Classical Beam Theory Research Articles

Vibration response of nanobeams subjected to random reactions

Nanobeams composed of materials exhibiting flexoelectric properties have been successfully used in advanced technological equipment, including electronic circuits and very sensitive sensors, owing to their remarkable unique effects. Therefore, it is essential to determine their mechanical behavior as a practical need. This study employs an analytical methodology to provide a precise solution to the vibration issue of nanobeams under random stationary loads. Under such circumstances, the nanobeam is influenced by both the temperature and moisture conditions simultaneously. Additionally, the beam is supported by a viscoelastic foundation that considers both the viscous resistance parameter and the elastic parameter. Analytical formulations are derived by integrating classical beam theory with nonlocal strain gradient theory in order to elucidate the impact of size effects on nanobeams. This research also demonstrates the reliability verification, which clearly validates the accuracy of the calculation formula used in this work. This study examines the impact of material parameters, viscoelastic foundation, temperature, and moisture on the displacement spectrum at the middle and entire length of the beam. The study also explores how the flexoelectric effect decreases the displacement in the nanobeam. Subsequently, we provide scientifically derived findings that have significant relevance for building practical nanobeam specifications.

Critical lateral‐torsional buckling loads of porous orthotropic rectangular beams containing warping effects

AbstractDue to their elemental porosity, porous materials display individual mechanical properties, rendering them essential components in various engineering applications. So, this study investigates the lateral‐torsional buckling (LTB) sensitivity of porous orthotropic rectangular beams subjected to a concentrated load, including the warping effects. Porosity‐dependent material properties vary along the beam's height via four different porosity distribution patterns. The constitutive equations are obtained using the principle of virtual work based on the classical beam theory containing the warping effect, and the governing equations are solved by performing the Ritz method to obtain the critical LTB load. The porous orthotropic beams and the warping effects add complexity to the analysis. So, the influence of some parameters, such as porosity coefficients, porosity distribution patterns, orthotropy, height‐to‐width ratio, slenderness ratio, and warping coefficients, on the LTB characteristics of porous orthotropic rectangular beams are investigated in detail. The novelty of this study lies in its comprehensive LTB analysis of orthotropic rectangular beams under specific effects, such as porosity and warping factors. The findings offer valuable insights into the LTB characteristics of orthotropic porous beams, contributing to an improved comprehension of their structural behavior and facilitating the design of engineering structures. This study contributes to the composite materials and structural stability analysis, offering potential applications in many engineering domains.

Static Model of the Underwater Soft Bending Actuator Based on the Elliptic Integral Function

Underwater soft manipulators are increasingly used for grasping underwater organisms and cultural relics due to their compliance and ability to protect delicate objects. Unlike rigid manipulators, these soft manipulators feature underwater soft bending actuators that deform under external forces, making their shape and output force challenging to predict. Consequently, classical beam theory is no longer applicable. To address these issues, this article models the underwater soft bending actuator as a cantilever beam and establishes its shape and output force using elliptic integral functions. Additionally, this article proposes a method for determining the unknown parameters of the driver shape and output force model using optimization techniques and introduces a solution process based on the particle swarm optimization framework. Given the role of tensile and bending stiffness in the actuator’s performance, this paper employs a parameter identification method based on length and bending angle information. Tests demonstrate that the proposed method effectively estimates the deformation and output force of the underwater soft bending actuators, confirming its accuracy.

Characterization of pressure-dependent nonlinear bending behavior of yarns and its application in modeling the compression of 2D woven fabrics

This paper investigates the pressure-dependent nonlinear bending behavior of yarns, which is essential for the application of the virtual fiber modeling (VFM) method in the mechanical analyses of fabrics. An experimental method, along with a theoretical model based on classical beam theory, is presented to characterize the varying bending stiffness of yarns under different pressures. A tailored beam user element is then developed, incorporating the nonlinear bending behavior and combined with a truss element to create a physics-based virtual fiber formulation. Utilizing this formulation, the original kinematic VFM method is extended for modeling the mechanical response of 2D woven fabrics under compression. The predicted results of the proposed model closely match the reported experiment, demonstrating the significance of introducing the nonlinear bending behavior of yarns. This method can be a valuable tool for the fabric compression process and generating realistic mesoscale geometries for textile composites.

