An Approximate Theory of Lateral Impact on Beams

1955 ◽  
Vol 22 (1) ◽  
pp. 69-76
Author(s):  
B. A. Boley
Keyword(s):  
Sudden Change ◽  
Approximate Theory ◽  
Lateral Impact ◽  
Governing Equations ◽  
Transverse Impact ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Lateral Contraction ◽  
Velocity Of Propagation ◽  
Variation Technique

Abstract The approximate theory derived in this paper describes, by means of a “traveling-wave” approach, the behavior of beams under transverse impact. Lateral impact is considered in detail, namely, one in which a section of the beam undergoes a sudden change in velocity or shear force. The theory considers the effects of shear deformations and of rotatory inertia according to Timoshenko’s model, and that of lateral contraction as suggested by Love. The governing equations and the boundary conditions are developed with the aid of an energy-variation technique. Numerical examples are given in which the behavior of the boundary layer near the point of impact is examined. For one of these the exact solution is available and is in agreement with the present approximate results. Some general considerations concerning the velocity of propagation also are discussed.

Response of a Submerged Cylindrical Shell to an Axially Propagating Step Wave

1965 ◽  
Vol 32 (4) ◽  
pp. 788-792 ◽  
Author(s):  
M. J. Forrestal ◽  
G. Herrmann
Keyword(s):  
Cylindrical Shell ◽  
Steady State ◽  
Wave Front ◽  
Transverse Shear ◽  
Shell Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Axial Coordinate

An infinitely long, circular, cylindrical shell is submerged in an acoustic medium and subjected to a plane, axially propagating step wave. The fluid-shell interaction is approximated by neglecting fluid motions in the axial direction, thereby assuming that cylindrical waves radiate away from the shell independently of the axial coordinate. Rotatory inertia and transverse shear deformations are included in the shell equations of motion, and a steady-state solution is obtained by combining the independent variables, time and the axial coordinate, through a transformation that measures the shell response from the advancing wave front. Results from the steady-state solution for the case of steel shells submerged in water are presented using both the Timoshenko-type shell theory and the bending shell theory. It is shown that previous solutions, which assumed plane waves radiated away from the vibrating shell, overestimated the dumping effect of the fluid, and that the inclusion of transverse shear deformations and rotatory inertia have an effect on the response ahead of the wave front.

Experimental Evaluation of Material Behavior in a Wire Under Transverse Impact

1967 ◽  
Vol 34 (2) ◽  
pp. 392-396 ◽  
Author(s):  
A. B. Schultz ◽  
P. A. Tuschak ◽  
A. A. Vicario
Keyword(s):  
Aluminum Alloy ◽  
Pure Copper ◽  
Closed Form Solution ◽  
Strain Curve ◽  
Simple Wave ◽  
Form Solution ◽  
Material Behavior ◽  
Governing Equations ◽  
Transverse Impact ◽  
Behavior Consistency

Results are reported from an experiment involving photographic observation of constant-velocity transverse impact on long wires of annealed pure copper, two pure aluminums, and an aluminum alloy. Predictions of deformation are made assuming the quasi-static stress-strain curve governs behavior. Consistency with experimental observations is examined. Predictions are based on a closed-form solution to the problem, which is shown to be a compounding of two simple wave solutions of the governing equations. Predictions are consistent with observations for the aluminum alloy even under conditions of moderate or high static prestrain. The two pure aluminums and the copper show consistency at low but not at high strain levels. Highest strain levels reached were in the range 0.06–0.14.

Transverse Impact of a Hyperelastic Stretched String

1985 ◽  
Vol 52 (1) ◽  
pp. 137-143 ◽  
Author(s):  
M. F. Beatty ◽  
J. B. Haddow
Keyword(s):  
Strain Energy Function ◽  
Similarity Solutions ◽  
Plane Motion ◽  
Governing Equations ◽  
Transverse Impact ◽  
Normal Impact ◽  
Wave Speeds ◽  
First Order ◽  
Special Case ◽  
Infinite String

Governing equations are derived for the plane motion of a stretched hyperelastic string subjected to a suddenly applied force at one end. These equations can be put in the form of a quasilinear system of first-order partial differential equations, which is totally hyperbolic for an admissible strain energy function. There are, in general, two wave speeds and two corresponding shock speeds. Special consideration is given to the jump relations across the shocks. Similarity solutions for a string moved at one end in loading or unloading are obtained for a general hyperelastic solid. The results are applicable to the familiar neo-Hookean or Mooney-Rivlin material, and the nature of the solution for another special hyperelastic material is discussed. These solutions are valid for a semi-infinite string, or until the first reflection occurs. It is shown that a special case of the similarity solution is valid for the normal impact of a stretched string by a constant speed, point application of load. Exact solution to the equations for the neo-Hookean model is derived in terms of elliptic integrals, and some numerical results are provided.

