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Marine Systems & Ocean Technology

, Volume 14, Issue 1, pp 1–11 | Cite as

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

  • N. DattaEmail author
  • Mohd. Atif Siddiqui
Article
  • 15 Downloads

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

δj, γj

Jth frequency parameter pair for Timoshenko beam

qj(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

References

  1. 1.
    A. Bokaian, Natural frequencies of beams under tensile axial loads. J. Sound Vib. 142(3), 481–498 (1990)CrossRefGoogle Scholar
  2. 2.
    Y.H. Lin, Vibration analysis of Timoshenko beams traversed by moving loads. J. Marin. Sci. Technol. 2(1), 25–35 (1994)Google Scholar
  3. 3.
    T.P. Chang, Deterministic and random vibration of an axially loaded Timoshenko beam resting on an elastic foundation, J. Sound Vib. 178(1), 55–66 (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    S.H. Farchaly, M.G. Shebl, Exact frequency and modeshape formulae for studying vibration and stability of Timoshenko beam system. J. Sound Vib. 180(2), 205–227 (1995)CrossRefzbMATHGoogle Scholar
  5. 5.
    R.T. Wang, Vibration of multi-span Timoshenko beams to a moving force. J. Sound Vib. 207(5), 731–742 (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    L. Majkut, Free and forced vibrations of Timoshenko beams described by single difference equation. J. Theor. Appl. Mech. 47(1), 193–210 (2009)Google Scholar
  7. 7.
    N. Datta, D. Kim, A.W. Troesch, Hydrodynamic impact-induced vibration characteristics of a uniform Euler–Bernoulli beam. International symposium on vibro-impact dynamics of ocean systems and related problems, Troy, Michigan, Oct (2008)Google Scholar
  8. 8.
    N. Datta, M.A. Siddiqui, Hydroelastic analysis of axially loaded Timoshenko beams with intermediate end fixities under hydrodynamic slamming loads. Ocean Eng. 127, 124–134 (2016)CrossRefGoogle Scholar
  9. 9.
    P. Rassinot, A.E. Mansour, Ship hull bottom slamming. J. Offshore Mech. Arct. Eng. 117(4), 252–259 (1995)CrossRefGoogle Scholar
  10. 10.
    O.M. Faltinsen, The effect of hydroelasticity on ship slamming. Philos. Trans. Royal Soc. A: Math. Phys. Eng. Sci. 355(1724), 575–591 (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    A. Korobkin, R. Gueret, Š. Malenica, Hydroelastic coupling of beam finite element model with Wagner theory of water impact. J. Fluids Struct. 22(4), 493–504 (2006)CrossRefGoogle Scholar
  12. 12.
    T.I. Khabakhpasheva, A.A. Korobkin, Elastic wedge impact onto a liquid surface: Wagner’s solution and approximate models. J. Fluids Struct. 36, 32–49 (2013)CrossRefGoogle Scholar
  13. 13.
    A.A. Korobkin, T.I. Khabakhpasheva, Regular wave impact onto an elastic plate. J. Eng. Math. 55(1–4), 127–150 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    O.M. Faltinsen, Water entry of a wedge by hydroelastic orthotropic plate theory. J. Ship Res. 43(3), 180–193 (1999)Google Scholar
  15. 15.
    I. Stenius, A. Rosén, J. Kuttenkeuler, Hydroelastic interaction in panel-water impacts of high-speed craft. Ocean Eng. 38(2), 371–381 (2011)CrossRefGoogle Scholar
  16. 16.
    T.I. Khabakhpasheva, A.A. Korobkin, Approximate models of elastic wedge impact. 18th International workshop on water waves and floating bodies, April (2003)Google Scholar
  17. 17.
    N. Datta, A.W. Troesch, Hydroelastic response of Kirchhoff plates to transient hydrodynamic impact loads. Mar. Syst. Ocean Technol. 7(2), 77–94 (2012)Google Scholar
  18. 18.
    N. Datta, Hydroelastic Response of Marine Structures to Impact-Induced Vibrations, (University of Michigan, Michigan, 2010)Google Scholar

Copyright information

© Sociedade Brasileira de Engenharia Naval 2018

Authors and Affiliations

  1. 1.Indian Institute of TechnologyKharagpurIndia
  2. 2.Buoyancy ConsultantsGoaIndia

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