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Authors

Christy Ushanth Navaratnam Alf Tørum Øivind A. Arntsen

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13

PRELIMINARY ANALYSIS OF WAVE SLAMMING FORCE RESPONSE DATA FROM TESTS ON A TRUSS STRUCTURE IN LARGE WAVE FLUME, HANNOVER, GERMANY

29th August 2013, Trondheim, Norway

Mail address Ph. +47 73 59 46 40 Location Høgskoleringen 7A Fax +47 73 59 70 21 Høgskoleringen 7A NO-7491 Trondheim Org. no. NO 974 767 880 NORWAY

Norwegian University of

Science and Technology

NTNU Department of Civil and Transport Engineering

REPORT

Title

PRELIMINARY ANALYSIS OF WAVE SLAMMING FORCE

RESPONSE DATA FROM TESTS ON A TRUSS STRUCTURE

IN LARGE WAVE FLUME, HANNOVER, GERMANY

Report no.

BAT/MB-R1/2013.

Date

2013 Authors

Christy Ushanth Navaratnam

Alf Tørum

Øivind A. Arntsen

Sign.

Knut V. Høyland

Head of division ISBN No.

ISBN 978-82-8289-000-7 (Paper Version)

ISBN 978-82-8289-001-4 (Electronic Version)

Number of pages

59

Client

Division of Marine Civil Engineering Availability

Open

Abstract

The foundation of offshore wind turbines are sometimes steel truss structures. In shallow water these

structures may be subjected to wave slamming forces due to plunging breakers. Previous researches show

that the impulsive forces from the plunging wave may be the governing factors in the design response of the

truss structure and the foundations.

Large scale (1:8) tests were carried out at the Large Wave Flume in Hannover Germany in order to

investigate the wave slamming forces on the truss structures and to improve the method to calculate forces

from the plunging breakers. This report presents some of the analysis of the data obtained from the test.

The obtained data were analysed based on total force response as well as the local force response. Total

force data have been analysed using Frequency Response Function (FRF) method whereas the local force

measurements have been analysed using Duhamel integral method as well as the Frequency Response

Function method.

The results showed that the measured slamming forces on the truss is very much less compared to the

calculated slamming forces based on Wienke & Oumeraci (2005). This could be due to the size effects and

wave form when the wave hits the structure. The slamming factor Cs was found to be smaller compared to

the value that was suggested by Wienke & Oumeraci (2005). The pattern of the slamming force distribution

was found to be triangular as found by Sawaragi and Nochino (1984) and Ros (2011). It was found that both

FRF and Duhamel integral methods give almost similar results.

It was found that there is a time phase shift in between the wave profile and the local force measurement

which needs to be investigated and further detailed analyse is recommended in order to come up with better

results.

Stikkord (Norwegian) Keywords (English)

Wave slamming

Response force

Breaking waves

i

PREFACE

This report presents some analyses of the data that have been obtained from the experiments on

a truss structure, which was carried out at the Large Wave Channel, Hannover, Germany in May

and June, 2013. The objective of this study to investigate and improve the method to calculate the

wave slamming forces from plunging breaking waves on truss structures through model on a large

scale.

The work described in this publication was supported by the European Community's 7th

Framework Programme through the grant to the budget of the Integrating Activity HYDRALAB

IV, Contract no. 261520.

This document reflects only the authors’ views and not those of the European Community. This

work may rely on data from sources external to the HYDRALAB IV project Consortium.

Members of the Consortium do not accept liability for loss or damage suffered by any third party

as a result of errors or inaccuracies in such data. The information in this document is provided "as

is" and no guarantee or warranty is given that the information is fit for any particular purpose. The

user thereof uses the information at its sole risk and neither the European Community nor any

member of the HYDRALAB IV Consortium is liable for any use that may be made of the

information.

Christy Ushanth Navaratnam

Alf Tørum

Øivind A. Arntsen

NTNU, August 2013

ii

TABLE OF CONTENTS

1 INTRODUCTION ........................................................................................................................... 1

2 LITERARURE REVIEW ................................................................................................................ 1

2.1 Morison’s Equation .................................................................................................................... 1

2.2 Wave Slamming Force ............................................................................................................... 2

2.3 Slamming Coefficients ............................................................................................................... 4

2.4 Curling Factor ............................................................................................................................. 5

2.5 Breaking Waves .......................................................................................................................... 6

3 METHODOLGY ............................................................................................................................. 7

3.1 Experimental set-up ................................................................................................................... 7

3.2 Data Analysing Methods ......................................................................................................... 10

3.2.1 Frequency Response Function (FRF) ........................................................................... 11

3.2.2 Duhamel Integral Method .............................................................................................. 12

4 ANALYSIS OF DATA .................................................................................................................. 14

4.1 Total Force Analysis ................................................................................................................ 14

4.1.1 Wave Test ‘2013061424’ with hammer test ‘2406201319’ and ‘Large-hammer-test-

2013_06_24_18_42_58’. ................................................................................................................. 14

4.2 Local Force Transducers ......................................................................................................... 24

4.2.1 Duhamel integral approach: ........................................................................................... 27

4.2.2 FRF Approach .................................................................................................................. 30

5 CONCLUSION AND RECOMMENDATIONS .................................................................... 46

6 REFERENCES ............................................................................................................................... 47

APPENDICES ......................................................................................................................................... 49

iii

LIST OF FIGURES

Figure 2.1: Definition sketch of von Karman’s model (Ros Collados, 2011) .................................... 2

Figure 2.2: Definition sketch of impact force on vertical cylinder (Wienke & Oumeraci, 2005) ... 3

Figure 2.3: Definition sketch of 2D impact distribution (Wienke & Oumeraci, 2005) .................... 3

Figure 2.4: Instrumented cylinder [cm]. (Tørum, 2013) ........................................................................ 4

Figure 2.5: Time histories of line forces according to different theories (Wienke & Oumeraci, 2005)

....................................................................................................................................................................... 5

Figure 2.6: Curling factor for different inclination of the pile (Wienke & Oumeraci, 2005) ........... 6

Figure 2.7: Breaker types based on Iribarren parameter (Judith & Marcel, 2012)............................. 7

Figure 3.1: Experimental set-up in the Large Wave Flume (Arntsen, 2013) ...................................... 7

Figure 3.2: Instrumented Structure .......................................................................................................... 8

Figure 3.3: Local force transducers .......................................................................................................... 9

Figure 3.4: West side (front). Approximate location of points, marked with big yellow stars, for

application of the 1.5 kg impulse hammer for the whole structure. Approximate location of points,

marked with small yellow stars for application of the 0.1 kg impulse hammer on the local force

cells. (Arntsen, 2013) ................................................................................................................................ 10

Figure 3.5: The derivation of the Duhamel integral (Ros Collados, 2011) ....................................... 12

Figure 3.6: Main steps involving in the Duhamel integral approach (Ros Collados, 2011) ........... 14

Figure 4.1: Hammer Impulse .................................................................................................................. 15

Figure 4.2: The total response of hammer impulse ............................................................................. 15

Figure 4.3: Spectrum of hammer and response forces ........................................................................ 16

Figure 4.4: Series of measured total responses ..................................................................................... 16

Figure 4.5: Deep water wave and wave at structure [wave period – 5.55s] ...................................... 17

Figure 4.6: Decomposition of total response ....................................................................................... 17

Figure 4.7: Dynamic Response ............................................................................................................... 18

Figure 4.8: Spectrum of dynamic response ........................................................................................... 18

Figure 4.9: Low-pass filtered slamming force variation (with the cut-off frequency of 200Hz) .. 19

Figure 4.10: Hammer impact at point 5. ............................................................................................... 19

Figure 4.11: Total hammer response ...................................................................................................... 20

Figure 4.12: Chosen hammer impulse ................................................................................................... 20

Figure 4.13: Spectrum of the hammer force ......................................................................................... 21

