Beam Problem Research Articles

Port-Hamiltonian Modeling and IDA-PBC Control of an IPMC-Actuated Flexible Beam

In this paper, the infinite-dimensional port-Hamiltonian modelling and control problem of a flexible beam actuated using ionic polymer metal composite (IPMC) actuators is investigated. The port-Hamiltonian framework is used to propose an interconnected control model of the mechanical flexible beam and the IPMC actuator. The mechanical flexible dynamic is modelled as a Timoshenko beam, and the electric dynamics of the IPMCs are considered in the model. Furthermore, a passivity-based control-strategy is used to obtain the desired configuration of the proposed interconnected system, and the closed-loop stability is analyzed using the early lumped approach. Lastly, numerical simulations and experimental results are presented to validate the proposed model and the effectiveness of the proposed control law.

A geometrically nonlinear structural formulation for analysis of beams with a new set of generalized displacements considering piezoelectric effects

AbstractIn this article, a nonlinear structural formulation that uses a new dimensionless set of generalized displacements is proposed for solving geometrically nonlinear beam problems, being validated by several applications given in literature where a cantilever beam is likely to undergo large displacements. By this approach, useful simplifications and insights are achievable in the analysis process, as the system matrices becoming linear and the reduction of required interpolation continuity degree. The formulation is firstly developed—in a Lagrangian perspective—and the equilibrium equations are then derived using Hamilton's principle. In the sequence, using finite element method, it is substantiated by comparison to examples given in literature in static, dynamic, and finally in an application where piezoelectric effects intervene, in order to assess its multiframework capabilities and deliver a convenient approach whereby beams constituted of smart or conventional materials can be efficiently studied.

Nonlinear solutions for the steady state oscillations of a clamped–free rotating beam

The rotating beam problem has been extensively used as a benchmark for testing nonlinear finite element implementations. The remarkable characteristic of this benchmark is the coupling between axial a transverse deformations due to the centrifugal forces. The rotating beam dynamics exhibits a centrifugal stiffening effect that can only be captured either by including a kinematic coupling between axial and transverse deformation or by using a nonlinear description of the elastic forces. This paper presents simplified models of the rotating beam that capture the centrifugal stiffening effect due to the inclusion of an axial to transverse kinematic coupling of the beam centreline. The simplified models allow a discussion on their accuracy and a meaningful analysis of the nonlinear oscillations present in the steady state rotation. The results of the simplified models are compared to finite element solutions obtained by using the absolute nodal coordinate formulation and a commercial FEM code.

Displacement-function modeling of thermo-mechanical behavior of fiber-reinforced composite structures

A new modeling scheme has been developed for the analysis of thermo-mechanical stresses in structural problems of fiber-reinforced composite materials with mixed mode of physical conditions. In contrast with conventional approaches, the coupled thermal and mechanical problems of structural mechanics are integrated here using a single scalar function of space variables, which is defined in terms of the displacement components of plane elasticity. The displacement-function is governed by a single partial-differential equation of equilibrium that takes into account the two orthogonal gradients of the resulting non-uniform temperature field through two different versions of the formulation. The temperature-dependent elastic field is determined using a two-step solution scheme that utilizes the two versions of displacement-function formulations in two sequential steps. The scheme is developed in such a flexible fashion that even the solution of the corresponding isotropic problems can readily be obtained with thermal and/or mechanical loading as a limiting case of the formulation. The application of the proposed modeling scheme is demonstrated through the solution of a number of problems of structural mechanics, namely plates and beams subjected to thermal as well as thermo-mechanical loadings with both uniform and mixed mode of physical conditions. An attempt is also made to verify the soundness and accuracy of the resulting thermoelastic fields by comparing them with the corresponding analytical as well as computational solutions available in the literature. Moreover, the correspondence between the versions of the displacement-function formulation is also verified by solving identical problems using different versions or different combinations of the versions of the formulation.

A robust and scalable unfitted adaptive finite element framework for nonlinear solid mechanics

In this work, we bridge standard Adaptive Mesh Refinement and coarsening (AMR) on scalable octree background meshes and robust unfitted Finite Element (FE) formulations for the automatic and efficient solution of large-scale nonlinear solid mechanics problems posed on complex geometries, as an alternative to standard body-fitted formulations, unstructured mesh generation and graph partitioning strategies. We pay special attention to those aspects requiring a specialized treatment in the extension of the unfitted h-adaptive Aggregated Finite Element Method (h-AgFEM) on parallel tree-based adaptive meshes, recently developed for linear scalar elliptic problems, to handle nonlinear problems in solid mechanics. In order to accurately and efficiently capture localized phenomena that frequently occur in nonlinear solid mechanics problems, we perform pseudo time-stepping in combination with h-adaptive dynamic mesh refinement and re-balancing driven by a-posteriori error estimators. The method is implemented considering both irreducible and mixed (u/p) formulations and thus it is able to robustly face problems involving incompressible materials. In the numerical experiments, both formulations are used to model the inelastic behavior of a wide range of compressible and incompressible materials. First, a selected set of benchmarks is reproduced as a verification step. Second, a set of experiments is presented with problems involving complex geometries. Among them, we model a cantilever beam problem with spherical hollows distributed in a simple cubic (SC) array. This test involves a discrete domain with up to 11.7M Degrees Of Freedom (DOFs) solved in less than two hours on 3072 cores of a parallel supercomputer.

