Formulas for Mechanical and Structural Shock and Impact


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Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015

The shock response and dynamic fracture of concrete gravity dams under impact load are the key problems to evaluate the antiknock safety of the dam. This study aims at understanding the effects of impact shock on the elastic response and dynamic fracture of concrete gravity dams.

Firstly, this paper uses acceleration records of a concrete gravity dam under impact to establish the correct way to determine the concrete gravity dam of the fundamental frequency and present cut sheets multi-degree-of-freedom dynamic modeling.

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Under strong impact loading, the constitutive relation of concrete gravity dam and the highest frequency of the impact are uncertain. So, the main advantage of this method is avoiding the use of elastic modulus in the calculation. The result indicates that the calculation method is a reliable computational method for concrete gravity dams subjected to impact. Finally, this paper puts forward suggestions for future research based on the results of the analysis.

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Recently, the Novosibirsk Hydroelectric Plant after a report about a planted explosive, Russia has declared a state of emergency. So, this event should cause the attention of researchers and administrative departments. Hydraulic dams are critical infrastructure in geotechnical engineering. They are often designed and built to store water for drinking and irrigation in adjacent areas, to add water recreation spaces, to create a water way for the short-distance transport of people and goods across deep canyons in mountainous regions, and to regulate the river during a flood event.

A water-filled dam can boost the local economy through various personal and business activities, leading to the establishment of a new community center such as a village or a town, and representing a major capital and long-term investment. On the other hand, the breaching and an accidental damage of a dam can lead to a catastrophic flood event and its chain effects such as engulfing downstream residential areas and washing away agriculture lands. Therefore, design and maintenance of dams are not only a serviceability issue but also a life-threatening matter to millions of people.

When extreme events such as earthquakes, tsunamis, hurricanes, and tornadoes took place, concrete dams can be subject to extensive shaking and wave impact. The March, , Japan Earthquake event testified the destructive power of the earthquake-induced tsunami. Many scholars have studied the high dam subjected to earthquake action. Among these are Zhou et al.


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Equally, if not more important, dams are also vulnerable targets for man-made explosion events, particularly with the advent of advanced long-range and precision missile technologies. Since the September 11 attacks by terrorists, there has been increasing public concern about the threat of bomb attacks on dam structures Federal Emergency Management Agency, Therefore, protection of dam structures against impact loads is a critical component of homeland security Lu et al. Indeed, as respectively studied by Lu et al. Currently, with the development of computational techniques and numerical simulation methods, as well as commercialization of nonlinear dynamic software e.

Many researchers have conducted comprehensive experimental and numerical investigations related to the effects of explosions on building structures Tian et al. In the modeling of transient loading, it is very critical to describe the propagation velocity of the stress waves correctly. In fact, the value of this velocity depends on the material elastic modulus that is given by the material constitutive relation. From the material point of view, concrete shows an increase in elastic modulus with the strain rate increases, a phenomenon called strain rate effect Georgin, The relationship between concrete strength and strain rate was extensively investigated by Bischoff and Perry , Georgin and Reynouard , Grassl and Tai, Y.

Because many problems of the concrete material constitutive parameters and constitutive model have not been a clear understanding, the numerical results obtained by different calculation models are very different.

Moreover, the modeling of the strain rate effect on concrete is not very easy to tackle. These problems, to a great extent, comprise the uncertainty existing in the macroscopic numerical simulation. One of the efficient methods to study the failure modes and mechanisms of structures is to carry out a large number of model experiments and obtain data from them.

Considering that the experimental study has its limitations, as well as great difficulties and expensive costs for the underwater explosion test, only a small amount of data can be obtained.


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  • For example, Lu et al. Overall, to the best of our knowledge, experimental investigations of concrete gravity dams under underwater shock wave effects have not yet been conducted to date. Concrete dams are thoroughly studied in this paper both experimentally and numerically to understand their behavior and failure modes. Specifically, this paper is to use acceleration records of the concrete structure under strong impact to establish the correct way to determine the concrete structure of the fundamental frequency and present cut sheets multi-degree-of-freedom dynamic modeling.

    The dams were numerically modeled to understand further their short-time failure process in the order of msec based on ABAQUS. The same test setup and test results as presented by Lu , Lu a , Lu b were used in this paper which will be briefly depicted as follows.

    Functional shock test per mil-std-810 - equivalent static load

    The dimension of the model dam and test layout are shown in Figure 1. The mechanical properties of the individually tested samples and their average values are given in Table 1. The main test results of Model test are shown in Table 2. The pressure, strain recorded and acceleration recorded are shown in Figures 2 , 3 and 4. As shown in the measured pressure-time curve, the impact load can be simplified to a triangular distribution.

    The failure mode and damage area are shown in Figure 5. Note that the accelerometers and pressure sensor were not synchronized during the test. As a result, the peak accelerations seem to occur before the application of the peak pressure. The maximum time delay among the five accelerometers is The time delay was likely caused by different periods when five hammers applied impact forces.