Figure 4.14: Spectrum of both hammer force and total force ........................................................... 21

Figure 4.15: Un-filtered slamming force ................................................................................................ 22

Figure 4.16: Low-pass filtered slamming force (with the cut-off frequency of 200Hz) ................. 22

Figure 4.17: Low-pass filtered slamming force (with the cut-off frequency of 40Hz) ................... 23

Figure 4.18: Hammer impulse and Response on FTLF02 ................................................................. 24

Figure 4.19: Time expanded view of above figure ............................................................................... 25

Figure 4.20: Spectrum of both hammer impulse and the response ................................................... 25

Figure 4.21: Spectrum of the response forces ...................................................................................... 26

Figure 4.22: The maximum response to a suddenly applied triangular force time history (Naess,

..................................................................................................................................................................... 27

Figure 4.23: Duhamel integral approach with impact duration 0.01s and rising time 0.00008s ... 28

Figure 4.24: Duhamel integral approach with impact duration 0.005s and rising time 0.00008s . 28




Figure 4.28: Un-filtered force variation on FTLF02 ........................................................................... 31

iv

Figure 4.29: Filtered force variation on FTLF02 [Cut-off frequency- 200Hz] ................................ 31



Figure 4.32: Time series of measures response of local force transducers [FTLF01-FTLF05] .... 33

Figure 4.33: Time series of measures response of local force transducers [FTLF06-FTLF10] .... 33

Figure 4.34: Time expanded view of local force transducer responses ............................................ 34

Figure 4.35: The dynamic part of the total response on FTLF02 ..................................................... 35

Figure 4.36: Power spectrum of the total response forces ................................................................. 36

Figure 4.37: Force variation on FTLF01 [Cut-off frequency 200Hz] ............................................... 36










Figure 4.47: The variation of the force intensity with the depth [based on FRF method] at the time

of maximum total response, t=128.3s ................................................................................................... 42

Figure 4.48: Duhamel integral approach for FTLF03 (response at about 128.3s) .......................... 42

Figure 4.49: Duhamel integral approach for FTLF04 (response at about 128.3s) ......................... 43



Figure 4.52: The variation of the force intensity with the depth [based on Duhamel integral

approach] at the time of maximum total response, t=128.3s ............................................................. 45

Figure 4.53: The response of FTLF02 (blue) and the wave at the structure (red) .......................... 45

v

LIST OF TABLES

Table 4.1: Characteristics of test ‘2013061424’ .................................................................................... 14

Table 4.2: Measured forces on each local force transducers [based on FRF method] ................... 41

Table 4.3: Measured forces on each local force transducers [based on Duhamel integral approach]

..................................................................................................................................................................... 44

1

1 INTRODUCTION

The foundation of offshore wind turbines are sometimes steel truss structures. In shallow water

these structures may be subjected to wave slamming forces due to plunging breakers. Previous

researches show that the impulsive forces from the plunging wave may be the governing factors

in the design response of the truss structure and the foundations.

Large scale (1:8) tests were carried out at the Large Wave Flume in Hannover Germany in order

to investigate the wave slamming forces on the truss structures and to improve the method to

calculate forces from the plunging breakers. This report presents some of the analysis of the data

obtained from the test.

2 LITERARURE REVIEW

[The following literature review has been directly extracted from the master’s thesis; Navaratnam

(2013)]

Many researches about wave slamming forces or breaking wave forces have been carried out and

still being carried out all over the world. In this chapter, findings from previous researches have

been described.

2.1 Morison’s Equation

The non-breaking wave forces acting on a vertical pile can be calculated using Morison’s equation

(Morison, et al., 1950) which is the summation of the quasi static inertia and drag forces.

𝑑𝐹 = 𝑑𝐹𝐷 + 𝑑𝐹𝑀 =

1

2 𝜌𝑤𝐶𝐷𝐷|𝑢|𝑢 𝑑𝑧 + 𝜌𝑤

𝜋𝐷2

4 𝐶𝑀

𝑑𝑢

𝑑𝑡 𝑑𝑧 (2.1)

Where ρw is the water density, CD is the drag coefficient, CM is the inertia coefficient, D is the

diameter of the pile, u is the water particle velocity, z is the water depth and t is the time. The

values of the drag and coefficients are depending on the Reynolds number, Keulagen Carpenter

number, roughness parameters and interaction parameters (Morison, et al., 1950). The total force

can be obtained by integrating the equation (2.1) along the height of the pile.

𝐹 = 𝐹𝐷 + 𝐹𝑀 = ∫1

2 𝜌𝑤𝐶𝐷𝐷|𝑢|𝑢 𝑑𝑧

𝜂

−𝑑

+ ∫ 𝜌𝑤

𝜋𝐷2

4 𝐶𝑀

𝑑𝑢

𝑑𝑡 𝑑𝑧

𝜂

−𝑑

(2.2)

Where, 𝜂 is the water surface elevation and the d is the total water depth.

The force coefficients CD and CM have been obtained with laboratory experiments. Different range

of values were found for a non-breaking wave for various flow conditions. Generally the Morison

equation is valid for small diameter members that don’t significantly modify the incident waves,

and it depends on the ratio of the wavelength to the member diameter. If this ratio is more than

5, the Morison equation is applicable (Chella, et. al., 2012).

When it comes to breaking wave attack, an additional force of short duration because of the impact

of the vertical breaker front and the breaker tongue has to be considered (Irschik, et. al., 2002).

So, an additional force term which is called ‘slamming force’ (FS) has to be added to the Morison

equation as given in the equation (2.3).

2

𝐹 = 𝐹𝐷 + 𝐹𝑀 + 𝐹𝑆 (2.3)

2.2 Wave Slamming Force

The first wave impact model and theoretical formulation of water impact force on rigid body was

derived by von Karman (von Karman, 1929). In his research, he considered a horizontal cylindrical

body with a wedged-shaped under surface as it strikes the horizontal surface of water and

calculated the force acting between the cylindrical body and the water. As it’s shown in the Figure

2.1, a cylinder is approximated by a flat plate of width c(t) which is equal to the immersed portion

of the cylinder at each instant of the impact. The force on this plate could be calculated by

considering the potential flow under the plate and integrating the pressures which can be found

by the Bernoulli’s equation and for this, the time history of the width of the plate should be known

as well.

Figure 2.1: Definition sketch of von Karman’s model (Ros Collados, 2011)

According to von Karman theory, the line force f(t) is given by the following equation,

𝑓(𝑡) = 0.5 𝐶𝑠 𝜌𝑤𝐷 𝐶𝑏2 (2.4)

𝐶𝑠 = 𝜋 (1 −

𝐶𝑏

𝑅𝑡) (2.5)

Where, Cs is the slamming factor, Cb is the wave celerity and D is the diameter of the cylinder and

R is the radius of the cylinder. The maximum line force occurs when the time t is zero (t=0, i.e.

beginning of the impact), and the slamming factor becomes 𝜋.

As this line force is two dimensional and was derived for an infinite length of cylinder based on

von Karman’s model, it should be integrated over the length of the impact area (Figure 2.2) of

cylinder assuming the same line force acting everywhere in the cylinder.

