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Purpose - The purpose of this paper is to present eight local elasto-plastic beam element formulations incorporated into the corotational framework for two-noded three-dimensional beams. These formulations capture the warping torsional effects of open cross-sections and are suitable for the analysis of the nonlinear buckling and post-buckling of thin-walled frames with generic cross-sections. The paper highlights the similarities and discrepancies between the different local element formulations. The primary goal of this study is to compare all the local element formulations in terms of accuracy, efficiency and CPU-running time.Design methodology approach - The definition of the corotational framework for a two-noded three-dimensional beam element is presented, based upon the works of Battini .The definitions of the local element kinematics and displacements shape functions are developed based on both Timoshenko and Bernoulli assumptions, and considering low-order as well as higher-order terms in the second-order approximation of the Green-Lagrange strains. Element forces interpolations and generalized stress resultant vectors are then presented for both mixed-based Timoshenko and Bernoulli formulations. Subsequently, the local internal force vector and tangent stiffness matrix are derived using the principle of virtual work for displacement-based elements and the two-field Hellinger-Reissner assumed stress variational principle for mixed-based formulations, respectively. A full comparison and assessment of the different local element models are performed by means of several numerical examples.Findings - In this study, it is shown that the higher order elements are more accurate than the low-order ones, and that the use of the higher order mixed-based Bernoulli element seems to require the least number of FEs to accurately model the structural behavior, and therefore allows some reduction of the CPU time compared to the other converged solutions; where a larger number of elements are needed to efficiently discretize the structure.Originality value - The paper reports computation times for each model in order to assess their relative efficiency. The effect of the numbers of Gauss points along the element length and within the cross-section are also investigated.

The object of this study was to investigate the inhomogeneity of density within a beam from a relationship between the dynamic Young’s moduli from the Euler-Bernoulli elementary theory of bending (En) and resonance mode numbers (n), which is plotted as the “E-n” diagram in this article. Rectangular beams with dimensions of 300 (L) × 25 (R) × 5mm (T) of Sakhalin spruce (Picea glehnii Mast.), Sitka spruce (Picea sitchensis Carr.), Japanese red pine (Pinus densiflora Zieb. et Zucc.) and white oak (Cyclobalanopsis myrsinaefolia Oerst.) were used for specimens. Small parts of beams were replaced with a small portion of another species to examine the influence of the inhomogeneity of density on En. A free-free flexural vibration test was undertaken and En was calculated by the Euler-Bernoulli theory. The resonance frequency of a specimen with inhomogeneity of density was simulated by modal analysis. The density distribution in the longitudinal direction of the specimen for which En did not decrease monotonically with n was obtained. From the modal analysis, the inhomogeneity of density was equivalent to a concentrated mass attached to a uniform beam. The pattern of the E-n diagram was changed by replacing a part of the specimen with another species. Specimens for which En did not decrease monotonically with n had a high density part because of indented rings, knots, or resin.

This chapter contains sections titled: Linear Bending Theory Nonlinear Bending Theory Combined Bending and Axial Load

Soft actuators are composed of a small chamber pneumatic network embedded in an elastic structure, which has great potential in medicine, industrial automation, aerospace, agricultural picking, and other fields. There are few modeling or analysis theories about soft actuators, especially in three-dimensional spaces. This paper uses the minimum potential energy method and the Euler-Bernoulli beam theory. Combined with material and structural characteristics, a design method for soft actuators capable of torsion and bending deformations in three-dimensional spaces is proposed. The accuracy and feasibility of the model theory were verified through the Finite Element Method (FEM) and grasping experiments. Furthermore, the influence of changes in material parameters, structural parameters, and external loads on the bending of the soft actuator was analyzed, providing a theoretical basis for the subsequent structural design of the soft actuator.

The morphological dynamics, instabilities, and transitions of elastic filaments in viscous flows underlie a wealth of biophysical processes from flagellar propulsion to intracellular streaming and are also key to deciphering the rheological behavior of many complex fluids and soft materials. Here, we combine experiments and computational modeling to elucidate the dynamical regimes and morphological transitions of elastic Brownian filaments in a simple shear flow. Actin filaments are used as an experimental model system and their conformations are investigated through fluorescence microscopy in microfluidic channels. Simulations matching the experimental conditions are also performed using inextensible Euler–Bernoulli beam theory and nonlocal slender-body hydrodynamics in the presence of thermal fluctuations and agree quantitatively with observations. We demonstrate that filament dynamics in this system are primarily governed by a dimensionless elasto-viscous number comparing viscous drag forces to elastic bending forces, with thermal fluctuations playing only a secondary role. While short and rigid filaments perform quasi-periodic tumbling motions, a buckling instability arises above a critical flow strength. A second transition to strongly deformed shapes occurs at a yet larger value of the elasto-viscous number and is characterized by the appearance of localized high-curvature bends that propagate along the filaments in apparent “snaking” motions. A theoretical model for the as yet unexplored onset of snaking accurately predicts the transition and explains the observed dynamics. We present a complete characterization of filament morphologies and transitions as a function of elasto-viscous number and scaled persistence length and demonstrate excellent agreement between theory, experiments, and simulations.