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    Based on the arrival time of peak accelerations, Hammers 4 and 5 were in contact with the model dam later than Hammers For that, the concrete dam is divided into I unites, the governing equation for the system with I freedom is as follows:. Displacement and acceleration using the vibration mode expand can be got. Therefore, three forms of the impact force-time curve were examined, as shown in Figure 6 a-c , with the same P max and impulse P m T r.

    The time of acceleration extremal point including the maximum and minimum values t m1 , t m2 , t m By the second type of the Equation 15 , the following equations can be obtained. The maximum acceleration shown in Table 2 is gradually decreased from top to bottom, which reflects the first mode characteristics of the dam. Experiment obtained vibration mode and natural frequency, so do not need too much freedom in the calculation. The mean stress calculated from Equation 12 thus represents the analytical hydrostat for the material, whereas Equation 11 represents the elastic Hugoniot.

    Quasi-static studies of amorphous alloys of various compositions have previously indicated elastic-perfectly plastic behavior 32 , In addition, shock wave calculations using the elastic perfectly-plastic assumption have matched experimental wave profile data in previous studies 17 , Therefore, this type of material response is an appropriate assumption for material behavior beyond the elastic limit for the amorphous alloys of this work. The experimentally obtained stresses and density compressions are plotted alongside the calculated Hugoniots and hydrostat to evaluate the strength of the material, as seen in Fig.

    It can be seen that while both sample types strain-soften beyond the elastic limit, SAM2X seems to dramatically lose shear strength at high loading stresses. On the other hand, SAM2X seems to retain some post-yield strength.

    Response to Mechanical Shock

    Both of these temperatures are significantly lower than the reported glass transition temperature of K for the chemical composition of SAM2X and SAM2X 34 , as well as the temperatures associated with long-range ordering and devitrification The shock response of both composites is therefore not influenced by any potential phase transitions induced by the temperature rise within the shock wave, since these likely do not occur under the range of impact stresses examined here.

    An assumption of the existence of steady waves in samples of both composites is implicit in the analysis presented above. However, further experiments to examine wave traces at a given impact stress for samples of widely varying thicknesses are needed to verify this assumption. Error bars represent calculated uncertainty in stress and density compression, propagated from uncertainty in density, particle velocity and shock velocity measurements.

    While the mechanistic phenomena behind the elastic limit and yielding of crystalline metals under shock loading is fairly well understood in terms of dislocation-mediated slip, the physical interpretation of the elastic limit in a brittle amorphous solid, such as the iron-based BMGs studied in this work, can be understood as the onset of relieving of shear strains from fracture by the joining and interaction of damaged zones and subsequent flow of the material In general, various mechanisms by which slip, and thereby yielding, may occur to accommodate shear strains have been proposed for brittle solids; these include plastic-brittle deformation comprising of slip zones with cracks and micro-cracks, brittle destruction deformation consisting of cracks and cleavages, and deformation from partial-melting zones A system of classification for the yielding of shocked solids based on the offset of the Hugoniot from the hydrostat or the isotropic loading state beyond the elastic limit has been described previously 28 , Following this system, materials may be classified into three main categories based on the comparison of their response to shock and isotropic loading when considered in the stress-density compression space: toughening solids whose shear strength increases with applied stress, perfect elasto-plastic solids which maintain a constant offset between their Hugoniot and hydrostat, and perfect elasto-isotropic solids which catastrophically lose shear strength above the HEL with the stress collapsing onto the isotropic state.

    Material behavior often lies between these two idealized extreme scenarios, and this is seen in the material response of SAM2X and SAM2X as is apparent in Fig. For both materials, the data points clustered around the inflection in the Hugoniot correspond to the HEL.

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    For the SAM2X samples, the data point corresponding to the peak state at the lowest impact stress of However at the higher impact stresses, both data points lie very close to the hydrostat, thus suggesting that there exists a certain threshold between 12 and 20 GPa beyond which SAM2X catastrophically loses all strength with the Hugoniot consequently nearly collapsing onto the isotropic stress state.

    Such catastrophic loss of shear strength has also been observed in high-HEL, brittle and hard materials such as boron carbide and silicon nitride On the other hand, peak state measurements for SAM2X compared to its calculated Hugoniot suggest that although the material strain-softens it seems to retain its strength even at impact stresses nearly one and a half times its HEL. This observed loss of post-yield strength in SAM2X and SAM2X is likely a result of yielding phenomena arising out of slip from propagation of cracks, cleavage and partial melting.

    Under intense loading scenarios such as the ones shock compression experiments present, the free volume in the amorphous matrix is quickly depleted and large stress concentrations are accumulated at those sites. Micro-voids coalesce into shear bands, which then provide pathways for the propagation of micro-cracks arising out of areas of intense shear localization. Evidence for partial strength loss beyond the elastic limit has also been seen in previous shock studies of various Zr-based metallic glasses 17 , 18 , 19 , 20 , 21 where maximum shock stresses ranged from just above the HEL to an order of magnitude higher than the HEL.

