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Exploring the physical and mechanical properties of rocks under dynamic and static load superposition is crucial for preventing and controlling underground engineering dynamic disasters. Therefore, a rock experimental technology based on a gas‒liquid composite dynamic and static load superposition cylinder is proposed. This experimental technology overcomes four technical difficulties: (1) simulating a preimposed static load with superimposed dynamic load and strain energy accumulation; (2) progressive automatic constant loading of creep load; (3) medium strain rate cyclic impact loading; and (4) integrated control and multiple information coupling collection. Through this technology, multiple strain rate dynamic and static loads, such as creep loads (< 10 –4  s −1 ), hydraulic loads (10 –4  ~ 10 0  s −1 ), and cyclic impact dynamic loads (10 0  ~ 10 2  s −1 ), can be superimposed on the rock. It can quickly compensate for the static load during rock collapse and instability to simulate the strain energy of the rapid release process of the surrounding rock. The experimental results verify that the peak strength and failure duration of rock are negatively correlated with the preimposed static load and positively correlated with the impact frequency. The cumulative damage and ultimate strain are positively correlated with the static load and negatively correlated with the impact frequency. With increasing static load or decreasing impact frequency, the rock failure mode transitions from “slope shear failure to vertical tensile failure to overall burst failure”. The energy generated by the preimposed static load mainly accumulates at the rock fracture tip, and the dynamic load can increase the brittleness of the rock. Highlights Multi strain rates dynamic load and static load superposition rock mechanics experimental technique. Pre-imposed static load superimpose medium strain rate cyclic impact dynamic load. Dynamic failure mechanical process simulation caused by strain energy rapid release in elastic zone surrounding rock. Integrated control and multiple information coupling collection during the dynamic and static loads superposition process.

Linear static response structural optimization has been developed fairly well by using the finite element method for linear static analysis. However, development is extremely slow for structural optimization where a non linear static analysis technique is required. Optimization methods using equivalent static loads (ESLs) have been proposed to solve various structural optimization disciplines. The disciplines include linear dynamic response optimization, structural optimization for multi-body dynamic systems, structural optimization for flexible multi-body dynamic systems, nonlinear static response optimization and nonlinear dynamic response optimization. The ESL is defined as the static load that generates the same displacement field by an analysis which is not linear static. An analysis that is not linear static is carried out to evaluate the displacement field. ESLs are evaluated from the displacement field, linear static response optimization is performed by using the ESLs, and the design is updated. This process proceeds in a cyclic manner. A variety of problems have been solved by the ESLs methods. In this paper, the methods are completely overviewed. Various case studies are demonstrated and future research of the methods is discussed.

Static loading can significantly alter the dynamics of unidirectional carbon-based composites (UCBCs), with modal parameters varying depending on the orientation of the carbon fibers. In this study, the sensitivity of modal parameters of UCBC structures under uniaxial static loading was investigated. The theoretical static load influential factor was derived from a linearized UCBC model and corresponded to the transformed decoupled response over the mass-normalized static load. Three rectangular UCBC specimens (carbon fiber orientation of 0°, 45°, and 90°) were prepared under fixed–fixed boundary conditions using a jig fixture. Uniaxial static loads between 0 N and 1000 N were applied, and the first three modes of the UCBC specimens were analyzed. An isotropic SUS304 specimen was used as a reference. The linearization assumption about the UCBC structure was preliminarily validated with the Modal Assurance Criterion (MAC). A high influential factor was found for the UCBC specimen when carbon fibers were aligned with the static load direction at the first two resonance frequencies. Therefore, the proposed influential factor is an efficient indicator for determining the sensitivity of the dynamic response of a UCBC structure over a static load case. The variations in the influential factors for the UCBC specimens were more pronounced than for the isotropic specimens.