Static analysis of composite beams on variable stiffness elastic foundations by the Homotopy Analysis Method

New analytical solutions for the static deflection of anisotropic composite beams resting on variable stiffness elastic foundations are obtained by the means of the Homotopy Analysis Method (HAM). The method provides a closed-form series solution for the problem described by a non-homogeneous system of coupled ordinary differential equations with constant coefficients and one variable coefficient reflecting variable stiffness elastic foundation. Analytical solutions are obtained based on two different algorithms, namely conventional HAM and iterative HAM (iHAM). To investigate the computational efficiency and convergence of HAM solutions, the preliminary studies are performed for a composite beam without elastic foundation under the action of transverse uniformly distributed loads considering three different types of stacking sequence which provide different levels and types of anisotropy. It is shown that applying the iterative approach results in better convergence of the solution compared with conventional HAM for the same level of accuracy. Then, analytical solutions are developed for composite beams on elastic foundations. New analytical results based on HAM are presented for the static deflection of composite beams resting on variable stiffness elastic foundations. Results are compared to those reported in the literature and those obtained by the Chebyshev Collocation Method in order to verify the validity and accuracy of the method. Numerical experiments reveal the accuracy and efficiency of the Homotopy Analysis Method in static beam problems.

Dynamic simulation for beam to beam frictionless contact using a novel region detection algorithm

The main goal of this work is to present the surface to surface frictionless contact dynamic formulation of flexible beam to beam based on the absolute nodal coordinate formulation (ANCF), and the region contact problem of flexible beam is simulated and analyzed. The​ high-order beam element of ANCF is employed to describe the warping deformation of the beam section in the contact process. Compared with ABAQUS, this modeling method only needs a small number of elements and has higher efficiency. In addition, a novel contact zone detection algorithm is proposed, which uses curved surface to describe the contact between beams, and can detect the contact state accurately when the beam diameter changes. On the other hand, the shape and position of the contact zone can be obtained using this algorithm, which has not been involved in previous studies. Next, combined with penalty method, the dynamic model of flexible beam to beam is established. Finally, three numerical examples are given to verify the effectiveness of the proposed approach in dealing with large zone contact problems of flexible beams.

Analysis of Timoshenko–Ehrenfest beam problems using the Theory of Functional Connections

The numerical solution of boundary value problems in solid mechanics is dominated by the Finite Element Method (FEM). The present study provides an alternative numerical approach using the Theory of Functional Connections (TFC) for solving problems of this type. Here TFC is used to solve multiple beam problems using the Timoshenko–Ehrenfest beam theory which accounts for transverse shear strain. The results obtained are then compared to those obtained with both the FEM and the analytical approaches. For cases where the analytical solution is continuous and smooth, TFC out performs FEM both in terms of the L2 norm of the residuals and the solution time. When the analytical solution is continuous and smooth, TFC results in an L2 norm on the order of machine-error level precision. However, when the higher order derivatives of the analytical solution are discontinuous (i.e. when the applied load is discontinuous), TFC under performs compared to FEM. This is due to TFC approximating the solution by continuous functions when the true solution is piece-wise continuous. In terms of computational expense, TFC consistently solved the differential equations faster than FEM across each type of problem analyzed in this paper.

Linear and nonlinear static bending of sandwich beams with functionally graded porous core under different distributed loads

In this investigation, linear and nonlinear bending analyses of sandwich beams with functionally graded cores are determined under different types of distributed loads. These sandwich beams are composed of two isotropic faces and a porous core with different gradients of internal pores. The governing formulation used to describe the beam’s linear and nonlinear behavior is constructed from Reddy's third-order shear deformation theory and nonlinear strain–displacement relations of von Kármán. The Gram-Schmidt orthogonalization procedure is adopted to generate numerically stable functions for the displacement field to solve the beam problems with various boundary conditions. Then, the Ritz method is utilized to find out linear and nonlinear bending results in conjunction with the iterative technique. The accuracy of our solutions is validated, and our numerical results agree well with some cases available in the literature. New results of the sandwich beams based on several effects of porosity coefficient, slenderness ratio, loading types, porous distributions of the core, etc., are presented in graphical and tabular forms, serving as a benchmark solution for future studies.

