خصوصیات غیر مخرب از میله های کراوات با استفاده از آزمایش دینامیکی، توده های اضافه شده و الگوریتم های ژنتیکی

The structural characterization of tie-rods is crucial for the safety assessments of historical buildings. The main parameters that characterize the behavior of tie-rods are the tensile force, the modulus of elasticity of the material and the rotational stiffness at both restraints. Several static, static–dynamic and pure dynamic nondestructive methods have been proposed in the last decades to identify such parameters. However, none of them is able to characterize all the four mentioned parameters. To fill this gap, in this work a procedure based on dynamic testing, added masses and genetic algorithms (GA) is proposed. The identification is driven by GA where the objective function is a metric of the discrepancy between the experimentally determined (by dynamic impact testing) and the numerically computed (by a fast and reliable finite element formulation) frequencies of vibration of some modified systems obtained from the tie-rod by adding a concentrated mass in specific positions. It is shown by a comprehensive numerical testing campaign in which several cases spanning from short, low-stressed, and almost hinged tie-rods to long, high-tensioned, and nearly clamped tie-rods, that the proposed strategy is reliable in the identification of the four unknowns. Finally, the procedure has been applied to characterize a metallic tie-rod located in Palazzo Paleotti, Bologna (Italy).

Metallic tie-rods are largely used in historical buildings and contribute to their overall structural response. When buildings undergo to repair, restoration or retrofitting, information about the status of tie-rods are of primary importance. For this reason, the structural characterization of tie-rods has been the focus of intensive research in the last decades (see [1] and the references therein). In this context, nondestructive identification procedures work towards the characterization of such systems by exploiting experimental data, structural models and optimization techniques. The nondestructive procedures currently available for the structural characterization of tie-rods can be grouped in static, static–dynamic and pure dynamic approaches. Static methods, see for example [2], [3] and [4], in spite of minor differences, are based on measures of displacement and/or strain at few cross-sections of the tie-rod due to applied static loads. Even if the data post-processing is quite straightforward [4], these methods are extremely sensitive to the experimental error in the measures of displacement. In addition, since tie-rods are usually positioned at considerable heights, the need of measuring vertical deflections with respect to a reference fixed base makes static methods difficult in practice. Mixed approaches try to identify the unknown parameters by combining static and dynamic measures [5] and [6]. The frequencies of vibration and the modal shapes of the tie-rod, which can be obtained by hammer impact testing and Fourier transforming the recorded accelerations, are usually considered as dynamic measures. Even if such methods can exploit additional dynamic information for the characterization, they are still affected by the shortcomings related to deflection measurements. Such drawbacks are avoided in pure dynamic procedures [1], [7], [8], [9], [10], [11], [12] and [13], where, in general, the difference between the experimental and the calculated natural frequencies of vibration is minimized in order to identify the unknown parameters. For example, in Lagomarsino and Calderini [11], an algorithm based on the first three frequencies measured by dynamic test, on an Euler–Bernoulli beam model and on the Line Search method, a gradient based searching algorithm, is presented. The unknown variables are the axial tensile force, the bending stiffness of the beam and an identical stiffness of the rotational springs at the ends. In a very recent work by Amabili et al. [1], a technique that employs four to six frequencies, a Timoshenko beam model and the Nelder–Mead minimization procedure (also known as the downhill search method), is proposed. The part of the rod inserted in the masonry wall is modeled as a beam on elastic foundation of unknown stiffness. The axial force and an identical foundation spring constant are the considered unknowns. However, if a beam structural model is assumed for the tie-rod, to verify the tie-rod structural efficiency a general identification procedure should be capable of identifying the tensile force, Young's modulus of the material and both boundary conditions. In fact if the stiffnesses of the restraints are not properly evaluated, the estimates of modulus of elasticity and tensile load are likely to be inaccurate. This might occur in arches or vaults where the tie-rod boundary conditions at the two sides can be quite different (see Fig. 1), and equal constraints should not be assumed unless verified by additional experimental tests that add complexity to the method.To the best of authors' knowledge, none of the works related to the structural characterization of tie-rods is able to characterize the tie-rod bending stiffness, the tie-rod tensile force, and the rotational stiffness of the springs at both tie-rod ends. In addition, the available identification methods are based on sensitivity or gradient-based searching algorithms. Therefore, the success of such methodologies is strictly dependent on the initial values of the target parameters that in historical buildings can be difficult to assess. To fill this gap, the aim of the present work is to propose an identification procedure: (i) capable of identifying the tie-rod Young's modulus, the applied tensile load, as well as different stiffnesses of the constraints; (ii) insensitive from the initial values assumed for the structural parameters; (iii) computationally straightforward (the numerical framework has to be easy to implement); (iv) feasible to execute (the experimental procedure has to be simple and rapid). To address these purposes, the following strategy has been followed. An optimization scheme driven by genetic algorithms (GA) has been adopted [14]. Even if often employed in inverse problems [15], [16], [17] and [18], GA have never been used in this context. As well known, differently from gradient-based optimization procedures, GA optimization results are minimally affected by the initial guess of the target variables. The basic idea of GA is to select an optimal individual from a population of individuals that represent potential solutions of the problem. In this study, each individual is a 4×1 vector of unknowns, i.e. Young's modulus, the tensile force and the rotational stiffnesses of both the elastically restrained ends. By applying genetic operators, the solution evolves generation-by-generation to the global optimum. Each individual is evaluated via an objective function here designed as the discrepancy between the experimental and numerical frequencies of vibration. Since the identification problem is symmetric, i.e. the numerical frequencies of vibration for the tie-rod are identical by exchanging the boundary conditions, different solutions can lead to the same fitness value. In other words, the evolutionary procedure fails in assigning the rotational stiffness to the proper tie-rod end since a change of boundary conditions at one end would yield to a frequency variation (and thus fitness value) identical to the same change of boundary conditions at the other end. The obstacle of identifying two different values for the end rigidities is overcome by exploiting the frequencies of vibration of one/some modified systems obtained from the original tie-rod by adding a concentrated mass in a non-symmetric position. The use of additional masses has been successfully adopted to identify structural parameters of beams such as mass density and flexural rigidity [19] and [20]. A fast and reliable specifically designed finite element (FE) formulation, based on the Euler–Bernoulli beam model, is introduced for the computation of the numerical frequencies. The model considers small-amplitude vibrations, linear strain and constant Young modulus regardless the value of the applied tensile force. It is shown by a comprehensive numerical testing that the proposed procedure is able to characterize the four unknown parameters by exploiting only the first few frequencies of vibration, succeeding where other methodologies have failed. Finally, an on-field application of the proposed procedure is presented. In particular, the strategy has been used to characterize the unknown parameters of a metallic tie-rod located in Palazzo Paleotti, Bologna (Italy). The authors believe that the use of common tools such as dynamic responses and genetic algorithms strengthens the proposed procedure. In fact, identifying additional parameters by slightly modifying the test setup, for example by adding a concentrated mass in the experimental determination of the frequencies of vibration, can be more effective than introducing new numerical/experimental procedures not yet fully validated. The paper is organized as follows. The FE formulation is presented and validated in Section 2. Section 3 is devoted to the description of the identification procedure based on GA. The proposed strategy has been tested numerically on several case studies in Section 4, while Section 5 shows its application on experimental data measured on a tie-rod located in Palazzo Paleotti, Bologna (Italy). Some final considerations conclude the paper.

