Equivalent Nodal Forces Research Articles

Study of the dynamic interactions between helix-stiffened cement mixing piles and soft soil under oblique incidence of SV waves in arbitrary space

Helix stiffened cement mixing (HSCM) piles are extensively utilized in soft soil sites due to their excellent seismic and pullout resistance. Initially, the equivalent nodal force formulas for each face under the oblique incidence of SV waves in any spatial direction are derived based on the viscous-spring artificial boundary. The method's applicability in studying pile–soil interactions is verified based on existing pile–soil tests. Subsequently, the HSCM pile–soil model is constructed using finite element software through a Python program. Various analyses of accelerations, relative displacements, soil reactions, and pile–soil dynamic p-y curves are conducted using two different directions of incidence angles (horizontal and vertical) and various types of seismic waves. Subsequently, the backbone line of the dynamic p-y curve is plotted, revealing the pile–soil interaction mechanism under spatial oblique incidence. Finally, Matlock's formula is modified to ensure its applicability to pile–soil interactions under spatially oblique incidence. The results demonstrate that loading the Kobe wave causes the peak acceleration in the x direction, the peak pile–soil relative displacement, and the peak pile-side soil reaction force to increase by 22.64 %, −18.90 % and 18.42 %, respectively, compared to those of the El Centro wave.

Longitudinal Seismic Response of Metro Tunnels Crossing a Fault with Multi-Slip Surfaces

There are multiple seismic fault zones near Xi’an in China, among which the Li Piedmont Fault has multiple slip surfaces. A 3D finite element dynamic Soil–Fault–Tunnel model was established based on the engineering background of the Xi’an Metro tunnel orthogonally crossing the Li Piedmont Fault. The input seismic loads used the Chi-Chi, El-Centro, and artificial seismic waves, and the latter was artificially synthesized based on seismic conditions and site conditions of actual engineering. The Chi-Chi seismic wave with larger PGV/PGA and wider acceleration-sensitive area is a near-field seismic wave, while the others are far-field seismic waves. The seismic loads were transformed into the equivalent nodal force on the boundary nodes of the model, and nonlinear dynamic calculation was carried out to obtain the longitudinal seismic response of the structure. The main results show that the fault amplifies the seismic response of the tunnel, and the tunnel at the position of the fault slip surfaces is more vulnerable to failure, especially near the slip surface between the hanging wall and the fault, where the maximum acceleration, soil pressure, and internal force of the tunnel structure occur. In addition, the seismic response of the tunnel and soil caused by near-field seismic motion is significantly stronger than that caused by far-field seismic motion.

Numerical investigations on the dynamic responses of metro tunnel-soil-ground building under oblique incidence of seismic wave

With the development of urban rail transits, the metro tunnels frequently underpass the above-ground buildings, and understanding the dynamic responses of metro tunnel-soil-ground buildings experiencing oblique incidence of seismic waves is of crucial importance. Considering that the seismic waves in the near-field seismic area are mostly obliquely incident to the ground at a certain angle, a two-dimensional numerical model for tunnel-soil interaction system and tunnel-soil-above ground structure was established by Abaqus software, in which the viscoelastic boundary and equivalent nodal forces were adopted to simulate the ground motion. The effects of tunnel burial depth, tunnel spacing, and tunnel frame spacing on the seismic responses of metro tunnel-soil and metro tunnel-soil-above ground building were analyzed under oblique incidence of SV wave with different incident angles. It was found that the tunnel depth has a significant effect on the lining dynamic bending moment and axial force peak, and its influence is closely related to the site category. The tunnel spacing has a minor impact on the dynamic response of the lining structure. The incident angle has a significant effect on the peak value of the dynamic shaft force of the lining. The application of the frame on the ground has a significant impact on the tunnel. The influence of above-ground buildings and tunnel burial depth should be considered when tunnel design is carried out.

