# Wave-induced flexural response of idealized non-uniform hull girder in random seas

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## Abstract

The aim is to analyze the wave-induced vertical vibration of a non-prismatic mathematical hull in a stochastic sea, by normal mode analysis. The hull has been generated mathematically, to represent two distinct Indian merchant vessels: DS (Tanker) and SCIM (Containership). The body-plan, deck waterline, bow and stern profiles, have been modeled as semi-superellipses. These render non-uniform distributions of mass and stiffness over the ship-length. The energy-based Rayleigh–Ritz method has been used to analyze the idealized hull girder natural frequencies and modeshapes. The non-uniform beam modeshape is a weighted series sum of prismatic beam-free vibration modeshapes. The 2D added mass of superelliptic sections is formulated, solving the radiation boundary value problem by the constant strength source distribution method. The hull girder is subject to the Pierson–Moskowitz sea spectrum in fully loaded condition. The diffraction force is formulated through the Khaskind’s relations. The flexural response of the girder is evaluated by the modal superposition method. The response spectra have been generated for various sea states and ship speeds. The magnitudes of the maximum flexural/shear stress for each vessel are generated. The probability of shear/tensile failure is also estimated, giving insights into the hull structural design.

## Keywords

Non-uniform beam Mathematical hull Rayleigh–Ritz method Ship springing Response spectrum## List of symbols

- \(x\)
Independent variable along length of the hull (m)

- \(y(x,z)\)
Offset from hull centreline (m)

- \(z\)
Independent variable along the depth of the hull (m)

- \(t\)
Independent variable in time (s)

- \(m(x)\)
Mass per unit length (kg/m)

- \(I(x)\)
Second moment of cross-section area about horizontal neutral axis (m

^{4})- \(E\)
Modulus of elasticity (N/m)

- LOA
Length overall of the hull (m)

- L
Length between perpendiculars (m)

- B
Moulded breadth (m)

- D
Moulded depth (m)

- T
Moulded draught (m)

- \(a(x)\)
Local half-breadth (m)

- \(b(x)\)
Local depth (m)

- \(p(x)\)
Rectellipse power (–)

- \(q(x)\)
Rectellipse power (–)

- \(\Delta\)
Displacement (ton)

- \({C_{\text{b}}}\)
Block coefficient (–)

- \(\lambda\)
Wavelength of ocean wave (m)

- \({\omega _e}\)
Wave circular frequency (rad/s)

- \(J\)
3D added mass/2D added mass (–)

- \(a_{{j,33}}^{{2{\text{D}}}}(x)\)
2D added mass coefficient in heave due to heave associated with

*j*th flexural mode (–)- \(z(x,t)\)
Vertical flexural displacement of beam (m)

- \({\Phi _j}(x)\)
*J*th non-uniform beam modeshape (–)- \({q_j}(t)\)
*J*th principal coordinate (m/s)- \({a_{jk}}\)
Weight of the contribution of \({\varphi _k}(x)\) to \({\Phi _j}(x)\) (–)

- \({\varphi _j}(x)\)
*J*th uniform beam modeshape (–)- \({\alpha _{jk}}\)
Generalized stiffness matrix in Rayleigh–Ritz method

- \({\beta _{jk}}\)
Generalized mass matrix in Rayleigh–Ritz method

- \({A_0}\)
Material cross-section area of midship section (m

^{2})- \({I_0}\)
Vertical second moment of inertia of the midship section (m

^{4})- \(\gamma\)
Spatial frequency of uniform beam vibration (1/m)

- \(\rho\)
Material density (kg/m

^{3})- \({\Psi _3}\left( {x,z,t} \right)\)
2D heave radiation potential (m

^{2}/s)- \(b_{{33}}^{{2{\text{D}}}}(x)\)
2D radiation damping coefficient in heave due to heave associated with

*j*th flexural mode (–)- \({G_{PQ}}\left( {y,z;\xi ,\eta } \right)\)
2D constant strength Green’s function between field point

*P*and source point*Q*- \(P(y,z)\)
Field point (–)

- \(Q(\xi ,\eta )\)
2D source point (–)

- \({r_{PQ,n}}\)
Distance between \(P\) and

*n*th source \(Q\) (m)- \({\omega _j}\)
*J*th natural frequency of non-uniform beam (rad/s)- \({\psi _j}(x,z)\)
*J*th auxiliary radiation potential (m)- \({\psi _0}\left( {x,z,t} \right)\)
2D incident wave potential (m

^{2}/s)- \({\psi _D}\left( {x,z,t} \right)\)
Wave diffraction potential (m

^{2}/s)- \(\zeta\)
Wave amplitude (m)

- \(k\)
Wave number of incident wave (1/m)

- \(S({\omega _{\text{e}}})\)
Sea spectrum ordinate

- \({\Psi _3}\left( {x,z,t} \right)\)
2D radiation potential due to heave (m

^{2}/s)- \({\text{G}}{{\text{M}}_{jk}}\)
Generalized mass in mode summation method

- \({\text{G}}{{\text{A}}_{jk}}\)
Generalized added mass in mode summation method

- \({\text{G}}{{\text{K}}_{jk}}\)
Generalized stiffness in mode summation method

- \(M(x,t)\)
Bending moment (N-m)

- \(V\left( {x,t} \right)\)
Shear force (N)

- \({\sigma _{xx}}\left( {x,t} \right)\)
Normal stress (N/m

^{2})- \({\sigma _{xx}}\left( {x,t} \right)\)
Shear stress (N/m

^{2})- \({T_z}\)
Average zero-up crossing period (s)

- H
_{1/3} Significant wave height (m)

- \(\theta\)
Heading angle (rad)

- \({\omega _{{\text{enc}}}}\)
Wave encounter frequency (rad/s)

- \(V\)
Ship speed (m/s)

- \(g\)
Acceleration due to gravity (m/s

^{2})- N
Number of waves (–)

- \(p({\sigma _{{\text{amp}}}})\)
Probability density of normal stress amplitude (–)

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