In-Plane Bending of a Short-Radius Curved Pipe Bend

1967 ◽  
Vol 89 (2) ◽  
pp. 271-277 ◽  
Author(s):  
N. Jones
Keyword(s):  
Power Series ◽  
Radial Displacement ◽  
Neutral Axis ◽  
Pipe Bend ◽  
Curved Pipe ◽  
Original Theory ◽  
Von Karman ◽  
Plane Bending ◽  
Von Kármán ◽  
Logical Extension

The analysis developed in this article is a logical extension, or generalization, of Von Karman’s original theory [1] on curved pipe bends. It can be shown, for long-radius pipe bends which have a negligible shift of the neutral axis, that the solution presented here reduces to that of Von Karman. It is evident from the results of this analysis that the stresses in and flexibility of curved pipe bends are virtually independent of γ(a/ρ) (even for γ approaching unity) and depend almost entirely on the simple Von Karman pipe factor λ(tρ/a2). Errors arising from premature truncation of the selected power series for the radial displacement are discussed, and a guide given to the number of terms necessary for a particular problem.

A Simple and Effective Pipe Elbow Element—Linear Analysis

1980 ◽  
Vol 47 (1) ◽  
pp. 93-100 ◽  
Author(s):  
K. J. Bathe ◽  
C. A. Almeida
Keyword(s):  
Radial Displacement ◽  
Linear Analysis ◽  
Pipe Bend ◽  
Pipe Elbow ◽  
Von Karman ◽  
Displacement Based ◽  
Von Kármán ◽  
Displacement Patterns ◽  
Bend Element

The formulation of a new, simple, and effective displacement-based pipe bend element is presented. The displacement assumptions are axial, torsional, and bending displacements that vary cubically along the axis of the elbow with plane sections remaining plane, and a generalization of the von Karman pipe radial displacement patterns to include the ovalization effects. The amount of ovalization varies cubically along the elbow with full compatibility between elbows. The pipe bend element has been implemented, and the results of various sample analyses are presented, which illustrate the effectiveness of the element.

A theoretical study of in-plane bending of a single unreinforced mitred-bend

1966 ◽  
Vol 1 (3) ◽  
pp. 264-276 ◽  
Author(s):  
N Jones ◽  
R Kitching
Keyword(s):  
Strain Energy ◽  
Close Agreement ◽  
Theoretical Study ◽  
Compatibility Conditions ◽  
Curved Pipe ◽  
Exact Analysis ◽  
Von Karman ◽  
Plane Bending ◽  
Experimental Values ◽  
Von Kármán

It can be shown that an exact analysis of the deformations or stresses in a single mitred-bend would involve some terms which rapidly decay along the pipe axes and others which would have a relatively long damping length. The overall flexibility and radial deformations of a single unreinforced mitred-bend have been estimated by minimizing the strain energy contributed by the long damping length terms in a manner similar to that used by von Kármán for curved pipe-bends. Finally, the short damping length terms are introduced in order to satisfy the equilibrium and compatibility conditions across the oblique intersection of the two pipes forming the mitred-bend. The theoretical values of overall flexibility, diameter changes, and stresses were found to be in close agreement with experimental values for a single right-angled unreinforced mitred-bend.

Type 1 energy analysis for the creep of smooth circular pipe bends under combined in-plane bending and internal pressure

1977 ◽  
Vol 12 (1) ◽  
pp. 1-7 ◽  
Author(s):  
J Spence
Keyword(s):  
Internal Pressure ◽  
Creep Behaviour ◽  
Constitutive Relationship ◽  
Constant Thickness ◽  
Pipe Bend ◽  
Curved Pipe ◽  
Linear Elastic ◽  
Special Cases ◽  
Plane Bending

Many pipeline systems, in addition to catering for thermal expansion conditions, have to carry internal pressure loading. An approximate theoretical analysis is given for the pure creep of a smooth curved pipe bend under fixed ratios of combined pressure and in-plane bending. The bend is circular in cross-section and of constant thickness. An energy method designated type 1, coupled with a simple n-power constitutive relationship, is used and implemented by direct minimization techniques. Selected deformation and stress results are presented as possible design factors for a number of the geometric and material parameters. For part of the geometric spectrum, large displacement effects are significant and lead to non-steady creep behaviour. Consequently there is not a direct analogy with elasticity, although linear elastic results can nevertheless be identified. Other special cases of loading or geometry are reducible from the general case.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

scholarly journals A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections

Micromachines ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 714
Author(s):  
Jiujiang Wang ◽  
Xin Liu ◽  
Yuanyu Yu ◽  
Yao Li ◽  
Ching-Hsiang Cheng ◽  
...  
Keyword(s):  
Analytical Modeling ◽  
Ultrasonic Transducer ◽  
Governing Equations ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Collapse Mode ◽  
Capacitive Micromachined Ultrasonic Transducer ◽  
Gap Height ◽  
Von Kármán ◽  
Analytical Expressions

Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán equations and the 2D plate equation, and solved by various methods such as Galerkin’s method and perturbation method. Analytical expressions from Timoshenko’s equation can be used for small deflections, while analytical expression from von Kármán equations can be used for both small and large deflections.

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