Published 25 April 2026 · last updated 28 April 2026

Hull Strength and Longitudinal Bending

Hull strength is the structural capability of a ship’s hull girder to resist the longitudinal bending moment, shear force, transverse loads and torsional loads experienced in still water and in waves. The longitudinal bending moment is the principal design load, comprising a still-water component ($M_{SW}$, from the longitudinal weight and buoyancy distribution along the length) and a wave-induced component ($M_W$, from the dynamic wave-induced buoyancy variation). The total bending moment is conventionally calculated as $M_T = M_{SW} + M_W$ for design check, with both hogging (sagging-down at the ends and bowed-up at midships, typical when the bow and stern are over wave crests) and sagging (sagging-down at midships and bowed-up at the ends, typical when midships is over a wave trough) considered. Hull strength design is governed by the IACS Common Structural Rules (CSR) for bulk carriers and oil tankers (in force from 2006, with significant updates including the harmonised CSR-H of 2015 and subsequent amendments), supplementary classification society rules for other vessel types, and the underlying SOLAS Chapter II-1 strength requirements. The principal strength check criteria are: yield (the section modulus must be sufficient to keep the maximum stress below the steel yield limit); buckling (compression-side plating and stiffeners must not buckle under the design loads); fatigue (the cumulative stress cycles over the design life must not exceed the fatigue endurance); ultimate strength (the hull girder must not collapse plastically before reaching the design ultimate load). Structural failures (notably the MOL Comfort mid-ship failure in June 2013, the Erika tanker failure in December 1999, the Prestige tanker failure in November 2002, the Derbyshire bulk carrier loss in September 1980, the MSC Napoli loss in January 2007 and many others) have driven the progressive strengthening of the IACS CSR and the related SOLAS provisions over four decades. Modern bulk carriers and tankers built to CSR-H 2015 have substantially better strength margins than the previous generation, although operational factors (cargo loading distribution, ship motions, hull fouling, corrosion, fatigue) continue to drive incident risk. ShipCalculators.com hosts the principal computational tools: the longitudinal bending moment calculator, the section modulus calculator, the hogging-sagging calculator, the shear force calculator, the yield stress check calculator, the buckling check calculator, the ultimate strength calculator and the IACS CSR compliance calculator. A full listing is available in the calculator catalogue.

Contents

Background

Why hull strength matters

The hull girder of a ship is the primary structural element that holds the vessel together longitudinally. The hull girder is subjected to:

Static loads: self-weight (lightship), cargo weight, ballast, fuel, stores, distributed along the length.
Buoyancy: the upward distributed force from displaced water.
Wave-induced loads: the dynamic wave-induced variation in buoyancy as the vessel moves through waves.
Torsional loads: moments about the longitudinal axis (significant for vessels with large open hatches like container ships).
Side loads: from waves and from internal cargo pressure.

The hull girder must resist all these loads simultaneously without yielding, buckling, fracturing, or progressively failing through fatigue. Failure of the hull girder is typically catastrophic: the vessel breaks in two, with usually total loss.

Hogging and sagging

The fundamental bending modes are:

Hogging: the bow and stern are supported by wave crests while the midships is over a wave trough. The hull bends with the bow and stern downward and the midships upward. The deck is in tension; the bottom is in compression.
Sagging: the midships is supported by a wave crest while the bow and stern are over wave troughs. The hull bends with the midships downward and the bow and stern upward. The deck is in compression; the bottom is in tension.

Both modes occur during normal voyage operation; the design must accommodate both.

The maximum hogging or sagging moment depends on:

Wave length relative to ship length: maximum hogging when wave length ≈ ship length (wave crests at bow and stern); maximum sagging when wave length ≈ ship length and midships is over crest.
Wave height: linear relationship for small waves; non-linear for large waves.
Speed and heading relative to waves: head and following seas produce maximum bending; beam seas produce minimal bending.

