Derivation of Wave Height and Peak Period Statistics Accounting for Physical Limitations

Volume 8B: Ocean Engineering ◽

10.1115/omae2014-23827 ◽

2014 ◽

Author(s):

Cécile Melis ◽

Guillaume Bonnaffoux

Keyword(s):

Lower Bound ◽

Normal Distribution ◽

Wave Height ◽

Joint Probability ◽

Peak Period ◽

Original Dataset ◽

Extreme Response ◽

Physical Limitations ◽

Log Normal ◽

Log Normal Distribution

When assessing the joint-probability of significant wave height and peak period, (Hs,Tp) measured over years at a given site, it is customary to fit a log-normal distribution to assess Tp dependence on Hs. The parameters of this distribution are then used either to compute N-year return period design curves in order to compute extreme response by means of short-term analysis, or response distributions, by means of response-based analysis. The main drawback of the Log-Normal distribution to represent the variability of Tp wrt. Hs is that its lower bound is zero, while physics tell us that wave steepness cannot be infinite, hence the lower bound, Tplim(Hs) should be greater than zero. If the distribution is kept unbounded, the resulting statistical fitting tends to predict occurrences of sea-states with (Hs,Tp) pairs having unphysical or unlikely steepness. This is particularly true in the range of 10–15s, where some ship-shaped units mooring systems responses are at their maximum. Attempts have been made in the past to introduce a lower bound to the log-normal distribution, for instance by Drago et al, [1], by shifting it by a predefined value of limit steepness. By doing so, some points of the original dataset had to be discarded as they were falling below the lower bound. An evolution of their methodology is proposed in this paper, which uses the points of the dataset in a relevant region which will be defined hereafter, and then uses this limit to shift the Log-Normal distribution. The obtained environmental contours are then compared against observed data to check which one fits most accurately the original set of measured (Hs,Tp) pairs.

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Assessment of Significant Wave Height – Peak Period Distribution Considering the Wave Steepness Limit

Volume 5: Ocean Engineering ◽

10.1115/omae2013-10227 ◽

2013 ◽

Author(s):

Michele Drago ◽

Giancarlo Giovanetti ◽

Claudia Pizzigalli

Keyword(s):

Normal Distribution ◽

Wave Height ◽

Significant Wave Height ◽

Wave Steepness ◽

Peak Period ◽

Near Surface ◽

Significant Wave ◽

Physical Limit ◽

Log Normal ◽

Log Normal Distribution

The physical limit of the significant wave steepness is generally exceeded when assessing the seastate climate and the extreme iso-probability contours, i.e. too short significant wave peak periods Tp are sometimes associated to a certain significant wave height Hs. The occurrence of not physically consistent Tp is clearly due to a fault in the generally made assumption of a log-normal distribution of the Tp, where the physical limit for the period would be Tp > 0, i.e. the existence limit of the log-normal distribution, which is well below the real physical limit for significant wave steepness. If this is not a problem for pipeline design, where stability and fatigue are dominated by longer peak periods associated at each significant wave height, loads overestimation could arise for near surface structures, e.g. riser, where the largest loads and fatigue, are caused by the shorter peak periods associated to a certain significant wave height. Hence, the possibility to define a Tp distribution which respects the physical lower bound of the limiting wave steepness has a significant relevance when dealing with design and installation of near surface structures. The present paper proposes a new methodology for the assessment of the Hs-Tp distribution which respects an a-priori defined wave steepness limit. This can be done basing on the definition of significant wave steepness Sp = 2πHs/gTp2 which, assessing a limiting steepness Sp, provide an physical lower bound Tplim for the peak period Defining a new variable Tp′ = Tp – Tplim and imposing that Tp′ follows a log-normal distribution, hence having a physical limit Tp′ > 0, is equivalent to assess a Tp distribution which respects the defined significant wave steepness limit Tp > Tplim. A test case compares results obtained with the ‘old’ and ‘new’ methodologies and shows the implication on the design loads. Moreover, another test case has been investigated to verify the performance and characteristics of the new methodology.

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A Universal Model for the Log-Normal Distribution of Elasticity in Polymeric Gels and Its Relevance to Mechanical Signature of Biological Tissues

Biology ◽

10.3390/biology10010064 ◽

2021 ◽

Vol 10 (1) ◽

pp. 64

Author(s):

Arnaud Millet

Keyword(s):

Elastic Modulus ◽

Normal Distribution ◽

Biological Tissues ◽

Force Microscopy ◽

Polymeric Gels ◽

Atomic Force ◽

Clear Explanation ◽

Log Normal ◽

Log Normal Distribution

The mechanosensitivity of cells has recently been identified as a process that could greatly influence a cell’s fate. To understand the interaction between cells and their surrounding extracellular matrix, the characterization of the mechanical properties of natural polymeric gels is needed. Atomic force microscopy (AFM) is one of the leading tools used to characterize mechanically biological tissues. It appears that the elasticity (elastic modulus) values obtained by AFM presents a log-normal distribution. Despite its ubiquity, the log-normal distribution concerning the elastic modulus of biological tissues does not have a clear explanation. In this paper, we propose a physical mechanism based on the weak universality of critical exponents in the percolation process leading to gelation. Following this, we discuss the relevance of this model for mechanical signatures of biological tissues.

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