Rackwitz Fiessler Research Articles

Reliability-based bond design for GFRP-reinforced concrete

Most of proposed models available so far associated with the evaluation of design embedded length of fiber reinforced polymer (FRP) rods in concrete are not reliability-based. This paper made an attempt, from the probabilistic standpoint, to determine the design embedded length of a glass fiber reinforced polymer (GFRP) rod in the case of splitting bond failure from concrete. The mathematical model put forward by Orangun et al. for evaluating the average bond strength of reinforcing bars in concrete for splitting failure has been adopted to develop the nonlinear limit state function corresponding to GFRP’s splitting bond failure in which five independent random variables, i.e. concrete strength, GFRP tensile strength, embedded length, GFRP diameter and computational uncertainty factor, are included. As the result of the probabilistic calibration procedures using the Rackwitz–Fiessler method, a non-dimensional factor K associated closely with the design embedded length of a GFRP rod in concrete is proposed to be 0.0306 with the suggested additional target reliability index of 1.10. The effects of some factors on GFRP’s design embedded length have been identified through the following parametric study. Although the calibration process is case-dependent to some extent, the proposed formula is thought to be acceptable for general bond design purposes of GFRP-reinforced concrete components.

Probabilistic assessment on flexural capacity of GFRP-reinforced concrete beams designed by guideline ACI 440.1R-06

The provisions in the guideline ACI 440.1R-06 (call ACI guideline thereafter) on the flexural design of concrete beams reinforced with glass fiber reinforced polymer (GFRP) rods are assessed from the probabilistic point of view by using the Rackwitz–Fiessler method. The assessment reveals that the design provisions in ACI guideline are really conservative. Among all resistance-related random variables, sectional width and height-to-width ratio have put significant influence on average reliability level in either failure mode. GFRP strength does not have any effect on reliability level in concrete crushing mode. Concrete strength shows a negligible effect in the failure mode of GFRP rupture. In the case of GFRP rupture, as the mean-to-nominal ratio of GFRP strength increases, the average reliability index at first increases and then decreases. Design of GFRP-reinforced concrete components subject to aggressive exposure conditions would achieve higher reliability level. As a result of the investigation on the relation between average reliability index and resistance reduction factor, a modified value of 0.81 for the factor is suggested.

The projection gradient algorithm with error control for structural reliability

Nowadays, probabilistic approaches are frequently used in the design of new civil engineering structures and the durability analysis of existing constructions. Hasofer–Lind’s reliability index is one of the most popular reliability measures due to its relevance and ease of use, and is now referred to in many structural design codes. This index can be determined by several algorithms dealing with minimization under constraint such as the well known Rackwitz–Fiessler algorithm based on the projection gradient method. The drawback of this algorithm lies in the estimation of the gradient vector of the limit-state function, which is often carried out by finite differences for non-explicit functions, resulting from the Finite Element Method for instance. If the perturbation chosen in the estimation of the gradient vector gives a variation of the output lower than the numerical accuracy of the limit-state function, the algorithm could lead to erroneous results or even not converge. In order to circumvent this problem, an original technique is suggested in this paper called “Projection gradient with error control”. The principle of the proposed technique is to attach to Rackwitz–Fiessler’s algorithm a procedure for judiciously determining the perturbation in the calculation of the gradient vector by finite differences, accounting for the numerical precision of the limit-state function. The efficiency and interest of the proposed procedure is emphasized through various examples.

Seismic failure probability of concrete slab on CFR dams with welded and friction contacts by response surface method

In this study, failure probability of the concrete slab on concrete-faced rockfill (CFR) dams with welded and friction contact is investigated under earthquake effects by reliability analysis. For this purpose, Torul CFR dam is selected as an example and numerical solutions are performed by considering combination of reliability analysis–finite element method. 1992 Erzincan earthquake acceleration record is used in the finite element analysis considering deconvolved-base rock input model. In this model, the ground motion to be applied to the foundation base rock is obtained by deconvolution of the free-field surface record. In the materially nonlinear analysis, Drucker–Prager model is used for concrete slab and multi-linear kinematic hardening model is utilized for rockfill. Geometrically nonlinearity is also taken into account. Viscous boundary conditions are defined in the finite element model for both foundation soil and reservoir water. The hydrodynamic pressure of the reservoir water is considered using 2D fluid finite elements based on the Lagrangian approach. Both welded contact and friction contact based on the Coulomb’s friction law are defined in the structural connections. Improved Rackwitz–Fiessler method is used with response surface method in the reliability analysis. The tensile and compression strengths of the concrete slab are utilized in the implicit limit state functions considering various thicknesses. The probability of failure of the most critical points in the concrete slab is obtained. According to this study, the probabilities of failure obtained from the CFR dam including friction contact are lower. When the welded contact is considered in joints, the probability of failure of the concrete slab is 1 due to tensile stress limit state and compression stress limit state only if concrete slab is linear. The most critical probability of failure of the concrete slab appears in the case that the concrete slab is linear and rockfill is materially nonlinear. The probability of failure of the concrete slab decreases if the nonlinearity of the concrete is considered. Also, hydrodynamic pressure decreases the reliability of the concrete slab.

Reliability indicator for layered composites with strongly non-linear behaviour

This paper set out a method to determine safety factors when designing composite laminate with strongly non-linear behaviour. The failure probability is used to assess the laminate reliability. The second moment-FORM (first order reliability method) method is chosen and the Nataf probability transformation is used in order to take account for a possible correlation between the different variables. The conception point is found from the Rackwitz–Fiessler method. Space conception parameters variability is then introduced in the reliability evaluation. Finally, the method is applied to a composite plate, to show up the conception and the space material parameters variability importance.

Probabilistic approach to corrosion risk due to carbonation via an adaptive response surface method

A study about a probabilistic approach to corrosion risk of reinforcements embedded in concrete due to carbonation is presented. The carbonation model is based on a single non-linear diffusion equation of the carbon dioxide. A global balance relationship between the carbon dioxide partial pressure and the solid calcium content in the hydrates of concrete is used in order to express the sink term of the equation and to render the solving easily tractable in a classical finite element analysis. The performance function invoked in the probabilistic approach is the deviation between the carbonation depth, i.e. the output of the carbonation model, and the concrete cover. The Hasofer–Lind reliability index is determined by the Rackwitz–Fiessler algorithm in which the performance function is replaced by a quadratic response surface in order to reduce computational cost and gain accuracy. An adaptive building of the numerical experimental design is proposed: points efficiently positioned with respect to the design point are re-used in the new iteration of the experimental design. In the case of explicit performance functions frequently reported in the literature comparisons with surface response techniques previously developed reveal the interest of the proposed technique. A practical application to a concrete girder shows that the reliability index decreases significantly with time.