In this paper, Functionally Graded Material (FGM) has been analyzed to examine bending and buckling of simply supported beams. Using Euler-Bernoulli beam theory (EBT), these beams that rested on Winkler-Pasternak elastic foundation are exposed to two types of loads that are axial compressive force and distributed transverse load. Here, based on power-law distributions, the properties of the material of FGM beam is assumed to be varied at the direction of the thickness. The derivation of the FGM beams’ governing equations was done using the total potential energy principle. The transverse deflection and the critical buckling of the FGM beam were determined using the Navier-type solution method with simple boundary conditions. A closure on the effects of the power-law exponent of FGM, and the spring constant with the shear constant of elastic foundation on the transverse deflection and critical buckling load was achieved. A validation study for numerical results was carried out here with previous results from the literature and they are said to be in excellent agreement. It is shown by the numerical results that critical buckling load is decreasing with increasing both, slenderness ratio and values of power-law exponent and vice versa for transverse deflection.

Nonlinear modeling of bolted joints is a highly challenging scientific task in structural dynamics. Considering the contact nonlinearity of joint interfaces, this study presents an analytical nonlinear vibration model for flange-bolted lap joints. Firstly, through co-simulations of finite element (FE) software Hyperworks and ANSYS, the detailed FE model of the flange is constructed, and the elastostatic properties are investigated numerically, indicating that the bending stiffness exhibits significant nonlinearity under positive and negative moments. Meanwhile, further studies indicate that the nonlinear bending stiffness is caused by interfacial contact nonlinearity. Subsequently, the nonlinear rotational springs with different bending stiffness are introduced to model the flexural behavior of the joint, and the nonlinear vibration model for the flange is further developed based on Euler–Bernoulli beam theory. The results indicate that the flange theoretically exhibits bi-linear flexural frequencies. Finally, the presented model is verified through impact experiments.

This paper presents the principle of virtual work (PVW) for piezoelectric semiconductors (PSs), which extends the piezoelectric dielectrics to involve the semiconducting effect. As an application of the PVW, a one-dimensional (1D) approximation theory for the extension and bending of PS nanowires is established by directly applying the PVW and Bernoulli–Euler beam theory with the aid of the second-order approximation of electrostatic potential. To illustrate the new model, the mechanical displacement, electrostatic potential, and concentration of electrons for extension and bending deformation of n-type ZnO nanowires are analytically determined. Additionally, numerical results show that, for n-type Zinc Oxide nanowires, the distribution of electrostatic potential is anti-symmetric along the thickness direction for extension deformation. In contrast, the bending deformation causes a symmetric distribution of electrostatic potential characterized by the zeroth-order and the second-order electrostatic potential. Furthermore, these two different deformations result in the redistribution of electrons. The electrostatic potential can be tuned by adjusting the amplitude of the applied mechanical load. Moreover, we find that the increase in doping level will reduce the magnitude of electrostatic potential due to the screening effect. The presented PVW provides a general approach to establishing structural theories and an effective way of implementing numerical methods.

This manuscript developed a comprehensive model and numerical studies to illustrate the effect of perforation parameters on critical buckling loads and static bending of thin and thick nanobeams for all boundary conditions, for the first time. Analytical closed-form solutions are presented for buckling loads and static deflections, respectively. Euler–Bernoulli beam theory is exploited for thin beam analysis, and Timoshenko beam theory is proposed to consider a shear effect in case of thick beam analysis. Nonlocal differential form of elasticity theory is included to consider a size scale effect that is missing in case of classical theory and macro-analysis. Geometrical adaptations for perforated beam structures are illustrated in simplest form. Equilibrium equations for local and nonlocal beam are derived in detail. Numerical studies are illustrated to demonstrate influences of long-range atomic interaction, hole perforation size, number of rows of holes and boundary conditions on buckling loads and deflection of perforated nanobeams. The recommended model is helpful in designing nanoresonators and nanoactuators used in NEMS structures and nanotechnology.