Tangential Spring Research Articles

Impacts of the mechanical connectors on seismic performance of the restored masonry Plaka Bridge using experimental and numerical studies

The use of clamp and dowel materials in the arch of the masonry bridges holds pivotal significance in enhancing the seismic resilience of these structures, as these elements contribute crucially to the structural integrity and dynamic response of these venerable structures. For this reason, the seismic performance of the restored historic Plaka Bridge, originally constructed in 1866 in Arta, Greece, and later restored in 2015 after flooding, was investigated considering the dowels and clamps in the arch section of this study. A combination of experimental and numerical analyses was conducted and verified to understand the in-situ interface behavior of the arch stones. In the first phase of the study, the mechanical properties of the dowels and clamps were characterized by the experimental tests. Compression and tension tests were conducted on the stone samples containing dowel bars and clamps respectively to determine the mechanical properties, The experimental outcomes were subsequently validated through Three-Dimensional (3D) Finite Difference Models (FDM). It was obtained that the dowels and clamps represent the bi-linear material model at the interfaces. Finally, the idealized interface behavior was constructed with normal and tangential spring stiffness properties in 3D-FD Models. These springs successfully simulate the interface delamination failure of the stones in all directions. In the final phase of the study, a 3D-FD model of the Plaka Bridge was constructed employing the previously validated normal and tangential stiffness elements in the arch of the bridge. A free-field non-reflecting boundary condition was imposed on the base and lateral boundaries of the bridge model. Seismic analyses were conducted utilizing the significant seismic events including 2023 Kahramanmaraş earthquakes. The results revealed significant alterations in the structural behavior of the bridge under considered earthquakes. Furthermore, the displacements, principal stresses and tensile-compressive damages in the main arch significantly reduced with the adopted stiffness values. This study proposes the undeniably realistic and efficient implementation of interface material modeling for the mechanical connections as opposed to the traditionally assumed mortar properties in literature. Furthermore, the combined behavior of the dowel, clamp, and Khorasan mortar proposes non-linear interface material model to the existing literature by invalidating commonly assumed linear interface behavior in masonry structures.

Numerical Simulation Study Considering Discontinuous Longitudinal Joints in Soft Soil under Symmetric Loading

In shield tunneling, the joint is one of the most vulnerable parts of the segmental lining. Opening of the joint reduces the overall stiffness of the ring, leading to structural damage and issues such as water leakage. Currently, the Winkler method is commonly used to calculate structural deformation, simplifying the interaction between segments and soil as radial and tangential Winkler springs. However, when introducing connection springs or reduction factors to simulate the joint stiffness of segments, the challenge lies in determining the reduction coefficient and the stiffness of the springs. Currently, the hyperstatic reflection method cannot simulate the discontinuity effect at the connection of the tunnel segments, while the state space method overlooks the nonlinear interaction between the tunnel and the soil. Therefore, this paper proposes a numerical simulation method considering the interaction between the tunnel and the soil, which is subjected to compression rather than tension, and the discontinuity of the joints between the segments. The model structure and external load are symmetrical, resulting in symmetrical calculation results. This method is based on the soft soil layers and shield tunnel structures of the Shanghai Metro, and the applicability of the model is verified through deformation calculations using three-dimensional laser scanning point clouds of sections from the Shanghai Metro Line 5. When the subgrade reaction coefficient is 5000 kN/m3, the model can effectively simulate the deformation of operational tunnels. By adjusting the bending stiffness of individual connection springs, we investigate the influence of bending stiffness reduction on the bending moment, radial displacement, and rotational displacement of the ring. The results indicate that a decrease in joint bending stiffness significantly affects the mechanical response of the ring, and the extent and degree of this influence are correlated with the joint position and the magnitude of joint bending stiffness.

Damage recovery in composite laminates through deep learning from acoustic scattering of guided waves

We propose an innovative deep learning (DL) regression strategy combined with guided wave modes to address inverse acoustic scattering problems effectively. This approach allows for accurate recovery of heterogeneous defect fields at the interfaces of composite laminates. The neural network (NN) model’s training process employs stochastic Gaussian fields as output, which are linked to the interfacial defect fields of the physical problem. Our method assumes prior knowledge of the material geometrical properties of the constituent layers. To model the interfaces, we utilize the Quasi-Static Approximation, a technique generating position-dependent interfacial stiffness matrices containing uncoupled normal and tangential springs. We validate our approach by assessing its performance in handling noisy input data and reduced models, as well as accounting model errors at the composite interface. The obtained results show that the proposed method has a remarkable generalization capability, allowing it to recover diverse defect field profiles with accuracy. Moreover, it exhibits robustness concerning noisy data and model errors. Lastly, thanks to the guided wave modes approach, the presented methodology not only maintains its capability to recover heterogeneous defect fields potentially in real-time but also extends the range of inspection to encompass a significantly larger structural area.