The electrical and magnetic signals generated by coal fracture can expose the mechanical properties and fracture behavior of coal, which are of great significance for underground mining engineering safety under high static stress and dynamic load disturbance conditions. This work used the split Hopkinson pressure bar experimental apparatus to perform impact dynamic experiments on coal samples with axial static load–dynamic load coupling and test the electric potential (EP) signal. We studied the characteristics of coal’s EP response under various dynamic and static load coupling conditions, discussed the mechanisms by which different variables affected EP response, and built an EP-based constitutive model of coal damage progression. The results revealed that under the coupling of axial static load–dynamic load, noticeable EP signals are stimulated in coal, and the change in EP signal is well correlated with the change in mechanical behavior. However, increasing dynamic load can excite a greater EP signal, and the peak EP grows linearly with stress. Peak EP first increases linearly as the axial static load increases, and when the axial static load reaches the critical threshold, peak EP and peak stress start to decrease progressively. Peak EP variation well corresponds to peak stress. Crack propagation and free electron escape can explain the effect of axial static load and dynamic load on coal EP signal at the micro level. On this basis, we developed an EP constitutive model of coal damage evolution under axial static load–dynamic load coupling. The model can well calculate the stress state of coal. The study’s findings provide a solid theoretical foundation for security monitoring of deep-underground engineering.

Rock indentation and fragmentation behavior under dynamic and quasi-static loads have significant impacts on the exploration drilling for oil, gas, and geothermal energy. In this study, the mechanical response of rock indentation subjected to dynamic and quasi-static loads was investigated in terms of the interaction between the button and the rock in percussion drilling. The effect of rock type, grain size, and mineralogical composition on rock fragmentation and penetration response was compared, and the cracking characteristics of the rock and crack propagation paths were analyzed. The results show that the damage characteristics of rocks under quasi-static load are similar to those under dynamic load, but the penetration performance under dynamic load is better than that under quasi-static load. The brittleness index can be used to assess the fragmentation performance of different rocks. When the brittleness index of the rock is less than 20, the rock fragmentation performance is particularly sensitive to the brittleness index. The mineral grain size will affect the period of rock failure, when the mineral grain size is less than 900 µm, there is no significant difference in crater depth or volume. The mineralogical composition has a significant effect on the penetration characteristics. When the content of feldspar is more than 5 times that of quartz, the fragmentation performance will not be much different. The relationship between the axial force and the penetration depth follows the power-law relationship, which can be used to describe the dynamic mechanical response of bit and rock in the process of percussion drilling.HighlightsThe mechanical response of rock indentation subjected to dynamic and quasi-static loads was investigated and compared.The influence of rock type, grain size, and mineralogical composition on rock fragmentation was evaluated.The rock fragmentation process under quasi-static load can be regarded as a slow motion of rock fragmentation under dynamic load.The relationship between the axial force and the penetration depth in the elastic deformation stage follows a power-law relationship.

Nonlinear dynamic structural optimization is a real challenge, in particular for problems that require the use of explicit solvers, e.g., crash. Here, the number of design variables is typically very limited. A way to overcome this drawback is to use linear auxiliary load cases which are derived from nonlinear dynamic analysis results in order to enable the application of linear static response optimization. The equivalent static load method (ESLM) provides a well-defined procedure to create such linear auxiliary load cases. The main idea here is that after the selection of a number of representative time steps, a set of equivalent static loads (ESLs) is computed for each time step such that the resulting displacement field in the linear static analysis is identical to the respective field in the nonlinear dynamic analysis. Each set of ESLs defines an auxiliary load case, which is used in the linear static response optimization. The crucial point is that the finite element (FE)-model for each auxiliary load case describes the undeformed initial geometry. This can lead to insufficient approximation quality in the linear static system for highly nonlinear problems. To overcome this drawback, a difference-based extension of the ESL method called DiESL has been developed for nonlinear dynamic response optimization problems. Here, the FE-model for each auxiliary load case describes the deformed nonlinear geometry at the respective time, and the corresponding ESLs create only the displacement field leading to the deformed state of the subsequent ESL time step. Consequently, responses in each linear auxiliary load case (corresponding to a time step) are computed as the accumulated sum of the previous linear auxiliary load cases. Furthermore, the linear static response optimization problem consists not only of one but of n T FE-models where n T is the number of selected time steps. Such a multi-model optimization (MMO) can be solved with commercial FE solvers. It turns out that the DiESL approach leads to a significant improvement of the nonlinear approximation quality and faster convergence to the optimum when compared to standard ESLM. This will be demonstrated and discussed based on selected test examples.

