Engineering Structures Research Articles

Mechanical properties and probabilistic models of wood and engineered wood products: A review of green construction materials

Wood and engineered wood products are green construction materials with excellent mechanical properties, widely utilized in various buildings and bridges. This paper reviews their short-term and long-term mechanical properties and probabilistic models. Short-term mechanical properties analysis reveals strong tensile (up to 165 MPa) and bending strengths (up to 264 MPa), contrasted by lower compressive strength (up to 103 MPa) and minimal shear/rolling strength (up to 14 MPa). These properties vary based on species type, lay-up angle, and dimensions, and are mutually influential. It also demonstrated that innovative techniques such as compression and cross-laminated lay-ups enhance the strength and sustainability of wood products, utilizing smaller logs effectively and reducing carbon emissions. For long-term mechanical properties, empirical models typically used for creep strength assessment fail to account for load duration variability, a gap partly addressed by damage accumulation models which require further development. The considerable time for long-term tests underlines the need for developing accelerated load tests to more effectively assess durability. As for probabilistic models, numerical methods and machine learning provide advanced probabilistic models by analyzing extensive data, offering significant improvements over traditional experimental methods.The global contribution of the work is mainly to enhance the understanding of wood and engineered wood products for properly design and extensive application of timber structures in civil engineering and promoting sustainable development.

A fatigue crack prediction method based on inductive semi-supervised learning and Lamb-wave monitoring for orthotropic steel bridge deck

Orthotropic steel bridge deck suffers from fatigue cracking due to its orthotropic configuration and cyclic vehicular load. Hence, fatigue crack monitoring and prediction are crucial for addressing this issue. The Lamb-wave technique has potential in monitoring the growth of fatigue cracks, whereas most of the existing signal processing methods show poor performance in the crack prediction. To improve the accuracy and reliability of fatigue crack prediction, an inductive semi-supervised learning (ISL) approach is proposed in conjunction with Lamb-wave monitoring. The presented method is first verified by conducting a fatigue test of orthotropic steel bridge deck loading by three load conditions which would lead to different types of cracks, where lightweight Lamb-wave monitoring devices were adopted. The approach was further applied in field monitoring to validate its effectiveness in real engineering structures. The results demonstrate that the proposed method can recognize the initiation of fatigue cracks by giving a predicted time interval in which the real initiation moment locates. Similar increasing trends are found between the predicted and real crack lengths during the growth of cracks, while a relative fluctuate is existing in the predicted results, and the prediction accuracy of crack lengths can be regarded as 2 mm. For bridge field monitoring, the real fatigue crack length was also well predicted by the proposed method, exhibiting an average error of 0.82 mm in comparison with the real values. In general, the research findings emphasize the adaptability of the proposed method for fatigue crack prediction in practical engineering.

Bridge modal identification method based on adaptive VMD-HT algorithm

ABSTRACT Bridge engineering structures play a crucial role in transportation. Traditional methods of road and bridge closures for inspection and evaluation, while effective, increase bridge inspection costs and disrupt normal traffic flow. Therefore, a rapid detection and evaluation method for bridge structural modal parameters in the operational state is more suitable for practical engineering applications. However, under real operating conditions, bridge vibration signals typically contain a large amount of noise and interference, and modal identification must overcome issues of non-stationarity and non-linearity in the vibration signals caused by vehicle loads and environmental changes to ensure the accuracy of the identification. This study replaces the traditional Empirical Mode Decomposition algorithm in the Hilbert-Huang Transform with the Variational Mode Decomposition algorithm. It proposes a new method that combines the VMD algorithm with the Hilbert Transform, referred to as the Variational Mode Decomposition-Hilbert Transform method. Compared to the EMD algorithm, it offers significant improvements in suppressing mode mixing and endpoint effects. A vehicle-bridge coupling model was constructed, and the bridge’s natural modal information was successfully extracted from the vibration response signals. The research results indicate that the improved algorithm offers higher accuracy and faster computational speed compared to traditional identification algorithms.

