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Dimensionless numbers for dynamic response analysis of clamped square plates subjected to blast loading

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Abstract

A dimensionless governing equation of clamped square plates was obtained by introducing dimensional analysis method to the basic governing equations of plates. The dimensionless governing equation includes three different influence aspects on structural dynamic response: the geometry of the structure, the dynamic resistance ability of material and the ratio of dynamic loads to the resistance ability of material. The dimensionless governing equation was applied to the dynamic response study of plates under blast loading, and a new dimensionless number was suggested for clamped square plates under explosion loads. The suggested dimensionless number has clear physical meaning, and the parameters included in the dimensionless number are easy to get. The dimensionless number suggested in this paper was applied to analysis the experimental data of clamped square plates under blast loading. Meanwhile, comparative analysis was proceeded which indicated the dimensionless number suggested in this paper can be effectively used to predict the dynamic response. The results showed the deflection–thickness ratio is in direct proportion to the dimensionless damage number. An empirical expression was obtained which can be used to the prediction of dynamic response of similar clamped plate under different blast loading condition.

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Abbreviations

\(C_{0}\) :

Sound velocity of environment

Dn :

Johnson damage number

\(D_{\mathrm{ex}}\) :

Dimensionless number suggested in this paper

\(E\) :

Elastic modulus

\(E_{\mathrm{e}}\) :

Explosive energy per unit volume

\(H\) :

Plate thickness

\(I\) :

Impulse

\(k\) :

Constant in Cuer formulation

\(L\) :

Plate length

\(M_{{x}}, M_{{y}}\) :

Bending moment per unit length

\(M_{{xy}}\) :

Torsion moment per unit length

\(m_{{X}}, m_{{Y}}\) :

Dimensionless bending moment

\(m_{{xY}}\) :

Dimensionless torsion moment

\(M_{0}\) :

Fully plastic bending moment per unit length \(M_{0}=\sigma _{0}H^{2 }/ 4\)

\(p\) :

Pressure

\(Q\) :

Total explosive energy

\(Q_{x}, Q_{y}\) :

Transverse shear per unit length

\(R\) :

Explosive distance

\(R_{0}\) :

Charge diameter

Rn :

Zhao’s response number

\(t\) :

Time

\(T\) :

Dimensionless time

\(u\) :

Transverse deflection

\(U\) :

Dimensionless transverse deflection

\(V\) :

Explosive volume

\(V_{0}\) :

Initial velocity

\(W\) :

Explosive mass

\(x\), \(y\) :

Coordinate

\(X\), \(Y\) :

Dimensionless coordinate

\(Z\) :

Scale distance

\(\alpha \) :

Constant in Cuer formulation

\(\nu \) :

Dimensionless number suggested by Li and Jones

\(\mu \) :

Mass per unit area

\(\tau \) :

Duration of the impulse

\(\sigma _{0}\) :

Yield stress

\(\phi _{q}\) :

Dimensionless number suggested by Nurick and Martin

\(\rho _\mathrm{s}\) :

Density of structure

\(\delta \) :

Largest deflection of plate

References

  1. Luccioni, B.M., Ambrosini, R.D., Danesi, R.F.: Analysis of building collapse under blast loads. Eng. Struct. 26(1), 63–71 (2004)

    Article  Google Scholar 

  2. The Blue Ribbon Panel on Bridge and Tunnel Security: Recommendations for bridge and tunnel security. FHWA, Washington (2003)

  3. Islam, A., Yazdani, N.: Performance of AASHTO girder bridges under blast loading. Eng. Struct. 30(7), 1922–1937 (2008)

    Article  Google Scholar 

  4. Zhang, D., Yao, S.J., Lu, F.Y., Chen, X.G., Lin, G.H., Wang, W., et al.: Experimental study on scaling of RC beams under close-in blast loading. Eng. Fail. Anal. 33, 497–504 (2013)

    Article  Google Scholar 

  5. Wang, W., Zhang, D., Lu, F.Y., Wang, S.C., Tang, F.J.: Experimental study and numerical simulation of the damage mode of a square reinforced concrete slab under close-in explosion. Eng. Fail. Anal. 27, 41–51 (2013)

    Article  Google Scholar 

  6. Bridgman, P.W.: Dimensional Analysis. Yale University Press, New Haven (1922)

    Google Scholar 

  7. Lin, C.C., Segel, L.A.: Mathematics Applied to Deterministic Problems in Natural Sciences. Macmillan, New York (1974)

    MATH  Google Scholar 

  8. Baker, W.E., Westine, P.S.: Similarity Methods in Engineering Dynamics. Hayden Books, El Paso (1973)

    Google Scholar 

  9. Johnson, W.: Impact Strength of Materials. Edward Arnold, London (1972)

    MATH  Google Scholar 

  10. Johnson, W., Sengupta, A.K., Ghosh, S.K., Reid, S.R.: Mechanics of high speed impact at normal incidence between plasticine long rods and plates. J. Mech. Phys. Solids 29, 413–445 (1981)

