Dynamic Axisymmetric Buckling of Shallow Conical Shells Subjected to Impulsive Loads

1965 ◽  
Vol 32 (1) ◽  
pp. 129-134 ◽  
Author(s):  
R. E. Fulton
Keyword(s):  
Conical Shell ◽  
Strain Energy ◽  
Maximum Displacement ◽  
Spherical Cap ◽  
Single Degree Of Freedom ◽  
Conical Shells ◽  
Simply Supported ◽  
Single Degree ◽  
Impulsive Loads ◽  
Axisymmetric Buckling

A theoretical investigation is made of the axisymmetric snap-through buckling of a shallow conical shell subjected to an idealized impulse applied uniformly over the surface of the shell. The shell is assumed to behave as a single-degree-of-freedom system, and a study is made of the strain energy at maximum displacement: i.e., zero velocity. Under certain conditions this equilibrium position becomes unstable and the shell can snap through (or buckle). Nonlinear strain displacement equations are used and solutions are obtained for clamped and simply supported boundaries at the edge of the shell. Results for the cone are compared with similar results for a shallow spherical cap having the same rise as the cone. This comparison indicates that the spherical shell can resist a larger impulse than the conical shell before buckling.

Impact of a Mass on a Damped Elastically Supported Beam

1948 ◽  
Vol 15 (2) ◽  
pp. 125-136
Author(s):  
W. H. Hoppmann
Keyword(s):  
Differential Equation ◽  
Elastic Foundation ◽  
Forced Vibration ◽  
Numerical Solutions ◽  
Coefficient Of Restitution ◽  
Tabular Form ◽  
Single Degree Of Freedom ◽  
Simply Supported ◽  
Single Degree ◽  
Elastically Supported

Abstract In this paper a study is made of the problem of the central impact of a mass on a simply supported beam on an elastic foundation with considerations of internal and external damping. The differential equation for the forced vibration of the beam is developed. It is solved for the case in which the force is a function of time and is concentrated at the center of the beam. Formulas are obtained for the deflections. An expression is developed for the coefficient of restitution which is essential in determining the deflections and the strains. Criteria are devised for determining the cases in which the beam may be considered as a single-degree-of-freedom system when damping and an elastic foundation are considered. The importance of these criteria is discussed. A numerical example illustrating the theory developed in the paper is worked out in detail. Results of computations for several numerical solutions are given in tabular form.

scholarly journals Effects of boundary conditions on bistable behaviour in axisymmetrical shallow shells

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170230 ◽  
Author(s):  
P. M. Sobota ◽  
K. A. Seffen
Keyword(s):  
Boundary Conditions ◽  
Strain Energy ◽  
Stable State ◽  
Spring Stiffness ◽  
Element Analysis ◽  
Single Degree Of Freedom ◽  
Shallow Shells ◽  
Single Degree ◽  
Thin Walled Structures ◽  
Energy Formulation

Multistable shells are thin-walled structures that have more than one stable state of self-stress. We consider isotropic axisymmetrical shallow shells of arbitrary polynomial shapes using a Föppl–von Kármán analytical model. By employing a Rayleigh–Ritz approach, we identify stable shapes from local minima in the strain energy formulation, and we formally characterize the level of influence of the boundary conditions on the critical geometry for achieving bistable inversion—an effect not directly answered in the literature. Systematic insight is afforded by connecting the boundary to ground through sets of extensional and rotational linear springs. For typical cap-like shells, it is shown that bistability is generally enhanced when the extensional spring stiffness increases and when the rotational spring stiffness decreases, i.e. when boundary movements in-plane are resisted but when their rotations are not; however, for certain other shapes and large in-plane stiffness values, bistability can be enhanced by resisting but not entirely preventing edge rotations. Our predictions are furnished as detailed regime maps of the critical geometry, which are accurately correlated against finite-element analysis. Furthermore, the suitabilities of single degree-of-freedom models, for which solutions are achieved in closed form, are evaluated and compared to our more accurate predictions.

scholarly journals Enhanced Single-Degree-of-Freedom Analysis of Thin Elastic Plates Subjected to Blast Loading Using an Energy-Based Approach

