Nonconservative Loads Research Articles

Asymptotic Numerical Method for dynamic buckling of shell structures with follower pressure

Asymptotic Numerical Method (ANM) is applied to non-linear dynamics of thin-shells subjected to conservative and non-conservative loads such as follower pressure. ANM is decomposed into several stages: the finite element discretization of the non-linear equations of motion of the shell dynamics, a homotopy transformation of the semi-discrete non-linear equations, a perturbation technique to expand the quantities into Taylor series according to the homotopy parameter and the time integration scheme to solve the series of linear problems resulting from the perturbation technique. ANM is applied here with the 7-parameter shell elements thanks to the Enhanced Assumed Strain (EAS) concept and implicit Newmark integration. In the case of non-conservative force, follower pressure also requires to be decomposed in either Taylor series or rational Padé approximants. The academic case of the cylindrical roof with dynamic snap-through phenomenon is investigated for the purpose of comparing ANM strategies and the classical Newton–Raphson (NR) method. Two engineering cases including an I-shaped thin-walled beam and a closed thin-shell cylinder under dynamic external follower pressure are also investigated. ANM turns out to be accurate, robust and efficient in terms of computation time, providing an alternative method to the well-established Newton–Raphson method.

An Investigation into the Impact of Time-Varying Non-Conservative Loads on the Seismic Stability of Concrete-Filled Steel-Tube Arch Bridges

When the arch rib of the mid-bearing through and lower-bearing through arch bridges undergoes out-of-plane deformation, it is usually subject to the resilience force provided by the flexible hanger, which is known as the “non-conservative force effect” of the suspender. In contrast to the static condition, in the dynamic scenario, the time-varying non-conservative force exerted by the flexible suspender becomes more complex due to dynamic changes in external load. Moreover, the difference in fundamental frequency and vibration period between the bridge system and arch rib may influence the stress distribution within the arch rib during ground motion. This paper investigates the impact of time-varying non-conservative forces on the dynamic stability of arch ribs in concrete-filled steel tube (CFST) bridges under seismic loads. Specifically, it examines the influence of different seismic waveforms, frequency disparities between bridge slabs and arch ribs, and suspender stiffness on the non-conservative effect. The results reveal significant disparities in the impact of non-conservative forces exerted by the suspender during seismic events with identical intensity but varying frequency characteristics. The influence of non-conservative forces on the dynamic stability of bridges escalates as deck stiffness increases, while it remains relatively unaffected by changes in suspender stiffness.

Nonlinear analysis of laminated composite shells with piezoelectric patches under displacement‐dependent pressure loads

AbstractIn this paper, the three‐dimensional stress field in nonlinear laminated composite shells with piezoelectric patches subjected to displacement‐dependent loads is analyzed. To solve the problem, the geometrically exact (GeX) hybrid‐mixed four‐node solid‐shell element has been developed. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms are taken exactly at the element nodes. The laminated solid‐shell element formulation is based on the choice of an arbitrary number of sampling surfaces (SaS) parallel to the middle surface and located at the Chebyshev polynomial nodes within the layers, in order to introduce the displacements and electric potentials of these surfaces as unknown functions. The outer surfaces and interfaces are also included into a set of SaS. Due to analytical integration utilized to evaluate the tangent stiffness matrix, the proposed GeX piezoelectric solid‐shell element under nonconservative loading exhibits superior performance in the case of coarse meshes and allows the use of only one load step in all considered benchmark problems. Therefore, it can be recommended for the large deformation analysis of doubly curved laminated composite shells with piezoelectric patches, since the SaS formulation allows one to find three‐dimensional solutions of electroelasticity with a prescribed accuracy.