The Comparison of Performance of Root Vetiver Grass Between Coconut Fiber and Eggshell Toward Slope Stabilization

Slope stability using vegetation has emerged as a more environmentally friendly and sustainable solution for slope stabilization, replacing traditional structural engineering methods and shotcrete cover. However, despite the application of Vetiver grass, failures in the form of deformation, cracks, and slope collapse still occur. The main objectives of this study are to identify the root properties and analyze the mechanical properties of Vetiver grass with coconut fiber and eggshell fertilizers. Laboratory testing was conducted to determine the mechanical properties of Vetiver grass roots using shear box and universal testing machine. As a result, the root diameter and root length with egg shell exhibit larger values than coconut fiber for 21 days (0.40 – 1.27 cm) and 35 days (11.5 – 61.4 cm). The direct shear test results for Vetiver root with eggshell showed highest shear strength which is 43.67 kN/m2 (21 days) and 168.17 kN/m2 (35 days) compared to Vetiver root with coconut fiber which is 42.94 kN/m2 (21 days) and 61.13 kN/m2 (35 days). For tensile strength, the Vetiver root with egg shell consist higher value which is 0.066 N/mm² (group 1) and 0.068 N/mm² (group 2) compared to Vetiver root with coconut fiber which is 0.059 N/mm² (group 1) and 0.060 N/mm² (group 2). Additionally, in group 1 and 2 of 35 days, eggshell fertilizer showed higher maximum tensile strength, measuring 0.071 N/mm² and 0.094 N/mm², respectively. While, coconut fiber fertilizer 1 and 2 recorded lower tensile strengths of 0.063 N/mm² and 0.062 N/mm², respectively. The significance of this study indicates that the eggshell fertilizers have greater impact towards Vetiver root than coconut fiber fertilizers and without fertilizer.

Effects of new nonlinear curvature models and non-local elasticity on buckling investigation of FG tapered nano-beams locating on elastic foundations

New nonlinear curvature models are proposed to show their effects on the compressive critical buckling load on functionally graded (FG) tapered nano-beams locating on elastic foundations in the presence of non-local elasticity. Fourth and fifth orders finite difference methods (bvp4c and bvp5c) are applied on the governing differential equation which is a highly non-linear, due to curvature and has variable coefficients, due to properties. Dimensional analyses have been applied on the governing system together with the problem conditions to produce their dimensionless forms. For different considered parameters, the critical buckling load is computed using the eigenvalue theorem. The effects of nonlinear curvature, variable Young modulus, variable moment inertia and foundation parameters are studied and displayed in tables and figures. The stability and convergence of the present solution are tested using more than one strategy as: comparison with available results, comparison between bvp4c and bvp5c with different mesh intervals and tolerances of truncation errors. These comparisons show that, the new curvature model affects the critical buckling with different ratios depending on variability of properties and foundation parameters. The new results show that the nonlinear curvature model reduces the critical buckling load in comparison with the classical linear model, which, leads to decreasing in factor of safety in design of structures. The new results show that the design of structures will be influenced with the new reduction of the compressive critical buckling load, since it may affect the factor of safety in design of structures. The new results can lead, generally, to more reduction of critical buckling load for large or small input parameters and it will affect other critical values such that frequencies. The introduced Tables and Figures of the present problem with comparisons of previously exact, analytical and numerical works establish the higher accuracy and stability of current findings using bvp4c or bvp5c with preferable of the later. This study is related to some practical applications such as aeronautical and structural engineering.

Boosting tree with bootstrap technique for pre-stress design in cable dome structures

Tensegrity structures, known for their rigidity derived from feasible pre-stresses, present unique challenges in structural engineering. Traditional force-finding methods, though comprehensive, rely heavily on intricate matrix computations, making them computationally intensive and often uncomfortable for considering external loads in practical engineering scenarios. This paper introduces a novel approach to compute pre-stresses in cable dome structures by integrating machine learning and probability theory, collectively termed the boosting tree with bootstrap technique (BTWBT). This method reduces the sample size to as few as 100 per iteration, while improving computational efficiency by randomly generating internal forces. By reframing the force determination as an inverse problem, it ensures that structural displacement converges to zero under feasible pre-stresses. The effectiveness of BTWBT is demonstrated across three distinct cable dome structures: the Geiger dome, Kiewitt dome, and rotating hyperboloid cable dome. Results show that BTWBT achieves the preset displacement requirement (maximum nodal displacement below 0.01 mm) with fewer iterations and reduced computational cost compared to traditional machine learning methods. BTWBT's capability to manage complex structural configurations with minimal data, while incorporating random internal force generation ranges, highlights its potential as a superior alternative for force determination in tensegrity structures.

