Rectangular Pressure Pulse Research Articles

Elastic effect on the final deflection of rigid-plastic square plates under pulse loading

Rigid-perfectly plastic (R-PP) models are widely employed in the study of structural impact dynamics. However, due to the complexity of dynamic elastic-plastic behavior and the limitations of numerical simulation, it is of great significance to evaluate the elastic effect on the structural response to pulse loading. This paper examines the validity of the R-PP theoretical solutions on large deflection of pulse-loaded square plates. First, by exploring the characteristic of “plastic membrane” and considering the effect of membrane forces and bending moments on energy dissipation, an energy ratio R and a characteristic time ratio are precisely defined. Then based on the R-PP complete solutions recently developed for square plates under rectangular pressure pulse, the discrepancies in final deflection and in saturated impulse between the R-PP theoretical predictions and the elastic-perfectly plastic (E-PP) simulation results are evaluated and elaborated. It is revealed that the discrepancy in final deflection is dependent on the dimensionless stiffness ζplate and the dimensionless pulse magnitude λ. Moreover, the discrepancies in final deflection are compared between square plates and beams via a thorough analysis of key parameters, indicating that most characteristics of the elastic effect are similar for both cases, i.e., the R-PP solutions for both square plates and beams can reliably provide theoretical predictions for engineering applications if energy ratios R>5, with no limitation on the pulse duration. With the same geometric and material parameters, however, the energy ratio R of square plates is about twice that of beams, the discrepancy in final deflection of plates is larger than that of beams, but the absolute value of the discrepancy in final deflection for the clamped square plates remains in the order of 1/(1.5R) and can be bounded by 1/[4λ(λ-1)].

Method based on the membrane theory for equivalence between arbitrarily shaped pressure pulse applied on circular plate and rectangular pressure pulse

Obtaining a theoretical solution for the response of plate structures to arbitrarily shaped pressure pulses is often difficult. For engineering purposes, arbitrarily shaped pressure pulses are often simplified to equivalent rectangular pressure pulses to calculate the theoretical responses of the plate structures. Therefore, the semi-empirical method proposed by Youngdahl is often used. Although the effectiveness of Youngdahl's method has been verified in small deformation cases, the method lacks a theoretical explanation of the equivalence basis for cases of large deformation. In this paper, a method based on the membrane theory (MEM) is established for equivalence between arbitrarily shaped pressure pulse applied on circular plate and rectangular pressure pulse, and the differences and similarities between the MEM and the Youngdahl's equivalent method (YEM) are analyzed. Using the membrane theory, the responses of five different pressure pulses are compared, revealing that the pulse shape has a significant influence on the theoretical deformation obtained by the membrane theory. Then, a load equivalent method based on the membrane theory is proposed. In particular, for a linearly decaying pressure pulse, an analytical solution of the equivalent load is provided, and the equivalent pressure and pulse duration of the MEM and YEM are calculated. Notably, the pressure and pulse duration obtained by the MEM under the large deformation theory significantly differ from those obtained by the YEM. A numerical simulation indicates that the response errors of the equivalent load relative to the original linearly decaying pressure pulse of the two methods are close, both within the acceptable range of engineering. The theoretical analysis shows that the product of the characteristic frequency of the membrane response and pulse duration — a dimensionless number — affects the relative magnitude of the equivalent pressure of the YEM and MEM. When the final deformation time occurs after the duration of the linearly decaying pressure pulse, the deformation of the circular plate under the equivalent load obtained by the YEM is greater than that obtained by the MEM.

Elastic effects on the dynamic plastic deflection of pulse-loaded beams

Beams under intense dynamic pulse loading may experience large elastic-plastic deformation. Despite numerous studies have been conducted based on the rigid-perfectly plastic idealization of material, it is of great interest in both theoretical modeling and practical application to characterize and evaluate the effect of elastic deformation on the dynamic behavior of structures. This paper is aimed to examine the range of applicability of rigid-plastic theoretical solutions on large deflection of beams in response to dynamic pulse loading. The energy ratio R and the characteristic time ratio are first precisely defined by referring to the characteristics of a “plastic string”, which represents the ultimate status of beams under very large deformation. Based on the recently published complete solutions on the dynamic response of fully-clamped and simply-supported rigid-plastic beams subjected to uniformly distributed rectangular pressure pulse, the discrepancies in final deflection and in saturated impulse of the rigid-plastic theoretical predictions from the elastic-plastic simulation results are evaluated and analyzed. It is revealed that if the energy ratio R>5 (with no limitation to the pulse duration), the rigid-plastic solutions can provide reliable predictions on final deflection and saturated impulse for the usage in engineering design. Then, the effects of the key dimensionless parameters related to the energy ratio R (e.g., the structural and material parameters as well as the loading one) on the discrepancies in final deflection and in saturated impulse are thoroughly examined. Finally, a unified dimensionless stiffness is identified to characterize the combined effect of structural and material parameters. For the cases of ζ≤1.581, which represent most of the practically used beams, 1/(2R) and 1/(3R) could provide upper bounds to the absolute value of the discrepancy in final deflection for fully-clamped and simply-supported beams, respectively.

