Force Equilibrium Equations Research Articles

Improved Theoretical Solutions for Estimating the Tunnel Response Induced by Overlying Excavation

As a result of China’s urbanization, it has been a common phenomenon that adjacent deep excavations were constructed near underground structures, which can have a series of detrimental effects on existing tunnels. Thus, it is crucial to assess the tunnel response induced by the overlying excavation, with the aim of maintaining the safety and serviceability of operating tunnels. The shield tunnel is idealized as an infinite beam lying upon a three parameter Kerr-model and the vertical force equilibrium equation of the tunnel element is established. Then, a theoretical solution is derived for capturing the soil–tunnel interaction. To prove the accuracy of the proposed method, the calculation results are compared with field measurements, along with the data of finite element studies. Thereafter, a parametric analysis will be conducted to assess some characteristic factors for tunnel responses caused by overlying excavations, such as tunnel-excavation horizontal distance, tunnel bending stiffness, and the buried depth of the tunnel. The results indicate that the increase in the bending stiffness and the buried depth of tunnel, as well as the tunnel-excavation horizontal distance, will significantly alleviate the tunnel deformation. However, the inner force will be increased when increasing the tunnel bending stiffness.

Longitudinal velocity profile of flows in open channel with double-layered rigid vegetation

Aquatic vegetation of different heights is widely scattered in natural rivers and is conducive to their environmental function while affecting the flow hydrodynamic conditions. A semi-analytical velocity model is constructed and used to study the longitudinal velocity profile in open channel flow through double-layered rigid vegetation. The double-layered vegetation flow is separated into three zones according to the velocity profile: 1) nearly uniform distributed velocity zone 1A in the lower region of the short vegetation layer, 2) a mixing layer zone B, 3) uniform distributed velocity zone 2A in the upper region of the tall vegetation layer. Two force equilibrium equations about the gravity-driving and vegetation drag are solved to obtain the uniform velocity distribution equations in zone 1A and 2A. The velocity of zone 1A and B is further modeled as a linear superposition of two concepts: the uniform velocity distribution term of zone 1A and a hyperbolic tangent profile. Meanwhile, longitudinal velocity and the lateral vorticity profiles of open channel flow through double-layered rigid vegetation are studied by laboratory flume tests of different vegetation arrangements exposed to two water depths and three slopes. The experimental results show that the longitudinal velocity increases with the slope increase. The verification of the velocity model is based on the instantaneous velocity measured by Acoustic Doppler Velocimetry (ADV), which shows acceptable agreement, indicating that the model can give a reference to the longitudinal velocity of multi-layered vegetation flow in some cases. The effects of wake vortices and boundary friction on the model are further explored in the discussions. The results presented in this study could contribute to the management of aquatic vegetation configurations and the restoration of freshwater ecology.

A Nonlinear Programming-Based Morphing Strategy for a Variable-Sweep Morphing Aircraft Aiming at Optimizing the Cruising Efficiency

This work develops a morphing decision strategy to optimize the cruising efficiency for a variable-sweep morphing aircraft, and a simple and practical guidance and control system is given as well. They can work in tandem to accomplish a cruise mission effectively. To make the morphing decision accurately, we take into account the equilibrium equations of forces; the variations in airspeed, altitude, and mass and the optimal configurations for different cruise conditions are solved based on the nonlinear programming method with the objective of minimum engine thrust. Considering that a large amount of computational resources are required to solve the nonlinear programming problems, we establish an offline database of the optimal configurations and design a database-based online morphing decision process. In addition, the proposed morphing decision strategy includes an anti-disturbance mechanism, which ensures that the optimal configuration can be given accurately without chattering under fluctuating airspeed measurements. Comparative results from the simulations finally validate the effectiveness of the proposed strategy.

