Analytical Solution of Thick Piezoelectric Curved Beams with Variable Curvature Considering Shearing Deformation

2017 ◽  
Vol 09 (01) ◽  
pp. 1750006 ◽  
Author(s):  
Yong Zhou ◽  
Timo Nyberg ◽  
Gang Xiong ◽  
Shi Li ◽  
Hongbo Zhou ◽  
...  
Keyword(s):  
Closed Form ◽  
Bending Moment ◽  
Arc Length ◽  
Curved Beams ◽  
Governing Equations ◽  
Displacement Fields ◽  
Closed Form Solutions ◽  
Shearing Deformation ◽  
Equivalent Modulus ◽  
Tangent Slope

In this paper, an analytical method based on Timoshenko theory is derived for obtaining the in-plane static closed-form general solutions of deep curved laminated piezoelectric beams with variable curvatures. The equivalent modulus of elasticity is utilized to take into account the material couplings in the laminated beam. The linear piezoelectric effect is considered to develop the static governing equations. The governing differential equations are formulated as functions of the angle of tangent slope by introducing the coordinate system defined by the arc length of the centroidal axis and the angle of tangent slope. To solve the governing equations, defined are the fundamental geometric properties, such as the moments of the arc length with respect to horizontal and vertical axes. As the radius is known, the fundamental geometric quantities can be calculated to obtain the static closed-form solutions of the axial force, shear force, bending moment, rotation angle, and displacement fields at any cross-section of curved beams. The closed-form solutions of the circle beams covered with piezoelectric layers under various loading cases are presented. The results show the consistency in comparison with finite results. Solutions of the non-dimensional displacements for the laminated circular and spiral curved beams with different lay-ups are available. The non-dimensional displacements with geometry and material parameters are also investigated.

Orthotropy Rescaling for the Fracture Problem in Anisotropic Piezoelectric Materials

2002 ◽  
Author(s):  
William S. Oates ◽  
Christopher S. Lynch
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Work Done ◽  
Elastic Dielectric

To date, much of the work done on ferroelectric fracture assumes the material is elastically isotropic, yet there can be considerable polarization induced anisotropy. More sophisticated solutions of the fracture problem incorporate anisotropy through the Stroh formalism generalized to the piezoelectric material. This gives equations for the stress singularity, but the characteristic equation involves solving a sixth order polynomial. In general this must be accomplished numerically for each composition. In this work it is shown that a closed form solution can be obtained using orthotropy rescaling. This technique involves rescaling the coordinate system based on certain ratios of the elastic, dielectric, and piezoelectric coefficients. The result is that the governing equations can be reduced to the biharmonic equation and solutions for the isotropic material utilized to obtain solutions for the anisotropic material. This leads to closed form solutions for the stress singularity in terms of ratios of the elastic, dielectric, and piezoelectric coefficients. The results of the two approaches are compared and the contribution of anisotropy to the stress intensity factor discussed.

Elastic Solutions for a Saturated Isotropic Half Space Subjected to a Fluid Line Sink

2013 ◽  
Vol 405-408 ◽  
pp. 275-284 ◽  
Author(s):  
John C.C. Lu
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Mass Conservation ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Cylindrical Coordinate ◽  
Line Sink ◽  
The Mathematical Model

The study derives the closed-form solutions of the long-term elastic consolidation subjected to the fluid line sink in a homogeneous isotropic elastic half space aquifer. The Hankel transform in a cylindrical coordinate system is employed to develop the analytical elastic solutions. Derivations of governing equations are based on the mathematical model of Biots theory of poro-mechanics, and the half space aquifer is modelled as a saturated porous stratum which is bounded by a horizontal surface. The total stresses of the aquifer obey Newtons second law and Hookes law. Besides, the mass conservation and Darcys law are introduced to formulate the governing equations of pore fluid flow. The software Mathematica is used to complete the symbolic integrations and obtain the closed-form solutions. The solutions can be applied in dewatering operations of compressible aquifer.

