Design of Foundations for Offshore Wind Turbines

Appendix BEuler‐Bernoulli Beam Equation

The readers are referred to Adhikari and Bhattacharya (2010), Bhattacharya and Adhikari (2011), and Arany et al. (2015a,b, 2016, 2017) for comprehensive understanding. The fundamentals are provided here.

The equation of motion using the Euler‐Bernoulli beam model for a beam with an axial force is

B.1

where EI(z) is the bending stiffness distribution along the axial coordinate z, μ(z) is the distribution of mass per unit length, P^* is the axial force acting on the beam due to the top head mass and the self‐weight of the tower, p(z, t) is the excitation of the beam, w(z, t) is the deflection profile.

Using constant equivalent values for the axial force, bending stiffness and mass per length, and considering free harmonic vibration of the beam with separation of variables w(z, t) = W(z) · e^iωt, the equation can be reduced to the following using the non‐dimensional parameters of Table B.1 and the dimensionless axial coordinate ξ = z/L:

B.2

Table B.1 Nondimensional variables.

Dimensionless group	Formula	Dimensionless group	Formula
Nondimensional lateral stiffness		Nondimensional axial force
Nondimensional rotational stiffness