Evaluation and prediction of bending properties of glulam beams made from young plantation-grown Eucalyptus nitens

ABSTRACT Interest in making glulam from plantation-grown Eucalyptus nitens has increased, but current structural timber standards were not designed for species grown in plantations. Additionally, the applicability of classical composite beam theory remains uncertain due to the unique features of this species. This study was designed to assess the hypothesis that the bending performance of glulam made from young plantation-grown E. nitens is reliable and predictable. We tested the hypothesis through experimental procedures and theoretical and statistical analyses. First, we assessed the bending performance of 20 glulam beams, which were manufactured in five configurations. Second, we compared our experimental results with predictive values to evaluate the accuracy of methods in current standards. Third, in our pursuit of improved predictive tools, we developed theoretical and parametric models using mechanical properties of the board elements. Our improved models showed a decreased deviation in both stiffness and strength values. A Monte Carlo simulation was used to determine the distribution for both stiffness and strength. The results validated that this product can be allocated a GL grade using current standards and the bending performance is predictable. The study enhances confidence in incorporating E. nitens into glulam products and offers valuable data for future specification modifications.

Examination of Beam Theories for Buckling and Free Vibration of Functionally Graded Porous Beams.

This paper examines the accuracy and effectiveness of various beam theories in predicting the critical buckling loads and fundamental frequencies of functionally graded porous (FGP) beams whose material properties change continuously across the thickness. The beam theories considered are classical beam theory (CBT), first-order shear deformation beam theory (FSDBT), third-order shear deformation beam theory (TSDBT), and the broken-line hypothesis-based shear deformation beam theory (BSDBT). Governing equations for those beam theories are formulated by using the Hamilton's principle and are then solved by means of the generalised differential quadrature method. Finite element simulation solutions are provided as reference results to assess the predictions of those beam theories. Comprehensive numerical results are presented to evaluate the influences of the porosity distribution and coefficient, slenderness ratio, and boundary condition on the difference between theoretical predictions and simulation results. It is found that the differences significantly increase as the porosity coefficient rises, and this effect becomes more noticeable for the rigid beam with a smaller slenderness ratio. Nonetheless, the results produced by the BSDBT are always the closest to simulation ones. The findings in this paper will contribute to the establishment of more refined theories for the mechanical analysis of FGP structures.

A systematic model for the mechanical behavior of thin-walled composite FGM pipelines subjected to strike-slip faults in geohazard area

This paper focuses on the mechanical performances of the buried pipelines with functionally graded materials (FGM) subjected to strike-slip faults. A refined displacement function is introduced to derive the fault displacement and the strain distributions. The yield fault displacement is obtained by using the classical beam theory and the first yielding scheme. A finite element model is developed to verify the present derivations, and the verification indicates that the numerical results are in close agreement with the present analytical predictions characterized by the yield displacement and the strains, respectively. Finally, a parametric analysis is carried out to assess the factors that impact the performances of FGM pipelines, such as the diameter-to-thickness ratio of the pipeline, volume fraction exponent, fault inclination angles, the properties of the soil, and so on.

Buckling analysis of thin-walled laminated composite or functionally graded sandwich I-beams using a refined beam theory

This study presents the buckling and lateral torsional buckling behaviors of thin-walled laminated composite or functionally graded sandwich I-beams. A refined beam model (RBT) based on the 3D Saint-Venant (SV) solution is used in this study to formulate the problem. This model allows for a realistic analysis of beams with arbitrary cross-sections and is free from the limitations of classical beam theories. The displacement models establish a consistent 1D beam theory that accurately captures the essence of the cross-section’s nature. The governing models consider cross-section deformations, including in and out of plane warpings and distortions derived from the 3D SV solution associated with the cross-section vibrational behavior. Furthermore, a user-friendly numerical tool called CSB (cross-section and beam analysis) is used to facilitate the implementation of this method. The numerical results are examined in detail and compared with previous works to investigate the influence of boundary conditions, angle-ply, shear effects, span-to-height ratio, and material distribution on the critical buckling loads of thin-walled I-beams. The results demonstrate that the RBT models accurately and efficiently analyze thin-walled I-beams under different loads and boundary conditions. Furthermore, some of the new results are presented as reference values for the future.

