Large displacements of slender beams in plane: Analytical solution by means of a new hypergeometric function

Iandiorio, Christian; Salvini, P

doi:10.1016/j.ijsolstr.2019.09.006

Cited by 14 publications

(11 citation statements)

References 21 publications

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“…The exact solution to Equation ( 2) can be given in elliptic functions known as the analytical solution of an elastica [26,27], as has been shown in earlier solutions [43,45,46]. The vibrations of such a beam has been studied with the EGM also in a similar procedure [4].…”

Section: The Large Deformation Of An Elastic Beammentioning

confidence: 98%

“…The nonlinear beam theory with the options for both physical and kinematic linearities is certainly the choice for large deformation, but the nonlinear equations have to be solved with approximate methods, as shown in a recent study of the flexure of a cantilever beam under a point load [16,[20][21][22][23][24]. It is highly nonlinear differential equation for the classical problem of an elastica, and the accurate solutions can be given in elliptic functions through extensive studies in history [1][2][3][25][26][27]. To obtain practical solutions to nonlinear differential equations, many approximate techniques have been tried for improved solutions and simpler procedures including the finite-element method and alternative formulation [16,[28][29][30][31][32][33][34][35][36][37][38][39].…”

Section: The Large Deformation Of An Elastic Beammentioning

confidence: 99%

See 1 more Smart Citation

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

Meng,

Lian,

Zhang

et al. 2023

Mathematical Problems in Engineering

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By integrating the physical and time domains with the Galerkin method and adopting approximate solutions satisfying the boundary conditions, a set of algebraic equations of nonlinear nature is obtained from the nonlinear differential equation of an elastic beam for the undetermined coefficients of solutions of the deflection. These coefficients are to be used with the approximate basis functions for the asymptotic and explicit solutions of the nonlinear differential equation of flexure of a beam widely known as an elastica. Taking advantage of powerful tools for the symbolic manipulation of algebraic equations, such a novel method and procedure offer a new and efficient approach and option with known linear solutions in dealing with increasingly complex nonlinear problems in practical applications of both static and dynamic nature.

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Section: The Large Deformation Of An Elastic Beammentioning

confidence: 98%

Section: The Large Deformation Of An Elastic Beammentioning

confidence: 99%

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

Meng,

Lian,

Zhang

et al. 2023

Mathematical Problems in Engineering

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“…Then giving the (sub)equilibrium equations: K (11) ij q (1) j + K (12) ij q (2) j − F (1) e,i = 0, K (21) ij q (1) j + K (22) ij q (2) j − F (2) e,i = 0. (56)…”

Section: Piezoelectric Problem Statementunclassified

“…An interesting example of nonlinear analytical approach for a cantilever beam by means of elliptic functions is given in Mattiasson, 1 where a dead load is applied at beam's tip and the neutral axis displacement is described in terms of dimensionless parameters. In Iandiorio and Salvin, 2 a closed form solution is obtained using a methodology that uses a new hypergeometric function of two variables and is applied for isotropic beams subjected to concentrated loads, where integration of elliptic functions is avoided. Tasora et al 3 present a geometrically exact formulation for solving three‐dimensional beams based on shear‐flexible Cosserat rod theory, where the beam is described by its centerline and its displacement and section rotation are parameterized by means of spline interpolation of quaternions.…”

Section: Introductionmentioning

confidence: 99%

A geometrically nonlinear structural formulation for analysis of beams with a new set of generalized displacements considering piezoelectric effects

Silva

Donadon

Cabral

2021

Numerical Meth Engineering

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In this article, a nonlinear structural formulation that uses a new dimensionless set of generalized displacements is proposed for solving geometrically nonlinear beam problems, being validated by several applications given in literature where a cantilever beam is likely to undergo large displacements. By this approach, useful simplifications and insights are achievable in the analysis process, as the system matrices becoming linear and the reduction of required interpolation continuity degree. The formulation is firstly developed-in a Lagrangian perspective-and the equilibrium equations are then derived using Hamilton's principle. In the sequence, using finite element method, it is substantiated by comparison to examples given in literature in static, dynamic, and finally in an application where piezoelectric effects intervene, in order to assess its multiframework capabilities and deliver a convenient approach whereby beams constituted of smart or conventional materials can be efficiently studied.

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“…Besides, (c) the rod axis is supposed to be inextensible. In light of these hypotheses, several past [34][35][36] as well as current [37][38][39][40] investigations can be found in literature dealing with different structural problems. In this paper, the additional assumption that the Young's modulus varies along the rod axis, accounting for the nonhomogeneity of the cantilever, is considered.…”

Section: Introductionmentioning

confidence: 99%

Lower bound estimate for buckling in axially graded cantilever rods

2020

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Functionally graded beams, bars and rods have been gaining a relevant consideration in engineering practice and research, taking into account variations of the material properties either in the transverse or in the longitudinal direction. Yet existing literature dealing with analytical study of structural instability for an arbitrary material longitudinal variation is still limited. In this paper, a variational approach to the buckling in axially graded cantilevers is developed within Euler-Bernoulli beam theory. Considering the large deflection static behavior and interpreting the first variation of the corresponding Action integral as a weak form of the associated Euler–Lagrange equation, the problem of analytically finding a lower bound estimate for the buckling load is investigated and solved for arbitrary variations of mechanical properties within an imposed condition on the maximal deflection of the free end. In particular, two examples of widely used material gradient forms have been considered and their lower bound buckling forces have been estimated in a closed form, compared with numerical results from literature developed within the linearized version of governing equations and validated using nonlinear finite element forecasts, showing promising results in terms of buckling prediction.

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Large displacements of slender beams in plane: Analytical solution by means of a new hypergeometric function

Cited by 14 publications

References 21 publications

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

A geometrically nonlinear structural formulation for analysis of beams with a new set of generalized displacements considering piezoelectric effects

Lower bound estimate for buckling in axially graded cantilever rods

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