Principle Of Minimum Potential Energy Research Articles

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.

Design Optimization of Flexural and Eigenvalue Characteristics of Perforated and Skewed Bio-Composite Functionally Graded Panels

The development of novel materials is of prime importance in the biomedical areas, especially for bone replacements and dental implants. This study uses finite element modeling of skewed functionally graded bio-composite structures with perforations to examine flexural and eigenfrequency responses. Here, the bio-composite structure comprises Titanium (Ti) and Hydroxyapatite (HAp). Voigt’s micromechanical material model via power-law distribution and Representative volume element (RVE) method is utilized to compute the in-homogenous material properties in the thickness direction. The strain field is based on the equivalent single-layer first-order shear deformation theory with five degrees of freedom, and the governing equations are obtained using principle of minimum potential energy. A computer algorithm is developed in a 2D finite element platform using eight-noded quadrilateral elements. The mesh stability of the skewed and perforated structure is confirmed through the smart mesh technique, and the present model’s accuracy is demonstrated by comparing it with the available reported results. Numerous examples reveal the significance of material gradation, skewness, aspect ratios, perforations, and support conditions on bio-composite functionally graded structures’ flexural and eigenfrequency responses. Furthermore, parametric optimization is conducted to minimize the flexural response and maximize the fundamental frequency by employing the response surface methodology (RSM), followed by determining optimal process parameters and valuable findings.

Analyze the temperature-dependent elastic properties of single-walled boron nitride nanotubes by a modified energy method

The elastic properties of boron nitride nanotubes (BNNTs) were investigated utilizing an enhanced energy method. By considering small deformations and applying the principle of minimum potential energy, the variations in atomic bonds and bond angles within the nanotube structure were determined. The modified model incorporated the contribution of inversion energy to the overall potential energy of the system, leading to the derivation of analytical expressions for the Young's modulus, shear modulus, and strain energy of both armchair and zigzag BNNTs under varying temperatures. The results indicate that compared to zigzag BNNTs, the impact of inversion energy on the elastic constants of armchair BNNTs is more significant, especially at small diameters (<1 nm). In thermal environment, this study demonstrates that the change in Young's modulus of BNNTs is lower than that of carbon nanotubes (CNTs), confirming the superior thermal stability of BNNTs over CNTs. Furthermore, molecular structure mechanics (MSM) and continuum mechanics models were employed to analyze the strain energy of BNNTs. The effects of different bonds, bond angles, and inversion angles on strain energy were analyzed in a thermal environment, revealing distinct differences between the two types of BNNTs. These findings provide more accurate theoretical guidance for thermal applications based on the stretching of BNNTs.

Ispitivanje poprečne unutarnje sile sandučastog nosača primjenom principa varijacije energije

This study investigated the transverse internal force of a box girder with a trapezoidal cross-section under eccentric loading. The effects of the box girder distortion and frame shear difference on the transverse internal force were considered. An energy variation method based on the minimum potential energy principle was adopted for transverse internal force analysis of the frames. The variable shear difference of the top slab was considered an unknown quantity, and a fourth-order governing differential equation was established. The transverse bending moment of the frame was obtained by solving for the shear difference. Two examples were used to verify the proposed method and analyse the differences between the transverse internal force results calculated using different methods. The effect of the distribution of the box-girder distorted transverse bending moment on each slab was investigated for various stiffness ratios. The results showed that the transverse internal force of the box girder calculated using the proposed method agreed well with the finite element results, with a maximum error of less than 9.68 %. When the stiffness ratio of the box girder increased, the distribution of the distorted transverse bending moment on the top slab increased, whereas that on the bottom slab decreased.

Simplified nonlinear model and 3D printing model experiment verification of longitudinal joints of shield tunnel segments

The segment joints represent vulnerable and core-bearing areas within shield tunnel structures. Their flexural stiffness, starting from the contact surface of longitudinal joints to the bolt hole, exhibits continuous changes under bending loads, typically demonstrating nonlinear mechanical characteristics. This study investigates the influence of joint contact surfaces on flexural stiffness, extending from the joint contact surface to damaged concrete in the compressed zone. A new simplified model for the variable nonlinear stiffness of longitudinal segment joints in shield tunnels is proposed, considering the flexural stiffness of the joint contact surface and the width of the damaged concrete zone in the segment ring while simulating the flexural stiffness decay induced by the opening of segment joints. Based on the minimum potential energy principle, a design calculation method for joint flexural stiffness is derived, establishing the nonlinear bending stiffness coupling relationship between the segment and the joint. Through similarity theory, 3D printing technology, and indoor loading tests, the theoretical model's accuracy regarding tunnel segment joints is verified. This theoretical model provides a basis for designing and assessing the performance of tunnel-segment joints.