3

Figure 2.2: Definition sketch of impact force on vertical cylinder (Wienke & Oumeraci, 2005)

As Figure 2.2 shows, the height of the impact area was found to be the multiplication of the curling

factor λ and the maximum breaking wave crest height ηb (Goda, et. al.,1966). So, the slamming

force Fs on the cylinder,

𝐹𝑠(𝑡) = 0.5 𝜌𝑤𝐷 𝐶𝑏

2 𝜋 (1 −𝐶𝑏

𝑅𝑡) λ 𝜂

𝑏 (2.6)

𝐹𝑠(𝑡) = 𝜋 𝜌𝑤𝑅 𝐶𝑏

2 (1 −𝐶𝑏

𝑅𝑡) λ 𝜂

𝑏 (2.7)

At the beginning of the impact with t=0 the equation (2.7) follows,

𝐹𝑠 = 𝜋 𝜌𝑤𝑅 λ 𝜂𝑏

𝐶𝑏2 (2.8)

From equation (2.4), the line force based on von Karman (1929),

𝑓(𝑡) = 𝜋 𝜌𝑤𝑅 𝐶𝑏2 (2.9)

Figure 2.3: Definition sketch of 2D impact distribution (Wienke & Oumeraci, 2005)

4

The line force given in equation (2.9) was obtained by considering the momentum conservation

during the impact. By taking into consideration not only the momentum conservation, but also

the flow beside the flat plate would result in the so-called ‘pile-up effect’, that is the deformation

of the water free surface (Figure 2.3). Because of this pile-up effect, the ‘immersion’ of the cylinder

occurs earlier. As a result, the duration of impact decreases and the maximum line force increases

(Wienke & Oumeraci, 2005).

According to Wagner (1932), the maximum line force is given as follows,

𝑓(𝑡) = 2𝜋 𝜌𝑤𝑅 𝐶𝑏2 (2.10)

The maximum line force calculated by Wagner’s theory is twice the maximum line force calculated

by von Karman’s theory. Generally this maximum line force is described as a function ‘Slamming

Coefficient’ Cs.

𝑓(𝑡) = 𝐶𝑆 𝜌𝑤𝑅 𝐶𝑏2 (2.11)

2.3 Slamming Coefficients

So, the general form of wave slamming force is given in the following equation.

𝐹𝑠 = 𝐶𝑆𝜌𝑤𝑅 λ 𝜂𝑏

𝐶𝑏2 (2.12)

According to von Karman (1929) and Goda et. al. (1966), Cs is π and Wagner’s theory suggests a

Cs value of 2π. Wienke & Oumeraci (2005) suggest a Cs value of 2π as they show that the

formulation of Wagner’s theory is more accurate even though Goda et. al (1966)’s description of

the impact is based on von Karman (1929). Ros Collados (2011) investigated the slamming

coefficient on a vertical cylinder in his master’s thesis and estimated a Cs value of 4.3 for a triangular

load case, and this value is between π and 2π. This experiment was done with a vertical cylinder

with a series of force transducers placed on it in different elevations as shown in Figure 2.4.

Figure 2.4: Instrumented cylinder [cm]. (Tørum, 2013)

5

The Cs values were found by considering the maximum impact force at the third transducer. It

should be noted that the impact duration time was set as 0.008s for all the cases, which was defined

at the same time as the triangular load.

Another experiment was carried out by Aune (2011) as part of his master’s thesis and he calculated

a Cs value of 4.77. But, this experiment was performed on a truss structure.

Wienke and Oumeraci (2005) obtained a time history of the impact line force. This is shown in

Figure 2.5. This shows that the value of the line force at the beginning of the impact (t=0), i.e. the

maximum line force that is calculated by their proposed model is equal to the value obtained from

the Wagner’s model.

2.4 Curling Factor

Wienke and Oumeraci (2005) investigated about the curling factor for the vertical and inclined

cylinders. The ratio of the impact force Fs to the line force f(t) provides the height area of the

impact ηb, where ηb is the maximum surface elevation of the breaking wave and the λ is the curling

factor. Figure 2.6 shows the variation of the cylinder factor with the different inclination of the

cylinder, i.e. yaw angle α.

For a vertical cylinder, the maximum curling factor is λ=0.46 and this is in agreement with the

values of curling factors cited in literature, for example, Goda, et. al. (1966) proposed a range of

curling factors λ=0.4-0.5 for plunging wave breakers.

Figure 2.5: Time histories of line forces according to different theories (Wienke & Oumeraci, 2005)

6

Figure 2.6: Curling factor for different inclination of the pile (Wienke & Oumeraci, 2005)

2.5 Breaking Waves

Waves breaking process is taken place in various different ways depending on the wave properties

and angle of bed slope (Judith & Marcel, 2012). Battjes (1974) showed that the Iribarren parameter

influences in the wave breaking process. The Iribarren parameter is difined as follows,

𝜉0 =

tan 𝛼

√𝐻0/𝐿0

(2.13)

where, tan 𝛼 is the steepness of the bed, 𝐻0 is the deep water wave height and 𝐿0 is the wave

length in deep water.

The Iribarren number 𝜉0 represents the ratio of the slope of the bed and the deep water wave

steepness. A distinction is made between spilling, plunging, collapsing and surging breakers based

on the value of 𝜉0 (Figure 2.7). The values of Iribarren number are indicative and the transition

between the various breaker types is gradual. Spilling breakers are generally found along the flat

bed. Plunging breaking occurs on a mild slope bed and the curling top is characteristic of such a

wave. When the curling top breaks over the lower part of the wave, a lot of energy is dissipated

into turbulence.

7

Figure 2.7: Breaker types based on Iribarren parameter (Judith & Marcel, 2012)

3 METHODOLGY

3.1 Experimental set-up

The experimental set-up at the Large Wave Flume, Hannover, Germany is shown in the Figure

3.1.

Figure 3.1: Experimental set-up in the Large Wave Flume (Arntsen, 2013)

The Figure 3.2 shows an isometric view of the instrumented structure. The structure is

instrumented as follows,

- 4 total force transducers, two at the top and the two at the bottom to measure the total force

on the structure

- 10 local force transducers at the front vertical legs of the structure

- 12 XY force transducers to measure the total force on six bracings

8

Figure 3.2: Instrumented Structure

Since there were only forces from total force transducers and the local force transducers were

analysed, the hammer points are shown only for these cases.

9

Figure 3.3: Local force transducers

10

Figure 3.4: West side (front). Approximate location of points, marked with big yellow stars, for application of the 1.5 kg impulse hammer for the whole structure. Approximate location of points, marked with small yellow stars for application of the 0.1 kg impulse hammer on the local force cells. (Arntsen, 2013)

3.2 Data Analysing Methods

[This section is directly extracted from the master’s thesis; Navaratnam (2013)]

A procedure used by Määtänen (1979) to resolve ice forces from measured response forces on

structures subjected to moving ice is applicable for wave slamming loads as well (Tørum, 2013).

The analysis method that Wienke and Oumeraci (2005) used was deconvolution method which is

similar to Duhamel integral method that was used by Ros Collados (2011). These deconvolution

11

and Duhamel integral approaches are more complex for truss structures and have not been used

so far for truss structure. So, the method used by Määtänen (1979), Frequency Response Function

method was used for both individual cylinders and truss structures. But, Duhamel integral method

also used for only individual cylinders in order to compare and check the influence of the analysis

methods.

The measured response force f(t) could be expanded into Fourier integral and in case of forced

vibration will be,

𝑓(𝑡) =1

2𝜋∫ 𝐻(𝜔)𝑆𝐹(𝜔)𝑒𝑖𝜔𝑡

∞

−∞

𝑑𝜔 (3.1)

Where, H(ω) is the frequency response function (FRF) and S(ω) is the linear spectrum of the

forcing function. The frequency response function H(ω) or the transfer function is a calibration

factor which is obtained by the pluck test by impulse hammer at several ponts. The Fourier

transform of equation (3.1) gives,

𝐻(𝜔)𝑆𝐹(𝜔) = ∫ 𝑓(𝑡) 𝑒−𝑖𝜔𝑡

∞

−∞

𝑑𝜔 = 𝑆𝑓(𝜔) (3.2)

Sf (ω) is the linear spectrum of the measured signal f(t). So, Sf (ω) can be solved from this above

equation as,

𝑆𝐹(𝜔) =

𝑆𝑓(𝜔)

𝐻(𝜔) (3.3)

Finally, the Inverse Fourier Transform (IFFT) of the above equation gives the requested wave

slamming force.