Modeling holes and voids in three dimensions using a single element within the Extended Finite Element framework

This paper introduces a highly efficient computational approach using the extended finite element method to model three-dimensional mechanics of holes and voids. The defect is encapsulated within a single element with a tailored exponential enrichment function that captures the behavior near the discontinuity boundary and can account for the pressure acting on the void’s surface through an equivalent nodal force vector. The method’s results were compared against analytical solutions and conventional finite element method results for problems involving single and multiple holes. The comparison demonstrated that the method provides accurate stress concentration estimates while adhering to computational constraints.

On the frequency and dynamic analyses of complex cylindrical shells submerged in fluid: An axisymmetric FEM/BEM method

For the ribbed and variable-thickness cylindrical shell with endplates in the fluid, this paper develops an efficient and applicable axisymmetric FEM/BEM method for frequency and dynamic analyses of such a structure, which is widely used in the aerospace, civil, and ocean engineering. Axisymmetric finite elements are employed for the discretization of cylindrical shells, endplates, and ribs to obtain the mass and stiffness matrices of the structure. Accordingly, axisymmetric boundary elements are developed for the fluid pressures on the structural surface. And they are transformed into the equivalent nodal forces to obtain the fluid-added mass matrix and the fluid–structure coupling equations of the system. The frequencies and modes are calculated by eigenvalue analysis. In the presented method, only the displacements on the structural meridians need to be solved, effectively reducing the computational effort. Since the mass and stiffness matrices and the system modes are obtained, the modal superposition method is used to solve the dynamical response of the system subject to force excitation, so that the complicated solution for fluid forces is avoided. The results show that the method is accurate and efficient and has significant engineering potential.

Machine learning‐based predictions of buckling behaviour of cold‐formed steel structural elements

AbstractDesigning thin‐walled structural members is a complex process due to the possibility of multiple instabilities. This study aimed to develop machine learning algorithms to predict the buckling behavior of thin‐walled channel elements under axial compression or bending. The algorithms were trained using feed‐forward multi‐layer Artificial Neural Networks (ANNs), with the input variables including the cross‐sectional dimensions, the thickness, the presence and location of intermediate stiffeners, and the element length. The output data included the elastic critical buckling load or moment, as well as a modal decomposition of the buckled shape into the pure buckling mode categories: local, distortional and global buckling. The Finite Strip Method (FSM) and the Equivalent Nodal Force Method (ENFM) were used to prepare the sample output for training. To ensure the accuracy of the developed algorithms, the ANN models were subjected to a K‐fold cross‐validation technique and featured optimized hyperparameters. The results showed that the trained algorithms had a remarkable accuracy of 98% in predicting the elastic critical buckling loads and modal decomposition of the critical buckled shapes.

Prestress design and geometric correction method of cable–truss structures based on equivalent equilibrium force model

Cable–truss structure is a typical form of cable–strut tensile structures, which consists of upper cable system, lower cable system and several vertical struts. As with the general cable–strut tensile structure, only when the reasonable geometric shape matches the feasible prestress distribution can the cable–truss structure obtain structural stiffness and bear the external load. Once the geometric shape is unreasonable, the cable–truss structure is just a mechanism without bearing capacity and cannot be stiffened by introducing prestress. Consequently, a reasonable geometry is the premise for cable–truss structure to become an effective load-bearing system. In this paper, an equivalent equilibrium force model, called EEFM, is proposed according to the geometric-mechanical characteristics of cable–truss structures. A cable–truss structure in this model is split into upper and lower cable systems with equivalent nodal forces. The geometric rationality can be easily evaluated from the equivalent nodal force relationship. According to the geometric topology and equilibrium equation of two cable systems, the self–stress mode of the structure with reasonable geometry can be directly obtained. For structures with unreasonable geometry, the node coordinates of one cable system are corrected based on the other cable system. Several cases are presented to verify the accuracy and validity of the proposed method. This method can be used for shape design, geometric correction and force finding, and has greater advantages for cable–truss structures with asymmetric or complex space configuration.