On the role of different nonlinear damping forms in the dynamic behavior of the generalized Beck’s column

The influence of internal and external nonlinear damping forms on the dynamics of a generalized Beck’s column, namely a visco-elastic cantilever beam, subjected to conservative and non-conservative loads at its free end, is investigated. A variational principle provides the equations of motion of the system, which are properly recast into an integro-differential form. The linear stability analysis of the system is then carried out and bifurcation points are detected in the space of parameters associated with the conservative and non-conservative loads. Starting from Hopf’s bifurcation points, a post-critical analysis, based on the Method of Multiple Scales is directly performed on the continuous system, avoiding any a-priori discretization. This method provides the bifurcation equations whose analysis reveals the double nature of nonlinear damping, which can be beneficial or detrimental in terms of stable or unstable bifurcated equilibria. It is found that both the internal and external forms of nonlinear damping can turn a supercritical instability of the system into a subcritical one, thus revealing another destabilizing effect of damping, beyond the very well-known one occurring in the linear field. Numerical simulations, grounded on a Galerkin discretization of the original system, confirm the analytical findings.

Three-dimensional nonlinear stress analysis of functionally graded shells subjected to displacement-dependent loads

The nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading, developed in the companion paper, is here extended to the stress analysis of functionally graded material (FGM) shells. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms can be taken exactly at element nodes. The FGM solid-shell element formulation is based on the choice of N sampling surfaces (SaS) parallel to the middle surface and located at Chebyshev polynomial nodes within the shell as well as on the outer surfaces, to introduce the displacements of these surfaces as basic unknowns. The use of Lagrange polynomials of degree N–1 in the distributions of displacements, strains, stresses, and elastic coefficients through the thickness of the shell leads to an efficient 3N-parameter shell formulation. To develop the FGM four-node solid-shell element based on the hybrid-mixed method, in which strains and stresses are utilized as primary variables, the Hu-Washizu variational principle is applied. Due to the analytical integration used to evaluate the tangent stiffness matrix, the developed FGM solid-shell element subjected to displacement-dependent loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered benchmark problems. It has been established that the difference between the second Piola-Kirchhoff and Cauchy stress tensors in non-conservative problems for FGM shells undergoing moderately large strains can be significant.

Buckling of pressurized cylindrical shells using the Asymptotic Numerical Method

In this paper, we present a numerical analysis of the buckling of elastic thin structures by using Asymptotic Numerical Method (ANM) and Padé approximants. We consider the case of cylindrical shells submitted to external pressure and we show how the perturbation technique can be applied for conservative and non-conservative loadings. A shell formulation well adapted to thin structures and which uses the Enhanced Assumed Strain (EAS) concept and a three-dimensional constitutive law without condensation is considered in this study. Numerical examples are presented with comparison between conservative and non-conservative loading in term of critical pressure.

Modeling nonlinear deformation of slender auxetic structures under follower loads with complex variable meshfree methods

The auxetic structure with negative Poisson’s ratio has broad application prospects in many engineering fields. Although it is often simplified as a linear elastic problem in theoretical models, the generally used slender structure, such as the reentrant honeycombs, will inevitably undergo large deformation resulting in significant non-conservative load effects in the service conditions. Therefore, a numerical framework for modeling nonlinear deformation of the auxetic structures under follower loads with the complex variable element-free Galerkin method is developed in this paper. The application of the complex variable meshfree method is to deal with the numerical difficulties caused by mesh distortion that may occur in large deformation problems, and at the same time, it can improve the construction efficiency of meshfree shape functions through the complex variable moving least-squares approximation. The Galerkin weak form of the incremental total Lagrangian formula for large deformation problems with the enforced essential boundary conditions using the penalty method is derived and then discretized in the complex variable meshless implementation. Five numerical examples are presented with detailed convergence study to demonstrate the accuracy of the proposed approach in dealing with non-conservative large deformation of the slender auxetic structures.

On the second Piola-Kirchhoff and Cauchy stress tensors in nonlinear shells subjected to displacement-dependent loads

In this article, a nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading using the sampling surfaces (SaS) method is developed. The term “geometrically exact” means that the parametrization of the middle surface is known and, therefore, the coefficients of the first and second fundamental forms and the Christoffel symbols are taken exactly at element nodes. The SaS formulation is based on the choice of N SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as fundamental shell unknowns. Such a choice of unknowns with the use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains, and stresses leads to an efficient higher-order shell formulation. To develop the hybrid-mixed four-node solid-shell element, which is free of shear and membrane locking, the Hu–Washizu variational principle is utilized. The proposed GeX hybrid-mixed solid-shell element subjected to follower loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered numerical examples. It is established that the difference between the second Piola-Kirchhoff and Cauchy stresses in some non-conservative problems for isotropic and composite shells undergoing finite rotations can be significant.