𝐹(𝑡) =1

2𝜋∫

𝑆𝑓(𝜔)

𝐻(𝜔) 𝑒𝑖𝜔𝑡

∞

−∞

𝑑𝜔 (3.4)

The above equations can easily be solved by computer programs such as Matlab, although they

look complicated. In this case Matlab has been used for the calculations and analyses.

Frequency Response Function (FRF)

As previously described, the frequency response function or transfer function was obtained by the

pluck test using impulse hammer. Plucking points are shown in The total response force due to an

impact by the impulse hammer can be sum of all the force transducers connected to the structure

assuming structure responding based on single degree of freedom (SDOF. The impact force is

directly measured by the impulse hammer. So, the ratio of the power spectrum of impulse force

to the response forces gives the transfer function or the frequency response function.

So, frequency response function is now,

12

𝐻(𝜔) =

𝑆𝑇𝑜𝑡𝑎𝑙,ℎ𝑎𝑚𝑚𝑒𝑟(𝜔)

𝑆𝐻𝑎𝑚𝑚𝑒𝑟(𝜔) (3.5)

Where, 𝑆𝑇𝑜𝑡𝑎𝑙,ℎ𝑎𝑚𝑚𝑒𝑟(𝜔) is the fast Fourier transform of the total response forces (power

spectrum) obtained by summing up all the transducer forces due to the impact by the hammer and

𝑆𝐻𝑎𝑚𝑚𝑒𝑟(𝜔) is the fast Fourier transform of the impact measurement obtained directly from

hammer.

𝑆𝑇𝑜𝑡𝑎𝑙,ℎ𝑎𝑚𝑚𝑒𝑟(𝜔) = ∫ 𝑓𝑇𝑜𝑡𝑎𝑙,ℎ𝑎𝑚𝑚𝑒𝑟(𝑡) 𝑒−𝑖𝜔𝑡

∞

−∞

𝑑𝑡 (3.6)

And,

𝑆𝐻𝑎𝑚𝑚𝑒𝑟(𝜔) = ∫ 𝑓𝐻𝑎𝑚𝑚𝑒𝑟(𝑡) 𝑒−𝑖𝜔𝑡

∞

−∞

𝑑𝑡 (3.7)

The frequency response function 𝐻(𝜔) is counter checked by multiplying it by𝑆𝐻𝑎𝑚𝑚𝑒𝑟(𝜔), this

should be equal to𝑆𝑇𝑜𝑡𝑎𝑙,ℎ𝑎𝑚𝑚𝑒𝑟(𝜔). So both the spectrum were checked in order to make sure

it has been done correctly.

Duhamel Integral Method

Duhamel integral approach has been used only for cylinder structures to compare with the results

with the FRF method. The theoretical description of the Duhamel integral method is briefly

described in this chapter. This method was used by Ros Collados (2011) in his master’s thesis.

Figure 3.5: The derivation of the Duhamel integral (Ros Collados, 2011)

13

The above figure (Figure 3.5) shows the differential response for a given impact p(τ). The total

calculated response can be obtained by integrating all the differential responses developed during

the loading history (Ros Collados, 2011).

𝑅𝑐(𝑡) = 𝑘

𝑚𝜔𝑑 ∫ 𝑝(𝜏)𝑠𝑖𝑛𝜔𝑑(𝑡 − 𝜏)𝑒−𝜉𝜔(𝑡−𝜏)𝑑𝜏

𝑡

0

(3.8)

Where, m is the oscillating mass, 𝜔𝑑 is the damped frequency of oscillation, p(τ) is the impact load

applied for very short time τ and 𝜉 is the damping coefficient and t is the time. It should be noticed

that for small values of damping ω ≈ 𝜔𝑑. Equation (3.8) is called as Duhamel integral equation

and this is being used to estimate the response of an undamped single degree of freedom (SDOF)

system subject to any form of dynamic loading p(τ). This equation can be simplified and written

as follows (Clough & Penzien, 1975)

𝑅𝑐(𝑡) = 𝐴(𝑡) sin 𝜔𝑑𝑡 − 𝐵(𝑡) cos 𝜔𝑑 𝑡 (3.9)

where,

𝐴(𝑡) =𝑘

𝑚𝜔𝑑∫ 𝑝(𝜏)

𝑒𝜉𝜔𝜏

𝑒𝜉𝜔𝑡cos 𝜔𝑑𝜏 𝑑𝜏

𝑡

0

(3.10)

𝐵(𝑡) =𝑘

𝑚𝜔𝑑∫ 𝑝(𝜏)

𝑒𝜉𝜔𝜏

𝑒𝜉𝜔𝑡sin 𝜔𝑑𝜏 𝑑𝜏

𝑡

0

(3.11)

The incremental summation procedure can be used to evaluate the above given integral equations.

The equation (3.10) can be written as below in order to describe the exponential decay behaviour

caused by damping. This is an approximate recursive form using simple summation.

𝐴𝑁 ≈ 𝐴𝑁−1𝑒−𝜉𝜔∆𝜏 +

∆𝜏 𝑘

𝑚𝜔𝑑𝑦𝑁−1𝑒−𝜉𝜔∆𝜏 , 𝑁 = 1,2,3, … (3.12)

where, 𝑦1 = 𝑝1 cos 𝜔𝑑𝑡1 , 𝑦2 = 𝑝2 cos 𝜔𝑑𝑡2, etc.

The same expressions will be applicable for 𝐵𝑁 but, now 𝑦𝑁 is in terms of sin 𝜔𝑑𝑡𝑁 , i.e. 𝑦1 =

𝑝1 sin 𝜔𝑑𝑡1 , 𝑦2 = 𝑝2 sin 𝜔𝑑𝑡2 and so on.

Finally, knowing all the calculated values of 𝐴𝑁 and 𝐵𝑁 for successive values of N, the

corresponding ordinates of the response will be obtained by using equation (3.9).

𝑅𝑐𝑁 = 𝐴𝑁 sin 𝜔𝑑𝑡𝑁 − 𝐵𝑁 cos 𝜔𝑑 𝑡𝑁 (3.13)

Although these expressions and procedure look more complex, it can be easy evaluated by the

Matlab program. A Matlab code written by Ros Collados (2001) was modified according to the

requirement. The main steps involving in this Duhamel integral method is shown in Figure 3.6.

This method was only used for individual local force meters and was not used for truss structure

sections.

14

Figure 3.6: Main steps involving in the Duhamel integral approach (Ros Collados, 2011)

This is an iterative process as the assumed impact force and the measured responses should be in

good agreement with each other. It means that the measures responses and calculated responses

should be coincided with each other or almost geometrically fit on to another for a particular

triangular impact force. Once these two responses are in agreement the impact force corresponds

to that response will be the wave slamming force.

4 ANALYSIS OF DATA

4.1 Total Force Analysis

Wave Test ‘2013061424’ with hammer test ‘2406201319’ and ‘Large-hammer-

test-2013_06_24_18_42_58’.

Wave test ‘2013061424’ has been chosen for all the analyses. The characteristics of the wave test

‘2013061424’ is given in Table 4.1.

Table 4.1: Characteristics of test ‘2013061424’

Wave Period 5.55s

Deep water wave height 1.70m

Wave height at structure 2.08m

Water depth at structure 4.3m

Wave type Regular

The hammer test ‘2406201319’ was done using large hammer and that was plucked at point 2 as

shown in Figure 3.4.

The following figure shows the hammer impulse of this test. There are two subsequent impacts

and the first impact has been chosen for this analysis. The hammer impulse data before and after

15

the impact has been adjusted to zero in order to get the clean spectrum. Figure 4.2 shows the total

response due to the hammer impact.