Seismic damage characteristics of high arch dams under oblique incidence of SV waves

This study investigates the seismic damage characteristics of high arch dams considering the oblique incidence of seismic waves. Based on the dynamic implicit finite element method, a viscous-spring artificial boundary is applied to consider radiation damping of the infinite foundation, the seismic wave input is converted into an equivalent nodal force of the artificial boundary, and the equivalent nodal loads of the wave input under oblique incidence of the SV wave is deduced. Taking JP Ⅰ arch dam as an example, combined with the concrete damaged plasticity (CDP) model, the nonlinear dynamic response of the dam under 25° oblique incidence of the SV wave is conducted considering the geometrical nonlinearity of contraction joints and the dynamic mechanical properties of concrete materials. Based on the incremental dynamic analysis (IDA) method, the relative displacement and transverse joint opening are selected as the damage measures for fragility analysis of the arch dam. According to the damage distribution from the analysis of a large number of seismic time-history responses, the damage levels of the arch dam are classified. Three limit states of slight damage, moderate damage and severe damage are proposed for the arch dam. The average responses of the 10 earthquakes are considered as the limit values to quantitatively describe the damage of the arch dam. By fitting the results of incremental dynamic analysis, the seismic probabilistic demand model and seismic fragility curves for the two damage measures to predict the probability of the arch dam reaching damage levels under different seismic actions have been established. The fragility curve shows that the probability of severe damage is almost zero under the maximum design earthquake (PGA = 0.1971 g) and less than 5% under the maximum credible earthquake (PGA = 0.2299 g). It is concluded that JP Ⅰ arch dam has a high degree of safety redundancy and fully satisfies the “two levels” design requirements.

Loading–shape interaction analysis of membrane structures subjected to deformation dependent hydrostatic loadings

Due to their flexibility, membrane structures loaded and/or supported by gas and/or fluid undergo large deformations when subjected to external loads, for example when a relatively flat membrane deflects under accumulative ponding. The most noteworthy feature during the deformation process is the loading–shape interaction since changes in the form of the structure induce changes in the loading profiles, and vice versa. Description of the hydrostatic loading is complex since the discontinuity of structural domain and fluid domain occurs with the deformation of membrane. In this paper, an interactive model dealing with accurate hydrostatic load description and nonlinear finite element analysis is employed to analysis the structural response. The hydrostatic loading is described in vectors by fluid level and loading direction in current mesh of the membrane. In case to conserve a specific volume, the free surface height is calculated with a Newton algorithm. Equivalent nodal forces evaluation on different elements are deduced especially when they are partially or fully loaded by judging the spatial relation between the element and the fluid domain, and integrating with shape functions and area coordinates in a local coordinate system. In case of multiple fluid domains, the hydrostatic load description procedure is proposed to be conducted for each fluid domain with care in identifying respective loading direction and then sum them up to get the total hydrostatic loading forces. Two numerical examples, (a) single fluid domain case: a flat membrane subjected to accumulative water loading, and (b) multiple fluid domains case: an air-inflated dam partially filled with water and loaded by lateral pressures of water, are presented to demonstrate correctness, performance and effectiveness of the proposed method in large displacement analysis of flexible membrane structures subjected to deformation dependent hydrostatic loading.

A 2.5D Finite Element Method Combined with Zigzag-Paraxial Boundary for Long Tunnel under Obliquely Incident Seismic Wave

Seismic waves propagation with an oblique angle to the tunnel axis will cause asynchronous tunnel motions and have a significant effect on the axial response. A high-precision 2.5D finite element method is established in the frequency domain to simulate the 3D seismic response of the tunnel. This method avoids the disturbance caused by the truncation of the tunnel in the longitudinal direction. Meanwhile, a 2.5D zigzag-paraxial boundary is derived to further improve the calculation efficiency of the 2.5D finite element model. Moreover, by combining the 2.5D finite element method, 2.5D zigzag boundary condition and seismic motion input methods, an obliquely incident substructure method for plane seismic waves is built by converting the plane seismic wave into equivalent nodal forces. The proposed 2.5D finite element method is verified by comparing with a reference solution. Finally, the 2.5D finite element method is applied to study the seismic response of the long lined tunnel. Parameter analyses illustrate that the wave propagation effect to the tunnel axis has a non-negligible influence on the axil deformation of long tunnels.

