Sensitivity calculations for the numerically integrated degenerate axisymmetric shell element

E x p l i c i t s e n s i t i v i t y exp ress ions f o r t h e v a r i a t i o n s o f s t r e s s e s , d i sp lacemen ts and n a t u r a l f r e q u e n c i e s w i t h r e s p e c t t o t r a n s v e r s e t h i c k n e s s o f t h e degenera te p a r a b o l i c ax i symmet r i c s h e l l e lement a r e d e r i v e d . The element i s cons ide red t o have u n i f o r m t h i c k n e s s and i s so f o r m u l a t e d as t o have i t s f o r m e x p l i c i t i n which. Thus, a l t hough s t i l l n u m e r i c a l l y i n t e g r a t e d , s e n s i t i v i t y c a l c u l a t i o n s and r e f o r m u l a t i o n s o f t h e e lement can be c a r r i e d o u t much more e f f i c i e n t l y a t some expense o f memory space. Severa l examples a r e used t o i l l u s t r a t e t h e proposed method and i t s a p p l i c a t i o n s i n minimum we igh t des ign o f t h i n s h e l l s o f r e v o l u t i o n . On ly ax i symmet r i c modes o f deformat i o n a r e cons ide red .