, non-linear (onset of plate buckling), and post-buckling (residual capacity). The specific curves depend on plate slenderness $\beta = (b/t)\sqrt{\sigma_y/E}$ and column slenderness $\lambda = (a/\pi r)\sqrt{\sigma_y/E}$ , where $b$ is the plate breadth between stiffeners, $t$ is the plate thickness, $a$ is the stiffener spacing, and $r$ is the radius of gyration of the plate-stiffener unit.

CSR BC&OT defines a set of standard LSC shapes (labeled I through VI) that each structural unit is assigned to based on its slenderness parameters. The incremental procedure is:

Apply an initial curvature increment $\Delta\kappa$ to the cross-section.
Compute the strain at each structural unit from the assumed neutral-axis position.
Look up the stress at that strain from the unit’s LSC.
Integrate the stress distribution across the section to get the net axial force; adjust the neutral-axis position until the net force equals zero (equilibrium).
Integrate the stress distribution times the arm from the neutral axis to get the total bending moment $M(\kappa)$ .
Repeat, increasing $\kappa$ until $M(\kappa)$ reaches a maximum and then decreases. That maximum is $M_U$ .

The CSR BC&OT requires:

\frac{M_U}{\gamma_{R1} \cdot \gamma_{R2}} \geq M_{SW} + \gamma_W \cdot M_W

where $\gamma_{R1} = 1.1$ is the hull-girder reliability factor, $\gamma_{R2} = 1.1$ is the material and fabrication uncertainty factor, and $\gamma_W = 1.2$ is the load factor applied to the rule wave bending moment. For a combined safety factor on the wave-bending side, the effective requirement is that $M_U \geq 1.21 \cdot (M_{SW} + 1.2 \cdot M_W)$ in round numbers.

The first-yield moment $M_Y$ (the moment at which the extreme fibre first reaches the yield stress) provides a lower bound: $M_Y = Z \cdot \sigma_y$ . For well-proportioned sections with compact plating, $M_U / M_Y \approx 1.1$ to $1.2$ . For sections with very slender plate panels (typical of older single-hull tankers), $M_U$ can fall below $M_Y$ because of premature buckling.

Fatigue: S-N curves and Miner’s rule

The hull girder is not only loaded to a maximum in the worst storm. Over a 25-year service life, the hull girder at midships experiences perhaps $10^8$ stress cycles from wave-induced bending, with amplitudes that follow a long-tail distribution. The majority of cycles are at low amplitude; a small fraction, during severe weather, are at large amplitude. The cumulative damage from all these cycles is assessed by Miner’s rule:

D = \sum_i \frac{n_i}{N_i}

where $n_i$ is the number of cycles at stress range $\Delta\sigma_i$ and $N_i$ is the number of cycles to failure at that stress range from the relevant S-N curve. Failure is predicted when $D \geq 1$ .

IACS CSR BC&OT Chapter 9 (Fatigue Assessment) uses a set of W-curve S-N relationships (DNV-style curves) fitted to welded joint test data. The critical structural details for fatigue are:

Longitudinal stiffener terminations at transverse frames and bulkheads, where the stiffener end bracket introduces a stress concentration.
Hatch corner cutouts in deck plating, where the stress concentration factor can reach 3 to 4 for poorly detailed corners.
Bilge keel attachment welds, which see both hull-girder and roll-induced stresses.
Bottom longitudinal welds in the flat-bottom region, where the combination of hull-girder bending and local water pressure produces biaxial fatigue.

The Miner’s rule fatigue calculator applies the basic damage accumulation formula. For detailed FEA-based hot-spot stress analysis of critical connections, the structural analysis codes (SESAM/GeniE for DNV, POSEIDON for LR) post-process the stress histories into hot-spot ranges and compute fatigue life directly.

Bulk carriers are particularly susceptible to hold-frame fatigue. The frame web-plate connections at the lower stool of transverse frames in ore-carrying holds are highly stressed under alternate-hold loading; CSR BC-specific fatigue requirements address this explicitly.

Bulk carrier specifics: alternate-hold loading and CSR BC

The alternate-hold loading condition places dense dry bulk cargo (iron ore at 3,000 kg/m³, heavy grade coal at 1,200 kg/m³) in every second hold and leaves the alternating holds empty or in ballast. For a typical 200,000 DWT Capesize ore carrier with nine holds, cargo in holds 1, 3, 5, 7, 9 and empty in 2, 4, 6, 8 produces a sagging SWBM at midships of roughly 10 to 14 GN m, often exceeding the uniform full-load SWBM by a factor of 1.5 to 2.

