Issue 29

D. De Domenico et alii, Frattura ed Integrità Strutturale, 29 (2014) 209-221; DOI: 10.3221/IGF-ESIS.29.18 210 the relevant literature (see e.g. [5–8]), both applicable and effective, especially when primary interest is in determining the limit (peak) load of FRP-strengthened RC elements. It is worth noting that all the phenomena arising just after a state of incipient collapse , such as delamination [9], debonding [10], damage in a wider sense, are not treatable and this consistently with the spirit of a limit analysis approach. Accurate treatment of such post-elastic phenomena is prosecutable only with more accurate step-by-step FE nonlinear analyses. Aware of such limitation, the methodology here proposed should then be viewed only as a preliminary design tool to gain a quick insight into the bearing capacity evaluation of the analyzed elements by determination of the peak load value, the prediction (but not the description) of failure mode as well as the detection of critical zones within the addressed FRP-strengthened RC elements. The numerical methodology here referred, already used by the authors to predict the limit-state solution of RC elements (see e.g. [11, 12]) and of pinned-joint orthotropic composite laminates (see e.g. [14, 15]), is quite versatile and is based on iterative linear FE analyses carried out on the structure endowed with spatially varying moduli and, if necessary, given initial stresses. Such quantities, viewed as pertaining to a fictitious material substituting the real one, are iteratively adjusted in such a way as to build , with reference to the assumed yield criteria, a collapse mechanism and an admissible stress field for the real structure so as to apply the kinematic and the static approaches of limit analysis, respectively. If a nonstandard nature of the constitutive behaviour has to be postulated, the peak load value of the analysed elements can, in fact, be numerically detected by predicting an upper and a lower bound to it. In the present study a very general multi-yield-criteria formulation of the above-mentioned limit analysis methodology is presented to appropriately describe the behaviour at collapse of structural elements of engineering interest strengthened by FRP techniques. Precisely, to simulate the behaviour at a state of incipient collapse of the three main constituent materials, concrete is described by a Menétrey–Willam-type yield criterion endowed with cap in compression, steel reinforcement bars are handled by the von Mises yield criterion, FRP strengthening laminates are governed by a Tsai–Wu- type criterion and the quoted methodology is applied in concomitance to the three yield criteria. To demonstrate the actual capabilities of the proposed approach, large-scale prototypes of a few FRP-strengthened RC beams, experimentally tested up to collapse [4, 16], are numerically investigated. T HEORETICAL BACKGROUND AND FUNDAMENTALS Constitutive models of concrete, steel and FRP oncrete is assumed as an isotropic, nonstandard material obeying a plasticity model derived from the Menétrey– Willam (M–W) failure criterion [17]. The latter provides a three parameter failure surface having the following expression: 2 ' ' ' ( , , ) 1.5 ( , ) 1 0 6 3 c c c f m r e f f f                           (1) where '2 '2 2 2 2 ' ' 2 2 2 2 4(1 ) cos (2 1) ( , ) ; 3 1 2(1 ) cos (2 1) 4(1 ) cos 5 4                  c t c t f f e e e r e m e f f e e e e e (2) Eq. (1) is expressed in terms of the three stress invariants , ,    known as the Haigh–Westergaard (H–W) coordinates (i.e. hydrostatic and deviatoric stress invariants and Lode angle); m is the friction parameter of the material depending, as shown in Eq. (2), on the compressive strength ' c f , the tensile strength ' t f as well as the eccentricity parameter e . The eccentricity e , whose value governs the convexity and smoothness of the elliptic function ( , ) r e  , describes the out-of- roundness of the M–W deviatoric trace and it strongly influences the biaxial compressive strength of concrete. The failure surface (1) is open along the direction of triaxial compression; therefore, to limit the concrete strength in high hydrostatic regime, a cap in compression closing the surface (1) is adopted. The cap is formulated in the H–W coordinates as follows: ( , ) ( , ) ( ) for / ( ) MW b a CAP a a b b a b                                    2 2 2 2 0 3 (3) where ( , ) MW a    is the explicit form of the parabolic meridian of the M–W surface easily obtainable from Eq. (1). The values a  and b  entering Eq. (3), namely the hydrostatic stress values corresponding to the intersection of the cap surface with the M–W surface and the hydrostatic axis, respectively, locate the cap position and can be calibrated C