# Impact-induced flexural response of axially loaded uniform Timoshenko beams with non-classical ends: a sensitivity study of the dynamic load factor

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## Abstract

A dynamic analysis of axially loaded Timoshenko beams with intermediate fixities is presented. The underwater part of a craft is modeled as a flexible beam, which rises out and slams against the water at a large vertical velocity, causing highly localized hydrodynamic impact pressure moving at high velocities across the beam, setting it into high-frequency vibrations. The beam natural frequencies depend on the slenderness ratio, axial load, end fixities, and structural damping. The natural frequencies and modeshapes (for total deflection and pure bending slope) are generated through Eigen analysis. Next, normal mode summation is used to analyze the impact-induced vibration response, which is generated for various impact speeds, deadrise angles, end fixities, and axial loads, of the beam. A parametric study is done to predict the maximum dynamic stresses on the structure. The sensitivity of the dynamic load factor (DLF) is studied with respect to the above parameter space. Conclusions are drawn leading to insights into sound structural designs.

## Keywords

Slamming loads Timoshenko beam Axial load Modal analysis Sensitivity study Non-classical ends## Abbreviations

- \(x\)
Independent space variable along the beam

- \(t\)
Independent variable in time

- \(z\left( {x,t} \right)\)
Dynamic flexural deflection of the beam

- \({z_{{\text{st}}}}\left( {x,t} \right)\)
Static flexural deflection of the beam

- \(\Phi \left( {x,t} \right)\)
Pure-bending slope of the beam

- \(L\)
Length of the beam

- \(\rho \)
Density of the beam material

- \(E\)
Elastic modulus of the beam material

- \(I\)
Second moment of area of the beam cross-section about the horizontal neutral axis.

- \(G\)
Shear modulus of the beam material

*µ*Shear correction factor (5/6 for rectangular cross-section)

*N*Axial load

*A*Cross-sectional area of the beam

- \({K_{\theta {\text{L}}}}\)
Spring constant on the left end

- \({K_{\theta {\text{R}}}}\)
Spring constant on the right end

- \({\phi _{\text{j}}}\left( x \right)\)
Beam modeshape

- \({\psi _{\text{j}}}\left( x \right)\)
Pure bending slope modeshape

*δ*_{j},*γ*_{j}*J*th frequency parameter pair for Timoshenko beam*q*_{j}(*t*)Principal coordinate

- \(F\left( {x,t} \right)\)
External transient load

- \({\omega _{n1}}\)
Fundamental natural frequency of the beam

- \({T_{{\text{n1}}}}\)
Fundamental natural period of the beam

- \(\tau \)
Non-D splash time

- \(V\)
Vertical impact velocity of slamming

- \(\beta \)
Deadrise angle of the craft

- DLF
Dynamic loading factor

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