A second order beam theory

Stephen, N.G.; Levinson, M.

doi:10.1016/0022-460x(79)90537-6

Cited by 109 publications

(30 citation statements)

References 8 publications

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“…On the other hand, the value k ¼ 5 6 provides a prediction always lower than the exact, at least over the range considered, with a maximum error of À2%. Experimental evidence in support of the value k ¼ 5ð1 þ nÞ=ð6 þ 5nÞ has recently been provided by Me´ndez-Sa´nchez et al [17], although they also note that a two-coefficient theory presented by Stephen and Levinson [18], and which incorporates both this, and Cowper's value [19] of k ¼ 10ð1 þ nÞ=ð12 þ 11nÞ, provides marginally better agreement with their experimental results.…”

Section: Numerical Examplementioning

confidence: 81%

The second spectrum of Timoshenko beam theory—Further assessment

Stephen

2006

Journal of Sound and Vibration

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A review of contributions and views on the second spectrum of Timoshenko beam theory (TBT) over the past two decades, together with some new results, are presented. It is shown that the Timoshenko frequency equation factorises not solely for hinged-hinged end conditions, as is often claimed, but also for guided-guided and guided-hinged; these new cases may be regarded as portions of a multi-span hinged-hinged beam. A higher-derivative Lagrangian that leads directly to the well-known fourth-order Timoshenko beam equation is reviewed. A simple relationship between the so-called Ostrogradski energy and the mechanical energy is derived for hinged-hinged end conditions. It is shown that the Ostrogradski energy is positive for the first spectrum but negative for the second; within some branches of physics, this would be sufficient to conclude that the second spectrum is ''unphysical''. A numerical example presented by Levinson and Cooke is re-examined using both TBT and exact plane stress elastodynamic theory. Agreement is excellent for the first spectrum. However, the second spectrum predictions are not in consistent agreement with any single mode of vibration. For long wavelength it is very close to the second asymmetric mode, but as wavelength shortens, it becomes closer to the second symmetric, then the third asymmetric modes. The conclusion remains unchanged: the second spectrum predictions of TBT should be disregarded. r

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Section: Numerical Examplementioning

confidence: 81%

The second spectrum of Timoshenko beam theory—Further assessment

Stephen

2006

Journal of Sound and Vibration

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“…This settles any remaining controversy [10,11] on the ''best'' choice for k among those derived by different and slightly ad hoc methods, and puts Timoshenko beam theory, with a canonical choice of the shear coefficient [12][13][14][15], on a firm footing.…”

Section: Examplementioning

confidence: 95%

Perturbation theory and the Rayleigh quotient

Chan

Stephen

Young

2011

Journal of Sound and Vibration

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a b s t r a c tThe characteristic frequencies o of the vibrations of an elastic solid subject to boundary conditions of either zero displacement or zero traction are given by the Rayleigh quotient expressed in terms of the corresponding exact eigenfunctions. In problems that can be analytically expanded in a small parameter e, it is shown that when an approximate eigenfunction is known with an error Oðe N Þ, the Rayleigh quotient gives the frequency with an error Oðe 2N Þ, a gain of N orders. This result generalizes a well-known theorem for N =1. A non-trivial example is presented for N =4, whereby knowledge of the 3rd-order eigenfunction (error being 4th order) gives the eigenvalue with an error that is 8th order; the 6th-order term thus determined provides an unambiguous derivation of the shear coefficient in Timoshenko beam theory.

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“…Moreover, this theory cannot model the cross sectional warping, an important element in modelling composites. The second-order theory [17] needs two factors, making this theory a problem-dependent one also. The third-order theory [18] has the basic disadvantage of C' continuity, while the fourth-order theory [20] has been formulated exclusively for beams with narrow rectangular cross sections and, hence, cannot be directly applied to other beams with arbitrary cross sections.…”

Section: Higher-order Theory For Transient Dynamicsmentioning

confidence: 99%

On the performance of higher order theories for transient dynamic analysis of sandwich and composite beams

Marur¹,

Kant

1997

Computers & Structures

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Abstract-Higher-order shear-deformable refined theories, based on isoparametric elements, are adopted for transient dynamic analysis of symmetric and unsymmetric sandwich and composite beam constructions. These shear-correction coefficient free theories model cross sectional warping using nonlinear variation of inplane displacements across the depth. They also incorporate transverse shear stress in the formulation. A special lumping scheme is employed for the evaluation of diagonal mass matrix, and a central difference scheme is used for carrying out the integration of the equation of motion, to obtain the response history. Through numerical experiments, the efficacy of higher-order models in predicting displacements and stress, resultants over from the first-order theory, with respect to time, is clearly brought out in this paper. 0 1997 Elsevier Science Ltd.

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A second order beam theory

Cited by 109 publications

References 8 publications

The second spectrum of Timoshenko beam theory—Further assessment

The second spectrum of Timoshenko beam theory—Further assessment

Perturbation theory and the Rayleigh quotient

On the performance of higher order theories for transient dynamic analysis of sandwich and composite beams

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