Marine Structures and Hull Integrity Formula Sheet

This formula sheet collects first-pass calculations used in marine structures and hull integrity review. It connects loading condition, hull girder strength, section modulus, shear flow, local pressure, stiffened panels, corrosion wastage, welded fatigue, crack-growth inspection intervals, vibration separation, and release evidence.

Use these relationships for screening, hand checks, survey interpretation, finite-element model review, repair assessment, and engineering discussion. They do not replace classification rules, flag requirements, approved loading manuals, structural repair manuals, material allowables, fatigue design codes, direct strength analysis, survey procedures, or vessel-specific class approval.

Basis and Conventions

State the basis before calculating:

1. vessel type, service area, loading condition, draft, trim, displacement, ballast state, fuel state, cargo distribution, and relevant restrictions;
2. whether the load case is still-water, wave, slamming, green water, tank pressure, docking, grounding, collision, lifting, towing, mooring, machinery, fatigue, corrosion, repair, or accidental;
3. whether the check is global hull girder strength, local strength, stiffness, buckling, fatigue, fracture, watertight integrity, vibration, or release after repair;
4. whether stresses are nominal, structural hot-spot, local notch, principal, equivalent, or finite-element element stresses;
5. which evidence supports the calculation: loading software, hydrostatic data, scantling model, finite-element model, thickness survey, NDT, strain measurement, sea trial, vibration survey, or service history.

The common mistake is to mix bases. A hull girder stress margin at one loading condition does not prove local pressure strength, fatigue life at a welded detail, corrosion reserve in a ballast tank, or watertight integrity after a repair.

Symbols

Symbol	Meaning	Typical unit
\Delta	displacement mass	kg or t
\nabla	displaced volume	\text{m}^3
x	longitudinal coordinate	m
w(x)	weight per unit length	N/m
b(x)	buoyancy per unit length	N/m
V(x)	shear force	N
M(x)	bending moment	N m
I	second moment of area	\text{m}^4
Z	section modulus	\text{m}^3
y	distance from neutral axis	m
\sigma	normal stress	Pa
\tau	shear stress	Pa
Q	first moment of area about the neutral axis	\text{m}^3
q	shear flow	N/m
p	pressure	Pa
A	loaded area	\text{m}^2
t	plate or shell thickness	m
b_p	unsupported plate breadth	m
E	elastic modulus	Pa
G	shear modulus	Pa
\nu	Poisson’s ratio	dimensionless
D	fatigue damage sum	dimensionless
K	stress intensity factor	MPa sqrt(m) or Pa sqrt(m)
a	crack size	m

Weight, Buoyancy, Shear, and Bending

Longitudinal net load:

Shear force from net load:

Bending moment from shear:

The constants are set by end conditions and equilibrium. In a complete hull girder calculation, total weight and total buoyancy should balance for the selected loading condition, and the resulting shear and bending diagrams should close consistently.

Still-water bending can be combined with wave bending conceptually as:

where M_d is design bending moment, M_{SW} is still-water bending, M_W is wave bending, M_T is thermal or locked-in contribution if relevant, M_A is an accidental or special load contribution, and \gamma values are the applicable factors from the selected design basis. Do not invent factors when class rules or project specifications define them.

Hull Girder Bending Stress

Elastic hull girder stress:

Section modulus:

Maximum elastic bending stress:

Required section modulus:

Reserve factor:

where c is the distance from neutral axis to the checked deck or bottom extreme fiber.

Worked Example: Hull Girder Section Modulus

A coastal vessel loading condition gives a design vertical bending moment of:

The effective section modulus at the checked section is:

The elastic stress is:

If the allowable stress for the stated basis is:

then:

and:

The section passes this simplified elastic hull girder screen with about 24 percent reserve. The result is not a complete structural approval: local buckling, shear, fatigue, corrosion deductions, openings, residual stress, class-rule details, and loading-manual restrictions still have to be checked.

Shear Flow and Web Shear

Beam shear formula:

Shear flow:

Stress from shear flow through a plate or web:

For multiple parallel webs or side-shell load paths, the split of shear flow must follow the structural model. A simple equal split is only a screening assumption.

Worked Example: Shear Flow Split Between Two Webs

A hull section has vertical shear:

At the checked web connection, the section property term is estimated as:

The total shear flow is:

If two similar longitudinal webs share the load, each web carries approximately:

For a web thickness of:

the average web shear stress is:

This is a useful hand check against a finite-element model. It does not prove the detail because cutouts, brackets, weld toes, shear lag, local panel buckling, and load introduction can raise local stresses.

Hydrostatic, Tank, and Local Pressure

Hydrostatic gauge pressure:

Absolute pressure:

Resultant force on an area under nearly uniform pressure:

Pressure with a simplified dynamic or rule factor:

where C_d is a dynamic, sloshing, impact, or rule coefficient when the design basis provides one, and \gamma_p is the selected load factor.