Free in-plane vibration of irregular laminated plate with curved edges based on boundary-type Chebyshev–Ritz method

The free in-plane vibration characteristics of laminated plate with arbitrarily geometrical shape and generally restrained edges, including curved and inclined edges, are investigated based on the boundary-type Chebyshev–Ritz method (BCRM). By adopting the divergence theorem, the double integral encountered in solving the energy equation of irregular laminates is converted to the integral on the boundary edge, which effectively reduces the computational complexity of the involved domain. The Chebyshev polynomials are utilized as test functions and the primitives of domain integrand functions are deduced according to the properties of Chebyshev polynomials. General boundary conditions on straight or even curved edges are achieved by introducing normal and tangential springs and assigning reasonable stiffness values. Natural frequencies and modes of laminated composite plates with straight edges (triangle, square, trapezoid, and hexagon), laminated composite plates with curved edge (ellipse), and irregular laminated composite plates are obtained. The method validation is carried out by comparison of the obtained solutions with literature and finite element results. The effects of variations in layering mode, number of layers, boundary conditions, and geometry on the eigenpairs of composite plate are extensively discussed.

A deep learning approach to inverse scattering analyses: Recovering interfacial defects in laminated structures

We propose a deep learning (DL) regression strategy to solve inverse acoustic scattering problems. This strategy is capable of recovering heterogeneous defect fields in any interface and direction of composite laminates. The training procedure of the neural network (NN) model uses stochastic Gaussian fields as output, which are related to interfacial damage fields of the physical problem. We assume prior knowledge of the material properties of the ultrasound incident field and the elastic layers’ properties. Furthermore, we model the interfaces using the Quasi-Static-Approximation, a method that generates position-dependent interfacial stiffness matrices, composed of a set of uncoupled normal and tangential springs. We validate our approach and evaluate its performance for noisy input data and reduced-order models. Additionally, we also consider model errors at the composite interfaces. The obtained results show that the presented method is promising due to its generalization capability in recovering different defect field profiles, its robustness in relation to noisy data and model errors and its low computational cost, once the DL model is trained, in comparison to traditional inverse approaches (iterative).

An improved model of refined zigzag theory with equivalent spring for mode II dominant strain energy release rate of a cracked sandwich beam

In this paper, an improved model, the refined zigzag theory with equivalent tangential spring (RZTES) in conjunction with the sub-laminate approach, for cracked sandwich beam (CSB) is developed. The kinematics of the refined zigzag theory (RZT) assumes that the axial displacements of a sandwich beam follow piecewise linear distribution across the thickness. This theory can accurately model the zigzag displacement in a sandwich beam without delamination. However, the kinematics of the RZT does not meet the deformations near the crack tip, which contain more complicated displacement distributions. It results in underestimations of both the compliance and mode II dominant strain energy release rate (SERR). The RZTES is developed to correct the inaccuracies due to this underestimation. In RZTES, an equivalent spring is inserted to decrease the local stiffness near the crack tip so that the compliance of the CSB is increased. Based on the procedure developed in this study, the equivalent spring constant can be determined according to the Young’s modulus and shear modulus of the core. Then a gap determined from the equivalent spring constant can be determined and be used to re-formulate the continuity conditions at the crack tip of a CSB. In order to validate RZTES, the CSB is calculated by commercial finite element method (FEM) software ANSYS with two-dimensional elements. The compliances are also validated by experiments. The results show that the RZTES can increase both the compliance and strain energy release rate. The case studies with various ranges of geometric parameters and material properties of the CSB are solved by RZT, RZTES and ANSYS. For the case with soft core or thick beam thickness, the calculation results by RZT need more corrections. By comparing RZT, both the compliances and SERRs calculated by RZTES are more accurate.

Using penalty function method in identification stiffness of elastic spring of the underground structure

This paper studies a method to identify the spring stiffness of the frame - soil structure. The model of the problem is a two-dimensional structure, linear elastic deformation, model of soil using the Winkler model (with normal, tangential spring). The problem will be solved by the penalty function method - the minimum of the objective function - combined with the finite element method. The numerical calculations show that the model, algorithm and calculation program are reliable. The program can be used to identify the spring stiffness of the frame - soil structure in two dimensions, serving to determine the actual working state of the structure, to propose solutions for reinforcement, repairing, improving bearing capacity, prolonging the life of the structure.