2020 ◽  
Vol 2020 ◽  
pp. 1-29
Author(s):  
M. D. Goel ◽  
T. Thimmesh ◽  
P. Shirbhate ◽  
C. Bedon
Keyword(s):  
Boundary Conditions ◽  
Blast Waves ◽  
Degree Of Freedom ◽  
Elastic Plates ◽  
Nonlinear Parameters ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Impulsive Loads ◽  
Technical Interest ◽  
Thin Elastic Plates

Single-degree-of-freedom (SDOF) models are known to represent a valid tool in support of design. Key assumptions of these models, on the other hand, can strongly affect the expected predictions, hence resulting in possible overconservative or misleading estimates for the response of real structural systems under extreme actions. Among others, the description of the input loads can be responsible for major design issues, thus requiring the use of more refined approaches. In this paper, a SDOF model is developed for thin elastic plates under large displacements. Based on the energy approach, careful attention is given for the derivation of the governing linear and nonlinear parameters, under different boundary conditions of technical interest. In doing so, the efforts are dedicated to the description of the incoming blast waves. In place of simplified sinusoidal pressures, the input impulsive loads are described with the support of infinite trigonometric series that are more accurate. The so-developed SDOF model is therefore validated, based on selected literature results, by analyzing the large displacement response of thin elastic plates, under several boundary conditions and real blast pressures. Major advantage for the validation of the proposed SDOF model is obtained from experimental finite element (FE) and finite difference (FD) models of literature, giving evidence of a rather good correlation and confirming the validity of the presented formulation.

SDOF Models for RC and FRC Circular Plates under Blast Loads

2011 ◽  
Vol 82 ◽  
pp. 440-445 ◽  
Author(s):  
Matteo Colombo ◽  
Paolo Martinelli
Keyword(s):  
Reinforced Concrete ◽  
Shock Tube ◽  
Pressure Wave ◽  
Fibre Reinforced Concrete ◽  
Circular Plates ◽  
Blast Loads ◽  
Single Degree Of Freedom ◽  
Simply Supported ◽  
Single Degree ◽  
Slabs On Ground

This work presents simplified models, in the form of single degree of freedom (SDOF)elasto-plastic systems, for the dynamic analysis of traditional reinforced concrete (RC) and fibre-reinforced concrete (FRC) circular plates under blast loads. Two cases have been examined inthis study: simply supported and resting on Winkler-type soil plates. Both cases intend toprovide a simplified tool for predicting the response respectively for specimens subjected toblast pressure wave inside shock-tube facilities and for slabs on ground under blast loads. Thesecond case also represents the loading conditions inside a new shock tube facility specificallyintended for the investigation of underground tunnel lining subjected to blast loads.

Nonlinear Stresses and Deflections of Beams Subjected to Random Time Dependent Uniform Pressure

1976 ◽  
Vol 98 (3) ◽  
pp. 1014-1020 ◽  
Author(s):  
P. Seide
Keyword(s):  
Accurate Determination ◽  
Equivalent Linearization ◽  
Uniform Pressure ◽  
Mean Square ◽  
Maximum Displacement ◽  
Single Degree Of Freedom ◽  
Simply Supported ◽  
White Noise Excitation ◽  
End Conditions

The nonlinear mean-square multimode response of beams subjected to uniform pressure uncorrelated in time is investigated. The method of equivalent linearization is used to obtain mean-square stresses and displacements in beams with arbitrary end conditions. Calculations are carried out for beams with both ends either simply-supported or clamped, for the case of white noise excitation. Although the maximum displacement can be obtained with the use of only a single-degree-of-freedom model, it is necessary to consider as many as 100 modal functions for accurate determination of the stresses. The maximum mean-square deflection of the clamped beam is found to be somewhat less than the simply-supported beam, whereas the maximum mean-square stresses are as much as twice as large.