Hydrostatic pressure is only one local load source. Slamming, green water, tank sloshing, cargo pressure, berthing, mooring fittings, crane foundations, sea chests, thruster tunnels, and machinery foundations can control local structure.

Worked Example: Bottom Panel Pressure

A bottom panel is checked at a seawater head of:

Using:

the gauge pressure is:

For a tributary area:

the static force is:

If the project basis applies C_d=1.4 and \gamma_p=1.2:

and:

The design force is almost 1.7 times the static force. That difference should be visible in the calculation record; otherwise a local panel can appear adequate under calm-water pressure while being weak under the actual rule or impact basis.

Plate Buckling Screen

Elastic critical compressive stress for an ideal flat plate:

where k depends on edge restraint, aspect ratio, and loading mode. This formula is a screen, not a replacement for stiffened-panel rules or nonlinear analysis.

Buckling reserve factor:

Thickness sensitivity:

Small corrosion losses can therefore have a large effect on plate buckling capacity.

Worked Example: Corroded Plate Buckling Sensitivity

A steel plate panel in compression has:

For an as-assessed thickness:

the elastic buckling stress is:

If the compressive demand is:

then:

If wastage reduces the effective thickness to 6 mm:

and:

The panel still passes the simplified screen, but the reserve has fallen sharply. This is why thickness survey data, corrosion allowance, coating condition, and inspection interval matter to buckling, not only to yielding.

Stiffener Column Buckling

Euler elastic buckling load:

where K is effective length factor and L is unsupported length.

Buckling utilization:

Real stiffeners may fail by local plate buckling, torsional buckling, tripping, lateral-torsional deformation, weld distortion, or interaction with attached plating. Euler buckling is only an ideal reference.

Worked Example: Stiffener Compression Screen

A longitudinal stiffener is idealized with:

The ideal buckling load is:

If the compressive demand from the checked load case is:

then:

The result is encouraging but incomplete. The real stiffener check should also consider attached plating, eccentricity, end bracket stiffness, cutouts, residual stress, corrosion, tripping, and whether the finite-element model captures the effective length correctly.

Stress Concentration and Hot-Spot Stress

Nominal-to-local stress relation:

Fatigue notch relation:

where K_t is theoretical stress concentration factor, K_f is fatigue stress concentration factor, and q_n is notch sensitivity.

For welded marine details, fatigue assessment often uses structural hot-spot stress or detail-category S-N curves rather than a simple notch factor. The selected stress definition must match the fatigue data.

Hot-spot extrapolation is often expressed conceptually as:

where \sigma_1 and \sigma_2 are stresses extracted at specified distances from the weld toe, and a and b follow the chosen method. Do not mix hot-spot stress with nominal S-N data unless the method permits it.

Corrosion Wastage and Remaining Thickness

Remaining thickness from a uniform corrosion-rate assumption:

Required initial thickness with corrosion allowance:

Guarded remaining thickness:

where t_0 is initial thickness, r_c is corrosion rate, t_s is service time, t_{req} is required remaining thickness, u_t is thickness measurement uncertainty, and k is the selected coverage factor.

Worked Example: Survey Thickness Release

A ballast-tank plate has nominal thickness:

The estimated corrosion rate is:

After 12 years:

The required remaining thickness for the checked basis is:

The apparent reserve is:

If the ultrasonic thickness uncertainty is u_t=0.10\,\text{mm}, the review uses k=2, and the vessel is released for three more years before the next focused survey:

The guarded reserve is:

The result is still positive, but the margin is small. A practical release would likely require coating repair, focused reinspection, limits on service profile, or earlier renewal if corrosion is localized rather than uniform.

Fatigue Stress Range and Miner Damage

Stress range:

Stress amplitude:

Mean stress:

Miner cumulative damage:

where n_i is applied cycles at stress range level i, and N_i is fatigue life at that stress range from the selected S-N curve.

Fatigue damage is not exact. It depends on stress definition, detail category, weld quality, corrosion, residual stress, sequence effects, inspection probability, sea-state scatter, vibration, and whether the load spectrum represents actual service.

Worked Example: Weld Detail Damage Accumulation

A welded bracket hot spot is screened over one inspection interval using three stress-range bins:

Stress range bin	Applied cycles n_i	S-N life N_i	Damage n_i/N_i
moderate sea states	1.8\times 10^6	6.0\times 10^7	0.030
heavy sea states	2.4\times 10^5	8.0\times 10^6	0.030
rare slam events	1.5\times 10^4	1.2\times 10^6	0.0125

The damage sum is:

If the project inspection philosophy limits damage accumulation to 0.25 before focused inspection, the interval passes the simplified screen:

The engineering comment is more important than the arithmetic. The review must confirm that the bracket stress is the correct hot-spot stress, the weld detail category matches the S-N curve, corrosion is represented, slam events are not undercounted, and NDT can actually find the expected crack before it becomes critical.