    Article  Google Scholar 

  11. Zhao, Y.P.: On some dimensionless numbers in fracture mechanics. Int. J. Fract. 83, 7–13 (1997)

    Google Scholar 

  12. Zhao, Y.P.: Suggestion of a new dimensionless number for dynamic plastic response of beams and plates. Arch. Appl. Mech. 68, 524–538 (1998)

    Article  MATH  Google Scholar 

  13. Fang, J., Zhao, Y.P.: Experimental investigation into the dynamic behavior of circular rings with arc-shaped supports. Int. J. Press. Vessel Pip. 62, 83–85 (1995)

    Article  Google Scholar 

  14. Zhao, Y.P., Fang, J., Yu, T.X.: Experimental and theoretical investigations into notched circular rings under impact loading. DYMAT J. 2, 135–142 (1995)

    Google Scholar 

  15. Li, Q.M., Jones, N.: On dimensionless number for dynamic plastic response of structural members. Arch. Appl. Mech. 70, 245–254 (2000)

    Article  MATH  Google Scholar 

  16. Nurick, G.N., Martin, J.B.: Deformation of thin plates subjected to impulsive loading—a review. Part I: Theoretical considerations. Int. J. Impact Eng. 8(2), 159–169 (1989)

    Article  Google Scholar 

  17. Jones, N.: Quasi-static analysis of structural impact damage. J. Constr. Steel Res. 33, 151–177 (1995)

    Article  Google Scholar 

  18. Jones, N.: Structural Impact. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  19. Sadovskyi, M.A.: Mechanical Action of Air Shock Waves of Explosion, Based on Experimental Data. Izd Akad Nauk SSSR, Moscow (1952)

    Google Scholar 

  20. Brode, H.L.: Blast wave from a spherical charge. Phys. Fluids 2, 217 (1959)

    Article  MATH  Google Scholar 

  21. Henrych, J.: The Dynamics of Explosion and Its Use. Elsevier, Amsterdam (1979)

    Google Scholar 

  22. Baker, W.E.: Explosion Hazards and Evaluation. Elsevier, New York (1983)

    Google Scholar 

  23. Cole, R.H.: Underwater Explosion. Princeton University Press, Princeton (1948)

    Book  Google Scholar 

  24. Jacob, N., Chung Kim Yuen, S., Nurick, G.N., Bonorchis, D., Desai, S.A., Tait, D.: Scaling aspects of quadrangular plates subjected to localised blast loads—experiments and predictions. Int. J. Impact Eng. 30(8–9), 1179–1208 (2004)

    Article  Google Scholar 

  25. Nurick, G.N., Teeling-Smith, R.G.: The deformation and tearing of thin circular plates subjected to impulsive loads. Int. J. Impact Eng. 11, 77–91 (1991)

    Article  Google Scholar 

  26. Nurick, G.N., Olson, M.D., Fagnan, J.R.: Deformation and rupture of blast loaded square plates—predictions and experiments. Int. J. Impact Eng. 13, 279–291 (1993)

    Article  Google Scholar 

  27. Nurick, G.N., Bonorchis, D.: The influnce of boundary conditions on the loading of rectangular plates subjected to blast loading—importance in numerical simulations. Int. J. Impact Eng. 36, 40–52 (2009)

    Article  Google Scholar 

  28. Rajendran, R., Lee, J.M.: Blast loaded plates. Mar. Struct. 22(2), 99–127 (2009)

    Article  Google Scholar 

  29. Wang, W., Zhang, D., Lu, F.Y., Wang, S.C., Tang, F.J.: Experimental study on scaling the explosion resistance of a one-way square reinforced concrete slab under a close-in blast loading. Int. J. Impact Eng. 49, 158–164 (2012)

    Article  Google Scholar 

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Acknowledgments

The research reported herein is supported by the National Natural Science Foundation of China (Project No. 11202236) which is gratefully acknowledged.

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  1. College of Science, National University of Defense Technology, Changsha, 410073, Hunan, People’s Republic of China

    Shujian Yao, Duo Zhang & Fangyun Lu

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  1. Shujian Yao
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Correspondence to Fangyun Lu.

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Yao, S., Zhang, D. & Lu, F. Dimensionless numbers for dynamic response analysis of clamped square plates subjected to blast loading. Arch Appl Mech 85, 735–744 (2015). https://doi.org/10.1007/s00419-015-0986-7

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