(1)
Constantine Memos, Areti Kiara and Constantine Vardanikas
Faculty of Civil Engineering, National Technical University of Athens,
Heroon Polytechniou 5, 157 80 Zografos, Greece
Tel.
0030-1-772 2851, Fax 0030-1-772 2853, E-mail: memos@hydro.ntua.gr
Abstract: Rubble-mound breakwaters are subject not only to wave action but, also, to other kinds of environmental loading such as earthquakes. During seismic excitation hydrodynamic pressures are being developed upon the inclined faces of the mound. A major investigation including experimental work was undertaken aiming at gaining insight in the behaviour of the above structures under seismic action. The present paper reports on the part of this investigation associated with the prediction of the values of the above-mentioned pressures and their characteristics. A boundary element code was developed, tailored to the needs of the rubble-mound type of breakwater. The model results were compared against the measured data, the comparison being favourable. A useful advance over current practice employing Westergaard’s formula is that the model can take into account the effect of the predominant frequency of the excitation which in turn can reveal dangerous resonance conditions. Otherwise, rubble-mound breakwaters were found to behave as rigid rather than as elastic bodies. However, as the strength of the excitation increases the maximum pressures among those acting on the faces of the mound tend to be located higher than the toe of the structure. This trend increases with increasing ground acceleration. Finally the code can detect areas where dangerous sub-pressures could be developed.
Keywords: breakwaters, earthquakes, hydrodynamic pressure, rubble-mound, seismic loading
Rubble-mound breakwaters are commonly used worldwide as protection structures of ports or coastal areas. Their design is normally based on the loading produced by the relevant design waves. However, there are also other environmental loads, and in particular those produced by earthquakes, that should be taken into account especially in areas of high seismicity and poor foundation soil.
During a seismic excitation hydrodynamic pressures are being developed at the inclined faces of the structure. These pressures tend to modify the added mass of the moving breakwater, hence the maximum horizontal accelerations along the inclined faces of the structure as well as its fundamental period of oscillation. Additionally, close examination of the detailed structure of the pressure distribution can reveal areas where sub-pressures could be developed. These negative pressures may be dangerous, since they tend to pull seawards the corresponding armour stones and affect thus adversely the structural integrity of the mound.
A limited number of analytical or numerical studies deal with the estimation of the hydrodynamic pressures on the faces of a dam-like structure. These studies refer to dams rather than to breakwaters, as e.g. those by Westergaard (1933), Chopra (1967), Chopra and Gupta (1981), Chen et al. (1999). Experimental investigations are very few: Zangar (1953) addresses the hydrodynamic pressures on dams; Wang et al. (1978) tackle rubble-mound breakwaters; Memos and Protonotarios (1992) have estimated the total loading due to hydrodynamic pressures on a breakwater with inclined faces.
The General Secretariat for Research and Technology of Greece funded an investigation aiming at gaining more insight in the seismic behaviour of rubble-mound breakwaters. Within this framework experimental work has been carried out at the National Technical University of Athens (N.T.U.A.) by employing the shaking table of the Engineering Seismology Laboratory. The present paper refers to a part of the above investigation, namely to the estimation of the hydrodynamic pressures exerted on the faces of the mound. To this effect a boundary element code was developed and relevant experimental data were collected and analysed.
Results on other aspects of the problem, in particular of a geotechnical nature, have been reported by Memos et al. (2000).
The experiments were carried out at the earthquake simulator of the Engineering Seismology Laboratory of the N.T.U.A. In order to study the seismic behaviour of rubble-mound breakwaters in association with their foundation, two series of experiments were conducted by testing two model breakwaters. Each one of the models was placed at a time inside a metallic rigid container, filled with water up to a certain level, and submitted to a total of 18 (model I) or 20 (model II) seismic excitations. The accelerations and hydrodynamic pressures developed during the tests were measured with waterproof, miniature acceleration and pressure transducers.
Model I sat on a rigid bed and had its armour stones placed with extra care so that safe interlocking was secured. Model II was placed on a layer of fine, loose to medium density sand and its armour layer was constructed as closely as possible to the usual practice in prototype construction technique. The present paper deals only with model II, the typical cross-section of which is shown in Fig. 1.
The model breakwater was subject to 20 horizontal pulses of increasing acceleration ranging from amax = 0.06g to amax = 1.55g. Two pressure transducers P1 and P2 were placed at the inclined face of the model, at –0.05m and – 0.15m from water level respectively (Fig. 1).
The hydrodynamic pressures were measured at tests with comparatively high maximum accelerations and were compared against the predictions of the numerical model, as it will be seen in the following. The validity of the mathematical model was checked for a breakwater with the features of the model breakwater. For more information on the experimental setup the reader is referred to Memos (2000).