Dynamic Plastic Response of Circular Plates With Transverse Shear and Rotatory Inertia

1980 ◽  
Vol 47 (1) ◽  
pp. 27-34 ◽  
Author(s):  
Norman Jones ◽  
J. Gomes de Oliveira
Keyword(s):  
Transverse Shear ◽  
Plastic Material ◽  
Yield Criterion ◽  
Circular Plates ◽  
Plastic Response ◽  
Governing Equations ◽  
Rotatory Inertia ◽  
Perfectly Plastic ◽  
Shear Effects ◽  
Transverse Shear Effects

The response of a simply supported circular plate made from a rigid perfectly plastic material and subjected to a uniformly distributed impulsive velocity is developed herein. Plastic yielding of the material is controlled by a yield criterion which retains the transverse shear force as well as bending moments and the influence of rotatory inertia is included in the governing equations. Various equations and numerical results are presented which may be used to assess the importance of transverse shear effects and rotatory inertia for this particular problem.

Predictable Model for Characteristics of One-Dimensional Solid-Gas-Liquid Three-Phase Mixtures Flow Along a Vertical Pipeline With an Abrupt Enlargement in Diameter

1999 ◽  
Vol 121 (2) ◽  
pp. 330-342 ◽  
Author(s):  
Natsuo Hatta ◽  
Masaaki Omodaka ◽  
Fumitaka Nakajima ◽  
Takahiro Takatsu ◽  
Hitoshi Fujimoto ◽  
...  
Keyword(s):  
Gas Flow ◽  
Sudden Change ◽  
Flow Characteristics ◽  
Point Of View ◽  
Solid Particles ◽  
Governing Equations ◽  
One Dimensional ◽  
Three Phase ◽  
Experimental Values ◽  
The One

This paper treats the numerical analysis of the rising process of a solid-gas-liquid three-phase mixture along a vertical pipeline with an abrupt enlargement in diameter. The system of governing equations used is based upon the one-dimensional multifluid model and the transitions of gas flow pattern are taken into account in the system of governing equations. For the case of a sudden enlargement in diameter in a coaxial pipeline, the procedure of the numerical calculation to obtain the flow characteristics in the pipeline section after a sudden change in diameter has been established here. Furthermore, in order to confirm the validity of the present theoretical model by the comparison between the calculated and experimental values, the experiments have been made using four kinds of lifting pipes, including the straight one. Thereby, it has been found that the numerical model proposed here gives good fit to the prediction of the flow rates of lifted water and solid particles against that of air supplied for the case of a sudden change in diameter. In addition, the flowing process for each phase has been investigated from a photographic point of view. As a result, we found that the moving process of the solid particles depends strongly upon the volumetric flux of gas-phase as well as the submergence ratio.

Hydroelastic Modeling of Wet Deck Slamming on Multihull Vessels

2021 ◽  
Vol 39 (03) ◽  
pp. 225-239
Author(s):  
Jan Kvalsvold ◽  
Odd M. Faltinsen
Keyword(s):  
Water Surface ◽  
Normal Modes ◽  
Two Dimensional ◽  
Sea Waves ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Inertia Effects ◽  
Elastic Stresses ◽  
The Impact ◽  
Impact Region

Slamming against the wet deck of a multihull vessel in head sea waves is studied analytically and numerically. The theoretical slamming model is a two-dimensional, asymptotic method valid for small local angles between the undisturbed water surface and the wet deck in the impact region. The disturbance of the water surface as well as the local hydroelastic effects in the slamming area are accounted for. The elastic deflections of the wet deck are expressed in terms of "dry" normal modes. The structural formu­lation accounts for the shear deformations and the rotatory inertia effects in the wet deck. The findings show that the slamming loads on the wet deck and the resulting elastic stresses in the wet deck are strongly influenced by the elasticity of the wet deck structure.