Aircraft landing gear assemblies comprise of various subsystems working in unison to enable functionalities such as taxiing, take-off and landing. As development cycles and prototyping iterations begin to shorten, it is important to develop and improve practical methodologies to meet certain design metrics. This paper presents an efficient methodology that applies high-fidelity multi-disciplinary design optimization techniques to commercial landing gear assemblies, for weight, cost, and structural performance by considering both structural and dynamic behaviours. First, a simplified landing gear assembly model was created to complement with an accurate slave link subassembly, generated based of drawings supplied from the industrial partner, Safran Landing Systems. Second, a Multi-Body Dynamic (MBD) analysis was performed using realistic input motion signals to replicate the dynamic behaviour of the physical system. The third stage involved performing topology optimization with results from the MBD analysis; this can be achieved through the utilization of the Equivalent Static Load Method (ESLM). Lastly, topology results were generated and design interpretation was performed to generate two designs of different approaches. The first design involved trying to closely match the topology results and resulted in a design with an overall weight savings of 67%, peak stress increase of 74%, and no apparent cost savings due to complex features. The second design focused on manufacturability and achieved overall weight saving of 36%, peak stress increase of 6%, and an estimated 60% in cost savings.

Castellated beams are widely applied in modern structural systems because they provide reduced self-weight while maintaining a higher moment of inertia compared to solid I-sections. However, the presence of web openings can create stress concentrations and excessive deformation, which may compromise structural integrity. To address these weaknesses, stiffeners are commonly introduced, yet the most effective configuration remains uncertain. This study investigates the effect of stiffener arrangements on the structural behavior of a 1.7-meter-long cantilever castellated beam under static loading, using numerical simulation in SolidWorks 2023. The base profile is a modified IWF 150.100.6.9 steel section, modeled in five variations: (a) castellated beam without stiffener, (b) castellated beam with stiffeners at both ends, (c) castellated beam with three stiffeners positioned at both ends and midspan, (d) castellated beam with full stiffeners applied to all web openings, and (e) the original IWF profile prior to castellated modification. All models assume ASTM A36 steel with yield strength (fy) = 250 MPa and ultimate tensile strength (Fu) = 410 MPa. The results indicate that adding stiffeners enhances beam performance by reducing maximum displacement and lowering stress concentrations. Among the castellated beams, the model with three stiffeners provided the most effective balance, achieving lower stress and smaller deformation while maintaining material efficiency compared to the fully stiffened model. Comparison with the original IWF section further highlighted the trade-off between structural response and weight reduction. These findings confirm that proper stiffener placement is crucial to optimize castellated beams for cantilever applications.

The orientation of carbon fibers significantly affects the dynamic properties of unidirectional carbon-based composites (UCBCs), with variations under different static loads. A previous study analyzed changes in the modal parameters of UCBC structures by using the static load influential factor (SLIF). This study introduces a revised SLIF, derived from a simplified formulation that accounts for shifts in resonance frequency and the in-phase relationship between static load and modal response. The revised SLIF is theoretically linked to the modal participation factor in UCBC structures. The dynamic behavior of UCBCs was studied across six modes—four bending and two torsional—using specimens with five carbon fiber orientations, from 0 to 90 degrees. The revised SLIF showed significant effects in two robust specimens, #1 and #2, and an isotropic SUS304 specimen subjected to uniaxial pre-static load, with resonance frequency variations under 0.16%. In contrast, the original SLIF gave negligible results in the fifth mode due to a damping term, which, when multiplied by the resonance frequency, led to an undetectable indicator. Therefore, the revised SLIF more effectively captures the static load’s impact on UCBC dynamic behavior compared with the original method.

Earthquake-resistant structures play a crucial role in mitigating damage and loss of life during seismic events, particularly in regions with high seismic risk. Castellated steel beams have emerged as a promising structural solution due to their enhanced strength-to-weight ratio, capacity to withstand dynamic loads, and improved energy dissipation characteristics. This study investigates the seismic performance of castellated beams derived from the IWF 150.75.5.7 steel profile, modified with an opening distance (e) of 42 mm and an opening angle (θ) of 60°. The castellated section achieves a depth of 210 mm and is analyzed under fixed support conditions using both static and harmonic loading. Numerical modeling and finite element simulations were performed in SolidWorks 2023 to evaluate stress, strain, stiffness, and deflection responses. The results show that castellated beams exhibit significantly reduced deformations compared to the solid IWF profile, with maximum reductions of 67.8% under static loads and 88.0% under harmonic loads. These findings highlight the effectiveness of castellated beams in enhancing stiffness and controlling deformation, confirming their potential application in the design of earthquake-resistant structures. Future studies should extend this work through experimental validation and parametric analysis to establish practical design guidelines.