On applications of generalized functions to beam bending problems

Yavari, Arash; Sarkani, Shahram; Moyer, E. T.

doi:10.1016/s0020-7683(99)00271-1

Cited by 75 publications

(61 citation statements)

References 9 publications

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“…where overbar means generalized derivative [36][37][38], while R (x) and (x) are generalized functions,…”

Section: Axial Frequency Responsementioning

confidence: 99%

“…being δ x − x j and δ (k) x − x j the Dirac's delta and its formal kth derivative at x = x j , respectively [36][37][38]. Frequency dependence in Eqs.…”

Section: Axial Frequency Responsementioning

confidence: 99%

“…In the same context, the exact dynamic stiffness matrix and load vector of the beam have been derived in a symbolic form, to be assembled for computing the frequency response of 2D frames with multiple dampers. For this purpose, the theory of generalized functions has been used [17,18,33,[36][37][38][39][40][41]. Solutions in Ref.…”

Section: Introductionmentioning

confidence: 99%

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Exact frequency response of bars with multiple dampers

2016

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The paper addresses the frequency analysis of bars with an arbitrary number of dampers, subjected to harmonically varying loads. Multiple external/internal dampers occurring at the same position along the bar, modelling external damping devices and internal damping due to damage or imperfect connections, are considered. In this context, the challenge is to handle simultaneous discontinuities of the response variables, i.e. axial force/displacement discontinuities at the location of external/internal dampers. Based on the theory of generalized functions, the paper will present exact closed-form expressions of the frequency response under point/polynomial loads, which hold regardless of the number of dampers. In addition, closed-form expressions will be derived for the exact dynamic stiffness matrix and load vector of the bar, to be used in a standard assemblage procedure for an exact frequency response analysis of 2D truss structures. Changes to consider a single damper at a given position are straightforward. Numerical applications show the advantages of the proposed method.

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“…where overbar means generalized derivative [36][37][38], while R (x) and (x) are generalized functions,…”

Section: Axial Frequency Responsementioning

confidence: 99%

“…being δ x − x j and δ (k) x − x j the Dirac's delta and its formal kth derivative at x = x j , respectively [36][37][38]. Frequency dependence in Eqs.…”

Section: Axial Frequency Responsementioning

confidence: 99%

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Exact frequency response of bars with multiple dampers

2016

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“…It has been generalised by Wittrick [4], who considered Euler-Bernoulli beams including axial compression and elastic foundations, as well as circular plates with a variety of discontinuous loads. Recent papers by Yavari et al [5][6][7][8] have provided a variety of research results, including application to Timoshenko beams, elastic foundations and to cases in which the bending and shear stiffness properties change abruptly.…”

Section: Introductionmentioning

confidence: 99%

Macaulay's Method for a Timoshenko Beam

Stephen

2007

International Journal of Mechanical Engineering Education

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The Macaulay bracket notation is familiar to many engineers for the defl ection analysis of a Euler-Bernoulli beam subject to multiple or discontinuous loads. An expression for the internal bending moment, and hence curvature, is valid at all locations along the beam, and the defl ection curve can be calculated by integrating twice with respect to the axial coordinate. The notation obviates the need for matching of multiple constants of integration for the various sections of the beam. Here, the method is extended to a Timoshenko beam, which includes the additional defl ection due to shear. This requires an additional expression for the shearing force, also valid at all locations along the beam.

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“…Recently, the theory of generalized functions was utilized for analyzing beams with internal and external discontinuities, [12]. In [12], it was shown that the equivalent distributed force for a point moment of order n can be expressed by the nth distributional derivative of the Dirac delta function.…”

Section: Introductionmentioning

confidence: 99%

Generalized solutions of beams with jump discontinuities on elastic foundations

Yavari

Sarkani

Reddy

2001

Archive of Applied Mechanics (Ingenieur Archiv)

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The bending solutions of the Euler±Bernoulli and the Timoshenko beams with material and geometric discontinuities are developed in the space of generalized functions. Unlike the classical solutions of discontinuous beams, which are expressed in terms of multiple expressions that are valid in different regions of the beam, the generalized solutions are expressed in terms of a single expression on the entire domain. It is shown that the boundaryvalue problems describing the bending of beams with jump discontinuities on discontinuous elastic foundations have more compact forms in the space of generalized functions than they do in the space of classical functions. Also, fewer continuity conditions need to be satis®ed if the problem is formulated in the space of generalized functions. It is demonstrated that using the theory of distributions (i.e. generalized functions) makes ®nding analytical solutions for this class of problems more ef®cient compared to the traditional methods, and, in some cases, the theory of distributions can lead to interesting qualitative results. Examples are presented to show the ef®ciency of using the theory of generalized functions.

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On applications of generalized functions to beam bending problems

Cited by 75 publications

References 9 publications

Exact frequency response of bars with multiple dampers

Exact frequency response of bars with multiple dampers

Macaulay's Method for a Timoshenko Beam

Generalized solutions of beams with jump discontinuities on elastic foundations

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