Timoshenko beam and plate non-stationary vibrations

The problems of Timoshenko beams and plates lateral vibrations under the influence of unsteady loads are considered. Both beam and plate is supposed to be unlimited. In case of the plate the problem has been simply studied. The approach to the solution was based on dominant function method and principle of superposition. Integral models of solutions with cores as dominant functions were built which could be analytically found with the help of the Fourier and Laplace integral transforms. Two original analytical methods for Fourier and Laplace transforms were offered and realized. The examples of calculations were given.

Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory

Beam theories such as the Timoshenko beam theory are in agreement with the elasticity theory. However, due to the different nonlocal averaging processes, they are expected to yield different results in their nonlocal forms. In the present work, the free vibration behavior of nonlocal nanobeams is studied using the nonlocal integral Timoshenko beam theory (NITBT) and two-dimensional nonlocal integral elasticity theory (2D-NIET) with different kernels and their results are compared. A new kernel, termed the compensated two-phase (CTP) kernel, is introduced, which entirely compensates for the boundary effects and does not suffer from the ill-posedness of previous kernels. Using the finite element method, the free vibration analysis is performed for different boundary conditions based on the first three natural frequencies. For both the NITBT and 2D-NIET with both the two-phase (TP) and CTP kernels, the nonlocal parameter has a softening effect on the natural frequencies for all the boundary conditions, without observing the paradoxical behaviors of the nonlocal differential theory. For both theories, the softening effect of the nonlocal parameter is more pronounced for the TP kernel compared to the CTP kernel. The sensitivity of the 2D-NIET to the nonlocal parameter is found to be higher than that of the NITBT. Also, the softening effects for different vibration modes are compared to each other for both theories and both kernels. The obtained results can be extended for various important beam problems with nonlocal effects and help obtain a better understanding of applicable nonlocal theories.

Load-bearing capacity and resonance stability of inelastic beams and plane trusses with initial defects

PurposeThe purpose of this paper is to contribute to the solution of the buckling and resonance stability problems in inelastic beams and wooden plane trusses, taking into account geometric and material defects.Design/methodology/approachTwo sources of non-linearity are analyzed, namely the geometrical non-linearity due to geometrical imperfections and material non-linearity due to material defects. The load-bearing capacity is obtained by the rheological-dynamical analogy (RDA). The RDA inelastic theory is used in conjunction with the damage mechanics to analyze the softening behavior with the scalar damage variable for stiffness reduction. Based on the assumed damages in the wooden truss, the corresponding external masses are calculated in order to obtain the corresponding fundamental frequencies, which are compared with the measured ones.FindingsRDA theory uses rheology and dynamics to determine the structures' response, those results in the post-buckling branch can then be compared by fracture mechanics. The RDA method uses the measured P and S wave velocities, as well as fundamental frequencies to find material properties at the limit point. The verification examples confirmed that the RDA theory is more suitable than other non-linear theories, as those proved to be overly complex in terms of their application to the real structures with geometrical and material defects.Originality/valueThe paper presents a novel method of solving the buckling and resonance stability problems in inelastic beams and wooden plane trusses with initial defects. The method is efficient as it provides explanations highlighting that an inelastic beam made of ductile material can break in any stage from brittle to extremely ductile, depending on the value of initial imperfections. The characterization of the internal friction and structural damping via the damping ratio is original and effective.

A gradient reproducing kernel based stabilized collocation method for the static and dynamic problems of thin elastic beams and plates

A gradient reproducing kernel based stabilized collocation method for the static and dynamic problems of thin elastic beams and plates

A novel hybrid water wave optimization algorithm for solving complex constrained engineering problems

Abstract In this work, a new hybrid optimization algorithm (HWW-NM), which combines the Nelder-Mead local search algorithm with the water wave algorithm, is introduced to solve real-world engineering optimization problems. This paper is one of the first studies in which both the water wave algorithm and the HWW-NM are applied to processing parameters optimization for manufacturing processes. HWW-NM performance is measured using the cantilever beam problem. Additionally, a problem for milling manufacturing optimization is posed and solved to evaluate HWW-NM performance in real-world applications. The results reveal that HWW-NM is an attractive optimization approach for optimizing real-life problems.

