Issue 51

A. Chikh, Frattura ed Integrità Strutturale, 51 (2020) 115-126; DOI: 10.3221/IGF-ESIS.51.09 119   0 4 1 , , 1, 3, 5... m q m Q m m     (17) In the case where static problems, we have the following equation:       K F   (18) where     , , t U W    and   K is the symmetric matrix given by   11 12 13 12 22 23 13 23 33 S S S K S S S S S S            (19) In the case of free vibration problem problems, the analytical solutions can be obtained by:         2 0 K M     (20) where   M is the symmetric matrix given by   11 12 13 12 22 23 13 23 33 m m m M m m m m m m            (21) For buckling problems, can be expressed as       0 K N    (22) in which: 2 3 11 11 12 11 13 1 11 4 2 2 22 11 1 0 2 2 3 2 2 2 23 1 11 33 1 55 1 11 11 1 12 2 13 1 3 22 1 4 2 2 4 2 23 5 33 1 6 , , ' , ' , ' ' , , , ' , , ' , p w s S A S B S k A D S E k k N k S k A F S k A A k A G m I m I m k A I m I I m I m k A I                                    (23) where     2 2 2 11 11 11 11 11 11 11 2 2 2 55 55 2 , , , , , 1, , ( ), , ( ), ( ) , ( ) h h h s h A B D E F G C z f z z zf z f z dz A C g z dz       (24)     2 2 2 1 2 3 4 5 6 2 , , , , , ( ) 1, , ( ), , ( ), ( ) h h I I I I I I z z f z z zf z f z dz     (25)