Journal of Marine Science and Engineering Article Influence of the Spatial Pressure Distribution of Breaking Wave Loading on the Dynamic Response of Wolf Rock Lighthouse Darshana T. Dassanayake 1,2,* , Alessandro Antonini 3 , Athanasios Pappas 4, Alison Raby 2 , James Mark William Brownjohn 5 and Dina D’Ayala 4 1 Department of Civil and Environmental Technology, Faculty of Technology, University of Sri Jayewardenepura, Pitipana 10206, Sri Lanka 2 School of Engineering, Computing and Mathematics, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK; [email protected] 3 Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands; [email protected] 4 Department of Civil, Environmental and Geomatic Engineering, Faculty of Engineering Science, University College London, Gower Street, London WC1E 6BT, UK; [email protected] (A.P.); [email protected] (D.D.) 5 College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK; [email protected] * Correspondence: [email protected] Abstract: The survivability analysis of offshore rock lighthouses requires several assumptions of the pressure distribution due to the breaking wave loading (Raby et al. (2019), Antonini et al. (2019). Due to the peculiar bathymetries and topographies of rock pinnacles, there is no dedicated formula to properly quantify the loads induced by the breaking waves on offshore rock lighthouses. Wienke’s formula (Wienke and Oumeraci (2005) was used in this study to estimate the loads, even though it was not derived for breaking waves on offshore rock lighthouses, but rather for the breaking wave loading on offshore monopiles. However, a thorough sensitivity analysis of the effects of the assumed Citation: Dassanayake, D.T.; Antonini, A.; Pappas, A.; Raby, A.; pressure distribution has never been performed. In this paper, by means of the Wolf Rock lighthouse Brownjohn, J.M.W.; D’Ayala, D. distinct element model, we quantified the influence of the pressure distributions on the dynamic Influence of the Spatial Pressure response of the lighthouse structure. Different pressure distributions were tested, while keeping the Distribution of Breaking Wave initial wave impact area and pressure integrated force unchanged, in order to quantify the effect of Loading on the Dynamic Response of different pressure distribution patterns. The pressure distributions considered in this paper showed Wolf Rock Lighthouse. J. Mar. Sci. subtle differences in the overall dynamic structure responses; however, pressure distribution #3, Eng. 2021, 9, 55. https://doi.org/ based on published experimental data such as Tanimoto et al. (1986) and Zhou et al. (1991) gave 10.3390/jmse9010055 the largest displacements. This scenario has a triangular pressure distribution with a peak at the centroid of the impact area, which then linearly decreases to zero at the top and bottom boundaries Received: 30 November 2020 of the impact area. The azimuthal horizontal distribution was adopted from Wienke and Oumeraci’s Accepted: 26 December 2020 work (2005). The main findings of this study will be of interest not only for the assessment of rock Published: 6 January 2021 lighthouses but also for all the cylindrical structures built on rock pinnacles or rocky coastlines (with Publisher’s Note: MDPI stays neu- steep foreshore slopes) and exposed to harsh breaking wave loading. tral with regard to jurisdictional clai- ms in published maps and institutio- Keywords: breaking wave loading; impact pressure distributions; offshore vertical cylinders; light- nal affiliations. houses; STORMLAMP project Copyright: © 2021 by the authors. Li- 1. Introduction censee MDPI, Basel, Switzerland. This article is an open access article Offshore rock lighthouses constructed during the Victorian era by the British engineers distributed under the terms and con- are constructed from masonry interlocking blocks. These towers mark submerged or low- ditions of the Creative Commons At- lying offshore rock pinnacles, otherwise difficult to observe from a vessel bridge. In addition tribution (CC BY) license (https:// to being indispensable permanent aids for navigation, these structures are of incalculable creativecommons.org/licenses/by/ cultural significance. The majority of these towers are subjected to direct wave impacts 4.0/). during severe storms. The offshore rocks typically consist of steep foreshores and pinnacles J. Mar. Sci. Eng. 2021, 9, 55. https://doi.org/10.3390/jmse9010055 https://www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2021, 9, 55 2 of 19 that are barely sufficient in area for the tower foundations. This shoal geometry forces the incoming waves to break either in close proximity to the base of the lighthouse or directly onto the tower, causing