History of hull strength incidents

Major hull strength incidents include:

Liberty ships, World War II era: approximately 4,700 Liberty ships built; approximately 200 had brittle fracture failures, including 12 that broke in half. Investigations revealed the importance of low-temperature fracture toughness and the design implications of welded construction.
Stockholm-Andrea Doria collision, 1956: the Andrea Doria capsize highlighted the importance of damage stability and the related hull strength considerations.
Hyundai 19, 1972: tanker break in two off South Africa.
Kowloon Bridge, 1986: bulk carrier break in two off Ireland.
Derbyshire, 1980: bulk carrier loss in Typhoon Orchid; investigation revealed inadequate hatch cover strength and progressive flooding from water on deck.
Erika, 1999: 12-year-old tanker structural failure off France; led to MARPOL Annex I tightening.
Prestige, 2002: 26-year-old tanker structural failure off Spain; further accelerated MARPOL Annex I tightening and single-hull tanker phase-out.
MSC Napoli, 2007: container ship structural failure off Cornwall, UK; led to enhanced container ship hull strength requirements.
MOL Comfort, 2013: 8,100 TEU container ship broke in two in monsoon weather in the Indian Ocean; drove enhanced container ship hull strength rules.

Each incident triggered IMO and IACS rule changes intended to prevent recurrence.

Loads and load combinations

Still water bending moment (SWBM)

The still water bending moment ($M_{SW}$) is the bending moment in calm water, driven by the longitudinal distribution of weight and buoyancy:

Weight distribution: lightship (steel weight), cargo, ballast, fuel, fresh water, stores, accommodations.
Buoyancy distribution: from the underwater hull volume distribution along the length, calculated from the Bonjean curves and the actual draught.

The longitudinal load at each section is $w(x) - b(x)$, the difference between weight and buoyancy. The shear force at each section is the integral from the bow:

$$ V(x) = \int_0^x [w(\xi) - b(\xi)] d\xi $$

The bending moment at each section is the integral of the shear force:

$$ M(x) = \int_0^x V(\xi) d\xi $$

The maximum SWBM typically occurs at midships for typical merchant ship loading distributions.

Wave-induced bending moment (VWBM)

The wave-induced bending moment is the additional bending moment from wave-induced buoyancy variation. The IACS CSR provides standardised wave bending moment values for design purposes:

Hogging wave moment ($M_{WH}$): typically calculated using a standard wave (proportional to ship length and load coefficient).
Sagging wave moment ($M_{WS}$): typically larger than hogging for typical hull forms.

The IACS CSR formulae are:

$$ M_{WH} = 0.19 C_W L^2 B C_B $$$$ M_{WS} = -0.11 C_W L^2 B (C_B + 0.7) $$

where $C_W$ is the wave coefficient (depends on length, latitude, area).

Combined design moment

The design bending moment is the algebraic sum:

$$ M_T = M_{SW} + M_W $$

Both hogging and sagging combinations are considered. The maximum positive (hogging) or negative (sagging) combined moment determines the design requirement.

Operational SWBM limits

The trim and stability booklet carried by every vessel includes maximum SWBM limits for each section along the length. The loading computer enforces these limits during cargo and ballast operations. Exceeding the SWBM limit at any point is non-compliant and must be corrected before voyage.

The Class society approves the SWBM envelope based on the design wave bending moment plus a safety margin.

Section modulus and strength check

Section modulus

The hull girder’s bending strength is characterised by the section modulus ($Z$):

$$ \sigma = \frac{M}{Z} $$

where $\sigma$ is the bending stress and $M$ is the bending moment. The section modulus is calculated from the hull cross-section’s geometry:

$$ Z = \frac{I}{c} $$

where $I$ is the second moment of area of the hull cross-section about the neutral axis (the location where bending stress is zero) and $c$ is the distance from the neutral axis to the deck (for deck section modulus) or to the bottom (for bottom section modulus).

