Principle Of Minimum Potential Energy Research Articles

Static and Transient Response Analysis of Arbitrary Laminated Composite and Sandwich Shells by the FSDT Meshless Method

This paper, for the first time, presents a novel meshless method based on the first-order shear deformation theory (FSDT) and moving-least squares (MLS) approximation to carry out static and transient response analysis on arbitrary laminated and sandwich shell structures. The novelty of the method lies in the implementation of MLS in the curvilinear shell formulation, and the derivation of meshless governing equations on the static and transient response of arbitrary laminated and sandwich shells according to the principle of minimum potential energy and Hamilton’s principle, respectively. The concept of a Convected coordinate system is adopted to deal with arbitrary curvilinear surfaces. Both the position and displacement vectors of shells are approximated by MLS, which is conceptually the same procedure as that in the isoparametric finite element method (FEM). The numerical integration of the stiffness matrices is conducted by the simple Gaussian integral in the Convected coordinate system. Because the moving-least squares (MLS) approximation does not satisfy the Kronecker delta condition, the full transformation method is used to modify the meshless governing equations. Numerical examples are calculated to investigate the static and transient response analysis of laminated and sandwich structures with different geometrical shell shapes, including square plates, spherical shells, doubly-curved shells, and cylindrical shells. The numerical tests show that the proposed method is reliable, and robust and produces accurate results compared to the analytical and numerical solutions taken from the previous literature.

Minimum energy principles in electrostatics

Abstract We employ the principle of minimum energy of the electrostatic field to derive Laplace's equation for electric potential in charge-free space. We also apply the principle of minimum electrical potential energy of a system of discrete charges to explain the possibility of the existence of their stable equilibrium configuration on the surface of a conductor.

New Insight into Quantity-Dependent Self-Assembly Pattern of Light Levitating Droplet Clusters.

It has been reported that the self-assembly pattern of light levitating droplet clusters above the hot gas-liquid interface is dependent on the quantity of droplets. However, the already-reported theoretical explanation of the quantity-dependent self-assembly pattern cannot work well when the quantity of the light levitating droplet exceeds 15. Herein, we propose a new theoretical perspective to understand the self-assembly of a light levitating droplet cluster by referring to the classical densest packing problem of identical rigid circles in a larger circle with the introduction of the minimum total potential energy principle. Amazingly, the theoretical results obtained by this new approach agree well with experimental results, even though the quantity of the light levitating droplet is up to 142. This study deepens our understanding of the quantity-dependent self-assembly pattern of the light levitating droplet clusters and provides significant inspiration for other analogous self-assembly phenomena.

Design of piezoelectric quasi— zero—stiffness metastructures for improved low—frequency vibration isolation

Abstract In this paper, a novel type of piezoelectric quasi–zero stiffness (QZS) metastructure is proposed and the improved vibration isolation performance is investigated. A piezoelectric QZS metastructure is composed of curved beams covered with piezoelectric macro–fiber composite patches, to which digital circuits are connected. Based on the principle of minimum potential energy and the mode superposition method, the constrain on the geometry and material parameters of the piezo–curved beams to achieve QZS is given. According to this design criterion, 1D, 2D and 3D piezoelectric QZS metastructures are designed, and their improved low–frequency vibration isolation characteristics are comprehensively analyzed through theoretical, numerical and experimental studies. It is demonstrated that by optimizing the parameters of the resonant transfer functions implemented in the digital circuits, high–level damping localized near the first resonant peak can be introduced into the curved beams. As a result, the resonant peaks of the metastructures can be reduced without compromising the isolation performance beyond those peaks.

Influence of Transverse Normal Strain on Buckling and Vibration of Sandwich Plates

This study presents the buckling and free vibration behavior of simply supported square and rectangular sandwich plates. A third-order shear deformation theory considering both the transverse shear and normal deformations is used. The condition of zero transverse shear stresses on the top and bottom surfaces of the plate is satisfied. Hence, the present theory does not require the shear correction factor generally associated with the first-order shear deformation theory. Governing equations and boundary conditions are obtained using the principle of minimum potential energy. The closed-form solutions of simply supported sandwich plates are obtained. The effect of the side-to-thickness ratio and plate aspect ratio on buckling and free vibration responses of simply supported square and rectangular sandwich plates is examined. Parametric effects of face sheet thickness ratios and load factor upon the critical buckling load for uniaxial and biaxial buckling analysis of square and rectangular sandwich plates for different staking sequences are studied. All numerical solutions are obtained using MATLAB® programs. The results obtained from the present theory are compared with those from classical plate theory, first-order shear deformation theory, higher-order shear deformation theories, and the exact three-dimensional elasticity solutions as applicable.

