“Computational Civil Engineering - "Intersections" International Journal

110 S. Oller, Al.H. Barbatsubsections, as shown in Figure 4. For each subsection, the damaged area,(x 3 ) (J) , isA A( x3) (J) = ∫ f ( x1,x2, x3) dA(13)A(J)and the distance between the neutral axes of each sub cross section and the globalneutral axis of the complete cross section is calculated. The global inertia of thedamaged cross section, I (x 3 ) (J) , is then defined bywhereIT( xI (J)3)(J)==∑44(J)j=1∑j=1I⎧⎨⎩⎡⎢⎢⎢⎣+⎡⎢⎣∫T+ A(x )∫∫A(J)A(J)A( J )31(J)1f ( x , x1⋅ Xf ( x , x2f ( x , x2, x22(J), x3, x3=) ⋅ x) ⋅ x3122⋅ xdA2dA⎤⎡ X) dA ⋅ ⎢⎥⎦ ⎢⎣X1 (J)∫A∫(J)22 (J)XA(J)f ( x⎤1,x2, x3) ⋅ x2⋅ x1dA⎥ +2f ( x ⋅ ⎥1,x2, x3) x1dA⎥⎦2 (J)XX2 (J) 1 (J)2X 1 (J)⎤⎥⎥⎦⎫⎬⎭(14)is the damaged inertia of subsection j, evaluated by means of Equation(14), A is the damaged area of the subsection j and X (J)are the distances(J)between the neutral axis of the subsections and the global neutral axis of the wholecross-section, which depend on the damage at the cross section. In the equations(13) and (14), the numerical integration has been carried out following its classicalform2A(x3m ( xiiiji)333(J))I ( x )I ( x )(J)(J)(J)====∫A( J )∫∫A( J )A( J )∫A( J )f ( x , x11112f ( x , xf ( x , xf ( x , x, x2223, x, x33, x) ⋅ dA = J3) ⋅ x) ⋅ x) ⋅ x2jjjdA = JdA = J⋅ xiacob⎡⋅ ⎢⎢⎣acobacobndA = Jn∑∑p= 1 q=1⎡⋅ ⎢⎢⎣⎡⋅ ⎢⎢⎣acobn∑∑p= 1 q=1nwnn∑∑p= 1q=1⎡⋅ ⎢⎢⎣p⋅ wnwwppnq∑∑p= 1 q=1[ f ( ξ , ξ , x )]⋅ w⋅ wwpqq12[ f ( ξ , ξ , x ) ⋅ ξ ]12[ f ( ξ , ξ , x ) ⋅ ξ ]⋅ wq1322[ f ( ξ , ξ , x ) ⋅ ξ ⋅ ξ ]1⎤⎥⎥⎦(J)3323jj⎤⎥⎥⎦⎤⎥⎥⎦(J)(J)ji⎤⎥⎥⎦(J)(15)