For typical merchant ships:

The neutral axis is approximately at 40 to 50% of the depth from the keel (for ships with similar deck and bottom plating thickness).
The deck section modulus is typically 0.7 to 0.9 of the bottom section modulus (because the deck is further from the neutral axis but typically thinner than the bottom).

Yield strength check

The maximum bending stress must be less than the steel yield strength divided by a safety factor:

$$ \sigma_{max} = \frac{M_T}{Z} < \frac{\sigma_y}{SF_{yield}} $$

For typical mild steel ($\sigma_y = 235$ MPa) with $SF_{yield} = 1.5$, the maximum allowable stress is approximately 156 MPa.

For high-tensile steel (HT32, HT36, HT40) the yield is correspondingly higher (315, 355, 390 MPa) allowing higher bending stress.

Buckling strength check

Compression-side plating and stiffeners must not buckle under the design loads. Buckling is a stability problem (not a strength problem); even if the steel has not yielded, plating may buckle plastically under combined compression and bending.

Buckling design is per the IACS CSR Chapter 8 (Buckling and Ultimate Strength). The principal checks:

Plate buckling: rectangular plates between stiffeners checked against critical buckling stress.
Stiffener buckling: longitudinal stiffeners checked against column buckling.
Stiffened panel buckling: combined plate-stiffener buckling.
Local buckling: web buckling, flange buckling.

Ultimate strength check

The ultimate strength is the maximum bending moment the hull girder can carry before plastic collapse. The ultimate strength must exceed the design moment by an additional safety factor:

$$ M_U > SF_{ultimate} \cdot M_T $$

with $SF_{ultimate}$ typically 1.10 to 1.20.

The ultimate strength calculation is non-linear, accounting for plastic redistribution of stress and progressive buckling. The IACS CSR Chapter 9 (Ultimate Strength) defines the calculation methodology.

Fatigue check

The fatigue strength must accommodate the cumulative stress cycles over the design life (typically 25 years). The fatigue check uses S-N curves (stress-cycle relationships) and Miner’s rule for cumulative damage.

Fatigue is particularly critical for:

Welded joints: stress concentrations at welds reduce the local fatigue endurance.
Cutouts and openings: stress concentrations around holes (cargo hold openings, watertight door openings, ventilation cutouts).
High-stress areas: midship section bottom and deck, especially near hatch corners on container ships.

The IACS CSR Chapter 10 (Fatigue Assessment) defines the fatigue check methodology.

IACS Common Structural Rules

History and scope

The IACS Common Structural Rules (CSR) were developed jointly by the eight IACS member classification societies (DNV, Lloyd’s Register, ABS, BV, NK, KR, RINA, CCS) to harmonise the strength requirements for bulk carriers and oil tankers. The CSR provides:

Common loads (still-water and wave-induced).
Common strength criteria (yield, buckling, ultimate, fatigue).
Common design methodology.
Common acceptance criteria.

The CSR replaced the previously separate rules of each Class society, eliminating “rule shopping” by shipyards seeking the lightest design.

CSR for bulk carriers (CSR-BC) and CSR for oil tankers (CSR-OT) were issued separately in 2006. The harmonised CSR-H combining both was issued in 2015 with subsequent updates (2017, 2019, 2021, 2024).

Application

The CSR applies to:

All new bulk carriers above 90 m length (since 2006).
All new oil tankers above 150 m length (since 2006).

The CSR does not directly apply to:

Container ships: covered by the Class society’s specific rules; harmonisation is being explored but not yet adopted.
LNG carriers, chemical tankers, ferries, cruise ships, naval vessels, fishing vessels: covered by Class-specific rules.

The CSR has been very influential on the rules for non-CSR ship types, with many of the principles and criteria being adopted into the Class-specific rules.

Principal CSR provisions

The CSR is structured as multiple Parts:

Part 1 (Hull Structure): hull girder strength, plate and stiffener strength, buckling, ultimate strength, fatigue.
Part 2 (Wear, Wastage and Corrosion): corrosion margins on plating thickness, periodic Class survey requirements.
Part 3 (Materials): steel grades, welded joint qualification.
Part 4 (Design Procedures): design methodology, finite element analysis requirements.
Part 5 (Survey and Maintenance): in-service survey requirements.