Analytical and numerical predictions of elastoplastic buckling behaviors of the subsea lined pipelines with ovality defects under hydrostatic pressure

The subsea pipelines may deteriorate and/or be cracked after a long period of service. A cost-effective trenchless rehabilitation technology is to install a thin-walled steel liner inside the pipelines. However, elliptical defects may occur in the liner due to issues such as insufficient inflation, deformation of the host pipelines, and joint misalignment in practical engineering. Thus, this study aims to develop a systematic analytical model and numerical model for the elastic and inelastic buckling behaviors of the thin-walled oval steel liner, respectively. An admissible radial displacement function is introduced to describe the single-lobe deformation of the liner. The governing equations are established to trace the equilibrium paths by applying the theory of thin-walled shells and the principle of minimum potential energy. A two-dimensional finite element model is developed for the elastic and inelastic buckling performance of the liner. The analytical solutions of elastic buckling are in good agreement with other closed-form solutions, numerical results, and respective test results available in the literature. Finally, parametric evaluations are carried out in terms of ovality, non-uniform liner-pipeline gap, internal pressure, and liner yield strength.

Three-dimensional buckling analysis of stiffened plates with complex geometries using energy element method

A novel numerical method, energy element method (EEM), is proposed for the three-dimensional (3D) buckling analysis of stiffened plates with complex geometries. The problem is formulated in a cuboidal domain, and any complex geometric stiffened plate is modeled by assigning cutouts within the cuboidal domain. The stiffened plate is considered as an energy body and is discretized using Gauss points with variable stiffness properties to simulate its energy distribution. Incorporating the extended interval integration, Gauss quadrature, variable stiffness properties, and Chebyshev polynomials, the strain energy of stiffened plates with complex geometries can be numerically simulated by putting the stiffness and thickness of Gauss points in the cutouts to zero in the cuboidal domain. Using the principle of minimum potential energy and Ritz solution procedure, the deformation and buckling behaviors of stiffened plates with complex geometries can be captured. As a result of the new formulations in EEM, new standard energy functionals and solving procedures have been developed. In addition, Gauss points are generated within the energy elements accounting for the geometric boundaries of the stiffened plate, which are characterized by level set functions. EEM is employed to investigate complex-shaped stiffened plates with straight or curvilinear stiffeners, and the results are compared to those obtained using FEM or mesh-free method. The precision, generalization, and stability of EEM are demonstrated.

Piezoelectric Interaction Analysis and Load‐Deflection Prediction for Nonlinear Bending and Snap‐Through of Post‐Buckled FG Piezoelectric Beams

Revealing the interaction effect in the nonlinear bending of post‐buckled functionally graded (FG) piezoelectric beams promotes the intelligent development of structural design. The nonlinear bending behavior of post‐buckled piezoelectric beams is studied. Based on the displacement field constructed by the refined beam model, the post‐buckling configuration is considered. The nonlinear geometric relationship and the constitutive relationship in the multi‐physical field are modeled. In the framework of the principle of minimum potential energy, the nonlinear static models of post‐buckled FG piezoelectric beams are established. The post‐buckling configuration is obtained by the extended two‐step perturbation method. To obtain the approximate solution for the static problems of the post‐buckled FG piezoelectric beam, the perturbation scheme of electric potential and generalized displacement is proposed. The detailed parametric analysis indicates the interaction mechanism in the nonlinear bending of the post‐buckled FG piezoelectric beam. The snap‐through process and its sensitive interval are further captured. The back propagation neural network (BPNN) technique developed for the nonlinear bending problem of post‐buckled FG piezoelectric beams achieves accurate prediction.

