“Computational Civil Engineering - "Intersections" International Journal

112 S. Oller, Al.H. Barbat0 =⇒beingnJn+1n+1∆σˆn+1n+1∆σt+∆t∆εˆ∂=t+∆tt+∆t( x3( x) ≅3n∆σˆ) = −t+∆t( xn n+1[ J ]∂) +−1n∆σˆn t +∆t[ ∆σˆ( x )]t+∆t∂εˆn+1( x , x , x ) = E ( x , x , x ) ⋅1n t+∆t[ ∆σˆ( x )]∂εˆthe Jacobian matrix.2330= E3012( x )3⎡ A(x3)⋅⎢⎢m1( x3)⎢⎣m2( x3)33⇒xTm ( xII11121⋅n+1n+1⋅ ∆εˆ3( x3( x3)))εˆ∆εˆt +∆tt +∆tIIt +∆t( x21222( x3( x3m ( x3) =) ⇒( x33( x3) ⎤)⎥⎥) ⎥⎦) + ...nεˆn+1t +∆tσ( xt+∆t3=) +nσn+1t+∆t∆εˆ+t +∆tn+1( x ) (17)For each time increment in which the predictor moment produces an unbalancedload increment greater than an adopted tolerance (equations 17 and 18), theprocedure considers an increment of the curvature in order to obtain a corrector ofgeneralized stresses which permits to reach the equilibrium state. The usedconvergence criterion states that the stable response is obtained for the crosssection if∆σ3t+∆t∑i∑i∆σˆ0( σˆ)i2i2≤ TOLwhere TOL is the tolerance adopted (TOL→ 0).(18)Box 2. Algorithm for the cross sectional damage integration.1. Loop over the time t + ∆t2. Loop over the cross section position x33. Compute the elastic generalized stresses −predictor− for each cross section.4. Compute the residual generalized internal stresses0intFor the first load step : J(x ) ≡ J ( x ) ; σˆ( x ) = 00int[ σˆ( x ) − σˆ( x )]∆σˆ ( x3) = 335. Balance equation verification on x 3 cross section :? ⎧0⇒ go to EXIT∆σˆ ( x3)= ⎨⎩≠0 ⇒ Continue6. Starting loop over Newton-Raphson n iteration process. Incremental generalized straincomputation and obtaining of its current value:n+1t+∆t∆εˆ( x ) = − J33n n+1[ ( x )]n+1 t+∆tn t+∆tn+1 t+∆tεˆ( x ) ˆ ( ) ˆ3 = ε x3+ ∆ε( x3)7. Damaged inertia computation at each x3cross section of pier k , using thecontinuum damage model showed in Box A-1 of the Annex:3−1n3∆σˆt+∆t( x )33