Effect of incident angle of P-wave on seismic response of underground cavern

This study investigates the effect of the P-wave incident angle on the seismic response of a rock mass surrounding an underground cavern in a hydropower station. Based on the viscoelastic artificial boundary theory and analysis of the incident and reflected states of P-waves at any angle in a semi-infinite medium, a comprehensive and effective analytical model of the oblique incidence of seismic waves is established. In the analysis model, conversion of the obliquely incident seismic wave time history and equivalent nodal force was implemented using the commercial software ABAQUS; the reliability of the method was verified using a numerical case. Using this method, the seismic responses of the rock mass surrounding an underground cavern under the action of incident P-waves at different angles were studied in terms of stress, displacement, and plastic damage. The results showed a significant effect of the oblique incidence of seismic waves on the dynamic response of the underground cavern.

Frequency domain-based analytical framework for seismic performance of viscously damped outrigger systems based on continuous Timoshenko beam theory

This paper proposes a frequency domain-based analytical framework for seismic performance of viscously damped outrigger systems. Based on a core–outrigger–damper–column simplified model, the global dynamic stiffness matrix is assembled from the core modeled as a Timoshenko beam, damped outriggers as complex rotational stiffness comprising outriggers, dampers, and perimeter columns, and inherent damping using Leung’s theory and modal damping construction. A general numerical method combining Wittrick-Williams algorithm and Newtonian iteration is developed to study the dynamic characteristic of such systems with multiple damped outriggers. The fast Fourier transformation (FFT) and the inverse Fourier transformation (IFFT) are then integrated with the principle of potential energy to obtain the equivalent nodal force and thus the time history response by transformations between time and frequency domains. Finally, the stochastic analysis is conducted via the transfer function resulting from the global stiffness matrix with the stochastic seismic excitation following Kanai-Tajimi spectrum. The proposed approach is verified by comparison with the finite element method through a case study of a tall building implemented with viscously damped outriggers. This study shows that the proposed analytical framework could serve as a powerful tool for evaluating the performance of viscously damped outrigger systems.

Predicting the buckling behaviour of thin-walled structural elements using machine learning methods

The design process of thin-walled structural members is highly complex due to the possible occurrence of multiple instabilities. This research therefore aimed to develop machine learning algorithms to predict the buckling behaviour of thin-walled channel elements subjected to axial compression or bending. Feed-forward multi-layer Artificial Neural Networks (ANNs) were trained, in which the input variables comprised the cross-sectional dimensions and thickness, the presence/location of intermediate stiffeners, and the element length. The output data consisted of the elastic critical buckling load or moment, while also providing an immediate modal decomposition of the buckled shape into the traditionally defined ‘pure’ buckling mode categories (i.e. local, distortional and global buckling). The sample output for training was prepared using a combination of the Finite Strip Method (FSM) and the Equivalent Nodal Force Method (ENFM). The ANN models were subjected to a K-fold cross-validation technique and the hyperparameters were tuned using a grid search technique. The results indicated that the trained algorithms were capable of predicting the elastic critical buckling loads and carrying out the modal decomposition of the critical buckled shapes with an average accuracy (R2-value) of 98%. The influence of the various channel parameters on the output was assessed using the SHapley Additive exPlanations (SHAP) method.