Mechanical Behavior Analysis of Fully Grouted Ground Anchor in Soft-Hard Alternating Stratum

Assuming that the ground anchor is connected with the rock–soil of the sidewall by a tangential linear spring, the load transfer model of the fully grouted ground anchor is established by using the spring element method, and the analytical solutions of the displacement, axial force, and shear stress distribution of the ground anchor in the upper and lower parallel strata foundation and sandwich foundation are given, respectively. Corresponding to the above two kinds of alternating strata, the mechanical behavior of the vertical fully grouted ground anchor in the soft–hard alternating stratum is analyzed using the four conditions in Case 1 and the six conditions in Case 2, respectively. Through the case analysis, it can be concluded that the mechanical behavior of the round anchor is greatly affected by the shear modulus of the shallow stratum, and is less affected by the shear modulus of the deep stratum. The depth of the stratum interface and the thickness of the interlayer have some influence on the mechanical behavior of the whole ground anchor but have little influence on the displacement and axial force distribution of the ground anchor. This paper has certain guidance and reference significance for the design of vertical fully grouted ground anchors in the alternating strata.

Mechanical Behavior Analysis of Fully Grouted Bolts under Axial Cyclic Load

Fully grouted bolts are widely used in engineering. In order to deeply understand the load-transfer mechanism of a fully grouted bolt, it is necessary to analyze and study its mechanical behavior under axial cyclic load. First of all, based on the idea of discretization and the force balance analysis of each mass spring element, this study proposes a method for analyzing the force of the bolt—the spring element method. Second, the load-transfer model of the fully grouted bolt is established by using the spring element method, assuming that the bolt and the sidewall rock and soil are connected by tangential linear springs. The analytical solutions for the displacement, axial force, and shear-stress distribution of the bolt before and after the damage of the sidewall spring are given. It is found that the analysis results of the analytical model proposed in this paper have a great relationship with λ, which is the square root of the ratio of sidewall spring stiffness k′u to bolt stiffness ku. Further analysis found that this model is more suitable for the two working conditions of λ ≈ 0 and λ ≈ 1, and the relationship between sidewall spring stiffness k′u and pull-out stiffness K of the bolt was established under these two working conditions. Finally, the rationality and accuracy of the analytical model proposed in this study are verified by an analysis of two typical test cases under the two working conditions of λ ≈ 0 and λ ≈ 1.

A variationally consistent hyperstatic reaction method for tunnel lining design

AbstractIn this technical note, a consistent finite element formulation of the Hyperstatic Reaction Method (HRM) for tunnel linings design is proposed by introducing a variational consistently linearized formulation. It permits to consider a nonlinear interaction between a lining structure and the surrounding ground. Recent advances of the HRM in regard to the consideration of the nonlinear response of the segmented tunnel lining exposed to design loads use an iterative algorithm for solving the nonlinear system of equations. In the proposed Variationally consistent Hyperstatic Reaction Method (VHRM), a distributed nonlinear spring model representing the interaction between the lining and the ground soils is considered in a variationally consistent format. Computing the tangential spring stiffness via consistent linearization, and using Newton‐Raphson iteration, requires significantly smaller number of iterations as compared to the original HRM model based on nodal springs. Furthermore, the method is applicable for simulations using solid finite elements (2D and 3D), as well as beam or finite shell elements, respectively.

Modeling of FRCM strengthening systems externally applied on curved masonry substrates

Aim of the present paper is the proposal of a simple modeling approach for the numerical study of the bond behavior of Fiber Reinforced Cementitious Matrix (FRCM) reinforcements externally applied to curved substrates. The model consists of a schematization of the specimen strengthened by FRCM throughout spring elements representing both the substrate, matrix and reinforcement, and, moreover, the reinforcement-matrix interface layers. Regarding the latter, both tangential and normal behavior is opportunely modeled by separated springs in order to reproduce phenomena related to the mode I and mode II behavior. A further important feature of the proposed model is the coupling between normal and tangential springs in order to consider the effect of normal stresses arising at the reinforcement/matrix interface because of the geometric curvature of the substrate on the de-bonding behavior of the reinforcing system. Numerical analyses are presented in the paper with reference to experimental case studies deduced from literature. In particular, the authors propose a parameter setting procedure based on the experimental outcomes and some theoretical considerations.