Low-velocity impact response and damage mechanism of cosine function cell-based lattice core sandwich panels

This study aims to investigate the impact response and damage mechanism of cosine-function cell-based (CFCB) lattice core sandwich panels. Several low-velocity impact tests were conducted to explore their advantages in impact resistance by comparing them with traditional body-centered cubic (BCC) lattice-core sandwich panels. The impact response and deformation patterns of CFCB lattice core sandwich panels with two different faceplate materials (aluminum alloy and carbon fiber reinforced plastic composite) were experimentally investigated. The CFCB lattice core sandwich panels exhibited higher impact resistance and energy absorption capacity than their BCC lattice core counterparts. Furthermore, sandwich panels with aluminum alloy faceplates provided better impact resistance capacity than those with CFRP faceplates. Subsequently, several numerical models were developed to explore the deformation mechanisms and energy absorption characteristics of CFCB lattice core sandwich panels. In addition, the effects of the core structural parameters on mechanical performance were numerically investigated. Results indicated that increasing the cell rod diameter and/or reducing the cosine period length could decrease the indentation depth and enhance crush force efficiency. Finally, based on the principle of minimum potential energy, a theoretical model was developed to predict the initial peak load of CFCB lattice core sandwich panels with isotropic faceplates. This study aimed to explore the impact response and provide design guidance for CFCB lattice core sandwich panels.

Analytical Solution to Estimate the Effect of Underlying Tunnel Excavations on the Existing Tunnel

ABSTRACTThe tunneling underlying inevitably leads to the displacement of adjacent soil, greatly influencing the deformation of the tunnel above. Most theoretical studies primarily concentrate on analyzing the mechanical equilibrium of individual tunnel sections, ignoring the energy generated by the system during tunnel deformation. Based on this, in view of the energy relationship, the overlying tunnel's deformation is simulated by using the Rayleigh–Ritz method. Further, its potential energy equation can be established based on the Vlasov foundation. The corresponding energy variational solution is solved according to the minimum potential energy principle, leading to an analytical solution for the shield tunneling‐induced overlying tunnel's response. The efficacy of the suggested method is verified by contrasting it with centrifuge experiments and field case studies derived from prior studies. Relative to the Winkler foundation model, which deviated from the suggested approach, the results derived from the suggested method show a closer correlation with the collected measurement data. Further parameter studies show that the vertical clearance and skew angle between two tunnels, the volume loss rate, and elastic modulus are significant factors affecting the tunnel behaviors due to tunneling underneath. The suggested theoretical model can be applied to forecast potential risks that an existing tunnel may encounter during the excavation of a new tunnel underlying in similar engineering projects.

Forced-Vibration Characteristics of Bowtie-Shaped Honeycomb Composite Sandwich Panel with Viscoelastic Damping Layer.

The incorporation of viscoelastic layers in laminates can markedly enhance the damped dynamic characteristics. This study focuses on integrating viscoelastic layers into the composite facesheet of the bowtie-shaped honeycomb core composite sandwich panel (BHC-CSP). The homogenization of the damped BHC-CSP is performed by employing the variational asymptotic method. Based on the generalized total energy equation, the energy functional of the representative unit cell of the damped BHC-CSP is asymptotically analyzed. The warping function, derived following the principle of minimum potential energy, provides a basis for obtaining the corresponding Euler-Lagrange equation to ascertain the equivalent elastic properties of the damped BHC-CSP. Utilizing the developed two-dimensional equivalent model, the free-vibration characteristics of the damped BHC-CSP are examined across diverse boundary conditions while delving into the impact of an external viscous damping layer on the natural frequency of the damped BHC-CSP. The results reveal that intensified boundary constraints effectively diminish the effective vibration region of the damped BHC-CSP, thereby enhancing its overall stability. The introduction of a PMI foam layer proves effective in adjusting the stiffness and mass distribution of the damped BHC-CSP. Resonance characteristics are explored through frequency and time-domain analyses, highlighting the pivotal roles of the excitation position and receiver point in influencing the displacement and velocity responses. Although the stiffness is improved by incorporating a PMI foam layer, its effect on the damping performance of the damped BHC-CSP is minimal when compared to the T-SW308 foam layer.

2D calculation method on global stability safety factor of diaphragm wall slurry trench based on minimum potential energy principle

In order to limit the phenomenon that the global stability safety factor (GSF) of the diaphragm wall slurry trench calculated by the two-dimensional (2D) wedge method is too small, the 2D calculation model on global instability of trench wall is established by assuming the whole critical failure surface (CFS) is a quarter elliptic cylindrical surface, and the determination method of CFS is given too. In view of the successful application of the minimum potential energy principle in slope stability checking calculation and on the basis of the new model established, the 2D minimum potential energy method for calculating the GSF is proposed according to the minimum potential energy principle. The correctness of the 2D calculation model built and the rationality of the 2D calculation method proposed are verified based on ten engineering cases, and the sensitivity of the GSF to various parameters is analyzed on the basis of six design examples. Finally, the new calculation method is applied to the global stability checking calculation of trench wall in practical engineering, and the result shows the new calculation method given in this paper is still conservative, so the result calculated by the new calculation method is advised to be multiplied by a modification factor of 1.67 in practical application.

