تجزیه و تحلیل حساسیت از ساختار هندسی ناپایدار تحت بارگذاری پالس مختلف

The dynamic response of a structure depends on both the structure and the applied load. A pulse approximation method proposed by Youngdahl has been proved to be applicable to stable structures and it has been adopted to deal with arbitrary loading pulse in analyses. However, for the structures possessing unstable load–displacement properties, the applicability of pulse approximation method needs to be validated. In our previous theoretical study, by employing the rigid-perfectly plastic idealization, the static and dynamic responses of tensor skin are obtained, revealing its softening (i.e. geometrically unstable) behavior. In the present paper, a similar theoretical approach is employed together with the pulse approximation method to examine the effect of various pulse shapes on the final deflection of the tensor skin. It is found that the final deflections are insensitive to pulse shape, and the differences resulted from different pulses are less than 10%. Hence, the pulse approximation method is also applicable to geometrically unstable structures such as tensor skin. Besides, for various pulse shapes, it is found that the final deflection of tensor skin increases with the total impulse as well as the effective pressure. To achieve a specified deflection, it only requires smaller total impulse for a more intensive impact loading. Furthermore, a damage diagram of tensor skin is constructed in terms of the failure impulse and the effective pressure under different pulse loading. By comparing various pulses with the same impulse level and effective pressure, the explosive pulse is more dangerous than other pulse shapes in producing failure of tensor skin.

On top of a static analysis, a dynamic analysis is even more important for a structure under impact or explosive loading conditions. In general, the dynamic response of a structure is an interaction process between the structure and the applied load. In the theoretical approach of dynamics response, both the structure and the applied load have to be idealized in order to make the analysis feasible. The rigid-perfectly plastic idealization is often adopted to predict the structural response if plastic deformation dominates the structural response [1]. For a given deformation mechanism, which is a kinematically admissible field, an upper bound of the collapse load can be obtained by energy method. For traditional structures, such as beams and plates under transverse loading, the load–displacement curves are stable; that is, the collapse load monotonically increases with displacement [1]. However, some structures exhibit unstable characteristics in their load–displacement relationship. In these cases, the load to initiate the plastic deformation is high, but the resistance of the structure decreases with the increase of its plastic deformation, so its load–displacement curve displays a kind of softening effect. For example, tensor skin is a kind of composite sandwich structure developed to improve the crashworthiness of helicopter subjected to water impacts [2]. From a rigid, plastic analysis of tensor skin [3], it is found that after the initial collapse, the static critical pressure first decreases then increases with the increasing central deflection. Hence, the tensor skin is a kind of unstable structure due to the nonlinear geometrical behavior in large deformation. The dynamic response of this tensor skin under water impact is of interest of the present study. To find out an accurate dynamic structural response requires the knowledge of the actual loading pulse. In the stage of structure design, however, simple pulse shapes such as rectangular pulse or triangular pulse are usually adopted in the analysis; while in reality various impact conditions may produce more complex-shaped pulses. Therefore, the influence of the pulse shape on the structure response should be studied. The earliest study about the effect of pulse shape can be traced back to 1953 when Symonds [4] found that the final deflection of a free beam subjected to a concentrated force pulse only depends on the total impulse I and peak load Pmax of the pulse. Later, Hodge [5] remarked that this conclusion was valid only for loading intensities far beyond the yield load; otherwise this simplification (by counting I and Pmax only) may introduce a large error. In order to eliminate pulse shape effects, two correlation parameters have been proposed by Youngdahl [6] and [7]. From his study, the dynamic response of a structure under a general loading pulse can be approximated by that under a rectangular pulse impulse Ie, with an effective load Pe and pulse duration 2tmean, see more details in Section 2. A number of studies by other researchers [8], [9], [10], [11] and [12] further confirmed that the pulse approximation method is able to eliminate pulse shape effects on the dynamic plastic bending response of various structural members, such as beams, circular plates and cylindrical shells. Later, it is found [13] that when damping effect is introduced, the pulse shape effect on the maximum deflection of a simply supported beam can be eliminated by introducing Ie and Pe. A theoretical foundation for rigid-plastic responses of common structural members was established using the bound theorems and it also reveals that Youngdahl's empirical estimation for the structural response time is, in general, a lower bound on the actual response time [14]. It is worth noted that, in the previous studies of the pulse approximation method, the structures concerned are all stable ones without any softening property. Hereby, an interesting question is raised: is the pulse approximation method also applicable to unstable structures, such as the tensor skin? In the present paper, the dynamic responses of the geometrically unstable structure, tensor skin, under various pulses will be investigated. First, the pulse approximation method and the theoretical model of tensor skin will be reviewed in Sections 2 and 3, respectively. Then the applicability of the pulse approximation method for tensor skin will be validated in Section 4. In Section 5, the effects of the total impulse and the effective pressure on the final displacement are analyzed. Finally, a damage diagram of the tensor skin is constructed for different pulse shapes in Section 6. By taking the tensor skin as a typical example, it is concluded that the pulse approximation method is also applicable to the geometrically unstable structures.

In this paper, the applicability of the pulse approximation method originally proposed by Youngdahl is examined for a geometrically unstable structure, tensor skin. It is found that this method is applicable not only to stable structures, but also to this geometrically unstable structure. According to the pulse approximation method, the dynamic response of tensor skin under an arbitrary pulse is approximately equivalent to that under a rectangular pulse with the same total impulse and the effective load intensity. The relative error in the final deflection is within 10%. The pulse approximation method can't predict the detailed deformation history, but can give a fairly accurate estimation of the overall dynamic response, e.g., the final deflection and the total response time. The dynamic response of a tensor skin under a triangular pulse, which by shape is very similar to the pulse produced by the impact between a plate and water surface, is quite close to that under an equivalent rectangular pulse. For all kinds of pulse shapes, as the load intensity View the MathML sourceq¯e/q¯s0 increases, the total impulses to reach a specific deformation will decrease. That is, for a given impulse, the final deflection will increase with the increase of load intensity. A damage diagram of tensor skin is constructed in terms of the effective pressure and the failure impulse for various pulse shapes. It is found that the explosive pulse is the most dangerous pulse among all pulses with the same impulse level and load intensity.