Amorphous mesoporous sulfur-rich 1T/2H-MoS2 nanospheres as high-capacity cathode materials for advanced magnesium ion batteries

Two-dimensional layered-structure MoS2 has been explored as the promising cathode materials for magnesium ion batteries benefited by its wide layer distance and satisfactory capacity. However, such strong interaction between Mg2+ and MoS2 brings about the sluggish diffusion kinetics and irreversible electrochemical reaction, leading to a decreased specific capacity and poor rate capability. How to adopt its structure engineering and interface engineering to boost the Mg2+ diffusion kinetics and enhance the electrochemical reversibility of MoS2 is still a huge challenge. Herein, the amorphous sulfur-rich 1T/2H-MoS2 mesoporous nanospheres (a-MoS2+x, 0<x<1) have been developed via a facile one-pot hydrothermal route. Acted as a cathode material for magnesium ion batteries, owing to the cooperate effects of inherent rich active sites, high conductivity and wide layer distance induced by the coexistence of sulfur-rich, 1T phase and nearly amorphous features, the as-prepared a-MoS2+x demonstrates an ultra-high discharge capacity (438.7 mAh g−1 for 50 cycles at 0.1 A g−1), promising rate capability (97.6 mAh g−1 at 1.0 A g−1), and good cyclic stability (110 cycles at 0.5 A g−1), superior to the crystalline MoS2 (c-MoS2) and other previously reported MoS2 counterparts. Besides, the kinetics results indicate that the a-MoS2+x manifests good charge transfer and diffusion kinetics, and its entire charge storage is a combination of surface-controlled capacitance behavior and diffusion-controlled battery behavior. Moreover, some ex-situ Raman, XPS and HRTEM characterization techniques are employed to disclose the Mg2+ storage mechanism. Such remarkable magnesium storage properties of a-MoS2+x demonstrates its promising application as cathode for magnesium ion batteries.

Enhancing mechanism of mechanical properties of lightweight and high-strength concrete prepared with autoclaved silicate lightweight aggregate

Lightweight aggregate concrete (LAC) possesses the merits of low density, excellent thermal insulation performance and outstanding seismic performance. However, the practical application in structural engineering is limited because its mechanical properties are weaker than those of ordinary aggregate concrete. It was found that the compressive strength of high-strength concrete prepared by autoclaved silicate lightweight aggregate (cloud concrete stone) was higher than that of ordinary aggregate high-strength concrete, and it showed significant lightweight and high-strength characteristics. To reveal the enhancing mechanism of the mechanical properties of high-strength concrete prepared by cloud concrete stone (CCS), this research examines the internal curing effect of CCS and the mechanical performances of the interfacial transition zones (ITZ). The influences of the elastic modulus of CCS, the shape of aggregate and fracture patterns on the mechanical properties of concrete are simulated via the discrete element method. The findings suggest that the internal curing effect of CCS delays the onset of capillary stress and mitigates autogenous shrinkage. The homogeneousness, elastic modulus, and fracture toughness of ITZ around CCS aggregate are superior than that ITZ around ordinary gravel in high-strength concrete. Furthermore, the small difference in elastic modulus between CCS aggregate and cement matrix is advantageous for preventing microcracks during compression tests. The round shape of the aggregate also mitigates the development of microcracks caused by the stress concentration effect. These three reasons contribute to the enhancement of mechanical properties in lightweight and high-strength concrete. The innovation of this research is that the intrinsic relationship between CCS aggregate and concrete’s mechanical properties is thoroughly revealed from the aspects of the characteristics of the interface transition zone, the morphology of the aggregate and the elastic modulus of aggregate. That provides a theoretical basis for the application of light aggregate in high-strength structural concrete.

Advancements in Rechargeable Zn-Air Batteries with Transition-Metal Dichalcogenides as Bifunctional Electrocatalyst.