3D novel zero Poisson’s ratio metamaterials with programmable thermal expansion and Young's modulus

The shape stability of the structure plays a very important role in deep space exploration engineering. This paper presents 3D novel zero Poisson’s ratio metamaterial (ZPRM) with programmable thermal expansion and Young's modulus through the combination of bi-material triangles and re-entrant hexagonal structures. The coefficient of thermal expansion (CTE) and Young's modulus of the ZPRM-1 are predicted by establishing theoretical analysis models based on the classical beam theory. The mechanical response behavior of the ZPRM −1 under small deformation is simulated by finite element analysis (FEA). 3D printing is adopted to design and prepare experimental specimens, and compression experiments are carried out to certify the accuracy of the theories and FEA of Young's modulus. The results of parameter analysis show that the ZPRM-1 can tune the CTE and Young's modulus in a wide range when the geometric parameters and material combination are changed. In addition, the mechanical properties of the ZPRM-1 under the combined field of temperature and mechanical loads are investigated to achieve the tunable Young's modulus from positive to negative. This study provides a feasible reference for temperature and mechanical sensitivity instruments.

Numerical investigation of smart material-based structures for vibration energy-harvesting applications

Abstract The present work deals with energy-harvesting devices, which are useful in scavenging power using piezoelectric materials. Utilizing classical beam theory and classical plate theory, finite element modelling has been carried out to optimize the performance of output power of a cantilever beam and a flexible rectangular plate. Harmonic oscillations and base excitation will be the two different forcing functions used to drive the system. Based on this, numerical investigations of the performance of output power for the piezoelectric cantilever beam and flexible rectangular plate with different cases are considered. The present work is also useful for designing the piezoelectric cantilever beams and plates to extract maximum output power within the frequency ranges from 0 to 200 Hz. Numerical investigations on the piezoelectric cantilever beam and flexible rectangular plate with different cases reveal that the performance of output power is influenced by factors like load resistance, applications of with and without host structures, and different design parameters like unimorph, bimorph, embedded, line- and cross-type piezoelectric patch arrangements.

Correction: Comparison of classical beam theory and finite element modelling of timber from fibre orientation data according to knot position and loading type

Correction: Comparison of classical beam theory and finite element modelling of timber from fibre orientation data according to knot position and loading type

Behavior of Modified Strut-and-Tie Model (STM) on Concrete Beams

Reinforced concrete components are generally designed to withhold shear and bending based on the assumption at strain varies linearly in a section where the applied force is a combination of shear with bending, torsion, or normal forces. The behavior of shear failure in reinforced concrete beams is very different from that in flexural failure. Shear failure is brittle without warning in the form of significant deflection, resulting in diagonal cracks in the beam then the shear force mechanism will be contributed by an arching action where this action can provide a reserve capacity large enough for the beam to carry the load. Nonlinear analysis using the strut-and-tie method is particularly useful for shear-critical structures where classical beam theory is not valid due to significant shear deformations. Strut-and-Tie Model research which is applied to concrete (25 MPa), also uses the optimal configuration FEM software tool from Strut-and-Tie which leads to the efficiency of Strut-and-Tie. The results of the optimization and modification of the Strut-and-Tie Model on concrete will also be applied to the experimental models tested until they fail so that the optimal conditions for numerical models will be obtained.Based on the analysis result of element model using ANSYS computational assistance program, the ultimate flexural capacity of the beam model will increase depending on the used STM model and inclination angle (Φ), at the STM type 2 high < 1000 mm, inclination angle (Φ) 45° having a decrease in ultimate shear capacity (Vu) of 49,31% against type 1, at the STM model 3 high < 1000 mm, with inclination angle (Φ) 45° having an increase ultimate shear (Vu) of 4,14% against type 1. Stress pattern performed bottle shape in line with diagonal strut. Ductility capacity will be decreased at inclination angle <45° of 27,11%, at inclination angle >45° will be decreased of 55,67%,

A novel metamaterial with instantaneously sign-switchable coefficient of thermal expansion and Poisson's ratio

