4. SELinux
4.1. Arquitectura
To verify the capabilities of the mass-spring-damper system, the general observations for the wave forces were compared with those from a laboratory experiment (Bradner 2008; Schumacher et al. 2008a; Schumacher et al. 2008b; Bradner et al. 2011). In the experimental study, a 1:5 scaled bridge deck model was tested in a large wave flume at the O. H. Hinsdale Wave Research Laboratory at the Oregon State University. The wave flume is 104 m in length, 3.66 m in width, and 4.57 m in depth. The span length of the bridge deck model is 3.45 m (corresponding to the width of the wave flume) and the width is 1.94 m. Six scaled AASHTO type III girders are evenly distributed underneath the deck slab. This model was placed on two linear guide rails with each one supporting one end of the span. Both the rigid setup and flexible setup (including soft springs setup and medium springs setup) were considered in the experimental study. For the flexible setup, in order to determine the restraining stiffness, a finite element analysis was conducted by typifying several different elevations of the bridge deck. Then, the structural vibration period was obtained for the corresponding deck elevation and a suitable support stiffness was subsequently chosen (two springs installed between the tested bridge model and the supporting frames) to match this period value.
The flexible setup for the bridge deck-wave interaction in the experimental study can be realized using the proposed mass-spring-damper system. However, several differences should be noted in the verification procedure: (a) 2D numerical simulations were conducted in the current study which may not fully capture all the characteristics observed in the experimental study; (b) since the span length of the bridge deck model is 3.45 m which is only slightly smaller than the width of the wave flume, 3.66 m, the 3D end effects may play significant roles in the experimental study; (c) the friction force between the bridge deck model and the supporting guide rails was not desired but cannot be avoided in the experimental study. However, this force was neglected and taken as 0 in the numerical simulations; and (d) all the AASHTO type III girders were simplified as rectangles in the numerical simulations. While these differences were noticed, it was expected that reasonable predictions of the general observations would be obtained. Other than the above discussed differences, all other parameters considered in the verification are exactly the same as those used in the experimental study, as listed in Table 5.1. Due to the reason that Bradner et al. (2011) only presented one figure (Fig. 12 in Bradner et al. 2011) to illustrate the results of the flexible setup and they normalized the time histories of the wave forces without giving the actual wave force and the corresponding wave height information, a direct comparison of the wave forces between the current method and the experimental measurements is not possible.
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Table 5.1 Parameters considered in the verification of the mass-spring-damper system with Bradner et al. (2011)
Parameter Value Geometry properties Girder height (m) 0.23 Girder spacing (m) 0.37 Deck thickness (m) 0.05 Overall height (m) 0.28 Span mass per unit length (kg) 562.3 Flexible setup properties
(soft springs setup)
Structural vibration period, 𝑇𝑠, (s) 0.95 Lateral restraining stiffness, 𝑘, (N/m) 31318
Damping ratio, 𝜉 0
Damping coefficient, 𝑐, (N·s2/m) 0
Wave properties
Still water depth, d, (m) 1.89 Wave height, H, (m)* 0.50 Wave period, T, (s) 2.5
Wave length (m) 8.6
Note: * Only one wave height of 0.50 m is considered here for the verification.
Based on the wave steepness H/gT2 and relative depth d/gT2 (Sarpkaya and Isaacson 1981), Stokes 2nd order wave theory is proper for the wave height of 0.50 m and its analytical expressions for the water particle velocities𝑢 and 𝑣, and the free surface profile 𝜂 are as follows (Lin 2008):
𝑢 =𝐻2𝑔𝑘𝜔 cosh 𝑘(𝑑+𝑦)cosh 𝑘𝑑 cos(𝑘𝑥 − 𝜔𝑡) +3𝐻162𝜔𝑘cosh 2𝑘(𝑑+𝑦)sinh4(𝑘𝑑) cos2(𝑘𝑥 − 𝜔𝑡) (5.10a) 𝑣 =𝐻2𝑔𝑘𝜔 sinh 𝑘(𝑑+𝑦)cosh 𝑘𝑑 sin(𝑘𝑥 − 𝜔𝑡) +3𝐻162𝜔𝑘sinh 2𝑘(𝑑+𝑦)sinh4
(𝑘𝑑) sin2(𝑘𝑥 − 𝜔𝑡) (5.10b)
𝜂 = 𝐻2cos(𝑘𝑥 − 𝜔𝑡) +𝐻162𝑘sinhcosh(𝑘𝑑)3(𝑘𝑑)(2 + cosh 2𝑘𝑑) cos 2(𝑘𝑥 − 𝜔𝑡) (5.10c) where 𝑘 is the wave number, 𝜔 is the wave frequency, 𝑡 is the simulation time, and𝑥 is the distance from the inlet boundary.
The numerical calculation domain is 40 m in length and 2.5 m in height. Using the wave properties listed in Table 5.1, the numerical wave profiles are obtained and compared with analytical results as shown in Fig. 10, demonstrating a good agreement with each other.
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The comparisons of the general characteristics of the numerically predicted wave forces between the flexible setup and rigid setup are demonstrated in Fig. 5.11, where three notable characteristics are observed: (a) while much smaller negative horizontal forces are found for the rigid setup, significant negative horizontal forces are observed for the flexible setup (soft springs setup). This is due to the consideration of the inertia forces of the bridge deck and will be discussed later; (b) a phase lag can be observed between the positive peak horizontal forces of the rigid setup and the flexible setup (soft springs setup) and this is also due to the inertia forces that are taken account in; and (c) there is no significant difference on the positive peak vertical forces. These observations follow the same trends as those documented by Bradner et al. (2011), indicating that the proposed mass-spring-damper system has a good capability to capture the general dynamic characteristics of the bridge deck-wave interaction problems. This system may be further adopted for other near shore and offshore structures, such as elastically mounted cylinder in a flowing fluid domain (Xu et al. 2014).
Fig. 5.11 Comparisons of the numerical wave forces between the flexible setup (soft springs setup) and rigid setup
In summary, a close match between the generated solitary waves and the prescribed ones demonstrated above is a premise for a reliable prediction of the wave forces. In addition, it has demonstrated that a simple mass-spring-damper model can capture the general dynamic characteristics in the bridge deck-wave interaction process. Therefore, this methodology is used in the following parametric study to systematically investigate the lateral restraining stiffness effect on the bridge deck-wave interaction.
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