Fusion of two stable elastic structures resulting in an unstable system

It is shown that a compound elastic structure, which displays a dynamic instability, may be designed as the union (or ‘fusion’) of two structures which are stable when separately analysed. The compound elastic structure has two degrees of freedom and is made up of a rigid rod connected with two springs to a smooth support, which evidences a jump in the curvature at the equilibrium configuration. Instability is proven in a linearized context and is related to the application of a non-conservative load of the follower type so that the instability disappears under dead loads. In the fully nonlinear range, the instability is also confirmed through numerical simulations. The obtained results may be useful in the design of new mechanical sensors, devices for energy harvesting, or architected materials. In addition, our findings have conceptual implications on piecewise-linear theories of mechanics such as for instance plasticity or frictional contact.

The Sub and Super-Tangential Nonconservative Load in Stability Problem of Nanobeams with Sprung Masses

In this study, the critical load and natural vibration frequency of Euler–Bernoulli single nanobeams based on Eringen’s nonlocal elasticity theory are investigated. Cantilever nanobeams with attached sprung masses were subjected to compressed concentrated and distributed follower forces. The parameter that determines the direction of nonconservative follower forces was given the positive and negative values, therefore, sub-tangential and super-tangential load were analyzed. The stability analysis is based on dynamical stability criterion and was carried out using a numerical algorithm for solving segmental nanobeams with many boundary conditions. The presented algorithm is based on the exact solutions of motion equations which are derived from equilibrium conditions for each separated segment of the nanobeam. Two comparison studies are conducted to ensure the validity and accuracy of the presented algorithm. The excellent agreement of critical load for Beck’s nano-column on Winkler foundation observed was confirmed as reported by other researchers. The effect of different values of the nonlocality parameter, tangency coefficient, spring stiffness coefficient, location of sprung mass and the greater number of attached sprung masses on a critical load of nanobeams compressed by nonconservative load are discussed. One of the presented results shows that significant differences between local and nonlocal theory appear when the beam subjected to follower forces loses its stability by flutter.

A complex variable EFG method for hyperelastic large deformation analysis under non-conservative loads

The finite element method is extremely popular in almost all fields of engineering and scientific research due to its powerful and accessible industrial software for computational modelling. However, alternatives including the meshless methods, have received increasing attention on the large deformation problems of hyperelastic materials that need to avoid mesh distortion. In this paper, based on the complex variable moving least-squares approximation, the complex variable element-free Galerkin (CVEFG) method for the large deformation problem of hyperelastic materials under non-conservative loads is established. In the present method, the Newton-Raphson method is used for solving the nonlinear equations. The algorithm for hyperelastic large deformation problems under non-conservative loads are established. Three numerical examples are used to verify the effectiveness of the CVEFG method in this paper. Numerical results show that the solution obtained using the CVEFG method is closer to that obtained by ABAQUS and is more efficient than the element-free Galerkin method.

Nonlinear instability problem for geometrically exact beam under conservative and non-conservative loads

In this work we propose a numerical solution procedure for nonlinear instability problems for geometrically exact beams (representing arbitrary large displacements and rotations) under either conservative or non-conservative load. Both static and dynamics frameworks are developed, with the latter needed for non-conservative case. The proposed model can also successfully deliver the analytic solution to linearized instability problems (with small pre-buckling displacements) for critical loads of Euler (no shear) and Timoshenko (with shear) cantilever beam, which is used for validation for both conservative and non-conservative loads. We also validated the proposed geometrically exact beam models against the analytic solutions to nonlinear problem (with finite rotations) and pointed out their advantage in providing a more sound interpretation of instability for both conservative and non-conservative loads. Finally, the proposed models performance is illustrated on more complex problems with corresponding numerical solutions obtained by using both the geometrically exact finite elements for Reissner beam (with shear) and for the Kirchhoff beam (no shear).