On the saturated impulse for a circular hollow beam under pressure pulse loading

The saturated impulse phenomenon, which may occur in a hollow beam with a circular cross-section, is examined when a rectangular pressure pulse loading is applied uniformly on the upper surface of the beam. A simplified 2DoF model of the dynamic response of a relatively long circular hollow section beam is developed to explore the role of the local and global deformations build up simultaneously during the beam response. In order to simplify the analytical formulation, the elastic deformations of the hollow beam are neglected, and a rigid-plastic method of analysis is applied. Finite element simulations using an elastic, perfectly plastic material model are carried out to verify the proposed analytical model and to reveal the influence of the elasticity on the beam response. The limit of the pressure pulse loading, beyond which saturated impulse is not defined for a hollow beam, is obtained numerically. The responses of hollow beams with lengths between 800 and 1200 mm, radius of 40 mm and several wall thicknesses are analysed. It is observed that the proposed 2DoF model can predict satisfactory the local and global saturated deflections while the saturated pressure pulse duration is underestimated due to the disregard of the stored elastic energy in the beam. It is suggested that the analytically predicted saturated impulse can be used as a lower bound estimate for the saturated impulse for a given beam geometry.

Large deformation of square plates under pulse loading by combined saturation analysis and membrane factor methods

Thin plates will experience large deformation under intense pulse loading, which may result in serious consequences. Numerous research work has been conducted since 1950s and some theoretical methods have been developed to predict the large deformation of plates based on Rigid-Plastic Idealization of material behavior. However, even with this idealization, it is often difficult to obtain complete solutions. Membrane Factor Method (MFM) has proved a powerful method to solve this problem, resulting in solutions taking into account of the effects of the transient phase and the interaction between membrane force and bending moment through a relatively simple procedure. On the other hand, Saturation Analysis (SA) based on the saturation phenomenon is more rational and accurate than the traditional analyses using total impulse of pulse loading. This paper combines these two approaches to study the dynamic plastic response of square plates under rectangular pressure pulse and linear decay pressure pulse loading, and then the results are compared with modal solutions and FE simulations. Meanwhile, the solutions of this combined method, which merely considers the modal phase are also provided to identify the effect of the transient phase. Finally, two sets of simple calculation formulae are proposed for the permanent deformation of plates so as to facilitate engineering applications.

Saturated impulse for fully clamped square plates under blast loading

Saturated impulse refers to the critical value, beyond which the deflection of a beam or plate under pulse loading will no longer increase with further applied load. The present paper investigates the saturated impulse of fully clamped square plates subjected to a blast loading that is assumed to be a uniformly distributed Linearly Rising Exponentially Decaying (LRED) pressure pulse. Considering both bending moment and membrane force, the transient dynamic plastic response is predicted by a rigid, perfectly plastic (R-PP) model characterized by travelling plastic hinge lines and time-dependent velocity field. The saturated duration, saturated impulse, saturated deflection, as well as the evolution of deformation mechanism are explored. By comparing three characteristic durations, i.e. the travelling duration of the plastic hinge lines, saturated duration and loading duration, the dynamic responses of a square plate are classified into three types such that a regime map on the loading parameters’ plane is constructed accordingly. The correlation is made between the results obtained from the elastic-plastic numerical simulation and the R-PP theoretical approximation. To facilitate engineering designs, a method of replacing the LRED pressure pulse by an equivalent rectangular pressure pulse is proposed to predict the maximum deflection of fully clamped square plates.

Saturated impulse for pulse-loaded rectangular plates with various boundary conditions

Saturated impulse refers to the critical value after which the deflection of the beam or plate under pulse loading will not increase with further applied load. This paper investigates this phenomenon for rectangular plates under rectangular pulse-loading with various symmetric boundary conditions, e.g. two opposite edges of the plate are fully clamped while the other two edges are simply supported or free. For rectangular plates with symmetric boundaries under rectangular pressure pulse, the saturated duration and saturated impulse are merely dictated by the deformation mechanism, not by the boundary conditions. Then the effects of the aspect ratio and the boundary condition on the saturation impulse for elastic-plastic plates are revealed through numerical simulation. Bilinear elastic-plastic rectangular plates, with fully clamped boundaries and simply supported boundaries, subjected to rectangular pulse-loading are taken as typical examples. Subsequently, the saturation phenomenon is summarized from square plates to rectangular plates. The correlation is made between the results obtained from elastic-plastic analysis and those predicted by the rigid-plastic approximation. Finally, based on numerical simulation, theoretically and practically meaningful empirical expressions on the saturation phenomenon are proposed to facilitate engineering designs.

