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BELL SYSTEM TECHNICAL JOURNAL 



The boundary condition for the extensional stresses will not be met; 

 however, they will be quite small in comparison to Xy if k is small, and may 

 be neglected. 



7.93. Thickness Flexures 



Consider a three dimensional plate having a thickness b lying along the y 

 direction. The following displacements are found to be of a form that can 

 be made to satisfy the equilibrium equations 7.2. 



u = U sin kx sin ly cos mz 

 V = V cos kx cos ^y cos mz 

 w = W cos kx sin /y sin mz ^ 



(7.37) 



Performing the operations indicated and substituting into the equilibrium 

 equations give the following result: 



A(k^ 4- ^' + w') + ^ (kU - ^V + mW) = pco' 



A{k' + f -\- m ) - ^{kU - tV + mW) = pw }■ 



Aik' + f + m') + ?^ (kU - IV + mW) = pco' 

 W 



(7.38) 



Subtract the second and third equations of (7.38) from the first: 



Bk , Bt ^ . Bk Bm ^ 

 then _ + _=Oand--— =0 



or 



V I ^ W m 



u=-k "°^ U = k 



(7.39) 



Putting these values back into 7.38, it is seen that the following relation- 

 ship must be satisfied. 



(A + B) (k^ + ^''-\- m") = pco2 



(7.40) 



Letting — = — - as in (7.39), another value for — may be obtained. 

 U k U 



The first and second equations of (7.38) will be satisfied for any ratio of 



W 

 Z7' 



so the 3rd equation is used. 



A{k^ + f + m') + -S yfe' + ^' -f mk 





pw 



(7.41) 