Nonlinear bending characteristics of multidirectional functionally graded porous panels using higher-order finite element solution with parametric optimization

The current study aims to investigate the nonlinear bending behavior of unidirectional (1D) and multidirectional (2D/3D) functionally graded porous shell panels. Here, the material properties of the multidirectional functionally graded (FG)-panels are estimated by employing the representative volume element method and Voigt’s micromechanical homogenization scheme. The nonlinear mathematical model of the FG-panel is developed by incorporating Green-Lagrange type nonlinear strains in higher-order-midplane kinematics. The minimum potential energy principle is used to obtain the final form of the equilibrium equation for the multidirectional FG shell panel and further solved by using Picard’s iterative-based 2D-isoparametric finite element method via quadrilateral Lagrangian elements. The efficacy and robustness of the present model are checked by comparing its results with previously reported literature. Numerous examples are solved to analyze the impact of material parameters, such as volume fraction index and porosity index, and geometrical parameters, such as aspect ratio and support conditions, on the nonlinear deformation behavior of multidirectional FG shell panels. In addition, parametric optimization is carried out to minimize the flexural response by using the response surface methodology and, hence, determining the optimal values of multidirectional porous FG-panel. The results demonstrate that, within the defined ranges of FG-panel, lower values of porosity index and volume fraction index yield lesser deflection.

Analysis of multi-scale failure behavior of SiCf/SiC composite clad tube under flexible clamping damage

The intricate spatial fiber winding structure of SiCf/SiC ceramic matrix composite (CMC) cladding tubes makes it difficult to investigate in depth the inherent evolutionary mechanism of damage failure under specific operating conditions. Therefore, in this study, based on X-ray computed tomography (X-CT) reconstruction and cross scale numerical simulation methods, combined with clamping failure experiments under specific conditions for the cladding tubes, we conduct an in-depth analysis of the material's damage, failure, and mechanical behavior in different clamped regions. The research indicates that variations in the clamped regions have a minimal impact on the overall mechanical feedback of the material, and the maximum clamping load remains essentially around 500 N. The tube damage exhibits a unique " regional progressive damage evolution mechanism," where stress differentials of 108.49–115.33% appear between the inner and outer surfaces in different regions, leading to variations in the initiation and propagation of cracks within these areas. Furthermore, based on the minimum potential energy principle and semi-geodesic line theory, the intricate macroscopic tube structure formed by the spatial stacking and winding of SiCf/SiC cladding tube fibers is accurately reconstructed, which is combined with macroscale and microscale local feature models to capture the fibre/matrix morphology of the material with dynamic mechanical feedbacks, and experimentally validated in this work. The results indicate that the numerical model can effectively characterize the complex mechanical damage morphology and progressive failure lows of SiCf/SiC CMC at multiscale. Simultaneously, it is observed that stress concentrations in the fiber crossover region result in a load-bearing capacity 112–283% higher than in other areas. This phenomenon often leads to the first occurrence of failure in this region after clamping overload, exhibiting more pronounced damage characteristics.

Theoretical analysis of inflated tube wrinkling behavior under pure bending

Inflatable tubes, owing to their lightweight and foldable characteristics, have important applications in soft robots and space expandable structures. The application of a bending load to the inflated tube might induce local instability on its compressed side, leading to the formation of wrinkles that could potentially compromise the structural strength. Therefore, it is necessary to have a comprehensive understanding of the wrinkling behavior exhibited by inflated tubes under bending load. Initially, a theoretical solution is developed for the critical wrinkling behavior of inflated tubes under pure bending, drawing upon the principle of minimum potential energy. According to the energy expression of the system, the critical wrinkling behavior of inflated tubes depends on dimensionless geometrical and internal pressure parameters. The critical wrinkling load and wrinkle pattern of the system are influenced by the thickness-radius ratio and the ratio of internal air pressure to Young's modulus. Subsequently, the theoretical solutions are validated through several finite element analysis examples and a systematic investigation is conducted into the influence of dimensionless geometrical and internal pressure parameters. Finally, the evolution of morphology in post-buckling is investigated through numerical results. The findings suggest that during post-buckling, the wrinkles may undergo a secondary bifurcation and evolve into a non-axisymmetric diamond-like pattern. These results hold significant implications for understanding the bending instability and failure mechanisms of inflated tubes.