Most materials expand when heated and contract when cooled. In contrast, negative thermal expansion materials exhibit the opposite behavior. In this paper, we present a metamaterial that expands both when heated and cooled. The unit cell configuration is crafted by embedding two isosceles triangular structures composed of high thermal expansion materials within a re-entrant hexagonal honeycomb framework. Initially, the bases of the two isosceles triangles are in contact. The metamaterial exhibits two deformation states under different external loads. Analytical formulas for the equivalent parameters were derived based on classical beam theory within the regime of small deformation. Finite element simulations were conducted to validate the accuracy of these analytical formulas. The influence of geometric parameters on the metamaterial is discussed in detail. The results demonstrate that the metamaterial can exhibit excellent multi-functional properties within a small range of deformation, including instantaneous sign-variable coefficients of thermal expansion and Poisson's ratios, along with significantly distinct tensile and compressive stiffnesses. This metamaterial consistently exhibits expansive characteristics regardless of variations in ambient temperature. Additionally, the material always expands laterally regardless of the sign of uniaxial loads. Based on these properties, the metamaterial can be applied to fasteners in the aerospace industry that require permanent locking, ensuring stability in various temperature and mechanical load conditions.

Comparison of classical beam theory and finite element modelling of timber from fibre orientation data according to knot position and loading type

Comparison of classical beam theory and finite element modelling of timber from fibre orientation data according to knot position and loading type

BROM: An extension of beam theory through model order reduction

Beam analysis played a crucial role in the design of structures throughout modern history. Classical beam theories rely on analytical derivations under certain a priori kinematic assumptions (and are accurate with low computational cost), but they are not amenable to, for example, orthotropic materials and arbitrary cross sections. On the other hand, full 3D linear elasticity theory solved via the Finite Element (FE) method provides solutions for a wide range of engineering problems at the expense of a high computational cost. Reduced-order models try to provide a theory as general as 3D linear elasticity but as efficient as classical beam theories by assuming the solution to be represented in a low-dimensional basis. Among the several options available in the literature, the domain decomposition and hyperreduction via Empirical Cubature Method (ECM), which we name ddROM [1], exhibits this compelling feature. However, ddROM applied to beam structures, may provide more Degrees of Freedom (DoF) per node than most standard commercial FE codes which only consider rigid body motion of the cross section. This paper presents a methodology, called beam Reduced Order Model (bROM), to integrate ddROM in standard FE codes by means of a condensation process followed by a regression procedure. Condensation gives the stiffness matrix, required by FE commercial codes, of a super-element of a given length. This is achieved by assembling several elements and finding the relationship between applied displacements and reactions (Lagrange multipliers). The condensation process is then used to create a database of stiffness matrices as a function of the beam length over which a regression is performed. Some numerical examples are first presented to validate the proposed methodology against the classical beam theory for the case of isotropic beams, and finally, two orthotropic beam showing the utility of the method in comparison with FE.

Beam theory of cellular microfilaments based on coarse-grained molecular force field

Microfilaments are the main component of the cell cytoskeleton and have a crucial influence on the morphology of living cells. The microfilaments provide mechanical support against applied loads and overcome competing forces that arise from various mechanisms, such as the elasticity of the cell membrane, binding reactions, and surface tension. However, modeling the mechanical properties of microfilaments is an open problem. In the present paper, we proposed a new continuum deformation energy density of microfilaments using the coarse-grained molecular force field and the local differential geometry of curves. In the new theory, the deformation energy density is a function of the microfilament’s strain and the curvature. This theory also unfolded the connections between the mechanical parameters and the coarse-grained force field. We found that the stretching stiffness and the bending stiffness are independent of the section of microfilaments. Microfilaments have a nonlinear stretching-bending term different from the classical beam theory. This means the classical beam theory cannot be directly applied to microfilaments with large deformations. Our theory provides the basis that the modified classical beam theory applies to the nonlinear mechanical behavior of cellular microfilaments.

Comprehensively characterizing heterogeneous and transversely isotropic properties of femur cortical bones