unwanted dynamic structural responses which threaten the stability of these structures. Historical anecdotes of former lighthouse keepers provide vital evidence of substantial past impact events. On some occasions, the accelerations due to structure responses could reach values equivalent to modest earthquakes (Raby et al. (2019) [1]). The most sensible method to assess the dynamics of multi-block structure responses due to extreme weather events is by means of a validated 3D distinct element model (DEM) (Pappas et al. (2018) [2]). The external force exerted on the surface of the structure is one of the crucial input parameters that governs the accuracy of the model predictions. However, there are no dedicated formulae to accurately quantify the loads induced by breaking waves for such peculiar configurations. Overall breaking wave forces acting on offshore monopiles can be estimated using Wienke’s formula (Wienke and Oumeraci (2005) [3]); however, this formula is sensitive to wave-induced pressure distribution on the surface of the structure. The wave-induced pressure distribution of an offshore monopile and an offshore lighthouse have significant differences, as wave kinematics for a breaking wave on a shoal are much more complex compared to offshore breaking waves in deeper water. Therefore, dynamic responses of lighthouses might be vastly different to those of offshore monopiles. Detailed numerical modelling of rock lighthouses are rare; therefore, the most probable synthetic pressure distributions were tested using the validated Wolf Rock DEM model (Raby et al. (2019) [1]). The main findings are of interest not only for the assessment of the survivability of rock lighthouses but also for all the cylindrical structures built on rock pinnacles or rocky coastlines (with steep foreshore slopes) and exposed to the harsh breaking wave loadings. The Wolf Rock lighthouse is located approximately 13.5 km south-southwest of the most south-westerly tip of mainland UK. It stands 35.33 m from the foundation to the gallery level (70 rock masonry courses), the lowest 11.98 m (20 courses) being solid except for a water tank. The lighthouse consists of characteristic concave elliptic frustum curve with a maximum diameter of 12.2 m at its base (1st course), which gradually decreases to 5.2 m (67th course). The lowest 20 courses are of a stepped design, with this feature considered as an effective effort to reduce wave runup (Raby et al. (2019), Boyle (1998) [1,4]). After the 67th course, the tower widens again to reach 6.4 m diameter at the 70th course with a recurved profile (radius = 1.15 m) intended to divert the up-rushing wave from the gallery. James Walker initiated the designing of Wolf Rock lighthouse in 1860 based on his prior design and construction experiences on similar rock towers such as Bishop Rock and the Smalls lighthouses (Nicholson (2015) [5]). The design (Figure1a) was completed by William Douglass in 1862 (Douglas (1870, 1871) [6,7]). In 1870, the lighthouse construction was completed after a slow and long construction process due to the severe constraints of working on such a small and low-lying rock exposed to Atlantic gales. This engineering masterpiece has withstood the forces of the ocean since then without requiring any major repairs, but with rigorous inspections and maintenance routines conducted by Trinity House. The UK and Irish General Lighthouse Authorities (GLAs), namely The Corporation of Trinity House, the Northern Lighthouse Board and the Irish Lights (responsible for England, Wales, the Channel Islands and Gibraltar; Scotland; and Northern Ireland and the Republic of Ireland, respectively), need to systematically study the conditions of these heritage structures, primarily focusing on the remaining operational life in light of climate change, with the increase in sea level and the possibility of increased storminess (frequency and magnitude). Therefore, the STORMLAMP project, funded by the UK Engineering and Physical Sciences Research Council (EPSRC), selected Wolf Rock lighthouse (Figure1b ) as one of the seven lighthouses investigated under the project (Raby et al. (2019) [1], Antonini et al. (2019) [8]). This study investigated the differences in the dynamic structure responses of offshore rock lighthouses due to different pressure distributions. The impact area was kept constant in this study, and different impact pressure distributions were tested J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 3 of 20 storminess (frequency and magnitude). Therefore, the STORMLAMP project, funded by the UK Engineering and Physical Sciences Research Council (EPSRC), selected Wolf Rock J. Mar. Sci. Eng. 2021, 9, 55 lighthouse (Figure 1b) as one