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Ação de Ondas em Estacas

Calcular forças de ação de ondas em estacas usando a equação de Morison.

Parâmetros da Estaca e do Local

Parâmetros da Onda

Documentação

Wave Parameters

A progressive wave presents a period T, a wavelength Lw (distance from one crest to another), a height H (distance between trough and crest) and a form η. The figure below illustrates the characteristics of the progressive wave, and Equation (1) presents the expression to determine the wave form as a function of abscissa x and time t.

Progressive wave parameters scheme

Figure 1: Schematic of progressive wave parameters.

$$ \eta(t)= \frac{H}{2} \cdot \cos(k_{w} \cdot x - \sigma_{w} \cdot t) $$

(1)

Where σw is the angular frequency of the wave, and kw is the wave number:

$$ k_{w}=\frac{2\pi}{L_{w}} $$

(2)

$$ \sigma_{w}=\frac{2\pi}{T} $$

(3)

Knowing the wave period T and the local depth h, the solution of the wave dispersion equation can provide the wavelength Lw. Therefore, the wave dispersion equation (4) is presented below, where g is the acceleration due to gravity.

$$ {\sigma_{w}}^2=g \cdot k_{w} \cdot \tanh \left (k_{w} \cdot h \right ) $$

(4)

Attempting to analytically solve Equation (4) for the wavelength results in Equation (5), which has no analytical solution. Its solution requires an iterative process of trials so that both sides of the equation converge to the same value.

$$ L_{w} = \frac{g \cdot {T}^2}{2 \cdot \pi} \cdot \tanh \left (\frac{2 \cdot \pi \cdot h}{L_{w}} \right ) $$

(5)

The literature presents simplifications through boundary conditions to approximate the trigonometric function value present in the equation (hyperbolic tangent). These boundary conditions are checks regarding the wave propagation regime (shallow, intermediate, or deep waters). These simplifications are not addressed here. Once these parameters are known, it is possible to begin determining the forces caused by waves on structures.

Forces on Piles

To determine the forces exerted on a pile, Morison's formula is widely used in the literature, presented in standards such as the Spanish engineering standard (ROM 2.0-11) and the British standard (BS 6349-1-2), for use in the design of port or maritime structures.

Morison's Formula

Morison's equation addresses the forces caused on the pile through two terms. The first term of Equation (6) deals with the drag force from fluid motion. The second term deals with the effects of acceleration caused by wave motion. To fit experimentally obtained results, two coefficients are used, CD and CM. These are respectively drag and inertia coefficients (obtained experimentally), ρ is the specific mass of the water or fluid, Dp the pile diameter, and uw the horizontal component of particle velocity under the wave. Morison's formula is presented below:

$$ F = C_D \cdot \frac{1}{2} \cdot \rho \cdot D_{p} \cdot \left | u_{w} \right | \cdot u_{w} \space - \space C_M \cdot \rho \cdot \frac{\pi \cdot {D_{p}}^2}{4} \cdot \frac{du_{w}}{dt} $$

(6)

The horizontal component of particle velocity during wave passage uw depends on the wave theory used. Linear theory is used here, so the equations describing this component and its time derivative follow below:

$$ u_{w} = \frac{H}{2} \cdot \frac{g \cdot T}{L_{w}} \cdot \frac{\cosh \left[ 2 \cdot \pi \cdot \left(h + z \right)/ L_{w} \right]}{\cosh \left( 2 \cdot \pi \cdot h / L_{w} \right)} \cdot \cos \left ( \frac{2 \cdot \pi \cdot t}{T} \right ) $$

(7)

$$ \frac{du_{w}}{dt} \simeq \frac{\partial u_{w}}{\partial t} = \frac{\pi \cdot g \cdot H}{L_{w}} \cdot \frac{\cosh \left[ 2 \cdot \pi \cdot \left(h + z \right)/ L_{w} \right]}{\cosh \left( 2 \cdot \pi \cdot h / L_{w} \right)} \cdot \sin \left ( - \frac{2 \cdot \pi \cdot t}{T} \right ) $$

(8)

Substituting the respective components into Equation (6), it is possible to separate and treat the terms in isolation. Thus, after separation and algebraic manipulation, we have Equations (9) and (10) for drag and inertia forces per unit length, respectively.

$$ F_D = C_D \cdot \frac{1}{2} \cdot \rho \cdot g \cdot D_{p} \cdot {H}^2 \left [ \frac{g \cdot {T}^2}{4 \cdot {L_{w}}^2} \cdot \left ( \frac{\cosh (k_{w} \cdot (z + h))}{\cosh(k_{w} \cdot h)} \right )^2 \right ] \cdot \left | \cos \left ( \frac{2 \cdot \pi \cdot t}{T} \right ) \right | \cdot \cos \left ( \frac{2 \cdot \pi \cdot t}{T} \right ) $$

