Search Results (203)

The calculation of internal forces is a critical aspect in the design of statically indeterminate structures. Local trapezoidal loads, as a common loading configuration in practical engineering (e.g., earth pressure, uneven surcharge), make it essential to investigate how to compute the internal forces of statically indeterminate structures under such loads by using the displacement method. The key to displacement-based analysis lies in deriving the fixed-end moment formulas for local trapezoidal loads. Traditional methods, such as the force method, virtual beam method, or integral method, often involve complex computations. Therefore, this study aims to derive a general formula for fixed-end moments in statically indeterminate beams subjected to local trapezoidal loads by using the integral method, providing a more efficient and clear theoretical tool for engineering practice while addressing the limitations of existing educational and applied methodologies. The integral method is employed to derive fixed-end moment expressions for three types of statically indeterminate beams: (1) a beam fixed at both ends, (2) an an-end-fixed another-end-simple-support beam, and (3) a beam fixed at one end and sliding at the other. This approach eliminates the redundant equations of the traditional force method or the indirect transformations of the virtual beam method, directly linking boundary conditions through integral operations on load distributions, thereby significantly simplifying the solving process. Three representative numerical examples validate the correctness and universality of the derived formulas. The results demonstrate that the solutions obtained via the integral method align with software-calculated results, yet the proposed method yields analytical expressions for structural internal forces. Comparative analysis shows that the integral method surpasses traditional approaches (e.g., force method, virtual beam method) in terms of conceptual clarity and computational efficiency, making it particularly suitable for instructional demonstrations and rapid engineering calculations. The proposed integral method provides a systematic analytical framework for the internal force analysis of statically indeterminate structures under local trapezoidal loads, combining mathematical rigor with engineering practicality. The derived formulas can be directly applied to real-world designs, substantially reducing computational complexity. Moreover, this method offers a more intuitive theoretical case for structural mechanics education, enhancing students’ understanding of the mathematical–mechanical relationship between loads and internal forces. The research outcomes hold both theoretical significance and practical engineering value, establishing a solving paradigm for the displacement-based analysis of statically indeterminate structures under complex local trapezoidal loading conditions. Full article

Long-range rotating wrap-around-fin rockets may exhibit non-convergent conical motion at high Mach numbers, causing increased drag, reduced range, and potential flight instability. This study employs the implicit dual time-stepping method to solve the unsteady Reynolds-averaged Navier–Stokes (URANS) equations for simulating the flow field around a high aspect ratio wrap-around-fin rotating rocket at supersonic speeds. Validation of the numerical method in predicting aerodynamic characteristics at small angles of attack is achieved by comparing numerically obtained side force and yawing moment coefficients with experimental data. Analyzing the rocket’s angular motion process, along with angular motion equations, reveals the necessary conditions for the yawing moment to ensure stability during angular motion. Shape optimization is performed based on aerodynamic coefficient features and flow field structures at various angles of attack and Mach numbers, using the yawing moment stability condition as a guideline. Adjustments to parameters such as tail fin curvature radius, tail fin aspect ratio, and body aspect ratio diminish the impact of asymmetric flow induced by the wrap-around fin on the lateral moment, effectively resolving issues associated with near misses and off-target impacts resulting from dynamic instability at high Mach numbers. Full article

This study investigates the development of axial force in micro anti-slide piles under soil movement during slope stabilization. Axial force arises from two primary mechanisms: axial soil displacement ($z_{\mathrm{s}}$) and pile kinematics. The former plays a dominant role, producing either tensile or compressive axial force depending on the direction of $z_{\mathrm{s}}$, while the kinematically induced component remains consistently tensile. A sliding angle of $\alpha =5°$ represents an approximate transition point where these two effects balance each other. Furthermore, the two mechanisms exhibit distinct mobilization behaviors: $z_{\mathrm{s}}$-induced axial force mobilizes earlier than both bending moment and shear force, whereas kinematically induced axial force mobilizes significantly later. The study reveals two distinct pile–soil interaction mechanisms depending on proximity to the slip surface: away from the slip surface, axial soil resistance is governed by rigid cross-section translation, whereas near the slip surface, rotation-dominated displacement accompanied by soil–pile separation introduces significant complexity in predicting both the magnitude and direction of axial friction. A hyperbolic formulation was adopted to model both the lateral soil resistance relative to lateral pile–soil displacement (p-y behavior) and the axial frictional resistance relative to axial pile–soil displacement (t-z behavior). Soil resistance equations were derived to explicitly incorporate the effects of cross-sectional rotation and pile–soil separation. A novel beam-spring finite element method (BSFEM) that incorporates both geometric and material nonlinearities of the pile behavior was developed, using a soil displacement-driven solution algorithm. Validation against both numerical simulations and field monitoring data from an engineering application demonstrates the model’s effectiveness in capturing the distribution and evolution of axial deformation and axial force in micropiles under varying soil movement conditions. Full article