Figure 4.1: Hammer Impulse

Figure 4.2: The total response of hammer impulse

It should be noted that the total response is the sum of forces from four total force transducers,

two at the top and two at the bottom. The spectrum of both hammer impulse and the total

response are shown in Figure 4.3 ,

0 0.5 1 1.5 2 2.5 3-2

0

2

4

6

8

10

12

Time [s]

For

ce [k

N]

Test: Hammer-2406201335 at Point 5 Hammer Force

8 8.5 9 9.5 10 10.5 11 11.5-1.5

-1

-0.5

0

0.5

1

1.5

2

Time [s]

Fo

rce

[kN

]

Test: Hammer-2406201319 at Point 2 Total Hammer Response

16

Figure 4.3: Spectrum of hammer and response forces

Figure 4.4 shows a series of measured total responses while Figure 4.5 shows the deep-water wave

and the waves measured at the structure, i.e. at the front section of the structure.

Figure 4.4: Series of measured total responses

As it can be seen in Figure 4.4, the total response varies considerably from wave to wave. The

response at a time approximately 128s has been chosen for the analyses.

0 10 20 30 40 50 60 700

10

20

30

40

50

60

70

80

90

Frequency [Hz]

Re

lativ

e V

alu

es

Test: Hammer-2406201319 at Point 2 Data Points: 78629-93145 Power Spectrum

Hammer Force

Total Force

70 80 90 100 110 120 130 140-10

-5

0

5

10

15

20

25

30

35

Time [s]

Fo

rce

[kN

]

Test: Hammer-2406201335 at Point 5 Total Measured Response

Total Response

Wave at Structure

17

Figure 4.5: Deep water wave and wave at structure [wave period – 5.55s]

As it was described in the previous chapter, the total force response is sum of all four transducers

and this total response consists of quasi-static forces and dynamic forces. Basically, the slamming

force comes from the dynamic forces. So, it’s important to decompose of the total responses.

Low-pass filtering does this decomposition, as it’s shown in Figure 4.6 filtered total force will be

the quasi-static force and the dynamic force will be obtained by subtracting quasi-static force from

the total response.

Figure 4.6: Decomposition of total response

70 80 90 100 110 120 130 140-1

-0.5

0

0.5

1

1.5

2

Time [s]

Wav

e H

eigh

t [m

]Test: 2013061424 Data Points: 499798-529637 Waves

Deep water wave

Wave at structure

128.2 128.4 128.6 128.8 129 129.2 129.4

-10

-5

0

5

10

15

20

25

30

Time [sec]

For

ce [k

N].

Test: 2013061424 Data Points: 499798-529637 Forces

Filtered Total Force

Total Force-Filtered

Total Force

18

Figure 4.7: Dynamic Response

Figure 4.8: Spectrum of dynamic response

128.3 128.4 128.5 128.6 128.7 128.8 128.9 129 129.1 129.2 129.3

-10

-5

0

5

10

15

20

25

Time [sec]

Fo

rce

[kN

].

Test: 2013061424 Data Points: 499798-529637 Forces

0 10 20 30 40 50 60 70 0

2000

4000

6000

8000

10000

12000

14000

Frequency [Hz]

Rela

tive V

alu

es

Test: 2013061424, Hammer-2406201319 at Point 2 Data Points: 499798-529637 Power Spectrum of Response Forces

19

Figure 4.9: Low-pass filtered slamming force variation (with the cut-off frequency of 200Hz)

According to the above figure, the measured slamming force for this test will be about 14kN.

Although the above figure shows many high forces for a cut-off frequency of 200Hz, it can be

eliminated if we use lower cut-off frequency for filtering. This will be illustrated in the following

section.

The same test with different hammer point (Hammer point 5 – test ‘2406201335’ )

The same wave test has been analysed using different hammer point that is point 5 shown in Figure

3.4.

Figure 4.10: Hammer impact at point 5.

0 0.5 1 1.5 2 2.5 3-2000

0

2000

4000

6000

8000

10000

12000

Time [s]

Forc

e [k

N]

Test: Hammer-2406201335 at Point 5 Hammer Force

127.6 127.8 128 128.2 128.4 128.6 128.8 129 129.2 129.4

-5

0

5

10

15

Data Points

Forc

e [kN

].

Test: 2013061424, Hammer-2406201319 at Point 2 Data Points: 499798-529637 Filtered IFFT(S(w)/H(w))=SS(w)

20

Figure 4.11: Total hammer response

Figure 4.12: Chosen hammer impulse

6.5 7 7.5 8 8.5 9 9.5

-1.5

-1

-0.5

0

0.5

1

1.5

Time [s]

Fo

rce

[kN

]

Test: Hammer-2406201335 at Point 5 Total Hammer Response

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 10-3

0

1

2

3

4

5

6

7

8

Time [s]

Fo

rce

[kN

]

Test: Hammer-2406201335 at Point 5 Hammer Impulse

21

Figure 4.13: Spectrum of the hammer force

Figure 4.14: Spectrum of both hammer force and total force

0 10 20 30 40 50 600.205

0.2055

0.206

0.2065

0.207

0.2075

0.208

Frequency [Hz]

Re

lativ

e V

alu

es

Test: Hammer-2406201335 at Point 5 Data Points: 66290-79194 Power Spectrum-Hammer Forces

0 10 20 30 40 50 60 700

20

40

60

80

100

120

Frequency [Hz]

Rel

ativ

e V

alue

s

Test: Hammer-2406201335 at Point 5 Data Points: 66290-79194 Power Spectrum

Hammer Force

Total Force

22

Figure 4.15: Un-filtered slamming force

Figure 4.16: Low-pass filtered slamming force (with the cut-off frequency of 200Hz)

127.4 127.6 127.8 128 128.2 128.4 128.6 128.8 129 129.2 129.4 -10

-5

0

5

10

15

Time [s]

Forc

e [kN

].

Test: 2013061424, Hammer-2406201335 at Point 5 Data Points: 499395-536089 IFFT(S(w)/H(w))=SS(w)

127.4 127.6 127.8 128 128.2 128.4 128.6 128.8 129 129.2 129.4 -10

-5

0

5

10

15

Time [s]

Fo

rce [

kN

].


23

Figure 4.17: Low-pass filtered slamming force (with the cut-off frequency of 40Hz)

According to Figure 4.16 & Figure 4.17, the measured slamming force is approximately 8-13 kN.

Calculated Slamming Force based on Wienke & Oumeraci (2005),

Slamming forces on two front vertical legs = 2 [0.5 ∗ 2𝜋 𝜌𝑤𝐷 λ 𝜂𝑏

𝐶𝑏2]

Wave celerity can be calculated by using the formula, 𝐶𝑏 = √𝑔(ℎ + 𝜂) = √9.81(2 + 1.4)

= 5.77 m/s

So, the slamming forces on two front vertical legs = 2 [0.5*2π*1000*0.14*0.5*1.4*5.772 ]

= 19.99 kN

Length of bracing within the wave impact area = 2 [λ 𝜂𝑏/𝑆𝑖𝑛𝜃]

where, 𝜃 is the inclination of the bracing.

= 2*[0.5*1.4/Sin27.8]

= 3m

Slamming forces on front bracing = 0.5*2π*1000*0.14*3*5.772

= 43.91 kN

So, total slamming force = 63.90 kN

127.4 127.6 127.8 128 128.2 128.4 128.6 128.8 129 129.2 129.4 -4

-2

0

2

4

6

8

10

Time [s]

Fo

rce

[kN

].


24

As we see from the above calculation, the calculated slamming force based on Wienke & Oumeraci

(2005) is about 64 kN. It should be noted that the calculated slamming force on the bracing is

much more larger than that on two front legs as the length of the bracing that exposed to the

breaking wave is larger.

So, according to the above calculations, the measured slamming force is very less than that we

obtained based on Wienke & Oumeraci (2005).

4.2 Local Force Transducers

The hammer test on local force transducer FTLF02 (Figure 3.3) was chosen for analysis.

