Issue 29

S. Terravecchia et al., Frattura ed Integrità Strutturale, 29(2014) 61-73; DOI: 10.3221/IGF-ESIS.29.07 68 * , * , ( ) ( ) ut ut u tc u gt g tc                     A C a 0 L U 0 A a            (25) ( ) ur g gr gr                 A 0 L G A C 0  (26) because the solution vector X has to be null. 3) If the solid is subjected to a combination of the rigid motions 1) and 2), the condition u g    L L L 0 has to be always valid. Example of coefficient calculation in presence of singularity The greater difficulties consist in removing the singularities when the cause is focused at the corner. The simple observation that the vector  u regards all the nodes of the boundary, including the vector C  u at the corner, allows to eliminate the higher order singularities after the first integration. Indeed, the use of the distribution theory permits to evaluate the effect on the boundary without limit operations and removing the strong singularities through the summing of effects, whereas the remaining ones are eliminated through the second integration. The singularities present in the fundamental solutions used for the plate analysis require shape functions of type (0) C . In order to show how operate to remove singularity in the coefficients of the solving system, we examine the load vector u L and specifically calculate the weighed normal derivative. a) b) Figure 3: a) Effect: weighed normal derivative , b yy S on the b  ; b) Cause: vertical displacement on the b  and c  and corner displacement , 1 y C U   . With referring to Fig.3, the weighed normal derivative , b yy S is , , b b yy b b yy S N g d     (27) where , b yy g is a normal derivative distribution due to: - vertical displacements modeled on boundary elements b and c , having maximum value 1 y U   ; - corner vertical displacement , 1 y C U   . In this case the SI of the normal derivative gives ' ' * * * , ( ) ' ' ( ) ' ' , ( ) d d b c b yy gt yy b b gt yy c c g tc yy g G N G N G                (28) and, suppressing for simplicity the subscript yy , through the process of weighing one obtains