This study presents an exact analytical investigation into the static response of helical single-walled carbon nanotube (SWCNT) beams based on Eringen’s differential nonlocal elasticity theory, which captures nanoscale effects arising from interatomic interactions. A key contribution of this work is the derivation of the governing equations for helical SWCNT beams, based on the nonlocal Euler–Bernoulli theory, followed by their exact analytical solution using the initial value method. To the best of the authors’ knowledge, this represents the first closed-form formulation for such complex nanostructures using this theoretical framework of nonlocal elasticity theory. The analysis considers both cantilevered and clamped–clamped boundary conditions, under various concentrated force and moment loadings applied at the ends and midpoint of the helical beam. Displacements and rotational components are expressed in the Frenet frame, enabling direction-specific evaluation of the deformation behaviour. Parametric studies are conducted to investigate the influence of geometric parameters—such as the winding angle ($\alpha$) and aspect ratio ($R/d$) and the nonlocal parameter ($R/\gamma$). Results show that nonlocal elasticity theory consistently predicts higher displacements and rotations than the classical local theory, revealing its importance for accurate modelling of nanoscale structures. The proposed analytical framework serves as a benchmark reference for the modelling and design of nanoscale helical structures such as nano-springs, actuators, and flexible nanodevices. Full article

Traditional analyses of ship roll damping and added moment of inertia rely on free roll decay and forced roll tests, but acquiring linear (small angles) and nonlinear (large angles) relationships demands extensive computational cases and parameter fitting, limiting efficiency. To address this, this study couples Support Vector Machine (SVM) and Neural Ordinary Differential Equation (NODE) networks: SVM solves for added moment of inertia, linear damping, and nonlinear damping, while NODE constructs a complete model for the roll motion equation. Using the DTMB5415 hull form, Computational Fluid Dynamics (CFD) simulations of forced roll build a “time-angle-moment” sample space, and the coupled model learns and predicts free roll decay under different initial angles. The results show that SVM effectively determines roll damping and added moment of inertia from constant-amplitude variable-frequency and constant-frequency variable-amplitude data, reducing required cases significantly. NODE’s simulation of free roll decay validates coefficient accuracy. Within a certain angle range, the SVM-NODE model meets rapid roll motion analysis needs, providing an innovative method for ship roll research and engineering. Full article

Aiming to address the seismic vulnerability of long-span continuous rigid-frame bridges in high-intensity seismic zones, this study proposes to use a novel annular steel wire rope damper spherical bearing (SWD-SB) to dissipate the input earthquake energy and reduce the seismic responses. Firstly, the structural configuration and mechanical model of the new isolation bearing are introduced. Then, based on the dynamic finite element formulation, the equation of motion of a continuous rigid-frame bridge with the new isolation bearings is established, where the soil-structure interaction is considered. In a practical engineering case, the dynamic responses of the Pingchuan Yellow river bridge with the SWD-SB bearings are calculated and analyzed under multi-level earthquakes including the E1 and E2 waves. The results show that, compared with the bidirectional movable pot bearings, the SWD-SB significantly reduces the internal forces and displacement responses at the critical locations of the bridge. Under the E2 earthquake, the peak bending moments at the basement of main piers and at the pile caps are reduced by up to 72.6% and 44.7%, respectively, while the maximum displacement at the top of the main piers decreases by about 34.6%. The overall structural performance remains elastic except the SWD-SB bearings, meeting the two-stage seismic design objective. This paper further analyzes the hysteretic energy dissipation characteristics of the SWD-SB, highlighting its advantages in energy dissipation, deformation coordination, and self-centering capability. The research results demonstrate that the steel wire rope isolation bearings can offer an efficient and durable seismic protection for long-span continuous rigid-frame bridges in high-intensity seismic regions. Full article