Goodman Screen for Non-Welded Details

For a non-welded metallic detail where a Goodman-style screen is appropriate:

where S_e is endurance or finite-life fatigue strength on the selected basis, \sigma_{UTS} is ultimate tensile strength, and N_f is fatigue design factor.

Worked Example: Machined Bracket Mean-Stress Check

A machined non-welded bracket has:

The Goodman utilization without a design factor is:

For N_f=2 the allowable right side is:

The screen passes narrowly:

This does not make the same method valid for welded hull details. Welded details often require stress-range S-N methods, corrosion corrections, hot-spot stresses, and inspection planning rather than an endurance-limit assumption.

Fracture and Inspection Interval

Stress intensity factor:

Critical crack size:

Paris-law crack growth:

with:

For constant Y and \Delta\sigma, an idealized growth life for m\neq 2 is:

Inspection interval with a conservatism factor:

where F accounts for uncertainty, consequence, accessibility, and probability of detection.

Worked Example: Inspection Interval From Crack Growth

A fracture screen for a repaired deck opening uses:

The critical crack size is:

Assume a crack-growth calculation from the detectable crack size to a_c gives:

Using an inspection factor:

the inspection interval in cycles is:

If the measured operating profile produces:

then:

A practical plan would choose a shorter scheduled interval, such as 3 years, and would also define the NDT method, access, detectable flaw size, repair acceptance criteria, and whether service restrictions apply.

Deflection and Alignment

Generic linear stiffness relation:

For a simply supported beam under central point load:

For a simply supported beam under uniform load:

Alignment-sensitive marine structures include shaft lines, engine beds, gearbox foundations, rudder stocks, thruster tunnels, cranes, davits, hatch covers, door frames, and seal interfaces. Strength margin alone is not enough when deflection can misalign machinery or compromise watertight closure.

Vibration and Resonance Separation

Single-degree natural frequency:

Excitation frequency from rotating machinery:

Blade-passing frequency:

Frequency separation:

where N_b is blade count. Acceptance limits depend on the vessel, forcing level, damping, fatigue sensitivity, noise requirements, and measurement evidence.

Worked Example: Foundation Frequency Screen

A local foundation has estimated stiffness:

and effective participating mass:

The natural frequency is:

A four-bladed propeller turns at 180 rpm:

so:

The separation is:

If the project target is at least 20 percent separation before sea-trial confirmation, the foundation is too close to blade-passing excitation. The next action is not simply to add material; the team should check the mode shape, damping, forcing level, fatigue-sensitive details, measurement plan, and whether stiffening or mass change moves the frequency away from another excitation.

Guarded Structural Margin

Guarded capacity:

Guarded demand:

Guarded margin of safety:

where u_C and u_D are uncertainty estimates for capacity and demand on the same basis.

Worked Example: Release Margin With Uncertainty

A repaired longitudinal detail has nominal allowable stress:

The nominal demand from the checked load case is:

The review assigns:

Therefore:

and:

The guarded margin is:

The repair has only 8.8 percent guarded stress margin. That may be acceptable for a temporary restricted release with inspection controls, but it is not a strong basis for unrestricted long-term service without additional evidence.

Validation Record

A credible marine structural calculation record should state:

1. loading condition, draft, trim, displacement, ballast state, cargo state, fuel state, and environmental basis;
2. stress basis: nominal, structural hot-spot, local notch, principal, equivalent, or finite-element stress;
3. section properties and whether corrosion deductions, openings, ineffective plating, and repairs are included;
4. local pressure basis, dynamic factors, load factors, and governing locations;
5. buckling method, boundary assumptions, effective length, imperfection assumptions, and corrosion thickness;
6. fatigue method, stress definition, S-N data, corrosion environment, cycle spectrum, detail category, and inspection plan;
7. fracture inputs, detectable crack size, NDT method, probability of detection, critical crack size, and inspection interval;
8. thickness survey locations, measurement uncertainty, trend history, coating condition, and corrosion rate estimate;
9. finite-element model scope, mesh checks, boundary conditions, load application, and correlation evidence;
10. release decision, restrictions, next inspection, repair traceability, and responsible approval basis.

Common Mistakes

Common mistakes include using the broad vessel-performance formula sheet as if it were a full structural assessment, checking only calm-water bending, ignoring shear flow around openings, treating hydrostatic pressure as the whole local load, and applying plate buckling formulas without corrosion deductions.

Other frequent errors are mixing nominal stress with hot-spot S-N data, assuming a positive Miner damage sum margin proves crack detectability, using thickness measurements without uncertainty or trend review, checking machinery foundation strength without vibration separation, and releasing a repair without updating the inspection plan and loading restrictions.

The strongest hull integrity calculations make assumptions visible. They connect loading, structure, material condition, inspection evidence, repair history, and operating restrictions so that future survey and life-extension decisions can be made from traceable engineering evidence rather than rediscovered assumptions.