Buckling and vibration analysis of axially functionally graded nanobeam based on local stress- and strain-driven two-phase local/nonlocal integral models

This paper presented an efficient finite element formulation for analysis of the buckling and free vibration of the axial functionally graded (AFG) Euler-Bernoulli nanobeam based on minimum total potential energy principle with linear constraints. The stress- and strain-driven two-phase local/nonlocal integral models are utilized to address small scale effects. The material properties of the AFG nanobeam vary exponentially along the axial direction. The nonlocal integral constitutive equation can be equivalently transformed into a differential equation with two constitutive boundary conditions. The governing equation and standard boundary conditions are derived via Hamilton's principle. The weak form associated with total potential energy is derived and the constitutive boundary conditions can be converted to external forces at the boundaries. By employing a numerical solution methodology on the basis of the finite element method (FEM) together with Lagrangian multiplier method (LMM), a unified finite element formulation based on stress- and strain-driven two-phase local/nonlocal elasticity is constructed to obtain the critical buckling load and vibration frequency. Meanwhile, the numerical results obtained through generalized differential quadrature method (GDQM) validate the efficiency and accuracy of the present results. The influence of gradient index, nonlocal parameter and local phase parameter on buckling and free vibration of AFG Euler-Bernoulli nanobeam is investigated in detail under different nonlocal models and boundary conditions. It is demonstrated that the gradient index has more effect on mode shapes than the buckling load and vibration frequency between the local stress- and strain-driven two-phase local/nonlocal models.

Research on the Optimal Design of Retaining Piles of a Wide Metro Tunnel Foundation Pit Based on Deformation Control

Engineers pay more and more attention to the economic benefits of foundation pit engineering. At present, the optimal design of the foundation pit supporting structure mainly focuses on strength and functional design, and there is no mature theoretical design method for deformation control. In this paper, a method for calculating the overall deformation of a foundation pit supporting structure based on the principle of minimum potential energy is proposed. Based on this method, the optimal design of the foundation pit of Guangzhou Baiyun District Comprehensive Transportation Hub Metro Station is realized. The deformation calculation results and optimization design scheme are validated by finite element numerical simulation and field monitoring data. The results show that the proposed theoretical algorithm predicts the pile deformation curves better than the finite element method, suggesting the proposed theoretical method is reasonable and the optimization scheme of the retaining pile is feasible. In the optimized design, the deformation of the foundation pit retaining pile is controlled by its push-back effect. The proposed deformation calculation method can realize the overall deformation calculation of the foundation pit supporting structure.

A single-axis force sensor based on a 3D-printed elastic ring

Force measurement is indispensable in vast engineering applications but conventional force sensors usually encounter large measurement errors or high costs. To address this problem, we develop a force sensor based on a 3D-printed elastic ring structure, which measures the external force by utilizing the displacement variation of the elastic ring. The nonlinear theory based on the minimum potential energy principle is proposed to describe the nonlinear relationship between the external force and the ring top displacement. The feasibility of the proposed force sensor scheme is validated through FEM simulations and experiments. The range of the force sensor can be adjusted by tuning the geometric parameters and the material parameters of the elastic ring, while the force sensor shows good measurement error at different measurement ranges. This force sensor possesses simple fabrication and low cost, showing good potential for various applications such as industrial automation, mechanical testing, and medical devices.

Multi-scale pore model construction and damage behavior analysis of SiCf/SiC composite tubes

SiCf/SiC composite materials are essential for spacecraft thermal protection, determining the safety of spacecraft. However, their complex structures and intricate fabrication processes have led to an unclear understanding of their failure mechanisms, limiting their application. In this study, circumferential compression experiments were conducted, and X-ray computed tomography (X-CT) was used to analyze the distribution characteristics of pore spaces. The experimental results indicate that damage manifests as secondary fracture phenomena based on temporal evolution characteristics. Specifically, after experiencing performance degradation of 24.73–60.96%, the material still maintained basic stability, exhibiting a performance recovery of 31.55–36.1% under the applied load until failure. To predict the dynamic damage behavior of materials at multiple scales, a multi-scale method combining statistical and microscopic approaches was proposed. A micro representative volume element (RVE) random fiber pore distribution model was established based on random processes and random medium theory. Finally, considering the CT data and the principle of minimum potential energy, a macroscopic finite element model of the micro-pores' structural characteristics in a three-dimensional wound tube was reconstructed. The results indicate that cracks initiate around pore defects and gradually propagate to form crack bands. The model closely matches the macroscopic damage and microscopic characteristics of the material. This work provides a new approach for the numerical simulation of pore defects in such materials.