In frame structures, connections play a vital role in governing both serviceability and ultimate strength. For pultruded fiber-reinforced polymer (PFRP) frames, connection design is even more critical due to the anisotropic and viscoelastic nature of the composite materials used in the primary elements (e.g., beams and columns) and their joints. This study presents a finite element model (FEM) to evaluate the influence of several connection parameters—namely, connection stiffening, bolt diameter, washer diameter, and clamping force—on the elastic behavior of beam-column joints composed of PFRP elements. The results demonstrate that stiffening the upper and lower connection angles significantly enhances joint performance. Increasing the bolt diameter improves moment capacity, reduces rotational deformation, decreases stress concentrations around bolt-hole edges, and increases both minor principal and compressive stresses beneath the bolt shank. Similarly, a larger washer diameter contributes to higher connection stiffness and reduces stress concentrations at bolt holes. Although the clamping force has a relatively modest effect on global connection behavior, it positively influences the through-thickness stress distribution in the angle beneath the bolt shank. Finally, regression equations were developed to quantify the relationship between rotation, moment, bolt diameter, washer diameter, and clamping force, providing a valuable tool for the design and optimization of PFRP connections in structural applications. Full article

This paper presents a combined theoretical, numerical, and experimental analysis to investigate the lateral plastic crushing behavior and energy absorption of circular and non-circular thin-walled rings between two rigid plates. Theoretical solutions incorporating both linear material hardening and power-law material hardening models are solved via numerical shooting methods. The theoretically predicted force-denting displacement relations agree excellently with both FEA and experimental results. The FEA simulation clearly reveals the coexistence of an upper moving plastic region and a fixed bottom plastic region. A robust automatic extraction method of the fully plastic region at the bottom from FEA is proposed. A modified criterion considering the unloading effect based on the resultant moment of cross-section is proposed to allow accurate theoretical estimation of the fully plastic region length. The detailed study implies an abrupt and almost linear drop of the fully plastic region length after the maximum value by the proposed modified criterion, while the conventional fully plastic criterion leads to significant over-estimation of the length. Evolution patterns of the upper and lower plastic regions in FEA are clearly illustrated. Furthermore, the distribution of plastic energy dissipation is compared in the bottom and upper regions through FEA and theoretical results. Purely analytical solutions are formulated for linear hardening material case by elliptical integrals. A simple algebraic function solution is derived without necessity of solving differential equations for general power-law hardening material case by adopting a constant curvature assumption. Parametric analyses indicate the significant effect of ovality and hardening on plastic region evolution and crushing force. This paper should enhance the understanding of the crushing behavior of circular and non-circular rings applicable to the structural engineering and impact of the absorption domain. Full article

The existence of fault zones in high-intensity earthquake areas has a serious impact on engineering structures, and the longitudinal response of tunnels crossing faults needs further in-depth research. To analyze the tangential contact effect between the surrounding rock and the tunnel lining, and the axial deformation characteristics of the tunnel structure, tangential foundation springs were introduced and a theoretical model for the longitudinal response of the tunnel under fault dislocation was established. Firstly, the tunnel was simplified as a finite-length beam. The normal and tangential springs were taken to represent the interaction between the soil and the lining. The fault’s free-field displacement was applied at the end of the normal foundation spring to simulate fault dislocation, and the differential equation for the longitudinal response of the tunnel structure was obtained. The analytical solution of the structural response was obtained using the Green’s function method. Then, the three-dimensional finite difference method was used to verify the effectiveness of the analytical model in this paper. The results show that the tangential contact effect between the surrounding rock and the lining has a significant impact on the longitudinal response of the tunnel structure. Ignoring this effect leads to an error of up to 35.33% in the peak value of the structural bending moment. Finally, the influences of the width of the fault zone, the soil stiffness of the fault zone, and the stiffness of the tunnel lining on the longitudinal response of the tunnel were explored. As the fault width increases, the internal force of the tunnel structure decreases. Increasing the lining concrete grade leads to an increase in the internal force of the structure. The increase in the elastic modulus of the surrounding rock in the fault area reduces the bending moment and shear force of the structure and increases the axial force. The research results can provide a theoretical basis for the anti-dislocation design of tunnels crossing faults. Full article