IACS UR S11 Part B (Bulk Carriers) requires that the section modulus be checked against the alternate-load SWBM explicitly. The loading instrument must include the alternate loading condition in its library and prevent the master from commencing alternate loading unless the calculated SWBM is within the approved envelope.

CSR BC Chapter 4 also prescribes enhanced transverse strength requirements for bulk carrier holds: the floor plates, hopper plates, topside tank structure, and transverse frames are sized against the lateral pressure from water flooding (damage stability scenario) as well as the structural loads from alternate loading. This is the lesson of the Derbyshire (1980) and subsequent bulk carrier losses.

Oil tanker specifics: double-hull and CSR OT

Oil tankers above 600 DWT built after MARPOL Annex I Reg.19 cut-off dates are double-hull: the cargo tanks are enclosed between the inner and outer hull, with wing and double-bottom ballast tanks separating the cargo from the sea. The double-hull cross-section has two closed-cell boxes (port and starboard wing tanks flanking the cargo tank centerline) connected at the bottom, which makes the shear flow calculation more complex than a single-box section.

CSR OT Part 1 Chapter 3 provides the tanker-specific still-water bending moment permissible envelopes for the range of loading conditions: full load at departure and arrival (different trim and SWBM), ballast departure and arrival, partial load, and tank inspection conditions. The alternate sagging and hogging maxima differ between the departure and arrival conditions because fuel and fresh water consumption changes the longitudinal weight distribution.

Tanker double hulls are divided by longitudinal bulkheads (typically one or two, producing two or three longitudinal bays of cargo tanks). These longitudinal bulkheads carry shear as well as bending, and their sizing per CSR OT is driven by the combined hull-girder shear demand and the local hydrostatic pressure of the oil cargo.

Section summary: hogging vs. sagging comparison

Parameter	Maximum hogging	Maximum sagging
Physical cause	Wave crests at bow and stern; or ballast with empty midship tanks	Wave crest at midships; or alternate-hold loading
VWBM sign (UR S11)	Positive	Negative
Deck stress	Compressive	Tensile
Keel stress	Tensile	Compressive
Governing structural limit	Deck buckling; keel yield	Deck yield; bottom buckling
CSR Smith-method check	$M_U^{hog}$ vs rule demand	$M_U^{sag}$ vs rule demand (usually governs)
Typical governing condition	Ballast tanker; lightly loaded departure	Loaded bulk carrier alternate-hold; VLCC full load arrival

The loading instrument and SWBM compliance

Under SOLAS II-1 Reg.5-1 (2009 amendments), all ships above 65 m keel-laid from January 2011 and certain categories of existing ships above 100 m must carry an approved loading instrument that can compute, in real time, the SWBM and shear force distribution for any proposed loading condition and compare them to the Class-approved permissible envelope. The loading instrument must be approved by the vessel’s Class society; it is not a generic program but a vessel-specific tool that encodes the hull form, tank and hold geometry, and the permissible envelope from the approved stability and hull-girder drawings.

The approved envelopes separate the harbour condition (relaxed limits, typically +20% above the sea-going limit, because the vessel is under close observation and can discharge/load quickly if a limit is approached) from the sea-going condition. The master’s obligation is to ensure the sea-going SWBM envelope is satisfied before departure. The loading computer checks both automatically when the departure condition is confirmed.

For ships required to carry the loading instrument, the marine stability booklet and loading computer article covers the regulatory basis, the Class approval process, and the operational duties in detail.

Buckling under combined loading

Buckling is a stability failure mode, not a material failure mode. A plate panel in the deck of a sagging ship can buckle at an average stress well below the yield stress if the plate slenderness ratio is high. The critical buckling stress for a long rectangular plate in uniaxial compression is:

\sigma_{cr} = k_c \frac{\pi^2 E}{12(1-\nu^2)} \left(\frac{t}{b}\right)^2

where $k_c$ is the plate buckling coefficient (dependent on aspect ratio and boundary conditions, approximately 4.0 for a long plate between stiffeners), $E = 206$ GPa for steel, $\nu = 0.3$ , $t$ is the plate thickness, and $b$ is the unsupported plate width between stiffeners.

For a deck panel with $b = 800$ mm and $t = 16$ mm, $\sigma_{cr} \approx 4 \times \pi^2 \times 206{,}000 / (12 \times 0.91) \times (16/800)^2 \approx 221$ MPa, which is close to the yield stress of mild steel. Increasing the stiffener spacing to 1,000 mm drops $\sigma_{cr}$ to $142$ MPa, well below yield, and the deck will buckle before yielding. CSR BC&OT Chapter 6 (Buckling Assessment) requires that the buckling utilization factor of every plate panel not exceed 0.8 under the combined hull-girder and local loading. The plate buckling calculator handles the CSR-aligned check.