Shortcomings of shear wall design as per IS:13920–2016

Reinforced concrete shear walls are almost routinely used in multi-storeyed buildings due to their increased strength, stiffness, and ductility considerations. Shear walls are typically stiffer than regular frame elements. Hence, are subjected to more lateral forces during response to earthquakes. Reinforced concrete walls subjected to compression along with flexure are similar to columns and hence need similar methodology. For a section subjected to compression and flexure, two equilibrium conditions namely force and moment equilibrium equations needs to be solved for any specific load combination. This being a cumbersome and complex proposition, design charts, similar to columns should be used for shear wall design. However, IS 13920:2016 provides closed form expressions for estimation of moment capacity of rectangular RC walls without boundary elements combining these two equilibrium equations by using certain assumption and simplifications which are not conforming to the fundamental philosophies of limit state method as per IS 456–2000. These closed form expressions have several limitations and constraints and the equations are not valid for all the probable positions of neutral axis. In the present work the shortcomings of the codal methodology of shear wall design have been highlighted and method of development of proper design aids for shear wall design have been outlined.

Design and Modeling of Multi-vertebrae Continuum Robot for Minimally Invasive Surgery

The constant curvature assumption is normally utilized in kinematic modeling of continuum manipulators due to its simplicity. Nevertheless, in practical applications, factors such as frictional disturbances may cause the curvature of a continuum manipulator to deviate from being constant. In this article, we design a novel multi-vertebrae continuum manipulator that utilizes the uniform joint deformation curvature of a superelastic NITI alloy rod. We propose kinematic modeling approach for a multi-vertebrae continuum manipulator. To improve the unevenness of joint angles and increase motion accuracy, we inserted elastic beams into the multi-vertebrae continuum manipulator. The design takes into account frictional forces, and we established geometric constraint equations and force equilibrium equations for the hinged continuum manipulator. We employed an identification method to establish the physical equations of the deformation section. By solving these equations and analyzing the geometric relationships, we obtained the mapping relationship between the driving screw force and the end displacement of the continuum manipulator. To prove the effectiveness of the proposed method, we performed MATLAB simulations. This paper serves as a reference for the design and modeling of future minimally invasive surgical manipulators.

MaterialSense: Estimating and utilizing material properties of contact objects in multi-touch interaction

In today’s digital art applications, users can play virtual musical instruments or paint on virtual canvas using multi-touch input. Existing touch input devices, however, do not consider that the result of artistic expression changes depending on the material properties of the tool that comes into contact with the art medium. This paper proposes a novel method called MaterialSense that estimates the material properties of objects in contact with a touch input surface. The technique uses a commercial touchpad with six load cells attached, and when up to two objects are in contact, solves the force equilibrium equation to estimate the stiffness and friction coefficient of each contacted object. By analyzing the time-series data of the measured 3-axis force and normal vector for each touchpoint, it can estimate the stiffness and kinetic friction coefficient of the contacted object. Estimated material properties enable novel and realistic artistic expressions in touch-based digital art applications, such as changes in the tone of virtual instruments or the effects of different painting brushes. We present two application scenarios using MaterialSense, along with an in-depth technical evaluation to verify the accuracy and precision of its estimates.

New macroscopic model for predicting the horizontal behavior of elastomeric bearings under end-plate rotation

In the scenarios of structure retrofit or bridge application, elastomeric bearings may be placed on top of columns or piers. These implementations result in a flexible boundary condition and the end-plates of the bearings may experience rotation. Experimental studies show that the end-plate rotation will affect the behavior of elastomeric bearings. In this paper, a new macroscopic model is developed for predicting the horizontal behavior of elastomeric bearings under end-plate rotation. The force equilibrium and deformation compatibility equations of the new model are derived, followed by the elaboration of its numerical implementation. Furthermore, the predictive capacity of this new model is validated by comparison with previous test data and numerical results. The application of this new model is demonstrated through a dynamic analysis example of a single-degree-of-freedom system. The results show that the macroscopic model will be promising to serve as an alternative approach to account for the effect of end-plate rotation.