(9)

$$ F_M = C_M \cdot \rho \cdot g \cdot \frac{\pi \cdot {D_{p}}^2}{4} \cdot H \cdot \left [ \frac{\pi}{L_{w}} \cdot \frac{\cosh (k_{w} \cdot (z + h))}{\cosh (k_{w} \cdot h)} \right ] \cdot \sin \left ( - \frac{2 \cdot \pi \cdot t}{T} \right ) $$

(10)

To obtain the maximum effort caused by the drag force, one can consider the maximum values for the cosine terms, i.e., when t is zero, at which point the wave crest is in contact with the pile. This yields Equation (11).

$$ F_D = C_D \cdot \frac{1}{2} \cdot \rho \cdot g \cdot D_{p} \cdot {H}^2 \left [ \frac{g \cdot {T}^2}{4 \cdot {L_{w}}^2} \cdot \left ( \frac{\cosh (k_{w} \cdot (z + h))}{\cosh(k_{w} \cdot h)} \right )^2 \right ] $$

(11)

Maximum drag force scheme

Figure 2: Schematic for maximum drag effort.

To obtain the maximum effort caused by the inertia force, one can consider the maximum value for the sine term, i.e., when t is negative at one quarter of the wave period (-T/4). This yields Equation (12).

$$ F_M = C_M \cdot \rho \cdot g \cdot \frac{\pi \cdot {D_{p}}^2}{4} \cdot H \cdot \left [ \frac{\pi}{L_{w}} \cdot \frac{\cosh (k_{w} \cdot (z + h))}{\cosh (k_{w} \cdot h)} \right ] $$

(12)

To better illustrate the responses of FD and FM, Figure 3 presents the approximate profile of load distribution along the pile. The profile is similar for both components (neglecting wave form and position). In Figure 3 the pile diameter was omitted.

Load distribution profile scheme

Figure 3: Schematic for load distribution profile on the pile.

Resultants

Since Equations (11) and (12) provide force per unit length of the pile, the resultant force can be obtained by integration. The method adopted by Mason and Dean neglects the wave surface, i.e., forces are calculated from the waterline down, these relations being valid for Dp/Lw < 0.05, something generally verified in engineering practice. Therefore, we have:

$$ F_{D_{resultante}} = C_D \cdot \rho \cdot g \cdot D_{p} \cdot {H}^2 \cdot \frac{1}{8} \cdot \left ( 1 + \frac{2 \cdot k_{w} \cdot h}{\sinh (2 \cdot k_{w} \cdot h)} \right ) $$

(13)

$$ F_{M_{resultante}} = C_M \cdot \rho \cdot g \cdot H \cdot \frac{\pi \cdot {D_{p}}^2}{8} \cdot \tanh (k_{w} \cdot h) $$

(14)

The application points of these forces, distant dD and dM from the base, are used for determining the acting moments, commonly applied during the design of these structures, and can be found approximately by expressions (15) and (16). Figure 4 shows a schematic of the resultant force and its application point.

$$ d_{D} = h \cdot \left(\frac{1}{2} + \frac{1}{\left ( 1 + \frac{2 \cdot k_{w} \cdot h}{\sinh (2 \cdot k_{w} \cdot h)} \right )} \cdot \left ( \frac{1}{2} + \frac{1 - \cosh(2 \cdot k_{w} \cdot h)}{2 \cdot k_{w} \cdot h \cdot \sinh(2 \cdot k_{w} \cdot h)} \right ) \right) $$

(15)

$$ d_{M} = h \cdot \left(1 + \frac{1 - \cosh(k_{w} \cdot h)}{k_{w} \cdot h \cdot \sinh(k_{w} \cdot h)} \right) $$

(16)

Resultant force scheme

Figure 4: Schematic of resultant force and application point.

References

  1. BS 6349. Maritime works - Part 1-2: General — Code of practice for assessment of actions. [S.l.]: BSI Standards Limited, 2017. ISBN 9780580988318.
  2. MASON, J. Obras Portuárias. 1. ed. Rio de Janeiro, RJ: Editora Campus, 1981.
  3. MORISON, J. R.; JOHNSON, J. W.; SCHAAF, S. A. The Force Exerted by Surface Waves on Piles. Journal of Petroleum Technology, v. 2, n. 05, p. 149–154, 05 1950. ISSN 0149-2136. Disponível em: <https://doi.org/10.2118/950149-G>.
  4. ROM 2.0-11. Recomendaciones para el proyecto y ejecución en Obras de Atraque y Amarre. [S.l.]: Puertos del Estado, 2012. (Recomendaciones para Obras Marítimas. Serie 2, Obras portuárias interiores).