The saturated impulse of fully clamped square plates subjected to linearly decaying pressure pulse

Saturated impulse refers to the critical value, beyond which the deflection of a beam or plate under pulse loading will no longer increase with further applied load. Previous studies have explored the saturated impulse for structures under rectangular pressure pulse loading. The present paper investigates the saturated impulse for fully clamped square plates subjected to linearly decaying pressure pulse, which is closer to the pulse produced by explosive loading. Both rigid-plastic model and elastic-plastic model are employed to study the saturated duration, saturated impulse, saturated deflection, etc. The results indicate that the saturated impulse exists if the decaying duration is greater than a certain value (i.e. the minimum decaying duration), and a saturation regime map is constructed accordingly. To facilitate engineering designs, an empirical expression for saturated duration is provided based on the elastic-plastic numerical results. In addition, a method is proposed to predict the deflection of fully clamped square plates, which replaces the linearly decaying pressure pulse with an equivalent rectangular pressure pulse. Inspired by Youngdahl's pioneering work, the proposed method has explicit physical meaning and wide range of application.

The Saturated Impulse for Fully Clamped Square Plates Subjected to Linear-Decay Pressure Pulse

Saturated impulse refers to the critical value beyond which the deflection of the beam or plate under pulse loading will no longer increase with further applied load. Previous effort has been made to study the saturate impulse for structures under rectangular pressure pulse loading. Through theoretical analysis, the present paper investigates the saturated impulse for fully clamped square plates subjected to linear-decay pressure pulse, which is closer to explosive loading pulse. The results indicate that the saturated impulse does exist if the decay duration is greater than a certain value (i.e. the minimum decay duration). The saturated impulse, saturated deflection and saturated duration all have nonlinearly relations with the dimensionless applied pressure, and grow with the increase of the decay duration.

Theoretical analysis of the static and dynamic response of tensor skin

Comprised of cover ply, tensor ply and carrying ply, tensor skin is a kind of composite sandwich structure developed to improve the helicopter's crashworthiness in water impacts. In this study, a theoretical model is proposed to analyze the static and dynamic response of a kind of tensor skin. The whole response of tensor skin is divided into three stages: an elastic deformation stage of the whole beam; an unfolding stage of the tensor ply; and a stretching stage of the tensor ply. At the beginning of impact, the whole beam undergoes elastic deformation until the breakage of the cover and carrying plies; then the tensor ply left is unfolded and stretched to absorb more impact kinetic energy.In the unfolding stage, by adopting the rigid, perfectly plastic material idealization, a deformation mechanism with stationary plastic hinges is proposed. It is found that the static critical pressure first decreases then increases with the increasing central deflection. The static critical pressure varies with the geometric parameters, but the total energy dissipated in the unfolding stage is independent of the geometric parameters. The residual kinetic energy at the end of unfolding stage will be dissipated by the plastic stretching. The dynamic responses of the tensor skin are analyzed for step loaded pressure and rectangular pressure pulse. It is verified that the theoretical predictions display very good agreement with the corresponding finite element simulations.

Numerical modelling of square tubular steel beams subjected to transverse blast loads

This paper presents the numerical simulations of thin-walled square hollow steel beams subjected to a uniform transverse blast load. The objectives of the numerical simulations were to gain an insight into the temporal distribution of the global and local deformation and the adiabatic temperature rise in the beams as a result of impulsive loading. Additionally, the finite element predictions using Ls-Dyna are compared to the experimentally observed global and local deformations. The full lengths of the beams were modelled using three material models based on the linear piecewise plasticity material model which incorporated strain hardening, with and without strain-rate hardening and with strain-rate hardening and temperature softening. The blast wave was simulated as a rectangular pressure pulse distributed over the top surface of the beams. Ls-Dyna and the material model used were found to predict the global and local deformation of the beams reasonably well. Incorporating strain-rate hardening was found to be important to be able to predict the global and local deformation of the beams. Thermal softening was found to play a small but not negligible role.

ONE-DIMENSIONAL WAVE PROPAGATION IN A FUNCTIONALLY GRADED ELASTIC MEDIUM

In this study the one-dimensional wave propagation in a functionally graded elastic slab is considered. It is assumed that the stiffness and density of the medium vary continuously in thickness direction and it is initially at rest and stress-free. The slab is subjected to a pressure pulse on one surface and a vanishing stress or displacement condition on the other. The solution is obtained in wave summation form. Propagation of a rectangular pressure pulse in a graded medium that consists of either nickel/zirconia or aluminium/silicon carbide is studied as examples. It is shown that there is considerable wave distortion in time and the distortion is much more pronounced in slabs with fixed/free boundary conditions. A simple approximate expression giving the peak stress is developed. Also it is demonstrated that the energy balance principle may be used as a convergence criterion in the calculation of stresses.

