## Application To Fixed Offshore Structures

The wave loads on fixed offshore structures are normally evaluated by means of one of two basic methods: a) Design wave method

Fig. 1-70 Jonswap and modified Pierson-Moskowitz wave spectra.

b) Spectral analysis method

The method to be used should be considered in each specific case with due regard to the design and purpose of calculation.

### Design Wave Method

Until now, most analysis regarding wave loads on fixed offshore structures has been based on a regular, sinusoidal design wave of given height and period. The wave height to be applied is evaluated from wave statistics for the area of interest, taken as the height of the 50 or 100 year wave, which may be determined with a reasonable degree of accuracy. The wave period is, however, more complicated to determine.

Normally, the principle that the period should correspond to the worst case loading is adopted as a guideline. This implies that the loads have to be calculated for several wave periods in order to establish the most severe case. A problem may appear, however, when considering total horizontal force on a large volume structure such as the one shown in Figure 1-71, The force is often found to increase continuously with increasing wave period towards a peak value far above the T = 20 seconds generally adopted as the upper limit for the period. The transfer function for horizontal force on the structure in Figure 1-71 is shown in Figure 1-72. The force is seen to attain its maximum magnitude at a wave period of about 30 seconds.

The difficulties of establishing a design criterion for the wave period are in this case obvious. To apply the maximum force with no regard for the period would lead to unrealistic results in many cases. Further investigation is necessary to clarify this problem, but until better knowledge of the subject is obtained, a 20 second limit derived from energy considerations seems appropriate.

Another effect which is important to consider is that maximum force and overturning moment may be obtained for different wave periods. This implies that the design wave period giving the most unfavorable conditions ought to be chosen separately foT each response. Generally, it may be stated that the overturning moment will achieve its maximum for lower periods than the horizontal force. The transfer function for overturning moment on the structure in Figure 1-71 is shown in Figure 1-73.

### Spectral Analysis Method

Spectral analysis of a fixed offshore structure can be justified only if the transfer functions for the responses can be established with a reasonable degree of accuracy. This implies that the nonlinear loads (for instance drag forces) have to be small com-

Fig. 1-71 Total horizontal force on large volume structure

Fig. 1-72 Transfer function-horizontal force.

pared to the linear loads (for instance inertia forces). If not, no well defined transfer function exists, and any result whatsoever could be obtained depending on the transfer function chosen. Linearization of nonlinear loads should be done with care and with due regard to their importance to the overall loads.

It should be emphasized that, even if a transfer function is established, the conditions may be very different from those applying to ships and other floating structures. The main differences are related to the shape of the transfer functions. Ship responses can normally be described by transfer functions having a well defined peak value at relatively low wave periods while this is not necessarily true for fixed offshore structures. The horizontal force on the structure in Figure 1-71 may be described by the transfer function in Figure 1-72, which is seen to increase continuously towards a peak value at about T — 30 seconds.

Combining this transfer function with the wave spectrum will give increasing values with increasing peak period of the spectrum far above T = 20 seconds. Consequently, careful consideration must be given to the location of the wave spectrum along the frequency axis when performing short term analysis.

F : HORIZONTAL FORCE H

a; WAVE AMPLITUDE

### 0.2 0.4 0 6 0 8

• jlra.d/s) -p.
• 1—I—tt—I-1-1-1--1-r

Fig. 1-73 Transfer function-overturning moment.

The choice of the period is to a great extent avoided, however, when carrying out a long term prediction according to the method normally used in ship design.11 The long term prediction is obtained by summing up a great number of short term statistically stationary conditions, each combined with a specified probability of occurrence. Even if the short term conditions obtained by locating the peak period of the spectrum above 20 seconds give unrealistic results, they will be associated with low probabilities of occurrence and consequently have little influence on the long term predictions.

The problems involved in spectral analysis are seen to be many, especially when only short term predictions are desired. Further investigatios are definitely needed to establish reasonable design criteria, Meanwhile, it seems appropriate to apply the 20 second limit to the peak period of the spectrum even in this case.

