scholarly journals Static Response of Double-Layered Pipes via a Perturbation Approach

2021 ◽  
Vol 11 (2) ◽  
pp. 886
Author(s):  
Daniele Zulli ◽  
Arnaldo Casalotti ◽  
Angelo Luongo
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Perturbation Approach ◽  
Perturbation Scheme ◽  
The Finite Element Method ◽  
Equilibrium Equations ◽  
Local Effects ◽  
Nonlinear Beam ◽  
Transverse Loads ◽  
Nonlinear Equilibrium

A double-layered pipe under the effect of static transverse loads is considered here. The mechanical model, taken from the literature and constituted by a nonlinear beam-like structure, is constituted by an underlying Timoshenko beam, enriched with further kinematic descriptors which account for local effects, namely, ovalization of the cross-section, warping and possible relative sliding of the layers under bending. The nonlinear equilibrium equations are addressed via a perturbation method, with the aim of obtaining a closed-form solution. The perturbation scheme, tailored for the specific load conditions, requires different scaling of the variables and proceeds up to the fourth order. For two load cases, namely, distributed and tip forces, the solution is compared to that obtained via a pure numeric approach and the finite element method.

A closed form solution for bending/stretching analysis of functionally graded circular plates under asymmetric loading using the Green function

2010 ◽  
Vol 224 (6) ◽  
pp. 1153-1163 ◽  
Author(s):  
A R Saidi ◽  
A Naderi ◽  
E Jomehzadeh
Keyword(s):  
Green Function ◽  
Closed Form ◽  
Volume Fraction ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Form Solution ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Circular Plates ◽  
Equilibrium Equations

In this article, a closed-form solution for bending/stretching analysis of functionally graded (FG) circular plates under asymmetric loads is presented. It is assumed that the material properties of the FG plate are described by a power function of the thickness variable. The equilibrium equations are derived according to the classical plate theory using the principle of total potential energy. Two new functions are introduced to decouple the governing equilibrium equations. The three highly coupled partial differential equations are then converted into an independent equation in terms of transverse displacement. A closed-form solution for deflection of FG circular plates under arbitrary lateral eccentric concentrated force is obtained by defining a new coordinate system. This solution can be used as a Green function to obtain the closed-form solution of the FG plate under arbitrary loadings. Also, the solution is employed to solve some different asymmetric problems. Finally, the stress and displacement components are obtained exactly for each problem and the effect of volume fraction is also studied.

scholarly journals A New Solution to Well-Known Hencky Problem: Improvement of In-Plane Equilibrium Equation

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 653
Author(s):  
Xue Li ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Shou-Zhen Li ◽  
Xiao-Ting He
Keyword(s):  
Closed Form ◽  
Equilibrium Equation ◽  
Circumferential Stress ◽  
Closed Form Solution ◽  
Form Solution ◽  
Circular Membrane ◽  
Equilibrium Equations ◽  
Power Series Method ◽  
Out Of Plane ◽  
Plane Equilibrium

In this paper, the well-known Hencky problem—that is, the problem of axisymmetric deformation of a peripherally fixed and initially flat circular membrane subjected to transverse uniformly distributed loads—is re-solved by simultaneously considering the improvement of the out-of-plane and in-plane equilibrium equations. In which, the so-called small rotation angle assumption of the membrane is given up when establishing the out-of-plane equilibrium equation, and the in-plane equilibrium equation is, for the first time, improved by considering the effect of the deflection on the equilibrium between the radial and circumferential stress. Furthermore, the resulting nonlinear differential equation is successfully solved by using the power series method, and a new closed-form solution of the problem is finally presented. The conducted numerical example indicates that the closed-form solution presented here has a higher computational accuracy in comparison with the existing solutions of the well-known Hencky problem, especially when the deflection of the membrane is relatively large.

Formulation for Static Behavior of the Viscoelastic Euler-Bernoulli Micro-Beam Based on the Modified Couple Stress Theory

2012 ◽  
Author(s):  
E. Taati ◽  
M. Nikfar ◽  
M. T. Ahmadian
Keyword(s):  
Size Effects ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Closed Form Solution ◽  
Modified Couple Stress Theory ◽  
Form Solution ◽  
Equilibrium Equations ◽  
Stress Theory ◽  
Modified Couple Stress ◽  
Micro Structures

In this work an analytical solution is presented for a viscoelastic micro-beam based on the modified couple stress theory which is a non-classical theory in continuum mechanics. The modified couple stress theory has the ability to consider small size effects in micro-structures. It is strongly emphasized that without considering these effects in such structures the solution will be wrong and not suitable for designing systems in micro-scales. In this study correspondence principle is used for deriving constitutive equations for viscoelastic material based on the modified couple stress theory. Governing equilibrium equations are obtained by considering an element of micro-beam. Closed-form solution for the static deflection of simply supported micro-beam is presented. Numerical results show that when the size of system is near the length scale parameter, the classical response will intensely be deviated from the correct solution observed in laboratories contrary to the modified couple stress which reflects the size effects.