This paper presents field investigations of a corrugated steel soil–steel arch structure with a span of 25.7 m and a rise of 9.0 m—currently the largest single-span structure of its kind in Europe. The structure, serving as a wildlife crossing along the DK16 expressway in northeastern Poland, was constructed using deep corrugated steel plates (500 mm× 237 mm) made from S315MC steel, without additional reinforcements such as stiffening ribs or geosynthetics. The study focused on monitoring the structural behavior during the critical backfilling phase. Displacements and strains were recorded using 34 electro-resistant strain gauges and a geodetic laser system at successive backfill levels, with particular attention to the loading stage at the crown. The measured results were compared with predictions based on the Swedish Design Method (SDM). The SDM equations did not accurately predict internal forces during backfilling. At the crown level, bending moments and axial forces were overestimated by approximately 69% and 152%, respectively. At the final backfill level, the SDM underestimated bending moments by 55% and overestimated axial forces by 90%. These findings highlight limitations of current design standards and emphasize the need for revised analytical models and long-term monitoring of large-span soil–steel structures. Full article

The model explicitly incorporates boundary conditions that account for the complex interplay between sections experiencing varying degrees of reduction. This interaction significantly influences the overall deformation behavior and force loading. The control effect is associated with boundary conditions determined by the unevenness of the compression, which have certain quantitative and qualitative characteristics. These include additional loading, which is less than the main load, which implements the process of plastic deformation, and the ratio of control loads from the entrance and exit of the deformation site. According to this criterion, it follows from experimental data that the controlling effect on the plastic deformation site occurs with a ratio of additional and main loading in the range of 0.2–0.8. The next criterion is the coefficient of support, which determines the area of asymmetry of the force load and is in the range of 2.00–4.155. Furthermore, the criterion of the regulating force ratio at the boundaries of the deformation center forming a longitudinal plastic shear is within the limits of 2.2–2.5 forces and 1.3–1.4 moments of these forces. In this state, stresses and deformations of the plastic medium are able to realize the effects of plastic shaping. The force effect reduces with an increase in the unevenness of the deformation. This is due to a change in height of the longitudinal interaction of the disparate sections of the strip. There is an appearance of a new quality of loading—longitudinal plastic shear along the deformation site. The unbalanced additional force action at the entrance of the deformation source is balanced by the force source of deformation, determined by the appearance of a functional shift in the model of the stress state of the metal. The developed theory, using the generalized method of an argument of functions of a complex variable, allows us to characterize the functional shift in the deformation site using invariant Cauchy–Riemann relations and Laplace differential equations. Furthermore, the model allows for the investigation of material properties such as the yield strength and strain hardening, influencing the size and characteristics of the identified limit state zone. Future research will focus on extending the model to incorporate more complex material behaviors, including viscoelastic effects, and to account for dynamic loading conditions, more accurately reflecting real-world milling processes. The detailed understanding gained from this model offers significant potential for optimizing mill roll designs and processes for enhanced efficiency and reduced energy consumption. Full article

This work systematically addresses the dual challenges of non-inertial dynamic coupling and kinematic constraint redundancy encountered in dynamic modeling of serial–parallel–serial hybrid robotic mechanisms, and proposes an improved Newton–Euler modeling method with constraint compensation. Taking the Skiing Simulation Platform with 6-DOF as the research mechanism, the inverse kinematic model of the closed-chain mechanism is established through $G_F$ set theory, with explicit analytical expressions derived for the motion parameters of limb mass centers. Introducing a principal inertial coordinate system into the dynamics equations, a recursive algorithm incorporating force/moment coupling terms is developed. Numerical simulations reveal a 9.25% periodic deviation in joint moments using conventional methods. Through analysis of the mechanism’s intrinsic properties, it is identified that the lack of angular momentum conservation constraints on the end-effector in non-inertial frames leads to system controllability degradation. Accordingly, a constraint compensation strategy is proposed: establishing linearly independent differential algebraic equations supplemented with momentum/angular momentum balance equations for the end platform. Co-Simulation results demonstrate that the optimized model reduces the maximum relative error of actuator joint moments to 0.98%, and maintains numerical stability across the entire configuration space. The constraint compensation framework provides a universal solution for dynamics modeling of complex closed-chain mechanisms, validated through applications in flight simulators and automotive driving simulators. Full article

