Diagram Loads on Pile Structures

The wave length \( L \), the maximum horizontal orbital velocity \( u_{max} \), the maximum horizontal orbital acceleration \( \left( \frac{\delta u}{\delta t} \right)_{max} \), as well as the displayed wave profile are calculated according to the linear wave theory of Airy/Laplace and follow the equations given in the lower part of the page for the purpose of clarity.

The force coefficients \( C_D \) and \( C_M \) are determined according to Shore Protection Manual (SPM), the formulae for this are also given in the lower part of the page.

Drag force \( f_D \), inertia force \( f_M \), and the total force \( f_{tot} \) are determined according to Morison (1950).
They thus follow the equation \( f_{tot} = f_D + f_M = \frac{1}{2} \cdot C_D \cdot \rho_w \cdot D \cdot |u| \cdot u + C_M \cdot \rho_w \cdot \frac{\pi \cdot D^2}{4} \cdot \frac{\delta u}{\delta t} \).

The following also applies: Reynolds number \( Re = \frac{u_{max} \cdot D}{\nu} \), sea water density \( \rho_w = 1025 \frac{kg}{m^3} \), acceleration due to gravity \( g = 9.81 \frac{m}{s^2} \) and kinematic viscosity \( \nu = 10^{-6} \frac{m^2}{s} \).

According to the linear wave theory of Airy/Laplace:
Wave length \( L = \frac{g \cdot T^2}{2 \cdot \pi} \cdot \tanh( \frac{2 \cdot \pi}{L} \cdot d) \),
Maximum horizontal orbital velocity \( u_{max} = \frac{H \cdot \pi}{T} \cdot \frac{\cosh \left(\frac{2 \pi}{L} \cdot (z+d) \right)}{\sinh \left(\frac{2 \pi}{L} \cdot d \right)} \),
Maximum horizontal orbital acceleration \( \left( \frac{\delta u}{\delta t} \right)_{max} = \frac{2 \cdot H \cdot \pi^2}{T^2} \cdot \frac{\cosh \left(\frac{2 \pi}{L} \cdot (z+d) \right)}{\sinh \left(\frac{2 \pi}{L} \cdot d \right)} \),
Wave profile \( \eta (t) = \frac{H}{2} \cdot \cos(- \frac{2 \cdot \pi}{T} \cdot t)\).

Force coefficients for the Morison equation according to Shore Protection Manual (SPM):
Drag force coefficient \( C_D = \begin{cases} 1.2 & \text{ for } Re < 2.0 \cdot 10^5 \\ \frac{23}{15} - \frac{Re}{6 \cdot 10^5} & \text{ for } 2.0 \cdot 10^5 < Re < 5.0 \cdot 10^5 \\ 0.7 & \text{ for } Re > 5.0 \cdot 10^5 \end{cases}\)

Inertia force coefficient \( C_M = \begin{cases} 2.0 & \text{ for } Re < 2.5 \cdot 10^5 \\ 2.5 - \frac{Re}{5 \cdot 10^5} & \text{ for } 2.5 \cdot 10^5 < Re < 5.0 \cdot 10^5 \\ 1.5 & \text{ for } Re > 5.0 \cdot 10^5 \end{cases}\)

The field of Loads on Structures is part of the module Coastal Dynamics and Engineering Design within the specialization Coastal and Ocean Engineering of the master programmes Civil and Environmental Engineering.