Torques and angular momenta of fluid elements in the octonion spaces

The paper focuses on applying the octonions to explore the influence of the external torque on the angular momentum of fluid elements, revealing the interconnection of the external torque and the vortices of vortex streets. J. C. Maxwell was the first to introduce the quaternions to study the physical properties of electromagnetic fields. The contemporary scholars utilize the quaternions and octonions to investigate the electromagnetic theory, gravitational theory, quantum mechanics, special relativity, general relativity, curved spaces, and so forth. The paper adopts the octonions to describe the electromagnetic and gravitational theories, including the octonionic field potential, field strength, linear momentum, angular momentum, torque, and force. In case the octonion force is equal to zero, it is able to deduce eight independent equations, including the fluid continuity equation, current continuity equation, and force equilibrium equation. Especially, one of the eight independent equations will uncover the interrelation of the external torque and angular momentums of fluid elements. One of its inferences is that the direction, magnitude, and frequency of the external torque must impact the direction and curl of the angular momentum of fluid elements, altering the frequencies of Karman vortex streets within the fluids. It means that the external torque is interrelated with the velocity circulation, by means of the liquid viscosity. The external torque is able to exert an influence on the direction of downwash flows, improving the lift and drag characteristics generated by the fluids.

A Design Methodology for the Seismic Retrofitting of Existing Frame Structures Post-Earthquake Incident Using Nonlinear Control Systems

A structural design methodology for retrofitting weakened frame systems following earthquakes is developed and presented. The design procedure refers to frame systems in their degraded strength and stiffness states and restores their dynamic performance using nonlinear control systems. The control law associated with the employed systems regards the gains between the negative state feedback and the control force, which consists of linear, nonlinear, and hysteretic portions. Structural optimization is introduced in designing the nonlinear control systems, and the controller gains are optimized using the fixed-point iteration to improve the frame system’s dynamic performance. The fixed-point iteration method relates to first-order PDE equations; hence, a new state-space formulation for weakened inelastic frame systems is developed and presented using the frame system’s lateral force equilibrium equation. The design scheme and optimization strategy differ from designing passive control systems, given that the nonlinear control system’s force consists of linear, nonlinear, and hysteretic portions. The utilization of the fixed-point iteration in the structural design area is by itself a novel application due to its robustness in addressing the gains of any type of nonlinear control system. This paper’s nonlinear control system chosen to exhibit the application is Buckling Restrained Braces (BRBs) since force consists of linear and hysteretic portions. The implementation of hysteretic control force is rare in structural control applications. In the case of BRBs, the fixed-point iteration optimizes the cross-sectional areas. Two system optimization examples of 3-story and 15-story inelastic frames are provided and described. The examples demonstrate the fixed-point iteration’s applicability and robustness in optimizing control gains of nonlinear systems and regulating the dynamic response of weakened frame structures.

A Substructure Condensed Approach for Kinetostatic Modeling of Compliant Mechanisms with Complex Topology.

Compliant mechanisms with complex topology have previously been employed in various precision devices due to the superiorities of high precision and compact size. In this paper, a substructure condensed approach for kinetostatic analysis of complex compliant mechanisms is proposed to provide concise solutions. In detail, the explicit relationships between the theoretical stiffness matrix, element stiffness matrix, and element transfer matrix for the common flexible beam element are first derived based on the energy conservation law. The transfer matrices for three types of serial–parallel substructures are then developed by combining the equilibrium equations of nodal forces with the transfer matrix approach, so that each branch chain can be condensed into an equivalent beam element. Based on the derived three types of transfer matrices, a kinetostatic model describing only the force-displacement relationship of the input/output nodes is established. Finally, two typical precision positioning platforms with complex topology are employed to demonstrate the conciseness and efficiency of this modeling approach. The superiority of this modeling approach is that the input/output stiffness, coupling stiffness, and input/output displacement relations of compliant mechanisms with multiple actuation forces and complex substructures can be simultaneously obtained in concise and explicit matrix forms, which is distinct from the traditional compliance matrix approach.

Morphing Wing Based on Trigonal Bipyramidal Tensegrity Structure and Parallel Mechanism

