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List of symbols

1P	The fundamental rotational frequency
A_b	Bolt stress area
A_f	Effective cross-sectional area for the compressed flanges
A_s	Surface area to be protected
A_mid	Area inscribed by the mid-thickness line
A	Cross-sectional area
CM_xoff	Offset of rotor nacelle assembly (RNA) along x from tower-top centerline
CM_yoff	Offset of RNA along y from tower-top centerline
CM_zoff	Offset of RNA along z from tower-top centerline
C_d	Drag coefficient
C_m	Added mass coefficient
C	Constant in the σ–N curve
DTR_b	Tower-base DTR
Table Continued

DTR_t	Tower-top DTR
DTR	Diameter-to-thickness ratio
D_b	Tower-base outer diameter (OD)
D_t	Tower-top OD
D_bc	Bolt circle diameter in the flange connection
D_fat	Fatigue damage
D_sh,m	Shell mid-wall diameter
D_sh	Shell OD
EUUtil	Shell buckling utilization ratio
E_b	Bolt material Young's modulus
E_f	Flange material Young's modulus
E	Young's modulus
F_d	Generic, design (factored) load within the load resistance factor design (LRFD) approach
F_j	Forcing associated with the j-th mode of vibration
F_k	Generic, characteristic load within the LRFD approach
F_p	Bolt preload
F_t	Bolt tension load
F_z	Shell tension load
F_si	Horizontal load from vessel impact
F_t,RD	Bolt strength load
F_u,A	Shell equivalent load in the failure mode A according to the flange segment-model
F_u,B	Shell equivalent load in the failure mode B according to the flange segment-model
F_u,C	Shell equivalent load in the failure mode C according to the flange segment-model
F_ult	Ultimate resistance load of the flange connection
F_xRNA	Force from the RNA along the x axis
F_yRNA	Force from the RNA along the y axis
F_z1	Threshold value of F_z tension load in Schmidt/Neuper's method
F_z2	Threshold value of F_z tension load in Schmidt/Neuper's method
F_zRNA	Aerodynamic force from the RNA along the z axis
F_zcr	Value of F_z tension load that would cause flange separation
Table Continued

GLUtil	Global buckling utilization ratio
G_f	Gust factor
H_w	Wave height
Htwr₂	Height of tower waist
I_xx	Mass second moment of inertia about the x axis
I_xz	Mass cross-moment of inertia about the x and z axes
I_yy	Mass second moment of inertia about the y axis
I_zz	Mass second moment of inertia about the z axis
I	Protective current density (A/m²)
J_xx	Cross-sectional area moment of inertia
K_α	Ratio of bolt stiffness to total connection stiffness
K_β	Ratio of flange stiffness to total connection stiffness
K_b	Bolt axial stiffness
K_f	Equivalent stiffness of the compressed flange pair
K_x	Equivalent soil–structure interaction (SSI) spring constant along x (=y) axis
K_z	Equivalent SSI spring constant along z axis
K_θx	Equivalent SSI rotational (about x (=y) axis) spring constant
K_θz	Equivalent SSI torsional (about z axis) spring constant
L_b	Bolt effective length
L_p	Pile embedment length
L	Tower length, or SSt length
M_d	Design (factored) bending moment load at the station of interest
M_p	Bending moment load resistance at the tower station of interest
M_x	Component of the bending moment along the x axis at the station of interest
M_y	Component of the bending moment load along the y axis at the station of interest
M_z	Torsion moment load along the z axis at the station of interest
M_Pl,2	Flange equivalent plastic bending (resistance) load in failure mode C according to the flange segment-model
M_Pl,3,MN	Shell equivalent plastic bending (resistance) load in failure modes B and C according to the flange segment-model, accounting for bending and tension interaction
M_Pl,3	Shell equivalent plastic bending (resistance) load in failure modes B and C according to the flange segment-model
M_xRNA	RNA aerodynamic moment along the x axis
Table Continued

