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Technical Briefs

# A Simple Model for Axial Displacement in a Cylindrical Pipe With Internal Shock Loading

[+] Author and Article Information
Neal P. Bitter

e-mail: nbitter@caltech.edu

Joseph E. Shepherd

California Institute of Technology,

Manuscript received April 22, 2013; final manuscript received July 30, 2013; accepted manuscript posted October 16, 2013; published online October 16, 2013. Assoc. Editor: Weinong Chen.

J. Appl. Mech 81(3), 034505 (Oct 16, 2013) (8 pages) Paper No: JAM-13-1167; doi: 10.1115/1.4025270 History: Received April 22, 2013; Revised July 30, 2013; Accepted October 16, 2013

## Abstract

This paper describes a simplified model for predicting the axial displacement, stress, and strain in pipes subjected to internal shock waves. This model involves the neglect of radial and rotary inertia of the pipe, so its predictions represent the spatially averaged or low-pass–filtered response of the tube. The simplified model is developed first by application of the physical principles of conservation of mass and momentum on each side of the shock wave. This model is then reproduced using the mathematical theory of the Green's function, which allows other load and boundary conditions to be more easily incorporated. Comparisons with finite element simulations demonstrate that the simple model adequately captures the tube's axial motion, except near the critical velocity corresponding to the bar wave speed $E/ρ$. Near this point, the simplified model, despite being an unsteady model, predicts a time-independent resonance, while the finite element model predicts resonance that grows with time.

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## References

Beltman, W., Burscu, E., Shepherd, J., and Zuhal, L., 1999, “The Structural Response of Cylindrical Shells to Internal Shock Loading,” ASME J. Pressure Vessel Technol., 121(3), pp. 315–322.
Beltman, W., and Shepherd, J., 2002, “Linear Elastic Response of Tubes to Internal Detonation Loading,” J. Sound Vib., 252(4), pp. 617–655.
Shepherd, J. E., and Akbar, R., 2009, “Forces Due to Detonation Propagation in a Bend,” Graduate Aeronautical Laboratories California Institute of Technology, Pasadena, CA, Technical Report No. FM2008-002.
Shepherd, J. E., and Akbar, R., 2009, “Piping System Response to Detonations. Results of ES1, TS1 and SS1 Testing,” Graduate Aeronautical Laboratories California Institute of Technology, Pasadena, CA, Technical Report No. FM2009-001.
Karnesky, J., Damazo, J. S., Chow-Yee, K., Rusinek, A., and Shepherd, J. E., 2013, “Plastic Deformation Due to Reflected Detonation,” Int. J. Solids Struct., 50(1), pp. 97–110.
Timoshenko, S. P., 1934, Theory of Elasticity, 1st ed., McGraw-Hill, New York.
Kolsky, H., 1963, Stress Waves in Solids, Dover, New York.
Haberman, R., 2004, Applied Partial Differential Equations, 4th ed., Pearson Education, Upper Saddle River, NJ.
Schiffner, K., and Steele, C. R., 1971, “Cylindrical Shell With an Axisymmetric Moving Load,” AIAA J., 9(1), pp. 37–47.

## Figures

Fig. 1

Simulation results for radial and axial displacement on the outer surface of the pipe at 225 μs; the shock wave front is located at 0.47 m and the bar wave front is at 1.1 m

Fig. 2

Simulation results for hoop and axial strains on the outer surface of the pipe at 225 μs

Fig. 3

Simulation results for radial and axial displacement wave systems shown at increments of 100 μs. For clarity, successive traces have the zero values offset by an amount proportional to the time increment.

Fig. 4

Selected axial displacements from Fig. 3 shown without offsetting the zero strain

Fig. 5

Extremum axial displacement (at x = υt) as a function of time showing the linear relationship

Fig. 6

Wave diagram for a shock wave traveling along a tube at speed υ

Fig. 7

Comparison of axial displacements from finite element simulations and the present simple model. Boundary condition is u(0, t) = 0. Simulation conditions are those summarized in Table 1.

Fig. 8

Comparison of axial displacement from finite element simulations and the present simple model. Boundary condition is Nx(0, t) = 0. Simulation conditions are summarized in Table 1.

Fig. 9

Comparison of axial strain profiles for several speeds of pressure load. For the case of υ / υb = 1.0, the model predicts a function of infinite height and zero width. The boundary condition is Nx(0, t) = 0, and strains are normalized by the static hoop strain Poa/Eh.

Fig. 10

Average axial strains in precursor and primary wave regions. Symbols correspond to finite element simulations and solid lines to the simple model. Boundary condition is Nx(0, t) = 0.

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