Corrosion margin, survey, and residual strength

The CSR corrosion addition for structural members ranges from 0.5 mm to 3.5 mm depending on member type, exposure, and tanker or bulk carrier application. The corrosion margin is the difference between the as-built (gross) scantling and the structural minimum (net) scantling. As the ship ages, actual corrosion consumes this margin. Class periodic surveys (five-year Special Survey, annual survey, two-and-a-half-year Intermediate Survey) require ultrasonic thickness gauging of plating at representative points. If gauging shows plating below the net scantling minimum, the section modulus of the corroded cross-section may fall below the UR S11 requirement, triggering a mandatory Class condition requiring steel renewal before the vessel returns to sea.

For ships approaching the end of their design life (20 to 25 years), residual strength analysis using the as-measured corroded cross-sections is sometimes carried out to support an application for extended trading. The analysis uses the same Smith method as the newbuild ultimate strength check, but with the actual measured plate thicknesses rather than the net scantling values.

Notable incidents and their regulatory consequences

Liberty ships (1942 to 1946)

Approximately 200 of the 2,710 Liberty ships built showed brittle fracture failures; 12 broke in half. The investigation identified notch brittleness in the low-temperature hull steel, propagation from weld defects, and stress concentrations at square hatch corners. The response introduced Charpy impact testing for structural steel, the concept of transition temperature, and the move from square to rounded hatch corners. These findings entered American Bureau of Shipping and Lloyd’s Register material specifications by 1948 and underpin the modern CSR W-series steel requirements.

Derbyshire (1980)

The Derbyshire, a 169,044 DWT ore-bulk-oil carrier, was lost in Typhoon Orchid south of Japan in September 1980 with all 44 crew. The 2000 re-investigation by the Marine Accident Investigation Branch and the re-analysis by Dr. R.W. Biles established that progressive flooding through inadequate forward ventilator covers and hatch covers preceded structural failure. The findings drove SOLAS Chapter XII (Additional Safety Measures for Bulk Carriers, 2002) and IACS UR S21 (Hatch Cover Strength), both of which substantially increased the required wave-pressure design loads for bulk carrier hatch covers.

Erika (1999) and Prestige (2002)

The Erika was a 12-year-old single-hull Maltese-flagged tanker that broke in two off Brittany in December 1999, spilling 19,800 tonnes of heavy fuel oil. The structural failure was attributed to corrosion-induced reduction of hull girder section modulus below the rule minimum, compounding a design that was already at the lower bound of the 1970s Class rules. The Prestige, a 26-year-old Bahamian-flagged single-hull tanker, suffered structural failure off Galicia in November 2002, releasing 63,000 tonnes of heavy fuel oil. Both incidents accelerated the MARPOL Annex I single-hull phase-out schedule (IMO Reg.20 amendment in 2003) and triggered the EU Erika I-II-III legislative packages, which extended port-state-control powers and banned single-hull tankers from EU ports for heavy-grade oil.

MOL Comfort (2013)

The MOL Comfort was an 8,110 TEU container ship that broke in two in the Indian Ocean in June 2013 in sea conditions that were within its certified operating envelope. The subsequent investigation by an expert committee, published in September 2014, attributed the failure to insufficient hull girder ultimate strength relative to the wave bending moment, combined with a fatigue crack propagation that may have reduced the effective section modulus ahead of the ultimate-strength event. The incident produced IACS Recommendation No. 142 (2015), which requires enhanced hull girder ultimate strength checks for large container ships. No equivalent to the CSR for container ships yet exists, but the IMO MSC and IACS are working toward harmonized container ship structural rules.

Operational control of the bending-moment envelope

Rule scantlings set the ship’s structural capacity, but the master controls the load applied to it on every voyage. The approved loading instrument, type-approved by Class under IACS UR L5, computes the still-water bending moment and shear force for the planned condition at each frame and displays them as a percentage of the permissible harbour and seagoing limits. The harbour envelope is wider than the seagoing envelope because the still-water moment in port need not be added to a wave moment; the seagoing limit is the value that, combined with the rule VWBM, keeps the deck and keel stress within the allowable.