M_yRNA	RNA aerodynamic moment along the y axis
M_zRNA	RNA aerodynamic moment along the z axis
N_d	Design (factored) normal load at the tower station of interest
N_i	Number of cycles at failure for the i-th load range level
N_p	Normal (axial) load resistance at the tower station of interest
N_DEL	Number of cycles at failure for the damage equivalent stress (DES) stress range
N_Pl,3	Shell equivalent plastic (resistance) load in failure modes B and C according to the flange segment-model
P–Δ	P–Δ effect
Q	Practical current capacity (Ah/kg)
R(f_d)	Probability distribution of the generic, design (factored) material resistance within the LRFD approach
S(F_d)	Probability distribution of the generic, design (factored) load within the LRFD approach
S_t	Strouhal number
S_a,i	i-th bin load-range
T_L	Expected lifetime of the component
T_c	Expected useful lifetime of the coating
T_d	Design (factored) shear load at the station of interest
T_x	Component of the shear load along the x axis at the station of interest
T_y	Component of the shear load along the y axis at the station of interest
U_cr	Critical wind velocity for vortex shedding
U_ref	Hub height reference wind speed in simple design load case (DLC) loads' analysis
$\dot{C}$	Corrosion rate
${\hat{N}}_{i}$	Number of cycles at failure for the i-th load range level, accounting for partial safety factors (PSFs)
$\hat{i}$	Unit vector along x axis
$\hat{j}$	Unit vector along y axis
U_w	Wave and current velocity
U_hub	Wind velocity at hub height
U	Wind velocity
f_a	Force per unit length due to wind aerodynamic drag
f_w	Force per unit length due to wave and current kinematics

ω_0,j	First natural frequency in rad associated with the j-th mode of vibration
a_si	Added mass coefficient during collision (1.4–1.6 sideway impact, 1.1 bow or stern collision)
a	Distance from bolt centerline to flange edge
b	Distance from bolt centerline to shell mid-wall centerline in the flange connection
c_f	Flange circular segment net length (segment-model)
c_j	Damping coefficient for the j-th mode of vibration
c_s	Shell circular segment length (segment-model)
c_c,j	Critical damping coefficient for the j-th mode of vibration
c_si	Stiffness of the impacting part of the vessel
d_b	Bolt hole diameter
d_w	Water depth
f₀	First natural frequency in Hz
f_d	Generic, design (factored) resistance within the LRFD approach
f_k	Generic, characteristic material resistance within the LRFD approach
f_y	Characteristic yield stress
f_obj	Objective function in the optimization problem formulation
f_u,b	Bolt characteristic ultimate strength
f_y,b	Bolt characteristic yield strength
f_y,f	Flange characteristic yield strength
f_y,s	Shell characteristic yield strength
g_cnt	Constraint function in the optimization problem formulation
g	Gravity acceleration
j	Protective current density (A/m²)
k_i	Interaction (axial-hoop stresses) factor in the local buckling utilization calculation
k_j	Stiffness associated with the j-th mode of vibration
k_w	Dynamic pressure factor to calculate hoop stresses—a function of cylinder dimensions and external pressure buckling factor per [2]
k_z	Exponent factor for the axial stress ratio, in the local buckling utilization calculation
k_lat	Lateral restraint spring constant
k_rot	Rotational restraint spring constant
Table Continued

m_a	Anode mass
m_j	Modal mass associated with the j-th mode of vibration
m_RNA	RNA mass
m_si	Displacement mass of impact vessel
m_twr	Tower mass, or SSt mass
m	Inverse exponent in the σ–N curve
nP	n-blades times the fundamental rotational frequency
n_b	Number of bolts in the flange connection
n_i	Number of cycles at the i-th load range
n_DEL	Reference number of cycles for the damage equivalent load (DEL) load range
n	Number of rotor blades
q_max	Maximum wind dynamic pressure
t_f	Flange thickness
t_s	Shell thickness
u(x_d)	States in the optimization problem formulation
v_si	vessel impact speed
x_d	States in the optimization problem formulation
x_j	Degree of freedom (DOF) variable associated with the j-th mode of vibration
y	Local tower station horizontal displacement
z_RNA	z coordinate of the RNA center of mass (CM)
z_hub	Hub height above mean sea level (MSL)
z	Altitude above MSL

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