Numerical simulation of multi-body floating piers to investigate pontoon stability

Abstract

The objective of this study is to develop a procedure to analyze the motions of a floating pier comprised of several pontoons that are modeled as rigid bodies and connected to each other by flexible and rigid connectors. Recently, the use offloating piers has increased because of their advantages, such as faster and higher-quality construction, seismic force isolation for a full-scale mooring system, low dependence on local soil conditions and tides, ability to relocate or reconfigure the pier modules during the operation period and 75-100 years of repair-free service. A floating pier consists of a pier, access bridge, mooring system and fender system, each of which comes in many variations to suit different usages and construction considerations. The typical loads used in the design of these piers are dead loads, live loads, mooring loads, fender loads and environmental loads induced by wind, currents and waves. For numerical simulation, three types of piers are used: passenger piers, light-cargo piers and semi-heavy-cargo piers. The selected piers consist of several large pontoons joined by pivots and have a pile-based mooring system. These piers are modeled by SAP2000software as two-dimensional frames that are linked together. As the first step, each type of pier is subjected to loading, and its general behavior is assessed. According to this behavior, the major load combinations are described for the design of piers and analyzed to determine the behavior of the modules. Lastly, according to the analysis results and the safe use and stability considerations, such as the maximum draft and longitudinal gradient, the dimensions of each module in each pier type are presented.

Keywords

coastal structures / numerical simulation / floating pier / rigid pontoons / stability

Introduction

Floating piers, which are of great importance in both military and civil fields, have been constructed across rivers and seas using floating pontoons rather than conventional piers and foundations in many countries.

Although the dynamic analysis of structures subjected to loads has a long history, the dynamic stability analysis of multi-body floating bodies (on a water surface) subjected to environmental loads, such as wind, waves and currents, as well as live and dead loads is a relatively new topic and few studies exist in the literature of Ref. [1]. Perhaps the first study investigating the dynamic response of an unmoored ship hull to a moving load was conducted by Wu [2]. The elastic vibration of a pontoon subjected to a moving load was investigated by the coupled dynamic response of the motion of a rigid body [3].

The dynamic stability of a single pontoon subjected to an external eccentric load or moment has been studied, and the sinking of any part of a pontoon due to the effect of concentrated load has been determined. In designing a conventional floating pier, the pier deck gradient should usually not exceed three to eight percent [4].

The static stability of a redundant pier system subjected to environmental loads, such as waves, wind, and currents, has also been studied [5-7], as have discrete-pontoon floating bridges. The finite element method was employed to idealize the stiffness of the bridge deck, and three-dimensional analysis was used to study the effect of environmental loads and moving loads [8]. A closed-form study of the dynamic response of floating bridges using beam theory and potential theory was performed, indicating that the water depth can be neglected in the design of floating bridges [9], which has been applied in modeling the floating pier.

The behavior of a floating pier under ship berthing impact was assessed using berthing analysis, and its significance to the design of berthing facilities was examined [10,11]. The results indicated that the flow induced by a large ship berthing in shallow water crucially affects all aspects of the ship berth. The behavior of a floating pier under lateral loads due to the impact of berthing vessels was studied [12]. This study also evaluated a new approach that considers the mobility of pontoons and flexibility of mooring piles in floating piers.

Researchers have recently been interested in studying the behavior of floating piers and bridges under exerted waves. The dynamic response of a floating bridge to wave excitations was studied [13].A closed-form study of the dynamic and structural motion responses of a floating pier in waves was performed, and a set of equations describing the response of the pontoon and pile column were derived and solved analytically and numerically considering the interaction among the wave, pontoon structure and pile column. One of the most important results of this effort was that the increase in the pontoon dimensions affects the heave motion oppositely to surge and pitch motions [14].The hydrodynamic behavior of multi-module floating piers with modules connected by hinges was also examined by computer-based modeling. Researchers have investigated the effect of waves incident on the pier and, by considering the operation conditions, proposed dimensions with regards to the height and angle of the incident waves [15].

One of the most recent investigations has studied the behavior of a floating wind turbine using a new method developed to directly apply Newton’s second law and the theorem of conservation of angular momentum to an entire floating turbine system [16].