Seismic response analysis of slope sites exposed to obliquely incident P waves

This study proposes a seismic input method for layered slope sites exposed to obliquely-incident seismic waves which transforms the waves into equivalent nodal forces that act on the truncated boundary of a finite element model. The equivalent nodal forces at the left and right boundaries are obtained by combining the free field response of a one-dimensional layered model with a viscoelastic boundary. The equivalent nodal forces at the bottom boundary are obtained by combining the incident wave field with the viscoelastic boundary. This proposed seismic input method for slope sites exposed to obliquely incident seismic waves is implemented with the aid of MATLAB software; it is applied to the seismic response analysis of slope sites in the commercial finite element ABAQUS software. The calculation results are compared with the reference solutions obtained by using the extended model to verify the correctness of the established seismic input method. The proposed seismic input method is then employed to investigate the influencing factors of the seismic response of layered slope sites exposed to oblique incidence P waves. The results show that the angle of incidence, location of the interface between soft and hard rocks, and impedance ratio have significant effects on the seismic landslide.

Spectral wave explicit weak-scattering approach for higher harmonic wave loads and ringing response of flexible monopile

This paper presents an accurate potential decomposition strategy for wave–structure interactions, known as the spectral wave explicit weak-scattering approach, where the entire original problem is divided into incident and scattered components. The incident component is evaluated using spectral models in advance; hence, only the scattered component is to be solved via a higher-order boundary element approach. The proposed coupled model is validated using published experimental and computational fluid dynamics results for wave interactions with a vertical cylinder. The results obtained are consistent with wave elevations and wave loads for regular waves and focused wave groups. A finite-element numerical model is applied to measure the severity of structural responses. Fluid–structure interactions are simulated by converting the obtained pressures into equivalent nodal forces, and new nodal velocities are converted into boundary conditions. The ringing response of a bottom-fixed monopile due to moderately steep focused wave groups is discussed based on various focal positions relative to the monopile, incident wave conditions, and damping ratios. The time–frequency characteristics of the ringing phenomenon are successfully revealed via wavelet analyses.

Vibroacoustic response of a high-speed train extruded profile under fluid-induced vibration

The wall pressure fluctuation caused by a turbulent boundary layer is an important source of noise and vibration in high-speed trains. Considering that railway vehicles are usually made of extruded profiles, a simple and reliable model is needed to predict the vibroacoustic characteristics of extruded profiles excited by a TBL in the early stages of design. A hybrid finite element and statistical energy analysis model is established for computing the vibroacoustic response of finite-sized panels excited by a TBL. In the hybrid model, the wall pressure fluctuation of the TBL is modelled in the wavenumber domain. The panels are modelled using FEs, and acoustic waves in the fluid space below the panels are modelled using SEA. The TBL wall pressure acting on the top surface of the panels is resolved into equivalent nodal forces with discretization of the structure using FEs, achieved by calculating the cross-spectrum matrix of the wall pressure in the spectral domain and using jinc functions with smooth boundary and symmetric properties as shape functions. The reciprocity between direct and reverberant fields couples the acoustic fields to the structure. Once the response of the hybrid model is determined, sound transmission through the panels can be calculated. By following the procedures outlined in this paper, the range of wall pressure model selection for TBL has extended and reduced computational time and provides a basis for the acoustic optimization design of extruded profiles.

Influence of multiple random variables on the safety of cracked tunnels under earthquake action

To explore the mechanical properties of cracked lining subjected to seismic loads based on wave theory and the extended finite element method, the dynamic viscoelastic boundary of a seismic wave and equivalent nodal force load were generated by MATLAB programming software to establish a simulation model of cracked lining structure. The internal force state change law of the lining structure was studied by varying the crack depth, crack length, secondary lining thickness, and other parameter values (including the layout between multiple cracks). Also, the safety factor of the lining crack section was obtained, and the functional relationship between the safety factor and parameter variables was established. Results show that the crack depth and secondary lining thickness were the main factors affecting the internal force of the crack section. Based on the least square method, the calculation formula and 95 % confidence interval between the minimum safety factor (Kmin) and each parameter of the crack section were obtained. Meanwhile, the Kmin prediction model was obtained via multiple nonlinear regression. When the crack depth value was 30 % of the lining thickness value, Kmin reduced 2.5 times. At 57 % crack depth of the lining thickness, the Kmin was less than the specification value, indicating that the lining structure’s safety reserve was low. Compared with the arrangement of the vertical distribution of the two cracks, the stress concentration generated when the two cracks were arranged in parallel would more likely affect the structure adversely. The findings can provide a reference for the safety study of cracked tunnels.