Material optimization of reinforced concrete structures in multiple limit states

AbstractA common limitation in current material optimization methods for reinforced concrete structures is that they often concentrate solely on either the serviceability limit state (SLS) or the ultimate limit state (ULS). In response, this paper introduces a novel approach to optimize the material layout in planar reinforced concrete structures in plane stress, accounting for both limit states simultaneously. The proposed method employs an iterative strategy, integrating criteria for serviceability and ultimate limit states while approximating the elastoplastic behavior of cracked reinforced concrete. Here, the criteria for the limit states are incorporated as limits on the strains. The methodology initiates with a finite element limit analysis for the preliminary layout. The minimum potential energy principle is utilized to determine the strains for the nonlinear material response, and subsequently, an iterative linearized method is applied for material optimization. The research showcases the successful application of this approach to simple examples and presents improved and optimized solutions that adhere to the limit criteria for more complex structural scenarios.

A method for predicting deformation field of deep foundation pit considering spatial effect

Urban underground spaces have experienced significant development and utilization due to the rapid progress of urban construction and continuous advancements in construction technology. The number of foundation-pit projects immediately adjacent to buildings, structures, and pipelines has been increasing. Ensuring the safety of construction and surrounding facilities, it is essential to predict deformation caused by foundation pit excavations. In this study, a simple calculation method was proposed for predicting deformation of an envelope structure and surface settlement caused by deep foundation excavation with spatial effects. Firstly, the principle of minimum potential energy was applied and the lateral displacement equation of the envelope structure was derived, considering spatial deformation effects. Secondly, based on linear elasticity theory, a surface settlement expression was formulated utilizing the plane strain boundary. Finally, the stratigraphic loss method was used to establish a connection between horizontal envelope deformation and surface settlement, allowing for calculations of surface settlement at any position behind the envelope structure. The effectiveness and practicability of the proposed method were demonstrated by comparison of two existing foundation-pit engineering examples.

Three-dimensional calculation method on local slurry wall stability of diaphragm wall trench based on minimum potential energy principle

Because the soft weak soil layer rich in confined groundwater is prone to local instability and failure in the construction of diaphragm wall trenches, in order to check the local slurry wall stability in the construction of diaphragm wall trenches, a new three-dimensional (3D) calculation method on the local stability safety factor (LSF) of slurry walls is proposed based on the minimum potential energy principle. After the new method is applied to two engineering cases, the value of the LSF solved is 0.82 for engineering case 1, and the values of the LSF solved are 0.84 and 1.05 when the phreatic water levels are 2 and 4.5 m, respectively, for engineering case 2. These results are very close to the calculation results in the existing references, so the rationality of the new method is verified. Then the sensitivity of the LSFs obtained from the new method to various parameters has been analyzed and clarified on the basis of ten design examples. Finally, the new method is used to check the local slurry wall stability in the construction of diaphragm wall trenches for a practical project, and the obtained values of the LSF for three concerned weak soil layers are 1.37, 1.19, and 0.82, respectively. These results are in agreement with the actual situation in the construction site. This reveals that both the 3D model established and the 3D calculation method proposed in this paper are suitable for application in practical projects.

Static response of vertically loaded pile in inhomogeneous soil

In this study, an analytical model is developed to obtain the static response of the vertically loaded pile in the inhomogeneous soil with arbitrary properties varying along the depth. The soil displacement is expressed by the product of the axial displacement function and the shape function attenuating with the distance from the pile periphery. Through the minimum potential energy principle, the governing equations are established for the axial displacement function and the shape function. The governing equation for the shape function is independent of soil properties and it can be explicitly solved. The governing equation for the axial displacement function contains variable coefficients due to the inhomogeneity of soil and it is solved using the weighted residual method. An iterative procedure is introduced to implement the solution. The proposed model is validated by comparing the present solutions with those obtained by the available methods. The parametric study shows that the pile head stiffness is affected by the distribution of soil shear modulus heavily, in which the average shear moduli are the same for these distributions. Moreover, the relationship between the pile head stiffness and the pile slenderness ratio for a floating pile is different from that for an end-bearing pile.