Consistent Treatment of Extensional Deformations for the Bending of Arches, Curved Beams and Rings

1977 ◽  
Vol 99 (1) ◽  
pp. 2-11 ◽  
Author(s):  
O. A. Fettahlioglu ◽  
J. Mayers
Keyword(s):  
Beam Theory ◽  
Dynamical Problem ◽  
Curved Beam ◽  
Curved Beams ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Circular Rings ◽  
Consistent Treatment ◽  
Discontinuity Conditions ◽  
Complete Solutions

A consistent treatment of the extensional deformations of thin circular rings subjected to general distributed and concentrated loadings is presented. Coupled Euler equations and consistent boundary condition combinations in terms of the radial and tangential midsurface displacements are obtained for the dynamical problem using Hamilton’s principle. Discontinuity conditions corresponding to discrete application of generalized forces are also provided. For static analysis, the governing equations are transformed into two uncoupled sixth-order differential equations in the radial and tangential displacements, respectively, and the complete solutions are obtained for general loading. Closed-form solutions for displacements and stress resultants are developed for illustrative and comparative purposes relative to specific full ring, curved beam, and arch examples. The results confirm that extensional deformations can play a significant role in the bending of thin curved beams, and arches and that such structures are more readily and consistently analyzed by the present treatise than by Winkler curved-beam theory with its potential for inconsistent approximations toward thinness and resulting possible violation of static equilibrium by the stress resultants calculated from the stress-displacement state.

scholarly journals A New Class of Plane Curves with Arc Length Parametrization and Its Application to Linear Analysis of Curved Beams

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1778
Author(s):  
Snježana Maksimović ◽  
Aleksandar Borković
Keyword(s):  
Analytical Solution ◽  
Special Functions ◽  
Exact Analytical Solution ◽  
Linear Analysis ◽  
Plane Curves ◽  
Arc Length ◽  
Curved Beams ◽  
Euler Beam ◽  
Closed Form Solutions ◽  
New Class

The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of L2(R) space and some of their interesting properties are discussed. The developed curves are used for the linear static analysis of curved Bernoulli–Euler beam. Due to the parametrization with arc length, the exact analytical solution can be obtained. These closed-form solutions serve as the benchmark results for the development of numerical procedures. One such example is provided in this paper.

A unified relaxed approach easing the practical application of a paradox in curved beams

2020 ◽  
Vol 234 (22) ◽  
pp. 4535-4542 ◽  
Author(s):  
HMA Abdalla ◽  
D Casagrande ◽  
A Strozzi
Keyword(s):  
Finite Element ◽  
Stress Reduction ◽  
Normal Force ◽  
Bending Moment ◽  
Neutral Axis ◽  
Mathematical Approach ◽  
Bending Stress ◽  
Practical Application ◽  
Curved Beams ◽  
Closed Form Solutions

The paper deals with an arising paradox in curved beams subjected to bending moment and normal force. This paradox consists in the fact that by laterally removing material from section zones close to the neutral axis, not only an obvious reduction of the beam mass can be obtained, but also an unexpected, though technically negligible, reduction of the bending stress. It has recently been shown that the relaxation of the demanding achievement of a concurrent mass and stress reduction may practically lead to interesting results, yet solvable numerically. In this paper we show that, under some mild assumptions, a remarkable simplification of the intrados stress functional is obtained. Hence, a unified approximate mathematical approach based on linearization is developed for the derivation of analytical closed-form solutions for the lateral grooved zones. A practical example of the application of the relaxed paradox to optimize a crane hook subjected to bending and normal force is illustrated and compared to finite element forecasts.

scholarly journals Large deformation behavior of functionally graded porous curved beams in thermal environment

2021 ◽  
Author(s):  
S. F. Nikrad ◽  
A. Kanellopoulos ◽  
M. Bodaghi ◽  
Z. T. Chen ◽  
A. Pourasghar
Keyword(s):  
Boundary Conditions ◽  
Volume Fraction ◽  
Thermal Environment ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Bending Moment ◽  
Functionally Graded ◽  
Equilibrium Path ◽  
Curved Beams ◽  
Governing Equations