Nonuniform bending theory of hyperelastic beams in finite elasticity

This paper deals with the equilibrium problem of slender beams inflexed under variable curvature in the framework of fully nonlinear elasticity. For the specific case of uniform flexion, the authors have recently proposed a mathematical model. In that analysis, the complete three-dimensional kinematics of the beam is taken into account and both deformations and displacements are considered large. In the present paper, the kinematics of the aforementioned model has been reformulated taking into account beams under variable curvature. Subsequently, focusing on the local determination of the curvature, new equilibrium conditions on cross sections are introduced in the mathematical formulation. The governing equations take the form of a coupled system of three equations in integral form, which is solved numerically through an iterative procedure. Therefore, for the generic class of hyperelastic and isotropic materials, explicit formulae for the displacement field, the stretches and stresses in every point of the beam, following both Lagrangian and Eulerian descriptions, are derived. The analysis allows studying a very wide class of equilibrium problems for nonlinear beams under different restraint conditions and subject to generic external load systems. By way of example, the Euler beam has been considered and the formulae obtained have been specialized for a specific neoprene rubber material, the constitutive constants of which have been determined experimentally. The shapes assumed by the beam as the load multiplier increases are shown through some graphs. The distributions of stretches and Cauchy stresses are plotted for the most stressed cross section. Some comparisons are made using a FE code. In addition, the accuracy of the obtained solution is estimated by evaluating a posteriori that the equilibrium equations are locally satisfied.

Improved results on the nonlinear feedback stabilisation of a rotating body-beam system

This article is dedicated to the investigation of the stabilisation problem of a flexible beam attached to the centre of a rotating disk. We assume that the feedback law contains a nonlinear torque control applied on the disk and nonlinear boundary controls exerted on the beam. Thereafter, it is proved that the proposed controls guarantee the exponential stability of the system under a realistic smallness condition on the angular velocity of the disk and general assumptions on the nonlinear functions governing the controls. We used here the strategy of Lasiecka and Tataru (1993) and Alabau-Boussouira (2005, 2010). This permits to improve the stability result shown in Chentouf and Couchouron (1999) in the sense that, on one hand, we deal with a general form of the nonlinear functions involved in the boundary controls. On the other hand, we manage to weaken the conditions on those functions unlike in Chentouf and Couchouron (1999), where the authors consider a special type of functions that are almost linear.

Numerical analysis of collapse modes in optimized design of alveolar Steel-concrete composite beams via genetic algorithms

Abstract The objective of this study is to present the formulation and applications of the optimization problem of steel-concrete composite alveolar beams. In addition to presenting the formulation, a comparative analysis of the predominant collapse modes is performed numerically via the finite element method. The optimization program was developed with the GUI platform available in Matlab 2016. Since this is a discrete problem, a Genetic Algorithm (GA) was used to solve the optimization model, implemented with the Matlab optimization toolbox. Numerical examples using finite elements model are presented to validate the solution and analyze its effectiveness, along with an assesment of the predominant collapse modes for a group of 12 beams. The results show that more efficient designs are obtained when optimization tools are applied.

The problem of plane bending a direct composite beam of arbitrary cross-section and the prerequisites for its approximate analytical solution

The approach to the reduction of the spatial problem of plane bending a composite discrete-inhomogeneous beam of arbitrary cross-section to the approximate two-dimensional bending problem of the equivalent multilayer beam has been discussed here. The result is represented in the form of relations for determination the characteristics of the equivalent multilayer structure by physical and mechanical materials characteristics of the original beam’s phases and the system of static, geometric and physical relations of the corresponding two-dimensional problem. The obtained equations are similar to the plane problem equations of the elasticity theory, but instead of stresses, they contain internal efforts consolidated to the main plane of the beam. The equations of the approximate two-dimensional problem were used to solve the problem of static bending a composite console of arbitrary structure with a load on the free end, taking into account the uniform change of the temperature field. The given system of equations and relations is the starting point for the construction of non-classical deformation models and solving a wide range of problems concerning the deformation of a direct composite beams.

Analytical Solution for the Boundary Value Problem of Elastic Beams Subjected to Distributed Load

Analytical solution for the boundary value problem (BVP) of elastic beams subjected to distributed load was investigated. Based on the study, dynamic application curves are developed for beam deflection. The partial differential equation of order four were analysed to determine the dynamic response of the elastic beam under consideration and solved analytically. Effects of different parameters such as the mass of the load, the length of the moving load, the distance covered by the moving load, the speed of the moving and the axial force were considered. Result revealed that the values of the deflection with acceleration being considered are higher than the system where acceleration of the moving load is negligible. These obtained results are in agreement with the existing results.

Analytical Solution for the Boundary Value Problem of Euler-Bernoulli Beam Subjected to Accelerated Distributed Load

The study of dynamic response of beam-like structures to moving or static loads has attracted and still attracting a lot of attention due to its wide range of applications in the construction and transportation industry especially when transverse by travelling masses. Hence, analytical solution for the boundary value problem (BVP) of elastic beams subjected to distributed load was investigated. The partial differential equation of order four were analysed to determine the dynamic response of the elastic beam under consideration and solved analytically. Effects of different parameters such as the mass of the load, the length of the moving load, the distance covered by the moving load, the speed of the moving and the axial force were considered. Result revealed that the values of the deflection with acceleration being considered increases than the system where acceleration of the moving load is negligible.