During a seismic excitation hydrodynamic forces, of alternating direction, develop on the inclined faces of the structure. The values of these forces depend mainly on the ratios ω/ω1 and ω/ωb, where ω represents the predominant frequency of the excitation, ω1 is the first cutoff frequency of the water body (ω1 = π c/ 2d, c = the speed of sound in water, d = the water depth) and ωb is the natural frequency of the mound. These ratios control, also, any resonance effects, as well as the shape of the pressure distribution along the wet surface of the breakwater.
Based on studies regarding the two-dimensional dam-reservoir interaction problems during earthquake loading, a boundary element model was developed at the N.T.U.A. The model is used for the computation of hydrodynamic pressures at the inclined faces of the breakwater that are developed during a horizontal, sinusoidal ground motion.
The model development is based on the work by Liu and Cheng (1984). However, there are some features that were additionally included in the present model pertinent to the kind of structure under consideration. These features are described as follows:
(1) The present model can incorporate not only vertical but also inclined faces of the structure.
(2) The natural frequency of the mound can only be expressed by semi-empirical relationships applicable to a non-homogeneous shear beam, in contrast to that of a homogeneous beam of concrete assumed by Liu and Cheng. The elastic behaviour of the former is thus expressed through the varying displacements at the various levels of the structure. The present model is able to accept as input the values of the horizontal displacement along the height of the structure, accommodating in this way the elasticity of the body. This represents a useful advance over past models.
(3) The structure of the present model permits coupling with corresponding geotechnical models describing the response of the mound. This coupling is quite important in trying to estimate the structure-water interaction during the shaking.
The main assumptions taken into account in the development of the model are that the effects of the viscosity of water and of gravity can be ignored. Also, that the water basin is considered infinite, a statement found to give acceptable accuracy for water basins with length at least 3 or 4 times greater than the sea depth at the location of the breakwater. The compressibility of water is not negligible.
The Helmholzt equation describes the velocity potential Φ in the water basin:
(1)
in which c = the speed of sound in water.
From the Bernoulli equation, the hydrodynamic pressure is related to the velocity potential:
(2)
in which ρ = the density of water. The ground motion is considered horizontal and sinusoidal, with displacement amplitude U0.
Each of the four boundary surfaces is described by an equation of known potential or its normal derivative or by radiation conditions.
Along the bottom of the water basin (see Fig. 2) the normal velocity is:
(3)
in which
is the unit outward normal along
the bottom defined by D, and
is the unit vector in the
x-direction.
Along the free surface the atmospheric pressure is a constant (zero):
(4)
Along the slopes of the breakwater the velocity is considered horizontal:
(5)
in which U ( z ) = the displacement of the breakwater at level z.
Along the fictitious boundary the horizontal velocity and its potential are described by the radiation conditions used by Liu and Cheng (1984):
(6)
(7)
From equations (1) to (7) a linear system [A] * [B] = [X], is formulated with complex coefficients A, B and X unknowns. The system is transformed into another linear system, of the same form but of double size, with real coefficients and unknowns. By solving the system, the hydrodynamic pressures at the midpoints of the boundary elements can be estimated.
Figure 2 gives a definition sketch and the coordinate system used for the hydrodynamic model. The information required for the model is:
l The geometry of the boundary elements.
l The time series of displacements at the end-points of the boundary elements along the faces of the breakwater.
l The ratio ω/ω1.
The model output consists of time series of hydrodynamic pressures at the midpoints of the boundary elements at the mound slopes.
Regarding displacements required as input, two assumptions are made. Firstly, that the breakwater is not to be considered as a rigid body and secondly, that it can be divided into rigid horizontal layers.
The model described above has been applied to conditions dictated by the experimental study mentioned in the previous section in order to be able to compare the results. These are presented in the following along with the relevant discussion. In order to investigate closer the type of resonance associated with the water mass supporting the propagation of compression waves due to the movement of the body, the elasticity of the structure has been ignored temporarily. Thus some numerical experiments were carried out assuming rigid rather than elastic body. Representative results of these numerical tests are also presented in the next section.
As mentioned in the previous section some numerical experiments were carried out assuming rigidity of the mound, in order to study the type of resonance associated with the behaviour of the water mass. Results from this exercise are presented in Fig. 3, which presents a dimensionless total force as related to the ratio ω/ω1. From this graph it becomes evident that the type of resonance under consideration can be avoided for small values of ω/ω1, less than 0.4, say. This result is in accordance with other investigators, e.g. Liu and Cheng (1984). In more practical terms this means that for earthquakes of typical dominant periods in the range 0.1s – 0.3s the above type of resonance can be avoided in water depths less than, say, 30m, which is well within the depths normally rubble-mound breakwaters are built. It should be reminded that for an elastic structure the additional requirement that the hydrodynamic pressures be non ω-dependant is that ω/ωb is also small, lower than about 0.5 (see e.g. Liu and Cheng (1984)).