Balancing mechanisms require the minimization of both the Shaking Moment ($ShM$) and Shaking Force ($\mathbf{ShF}$), a complex multi-criteria challenge often tackled using single-objective algorithms. However, these methods face difficulties in navigating competing objectives. In contrast, multi-objective algorithms provide a more efficient and adaptable framework, while Fully Cartesian Coordinates ($FCC$) simplify the balancing equations compared to conventional Cartesian formulations. This study focuses on optimizing the dynamic balance of a six-bar Watt linkage using $FCC$. A wide set of optimization methods is analyzed and compared, and among them, the S-Metric Selection Evolutionary Multi-objective Optimization Algorithm (SMS-EMOA) demonstrates superior performance. This algorithm achieves the most significant hypervolume value in only 10.44 min of execution. The results indicate that multi-objective algorithms outperform single-objective approaches, offering faster and more diverse optimization solutions. Additionally, this study introduces an analytical method that enables the straightforward identification of removable counterweights, achieving an equally effective balance while minimizing the number of counterweights required. Full article

The authors have designed a gyro-system for orientation (guidance) and stabilization, with two gimbals and a rotor in magnetic suspension (AMB—Active Magnetic Bearing) usable for self-guided rockets. The gyro-system (DGMSGG—double gimbal magnetic suspension gyro-system for guidance) orients and stabilizes the target coordinator’s axis (CT) and, at the same time, the AMB–rotor’s axis so that they overlap the guidance line (the target line). DGMSGG consists of two decoupled systems: one for canceling the AMB–rotor translations along the precession axes (induced by external disturbing forces), the other for canceling the AMB–rotor rotations relative to the CT-axis (induced by external disturbing moments) and, at the same time, for controlling the gimbals’ rotations, so that the AMB–rotor’s axis overlaps the guidance line. The nonlinear DGMSGG model is decomposed into two sub-models: one for the AMB–rotor’s translation, the other for the AMB–rotor’s and gimbals’ rotation. The second sub-model is described first by nonlinear state equations. This model is reduced to a second order nonlinear matrix—vector form with respect to the output vector. The output vector consists of the rotation angles of the AMB–rotor and the rotation angles of the gimbals. For this purpose, a differential geometry method, based on the use of the output vector’s gradient with respect to the nonlinear state functions, i.e., based on Lie derivatives, is used. This equation highlights the relative degree (equal to 2) with respect to the variables of the output vector and allows for the use of the dynamic inversion method in the design of stabilization and guidance controllers (of P.I.D.- and PD-types), as well as in the design of the related linear state observers. The controller of the subsystem intended for AMB–rotor’s translations control is chosen as P.I.D.-type, which leads to the cancellation of both its translations and its translation speeds. The theoretical results are validated through numerical simulations, using Simulink/Matlab models. Full article

A variational formulation and variationally consistent boundary conditions were derived for a coupled system of two boron nitride nanotubes (BNNTs), with the piezoelectric and surface effects taken into account in the formulation. The coupling between the nanotubes was defined in terms of Winkler and Pasternak interlayers. The equations governing the vibrations of the coupled system were expressed as a system of four partial differential equations based on nonlocal elastic theory. After deriving the variational principle for the double BNNT system, Hamilton’s principle was expressed in terms of potential and kinetic energies. Next, the differential equations for the free vibration case were presented and the variational form for this case was derived. The Rayleigh quotient was formulated for the vibration frequency, which indicated that piezoelectric and surface effects led to higher vibration frequencies. Next, the variationally consistent boundary conditions were formulated in terms of moment and shear force expressions. It was observed that the presence of the Pasternak interlayer between the nanotubes led to coupled boundary conditions when a shear force and/or a moment was specified at the boundaries. Full article