Figure 4.18: Hammer impulse and Response on FTLF02

0.88 0.89 0.9 0.91 0.92 0.93 0.94

0

100

200

300

400

500

600

Time [s]

Fo

rce

[N]

Test: FTLF02-2 Forces

Hammer Impulse

Response Force

25

Figure 4.19: Time expanded view of above figure

If we look into Figure 4.19 response occurs ahead of the impact, apparently this is impossible. It

was found that the reason for such time lag is related to the hardware system during

analogue/digital conversion. This time lag will not affect the results although it makes confusion

to the reader.

The spectrum of the hammer impulse and the total response is shown in the following figures.

Figure 4.20: Spectrum of both hammer impulse and the response

0.873 0.874 0.875 0.876 0.877 0.878 0.879

0

100

200

300

400

500

600

Time [s]

Fo

rce

[N]

Test: FTLF02-2 Forces

Hammer Impulse

Response Force

0 200 400 600 800 1000 1200 1400 1600 1800 2000 0

0.5

1

1.5

2

2.5

3 x 10 4

Frequency [Hz]

Re

lative

Va

lue

s

Test: FTLF02-2 Data Points: 8315-10502 Power Spectrum-Hammer Forces

Hammer Impulse Response Force

26

Figure 4.21: Spectrum of the response forces

Comments on the spectrum shown in Figure 4.21,

- Decreasing high force until 600Hz due to the clean response signal (similar to the spectrum for

the hammer impulse signal)

- As we see on Figure 4.21, there are two clean peaks at 628 Hz and 1202Hz, this could be due to

the small oscillation in between two peaks in Figure 4.18.

According to Figure 4.19,

The impact duration (td) = 0.0029 s

The natural period of oscillation (tn) = 0.0016s (based on 628Hz)

Ratio td/tn = 1.8125

0 200 400 600 800 1000 1200 1400 1600 1800 2000 0

0.5

1

1.5

2

2.5

3 x 10

4

Frequency [Hz]

Rela

tive V

alu

es

Test: FTLF02-2 Data Points: 8315-10432 Power Spectrum-Response Forces

27

Figure 4.22: The maximum response to a suddenly applied triangular force time history (Naess,

2011)

If the hammer impact shown in Figure 4.19 can be approximated to a symmetric triangular impact,

the response ratio has to be about 1.2 for a time ratio of 1.8., but if we look in to the Figure 4.19,

the measured response ratio (measured response/impact force) is about 1.08 (650N/600N). This

difference could be due to the difference in theoretical impact duration [~0.01s = (0.5*0.14/6)]

and the measured impact duration of hammer impact (about 0.003s) from Figure 4.19.

Duhamel integral approach:

4.2.1.1 Test 2013061424 for Local Force Transducer FTLF02

The Duhamel integral approach has been used to the local force transducer FTLF02 shown in

Figure 3.3, in this case the maximum response of this local force transducer has been chosen for

the analysis and the results are shown in the following figures. An impact duration of 0.01s has

been considered for this analysis as the theoretical impact duration is 0.013s.

It should be noted that the rising time of the impact is denoted as Tp although Tp is used fir peak

period.

28

Figure 4.23: Duhamel integral approach with impact duration 0.01s and rising time 0.00008s

When impact duration is 0.005s, the following figure shows the result,


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014-100

0

100

200

300

400

500

600

Time [s]

Re

lative

re

sp

on

se

[N

]

Triangular Impulse

Calculated Response

Measured ResponseF0= 303 N

Tp= 0.00008s

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014-100

0

100

200

300

400

500

600

Time [s]

Re

lativ

e r

esp

on

se [N

]

Triangular Impulse

Calculated Response

Measured ResponseF0=315 N

Tp=0.00008s

29

When the rising time increases,



0 0.002 0.004 0.006 0.008 0.01 0.012 0.014-100

0

100

200

300

400

500

600

Time [s]

Re

lativ

e r

esp

on

se [N

]

Triangular Impulse

Calculated Response

Measured ResponseF0= 308 N

Tp= 0.0001s

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014-100

0

100

200

300

400

500

600

Time [s]

Re

lativ

e r

esp

on

se [N

]

Triangular Impulse

Calculated Response

Measured ResponseF0=365 N

Tp= 0.0005s

30


4.2.1.2 Measured Slamming Factor

As we see in the above figures, when the rising time increases, the impact force also increases and

finally it becomes very hard to fit the measured response curve as it lags behind it.

According to Figure 4.23, the measured Cs value will be,

Measured slamming factor [Cs] = Fs/[0.5 𝜌𝑤𝐷 ℎ 𝐶𝑏2]

= 303/ [0.5*1000*0.14*0.04*5.772]

= 3.33

As we see from the above calculation, the slamming factor is very small compared to the slamming

factor 2π that Wienke & Oumeraci (2005) used.

Since the maximum total response occurred at the FTLF02 and the measured slamming force also

found to be highest compared to the slamming force that has been obtained from the other local

force transducers (Figure 4.32 & Figure 4.33 ), the above slamming factor will be larger than the

slamming factor that can be obtained from other local forces

FRF Approach

Now we consider the Frequency Response Function method for the same local force transducer

FTLF02. Hammer tests that were done on FTLF02 has been used in all the analysis in order to

compare the results of both the analysing methods, Duhamel integral approach and frequency

response function method.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

100

200

300

400

500

600

Time [s]

Re

lative

re

sp

on

se

[N

]

Triangular Impulse

Calculated Response

Measured ResponseTp=0.0008s

31

Figure 4.28: Un-filtered force variation on FTLF02

Figure 4.29: Filtered force variation on FTLF02 [Cut-off frequency- 200Hz]

100.35 100.4 100.45 100.5 100.55 100.6 100.65 100.7 100.75 100.8 100.85-600

-400

-200

0

200

400

600

800

Time [s]

Fo

rce

[N].

Test: 2013061424-FTLF02, FTLF02-2 Data Points: 227694-237335 IFFT(S(w)/H(w))=SS(w)

100.35 100.4 100.45 100.5 100.55 100.6 100.65 100.7 100.75 100.8 100.85-50

0

50

100

150

200

Time [s]

Fo

rce

[N

].

Test: 2013061424-FTLF02, FTLF02-2 Data Points: 227694-237335 Filtered IFFT(S(w)/H(w))=SS(w)

32



As it can be seen in above figures, if we expect the slamming force to be very clean as theoretical

then the Figure 4.29 is the most suitable situation where the slamming force is about 180N that is

less than that we obtained from the Duhamel integral approach (303N). Although it can always

be expected to get force variation shown in Figure 4.29, still the slamming force shown in Figure

4.31 is acceptable and in this case the slamming force is about 315N which is almost similar to

what we obtained from the Duhamel integral approach.

100.35 100.4 100.45 100.5 100.55 100.6 100.65 100.7 100.75 100.8 100.85-100

-50

0

50

100

150

200

250

300

Time [s]

For

ce [N

].


100.35 100.4 100.45 100.5 100.55 100.6 100.65 100.7 100.75 100.8 100.85-50

0

50

100

150

200

250

300

350

Time [s]

For

ce [N

].