On the Coupled Flap-Lag Bending Vibration of Propeller Blades: An Exact Dynamic Finite Element Formulation

2000 ◽  
Author(s):  
S. M. Hashemi ◽  
M. J. Richard
Keyword(s):  
Finite Element ◽  
Dynamic Stiffness ◽  
Closed Form Solution ◽  
Form Solution ◽  
Element Formulation ◽  
The Finite Element Method ◽  
Weighted Residual Method ◽  
Bending Vibration ◽  
Dynamic Finite Element ◽  
Propeller Blades

Abstract An exact Dynamic Finite Element (DFE) approach for the coupled Flap-Lag vibration of blades is presented. DFE can be considered as a combination of the Finite Element Method (FEM) and the Dynamic Stiffness Matrix (DSM) formulations. The weighted residual method is used and the weighting and shape functions are chosen referring to the appropriate closed form solution of the non-coupled member equations. Based on the DFE approach, the natural frequencies for a scaled propeller blade are calculated and they are compared to the experimental data and other existing results.

Thermoelastic Stability of Imperfect Functionally Graded Plates Based on the Third Order Shear Deformation Theory

2006 ◽  
Author(s):  
B. Samsam Shariat ◽  
M. R. Eslami ◽  
A. Bagri
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Form Solution ◽  
Functionally Graded ◽  
Functionally Graded Plates ◽  
Equilibrium Equations ◽  
Third Order ◽  
The Third

Thermal buckling analysis of rectangular functionally graded plates with initial geometric imperfections is presented in this paper. It is assumed that the non-homogeneous mechanical properties vary linearly through the thickness of the plate. The plate is assumed to be under various types of thermal loadings, such as the uniform temperature rise and nonlinear temperature gradient through the thickness. A double-sine function for the geometric imperfection along the x and y-directions is considered. The equilibrium equations are derived using the third order shear deformation plate theory. Using a suitable method, equilibrium equations are reduced from 5 to 2 equations. The corresponding stability equations are established. Using these equations accompanied by the compatibility equation yield to the buckling loads in a closed form solution for each loading case. The results are compared with the known data in the literature.

scholarly journals Best theory diagrams for multilayered plates considering multifield analysis

2017 ◽  
Vol 28 (16) ◽  
pp. 2184-2205 ◽  
Author(s):  
M Cinefra ◽  
E Carrera ◽  
A Lamberti ◽  
M Petrolo
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Equilibrium Equations ◽  
Material Parameters ◽  
Stress Components ◽  
Minimum Number ◽  
Multilayered Plates ◽  
Carrera Unified Formulation ◽  
Definition Of

This work presents the best theory diagrams (BTDs) for multilayered plates involved in multifield problems (mechanical, thermal and electrical). A BTD is a curve that reports the minimum number of terms of a refined model for a given accuracy. The axiomatic/asymptotic technique is employed in order to detect the relevant terms, and the error is computed with respect to an exact or quasi-exact solution. The models that belong to the BTDs are constructed by means of a genetic algorithm and the Carrera Unified Formulation (CUF). The CUF defines the displacement field as an expansion of the thickness coordinate. The governing equations are obtained in terms of few fundamental nuclei, whose form does not depend on the particular expansion order that is employed. The Navier closed-form solution has been adopted to solve the equilibrium equations. The analyses herein reported are related to plates subjected to multifield loads: mechanical, thermal and electrical. The aim of this study is to evaluate the influence of the type of the load in the definition of the BTDs. In addition, the influence of geometry, material parameters and displacement/stress components are considered. The results suggest that the BTD and the CUF can be considered as tools to evaluate any structural theory against a reference solution. In addition, it has been found that the BTD definition is influenced to a great extent by the type of load.