Conclusions—Fixed Structures

The analysis of wave forces may be based on the design wave method or on the spectral analysis method. Spectral analysis is

### 0.2 0.4 0 6 0 8

• jlra.d/s) -p.
• 1—I—tt—I-1-1-1--1-r

in principle considered to give more reliable results than the design wave method, provided that the transfer functions for the responses in question can be established with a reasonable degree of accuracy.

Long term analysis is generally perferred. If, however, only short term analysis is carried out, the location of the spectrum along the frequency axis should be properly justified.

In the cases where the above assumptions are not fulfilled, the design wave method should be applied.

### FUTURE ASPECTS

It has been attempted here to give a view of the methods that are commonly used today in calculating wind, wave, and current forces on offshore structures. As the reader has most certainly noticed, there are certain problems involved in applying the spectral analysis method to fixed offshore structures due to nonlinearities in the loads, which may be of great influence. Methods involving nonlinear statistics for practical application are not yet available.

As the forces and moments are statistical in nature, however, the need for statistical methods which take the nonlinearities into account is evident. It is hoped that strong efforts in clarifying this field will be made in the near future, and Det norske Veritas hopes to participate actively in this research in the near future.

Nomenclature

A Projected area of a member taken as the projection on a plane normal to ihe direction of the considered force. Aj Parameter of P(V'K).

a Elementary wave amplitude.

C Shape coefficient (3-dimensional flow),

Co= Shape coefficient (2-dimensional flow).

CD Drag coefficient,

CH Height coefficient.

D Diameter, d Cross sectional dimension normal to the direction of the considered force.

E Parameter of the short term Rayleigh distribution.

E(f) Wave spectrum as a function of frequency f (hz).

Fh Total horizontal force on a large volume structure.

Fw Wind force acting perpendicular to an area, A.

Fy Wind force acting in direction of the wind, t( O.) Directionality function, f Frequency (hz), fm Peak frequency of the wave spectrum (hz ).

g Acceleration due to gravity.

H Individual crest-to-trough wave height.

Hj Still water depth.

Hv Visually observed wave height.

Hy^ Significant wave height.

Hmax Most probable largest wave height, i Wave frequency number.

L Length of member.

AL Length of element or portion of structure considered.

1 Wave train number.

My Total overturning moment on a large volume structure.

P(a) The long term Weibull distribution of the response amplitude a

Ps[cr) The short term Rayleigh distribution of the response amplitude o

P[Ve ) The long term Weibull distribution of Ve

P(H) Long term Weibull distribution of individual wave heights.

P(HV) Long term Weibull distribution of visually observed wave heights.

7 Reynold's number.

SR2 Area beneath the response spectrum curve.

Sw2 Area beneath the wave spectrum curve.

¡Sr (w)2 Response spectrum. Swioj)2 Two-dimensional wave spectrum. Sw a)2 Three-dimensional wave spectrum. T Zero-uncrossing period.

T Average apparent zero-upcrossing period.

Tm Peak period of the wave spectrum.

Tr Average apparent zero-upcrossing period of the response amplitude a ■ t Time interval.

V Water particle velocity due to wave and current.

V Water particle acceleration due to wave,

Vcy Total current velocity y meters above sea bottom.

Vy Tide-induced current velocity at sea surface,

Vyy Wind-induced current velocity at sea surface.

Vz Sustained wind velocity z meters above sea surface.

CVzJq Gust wind velocity z meters above sea surface.

V10 Sustained wind velocity at 10 meters above sea surface.

V10 Sustained wind velocity at 1.0 meters above sea surface acting normal to an area, A.

V Transfer function.

y Distance above sea bottom, z Distance above sea surface.

a Angle between main direction of propagation of short crested wave system and the elementary long crested wave systems.

a( Angle between velocity vector and axis (or surface) of member, (3 Heading angle.

13] Angle between acceleration vector and axis (or surface) of member. ^ Wave length.

m Circular wave frequency.

V. Elementary response amplitude, f Phase angle,

£ Density of fluid.

7i Kinematic viscosity.

7 Peakedness parameter.

### REFERENCES

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