Mechanics of inner core debonding of composite sandwich beam with CFRP hexagonal honeycomb

The inner core debonding of composite sandwich beams with carbon fiber reinforced polymer (CFRP) hexagonal honeycomb core is investigated theoretically and experimentally. Based on the principle of minimum potential energy, a theoretical model is established to reveal the core debonding mechanism with Timoshenko’s beam theory and Extended High-Order Sandwich Panel Theory (EHSAPT) for composite face sheets and CFRP hexagonal honeycomb core, respectively. The theoretical approach of core debonding simulation is based on the framework of the cohesive zone model (CZM). By constructing the constitutive relation of the core interface as the discontinuous stress and displacement boundary conditions, the vertical core debonding can be simulated overcoming the singularity of equilibrium equations. The crack propagation process is decomposed into three stages with the consideration of the double crack propagation mechanism. In experiments, the regular and reinforced hexagonal honeycomb sandwich beams are designed and fabricated, and the quantification of the defect degree of the core bonding interface is achieved by equivalent energy release rate obtained from the homogenized hexagonal honeycomb. The double cantilever beam (DCB) experiments are performed to determine the equivalent energy release rate and reveal the core debonding mechanism of CFRP hexagonal honeycomb sandwich beams. Comparing the load–displacement curves and debonding deformation patterns indicates that the established theoretical model can effectively predict the core debonding mechanical response. Finally, parametric analysis is carried out to discuss the impact of structural dimension parameters, defect location and defect degree on the debonding modes of CFRP hexagonal honeycomb sandwich beams.

A new model for spatial rods incorporating surface energy effects

A new non-classical model for spatial rods incorporating surface energy effects is developed using a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equations and complete boundary conditions. The newly developed spatial rod model contains three surface elasticity constants to account for surface energy effects. The new model recovers the classical elasticity-based Kirchhoff rod model as a special case when the surface energy effects are not considered. To illustrate the new spatial rod model, two sample problems are analytically solved by directly applying the general formulas derived. The first one is the buckling of an elastic rod of circular cross-section with fixed-pinned supports, and the other is the equilibrium analysis of a helical rod deformed from a straight rod. An analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, and two closed-form expressions are obtained for the force and couple needed in deforming the helical rod. These formulas reduce to those based on classical elasticity when the surface energy effects are suppressed. For the buckling problem, it is found that the critical buckling load predicted by the current new model is always higher than that given by the classical elasticity-based model, and the difference between the two sets of predicted values is significantly large when the radius of the rod is sufficiently small but diminishes as the rod radius increases. For the helical rod problem, the numerical results reveal that the force and couple predicted by the current model are, respectively, significantly larger and smaller than those predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase of the rod radius.

Study on the effect of tunneling and secondary grouting on the deformation of existing pipelines in bedrock raised strata

Shield crossing the bedrock raised strata easily leads to large deformation of the soil, resulting in deformation and damage of the upper pipelines. To investigate the law of pipeline deformation caused by shield underpassing in bedrock raised strata, the changes in the action range of the additional thrust of the cutter head (p1), the friction resistance of the shield shell (p2), and the additional grouting force (p3) were analysed. A convergence model of the excavation face suitable for bedrock raised strata was proposed. The calculation formula of soil displacement at the axis of the pipeline was derived by using Mindlin’s solution and stochastic medium theory. The solutions of the vertical displacement, bending moment and strain of the pipeline were obtained by the energy variational method and the principle of minimum potential energy. Pipeline deformation analysis and reliability verification were carried out by an engineering case in Hangzhou, China. The influence of bedrock distribution and secondary grouting behind the pipe wall on pipeline deformation was studied. The results show that the calculated results are similar to the measured data with a high degree of coincidence, and the degree of agreement is higher compared to the existing method. When the shield tunneling direction is well controlled, the settlement and bending moment of the pipeline caused by the shield crossing the raised bedrock stratum will be reduced, and the reduction increases with increasing bedrock length and intrusion depth. The secondary grouting behind the pipe wall can reduce the settlement of the pipe line, but it is necessary to reasonably control the three parameters of grouting amount per unit length (Vinj), grouting ring length (Lh), and grouting ring angle (δ). Lh should not be too long and can be controlled below 30 m, δ should be controlled within 60°, and Vinj can be adjusted according to the deformation size of the pipeline to prevent excessive or insufficient correction.