Analisis Perpindahan Besar Akibat Gaya Non-Konservatif

Structural applications based on large displacement analysis are widespread in various fields, such as the application of aircraft structures, membranes, cables, pipelines and risers. In this study, the author reviews the structural responses based on large displacement theory and non-conservative load models, assuming linear sections and materials. The parameters reviewed are the variation of the load factor and the variation of the slope angle of the load P at the end of the span. Solving non-linear differential equations with non-conservative force, using numerical integration method. The simulation results with the variation of the load factor with the angle of inclination, vertical and horizontal displacements are obtained, where the final orientation of the acting force corresponds to the deformation of the structure, and the maximum deformation occurs at a load angle of 90o, the deformation decreases for the angle orientation is getting smaller.

Asymmetric Design Sensitivity and Isogeometric Shape Optimization Subject to Deformation-Dependent Loads

We present a design sensitivity analysis and isogeometric shape optimization with path-dependent loads belonging to non-conservative loads under the assumption of elastic bodies. Path-dependent loads are sometimes expressed as the follower forces, and these loads have characteristics that depend not only on the design area of the structure but also on the deformation. When such a deformation-dependent load is considered, an asymmetric load stiffness matrix (tangential operator) in the response region appears. In this paper, the load stiffness matrix is derived by linearizing the non-linear non-conservative load, and the geometrical non-linear structure is optimally designed in the total Lagrangian formulation using the isogeometric framework. In particular, since the deformation-dependent load changes according to the change and displacement of the design area, the isogeometric analysis has a significant influence on the accuracy of the sensitivity analysis and optimization results. Through several numerical examples, the applicability and superiority of the isogeometric analysis method were verified in optimizing the shape of the problem subject to deformation-dependent loads.

Designing of the blades of aircraft propellers by the finite element method, taking into account the strength of structure

The blades of contemporary turboprop engines have a complex spatial configuration. They can be classified as shells. Methods for the shells calculation are well known. A number of computer programs have been created on their basis. However, these programs do not take into account the peculiarities associated with the mutual influence of deformations of the blade and the aerodynamic and inertial loads acting on it. The aim of this work is to develop a method of finite element calculation of aircraft propeller blades taking into account aeroelastic effects and to create a computer program on its basis that is available to a wide range of designers and engineers. The finite element method is used in a geometrically nonlinear formulation. As the initial one, the equilibrium equation is used, which includes a complete nonlinear stiffness matrix and takes into account both conservative and non-conservative loads. The blade of one of the serial propellers was calculated. The effect of deformations on the magnitude of the aerodynamic load and, as a result, on the stresses in the design sections was found and analyzed. The proposed technique and the program compiled on its basis can be used in the design of aircraft propeller blades.

On large deformation and stability of microcantilevers under follower load

In the present study, a novel geometrically exact nonlinear beam model based on the nonlinear displacement field, centerline-inextensibility condition, Euler-Bernoulli beam hypothesis, and modified couple stress theory is established to investigate the extremely large deformations and stability characteristics of size-dependent microcantilevers subjected to a tip-concentrated non-conservative (follower type) load with arbitrary inclination angle. The newly developed mathematical formulation explains the size effect phenomenon in the microcantilevers with only one material length scale parameter. The nonlinear shooting method for two-point boundary value problems is utilized to assess the deformed configuration of microsystem based on the quasistatic form of nonlinear mathematical model. Afterwards, the linearized dynamical mathematical model around the nonlinear deformed configurations of microsystem is numerically solved using the Galerkin technique to examine the stability of obtained responses. A comprehensive investigation is then conducted to highlight the role of size-dependency in the critical follower force at which the microsystem undergoes a flutter instability, as well as the nonlinear large deformation of microsystem in the pre-flutter region.

Flutter Instability and Ziegler Destabilization Paradox for Elastic Rods Subject to Non-Holonomic Constraints

Abstract Two types of non-holonomic constraints (imposing a prescription on velocity) are analyzed, connected to an end of a (visco)elastic rod, straight in its undeformed configuration. The equations governing the nonlinear dynamics are obtained and then linearized near the trivial equilibrium configuration. The two constraints are shown to lead to the same equations governing the linearized dynamics of the Beck (or Pflüger) column in one case and of the Reut column in the other. Although the structural systems are fully conservative (when viscosity is set to zero), they exhibit flutter and divergence instability. In addition, the Ziegler's destabilization paradox is found when dissipation sources are introduced. It follows that these features are proven to be not only a consequence of “unrealistic non-conservative loads” (as often stated in the literature); rather, the models proposed by Beck, Reut, and Ziegler can exactly describe the linearized dynamics of structures subject to non-holonomic constraints, which are made now fully accessible to experiments.