Modeling Complex Contact Conditions and Their Effect on Blade Dynamics

Abstract Dry friction devices such as underplatform dampers are commonly included in turbine bladed disks designs to mitigate structural vibrations and avoid high cycle fatigue failures. The design of frictionally damped bladed disks requires adequate models to represent the friction contact. A widely used approach connects contact node pairs with normal and tangential springs and a Coulomb friction law. This simple model architecture is effective in capturing the softening behavior typically observed on frictionally damped structures subjected to increasing forcing levels. An unexpected hardening behavior was observed on the frequency response functions (FRFs) of a two-blades-plus-damper system tested by the authors in a controlled laboratory environment. The reason behind this unexpected behavior will be carefully analyzed and linked to the damper kinematics and to the dependence of contact elasticity on the contact pressure. The inadequacy of contact models with constant spring values will be discussed and alternatives will be proposed. The importance of being able to represent complex contact conditions in order to effectively predict the system dynamics is shown here using a laboratory demonstrator; however, its implications are relevant to any other case where large contact pressure variations are to be expected. The nonlinear steady-state simulations of the blades-plus-damper system will be carried out using an in-house code exploiting the multiharmonic balance method in combination with the alternating frequency time method.

On the effect of the loading apparatus stiffness on the equilibrium and stability of soft adhesive contacts under shear loads

The interaction between contact area and frictional forces in adhesive soft contacts is receiving much attention in the scientific community due to its implications in many areas of engineering such as surface haptics and bioinspired adhesives. In this work, we consider a soft adhesive sphere that is pressed against a rigid substrate and is sheared by a tangential force where the loads are transferred to the sphere through a normal and a tangential spring, representing the loading apparatus stiffness. We derive a general linear elastic fracture mechanics solution, taking into account also the interaction between modes, by adopting a simple but effective mixed-mode model that has been recently validated against experimental results in similar problems. We discuss how the spring stiffness affects the stability of the equilibrium contact solution, i.e. the transition to separation or to sliding.

Elastic-slip interface effect on dynamic response of underwater convey tunnel in saturated poroelastic soil subjected to plane waves

In engineering construction, the elastic-slip interface often exists around the tunnel. An elastic-slip interface model is proposed to analyze the dynamic response of underwater convey tunnel, and the surrounding soil is assumed to be saturated poroelastic medium. The reflected, refracted and scattered waves in the free surface water, the surrounding soil medium and the tunnel are expressed by wave function expansion method. Based on Biot’s dynamic theory, an elastic-slip interface is developed to analyze the scattering of P and SV waves around the tunnel. The interface coefficients (normal and tangential spring coefficients and slip coefficient) are introduced to analyze the elastic-slip interface effect on the dynamic stresses under different wave frequencies. It can be concluded that the effect of normal spring coefficient is greater than that of tangential spring coefficients. The interface effect in the region of low frequency is greater than that of high frequency. The interface effect under different surface water depths and embedded depths of tunnel is also examined. Comparison with existing numerical results validates this dynamic model.

Dynamic characteristics of a shrouded blade with impact and friction

A simplified computational model of a twisted shrouded blade with impact and friction is established. In this model, the shrouded blade is simulated by a flexible Timoshenko beam with a tip-mass, and the effects of centrifugal stiffening, spin softening, and Coriolis force are considered. Impact force is simulated using a linear spring model, and friction force is generated by a tangential spring model under sticking state and a Coulomb friction model under sliding state. The proposed model is validated by a finite element model. Then, the effects of initial gap and normal preload, coefficient of friction, and contact stiffness ratio (the ratio of tangential contact stiffness to normal contact stiffness) on system vibration responses are analyzed. Results show that resonant peaks become inconspicuous and impact plays a dominant role when initial gaps are large between adjacent shrouds. By contrast, in small initial gaps or initial normal preloads condition, resonant speed increases sharply, and the optimal initial normal preloads that can minimize resonant amplitude becomes apparent. Coefficient of friction affects the optimal initial normal preload, but it does not affect vibration responses when the contact between shrouds is under full stick. System resonant amplitude decreases with the increase of contact stiffness ratio, but the optimal initial normal preload is unaffected.

The Effects of Tangential Ground–Lining Interaction on Segmental Lining Behavior Using the Beam-Spring Model

The shield tunneling method is widely used, especially in urban areas, since it is efficient for minimizing disturbances to surroundings. Although segmental lining is commonly used in this method, in both the research and practice of tunnel lining design, the interaction between the ground and lining in the tangential direction remains unclear; that is, the mobilizing shear stress due to load models and the degree of the bond in the tangential direction. Therefore, to clarify the effects and mechanism of the tangential ground–lining interaction on segmental lining behavior, a parameter study was carried out, taking tangential spring stiffness, load models, soil stiffness, and shallow and deep tunnels as parameters. The interaction conditions were based on the existing literature. It was found that (1) the tangential spring has small effects on lining behavior, (2) the load model significantly affects the sectional forces, (3) the initial tangential earth pressure and slip ground–lining boundary provide more safety from a design viewpoint, and (4) in the case of shallow tunnels in soft ground, tensile stress appears in the lining. Therefore, it is important to take the tangential ground–lining interaction conditions into consideration during tunnel lining analysis.