This review covers recent progress on transition metal dichalcogenides (TMDs) as bifunctional electrocatalysts for Zinc-air batteries (ZABs), emphasizing their suitable surface area, electrocatalytic active sites, stability in acidic/basic environments, and tunable electronic properties. It discusses strategies like defect engineering, doping, interface, and structural modifications of TMDs nanostructures for enhancing the performances of ZABs. Zinc-air batteries are promising energy storage devices owing to their high energy density, low cost, and environmental friendliness. However, the development of durable and efficient bifunctional electrocatalysts is a major concern for Zn-air batteries. In this review, we summarize the recent progress on transition metal dichalcogenides (TMDs) as bifunctional electrocatalysts for Zn-air batteries. We discuss the advantages of TMDs, such as high activity, good stability, and tunable electronic structure, as well as the challenges, such as low conductivity, poor durability, and limited active sites. We also highlight the strategies for fine-tuning the properties of TMDs, such as defect engineering, doping, hybridization, and structural engineering, to enhance their catalytic performance and stability. We provide a comprehensive and in-depth analysis of the applications of TMDs in Zn-air batteries, demonstrating their potential as low-cost, abundant, and environmentally friendly alternatives to noble metal catalysts. We also suggest future directions like exploring new TMDs materials and compositions, developing novel synthesis and modification techniques, investigating the interfacial interactions and charge transfer processes, and integrating TMDs with other functional materials. This review aims to illuminate the path forward for the development of efficient and durable Zn-air batteries, aligning with the broader objectives of sustainable energy solutions.

High-modulus engineered cementitious composites: Design mechanism and performance characterization

Engineered cementitious composites features with high tensile performance while relatively weak compressive stiffness. This study endeavors to address the inherent trade-off between tensile properties and elastic modulus in conventional Engineered Cementitious Composites (ECC). Utilizing the distinctive characteristics of iron ore aggregates, known for their relatively stiff nature, smooth surface and rounded shape, an Iron Sand-based ECC (IS-ECC) emphasizing both high elastic modulus and ductility is formulated. Guided by micromechanical design theory and multiscale homogenization model, this study systematically explores the impacts of aggregate types, water-to-binder (w/b) ratios, sand-to-binder (s/b) ratios, and sand particle sizes on ECC properties. Compared with Quartz Sand-based ECC (QS-ECC), IS-ECC exhibits notably enhanced matrix fluidity and elastic modulus, reduced matrix toughness, and more robust strain-hardening behavior. The proposed three-level multiscale homogenization model accurately predicts the elastic modulus of ECC and provides insights into the underlying mechanism contributing to the enhanced elastic modulus of IS-ECC. With a resulting high elastic modulus of 33.3–48.6 GPa and superior tensile properties, IS-ECC holds promise for widespread applications in structural engineering.

Structural Design Considerations for a Large Single Span Reinforced Concrete Monolithic Beam-Column Structures

Large-span reinforced concrete structures present unique design challenges that necessitate meticulous consideration of load-bearing capacity, deflection control, reinforcement, and material choices. For spans exceeding 10 meters, structural engineers must carefully apply the principles and recommendations of Eurocode 2 to ensure that the structure is safe, functional, and durable. This paper explores the critical aspects of designing large single spans in reinforced concrete beam- column systems using Eurocode, with a focus on load distribution, reinforcement detailing, deflection limits, and material properties. Practical insights into the implementation of these guidelines are provided through examples, offering structural engineers in Nigeria a comprehensive approach to achieving safe and cost- effective long-span structures.

Fatigue Crack Propagation Life of Metallic Materials Under Random Loading: A Coupling Analysis Method in the Frequency Domain

ABSTRACTThis paper proposes an equivalent spectrum method to predict the fatigue crack propagation (FCP) life of metallic materials subjected to random loading. To adequately account for the coupling effects between crack propagation and the random response of structures, a coupling analysis model is introduced. The stress intensity factor (SIF) can be estimated based on the power spectral density (PSD) of an equivalent displacement. Random vibration fatigue tests were conducted to evaluate the proposed model on aluminum alloy specimens. Results indicate significant variations in natural frequency with crack length. The predicted results are compared with the experimental values, demonstrating satisfactory prediction accuracy of the proposed coupling analysis model. This model enables the assessment of coupling effects between crack length and random response, facilitating more precise predictions of FCP life in metallic materials and guiding the expanded application of damage tolerance criteria in structural engineering.