The development of morphing wings is in the pursuit of lighter weight, higher stiffness and strength, and better flexible morphing ability. A structure that can be used as both the bearing structure and the morphing mechanism is the optimal choice for the morphing wing. A morphing wing composed of a tensegrity structure and a non-overconstrained parallel mechanism was designed. The self-balancing trigonal bipyramidal tensegrity structure was designed based on the shape-finding method and force-equilibrium equation of nodes. The 4SPS-RS parallel mechanism that can complete wing morphing was designed based on the configuration synthesis method. The degree of freedom and inverse solution of the parallel mechanism was obtained based on the screw theory, and the Jacobian matrix of the parallel mechanism was established. The stiffness model of the tensegrity structure and the 4SPS-RS parallel mechanism was established. The relationship between the deformation of the 4SPS-RS parallel mechanism and sweep angle, torsion angle, spanwise bending, and span was obtained. Through the modular assembly and distributed drive, the morphing wing could perform smooth and continuous morphing locally and globally. In the static state, it has the advantages of high stiffness and large bearing capacity. In the process of morphing, it can complete morphing motion with four degrees of freedom in changing sweep, twist, spanwise bending, and span of the wing.

Bending stiffness of bamboo-concrete composite (BCC) beams under short-term loads

Bamboo-concrete composite (BCC) beams are newly-innovated products used for floor system which synergistically utilizes bamboo and concrete components. These composite beams at current technology belong to the beam characterized with partial composite action due to the existence of relative slip between bamboo segment and concrete slab. Existing investigations have demonstrated that the load bearing capacity of BCC beams depends on their bending stiffness; however, very limited prediction models are available to accurately estimate the bending stiffness of BCC beams. At present, the bending stiffness of BCC beams is predicted with the help of γ-method in Eurocode 5-Part 1-1, and available studies have demonstrated that this method overestimates the bending stiffness of BCC beams. To this end, this paper presents a novel explicit model characterized with mechanism and data mining to predict the bending stiffness of BCC beams under short-term loads. Three steps were performed to achieve the aforementioned research objective. The first step was to introduce the axial force transfer coefficient into the equilibrium equations of internal force for bamboo segment and concrete slab. Secondly, the bending stiffness of BCC beams was established by taking into account the effect of relative slip with the help of existing experimental results from the interfacial shear test and flexural test. Finally, the proposed model was validated by comparing with flexural performance results of BCC beams collected from available literature as well as those obtained from γ-method. The comparative results show that: i) the predicted bending stiffness of BCC beams is much closer to tested ones compared with γ-method; ii) the errors of proposed model predicting the bending stiffness for most BCC beams are within 10% of test results; however, nearly half of the predicted ones using γ-method have errors exceeding 10% of test results, both of which verify that the mechanism and data mining-dual driven model receives a better prediction for short-term bending stiffness of BCC beams.

Effects of a tilted flat-ended punch on the receding contact between a graded and a homogeneous layer

This paper considers the receding contact problem between an exponentially graded layer and a homogeneous layer under the indentation of a tilted rigid flat-ended punch. In the presence of an offset load deviated from the symmetry axis of the rigid punch, the contact pressures both under the rigid punch and along the receding contact interface between the layers lose symmetry. Depending on the magnitude of eccentricity, either complete or incomplete contact may take place under the rigid flat-ended punch, whereas a receding type of contact always occurs along the interface separating the graded and the homogeneous layers. Fourier integral transforms help to convert the governing equations and mixed boundary conditions of the double-contact problem into dual Fredholm integral equations of the second kind with Cauchy-type singular kernels. Gauss–Chebyshev and Gauss–Jacobi numerical quadratures are subsequently employed to discretize and collocate the dual singular integral equations, together with the four force and moment equilibrium equations, for the complete and incomplete contact, as well as the transitional status corresponding to the critical load eccentricity. Extensive parametric studies indicate that the critical load eccentricity depends on the property gradation of the graded layer and the layer-thickness ratio, but not on the magnitude of the indentation load. The extent of contact both under the rigid flat-ended punch and along the receding contact interface is also independent of the magnitude of the indentation load. The results suggest reasonable departures from the contact properties of a homogeneous half-plane under a tilted rigid flat-ended punch, indicating the valuable information on optimizing the double-contact properties in terms of the property gradation and layer-thickness ratio under a given eccentricity.

