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Article

Vertical Bending Moment in Extreme Regular Waves—Benchmarking of Numerical Codes Against Model Tests

by
Ole Andreas Hermundstad
1,*,
Guillaume de Hauteclocque
2,
Sopheak Seng
2,
Masayoshi Oka
3,
Chong Ma
3,
Benjamin Bouscasse
4,
Roberto Vettor
5,
Shan Wang
5,
Ivan Sulovsky
6,
Jasna Prpic-Orsic
6,
Kei Sugimoto
7 and
Tormod R. Landet
8
1
SINTEF Ocean AS, 7052 Trondheim, Norway
2
Bureau Veritas, 75013 Paris, France
3
National Maritime Research Institute, Tokyo 181-0004, Japan
4
LHEEA, UMR 6598, CNRS, École Centrale Nantes, Nantes Université, 44000 Nantes, France
5
Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal
6
Faculty of Engineering, University of Rijeka, 58, 51000 Rijeka, Croatia
7
ClassNK, Tokyo 102-8567, Japan
8
Det Norske Veritas—DNV, 1363 Høvik, Norway
*
Author to whom correspondence should be addressed.
J. Mar. Sci. Eng. 2026, 14(5), 481; https://doi.org/10.3390/jmse14050481
Submission received: 2 December 2025 / Revised: 20 January 2026 / Accepted: 25 February 2026 / Published: 2 March 2026
(This article belongs to the Special Issue Numerical Modeling of Fluid-Structure Interactions in Ocean Engineering)
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Abstract

A benchmark study of 10 different numerical methods for ship motion and load assessment is presented. Pitch motions and midship vertical bending moments are compared to model test results for a containership at zero speed in head regular waves. The wave steepness is varied from 2.1% to 10.5%. The model tests show that pitch and the vertical bending moment (VBM) display nonlinear behavior even for low-steepness waves. It is demonstrated that computational fluid dynamics (CFD) methods can reproduce the ship responses with good accuracy, even in very steep waves, involving green water and parts of the ship going in and out of water. Weakly nonlinear potential-theory methods tend to overestimate the pitch motions and the sagging moments as the wave steepness increases. For the vertical bending moment in steep waves, the 3D panel methods did not give significantly better results than those obtained with the nonlinear strip theories.
Keywords: ships; nonlinear wave loads; VBM; regular waves; steepness; benchmark study ships; nonlinear wave loads; VBM; regular waves; steepness; benchmark study

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MDPI and ACS Style

Hermundstad, O.A.; Hauteclocque, G.d.; Seng, S.; Oka, M.; Ma, C.; Bouscasse, B.; Vettor, R.; Wang, S.; Sulovsky, I.; Prpic-Orsic, J.; et al. Vertical Bending Moment in Extreme Regular Waves—Benchmarking of Numerical Codes Against Model Tests. J. Mar. Sci. Eng. 2026, 14, 481. https://doi.org/10.3390/jmse14050481

AMA Style

Hermundstad OA, Hauteclocque Gd, Seng S, Oka M, Ma C, Bouscasse B, Vettor R, Wang S, Sulovsky I, Prpic-Orsic J, et al. Vertical Bending Moment in Extreme Regular Waves—Benchmarking of Numerical Codes Against Model Tests. Journal of Marine Science and Engineering. 2026; 14(5):481. https://doi.org/10.3390/jmse14050481

Chicago/Turabian Style

Hermundstad, Ole Andreas, Guillaume de Hauteclocque, Sopheak Seng, Masayoshi Oka, Chong Ma, Benjamin Bouscasse, Roberto Vettor, Shan Wang, Ivan Sulovsky, Jasna Prpic-Orsic, and et al. 2026. "Vertical Bending Moment in Extreme Regular Waves—Benchmarking of Numerical Codes Against Model Tests" Journal of Marine Science and Engineering 14, no. 5: 481. https://doi.org/10.3390/jmse14050481

APA Style

Hermundstad, O. A., Hauteclocque, G. d., Seng, S., Oka, M., Ma, C., Bouscasse, B., Vettor, R., Wang, S., Sulovsky, I., Prpic-Orsic, J., Sugimoto, K., & Landet, T. R. (2026). Vertical Bending Moment in Extreme Regular Waves—Benchmarking of Numerical Codes Against Model Tests. Journal of Marine Science and Engineering, 14(5), 481. https://doi.org/10.3390/jmse14050481

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