On bulk carriers the controlling case is often alternate-hold loading, where loading heavy ore in alternate holds raises the still-water moment far above a homogeneous condition, and the loading manual specifies which multi-port and alternate sequences are permitted. Exceeding the envelope risks deck buckling in hogging or bottom buckling in sagging long before the hull-girder ultimate capacity is reached. The instrument therefore acts as the daily enforcement of the section-modulus margin that the rules establish at the design stage, and Class verifies it at the annual survey with a set of test loading conditions.

Limitations

IACS UR S11 VWBM values are calibrated for unrestricted ocean service (North Atlantic reference). Ships operating continuously in sheltered waters (restricted area Class notation, inland waterways notation) may apply reduced wave moments approved by Class, but SWBM limits remain unchanged because calm-water loading is independent of sea area.

The Smith method for ultimate strength assumes that structural units behave in accordance with the tabulated LSC shapes in CSR BC&OT. For unusual cross-sections (ships with very large hatches spanning most of the breadth, ships with composite or hybrid materials) the standard curves may not apply; finite element method non-linear analysis is required instead.

CSR BC&OT applies to ships with $L \geq 90$ m (bulk carriers) and $L \geq 150$ m (tankers). Smaller vessels fall under each Class society’s own rules, which may not match the CSR methodology; the SWBM and VWBM formulas from UR S11 extend down to $L = 90$ m but UR S7 applies from $L = 90$ m to $L = 500$ m.

Fatigue assessment per CSR Chapter 9 uses a reference design life of 25 years. Ships operated beyond 25 years without a dedicated residual-life fatigue reassessment carry structural risk that is not covered by the original design basis.

The loading instrument enforces the SWBM envelope for planned conditions. Rapid weather-driven changes in the wave bending moment at sea (for example, pitching heavily in a confused sea after course changes) are not monitored in real time; the master’s judgment and Class-recommended avoidance of extreme sea states remain the only safeguard for dynamic overloading.

Frequently asked questions

What is the difference between the still-water bending moment and the wave-induced bending moment?

The still-water bending moment (SWBM) arises from the static mismatch between the longitudinal weight distribution and the buoyancy distribution in calm water; it is load-dependent and must be controlled by cargo planning within the Class-approved envelope. The wave-induced vertical bending moment (VWBM) is the additional dynamic moment from wave-induced buoyancy variation as the hull pitches and heaves; IACS UR S11 prescribes the rule design values of VWBM as a function of ship length, beam, block coefficient, and a wave coefficient that accounts for the wave climate on unrestricted ocean service. The design total vertical bending moment is the algebraic sum of the two.

How does IACS UR S11 define the minimum required midship section modulus?

IACS UR S11 (Revision 7, 2022 consolidation) requires that the hull girder section modulus at the midship region not be less than the value derived from setting the design total bending moment equal to the permissible stress, where the permissible stress is 175 MPa for mild steel (yield 235 MPa). The formula is Z_req = M_T / sigma_perm, where M_T is the sum of the SWBM envelope limit and the rule VWBM. UR S7 (Minimum Longitudinal Strength Standards) sets an absolute floor on section modulus that is independent of loading condition.

What is the Smith method for hull girder ultimate strength?

The incremental-iterative method attributed to C.S. Smith (1977) progressively increases the applied curvature to the hull cross-section and at each increment derives the load-shortening response of every plate-stiffener unit using tabulated or computed column-buckling and post-buckling curves. The bending moment carried by the section is integrated from the stress distribution until the moment reaches a maximum and then falls; that maximum is the ultimate bending moment M_U. IACS CSR BC&OT Part 1 Chapter 5 Section 5 mandates this method for the hull-girder ultimate strength check of new bulk carriers and oil tankers.

What SWBM limits must the loading instrument enforce?

Class approves a still-water bending moment envelope that defines permissible SWBM hogging and sagging limits at every frame station along the ship length. The onboard loading instrument (per SOLAS II-1 Reg.5-1 for ships above 65 m and IACS UR S1 for Class compliance) computes the SWBM distribution for any proposed loading condition and compares it to the approved envelope. The instrument must flag any condition where the calculated SWBM exceeds the limit and must not permit commencement of the voyage in a non-compliant condition.

Why does alternate-hold loading govern bulk carrier design more than full-load?

Alternate-hold loading places dense cargo (iron ore, heavy coal) in every other hold and leaves the intermediate holds empty. The resulting weight distribution creates a dramatic sagging bending moment at midships because the loaded holds weigh far more per frame bay than the empty ones directly between them. IACS UR S11 Part B and CSR BC Part 1 Chapter 4 treat alternate hold loading explicitly as a design loading condition that often produces the maximum SWBM. The required section modulus for many Capesize bulk carriers is set by the alternate-load case rather than by the full-loaded departure condition.