Efficient formulation of a two‐noded geometrically exact curved beam element

AbstractThe article extends the formulation of a 2D geometrically exact beam element proposed by Jirásek et al. (2021) to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first‐order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The article develops the theoretical framework based on the Navier–Bernoulli hypothesis, with a possible extension to shear‐flexible beams. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant. The efficiency and accuracy of the developed scheme are documented by seven examples that cover circular and parabolic arches, a spiral‐shaped beam, and a spring‐like beam with a zig‐zag centerline. The proposed formulation does not exhibit any locking. No excessive stiffness is observed for coarse computational grids and the distribution of internal forces is not polluted by any oscillations. It is also shown that a cross effect in the relations between internal forces and deformation variables arises, that is, the bending moment affects axial stretching and the normal force affects the curvature. This coupling is theoretically explained in the Appendix.

Bayesian-based method for the simultaneous identification of structural damage and moving force

The input force is usually difficult to measure directly in the damage identification due to the system uncertainty and noise uncertainty. A Bayesian method for the simultaneous identification of damage and moving force is proposed. The moving force on the structure are transformed into equivalent nodal forces through the element function, and these equivalent nodal forces are expanded by using Chebyshev orthogonal polynomials. The posterior probability density function of the damage parameter and the force orthogonal parameter under Bayesian theory is deduced in time-domain. The prior information of the Chebyshev orthogonal polynomials adopts the uniform prior, and the damage parameters adopt the Gaussian prior in accordance with the characteristics of each parameter. The Laplace approximation method for time domain data is used to determine the hyperparameters of elemental damage and moving force. The first-order and second-order sensitivities of the dynamic response with damage and force parameters are derived to update the hyperparameters and to accelerate gradient-based system identification. The uncertainty of the identified results of damage parameters is quantified by using an approximate Gaussian distribution. Two numerical cases of a simply supported beam and the Tanjiang Bridge on the Shen-Mao Railway show that the proposed method can simultaneously identify the elemental damage and moving forces with uncertainty quantified.

Seismic Responses and Damage Mechanisms of Deep-Buried Underground Powerhouse in Layered Rock Mass

Layered rock mass generally has adverse effects on the seismic stability of deep-buried underground powerhouses. Aiming at the complexity of seismic wave field in deep-buried underground powerhouses, the obliquely input methods of P and SV waves considering the incident direction and multi-incident surfaces are constructed in this study, respectively. It can convert the seismic wave input problem into the problem of solving equivalent nodal force acting on the artificial boundaries. Based on the characteristics of dynamic interaction between interlayers in layered rock mass, an explicit dynamic contact force method considering the seismic deterioration effect and bond-slip characteristics of interface is also presented to simulate various contact states such as bond, separation, and sliding. The combined application of the above methods to analyze the seismic response of underground powerhouse at Azad Pattan hydropower station shows that they could accurately simulate the seismic damage evolution process of deep-buried underground powerhouses in layered rock mass. The numerical results indicate that the obliquely incident seismic waves contribute to a larger seismic reaction of underground powerhouses, which largely lies in the amplitudes of the displacement and stress fluctuations. After considering the dynamic contact, the obvious seismic deterioration effect and interlaminar dislocation displacement occur between soft and hard rock, and the sidewalls suffer more severe deterioration degree than the top arch. The seismic damage area of lining structure is mainly distributed in the place where the soft rock strata pass through and the upper structure with larger free surface. Additionally, two major damage modes of underground powerhouses in layered rock mass, namely, damage due to structural deformation and interlaminar dislocation, are derived from the numerical results and can be reasonably explained by the corresponding damage mechanisms.