Shells With Edge Loads of Rapid Variation—II

1965 ◽  
Vol 32 (1) ◽  
pp. 87-98 ◽  
Author(s):  
C. R. Steele
Keyword(s):  
Stiffness Matrix ◽  
Thin Shell ◽  
Wave Length ◽  
Elastic Shell ◽  
General Character ◽  
Membrane Theory ◽  
Rapid Variation ◽  
Flexibility Matrix ◽  
Shear Beam ◽  
Interior Stress

The previously obtained solution for the thin elastic shell with edge loads whose wave-length is smaller than the shell radii of curvature is shown to provide a transition from “thin-shell” behavior to “flat-plate” behavior. The general character of the interior stress distribution is discussed in this paper. An examination is made of the properties of the “stiffness” matrix, which give the edge stresses due to prescribed edge displacements, and the properties of the inverse (“flexibility”) matrix. A solution is provided for the paradax of the shell of negative curvature with “shear-beam” supported edges, for which the membrane theory indicates a spectrum of edge loads that produce infinite stress.

scholarly journals The effect of junctions on the propagation of electric waves along conductors

1913 ◽  
Vol 88 (601) ◽  
pp. 103-110
Keyword(s):  
Cylindrical Shell ◽  
Wave Length ◽  
General Character ◽  
Original Form ◽  
Annular Space ◽  
Approximate Methods ◽  
Positive Direction ◽  
Positive Side ◽  
The Waves ◽  
Electric Waves

Some interesting problems in electric wave propagation are suggested by an experiment of Hertz. In its original form waves of the simplest kind travel in the positive direction (fig. 1), outside an infinitely thin conducting cylindrical shell, AA, which comes to an end, say, at the plane z = 0. Co-axial with the cylinder a rod or wire BB (of less diameter) extends to infinity in both directions. The conductors being supposed perfect, it is required to determine the waves propagated onwards beyond the cylinder on the positive side of z , as well as those reflected back outside the cylinder and in the annular space between the cylinder and the rod. So stated, the problem, even if mathematically definite, is probably intractable; but if we modify it by introducing an external co-axial con­ducting sheath CC (fig. 2), extending to infinity in both directions, and if we further suppose that the diameter of this sheath is small in comparison with the wave-length (λ) of the vibrations, we shall bring it within the scope of approximate methods. It is under this limitation that I propose here to consider the present and a few analogous problems. Some considerations of a more general character are prefixed.

scholarly journals Crack Detection through the Change in the Normalized Frequency Shape

Vibration ◽  
2018 ◽  
Vol 1 (1) ◽  
pp. 56-68
Author(s):  
Mustapha Dahak ◽  
Noureddine Touat ◽  
Tarak Benkedjouh
Keyword(s):  
Stiffness Matrix ◽  
Electrical Discharge ◽  
Inverse Method ◽  
Crack Detection ◽  
Natural Frequencies ◽  
Electrical Discharge Machining ◽  
Element Model ◽  
Experimental Application ◽  
Flexibility Matrix ◽  
Crack Position

The objective of this work is to use natural frequencies for the localization and quantification of cracks in beams. First, to study the effect of the crack on natural frequencies, a finite element model of Euler–Bernoulli is presented. Concerning the damaged element, the stiffness matrix is calculated by the theory of fracture mechanics, by inverting the flexibility matrix. Then, in order to detect damage, we are going to show that the shape given by the change in the natural frequencies is as function of the damage position only. Thus, the crack is located by the correlation between the shape of the measured frequencies and those obtained by the finite elements, where the position that gives the calculated shape which is the most similar to the measured one, indicates the crack position. After the localization, an inverse method will be applied to quantify the damage. Finally, an experimental application is presented to show the real applicability of the method, in which the crack is introduced by using an Electrical Discharge Machining. The results confirm the applicability of the method for the localization and the quantification of cracks.

