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An extension of Boley’s continuum mechanics-based successive approximation method is presented for rectangular beams composed of two isotropic linear elastic layers. The solution is cast into the form of tables, in complete analogy to the tables originally presented by Boley and Tolins for single-layer strips. The first column in these tables corresponds to the classical Bernoulli–Euler theory of beams. The further columns represent comparatively fast converging correction terms of an increasing refinement. Our two-layer formulation automatically satisfies the stress continuity conditions at the interface of the two layers. Enforcing displacement continuity at the interface between the layers, we derive results that do satisfy the equilibrium field equations, the stress continuity conditions at the interface and the stress boundary conditions at the upper and lower edges. When converged, the field constitutive relations and the displacement continuity at the interface between the two layers are also satisfied. We present a compact formulation, which allows writing down the results for more than the three successive steps considered by Boley and Tolins. The elasticity solutions presented subsequently can be used as novel analytic benchmarks for comparison with refined structural mechanics beam theories. Interior solutions for beams with a finite axial extent can be obtained by assigning approximate boundary conditions at the lateral ends. Comparisons to finite element computations for a clamped–clamped beam give strong evidence for the correctness of our analytic results.

Automatic panel benders of Salvagnini Maschinenbau GmbH enable high-quality and high-efficient production of sheet metal components. To achieve the steadily increasing requirements on precision, a model-based adaptive concept has been developed controlling the complete production process as a digital twin, from CAD data to the final sheet metal part. First, an overview of the underlying simulation models with different levels of detail is given. The models consider the elastoplastic deformation of the metal sheets as well as the elastic machine components and mechatronic models of the powertrain. Secondly, an overview is given of the adaptive production concept allowing the real-time adaption of the machine to changing process parameters like material properties. Finally, all production steps are controlled by a digital twin based on this model-based adaptive strategy. The paper demonstrates the successful transfer of scientific results to an industrial application.

The present paper is concerned with bending of extensible and shear-deformable rods, where we treat the case of plane deformations. Our emphasis is laid upon infinitesimal deformations superimposed upon finite deformations. In this framework, we focus on deriving generalized reciprocity theorems that must hold with respect to the superimposed infinitesimal deformations, when produced by different types of loadings. Besides imposed distributed and concentrated force and moment loadings, we consider the action of eigenstrains. Moreover, we also take into account the presence of jumps in kinematic entities, particularly concerning jumps of axial rod positions and of cross-sectional rotations. We first derive local and global universal reciprocity relations, which must hold for rods made of any material. The global universal relation then is specialized to the case of hyperelastic rods. From the latter relation, we prove that reciprocity theorems known from the linear theory, such as the theorems of Betti, Maxwell and Land, as well as Maysel’s formula, formally do hold for infinitesimal deformations superimposed upon finite deformations. As a formulation that appears to be novel also with respect to the purely linear theory, we present a combination of Maysel’s formula with the theory of Land, the former dealing with eigenstrains, the latter involving jumps in kinematic entities.

The search for efficient explicit time integration schemes is a relevant topic in the current literature on dynamic mechanical systems. In this paper, we describe a strategy of utilizing the balance relations of mechanics in their integral form, so-called general laws of balance, where the time-evolution of the integrands is approximated by established computational techniques of the discrete-mechanics-type. In a Picard-type iteration, the outcomes are used for repeating the procedure several times, leading to an increased accuracy. The advantages of the present explicit approach are discussed in the context of linear and nonlinear motions of the mathematical pendulum. We utilize the modern symbolic procedures to obtain the time integration formulae and compare the results of our methods with exact solutions and with the results of higher-order implicit methods and also with a recent explicit formulation from the literature. (VLID)4848919 Submitted version

Um den steigenden Anforderungen an moderne Produktionsmaschinen hinsichtlich Präzision und Qualität bei optimaler Ressourceneffizienz gerecht zu werden, sind neue Konzepte zur Reduktion des Material- und Energieverbrauchs bei deren Herstellung notwendig. In dieser Arbeit wird eine empirisch begründete Strategie zur Gewichts- und Energiereduktion in Blechbiegeautomaten vorgestellt, bei der mittels Topologie-Optimierung eine Leichtbau-Ausführung geometrisch komplexer und bewegter Maschinenteile erzielt wird. Diese Strategie wurde bei dreidimensionalen Maschinenrahmen der Maschinenfamilie „Lean Panel Bender“ der Salvagnini Maschinenbau GmbH erfolgreich angewendet und erbrachte gegenüber älteren Ausführungen große Einsparungen an Gewicht, Fertigungs- und Transportkosten sowie beim Energieverbrauch. So ist eine sowohl kostengünstige als auch hochpräzise Herstellung komplex geformter Blechprodukte möglich, sowohl bei Losgröße 1 als auch in der Serie

Nachfolgend werden einige wissenschaftliche Aspekte der Echtzeit-Materialerkennungsstrategie MAC diskutiert, die in der neuen Maschinenfamilie LEAN PANEL BENDER der Firma Salvagnini Maschinenbau realisiert ist und die eine hochprazise und hocheffiziente Herstellung von komplex geformten Blechprodukten sowohl bei Losgrose 1 als auch in der Serie erlaubt.