AbstractThe in-plane thermoelastic response of curved beams made of porous materials with different types of functionally graded (FG) porosity is studied in this research contribution. Nonlinear governing equations are derived based on the first-order shear deformation theory along with the nonlinear Green strains. The nonlinear governing equations are solved by the aid of the Rayleigh–Ritz method along with the Newton–Raphson method. The modified rule-of-mixture is employed to derive the material properties of imperfect FG porous curved beams. Comprehensive parametric studies are conducted to explore the effects of volume fraction and various dispersion patterns of porosities, temperature field, and arch geometry as well as boundary conditions on the nonlinear equilibrium path and stability behavior of the FG porous curved beams. Results reveal that dispersion and volume fraction of porosities have a significant effect on the thermal stability path, maximum stress, and bending moment at the crown of the curved beams. Moreover, the influence of porosity dispersion and structural geometry on the central radial and in-plane displacement of the curved beams is evaluated. Results show that various boundary conditions make a considerable difference in the central radial displacements of the curved beams with the same porosity dispersion. Due to the absence of similar results in the specialized literature, this paper is likely to provide pertinent results that are instrumental toward a reliable design of FG porous curved beams in thermal environment.

Stability and Large Deformation of 2-D Laminate Circular Thin Curved Beams

2014 ◽  
Vol 644-650 ◽  
pp. 5146-5150
Author(s):  
Chiu Wen Lin ◽  
Han Ming Tseng ◽  
Tso Liang Teng
Keyword(s):  
Stability Analysis ◽  
Large Deformation ◽  
Static Loading ◽  
Shear Force ◽  
Radius Of Curvature ◽  
Curved Beam ◽  
Curved Beams ◽  
Governing Equations ◽  
Laminate Theory ◽  
Tangent Slope

In this research, both un-deformed or Lagrangian state and deformed or Eulerian state are used to derive for stability analysis and large deformation. By choosing the deformed radius of curvature and deformed angle of tangent slope as parameters, the governing equations of laminated curved beam under static loading are transformed into a set of equations in terms of angle of tangent slope. All the quantities of axial force, shear force, radial and tangential displacements of circular thin curved beam are expressed as functions of angle of tangent slope by using laminate theory. The buckling load and large deformation analytical solutions of circular thin curved beam under a pair of forces are presented as well.

Determination of Pipe Whip Restraint Location to Prevent Plastic Hinge Formation in High Energy Piping Systems

2011 ◽  
Author(s):  
Sivadol Vongmongkol ◽  
Asgar Faal-Amiri ◽  
Hari M. Srivastava
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Power Plants ◽  
Plastic Hinge ◽  
Bending Moment ◽  
High Energy ◽  
Piping System ◽  
Secondary Effects ◽  
Element Analysis ◽  
Closed Form Solutions

The purpose of this study is to determine the Pipe Whip Restraint (PWR) location that would prevent the formation of a plastic hinge due to secondary effects of a postulated pipe break load in a high energy line(1). The prevention of a plastic hinge formation at the PWR location is important since its secondary effects could lead to additional interactions with safety related equipment, structure, and component that are essential to safely shutdown the nuclear power plants. The proper location of the PWR can be found by using the relationship between bending moment-carrying capacity of the pipe and the applied thrust force. Several closed-form solutions obtained from several literatures were studied and used to calculate bending moment-carrying capacities of a piping system and ultimately used to determine a plastic hinge length. The plastic hinge formation is also determined analytically by using the Finite Element Analysis (FEA) method. ANSYS LS-DYNA® [8] Explicit Finite Element code is used in modeling the pipe whip models, which includes the piping system and pipe whip restraint. Comparisons are made between the analytical (FEA) results and the results from several closed-form solutions.

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