In this paper a numerical procedure based on genetic algorithms aimed at identifying Young's modulus, the tensile force, and different stiffnesses at the rotational restraints of tie-rods has been proposed. In particular, it has been shown that GA fed with the first few frequencies of vibration of tie-rod systems obtained from the original one by adding a concentrated mass at various locations along the length, allow to identify the target parameters. The required data are the frequencies of vibration that can be obtained in a total nondestructive manner from dynamic testing. Once the testing phase is completed, the added mass is simply removed and the structure comes back to its original state. For this reason, the required equipment is not more cumbersome than that of a typical dynamic test (impact hammer, accelerometers, PC for data acquisition). The only difference is that an added mass should be considered in the experimental apparatus; however, this does not increase the complexity of the instrumentation. The wide numerical identification campaign has revealed that the procedure is reliable also considering very high levels of noise in the pseudo-experimental frequencies. In addition, it has been shown that the fitness function could be strengthened by adding experimental measures obtained from additional tests. Finally, the proposed technique has been applied to characterize a metallic tie-rod located in Palazzo Paleotti, Bologna (Italy). Among the specific benefits of the proposed identification strategy are (i) the use of the added mass in a non-symmetric position allows to identify two different values of the ends rotational stiffness that in literature are generally assumed to be identical; (ii) GA provide convergence regardless the initial guess on the target parameters while sensitivity or gradient based identification schemes do not; (iii) the adopted FEM formulation and the GA scheme are easy to implement especially by using software package like MATLAB where the main GA functions are coded and open-ended to address the user needs.