The CSR is a substantial document (approximately 1,500 pages); design and review against the CSR is a major engineering activity in ship design.

URS rules

In addition to the CSR, the IACS publishes Unified Requirements (URs) that are harmonised across all member societies:

UR S11: Longitudinal strength of hull girder.
UR S18: Cargo handling deck strength on container ships.
UR S19: Bulk carrier hold strength.
UR S21: Hatch cover strength on bulk carriers (post-Derbyshire).
UR S26: Strength of structural elements in cargo hold area.

The URS rules apply to all Class-classed ships, regardless of CSR or Class-specific rule application.

Design methodology

Direct calculation

For modern ship design, the hull strength check involves:

Define the loads: still-water and wave-induced bending moments per CSR or Class rules.
Calculate the hull girder section modulus: from the design cross-section.
Check yield, buckling, fatigue, ultimate strength: per CSR or Class rules.
Iterate: if any check fails, modify the cross-section (additional plate thickness, additional stiffeners, modified stiffener spacing) and repeat.

The calculation is typically performed using specialised marine structural analysis software:

Mars 2000 (Bureau Veritas).
NAUTICUS Hull (DNV).
POSEIDON ND (Lloyd’s Register).
SafeHull (ABS).
NK PrimeShip Hull (NK).

These tools embed the CSR methodology and produce direct compliance reports.

Finite element analysis (FEA)

For detailed analysis of high-stress areas (hatch corners, bracket details, cargo hold corner, stiffener intersections), finite element analysis (FEA) is the modern standard. FEA captures local stress concentrations that simplified beam-theory calculations cannot represent.

Modern FEA models for large ships (post-Panamax container ships, VLCCs) typically include 100,000 to 1 million elements covering the full hull. The analysis is computationally intensive but provides the highest resolution stress data.

Direct calculation vs partial safety factor methods

The CSR uses a partial safety factor (PSF) approach: each load and each strength contribution is multiplied by a partial safety factor. This contrasts with the older deterministic safety factor approach (a single global factor). The PSF approach allows more rational treatment of uncertainty in different load and strength components.

The PSF approach is now standard in modern marine structural design and aligns with similar developments in civil engineering (Eurocodes) and aerospace.

In-service strength considerations

Corrosion

Steel hulls corrode over their service life. The CSR corrosion margin specifies additional plating thickness above the structural minimum to account for corrosion through the design life:

Cargo hold spaces (bulk carriers): typically 2 to 6 mm corrosion margin.
Cargo tanks (oil tankers): typically 3 to 7 mm corrosion margin.
Ballast tanks: typically 2 to 6 mm corrosion margin.

The corrosion margin is consumed over the design life; periodic surveys check the actual remaining thickness against the structural minimum.

Periodic surveys

Class periodic surveys (typically every 5 years for the Special Survey, with annual and intermediate surveys in between) check:

Plating thickness at representative points.
Stiffener integrity.
Coating condition.
Cargo hold inspection.
Ballast tank inspection.

If thickness measurements show plating below the structural minimum (the CSR-specified value), repairs are required (steel renewal) before the vessel can continue in class.

Fatigue cracking

Welded joints in high-stress areas can develop fatigue cracks over time. Common locations:

Hatch corners (especially container ships).
Cargo hold corner brackets.
Frame-to-deck connections in the forward bottom area.
Stiffener-to-bracket connections.

Fatigue cracks are inspected during periodic surveys and repaired by re-welding or by adding doubler plates.

Ice strengthening

Vessels operating in ice-covered waters (Polar regions, Baltic, St. Lawrence, etc.) require additional ice strengthening:

Increased plating thickness in the ice belt (the area around the waterline subject to ice impact).
Additional frame spacing.
Stronger stiffeners.