A New Solution to Coulomb Friction in Mechanism Bearings: Theory and Application

1981 ◽  
Vol 103 (4) ◽  
pp. 764-775 ◽  
Author(s):  
I. Imam ◽  
M. Skreiner ◽  
J. P. Sadler
Keyword(s):  
Coulomb Friction ◽  
Circuit Breaker ◽  
Closed Form Solution ◽  
Form Solution ◽  
Design Tool ◽  
Operating Conditions ◽  
Equilibrium Equations ◽  
Simplified Approach ◽  
Novel Approach ◽  
In Series

A new approximate closed form solution is presented for analyzing Coulomb friction in mechanism bearings. This is accomplished by linearizing the nonlinear terms of the equilibrium equations by a novel approach. The method, based on the traditional friction circle concepts, is particularly well suited to computers and proceeds to the solution without the need for any numerical methods or iterative steps as has previously been required. Further, only a pair of simultaneous equations is solved at a time, thus eliminating the need for a complete matrix inversion. It is believed that this simplified approach will be more readily usable in actual design situations where consideration of bearing friction is of considerable importance. A number of analytical validity checks were applied that successfully verified the adequacy of this new approach. As an example, the theory is applied to various models of a class of circuit breaker mechanisms, consisting of five four-bar linkages in series under different operating conditions. In all cases, the analytical results compare very favorably with the test data, thus establishing the method as a valuable practical design tool.

scholarly journals Analytic Simulations of Packaging Cardboards with Non-Rigid Adhesives

2019 ◽  
Vol 7 (6) ◽  
pp. 01-15
Author(s):  
R. Hussein
Keyword(s):  
Boundary Conditions ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Equilibrium Equations ◽  
The Real ◽  
Usual Assumption ◽  
Perfect Bonding

The understanding of the cardboard performance is necessary to the design of packaging containers and the protection of their contents for safe deliveries. The use of adhesives is unavoidable in the manufacturing of the cardboards. Like all materials, the adhesives have finite stiffness but when used in the literature, they are assumed perfectly rigid. This study changes this assumption by using the real properties of adhesives. A closed-form solution for cardboard panelsassembled withnon-rigid adhesives, and subjected to edgewise loading is presented. The solution satisfies the equilibrium equations of the layers, the compatibility equations of stresses and strains at the interfaces, and the boundary conditions. To investigate the effects of the finite values of adhesivestiffness on the responses, numerical evaluations are conducted. The results obtained have shown that the adhesive stiffness has a strong effect on the performance. Beyond a certain level of stiffness, the usual assumption of perfect bonding used in classical theories is acceptable. This could provide an answer to what constitutes perfect bonding in terms of the ratio of the fluted layer, or simply flute, stiffness to the bonding stiffness.

A new approach for closed-form analytical solution of two-dimensional symmetric double contacts and the comparison with finite element method

2017 ◽  
Vol 232 (8) ◽  
pp. 1025-1035 ◽  
Author(s):  
Parisa Ghanati ◽  
Saeed Adibnazari ◽  
Mohammad Alrefai ◽  
Azadeh Sheidaei
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Closed Form ◽  
Flat Surface ◽  
Closed Form Solution ◽  
Form Solution ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Contact Zones ◽  
Element Method

In this paper, a new procedure is developed for the solution of a general two-dimensional uncoupled symmetric double contact problem with smooth contact zones in which the indenter geometry is described by a piecewise biquadratic function. This procedure gives an approximate closed-form solution for any smooth indenter profile. In order to evaluate the accuracy of this approach, it is applied to the symmetric indentation of a flat surface by two rigidly interconnected parabolic indenters and results are compared with the exact unclosed-form solution. Moreover, this procedure is applied to the symmetric indentation of a flat surface by two rigidly interconnected cylinders to compare the results with the finite element solution obtained by the finite element method software, ABAQUS. The results showed that in comparison with the finite element method, this procedure is a fast and highly accurate method with low complexity that makes feasible the possibility of determining approximate closed-form solution for a wide range of indenter geometries with a concavity between two symmetric contact zones; hence it can be useful in practical issues.

scholarly journals Closed-Form Solution of a Peripherally Fixed Circular Membrane under Uniformly Distributed Transverse Loads and Deflection Restrictions

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Teng-fei Wang ◽  
Xiao-ting He ◽  
Yang-hui Li
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Closed Form Solution ◽  
Axisymmetric Deformation ◽  
Form Solution ◽  
Circular Membrane ◽  
Rigid Plate ◽  
Membrane Stress ◽  
Transverse Loads ◽  
First Time

The problem of axisymmetric deformation of a peripherally fixed and uniformly loaded circular membrane under deflection restrictions (by a frictionless horizontal rigid plate) was analytically solved, where the assumption of constant membrane stress adopted in the existing work was given up, and a closed-form solution of this problem was presented for the first time. The numerical analysis shows that the closed-form solution presented here has higher calculation accuracy than the existing approximate solution.

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