Completely hierarchical two-dimensional curved beam element for dynamics

Three-dimensional curved beam element based on p-Version for dynamics

Strut-and-tie and finite element modelling of unsymmetrically-loaded deep beams

p-Version axisymmetric shell element for geometrically nonlinear analysis

In order to enhance currently used beam theories in $$\mathbb {R}^2$$ and $$\mathbb {R}^3$$ to include mechanisms of dissipation and memory, it is necessary to establish if the mathematical models for these theories can be derived using the conservation and the balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the beam mathematical models. Thermodynamic consistency of the currently used beam models will permit use of entropy inequality to establish constitutive theories in the presence of dissipation and memory mechanism in the currently used beam theories. This is the main motivation for the work presented in this paper. The currently used beam theories are derived based on kinematic assumptions related to the axial and transverse displacement fields. These are then used to derive strain measures followed by constitutive relations. For linear beam theories, strain measures are linear functions of displacement gradients and stresses are linear functions of strain measures. Using these stress and strain measures, energy functional is constructed over the volume of the beam consisting of kinetic energy, strain energy and potential energy of loads. The Euler’s equation(s) extracted from the first variation of this energy functional set to zero yields the differential equations describing the evolution of the deforming beam. Alternatively, principle of virtual work can also be used to derive mathematical models for beams. For linear elastic behavior with small deformation and small strain, the two approaches yield same mathematical models. In this paper we examine whether the currently used beam mathematical models with the corresponding kinematic assumption (i) can be derived using the conservation and balance laws of classical continuum mechanics or (ii) are the conservation and balance laws of non-classical continuum mechanics necessary in their derivation. In order to ensure that the mathematical models for various beam theories result in deformation that is in thermodynamic equilibrium we must establish the consistency of the beam theories with regard to the conservation and the balance laws of continuum mechanics, classical or non-classical in conjunction with their corresponding kinematic assumptions. Currently used Euler–Bernoulli and Timoshenko beam mathematical models that are representative of most beam mathematical models are investigated. This is followed by details of general and higher-order thermodynamically consistent beam theory that is free of kinematic assumptions and other approximations and remains valid for slender as well as deep beams. Model problem studies are presented for slender as well as deep beams. The new formulation presented in this paper ensures thermodynamic equilibrium as it is derived using the conservation and the balance laws of continuum mechanics and remains valid for slender as well as non-slender beams.

This paper proposes a numerical solution to deep beams using the layerwise displacement theory. Most of the methods for performing structural analyses of deep beams have geometric and boundary conditions limitations, as well as modeling inconveniences. This paper provides a finite element solution for deep beams based on a layerwise displacement field considering the full stress/strain tensors. In this formulation, the cross section is discretized in a pre-defined number of independent virtual layers, with linear interpolation within the thickness direction. To validate the model developed, two numerical examples are analyzed. The first is a reinforced concrete deep beam with two supports, loaded over the top face, validated by finite element analysis based on solid-element ABAQUS™ software. Next, an isotropic deep beam with both ends cantilevered is analyzed and the outcome is compared to the literature. The results of both numerical examples are accurate and can estimate the complete state of stress over all domains of the element. Moreover, the layerwise formulation does not suffer from shear and membrane locking, and it may use fewer computational resources than equivalent 3D finite element analyses.

This study aims to experimentally investigate the performance of steel fiber-reinforced concrete deep beams with various sizes of square web-opening applied to shear and shear-bending control loading scenarios. Twelve specimens were distributed to two groups according to loading control (shear & shear-bending). Each group consist of three beams prepared of fibrous concrete and others prepared of normal concrete. The three specimens consists of one that is solid as a reference and others containing various-sized square web openings (200 mm & 300 mm). The test results show that all the tested beams failed as shear failure except for the solid beams under shear-bending control, which failed as flexural failure. The presence of square openings with sharp corners and edges leads to high stress concentration, especially when these openings are located in the natural load transmission path. Steel fibers maintain the linear behavior in deep beams with openings until late loading stages, as evident in load-deflection curves. Fibrous beams with openings remained nearly identical. Unlike normal concrete beams, there was a large difference in the load-deflection curves since the early stages of loading. The tested beams failed after yielding of steel reinforcement, as the steel strain of fibrous specimens was lower than that of normal concrete. Steel fibers contributed to reducing stresses in reinforcing steel. Fibrous beams exhibit higher stiffness than normal beams, largely due to the presence of steel fibers. These fibers provide additional tensile strength, resisting shear cracks. Additionally, fibers increase the ductility index and absorption energy, redistributing stresses and provide warning before failure

The present work fills a gap on the postbuckling behavior of multilayer functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and spherical shell panels resting on elastic foundations subjected to central pinching forces and pressure loadings. Based on a higher-order shear deformation theory and the von Karman’s nonlinear strain–displacement relations, the governing equations of the FG-GPLRC cylindrical and spherical shell panels are established by the principle of virtual work. The non-uniform rational B-spline (NURBS) based isogeometric analysis (IGA), the modified arc-length method and the Newton’s iteration method are employed synthetically to obtain nonlinear load–deflection curves for the panels numerically. Several comparative examples are performed to test reliability and accuracy of IGA and arc-length method in present formulation and programming implementation. Parametric investigations are carried out to illustrate the effects of dispersion type of the graphene platelet (GPL), weight fraction of the GPL, thickness of the panel, radius of the panel and parameters of elastic foundation on the load–deflection curves of the FG-GPLRC shell panels. Some complex load–deflection curves of the FG-GPLRC cylindrical and spherical shell panels resting on elastic foundations may be useful for future references.

