In recent decades, the analysis and evaluation of the cracked structures were hot spots in several engineering fields and has been the subject of great interest with important and comprehensive surveys covering various methodologies and applications, in order to obtain reliable and effective methods to maintain the safety and performance of structures on a proactive basis. The presence of a crack, not only causes a local variation in the structural parameters (e.g., the stiffness of a beam) at its location, but it also has a global effect which affects the overall dynamic behavior of the structure (such as the natural frequencies). For this reason, the dynamic characterization of the cracked structures can be used to detect damage from non-destructive testing. The objective of this paper is to compare the accuracy and ability of two methods to correctly predict the results for both direct problem to find natural frequencies and inverse problem to find crack’s locations and depths of a cracked simply supported beam. Several cases of crack depths and crack locations are investigated. The crack is supposed to remain open. The Euler–Bernoulli beam theory is employed to model the cracked beam and the crack is represented as a rotational spring with a sectional flexibility. In the first method, the transfer matrix method is used; the cracked beam is modeled as two uniform sub-segments connected by a rotational spring located at the cracked section. In the second method which is based on the Rayleigh’s method, the mode shape of the cracked beam is constructed by adding a cubic polynomial function to that of the undamaged beam. By applying the compatibility conditions at crack’s location and the corresponding boundary conditions, the general forms of characteristic equations for this cracked system are obtained. The two methods are then utilized to determine the locations and depths by using any two natural frequencies of a cracked simply supported beam obtained from measurements as inputs. The two approaches are compared with a number of numerical examples for simply supported beams including one crack. The theoretical results show that the accuracy of the Rayleigh’s method to predict natural frequencies decreases for higher modes when crack depth increases. It is also found that for the inverse problem, the transfer matrix method show a good agreement with those obtained from previous works done in this field.
In order to improve the fragile nature of concrete, and its low tensile strength, and with a view to giving it the desired properties, which serve to build more durable structures at less cost, the association of a self-consolidating concrete with fiber, is considered a wise combination. However, given the limited amount of research on the response of SFSCC structures, designers and engineers do not use this material with confidence. In the present work, an experimental companion was conducted, in the interest of examining, the combined effect of fibers and stirrups include low and high rate of steel fiber, on the behavior of SFSCC beams. This choice allowed working on economically viable SFSCC. Beams were made also with ordinary concrete and others with self-consolidating. Thirty-six beams were of identical cross-section 10x20cm and length of 120cm; carried out with or without longitudinal and transverse reinforcement. Before proceeding with the main part of the research program, the concrete mixtures were characterized first in the fresh state by the following tests: Slump Flow, Time Flow T500; J-Ring, L-Box, V-Funnel and Sieve stability, and then in the hardened state: compressive and tensile strengths. In the light of the results obtained, it was found that adding steel fibers to fresh self-consolidating concrete decreased its workability and fluidity, but improved its hardening properties. Subsequently, the addition of the steel fibers increased the flexural capacity of the beams significantly, and improved their ductility. Also, an addition of the steel fibers in an adequate percentage, in this case at 0.9%, made it possible to replace the shear reinforcements, and can lead to changing the mode of failure from a collapse by brittle shear, to a mechanism of ruin in ductile bending.
The dynamic behavior of bridges under the effect of moving loads simulating the vehicle moving along the bridge structure idealized by an Euler beam is analyzed. We will present the dynamic behavior of beams under the stress of moving loads (or masses) by the analytical and semi-analytical approaches. When the mass of the bridge structure is comparable to that of the vehicle, the mobile source requesting the bridge is simulated by a mass. In most practical cases, the mobile force used is due to the effects of the gravitational moving masses: . When the moving mass is small compared to the beam mass, the obtained solution under the effect of moving force is approximately correct for the solution obtained with the moving mass. Otherwise, the problem of the moving mass is imperative. To do this, we wrote a program in Matlab language which reflects the dynamic behavior of beams under the effect of moving charges, which gives the following results "The frequencies and modes of vibration, the dynamics deflection of the beam requested by moving force, the dynamic response (DAF: dynamic amplification factor) of the beam requested by a moving force, over the whole length of the beam, for all times and for different speeds. The numerical example that we look to see for study the dynamic behavior of this type of bridge under moving loads is that of a thin beam unamortised on simple support and length of 50m, under the solicitation of moving force and mass at a constant speed and varies from 0 to 100 m / s (M. A. Foda, 1997), depending on the relationship between the vehicle mass and the mass of the bridge that will allow us to see the contribution of the choice of modelling type on the total response and then the vibration of bridge, also we will study the effect of type of simulation of the load by moving force or mass on the dynamic amplification factor and comparing our results with those from the literature.
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