“Computational Civil Engineering - "Intersections" International Journal

68 C. Chioreanby an iterarive process. In this way, the arbitrary cross-sectional shape andreinforcement layout, the effect of concrete tensile cracking, the nonlinearcompressive response of concrete with different levels of confinement areaccuratelly included in the analysis.2.1 Cross-section analysisConsider the cross-section subjected to the action of the external bending momentsabout each global axes and axial force as shown in Fig. 1. Under the aboveassumptions the resultant strain distribution corresponding to the curvatures aboutglobal axes Φ = [ Φ z Φ y] and the axial compressive strain u can be expressed in[ z y]point r = in a linear form as:ε =u + Φzy + Φyz = u + ΦrT(1)Neutral axisCompresion sideTension sideyηθzΦzu 0Interior boundaryExterior boundaryξεcuΦ =Φ2 2y + Φ zΦ yMM= tanαΦy= tanθΦzyzFig.1. Cross-section analysisEquilibrium is satisfied when the external forces are equal to the internal forces.The basic equations of equilibrium for the axial load N and the biaxial bendingmoments , are given in terms of of the stress resultants as:M z , exty extM ,⎡ 1 ⎤ T( , , ) 0(2)∫σε 0 Φ y Φ z ⋅ ⎢ T ⎥dA − Sext=⎣r⎦where the vector S = [ M M ]extAN y,ext z,ext. The Eqs. (2) are solved numericallyusing the Newton-Raphson method, and results in three recurrence relationships toobtain the unknowns u and Φ and then flexural EI and axial EA rigidity modulus