The Numerical Simulation for Vibroacoustic Phenomena of a Slender Elastic Shell

2012 ◽  
Vol 479-481 ◽  
pp. 1365-1370
Author(s):  
Zhi Xi Yang ◽  
Sheng Hua Qiu
Keyword(s):  
Numerical Simulation ◽  
Finite Element ◽  
Thin Shell ◽  
Acoustic Pressure ◽  
Pressure Field ◽  
Elastic Shell ◽  
Numerical Examples ◽  
Longitudinal Acoustic ◽  
3 Dimensional ◽  
Variational Formulas

The vibroacoustic phenomena for the slender elastic thin shell filled with water by finite element method is introduced in this paper. The unsymmetric (u, p) variational formulas and finite element procedures are implemented for 3 dimensional structures of vibroacoustic environment based on the displacement field u and the fluid acoustic pressure field p. As illustrated by numerical examples, the longitudinal acoustic pressure eigenmodes will be occurred besides the transverse bendable eigenmodes of the slender shell, nonetheless the eigenvalues and the order of eigenmodes for the fluid acoustic pressure field can only be determined by the flexibility and geometry stiffness of the slender shell.

CHARACTERIZATION OF THE MEMBRANE THEORY OF A CLAMPED SHELL: THE HYPERBOLIC CASE

1996 ◽  
Vol 06 (02) ◽  
pp. 169-194 ◽  
Author(s):  
JYRKI PIILA
Keyword(s):  
Convergence Rate ◽  
Thin Shell ◽  
Theory Model ◽  
Hyperbolic Surface ◽  
Membrane Theory ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Mathematical Properties ◽  
Deformation State

We study the membrane-dominated deformation state of a thin shell, the midsurface of which is located on a hyperbolic surface of revolution whose boundary is a curvilinear polygon. The shell is loaded by a surface tractions on both faces, and clamped all along the edge. Proceeding from the classical shell model of Koiter–Sanders–Novozhilov, an asymptotic membrane theory model is constructed to describe the limit behavior of the shell as the thickness tends to zero. The mathematical properties of the membrane theory are analyzed and the convergence rate estimate between the two models is proven.

scholarly journals Exact Stiffness for Beams on Kerr-Type Foundation: The Virtual Force Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Suchart Limkatanyu ◽  
Woraphot Prachasaree ◽  
Nattapong Damrongwiriyanupap ◽  
Minho Kwon ◽  
Wooyoung Jung
Keyword(s):  
Stiffness Matrix ◽  
Element Stiffness Matrix ◽  
Flexibility Matrix ◽  
Force Equation ◽  
Virtual Force ◽  
Proposed Model ◽  
The Matrix ◽  
Natural Element ◽  
The Individual ◽  
Interpolation Functions

This paper alternatively derives the exact element stiffness equation for a beam on Kerr-type foundation. The shear coupling between the individual Winkler-spring components and the peripheral discontinuity at the boundaries between the loaded and the unloaded soil surfaces are taken into account in this proposed model. The element flexibility matrix is derived based on the virtual force principle and forms the core of the exact element stiffness matrix. The sixth-order governing differential compatibility of the problem is revealed using the virtual force principle and solved analytically to obtain the exact force interpolation functions. The matrix virtual force equation is employed to obtain the exact element flexibility matrix based on the exact force interpolation functions. The so-called “natural” element stiffness matrix is obtained by inverting the exact element flexibility matrix. One numerical example is utilized to confirm the accuracy and the efficiency of the proposed beam element on Kerr-type foundation and to show a more realistic distribution of interactive foundation force.

Bi2Te3: Structural Modulations in Epitaxially Grown Superlattices and Bulk Materials

2005 ◽  
Vol 886 ◽  
Author(s):  
Nicola Peranio ◽  
Oliver Eibl ◽  
Joachim Nurnus
Keyword(s):  
Thin Films ◽  
Growth Parameters ◽  
Bulk Material ◽  
Displacement Vector ◽  
Wave Length ◽  
Transport Coefficients ◽  
General Character ◽  
Threading Dislocations ◽  
Bulk Materials ◽  
Structural Modulation