Comprehensive characterization of the transversely isotropic mechanical properties of long bones along both the longitudinal and circumferential gradients is crucial for developing accurate mathematical models and studying bone biomechanics. In addition, mechanical testing to derive elastic, plastic, and failure properties of bones is essential for modeling plastic deformation and failure of bones. To achieve these, we machined a total of 336 cortical specimens, including 168 transverse and 168 longitudinal specimens, from four different quadrants of seven different sections of 3 bovine femurs. We conducted three-point bending tests of these specimens at a loading rate of 0.02 mm/s. Young’s modulus, yield stress, tangential modulus, and effective plastic strain for each specimen were derived from correction equations based on classical beam theory. Our statistical analysis reveals that the longitudinal gradient has a significant effect on the Young’s modulus, yield stress, and tangential modulus of both longitudinal and transverse specimens, whereas the circumferential gradient significantly influences the Young’s modulus, yield stress, and tangential modulus of transverse specimens only. The differences in Young’s modulus and yield stress between longitudinal specimens from different sections are greater than 40%, whereas those between transverse specimens are approximately 30%. The Young’s modulus and yield stress of transverse specimens in the anterior quadrant were 18.81%/15.46% and 18.34%/14.88% higher than those in the posterior and lateral quadrants, respectively. There is no significant interaction between the longitudinal gradient and the circumferential gradient. Considering the transverse isotropy, it is crucial to consider loading direction when investigating the impact of circumferential gradients in the anterior, lateral, medial, and posterior directions. Our findings indicate that the conventional assumption of homogeneity in simulating the cortical bone of long bones may have limitations, and researchers should consider the anatomical position and loading direction of femur specimens for precise prediction of mechanical responses.

Artificial neural networks coupled with numerical approach for the stability prediction of non-uniform functionally graded microscale cylindrical structures

Functionally graded materials (FGMs) have garnered significant attention in research due to their potential for creating innovative materials with diverse applications. This study focuses on investigating the behavior of steel-concrete functionally graded structures under axial loading, specifically exploring nonlinear buckling. The research considers cylindrical structures with simply-supported boundary conditions and analyzes four different configurations with varying thickness. The governing equations are derived using the modified couple stress theory, incorporating assumptions of nonlinear von-Kármán theory coupled with classical beam theory. Numerical solutions are obtained using the general differential quadrature (GDQ) method. A notable aspect of this work is the use of Artificial Neural Networks (ANN) to predict nonlinear buckling forces, greatly enhancing the accuracy and efficiency of the predictions. This approach offers a novel method for analyzing concrete structures. The results are validated by comparing them with data from other published works, using dimensionless parameters for consistent comparisons. The findings provide a comprehensive analysis of various parameters, including wall thickness function and porosity, on the buckling stability of the cylindrical structure. Both numerical and ANN approaches are employed to report the effects. The study concludes that the uniform thickness cylinder exhibits the highest resistance to axial loading, resulting in buckling.

Computational study of a homogenized nonlinear generalization of Timoshenko beam proposed by Turco et al.

&lt;p&gt;Mechanical metamaterials are most often assemblies of stocky beam elements connected through rigid connections, hinges, or flexural joints. The description of these materials through classical beam theories is challenging because of the wide variety of complex phenomena observed in the severe deformation regime mechanical metamaterials must undergo and because most classical beam theories can only be applied to elements with sufficiently high slenderness. In the spirit of Hencky, Turco et al. (2020) has recently formulated an intrinsically discrete nonlinear elastic model suitable for the design of mechanical metamaterials. The objective of this contribution was to present a numerical study of the nonlinear generalization of the Timoshenko beam that results from the asymptotic homogenization of the discrete model introduced by Turco et al. The present numerical study took into account several loading cases and elucidated the sensitivity of the homogenized continuum with respect to axial, bending, and shear stiffness parameters, as well as to load imperfections, in terms of mechanical behavior, including buckling onset and post-critical behavior. It was found that the predictions obtained with the homogenized model in the large deformation regime matched excellently with those of the discrete model proposed by Turco et al.&lt;/p&gt;

Generic and simplified approaches for the structural analysis of precast concrete sandwich panels

This paper presents a comparison between three numerical models developed for the structural analysis of composite precast concrete sandwich panels under lateral loading. The first model is based on a full finite element analysis where “slip” between the layers is obtained from the elastic deformability of the various components. The second model is based on a simplified finite element analysis where the effect of the shear connectors is smeared and modeled through an effective shear stiffness of the insulation layer. The third approach provides closed-form expressions for evaluating the deflection and is based on the classical sandwich beam theory where the shear connectors are considered through the effective stiffness of the insulation, similar to the second model. The structural response obtained from the three models was compared for various panel configurations and good correlation was obtained. A reasonable agreement with test results from the literature was also demonstrated. Therefore, both the simplified finite element model and the classical sandwich beam theory (closed-form solution) are recommended for the elastic structural analysis and for estimating the degree of composite action at the serviceability limit state of precast concrete sandwich panels.