Effective reduction of iron impurities in recycled aluminium-silicon alloy type LM-6 through sedimentation and filtration with synthesized Mn, Cr and Zr transition metal powders

Abstract In secondary aluminium recycling, impurity reduction, especially of iron, is a significant challenge as it deteriorates the material quality in the aluminium melt. Recycled aluminium alloys are widely used in various industries, including automotive, marine, and structural engineering. Specifically, LM-6 aluminium alloy is commonly used in automotive engine parts and transmission cases. This research focused on reducing the excessive iron content in LM-6 alloy scrap. Elements like manganese (Mn), chromium (Cr) and zirconium (Zr) were added to the melt in form of synthesized powders to form intermetallic compounds. These powders will also act as grain refiners, neutralizers and reducing the detrimental effects on the alloys. The powders were synthesized using the ball milling method with a ball to powder ratio of (5:1) to enhance mixing and amalgamation with the melt. The melt was held at an optimized temperature of 640 ± 10 °C to promote the formation of intermetallic sludge, encouraging the sedimentation of aluminium-iron (Al-Fe) intermetallic phases. XRD confirmed the formation of AlMn, AlCr, AlZr with other phases. After sedimentation, a glass fabric mesh with a pore size of 200 μm was used to effectively filter out β-iron during the filtration process. This technique reduced iron impurity in the melt by 50%–60%. As a result, the mechanical properties of the recycled aluminium improved significantly: hardness increased by 14%, and tensile strength increased by 36%. The wear and corrosion resistance of the material also improved due to the incorporation of the synthesized powders. SEM analysis revealed the plate-like formation of iron structures and confirmed the presence of the manganese, chromium, and zirconium powders in the metallographic analysis.

Shakedown analysis of incompressible materials under cyclic loads: A locking-free CS-FEM-Q5 numerical approach

Volumetric locking may occur in plastic analysis of incompressible materials using low-order finite elements due to incompressibility constraints. This study presents a locking-free smoothed five-node quadrilateral element-based approach for plastic analysis in structural engineering. The proposed Q5-element employing four cell-based smoothing domains effectively alleviates the volumetric locking issues, here in the problems under plane strain conditions. The resulting large-scale optimization problem is formulated in a conic programming form, enabling efficient use of the interior-point optimizer. Numerical investigations demonstrate the method’s effectiveness in alleviating volumetric locking, accurately predicting collapse and shakedown limits, and generating interaction diagrams for load-carrying capacity and structural collapse mechanisms.

Optimal Electrocatalyst Design Strategies for Acidic Oxygen Evolution.

Hydrogen, a clean resource with high energy density, is one of the most promising alternatives to fossil. Proton exchange membrane water electrolyzers are beneficial for hydrogen production because of their high current density, facile operation, and high gas purity. However, the large-scale application of electrochemical water splitting to acidic electrolytes is severely limited by the sluggish kinetics of the anodic reaction and the inadequate development of corrosion- and highly oxidation-resistant anode catalysts. Therefore, anode catalysts with excellent performance and long-term durability must be developed for anodic oxygen evolution reactions (OER) in acidic media. This review comprehensively outlines three commonly employed strategies, namely, defect, phase, and structure engineering, to address the challenges within the acidic OER, while also identifying their existing limitations. Accordingly, the correlation between material design strategies and catalytic performance is discussed in terms of their contribution to high activity and long-term stability. In addition, various nanostructures that can effectively enhance the catalyst performance at the mesoscale are summarized from the perspective of engineering technology, thus providing suitable strategies for catalyst design that satisfy industrial requirements. Finally, the challenges and future outlook in the area of acidic OER are presented.

Stress-strain behaviour of pre-damaged concrete confined with natural fibre reinforced polymer

In recent years, using environmentally friendly construction materials has become a new trend in structural engineering. This research presents the application of natural fibre reinforced polymer (NFRP) composites for repairing damaged structural concrete members. The compressive behaviour of pre-peak damaged concrete confined with two types of NFRP composites was investigated. The tensile properties of jute fibre reinforced polymer (JFRP) and sisal fibre reinforced polymer (SFRP) were examined and compared with those of carbon fibre reinforced polymer (CFRP) using coupon tests. Subsequently, a series of pre-damaged concrete cylinders, confined with JFRP, SFRP, and CFRP, were subjected to uniaxial compression tests. The results showed that JFRP and SFRP are effective in providing confinement to pre-peak damaged concrete. The strength and ductility of pre-peak damaged concrete confined with NFRP increased with the number of NFRP layers. The cost-effectiveness of FRP-confined concrete was also evaluated, with JFRP being the most economically efficient. The eco-strength efficiency (ESE), defined as the increased compressive strength per unit of CO2 emissions, shows that JFRP is the most effective. This highlights JFRP's potential for sustainable construction. Finally, the compressive strength models were proposed. The models demonstrate improved accuracy in predicting the peak compressive strength of pre-peak damaged concrete confined with NFRP compared to existing models.