Einschwingverhalten infolge dynamischer, strukturbedingt plastischer Vorgänge und den zugehörigen Grundfrequenzen der elastischen Schwingungen

The relationship is determined between saturated duration of rectangular pressure pulses applied to rigid, perfectly plastic structures and their fundamental periods of elastic vibration. It is shown that the ratio between the saturated duration and the fundamental period of elastic vibration of a structure is dependent upon two factors: the first one is the slenderness or thinness ratio of the structure; and the second one is the square root of ratio between the Young’s elastic modulus and the yield stress of the structural material. Dimensional analysis shows that the aforementioned ratio is one of the basic similarity parameters for elastic-plastic modeling under dynamic loading.

Prediction of structural dynamic plastic shear failure by Johnson’s damage number

It is proved that Johnson’s damage number is the sole similarity parameter for dynamic plastic shear failure of structures loaded impulsively, therefore, dynamic plastic shear failure can be understood when damage number reaches a critical value. It is suggested that the damage number be generally used to predict the dynamic plastic shear failure of structures under various kinds of dynamic loads (impulsive loading, rectangular pressure pulse, exponential pressure pulse, etc.,). One of the advantages for using the damage number to predict such kind of failure is that it is conveniently used for dissimilar material modeling.

Saturated duration of rectangular pressure pulse applied to rectangular plates with finite-deflections

The inequities in health care and housing access experienced by low-income women in the United States are a continuing concern. This article addresses the interrelationships between housing and health as experienced by low-income clients so that health care practitioners can begin to build active and effective health-promoting partnerships with clients, their families, and their communities. A case study is presented that describes the actual experience of a woman living in a low-income housing development and its effect on her health and access to health care. The importance of the role of midwives in addressing the health care and advocacy needs of women in substandard housing is highlighted.

Saturation impulses for dynamically loaded structures with finite-deflections

The concept of ''Saturation Impulse'' for rigid, perfectly plastic structures with finite-deflections subjected to dynamic loading was put forward by Zhao, Yu and Fang (1994a). This paper extends the concept of Saturation Impulse to the analysis of structures such as simply supported circular plates, simply supported and fully clamped square plates, and cylindrical shells subjected to rectangular pressure pulses in the medium load range. Both upper and lower bounds of nondimensional saturation impulses are presented.

Large dynamic plastic deflection of a simply supported beam subjected to rectangular pressure pulse

This paper concerns the dynamic behaviour of rigid, perfectly plastic simply supported beams with finite deflections by considering both bending moments and axial forces. In the case of rectangular pressure pulse loading, approximate rigid-plastic theoretical solutions are obtained for both medium load and high load; to this end a parabolic law between bending moment and axial force as yield condition is approximated by a circumscribing or an inscribing rectangle. The concept of Saturation Impulse is put forward for the dynamic response of rigid, perfectly plastic structures with finite deflections subjected to rectangular pressure pulse. The value of dimensionless saturation impulse for both simply supported and fully clamped beams are given.

The dynamic plastic behavior of a simply supported circular plate in a damping medium with finite-deflections

A theoretical analysis is presented for the dynamic plastic behavior of a simply supported rigid, perfectly plastic circular plate in damping medium with finite-deflections subjected to a rectangular pressure pulse. Analytical solutions of every moving stage under both medium and high loads are developed.

The stresses produced in an elastic half-space by a pressure pulse applied uniformly over a circular area: role of the pulse duration

The application of a pressure pulse over a circular area at the surface of an elastic half-space produces a complex wave system. An exact solution for the stresses at interior points of the solid was given previously for the case of a step of pressure. This solution is applied to obtain a numerical evaluation of the stress distribution along the axis of symmetry, in the case of rectangular pressure pulses of different durations. The patterns due to longitudinal compression and rarefaction waves, shear edge wave, head waves and Rayleigh wave are distinguished. Two examples are considered, depending on whether the pulse is still applied at the arrival time of the rarefaction wave on the axis or not. Short pulses are shown to generate tensile stresses which can explain surface damage for brittle materials.

A New Versatile Apparatus for Relaxation Kinetics with Shock Waves

AbstractA new shock wave apparatus uses the discharge of a high voltage capacitor through an organic electrolyte to generate rectangular pressure pulses of typically 8 MPa amplitude and 80 μsec duration. The apparatus permits the study of fast reactions with relaxation times between a few microseconds and about fifty microseconds in any solvent. Measurements on proton transfer reactions in acetonitrile and water are reported.