Cartesian and piecewise parametric large deflection solutions of tip point loaded Euler–Bernoulli cantilever beams

Tari, Hafez; Kinzel, Gary L.; Mendelsohn, Daniel

doi:10.1016/j.ijmecsci.2015.06.024

Cited by 23 publications

(11 citation statements)

References 21 publications

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“…), as follows analytically from Equation (11). Given its nonlinear character and complicated dependence on arc length, we know of no closed form analytical solution to Equation (13).…”

Section: Numerical Solution Of the Exact Euler-bernoulli Equationmentioning

confidence: 99%

“…Of the investigations of large deflections of cantilever beams known to the authors (see [9] [10] [11] and references therein), nearly all concerned beams of uniform cross section, since elastica theory in these cases can be solved in closed form in terms of incomplete elliptic integrals [12], as first done in [13]. However, many modern physical applications utilize tapered cantilever beams.…”

Section: Introductionmentioning

confidence: 99%

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Bending of a Tapered Rod: Modern Application and Experimental Test of Elastica Theory

Silverman¹,

Farrah²

2018

WJM

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A tapered rod mounted at one end (base) and subject to a normal force at the other end (tip) is a fundamental structure of continuum mechanics that occurs widely at all size scales from radio towers to fishing rods to micro-electromechanical sensors. Although the bending of a uniform rod is well studied and gives rise to mathematical shapes described by elliptic integrals, no exact closed form solution to the nonlinear differential equations of static equilibrium is known for the deflection of a tapered rod. We report in this paper a comprehensive numerical analysis and experimental test of the exact theory of bending deformation of a tapered rod. Given the rod geometry and elastic modulus, the theory yields virtually all the geometric and physical features that an analyst, experimenter, or instrument designer might want as a function of impressed load, such as the exact curve of deformation (termed the elastica), maximum tip displacement, maximum tip deflection angle, distribution of curvature, and distribution of bending moment. Applied experimentally, the theory permits rapid estimation of the elastic modulus of a rod, which is not easily obtainable by other means. We have tested the theory by photographing the shapes of a set of flexible rods of different lengths and tapers subject to a range of impressed loads and using digital image analysis to extract the coordinates of the elastica curves. The extent of flexure in these experiments far exceeded the range of applicability of approximations that linearize the equations of equilibrium or neglect tapering of the rod. Agreement between the measured deflection curves and the exact theoretical predictions was excellent in all but several cases. In these exceptional cases, the nature of the anomalies provided important information regarding the deviation of the rods from an ideal Euler-Bernoulli cantilever, which thereby permitted us to model the deformation of the rods more accurately.

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“…), as follows analytically from Equation (11). Given its nonlinear character and complicated dependence on arc length, we know of no closed form analytical solution to Equation (13).…”

Section: Numerical Solution Of the Exact Euler-bernoulli Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bending of a Tapered Rod: Modern Application and Experimental Test of Elastica Theory

Silverman¹,

Farrah²

2018

WJM

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“…Only in few cases, e.g., Ohtsuki & Ellyin (2001), the approach was extended to the analysis of more complex structures, such as square frames. Recently, Zhang & Chen (2013), Tari et al (2015) and Cammarata et al (2019) extended the method to a wider range of geometric and loading configurations. Finally, Wang et al, (2018) introduced a new procedure based on the analysis of rocking effects to reproduce the behavior of post-tensioned precast piers.…”

Section: Introductionmentioning

confidence: 99%

A simple numerical approach for the pushover analysis of slender cantilever bridge piers taking into account geometric nonlinearity

Bernardini

Ruta

et al. 2022

Asian J Civ Eng

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The response of slender bridge piers to horizontal actions may be significantly influenced by geometric nonlinearities. In such conditions, the use of sophisticated models implemented in complex structural analysis software can be economically disadvantageous, especially in the preliminary design phases. This paper proposes a simple numerical procedure to compute the nonlinear pushover response of cantilever bridge piers subject to horizontal loads. The procedure is based on an iterative approach to enforce the element equilibrium under large displacements, efficiently accounting for P-Delta effects induced by vertical loads. Evaluation of the bending moment–curvature response of the pier base cross section is required and used as basic input data. For fast preliminary analyses, sectional response can be manually computed in simplified linearized form, thus completely eliminating the need to use structural analysis software. Indeed, the entire procedure can be implemented in standard programming codes, such as PythonTM or Matlab®, and used to evaluate the pushover response of piers with arbitrary cross section. Comparison with experimental test results and solutions based on Finite Element models shows that proposed procedure can be used to get a fast, yet accurate, estimate of the entire force–displacement curve and, in particular, of the pier ultimate displacement.

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“…Bisshopp and Drucker (1945) used variational method to analyze large deformation problems. Tari (2013) and Tari et al (2015) used the automatic Taylor expansion technique (ATET) to solve the large deformation problem of cantilever beam. Ho and Chen (1998) and Chen and Ho (1999) used the differential transformation method to study the vibration of a nonuniform beam with elastic constraints at both ends.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of large deflection beams by using the Laplace Adomian decomposition method

Lin¹,

Tseng²,

Chen³

2021

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PurposeThis paper presents the problems using Laplace Adomian decomposition method (LADM) for investigating the deformation and nonlinear behavior of the large deflection problems on Euler-Bernoulli beam.Design/methodology/approachThe governing equations will be converted to characteristic equations based on the LADM. The validity of the LADM has been confirmed by comparing the numerical results to different methods.FindingsThe results of the LADM are found to be better than the results of Adomian decomposition method (ADM), due to this method's rapid convergence and accuracy to obtain the solutions by using fewer iterative terms. LADM are presented for two examples for large deflection problems. The results obtained from example 1 shows the effects of the loading, horizontal parameters and moment parameters. Example 2 demonstrates the point loading and point angle influence on the Euler-Bernoulli beam.Originality/valueThe results of the LADM are found to be better than the results of ADM, due to this method's rapid convergence and accuracy to obtain the solutions by using fewer iterative terms.

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Cartesian and piecewise parametric large deflection solutions of tip point loaded Euler–Bernoulli cantilever beams

Cited by 23 publications

References 21 publications

Bending of a Tapered Rod: Modern Application and Experimental Test of Elastica Theory

Bending of a Tapered Rod: Modern Application and Experimental Test of Elastica Theory

A simple numerical approach for the pushover analysis of slender cantilever bridge piers taking into account geometric nonlinearity

Numerical solution of large deflection beams by using the Laplace Adomian decomposition method

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