33

Figure 4.32: Time series of measures response of local force transducers [FTLF01-FTLF05]

Figure 4.33: Time series of measures response of local force transducers [FTLF06-FTLF10]

70 80 90 100 110 120 130 140-0.5

0

0.5

Forc

e [

kN

]

FTLF01

70 80 90 100 110 120 130 140-1

0

1

Forc

e [

kN

]

FTLF02

70 80 90 100 110 120 130 140-0.5

0

0.5

Forc

e [

kN

]

FTLF03

70 80 90 100 110 120 130 140-1

0

1

Forc

e [

kN

]

FTLF04

70 80 90 100 110 120 130 140-0.5

0

0.5

Time [s]

Forc

e [

kN

]

FTLF05

70 80 90 100 110 120 130 140-0.5

0

0.5

Forc

e [

kN

]

FTLF06

70 80 90 100 110 120 130 140-1

0

1

Forc

e [

kN

]

FTLF07

70 80 90 100 110 120 130 140-1

0

1

Forc

e [

kN

]

FTLF08

70 80 90 100 110 120 130 140-0.1

0

0.1

Forc

e [

kN

]

FTLF09

70 80 90 100 110 120 130 140-0.1

0

0.1

Time [s]

Forc

e [

kN

]

FTLF10

34

Figure 4.34: Time expanded view of local force transducer responses

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.2

0

0.2

Time [s]

Forc

e [

kN

]

FTLF01

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.2

0

0.2

Time [s]

Forc

e [

kN

]

FTLF02

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.2

0

0.2

Time [s]

Forc

e [

kN

]

FTLF03

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.5

0

0.5

Time [s]

Forc

e [

kN

]

FTLF04

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.2

0

0.2

Time [s]

Forc

e [

kN

]

FTLF05

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.2

0

0.2

Time [s]

Forc

e [

kN

]

FTLF06

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.5

0

0.5

Time [s]

Forc

e [

kN

]

FTLF07

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.5

0

0.5

Time [s]

Forc

e [

kN

]

FTLF08

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.1

0

0.1

Time [s]

Forc

e [

kN

]

FTLF09

128.1 128.2 128.3 128.4 128.5 128.6 128.7-0.1

0

0.1

Time [s]

Forc

e [

kN

]

FTLF10

35

Figure 4.32 and Figure 4.33 show the time series of the measured response from each local force

transducer for test ‘2013061424’. As we see in these figures, the measured response is different in

each local force transducer at a particular time. Figure 4.34 clearly shows that the maximum

response occurs at different time points in each local force transducers. For example, consider

FTLF03 and FTLF07, both are located at the same height from the still water line but on different

legs, FTLF03 gives the maximum response at 128.365s whereas FTLF07 gives the maximum

response at about 128.38s. This confirms that the wave is not so uniform across the channel.

Although this time difference is small, this will be significant when it’s compared to the impact

duration. FTLF01 and FTLF05 are located at about 1.45m above the still water line, the crest

height of this wave is about 1.4m, but we still get some response on FTLF01 and FTLF05, this

could be due to the wave run-up also another reason might be the wave recorded close to channel

wall and at some distance from the structure.

It’s interesting to note that Figure 4.34 shows that the forces on FTLF01, FTLF02 and to some

extent FTLF03, FTLF05 and FTLF06 are somewhat ahead of the forces on the other local force

transducers, except FTLF09 and FTLF10. This may be due to curling over of the wave crest, as

indicated in Figure 2.4.

As we can see in Figure 4.34, there are two peaks in some local force transducers, this might be

due to the wave run-up, but it should be investigated further.

The maximum response on the total force analysis was occurred at about 128.3 sec of the test

‘2013061424’ (Figure 4.4). So, the responses at this time in each local force transducers will be

considered for the analysis in order to compare the total forces from the total force transducer

with the total force from the local force transducers. Please note that the hammer test on the

FTLF02 is only considered for all the analysis assuming the hammer response behaviour is the

same for all local transducers.

Figure 4.35: The dynamic part of the total response on FTLF02

128.3 128.4 128.5 128.6 128.7 128.8

-60

-40

-20

0

20

40

60

80

Time [sec]

Fo

rce

[N].

Test: 2013061424-FTLF02 Data Points: 505847-517540 Dynamic Response Forces

36

Figure 4.36: Power spectrum of the total response forces

Figure 4.37: Force variation on FTLF01 [Cut-off frequency 200Hz]

It should be noted that the values (x-values) shown in the boxes in each figures are round off to

first decimal.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.5

1

1.5

2

2.5x 10

4

Frequency [Hz]

Re

lativ

e V

alu

es

Test: 2013061424-FTLF01, FTLF02-2 Data Points: 506250-517944 Power Spectrum of Response Forces

128.05 128.1 128.15 128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55-40

-20

0

20

40

60

80

100

X: 128.4

Y: 82.89

Time [s]

Fo

rce

[N].


37

Similarly the force variation from the other local force transducers are given below.



128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55 128.6 128.65-30

-20

-10

0

10

20

30

40

50

60

70

X: 128.4

Y: 66.26

Time [s]

Fo

rce

[N].


128.15 128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55 128.6-60

-40

-20

0

20

40

60

80

100

120

X: 128.4

Y: 113.7

Time [s]

Fo

rce

[N].


38



128.15 128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55 128.6-50

0

50

100

150

X: 128.4

Y: 144.4

Time [s]

Fo

rce

[N].


128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55 128.6 128.65-30

-20

-10

0

10

20

30

40

50

60

X: 128.4

Y: 50.03

Time [s]

Fo

rce

[N].


39



128.15 128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55 128.6-30

-20

-10

0

10

20

30

40

50X: 128.4

Y: 42.78

Time [s]

For

ce [N

].


X: 128.4

Y: 37.96

128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55 128.6 128.65-50

0

50

100

150

200

X: 128.4

Y: 164.9

Time [s]

Fo

rce

[N].


40



128.15 128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55 128.6-100

-50

0

50

100

150

200

250

X: 128.4

Y: 207.7

Time [s]

Fo

rce

[N].


128.05 128.1 128.15 128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55-20

-15

-10

-5

0

5

10

15

20

25

X: 128.4

Y: 20.3

Time [s]

Fo

rce

[N].


41


As it can be seen in all the above figures, the maximum force occurred at around 128.355s and the

local force transducers FTLF09 and FTLF10 give very less or almost no force at this time as its

located away from the slamming point. The following table shows the summary of the forces of

each transducers. Please note that the z is the height above the still water line, which means that z

is zero at the still water line.

Table 4.2: Measured forces on each local force transducers [based on FRF method]

FTLF Z (mm) Meas. Force

(N)

Front Leg 1 1 1450 82.89

2 1300 66.26

3 1150 113.7

4 1063 144.4 407.25

Front Leg 2 5 1450 50.03

6 1300 37.96

7 1150 164.9

8 1063 207.7

9 377 20.3

10 250 20.3 501.19

908.44

128.05 128.1 128.15 128.2 128.25 128.3 128.35 128.4 128.45 128.5 128.55-20

-15

-10

-5

0

5

10

15

20

25

X: 128.4

Y: 20.33

Time [s]

Fo

rce

[N].


42

Based on the above table, the force intensity has been plotted against the location each transducers

which is shown in the following figure.

Figure 4.47: The variation of the force intensity with the depth [based on FRF method] at the time of maximum total response, t=128.3s

The response from all individual local force transducers at 128.3s has been analysed using Duhamel

integral approach. Few of them (those give maximum forces) shown in the following figures.

Figure 4.48: Duhamel integral approach for FTLF03 (response at about 128.3s)

0

200

400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6

Z [m

m]

fs [N/mm]

Front Leg 1

Front Leg 2

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014-50

0

50

100

150

200

Time [s]

Re

lativ

e r

esp

on

se [N

]

Triangular Impulse

Calculated Response

Measured ResponseF=99 N

43



0 0.002 0.004 0.006 0.008 0.01 0.012 0.014-50

0

50

100

150

200

250

300

350

Time [s]

Re

lativ

e r

esp

on

se

[N

]

Triangular Impulse

Calculated Response

Measured ResponseF= 186N

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014-50

0

50

100

150

200

250

300

350

Time [s]

Re

lative

re

sp

on

se

[N

]

Triangular Impulse

Calculated Response

Measured ResponseF=182N

44


Table 4.3: Measured forces on each local force transducers [based on Duhamel integral approach]

FTLF Z (mm) Meas. Force

(N)

Front Leg 1 1 1450 81

2 1300 72

3 1150 99

4 1063 186 438

Front Leg 2 5 1450 57

6 1300 59

7 1150 182

8 1063 265

9 377 39

10 250 31 633

1071

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014-100

0

100

200

300

400

500

Time [s]

Re

lative

re

sp

on

se

[N

]

Triangular Impulse

Calculated Response

Measured ResponseF= 265N

45

Figure 4.52: The variation of the force intensity with the depth [based on Duhamel integral approach] at the time of maximum total response, t=128.3s

According to both the analysing methods, the maximum slamming force seems to have occurred

at about 1.05m above the still water line or at the local force transducers FTLF04 and FTLF08.