In this study, the different types of floating piers will first be introduced, after which the advantages and disadvantages of these types will be discussed. After selecting the best type, the operational restriction of floating piers and the operational loads exerted on the pier will be introduced. Closed-form analysis is then performed to understand the reactions of the multi-body floating pier under exerted loads. Lastly, based on these limitations and the obtained equations, a case study of three types of piers will be performed. Tables for the optimum design of floating piers are presented at the end of the paper.

Floating pier operational restriction

Regardless of stability and balance issues, to facilitate the safe transport of goods and provisions using a pier, the gradients and drafts of the various pontoons comprising a floating pier should not exceed the specified limits. The maximum longitudinal and transversal gradient in the floating pier can be varied between 3 and 10 such that the exact gradient depends on the pier use and vehicle types traveling on the pier. The maximum allowable longitudinal and transversal gradient is 5 percent for passenger piers and 8 percent for cargo piers. The pontoon draft under its own weight is approximately 0.2 to 0.3 of each pontoon height. The maximum allowable draft for each pontoon is 50 cm for passenger piers, whereas there is no draft limitation for cargo piers [4]. It is important to note that cases in which each point of the pontoon is above the water will be rejected.

Model description

Pontoons can be made of concrete, rubber or steel. In this study, steel pontoons are considered. A typical floating pier consists of five principal elements: a relatively long floating system that accommodates mooring vessels and cargo handling equipment and provides space for cargo and passenger traffic, an access bridge, a mooring system, a fender system and mooring accessories. Pontoons can be divided into three general categories: flat pontoons, which are simple cubes; catamarans, which are comprised of two floating pieces connected by a deck; and semisubmersibles, which are comprised of several floating columns connecting the submerged pontoons to the deck. According to the shape, catamaran and semisubmersible pontoons are more resilient to waves and currents than flat pontoons because their shape allows the waves and currents to flow inside them. However, catamaran and semisubmersible pontoons are less stable to external loads, such as moving loads, than flat pontoons [17].In this study, flat pontoons are investigated.

There are four principal structural schemes for floating piers [18]:

A. One long pontoon

B. Several large pontoons joined by pivots.

C. A series of small pontoons spanned by single-span decks.

D. A series of small pontoons spanned by a continuous deck

From a capital cost perspective, the last two alternatives are the least desirable because of the extra weight of the deck. From a material consumption perspective, the continuous deck span (A) is more economical than a pier consisting of a number of single-span decks (C). The most economical solution providing the pier with maximum stability is that of several large pontoons joined by pivots (B), whereas both (A) and (B) are the most desirable from a cost perspective. For the foregoing reasons, the scheme involving several large pontoons joined by pivots (B) is investigated. The selected model is shown in Fig. 1.

The mooring system prevents the floating pier and the access bridge from moving out of their design location. There are two types of mooring systems: dolphin and mooring cable [4]. Because of the benefits of dolphin-type mooring systems, such as the lack of need for advanced construction technology, this type of mooring system is selected. The pontoon connection to pile is composed of a guide frame rigidly attached to the pontoon and featuring rollers on its sides to allow for free vertical movement of the pontoon along the pile during waves and water level changes. Mooring piles restrain the movements of the pontoon in the horizontal plane [12].

Loading

Environmental loads

Wind, waves and currents constitute the principal environmental loads on floating piers. Wind acts the parts of the pier that project from the water, especially those near water level. The design wind force is determined by a storm with a return period of 50 years [4]. In floating pier design, two components of wind force are considered: perpendicular and parallel to the pier. The wind force may be expressed as

(1)P_w=1.3{\rho }_w{C}_1{C}_2{\displaystyle \sum A},

In Eq. (1), P_w is the wind force exerted on the pier, and C₁ is the pier length coefficient, the recommended value of which is given in Table 1. C₂ is the guest factor, which varies from 1.35 to 1.45 on average. Lastly, {\displaystyle \sum A} is the pier freeboard exposed to the wind in m².

Three major types of current are considered in the floating pier design [19]:

a. Natural current in rivers

b. Tidal currents

c. Wind-driven currents in natural harbors

Currents a and b are usually determined based on available data. The wind-driven current at the still water level is usually taken as 1 percent of the sustained wind at 10 m above the face of the still water. The force of the current exerted on the pier system in both longitudinal and transversal directions P_c in kN can be obtained using

(2)P_c=C{V}_{\mathrm{c}}^{\mathrm{2}}{\displaystyle \sum A}.