Thin Circular Conical Shells Under Arbitrary Loads

1955 ◽  
Vol 22 (4) ◽  
pp. 557-562
Author(s):  
N. J. Hoff
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conical Shell ◽  
Strain Energy ◽  
Variational Calculus ◽  
Radius Of Curvature ◽  
Median Surface ◽  
Conical Shells ◽  
Arbitrary Loading ◽  
Circular Conical Shells

Abstract Equations defining the displacements of the median surface of a conical shell under arbitrary loads are developed. In their derivation only the essential parts of the strain energy are considered and three simultaneous partial differential equations are obtained through the use of the variational calculus. When the minimum radius of curvature of the median surface of the cone is made to approach a constant value, the cone goes over into a cylinder. At the same time the equations here developed for the cone are transformed into the Donnell equations of the theory of cylindrical shells. It is shown how eigenfunctions of the homogeneous equations can be constructed and how particular solutions can be found for any arbitrary loading.

A Simplified Analysis Method for Elasto-Plastic Seismic Response Using the Energy Balance Equation of a Maximum Hysteresis Loop

2015 ◽  
Author(s):  
Gaku Nakamura ◽  
Yukihiko Okuda ◽  
Shoichi Ebato ◽  
Hiroshi Niwa ◽  
Tadashi Iijima ◽  
...  
Keyword(s):  
Energy Balance ◽  
Seismic Response ◽  
Degree Of Freedom ◽  
Integration Scheme ◽  
Analysis Method ◽  
Maximum Displacement ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Simplified Analysis ◽  
Vibration Tests

With increasing magnitude of design earthquake ground motions, it is necessary to develop methods of evaluating the seismic safety margin that are more exact than the current methods. However, a standard nonlinear analysis method requires step by step calculations of the numerical time integration scheme to obtain the seismic response. The authors present a new simplified analysis method of elasto-plastic seismic response. The proposed method is formulated by the energy balance between the input energy and the dissipated energy of an equivalent single degree of freedom model for actual equipment. Assuming the harmonic resonance of the single degree of freedom model, the maximum displacement response can be estimated conservatively. To verify the proposed method, static tests and vibration tests with cantilever-type specimens were performed. The vibration tests were conducted with sine, sweep down sine and random waves to verify the conservativeness of the proposed method. Comparisons of the maximum displacement between the tests and the proposed method show the conservative estimation of the displacement by the proposed method.

Nonlinear Analysis of Concrete-Filled Steel Tubular Columns under Blast Loading

2014 ◽  
Vol 941-944 ◽  
pp. 765-769
Author(s):  
Xue Ye Cao ◽  
Yan Li ◽  
Jun Hai Zhao
Keyword(s):  
Integration Method ◽  
Blast Load ◽  
Strength Theory ◽  
Single Degree Of Freedom ◽  
Unified Strength Theory ◽  
Simply Supported ◽  
Tubular Columns ◽  
Single Degree ◽  
Distributed Load ◽  
Ultimate Moment

The plastic ultimate moment of concrete-filled steel tubular and the ultimate displacement of the simply supported beam under uniformly distributed load is established based on unified strength theory. Considered nonlinear impact of mass and stiffness changed in the process of the reaction, the dynamic response of concrete-filled steel tubular columns under blast load were analyzed by the equivalent single degree of freedom model and step by step integration method. Compared the results of this method with relevant literatures, the consequence is good. It can be seen that from the results, this method was satisfied for the requirement of the analytical precision, it can be referred for the research and the safety of concrete-filled steel tubular columns under blast load.

The Optimization of Damping of Four Configurations of a Vibrating Uniform Beam

1963 ◽  
Vol 85 (3) ◽  
pp. 259-264 ◽  
Author(s):  
Alex Henney ◽  
J. P. Raney
Keyword(s):  
Digital Computer ◽  
Empirical Formula ◽  
Response Curves ◽  
Maximum Displacement ◽  
Single Degree Of Freedom ◽  
Displacement Response ◽  
Single Degree ◽  
Uniform Beam ◽  
Computer Response ◽  
Analytical Expressions

Approximate analytical expressions for optimum damping for four configurations of forcing and damping a uniform beam were evaluated and the displacement responses for optimum damping values obtained from these expressions were calculated by a digital computer. It was found that the responses were accurately optimized. The sensitivity of maximum displacement response to deviation from optimum damping was investigated, but no analytical expressions relating change to deviation were obtained. An empirical formula, based upon consideration of the response of the beam as a single-degree-of-freedom system, was shown to be a good approximation of results obtained from the computer response curves. The four configurations investigated were relatively insensitive to changes in damping from optimum.

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