Form function matrix is created by introducing high order displacement interpolation function in the node. Based on the virtual work principle and dynamic finite element theory, the spatial element stiffness matrix, mass matrix and earthquake mass matrix of a thin-walled box girder having 9 freedom degrees at each node are deduced. The D’Alembert vibration equation is also established. Newmark-β method is used through MATLAB to solve the seismic response of a long-span continuous curved box girder bridge under El-centro seismic waves. Meanwhile the spatial finite element model of the whole bridge is established by ANSYS. The results indicate that the dynamic responses of pier columns exhibit spatiality. The dynamic response of a bridge structure under 2D coupling horizontal seismic excitation is much bigger than that under 1D horizontal seismic excitation. The critical angle of seismic waves is 50° for radial displacement response. Theoretical calculation results are in agreement with the finite element analysis results. The deduced element matrix not only can be used to calculate the seismic response of long-span curved beam bridge structures but also can provide significant references for the structures in vibration response caused by moving traffic.

Wave Propagation In Multiply Connected Deep Waveguides

Instead of using the previous straight beam element to approximate the curved beam, in this paper, a curvilinear coordinate is employed to describe the deformations, and a new curved beam element is proposed to model the curved beam. Based on exact nonlinear strain-displacement relation, virtual work principle is used to derive dynamic equations for a rotating curved beam, with the effects of axial extensibility, shear deformation and rotary inertia taken into account. The constant matrices are solved numerically utilizing the Gauss quadrature integration method. Newmark and Newton-Raphson iteration methods are adopted to solve the differential equations of the rigid-flexible coupling system. The present results are compared with those obtained by commercial programs to validate the present finite method. In order to further illustrate the convergence and efficiency characteristics of the present modeling and computation formulation, comparison of the results of the present formulation with those of the ADAMS software are made. Furthermore, the present results obtained from linear formulation are compared with those from nonlinear formulation, and the special dynamic characteristics of the curved beam are concluded by comparison with those of the straight beam.

Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

On base of the principle of virtual work, the inverse dynamic model of the 2-DOF parallel platform is set up. The dynamic formulation is a system of equations including two equations and three unknown quantities referring to three driving forces. The optimum solution of the driving forces is carried out by employing the generalized inverse matrix with minimizing an objective function which is the square of the sum of three driving forces. The parameters of drivers are designed.

Single-degree-of-freedom lumped parameter model, conventional finite element method, and distributed parameter model have been developed to design, analyze, and predict the performance of piezoelectric energy harvesters with reasonable accuracy. In this article, a spectral finite element method for bimorph piezoelectric beam energy harvesters is developed based on the Timoshenko beam theory and the Euler–Bernoulli beam theory. Linear piezoelectric constitutive and linear elastic stress/strain models are assumed. Both beam theories are considered in order to examine the validation and applicability of each beam theory for a range of harvester sizes. Using spectral finite element method, a minimum number of elements is required because accurate shape functions are derived using the coupled electromechanical governing equations. Numerical simulations are conducted and validated using existing experimental data from the literature. In addition, parametric studies are carried out to predict the performance of a range of harvester sizes using each beam theory. It is concluded that the Euler–Bernoulli beam theory is sufficient enough to predict the performance of slender piezoelectric beams (slenderness ratio > 20, that is, length over thickness ratio > 20). In contrast, the Timoshenko beam theory, including the effects of shear deformation and rotary inertia, must be used for short piezoelectric beams (slenderness ratio < 5).

This paper proposes a method to compute positioning error of a parallel mechanism having passive spherical joints with clearances in motion without numerical integration nor iterative calculation. A model of passive spherical joints, in which the relationship between the relative displacement between the joint elements and the joint force obtained without taking the clearance into consideration is defined, is presented. Based on this model and the principle of virtual work, a procedure to calculate the output position error caused by joint clearance is proposed. Experiments for a DELTA parallel robot with joint clearances were conducted to evaluate the effectiveness of the proposed method comparing the measured output position error with the computational results. The applicability of the proposed method was discussed with a time variable for the relative displacement of the joint elements to converge, when the joint reaction force changes, defined as stabilization time. An index, representing an approximation of the stabilization time, was defined under consideration of rebounding and sliding motions between joint elements. The validity of the index was confirmed by investigating the correspondence between calculated index and the stabilization time obtained with experiments when changing the acceleration of the robot.

Large deformations of cord-reinforced multilayered shells