In Eq. (2), the C factor varies between 0.5 and 1.0 kN·s²/m⁴, where the larger values are for piers; V_c is the current velocity in m/s; and {\displaystyle \sum A} is the area exposed to the underwater current in m².

Several wave theories have been developed over the past 200 years. Most coastal engineering design procedures, however, apply linear wave theory, providing a first-order approximation to the complete mathematical description of a wave. Nonlinear wave theories provide a higher order of approximation. Linear wave theory is quite satisfactory for the design of floating piers [4]. The wave pressure acting on a floating pier can be obtained from the 3D wave diffraction theory. The wave pressure acting on pontoons can be derived from Eq. (3), where Φ = Φ_i + Φ_s [20].

(3)P=-\rho gz-\rho (d\mathit{\Phi }/dt).

Φ_i indicates the incident wave velocity potential, and Φ_s indicates the scattered wave velocity potential. In Eq. (3), ρ is the seawater density.

The wind and current forces will be modeled as concentrated lateral loads acting on fenders and bollards. Because floating piers are usually constructed in a harbor, the wave force effect is negligible.

Dead and live loads

The pier dead load, equal to the weight of the pontoons and other equipment, is assumed to be 5000 N/m².The live load is determined based on the pier operational type: 1500 N/m² for passenger piers, 5000 N/m² for light-cargo piers and 10000 N/m² for semi-heavy-cargo piers.

Numerical simulation

Model description

The three operational types of floating piers are defined in Table 2. Based on construction considerations, each pontoon length is chosen to be multiple of three. Thus, the pontoons lengths are 9, 12, 15, 18, 21, 24, 27, 30, 45 and 60 m. The total number of pontoons in each model is shown in Table 3.

Modeling

The aim of this study is to investigate the pontoon displacement and gradient under defined loads to determine their optimal dimensions. The model is comprised of several pontoons that are modeled as rigid bodies and connected to each other by flexible connectors. Because the pontoons will be considered to be rigid bodies, there is no need to calculate the internal forces in the pontoons or mooring dolphins. Figure 2 shows the SAP2000 model. The mooring dolphins create a horizontal force on the pontoons. In this study, the pontoons are modeled using two-dimensional plane frames. The entire pontoon is considered to be a rigid body resting on an elastic foundation, where the water buoyancy is represented by the intensity of distributed springs modeling the elastic foundation [1].

Results

A multi-module floating pier under the critical load combination described above has been investigated. The main parameters that studied are the maximum draft, the maximum longitudinal and transversal gradient and the contact of the bottom of the pontoon with the water. The results show that the transversal gradient always is less than 1.5 percent, indicating that the transversal gradient is not an important criterion in floating pier design and that this gradient need not be a concern in operation.

The relationship between the longitudinal gradient and the pontoon width is shown in Figs. 3-5.

As an example, the draft and longitudinal gradient of a pontoon with a width of 6 m comprising a semi-heavy-cargo pier are shown in Table 4.

Discussion

The longitudinal gradient and the pontoon’s draft in water vary by pontoon width and load combination, as shown in the previous section. As expected, as the width increased, the pontoon longitudinal gradient decreased but the pontoon draft increased due to the increase in pontoon weightfor passenger and light-cargo piers. However, for the semi-heavy-cargopier, increasing the width has little effect on the longitudinal gradient.

The best height for an individual pontoon is derived by stability analysis of a single pontoon. The minimum width-to-height ratio of a single pontoon is \sqrt{\mathrm{6}}/\mathrm{2} to satisfy the stability of a single pontoon [4]. As a final result, the best dimensions for all three pier types with various widths are given in Table 5. Note that the freeboard adds to the pontoon height.

Conclusiont

In this study, the various modes and loads exerted on floating piers were introduced and used to determine the best model. After analyzing the selected model, the results were presented in tables and graphs. Lastly, for various pontoon widths, the optimal dimensions satisfying the design restrictions were provided. Designers can use the dimensions obtained in this study to apply appropriate safety factors in the design of floating piers.

As seen from the results, in the semi-heavy-cargo piers, at least one point of the pontoon levitates above the water, perhaps because of the mobile crane that allows passing on the pier. As can be observed, although some of pontoon lengths meet the allowable longitudinal gradient criteria, they are rejected considering the connection between the pontoon bottom and the water surface. An obvious example can be seen in Table 4.

One important note is that the transversal gradient is always below 2 percent, which satisfies the gradient limitations for all three floating pier types.

References

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