Machine-learning assistant DFT study of half-metallic full-Heusler alloy N2CaNa: Structural, electronic, mechanical, and thermodynamics properties

We studied the structural, electronic, mechanical, and thermodynamic properties of N2CaNa full Heusler alloys using density functional theory (DFT). Results for the structural analysis establish structural stability with a minimum formation energy of 29.9 eV. The compound is brittle and mechanically stable, having checked out with the Pugh criteria. The B/G ratio of bulk modulus B to shear modulus G for N2CaNa is 4.766, hence the material is ductile. N2CaNa alloy is ductile in nature. The Debye model correctly predicts the low-temperature dependence of heat capacity, which is proportional to Debye’s T3 law. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures, suggesting the thermodynamic stability of the compounds at moderate temperatures. The results demonstrate potential N2CaNa for applications in spintronics, structural engineering, and other fields requiring materials with tailored properties.

Steel Structures to Withstand the Elements: What Structural Engineers Need to Know about Corrosion

While regimented design processes for load-induced effects in structures are ubiquitous, similar design processes for considering corrosion resistance are lacking. This is a critical gap in the structural engineering profession as material degradation is the most common cause of diminishing structural condition for bridges and other infrastructure exposed to the elements. This results in both safety and financial consequences. This paper addresses this gap by reviewing basic principles governing corrosion, how these corrosion principles translate to real-world environments, commonly available corrosion protection systems, long-term field data assessing corrosion in varied quantified environments and associated conclusions, and practical design and maintenance strategies for improving corrosion resistance. These concepts are connected through a proposed framework for considering corrosion as a limit state that can be applied to all structures. Detailed consideration of uncoated weathering steel (UWS) bridges is provided as a pilot material and structure type for considering corrosion as a limit state. Thoughtful application of these concepts can be used to optimize corrosion resistance, improving life-cycle costs and service lives of civil engineering structures.

Acoustic emission monitoring of a large-scale 50-year-old prestressed concrete bridge girder during an eccentric three-point bending test

Deterioration of reinforced and prestressed concrete structures is one of the main challenges in structural engineering. As the majority of our infrastructure is built around 1960-1980, most structures have reached or will soon reach their design lifetime, giving rise to more structural problems and costs in the coming decades. It is therefore important to develop efficient assessment schemes allowing to assure the safety and reliability of these existing structures and to estimate their residual lifetime. Within the framework of the Bridge|50 research project, a 50-year-old prestressed concrete bridge showing signs of corrosion damage was decommissioned. The bridge girders were recovered and have been subjected to large-scale load tests with varying loading patterns. This paper presents the results acquired during acoustic emission (AE) monitoring of one of the I-shaped girders during a monotonic three-point bending test with an eccentric point load. The results show that the failure mode and location could be detected based on AE activity and zonal AE localization. The onset of concrete cracking was observed by a decrease in peak frequency of the AE signals. Average frequency (AF) and rise angle (RA) value analysis was used to characterize the crack mode, which shifted from tensile to shear mode during bending.

Snap-through and indirect reduced-order modelling

Snap-through is a nonlinear jump phenomenon that is increasingly being designed into engineering structures. The rich and varied dynamics of snap-through present a significant bottleneck in the design process as costly dynamic simulations are required to ensure predictable behaviour. The purpose of this research is to investigate the use of a recent projection-based, reduced-order modelling technique, namely, implicit condensation and expansion with inertial compensation, to reproduce the snap-through behaviours found in bistable finite element systems in analogous models with minimal degrees of freedom. In this work, the free response of a simple mass-spring oscillator is used to differentiate different types of snap-through. This provides a framework for discussing the dynamics of a family of buckled beams. The ability of the reduced-order models to faithfully reproduce these dynamics is investigated by testing the existence and stability of predicted periodic orbits. It is shown that the minimum projection basis required to capture snap-through accurately can be found through considering the minimum energy required for different types of snap-through. For bistable beams, this can be further simplified to just requiring an eigenvalue analysis at the most unstable equilibrium.