Determination of the Plastic Stress–Strain Relationship of a Rupture Disc Material with Quasi-Static and Dynamic Pneumatic Bulge Processes

Rupture discs, manufactured using a hydraulic or pneumatic bulge process, are widely used to protect vessels from over-pressuring. The stress–strain relationship of the material in the bulge process plays a major role in understanding the performance of rupture discs. Moreover, both the theoretical analyses and numerical simulations of rupture discs demand a reliable stress–strain relationship of the material in a real bulge process. In this paper, an approach for determining the plastic stress–strain relationship of a rupture disc material in the bulge process is proposed based on plastic membrane theory and force equilibrium equations. Pressures of compressed air and methane/air mixture explosions were used for the bulge pressure to accomplish the quasi-static and dynamic bulge processes of tested discs. Experimental apparatus for the quasi-static bulge test and the dynamic bulge test were built. The stress–strain relations of 316L material during bulge tests were obtained, compared, and discussed. The results indicated that the bulge height at the top of the domed disc increased linearly with an increase in bulge pressure in the quasi-static and dynamic bulge processes, and the effective strain increased exponentially. The rate of pressure rise during the bulge process has a significant effect on the deformation behavior of disc material and hence the stress–strain relationship. At the same bulge pressure, a disc tested with a larger pressure rise rate had smaller bulge height and effective strain. At the same effective stress at the top of the domed disc, discs subjected to a higher pressure rise rate had smaller effective strain, and hence they are more difficult to rupture. Hollomon’s equation is used to represent the relationship between the effective stress and strain during bulge process. For pressure rise rates in the following range of 0 (equivalent to quasi-static condition), 2–10 MPa/s, 10–50 MPa/s, and 50–100 MPa/s, the relation of stress and strain is σe = 1259.4·εe0.4487, σe = 1192.4·εe0.3261, σe = 1381.2·εe0.2910, and σe = 1368.4·εe0.1701, respectively.

A method for calculating lateral response of offshore rigid monopile in sand under slope effect

This paper develops a method for calculating nonlinear lateral response of offshore rigid monopile in sand under slope effect. The method modifies three-dimensional wedge failure model to evaluate the ultimate soil resistance in front of the laterally loaded rigid monopile near sloped ground. The modified model considers the ultimate horizontal frictional resistance, the slope effect, the vertical friction force, the mutual influence of the forces within the wedge. And the base horizontal shear resistance is considered in the method. In addition, the load-displacement curves of the rigid monopile under five different distribution modes of the soil reaction along the pile are calculated by the force and moment equilibrium equations. The method in this paper is verified by comparing its results with those of field test, 3D FE model, 1g laboratory model test and existing analytical methods. Finally, this method is used to study the rotation behavior and lateral response of the offshore rigid monopile.

Analytical solution for predicting the interaction stress of axially loaded concrete-filled double-tube columns

Concrete-filled double-tube (CFDT) members are promising steel-concrete composite structural forms. Existing studies mainly focus on members’ load-deformation performance. The interaction stress between components, which is a fundamental cause for change in load-deformation performance, receives much less attention. This study presents an analytical solution for interaction stress between components of axially loaded CFDT columns. The stress–strain relationships and force equilibrium equations of the inner concrete, inner steel tube, sandwiched concrete, and outer steel tube were analysed using elasticity method first. Subsequently, the deformation compatibility equations were established concerning the interactions between the components. Based on the above analysis, an analytical solution was derived. Equivalent lateral deformation coefficients were used to substitute Poisson’s ratio of concrete to account for the nonlinear behaviour of materials during loading. The proposed model shows reasonable agreement with experimental results. Finite element analysis further shows the accuracy of the interaction stress model till an axial compressive strain of 0.01 of the composite column. The derived interaction stress model was successfully applied in predicting the load-bearing capacities of CFDT columns through validation of 39 specimens from five independent studies. The average ratio of the predicted-to-experimental bearing capacities was 1.054, with a coefficient of variation of 0.077. The accuracy of models in Eurocode 4, ACI 318-11, and GB 50936 were also evaluated. Thereby the proposed model provides insights into the interaction analysis between components and mechanism-based load-bearing capacity prediction of CFDT columns.