Stability and free vibration analysis of compound column with piezoelectric rod

This paper deals with the divergence and flutter instability of a slender geometrically nonlinear two-member column under eccentric partially follower load. The external force direction is governed by follower parameter η, which changes over the interval 0 ≤ η ≤ 1, hence both the conservative (η = 0) and nonconservative load may be taken into account. To eliminate the static column deflection, the host aluminium member is associated with an auxiliary piezoceramic rod subjected to an electric field. The governing equations of motion and boundary conditions are derived using extended variational Hamilton’s principle. Constitutive equations expressing the piezoelectric coupling between the electric and elastic fields, the von Kármán theory of moderately large rotations but small strains and the Euler–Bernoulli beam theory are used in the formulation. Due to the nonlinearity of the governing equations, their solutions are attempted by a regular perturbation technique leading to asymptotic expansions of displacements, internal forces and the natural vibration frequency. The stability analysis of the column is investigated by using the kinetic criterion, i.e. by studying the evolution of the eigenfrequencies for an increasing external load. Both the divergence and flutter instability phenomena are detected and the latter exists only for the pure tangential load.It has been found that the courses of frequency curves, which are strictly dependent on the load eccentricity and follower parameter, may be modified by the piezoelectric actuation. As a result, both the divergence and flutter forces of the column are affected by the electric field. When η ≤ 0.5, the tensile piezoelectric force increases the vibration frequency, whereas the compressive force acts in the opposite way. For such η, the limit points of column instability are manifested, which are reflected by the additional branches of the eigenfrequency curves existing over a narrow range of the external load.For the pure follower load, the piezoelectric force may increase or decrease the frequency, which depends on both the external force level and its eccentricity. On the other hand, the piezoelectric actuation may be efficiently used to increase the flutter force. Specific mode shapes of the column are observed, which are characterized by an uneven number of nodes for particular members. Additionally, it is observed that the sequence of vibration modes with the growing values of the external load is reorganized. Finally, the present analysis reveals that the piezoelectric actuation not only counteracts the column bending, but also modifies the transversal vibration frequency of the system and enhances its critical forces for the conservative or nonconservative external load.

Effect of intermediate support on critical stability of a cantilever with non-conservative loading: Some new results

The introduction of intermediate support can significantly alter the stability characteristics of an elastic structure subjected to non-conservative loading; this follows from investigations carried out almost three decades ago. A majority of these investigations however assume the non-conservative loading to have the form of a follower force. A new type of non-conservative loading in the from of a dynamic moment was recently introduced in the literature using both theory and experiments and it behooves us to investigate the effect of intermediate support on structures with such type of loading. The dynamic moment is proportional to the curvature of a point on the structure; critical stability is therefore investigated in the two-parameter space defined by the locations of the intermediate support and the point of curvature measurement. For a cantilever beam with terminal dynamic moment, the investigations reveal a rich set of instability transitions not observed heretofore; these include multiple stability transitions between divergence and flutter and between different modes of flutter, transitions occurring with and without jumps in the critical load, orderly and random flutter-to-flutter transitions, and multiple instability transitions involving jumps in the critical load. The jump in the critical load is also observed for the cantilever beam with follower force; this jump, which was reported to be absent in earlier work, results from the veering phenomenon.

Stability of nanobeams under nonconservative surface loading

A universal algorithm for analyzing the stability of Euler–Bernoulli nanobeams with any support conditions, subjected to arbitrary conservative and nonconservative loads, has been shown. The analysis was carried out using exact solutions in each of the prismatic nanobeam segments. The study of the determinant of a homogeneous system of equations resulting from boundary conditions and continuity conditions at the contact points of the nanobeam elements was the basis for the analysis of its critical loads. The presented general algorithm was used to analyze the impact on critical loads of prestress nanobeams caused by conservative and nonconservative external surface loads.