    Since the maximum shock stress that SAM2X was subjected to is less than the maximum impact stress for SAM2X, a further loss of strength for SAM2X than is suggested by the current data cannot be completely ruled out. Even then, such dramatically different responses at high Hugoniot pressures within a comparable range of impact stresses for samples that are nearly identical in their make-up, except for a very small amount of crystallinity, cannot be ignored.

    X-ray diffraction patterns of the samples can be seen in Fig. On the other hand, samples of SAM2X were made by sintering powders at K, just below the onset of this crystallization, but above the temperature at which structural relaxation and some long-range ordering occurs.

    Metal carbides and borides are known to have high hardness values, and the presence of the nanocrystallites dispersed in the amorphous matrix within SAM2X are likely responsible for strengthening this material - in both increasing its elastic limit compared with SAM2X, as well as retention of post-yield strength - by acting as barriers to the propagation of failure fronts in the form of shear bands and cracks.

    In another work, the precipitation of the Fe 23 B 6 phase in a Fe-Dy-B-Nb metallic glass was also seen to enhance fracture strength and Vickers hardness, in addition to improving the thermal stability and glass forming ability of the composition Therefore, while a small proportion of crystalline precipitate is not large enough to cause any observable distinction in the ambient and quasi-static mechanical response of SAM2X and SAM2X, it proves to be significant in influencing the response of the shocked samples at very high strain rates of the order of 10 6 per second. The sharper peak of sample SAM2X is an indirect measure of the higher extent of crystallinity in this sample.

    In conclusion, amorphous steels, the high-strain rate mechanical response of which was hitherto unexplored, demonstrate high strength under shock wave compression, the magnitude of the elastic limit of one amorphous steel composite being 1.

    Further, a minor addition of nanocrystallinity to the amorphous matrix of the iron-based BMG studied in this work results in a significant improvement in yield strength and post-yield shear strength retention, as seen in the response of the partially crystalline SAM2X when compared with that of the X-ray amorphous SAM2X This result is in contrast with a previous work on Zr-BMG Vitreloy 1 and its in situ dendritic phase composite, where no difference was seen in the shock response of the monolithic BMG and its composite While the shock response of the amorphous steels of this work is qualitatively similar to that of previously studied Zr-based compositions, namely large amplitude elastic waves and loss of post-yield strength, a significant enhancement in response based on the extent of devitrification is observed in both of these attributes.

    This work therefore reveals in situ reinforced metallic glass composites as promising candidates for use in high-strength applications. Furthermore, they demonstrate that controlled devitrification is a potentially viable adjustable parameter for synthesis of amorphous metallic materials with properties that can be tailored as desired. Samples were prepared using spark plasma sintering 39 , 40 , 41 , 42 , described previously 34 , Shock waves were generated in the sample by the plate impact technique in which a flyer plate is launched at high velocities using a propellant gun onto a target sample.

    All target and flyer plates were lapped flat and parallel to within 2 microns to ensure a normal shock wave. Flyer plates were epoxied into the sabot using Hysol Loctite E HP epoxy and were cured under a flat steel weight overnight. Particle velocity as well as projectile impact velocity was measured using a custom- built Photonic Doppler Velocimetry PDV system. In the experiments of this work, only two probes were used, one focused on the flyer plate for projectile velocity measurement, and the other focused at nominally the center of the rear surface non-impact side of the metallic glass sample for particle velocity measurement.

    Stress Due to Impact Load - Strain Energy - Strength of Materials -

    As part of target preparation before each experiment, PDV probes are set into the target holder at a height that results in the least possible return loss, monitored by a hand-held optical return loss meter JDSU SmartClass ORL Once targets were completely assembled, the probes were affixed to the PDV system while making sure that the fiber ends were clean by examining through a fiber scope.

    Attenuators on the PDV system were adjusted to tune the laser return from the target. Finally, recorded fringe data were reduced to particle velocity wave profiles using the PlotData software from Sandia National Labs. How to cite this article : Khanolkar, G. Many thanks to Dr. I-Chung Chen and Mr. Lindo of the University of Southern California for their help with material characterization.

    Author Contributions G. All authors reviewed the manuscript. National Center for Biotechnology Information , U.

    Formulas for Mechanical and Structural Shock and Impact Formulas for Mechanical and Structural Shock and Impact
    Formulas for Mechanical and Structural Shock and Impact Formulas for Mechanical and Structural Shock and Impact
    Formulas for Mechanical and Structural Shock and Impact Formulas for Mechanical and Structural Shock and Impact
    Formulas for Mechanical and Structural Shock and Impact Formulas for Mechanical and Structural Shock and Impact
    Formulas for Mechanical and Structural Shock and Impact Formulas for Mechanical and Structural Shock and Impact
    Formulas for Mechanical and Structural Shock and Impact Formulas for Mechanical and Structural Shock and Impact

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