The collapse of square section tubes in bending
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c o l l a p s e of thin w a l l s t r u c t u r e s is r e c e i v i n g i n c r e a s e d a t t e n t i o n b e c a u s e of t h e i r u s e f u l n e s s f o r e n e r g y a b s o r p t i o n a p p l i c a t i o n s . When s u c h s t r u c t u r e s c o l l a p s e u n d e r a load, the l a r g e d e f l e c t i o n s a r e a c c o m modated usually by extensive local plastic deformation. B e c a u s e of the i n h o m o g e n e i t y of the d e f o r m a t i o n , a n a l y t i c methods a r e g e n e r a l l y of l i t t l e p r e d i c t i v e value in t h e s e s i t u a t i o n s . Both the m e c h a n i c a l c h a r a c t e r i s t i c s of the m a t e r i a l and the g e o m e t r y of the s t r u c t u r e affect the c o l l a p s e p r o c e s s and it is n e c e s s a r y t h e r e f o r e to identify the c o n t r i b u t i o n s of t h e s e two f a c t o r s s e p a r a t e l y . T h i s was c a r r i e d out r e c e n t l y for the i m p o r t a n t c a s e of a x i a l e n e r g y a b s o r b e r s . ~ ' 2 The c o l l a p s e p r o c e s s e s in f a b r i c a t e d t u b u l a r s e c t i o n s have also been investigated, primarily from a structural v i e w p o i n t 3-6 but in one c a s e a l s o with r e g a r d to the material contribution. 7 T h e c o l l a p s e of thin w a l l s t r u c t u r e s b y bending is an i m p o r t a n t c a s e of s t r u c t u r a l c o l l a p s e . It d i f f e r s f r o m that of the a x i a l c o l l a p s e of t u b e s b e c a u s e the d e f o r m a t i o n is confined to one r e g i o n only in the s t r u c t u r e w h e r e a s in the c a s e of a x i a l c o l l a p s e , the d e f o r m a t i o n although l o c a l i z e d at any one t i m e , p r o p a g a t e s c y c l i c a l l y along an u n d e f o r m e d s h e l l . In tube bending, the d e f o r m a t i o n o c c u r s in one p r o g r e s s i v e l y d e g e n e r a t i n g r e g i o n . T e n s i l e s t r e n g t h was found to be the p r i m e m a t e r i a l p a r a m e t e r u s e f u l in p r e d i c t i n g the tube r e s p o n s e in a x i a l c o l l a p s e . ~,2 Since the new m a t e r i a l s of c u r r e n t t e c h n o l o g i c a l i n t e r e s t for s t r u c t u r a l a p p l i c a t i o n s , e . g . d u a l - p h a s e s t e e l s , have e x c e p t i o n a l c o m b i n a t i o n s of m e c h a n i c a l p r o p e r t i e s , it is d e s i r a b l e to identify the r o l e of m a t e r i a l p r o p e r t i e s in the b e n d i n g c o l l a p s e of t u b e s . In the c a s e of the l a r g e s c a l e c o l l a p s e of thin w a l l s t r u c t u r e s b y bending, v e r y l i t t l e w o r k has b e e n publ i s h e d . The g e o m e t r y c h a n g e s in c e n t r a l l y l o a d e d c y l i n d r i c a l t u b e s have been s t u d i e d 8,9 and a n a l y s e s have i n d i c a t e d that the l i m i t i n g p l a s t i c m o m e n t of t u b e s s u b j e c t to bending is p r o p o r t i o n a l to ~ot w h e r e r o is the i n i t i a l r a d i u s and t
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