Analytical modelling of retrofitted reinforced concrete members with flexible bond

Abstract

The model for physically and geometrically non-linear analysis of retrofitted cross-sections and elements is a consequent continuation of the authors’ former work [2]. Instead of the then assumed rigid bond between the different segments of the cross-section, the modelling includes flexible bond also. The mathematical model is based on a kinematic formulation of the mechanical problem by the Lagrange principle of Minimum of total potential energy. Stresses and strain energies are functions of the strains. The strain energy of a cross-section is determined by a contour integral related to the generalised function F to describe the material behaviour in an integral form. Thus a discretisation of the cross-section is unnecessary. Minimising the total potential energy of the entire member as a function of the displacements u, v und w enables the computation of deformations and stresses of composite structural elements, whereas the compatibility of these displacements and deformations must be taken into account. According to Lagrange principle flexible bond between the subsections will be considered in the model by means of energy expressions. Different bond effects like friction and adhesion as well as special bond components e.g. headed studs are included into model by separate shear force—slip relations. that one is actual which corresponds to the minimum value of total potential energy of the system Π Π Π = + → i a Minimum, (1) where Π i is the strain energy and Π a is the potential energy of the external forces. The deformation states are kinematically admissible when they fulfil the geometrical compatibility conditions and the statical boundary conditions. 2.1 Compatibility of deformations of cross-sections According to the Bernoulli hypothesis crosssections normal to the axis of the element remains plane during the deformation process. Thus the strain at an arbitrary point of the cross-section is defined by the linear function e e κ κ x y z y z y z ( , ) = + + 0 (2) where e 0 is the strain in the origin. The parameters κ y and κ z are the curvature of the “strain plane” related to the axis y and z of the Cartesian coordinate system. Rotating the y,zcoordinate-system by the angle φ we get a new coordinate-system η and ζ. The relation