ABSTRACTMultiquantum well structures of Bi2Te3 are predicted to show an enhancement of the thermoelectric figure of merit ZT. Electron-conducting Bi2Te3 thin films and superlattices (SL) with a period of 12 nm were epitaxially grown on BaF2 substrates by molecular beam epitaxy. The microstructure was investigated by transmission electron microscopy. The SL could be imaged with strong contrast yielding a period of 12.0±0.5 nm. The SL is slightly bent with an amplitude of 30 nm and a wave length of 400 nm. Threading dislocations were found with a density of 2·109 cm−2. The SL interfaces are strongly bent close to threading dislocations, undisturbed regions have a maximum lateral size of 500 nm. A structural modulation (nns) with a wave length of 10 nm was found in Bi2Te3 thin films, superlattices and bulk materials. The structural modulation is found to be of general character for Bi2Te3 materials and is superimposed to the average structure. It was analysed in detail by stereomicroscopy in bulk material yielding a pure structural modulation with a displacement vector parallel to [5,-5,1] and a wave vector parallel to (1,0,10). The investigations showed the presence of none, one or two (nns). The number of (nns) and thereby the thermoelectric properties might be controlled by the growth parameters. Phonons should be scattered on the sinusoidal strain field of the (nns) yielding (i) a significantly decreased thermal conductivity, (ii) a reduced dimensionality and (iii) anisotropic transport coefficients in the basal plane.

Thin-Shell Thickness of Two-Dimensional Materials

2015 ◽  
Vol 82 (12) ◽  
Author(s):  
Enlai Gao ◽  
Zhiping Xu
Keyword(s):  
Thin Shell ◽  
Shell Thickness ◽  
2D Materials ◽  
Wide Spectrum ◽  
Elastic Shell ◽  
Bending Stiffness ◽  
Interlayer Distance ◽  
Structural Deformation ◽  
Two Dimensional ◽  
Number Of Layers

In applying the elastic shell models to monolayer or few-layer two-dimensional (2D) materials, an effective thickness has to be defined to capture their tensile and out-of-plane mechanical behaviors. This thin-shell thickness differs from the interlayer distance of their layer-by-layer assembly in the bulk and is directly related to the Föppl–von Karman number that characterizes the mechanism of nonlinear structural deformation. In this work, we assess such a definition for a wide spectrum of 2D crystals of current interest. Based on first-principles calculations, we report that the discrepancy between the thin-shell thickness and interlayer distance is weakened for 2D materials with lower tensile stiffness, higher bending stiffness, or more number of atomic layers. For multilayer assembly of 2D materials, the tensile and bending stiffness have different scaling relations with the number of layers, and the thin-shell thickness per layer approaches the interlayer distance as the number of layers increases. These findings lay the ground for constructing continuum models of 2D materials with both tensile and bending deformation.

Analysis of Reinforced Concrete Column Using a Beam-Column Element with a Meshed Section

2011 ◽  
Vol 255-260 ◽  
pp. 1954-1958
Author(s):  
Ling Yuan Zhou ◽  
Qiao Li
Keyword(s):  
Reinforced Concrete ◽  
Stiffness Matrix ◽  
Concrete Beam ◽  
Reinforced Concrete Beam ◽  
Beam Element ◽  
Reinforced Concrete Column ◽  
Flexibility Matrix ◽  
Stiffness Matrices ◽  
Column Element ◽  
Interpolation Functions

A efficient 3D reinforced-concrete beam element based on the flexibility method and distributed nonlinearity theory is proposed, The sections of the beam element are divided into the plane isoparametric elements in this formulation, the section stiffness matrices are calculated through the integration of stress-strain relations of concrete including reinforcing steel effect in the section. The flexibility matrices of the sections are calculated by inverting the stiffness matrices, and the element flexibility matrix is formed through the force interpolation functions. The element stiffness matrix is evaluated through the element flexibility matrix. Finally, the buckling behaviors of a reinforced concrete beam under various eccentric loads are analyzed with the proposed formulation to illustrate its accuracy and computational efficiency.

scholarly journals II. On the theory of resonance

1871 ◽  
Vol 19 (123-129) ◽  
pp. 106-107
Keyword(s):  
General Theory ◽  
Wave Length ◽  
General Character ◽  
Rigid Walls

An attempt is here made to establish a general theory of a certain class of resonators, including most of those which occur in practice. When a mass of air or other gas is enclosed in a space bounded nearly all round by rigid walls, but communicating with the external air by one or more passages, there are certain natural periods of vibration or resonant notes whose determination is a matter of interest. If the dimension of the airspace is small compared to the wave-length of the vibration, the dynamics of the motion is, in its general character, of remarkable simplicity.

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