AM 5430:

Stochastic Processes in Structural Mechanics

Assignment 4

March 5, 2012

Last date for submission: April 23, 2014.

Instructions:

• Please submit a report where all the steps involved in analytical calculations are written

clearly and legibly.

• Email all your MATLAB codes together in a zipped format to gupta.sayan@gmail.com.

The name of the zipped file should be your full roll no, example, AM10D001.zip.

• Attach print outs of your plots along with your report. Please print about 4 plots in

one A4 page to save paper and ensure readability.

• Provide axis labels, captions and figure numbers to all your plots.

• Please ensure that you use different line styles in your plots so that the plots are

discernible in b/w print outs.

Marks: 20

Modern wind-turbines are enormous structures and are usually built off the coast, in open

seas. These wind turbines could be either floating type, which involves partially submerged

masts, or fixed type, which contains pier foundations with the masts fixed to the piers. In

either case, in the design of these masts, a crucial aspect of the load calculation is the force

due to the ocean waves. As the force is proportional to the height of the sea-waves, the

problem on hand is to characterize the sea-waves, which are random, in probabilistic terms.

It is common practice to model the sea surface elevation at a fixed point as a Gaussian

process, X(t), which during a limited period of time ranging from 20min to 3 hours, can

be considered to be stationary. The statistical properties of the sea surface elevation under

stationary conditions are called a sea state. The sea state is characterized in terms of the

PSD, S(ω), the mean sea level, m, referred to as the still water level usually set to zero

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and the water depth d. An apparent wave is defined as the part of the sea record between

two successive up-crossings of the still water level, m and the wave crest height, Ac, is the

maximum of the apparent wave, i.e., the maximum value the process X(t) attains between

two successive up-crossings of the level m. In reality, the exact spectral density S(ω) is

usually not known; instead, the sea state is characterized by the significant wave-height, Hs,

given by,

σ2

X=

H2s

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and the zero-crossing wave period, Tz, i.e., the inverse up-crossing intensity of the still water

level.

The intensity of waves is by definition equal to the intensity of up-crossings of the still water

level m by the process X(t). The crossing intensity µ+

X(m) can be easily calculated using

Rice’s formula. However, the crossing intensity of waves with crest higher than v is more

difficult to compute. It can be shown that

P [Ac > h|S] ≤µ+

X(h|S)

µ+

X(m|S)

, h ≥ m (1)

where, µ+

X(h|S) is the up-crossing intensity of the level h during the sea state S.

1. Show that for any Gaussian sea state,

P [(Ac −m) > h|S] ≤ exp[−8(h

Hs

)2].

2. If at a location of a buoy, the sea has Tz = 10s and Hs = 10 m, estimate the probability

that a crest height is above 10 m.

3. Consider now a deep water location having Pierson-Moskowitz spectrum of the form

SPM(ω) =H2

sθ

4π(ω/θ)−5 exp

(

−1

π(ω/θ)−4

)

,

with Hs = 7 m and Tz = 10 s, plot the excedance probability obtained analytically as

a function of wave height h, for the sea state S. Here, θ = 2π/Tz.

4. In the same plot, compare the predictions of the exceedance probability obtained from

Monte Carlo simulations. [Here, simulate N samples of time histories using spectral

decomposition method and find the probabilities of exceedance. Consider N = 1×10s,

where s is at least 3.

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