Three-Dimensional and Shell-Theory Analysis of Axially Symmetric Motions of Cylinders

1956 ◽  
Vol 23 (4) ◽  
pp. 563-568
Author(s):  
George Herrmann ◽  
I. Mirsky
Keyword(s):  
Shear Deformation ◽  
Shell Theory ◽  
Plate Theory ◽  
Wave Length ◽  
Three Dimensional ◽  
Approximate Theory ◽  
Axially Symmetric ◽  
Rotatory Inertia ◽  
Phase Velocities ◽  
Transition Wave

Abstract The frequency (or phase velocity) of axially symmetric free vibrations in an elastic, isotropic, circular cylinder of medium thickness is studied on the basis of the three-dimensional linear theory of elasticity and several different shell theories. To be in good agreement with the solution of the three-dimensional equations for short wave lengths, an approximate theory has to include the influence of rotatory inertia and transverse shear deformation, for example, in a manner similar to Mindlin’s plate theory. A shell theory of this (Timoshenko) type is deduced from the three-dimensional elasticity theory. From a comparison of phase velocities it appears that, to a good approximation, membrane and curvature effects on one hand, and on the other hand, flexural, rotatory-inertia, and shear-deformation effects are mutually exclusive in two ranges of wave lengths, separated by a “transition” wave length. Thus, in the full range of wave lengths, the associated lowest phase velocities may be determined on the basis of the membrane shell theory (for wave lengths larger than the transition wave length) and on the basis of Mindlin’s plate theory (for wave lengths smaller than the transition wave length).

Estimation of Response and Loss Factor of Damped Sandwich Beam

1991 ◽  
Author(s):  
Š. Markuš ◽  
T. Nánási
Keyword(s):  
Loss Factor ◽  
Sandwich Beam ◽  
Normal Modes ◽  
Governing Equations ◽  
Shear Deformations ◽  
Energy Estimation ◽  
The Core ◽  
Previous Formulation ◽  
Real Domain ◽  
Base Functions

Abstract Classical governing equations of a three-layered sandwich beam are generalized with the aim to avoid the necessity of identical arrangement of both face layers at the end of the beam. In this way more shear deformations in the core are allowed for. This may significantly contribute to the increased dissipation of vibratory energy. Estimation of the response of the damped sandwich beam and the loss factor are based on truncated integral modal transformation in which the base functions are the modal data from the associated undamped system. Simple formula expressing the modal loss factor in terms of the portion of strain energy due to shear deformations in the core is developed. Presented approach is in fact a reinterpretation of the concept of damped normal modes in such a way that numerical computations are to be carried out only in the real domain while the previous formulation required numerical treatment in the complex domain.

Joule Heating Induced Temperature Gradient Focusing in a Microfluidic Channel With a Sudden Change in Cross Section

2008 ◽  
Author(s):  
Gong Yue Tang ◽  
Chun Yang
Keyword(s):  
Temperature Gradient ◽  
Joule Heating ◽  
Stokes Equations ◽  
Sudden Change ◽  
Microfluidic Channel ◽  
Navier Stokes ◽  
Governing Equations ◽  
Temperature Gradient Focusing ◽  
Poisson Boltzmann Equation ◽  
Energy Equations

Temperature gradient focusing is a recently developed technique for spatially focusing and separating ionic analytes in microchannels. The temperature gradient required for temperature gradient focusing can be generated either by an imposed temperature gradient or by Joule heating resulted from an applied electric field that also drives buffer flow. In this study, a numerical model describing the Joule heating induced temperature development and temperature gradient focusing is developed. The model consists of a set of governing equations including the Poisson-Boltzmann equation, the Laplace equation, the Navier-Stokes equations, the energy equations and the mass transport equation. As the thermophysical and electrical properties including the liquid dielectric constant, viscosity and electric conductivity are temperature-dependent, these governing equations are coupled, and therefore the coupled governing equations are solved numerically by using a CFD based numerical method. The numerical simulations agree well with the experimental results, suggesting that the valid mathematical model presented in this study.

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