The slamming force distribution is triangular as found by Sawaragi and Nochino (1984) and Ros

(2011) and not uniform as used by Goda et al. (1966) and Wienke and Oumeraci (2005).

Figure 4.53: The response of FTLF02 (blue) and the wave at the structure (red)

0

200

400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7

Z [m

m]

fs [N/mm]

Front Leg 1

Front Leg 2

46

As we see in Figure 4.53, the maximum wave crest occurs at about 122.9s and the maximum

response occurs at 122.75s, this indicates that there is a time phase shift which is still unclear and

needs to be investigated further.

5 CONCLUSION AND RECOMMENDATIONS

There were few data analysed and presented in this report. Initially the total force on the structure

was analysed using Frequency Response Function (FRF) method. The wave test ‘2013061424’ was

chosen for all the analysis. In the total force analysis it was observed that the measured slamming

force is very less compared to the calculated slamming force based on Wienke & Oumeraci (2005).

In the analysis of the local force measurement, there were two methods used such as Duhamel

integral approach and the frequency response function method. Both of the analysing methods

seem to be promising as they both give almost similar result for a particular test. The slamming

factor, Cs is small (3.3) according to the measured slamming force whereas the slamming factor

used in the Wienke & Oumeraci (2005) method was 2π, which is much larger than the obtained

slamming factor. The forces acting on different level of each front vertical legs have been found

and plotted against the depth. This shows almost the similar pattern that Ross (2011) obtained in

his tests. It was also observed from the local forces that the same wave hits the structure at different

times but the time difference is very small.

Apelt and Piorewicz (1987) who point out that there is a size effect (varying D/H). Their results

indicate that we should have smaller forces than obtained by Wienke and Oumeraci (2005). Wienke

and Oumeraci (2005) had a diameter of 0.70 m, while we had diameters D = 0.14 m. According

to Apelt and Piorewicz (1987), we should have relatively smaller forces than Wienke and Oumeraci

(2005).

There was a time lag observed between wave recording and the response data, still it’s unclear and

has to be investigated. Also, there were two peak points observed on the local force transducer

recording, this could be due to the wave run-up on the structure, but still it needs to be further

investigated.

Further detailed analysis with different analysis methods would be recommended to overcome

such problems and to come up with better conclusions.

47

6 REFERENCES

Aashamar, M. (2012). Wave slamming forces on truss support structures for wind turbines.

NTNU, Department of Civil and Transport Engineering. Trondheim: Norwegian

University of Science and Technology.

Apelt, C., & Piorewicz, J. (1987). Laboratory Studies of Breaking Wave Forces Acting on

Vertical Cylinders in Shallow Water. Coastal Engineering, 11, 263-282.

Arntsen, Ø. (2013). WAVE SLAMMING FORCES ON TRUSS STRUCTURES IN SHALLOW

WATER. Trondheim.

Aune, L. (2011). Bølgjeslag mot jacket på grunt vatn (Wave slamming forces on jacket in

shallow water). Trondheim: Department of Structural Engineering.

Battjes, J. (1974). Surf similarity. Proceedings of 14th International Conference on Coastal

Engineering, 1, pp. 467-479. Copenhagen.

Chella, M., Tørum, A., & Myrhaug, D. (2012). An Overview of Wave Impact Forces on

Offshore Wind. Energy Procedia, 20, 217 – 226.

Clough, R., & Penzien, J. (1975). Dyanamics of structure (2nd ed.). International Editions.

Retrieved 1993

Endresen, H., & Tørum, A. (1992). Wave forces on a pipeline through the surf zone. Coastal

Engineering, 18, 267-281.

Goda, Y., Haranka, S., & Kitahata, M. (1966). Study of impulsive breaking wave forces on

piles. Port and Harbour Research Institute, 5(6), 1-30.

Irschik, K., Sparboom, U., & Oumeraci, H. (2002). Breaking wave characteristics for the

loading of a slender pile. Cardiff: ASCE.

Judith, B., & Marcel, S. J. (2012). Coastal Dynamics I. Delft: VSSD, Delft University of

Technology.

Morison, J., O'brien , M., Johnson, J., & Schaaf, S. (1950). The forces exerted by surface waves

on piles. Jounal of Petroleum Technology, Petroleum transactions, AMIE, 189, 149-

154.

Naess, A. (2011). An introduction to random vibrations. Trondheim: Centre for Ships and

Ocean Structures, NTNU.

Navaratnam, C. (2013). Wave slamming forces on truss structure for wind turbine structures.

Trondheim: Norwegian University of Science and Technology.

Ros Collados, X. (2011). Impact forces on a vertical pile from plunging breaking waves.

Trondheim: Norwegian University of Science and Technology.

48

Sawaragi, T., & Nochino, M. (1984). Impact forces of nearly breaking waves on a vertical

circular cylinder. Coastal Engineering, 27, 249 – 263.

Tanimoto, K., Takahashi, S., Kaneko, T., & Shiota, K. (1986). Impulsive breaking wave forces

on an inclined pile exerted by random waves. Proceedings of 20th International

Conference on Coastal Engineering, (pp. 2288-2302).

Tørum, A. (2013). Analysis of force response data from tests on a model of a truss structure

subjected to plunging breaking waves. Department of Civil and Transport Engineering.

Trondheim, Norway: Norwegian University of Science and Technology.

von Karman, T. (1929). The impact on seaplane floats during landing. Washington: National

Advisory Committee for Aeronautics.

Wagner, H. (1932). Über stoss-und gleitvorgänge an der oberfläche von flüssigkeiten.

Zeitshrift für Angewandte Mathmatik und Mechanik, 12(4), 193-215.

Wienke, J., & Oumeraci, H. (2005). Breaking wave impact force on a vertical and inclined

slender pile—theoretical and large-scale model investigations. Coastal Engineering ,

52, 435 – 462.

49

APPENDICES

- 14 -

Attachment I - Dimensions of the local force transducer sections (Pole 1,2,3)

- 15 -

Pole

320

150

42

42

42

42

150

87

151

40

1

40

40

40

139.7

820

896

38

38 42 50

42 337

130

108

45

108

140

z = 770 mm

z = 1063mm

z = 1150mm

z = 1300mm

z = 1450mm

Z=0 mm at MWL Drawing not in scale

180

180

z = 950mm

113

1180

z = 1770mm

z = 1950mm

140

40

40

Joint steel

2

2 40

Joint steel

50

Truss structure

Pole 1 Pole 2

Pole 3

Torque on connection rods: 81 Nm each

- 16 -

Pole

320

150

42

42

42

42

150

87

151

40

2

40

40

40

139.7

820

896

38

38 42 50

42 337

130

108

45

108

140

z = 770 mm

z = 1063mm

z = 1150mm

z = 1300mm

z = 1450mm


180

180

z = 950mm

113

1180

z = 1770mm

z = 1950mm

140

40

40

Joint steel

Joint steel

2

2 40

50 207

Truss structure

Pole 1 Pole 2

Pole 3


- 17 -

Pole

113

127

618

3

40

139.7

820

896

38

38 42 50

42 130

130

85

467

140

z = -510 mm

z = 250mm

z = 377mm


180

180

92

1180

z = 490 mm

z = 670mm

140

40

40

Joint steel

Joint steel

2

2 40

50 z = 398 mm 40

42

42

z = -330 mm

z = -238 mm

488


Truss structure

Pole 1 Pole 2

Pole 3

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