Systematic failure mechanism of an FGMs polyhedral arched liner under a fire disaster environment

This paper investigated the failure performance of an arched polyhedral liner yielding a fire disaster environment. The functionally graded materials (FGMs) are used in the liner to increase the ability of fire resistance. A concentrated load may occur at the invert of the arched liner due to the debris, which is considered in the study. The polyhedral arched liner deforms in a single-lobe shape, and this deformation is expressed by a trigonometric function. The nonlinear equilibrium equations are presented by applying the thin-walled shell theory and the principle of minimum potential energy, respectively. Based on the above investigations, two innovative contributions of the present study are addressed: (1) a novel FGMs polyhedral liner is introduced to rehabilitate the cracked non-circular tunnel in the combination of the mechanical loading and fire environment; and (2) the critical buckling load, as well as the load-displacement nonlinear equilibrium curves, is derived explicitly by solving the nonlinear equilibrium equations. The effect of the fire disaster on the buckling load is quantized to describe the destabilization behavior of the arched liner in a fire environment, respectively. Then, the present study is compared efficiently with other closed-form formulations when the polyhedral FGM arched liner simplifies to a circular homogeneous liner. It is shown that a smaller number of polyhedral edges and a higher content of ceramic are beneficial to resist external loadings and fire disasters in engineering practices. Finally, the effects of geometric parameters, volume fraction exponents, and temperature variations on the nonlinear buckling behavior of the polyhedral arched liner are evaluated.

3D meshless modeling of piezoelectric structure based on the radial point interpolation method

Piezoelectric materials find widespread applications in high-precision actuators and sensors. However, the traditional finite element method falls short in meeting the simulation needs of piezoelectric structures due to complexities in mesh generation and precision requirements for accurate simulations. This paper focuses on adapting and generalizing the meshless modeling technique based on the radial point interpolation method to address three-dimensional simulations of piezoelectric structures and accurately model positive and inverse piezoelectric behaviors. The piezoelectric body is discretized into point cloud data, and displacement and potential are approximated by interpolating values through the radial point interpolation method. Subsequently, a 3D meshless model for general piezoelectric structures is formulated based on the principle of minimum potential energy. The influences of key factors on interpolation accuracy and computational cost were discussed. Numerical examples, featuring a 3D piezoelectric bimorph beam as actuators and sensors, validate the efficacy of this extended approach. Results affirm that the three-dimensional meshless method, rooted in the radial point interpolation technique, achieves superior accuracy in predicting piezoelectric behavior compared to the traditional finite element method. This adaptation not only enhances the applicability of meshless modeling but also represents a significant advancement in accurately simulating complex three-dimensional piezoelectric structures.

Analytical modeling of stress concentration in composite box girders

In several structures/structural forms, including box beams, T-beams, U-beams, box girders, and tubular buildings, the stress concentration caused by the shear lag effect reduces their load-resisting capacity. The energy method has been proven to be more effective in analyzing the shear lag effect in these structures. Utilizing the minimum potential energy principle, this study illustrates a simplified approach to assessing stress concentration in composite box girders. Based on boundary conditions and shear loading conditions, closed form solutions are obtained for normal stresses and deflection. The present research results are verified with finite element analysis results and literature, demonstrating the robustness of the methodology. A composite box girder can be analyzed using the present approach under any support and loading conditions, taking material properties into account. To increase composite box girders’ load carrying capacity, additional layers of material are generally added to the flange. When retrofitting a box girder in this way, cross-sectional flexural rigidity is trivially improved. The additional layer of material on the flange of the box girder may reduce stress concentrations caused by the shear lag effect when it improves the shear flow capacity (SFC) of the flange. Shear flow parameter (Γ) measures flange shear flow capacity. The flange SFC increases significantly for a core/layer with a ratio of elastic modulus to shear modulus (E c /G c ) up to 1.26. When E c /G c rises even further, the flange SFC decreases. A composite box beam may have enhanced flange stiffness, with cores/layers having E c /G c ratios up to 2.86 and stress concentrations similar to that of a box beam without a core/layer. In composite box beams with a low aspect ratio, considering only flexural deformation in the core/layer augments the stress concentrations significantly. Deflection and negative shear lag are reduced significantly when the composite box beam flange has a high SFC.