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By J. H Poynting and J. J. Thomson

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Extra info for A text-book of physics: Sound (Fourth Edition)

Example text

Inconsistencies, however, appeared in many of these theories and were reported in much of the literature on shell theory during the middle of the 20th century (Leissa 1973a). For example, the strain –displacement relations used by Naghdi and Berry (1964) are inconsistent with regard to rigid body motion. Other theories including Love (1892) and Timoshenko and Woinowsky-Krieger (1959), although free from rigid body motion inconsistencies, introduced unsymmetrical differential operators, which contradicts the theorem of reciprocity and yields imaginary numbers for natural frequencies in a free vibration analysis.

37 38 Vibration of Laminated Shells and Plates The bending and twisting moment resultants and higher order shear resultant terms are 2 3 2 3 Ma sa 6 7 ðh=2 6 7 6 7 6 7 6 Mab 7 ¼ 6 sab 7ð1 þ z=Rb Þz dz 4 5 5 2h=2 4 Pa saz 2 Mb 3 2 sb 3 ð2:40Þ 7 ðh=2 6 7 6 6M 7 6s 7 6 ba 7 ¼ 6 ab 7ð1 þ z=Ra Þz dz 5 5 4 2h=2 4 Pb sbz where Pa and Pb are higher order shear terms. It should be noted here that the actual contribution of these terms to the accuracy of the shell equations needs to be determined. These terms will be kept here.

Rn þ hk > 2 > 22Rn ðhk 2 hk21 Þ 2 Rn ln > > Rn þ hk21 > > > > > N ðhk N 3 X X > z dz 1 > ðkÞ ðkÞ 3 3 >  ij  ij ¼ Rn {ðRn þ hk Þ 2 ðRn þ hk21 Þ } Eijn ¼ Q Q > > 1 þ z=Rn 3 > > k¼1 hk21 k¼1 >  ! > > > 3 R þ h > n k > þ Rn {ðRn þ hk Þ2 2 ðRn þ hk21 Þ2 } 2 3R2n ðhk 2 hk21 Þ 2 R3n ln ; Rn þ hk21 2 N ðhk X  ðkÞ Q ij   N X dz Rn þ hk  ðkÞ ¼ Rn Q ij ln Rn þ hk21 1 þ z=Rn k¼1 ð2:44Þ n ¼ a; b One can also use the truncated equations (Leissa and Chang 1996), for better numerical stability. The term 1=ð1 þ z=Rn Þ is expanded in a geometric series as 1 z z2 ¼12 þ 2 þ ðHigher order termsÞ 1 þ z=Rn Rn Rn ð2:45Þ The terms having orders higher than ðz=Rn Þ2 are truncated.

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A text-book of physics: Sound (Fourth Edition) by J. H Poynting and J. J. Thomson

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