During the experiments time series of the dynamic pressure were taken at the points shown in Fig. 1. In order to be able to compare the experimental results with the mathematical model, the following procedure was applied: All runs of the experiments were recorded on videotape. After each shaking the cross-section of the breakwater was reproduced on paper with the aim of the grid drawn on the transparent window located at one side of the container. Then a computational grid was related to the deformed cross-section of each test in such a way that the location of the pressure transducers coincided with the midpoint of a boundary element. The location of the transducers was inferred from their original location by assuming linear dependence of the total settlement with height. By double integrating the accelerations measured by the corresponding accelerometers at various levels of the structure the horizontal displacements were obtained at the nodes of the boundary elements at the breakwater faces. These displacements were then input into the hydrodynamic code.
The code was tested with input sinus waves giving satisfactory results as verified by comparison with results by Liu and Cheng (1984). In the experiment however, a synthetic seismic spectrum was used as input excitation in order to represent more closely real life earthquakes. Thus the response spectra gave a range rather than a single value of dominant periods. The actual value of the period to serve as input in the code was selected from this range by performing a fine-tuning of the model.
Representative results of both the mathematical model and the experiments are given in terms of maximum hydrodynamic pressures along the depth of the structure (Fig. 4). For the sake of comparison the analytical Westergaard’s solution for a rigid body is also presented in the graphs of Fig. 4. It should be noted that the presented graphs do not provide a pressure distribution at any particular instant, since the maximum values of the pressures at various levels do not act simultaneously, due to the elasticity of the shaken body.
As expected, the hydrodynamic pressures are relatively low and can be measured easily by available equipment only for quite large accelerations, normally infrequent in nature. Nevertheless comparison of the model with the experimental results at these high excitations should be viewed rather as a checking procedure for the mathematical model, which can then be applied to any level of shaking.
The comparison between model and experiment is good, taking into account the many uncontrollable factors involved in measuring the dynamic pressure at the face of a rubble-mound. Inspection of results from 11 tests shows a systematic overprediction of the maximum pressure especially at the lower section of the structure. This can be attributed to the fact that the model produces results assuming that the ground movement obeys a sinusoidal law the period of which is determined as explained at the beginning of the present section. However, at the experiments the imposed excitation did not follow such a law, since, as mentioned above, the aim was to represent as closely as possible a true event displaying a spectrum of frequencies. An additional factor for the observed discrepancy may relate to the fact that the porosity of the breakwater was not accounted for in the mathematical model.
Model and experimental results suggest that the maximum hydrodynamic pressures increase with depth as Westergaard’s solution suggests, except for very high accelerations, for which the maximum pressure does not occur at the toe of the structure. A general feature of the behaviour of the maximum pressures as the acceleration increases is that the former redistribute themselves giving higher values at the upper section and lower at the lower section of the mound as compared with the corresponding values associated with a weaker shaking. In any case, however, the maximum pressures distribute themselves along the breakwater faces in a manner similar to that of a rigid rather than an elastic body. Another conclusion that can be drawn on the basis of the above results is that Westergaard’s solution overpredicts the hydrodynamic pressures for accelerations larger than, say, 0.6 g.
Rubble-mound breakwaters are subject to hydrodynamic pressures at both inclined faces during seismic excitation. These pressures although relatively small especially for short structures, tend to modify the added mass of the moving structure and hence its acceleration and its fundamental period of oscillation. Also, development of negative pressures at the faces of the mound can give rise to forces tending to pull out stones from the armour layer. A boundary element algorithm was developed under fairly general assumptions capable of predicting the hydrodynamic pressures at the faces of the breakwater by taking into account the predominant excitation period. In addition, an experimental investigation was undertaken to study the behaviour of the rubble-mound breakwater under seismic action and to collect data for the verification of the algorithms. Comparisons of model results with experimental data were quite encouraging. This is quite important since use of Westergaard’s formula, which is a current practice, does not take into account the period of excitation that is quite crucial in determining resonance conditions. These latter modify appreciably the value of the hydrodynamic pressures at the slopes of the mound. It is planned to extend the present research in predicting the possible negative pressures and their effect to the integrity of the structure.
Acknowledgements
The present report is based on research partially funded by the General Secretariat for Research and Technology of Greece, whose support is gratefully acknowledged. The experimental study would have been impossible without the help of P. Carydis, Ch. Mouzakis of the Engineering Seismology Laboratory of NTUA. The contribution of V. Grossomanides to the development of the code is also thankfully acknowledged.
References
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Fig. 1 Cross-section of model breakwater with location of pressure transducers (length in m)
Fig. 2 Definition sketch and boundary element grid
Fig. 3 Total force on breakwater face as related to excitation frequency
Fig. 4 Comparison between model, experiment and Westergaard’s solution