The IACS Polar Class system (PC1 through PC7) defines five levels of ice strengthening, plus the Finnish-Swedish Ice Class system (1A Super, 1A, 1B, 1C) widely used in the Baltic.

Notable structural failures and lessons

Liberty ship brittle fracture (1942-1946)

The Liberty ship brittle fracture experiences led to fundamental changes in steel selection for marine application:

Steel grade specification by impact (Charpy) test temperatures.
Notch toughness requirements for welded structures.
Better understanding of weld stress concentrations.

These changes are reflected in modern Class society materials specifications.

Derbyshire (1980)

The Derbyshire investigation revealed inadequate hatch cover strength under wave-induced pressure. The findings led to:

SOLAS Chapter XII (2002): additional safety measures for bulk carriers.
IACS UR S21: hatch cover strength requirements.
Higher freeboard for bulk carriers.

Erika (1999) and Prestige (2002)

The Erika and Prestige tanker structural failures led to:

Accelerated single-hull tanker phase-out (MARPOL Annex I 13G amendment).
Tightened double hull requirements for new tankers.
EU Erika I, II, III legislation packages: stricter port-state-control, ban on single-hull oil-cargo movement in EU ports.

MOL Comfort (2013)

The MOL Comfort container ship structural failure led to:

IACS Recommendation No. 142 on container ship hull girder strength.
Class-specific enhanced rules for large container ships.
CSR-Container Ship discussion (still in development as of 2024).

MSC Napoli (2007)

The MSC Napoli structural failure led to:

Enhanced fatigue assessment requirements for container ships.
Improved hatch coaming and corner detail rules.

Implications for design and operations

Design

For newbuild design, hull strength considerations affect:

Steel weight: stronger structures use more steel, increasing capital cost and reducing deadweight.
Steel grade selection: high-tensile steel allows lighter structures but more difficult fabrication and higher fatigue cracking risk.
Cross-section optimisation: balance of plate thickness, stiffener spacing, and stiffener size.

Modern designs typically use:

Medium-tensile steel (HT32, HT36) for the bottom and deck areas.
Mild steel (Grade A, B) for less-stressed areas.
High-tensile steel (HT40) for very-high-stress areas.

Operations

For in-service operations, hull strength considerations include:

Cargo loading distribution: must satisfy SWBM limits at all points in the loading sequence.
Maintenance: regular hull inspection and steel renewal at periodic surveys.
Voyage routing: avoiding extreme weather that could exceed design wave loads.
Trim and stability booklet compliance: continuous compliance during all operations.

References

IACS. Common Structural Rules for Bulk Carriers and Oil Tankers (CSR BC and OT, harmonised CSR-H). International Association of Classification Societies, 2006 / 2015 / 2024 editions.
IACS. UR S11: Longitudinal strength of hull girder. International Association of Classification Societies, 2024 edition.
IACS. UR S21: Strength of hatch cover and hatch coamings. International Association of Classification Societies, 2024 edition.
IACS. Recommendation No. 142: Container ships, hull girder strength assessment. International Association of Classification Societies, 2014.
SOLAS Chapter II-1: International Convention for the Safety of Life at Sea, 1974, as amended. International Maritime Organization, 1974 with subsequent amendments.
SOLAS Chapter XII: Additional safety measures for bulk carriers. International Maritime Organization, 2002 with subsequent amendments.
MARPOL Annex I: Regulations for the Prevention of Pollution by Oil. International Maritime Organization, 1973/1978 with subsequent amendments.
DNV. DNV Rules for Classification of Ships, Pt 3 Hull. DNV, 2024 edition.
Lloyd’s Register. Rules and Regulations for the Classification of Ships, Part 3 Ship Structures. Lloyd’s Register Group, 2024 edition.
ABS. Rules for Building and Classing Steel Vessels. American Bureau of Shipping, 2024 edition.
Hughes, O. F. and Paik, J. K. Ship Structural Analysis and Design. SNAME, 2010.