The effects of couple stress lubricants and surface roughness on squeeze EHL motion between porous medium layer and elastic ball

This paper investigated the effects of porous media layer (PML) and couple stress (CS) lubricants at pure squeeze action in an isothermal elastohydrodynamic lubrication (EHL) point contact with surface roughness (SRN) under constant load condition. The modified transient stochastic Reynolds equation was derived in polar coordinates by means of the Christensen’s stochastic theory and the Stokes’s CS fluid theory. The Gauss-Seidel method (GSM) and the finite difference method (FDM) were both used to solve simultaneously modified transient stochastic Reynolds, force equilibrium, elasticity deformation, and rheology equations. The simulation results revealed that the greater permeability ( K) and porous layer thickness (Φ) are, the smaller central pressures ( Pc) are, and the smaller film thicknesses ( H) are. The times required for achieving maximum central pressures (Pcmax) decrease with increasing K and Φ. Due to squeeze and elastic deformation effects, the slope values of central velocity ( Vc) change from negative to positive. The magnitude of Vc and Vmin decreased with decreasing K and Φ. The Hcmin of circular type RN are greater than that of radial type RN. The Pcmax of radial type RN are slightly greater than that of circular type RN.

One underlying mechanism for two piezoelectric effects in the octonion spaces

The paper aims to apply the algebra of octonions to explore the contributions of external derivative of electric moments and so forth on the induced electric currents, revealing a few major influencing factors relevant to the direct and inverse piezoelectric effects. J. C. Maxwell was the first to adopt the algebra of quaternions to describe the physical quantities of electromagnetic fields. The contemporary scholars utilize the quaternions and octonions to research the physical properties of electromagnetic and gravitational fields. The application of octonions is able to study the physical quantities of electromagnetic and gravitational fields, including the octonion field strength, field source, linear moment, angular moment, torque and force. When the octonion force is equal to zero, it is capable of achieving eight independent equations, including the force equilibrium equation, fluid continuity equation, current continuity equation, and second-precession equilibrium equation and others. One of inferences derived from the second-precession equilibrium equation is that the electric current and derivative of electric moments are able to excite each other. The external derivative of electric moments can induce the electric currents. Meanwhile the external electric currents are capable of inducing the derivative of electric moments. The research states that this inference can be considered as the underlying mechanism for the direct and inverse piezoelectric effects. Further the second-precession equilibrium equation is able to predict several new influencing factors of direct and inverse piezoelectric effects.

Eight equilibrium and continuity equations within the material media

This paper aims to apply the octonions to explore the torques and forces and so forth in the electromagnetic and gravitational fields, investigating the influences of material media on the equilibrium and continuity equations. The contemporary scholars utilize the quaternions and octonions to research the electromagnetic fields and gravitational fields and so forth. In this paper, the octonions are capable of surveying the electromagnetic and gravitational fields within material media, including the octonion field strength, field source, angular momentum, torque and force. Further, the octonion field strength and angular momentum can be combined together to become the octonion composite field strength, deducing the octonion composite torque and force and others. When the octonion composite force is equal to zero under certain circumstances, it is able to achieve eight equations independent of each other, including the force equilibrium equation, fluid continuity equation, current continuity equation and so on. The above reveals that the material media and octonion field strength can make a contribution to the eight equilibrium and continuity equations. And it is beneficial for deepening the understanding of equilibrium and continuity equations within material media.

Wearable Multifunctional Additive Hand System for Enhancing the Workspace and Grasping Capability of the Human Hand

In this study, a wearable multifunctional additive hand (MAH) system is proposed for enhancing the workspace and grasping capability of the human hand. The MAH system consists of a main body and three fingers, and each finger has two joints. To optimize the performance, the following five parameters are considered: finger length, number of fingers, number of joints, base frame size, and joint angle range. The MAH system is designed to assist the human hand in the inward and outward modes. In the inward mode, the finger joints of the proposed system rotate in the same direction as the finger flexion of the human hand, whereas in the outward mode, the robotic fingers of the MAH system rotate in opposite directions. The proposed system provides assistive torque and expands the workspace of the human hand in the inward mode. When the additive finger joints rotate outward, they grasp an object as a third hand. The validity of the proposed system is verified analytically by changing the design parameters, considering the workspace expansion and joint torque reduction. An alpha shape is introduced to calculate the expanded workspace volume using the proposed system. The joint torque was estimated by utilizing kinematics and the force equilibrium equation, assuming that the human hand with the MAH system holds a postulated object.