A new contact potential based three-dimensional discontinuous deformation analysis method

The three-dimensional discontinuous deformation analysis (3D-DDA) method was developed for the deformation simulation of rock block system cut by the natural discontinuities in rock mass engineering. In the conventional DDA, open-close iteration is used to deal with the contact constraints, which needs to apply or remove the normal or tangential springs repeatedly to meet the equilibrium equations at each time step. DDA provides a time step adjustment strategy to meet the fast convergence of open-close iterations, but when solving large-scale problems, the adjusted time step often reaches a very small order of magnitude, which makes the calculation time-consuming increase sharply. In the framework of the original 3D-DDA, a new contact potential based three-dimensional discontinuous deformation analysis method (3D-CPDDA) is developed. The proposed method not only retains the advantage of the original DDA method in defining local displacement functions on a single patch, but also integrates the simplicity and rapidity of potential based contact processing. The improved method is easier to be implemented in the parallel way, which can further improve the computational efficiency. Numerical examples have confirmed the correctness and feasibility of the proposed procedure.

Effect of contact force modeling parameters on the system hydrodynamics of spouted bed using CFD-DEM simulation and 2k factorial experimental design

In this study, the effects of contact force modeling parameters, which included the particle–particle friction coefficient (A), spring constant (B), ratio of the tangential spring constant to normal spring constant (C), normal restitution coefficient (D), and tangential restitution coefficient (E) presenting in the linear spring-dashpot contact model on the hydrodynamics profile of spouted bed reactor, were investigated via computational fluid dynamics coupled with discrete element method (CFD-DEM) using Multiphase Flow with Interphase eXchange (MFIX). First, the base case simulation was validated with the experimental data. Next, the 2k factorial experimental design and an analysis of variance (ANOVA) were conducted to identify the significant main and interaction contact force modeling parameters. Four response variables were observed: the translation kinetic energy of particles, the rotational kinetic energy of particles, the bed expansion, and the standard deviation of pressure drop. Based on these results, the particle–particle friction coefficient, spring constant, and normal restitution coefficient were the major contact force modeling parameters that affected the system hydrodynamics in the spouted bed system. In addition, the effects of main and interaction contact force modeling parameters were summarized, which enhanced our knowledge of the modeling parameters on system hydrodynamics. Contact force modeling parameters must be carefully selected when simulating the spouted bed reactor or related gas-solid multiphase flow process, such as fluidized bed reactor.

Effective spring boundary conditions for modelling wave transmission through a composite with a random distribution of interface circular cracks

In the present study frequency dependent effective spring boundary conditions are formulated to simulate wave propagation through an interface with distributions of micro-cracks of various sizes. The frequency dependent formulae for spring stiffnesses are obtained for circular cracks employing the boundary integral equation method, the ensemble averaging technique and Betti’s reciprocity theorem. The frequency dependent analytical relations for normal and tangential spring stiffnesses are expressed in terms of the elastic properties of contacting materials and properties of distributed cracks. In the case of equally sized cracks, the formulae for stiffnesses have an analytical form. The provided analysis of the influence of frequency on spring stiffnesses shows that for each crack size a number of sharp and narrow peaks can be observed in certain frequency ranges. The novelty of this study also lies in the consideration of different sized interface cracks. The influence of the standard deviation of radii of circular cracks on spring stiffnesses is analysed using the lognormal distribution. It is shown that there is a relatively low influence of size variation on spring stiffnesses, if the standard deviation in the distribution is not extremely large.

Wave Analysis of Planar Deployable Structures with Revolute Clearance Joints Based on Spectral Element Method

Clearance is inevitable in the deployable mechanisms due primarily to the kinematic function requirements. This phenomenon affects the dynamic performances of deployed structures negatively. In this paper, the wave analysis of dynamic characteristics of planar structures with revolute clearance joints is developed by spectral element method. First, the spectral element model of revolute clearance joints is established. The radial and tangential springs and damping coefficients of revolute clearance joints are evaluated based on the contact model of elastic foundation. Then, the wave equations of two beams connected by a revolute clearance joint are derived, and extended to the case of multiple beams connected by revolute clearance joints. Finally, the dynamic response is analyzed for planar structures with single revolute clearance joint and multiple revolute clearance joints under the impact load. The wave propagation rules in planar structures with revolute clearance joints are revealed.