“Computational Civil Engineering - "Intersections" International Journal

108 S. Oller, Al.H. Barbatstate is reached by increasing the curvature ∆ χ x ) and axial strainN∆ε ( x 3)of the beam.00⎧NN ( x3) M 1 ( x3)⎪ε(x1,x2, x3) = ε ( x3) + χ1(x3) ⋅ x2+ χ 2 ( x3) ⋅ x1= + x0 0 0 0⎪E A E I11⎪0 0−10⎪⎡EA 0 0 ⎤ ⎧ N ( x3) ⎫⎪⎢0 0 ⎥ ⎪ 0 ⎪= { 1 x }⎪2 x1⋅ ⎢ 0 E I110 ⎥ ⋅ ⎨M1 ( x3) ⎬⎪⎢0 0 ⎥ ⎪ 0⎩M 2 ( x3) ⎪⎣0 0 E I 22 ⎦⎭⎪000⎪0N ( x3) M 1 ( x3) M 2 ( x3)⎨σ(x1,x2, x3) = E ⋅ ε(x1,x2, x3) = + x2+ x000 1⎪A I11I 22⎪0−10⎪⎡A0 0 ⎤ ⎧ N ( x ) ⎫3⎪⎢ 0 ⎥ ⎪ 0 ⎪= { 1 x2x1}⋅⋅⎪⎢ 0 I110 ⎥ ⎨M1 ( x3) ⎬⎪⎢0 ⎥ ⎪ 0 ⎪⎣0 0 I 22 ⎦ ⎩M 2 ( x3)⎭⎪−1⎪T 0 0= x ⋅ [ J ] ⋅ σˆ( x3)⎪⎩( 1 − d(x , x , )( 3, x3) ⋅dA⎫⎤⎥⎪, x3) ⋅ x2dA⎬⎥→ 0⎪⎥, x3) ⋅ x1dA⎪⎥⎭⎦M 2 ( x+0 0E IThis procedure is iterative and starts with the linearization of the followingunbalanced equilibrium equation at each cross section of the beam:∆σˆ( x⎧∆N⎪⎨∆M⎪⎩∆M31) =0int[ σˆ( x ) − σˆ( x )]⎡( x ⎫3 ) ⎢⎧N⎪⎢⎪( x ⎬ = ⎨3 )⎪ ⎢M⎪( x3) ⎭ ⎢⎩M⎣23001023⎧( x ⎫3 )⎪⎪( x ⎬ − ⎨3 )⎪ ⎪( x3) ⎭ ⎪⎩→ 0∫∫∫σ(x , xσ(x , xσ(x , xwhere the stress at each point of the cross section is obtained by using theconstitutive damage model briefly described by the following equations:Nσ( x1,x2, x3) = E(x1,x2, x3) ⋅ ε(x1,x2, x3) = E(x1,x2, x3) ⋅ [ ε ( x3) + χ 1 ( x3) ⋅ x2+ χ 2 ( x3) ⋅ x1](11)0with : E(x , x , x ) = f ( x , x , x ) ⋅ E123123According to the Annex, the evolution of the local damage internal variable at eachpoint of each cross section of the Bernoulli beam, obtained from the local damage3 ) c A 1−maxconstitutive model, is f ( x1,x2, x3) =1 2x = ec, withc0 ≤ cmaxmax≤ c . The values c and c are the maximum and current tension strengthat each point of the solid, A is a parameter depending of the fracture energy andAAA111222max20322c(d )( ))x1(9)(10)