The contour motion of the belt drive, i.e., the motion with the constant trajectory, is addressed. The belt is considered as a closed Cosserat line whose particles have translational and rotational degrees of freedom. The problem is considered in the framework of geometrically nonlinear formulation with no restrictions on the smallness of displacements and rotations. The spatial (Eulerian) coordinate which is the arc coordinate in the actual configuration is introduced. The belt is divided into four segments: two contact segments on the pulleys and two free spans. The friction forces are assumed to obey the Coulomb law. The study is limited to the stationary case with the constant angular velocities of the pulleys and the equations in components are derived for both contact and free spans. In the contact segment two assumptions are employed to eliminate the unknown contact pressure and friction: (1) the full contact, i.e., coincidence between the pulley and the belt and (2) the stick condition, i.e., the belt velocity is related to the pulley angular velocity. A nondimensional coordinate is introduced in the segments to obtain the boundary value problem with fixed boundaries. The boundary coordinates of the contact zones are the integration constants of the derived problem along with the other constants. (VLID)4851961

In the framework of the linear theory of elastodynamics, we consider feed-forward control of structural displacements by eigenstrains. In this so-called displacement-tracking problem, one seeks for an eigenstrain actuation, such that the displacements of the structure do follow desired paths, despite imposed forces and possibly non-trivial initial and boundary conditions are present. The goal of our presentation is the solution of the displacement-tracking problem in an extended framework, namely with respect to the presence of singular waves, particularly shock and acceleration waves. For a solution, we start by formulating a uniqueness theorem for the null input displacement characterisation of elastodynamics. From the latter theorem, we present necessary conditions for the desired displacement field to be tracked, together with sufficient conditions for the eigenstrain actuation, in order that the displacements are controlled also when shock and acceleration wave fronts result from imposed force loadings in the uncontrolled case. It turns out that the actuation stresses must satisfy certain quasi-static equilibrium conditions, together with certain relations of jump at the propagating singular wave fronts. The first part of this result is in coincidence with previous findings of our group concerning the control of structural vibrations by eigenstrains, but in the absence of propagating singular wave fronts. The second part, the jump relations that must be satisfied by the eigenstrains across the singular fronts, represent novel findings. As an illustrative case, one-dimensional singular waves propagating in a straight rod are used in order to give evidence for the appropriateness of our results.

This paper is concerned with the dynamics of hyperelastic solids and structures. We seek for a smart control actuation that produces a desired (prescribed) displacement field in the presence of transient imposed forces. In the literature, this problem is denoted as displacement tracking, or also as shape morphing problem. One talks about shape control, when the displacements to be tracked do vanish. In the present paper, it is assumed that the control actuation is provided by imposed eigenstrains, e.g., by the electric field in piezoelectric actuators, or by thermal actuators, or via analogous physical effects, such as magneto-striction or pre-stress. Structures with a controlled eigenstrain-type actuation belong to the class of smart structures. The action of the eigenstrains can be conveniently characterized by actuation stresses. Our theoretical derivations are performed in the framework of the theory of small incremental dynamic deformations superimposed upon a statically pre-deformed configuration of a hyperelastic solid or structure. We particularly ask for a distribution of incremental actuation stresses, such that the incremental displacements follow exactly a prescribed trajectory field, despite the imposed incremental forces are present. An exact solution of this problem is presented under the assumption that the actuation stresses can be tailored freely and applied everywhere within the body. Extending a Neumann-type solution strategy, it is shown that the actuation stresses due to the distributed control eigenstrains must satisfy certain quasi-static equilibrium conditions, where auxiliary body-forces and auxiliary surface tractions are to be taken into account. The latter auxiliary loading can be directly computed from the imposed forces and from the desired displacement field to be tracked. Hence, despite the problem is a dynamic one, a straightforward computation of proper actuator distributions can be obtained in the framework of quasi-static equilibrium conditions. Necessary conditions for the functioning of this concept are presented. Particularly, it must be required that the intermediate configuration is infinitesimally superstable. Previous results of our group for the case of shape control and displacement tracking in linear elastic structures are included as special cases. The high potential of the solution is demonstrated via Finite Element computations for an irregularly shaped four-corner plate in a state of plain strain.

In the present paper, we consider nonlocal linear elastic Kirchhoff plates, where we restrict to thin, isotropic, and homogeneous plates under the action of static transverse forces, utilizing the differential equation form of the Eringen nonlocal continuum theory. For the case of simply supported plates of polygonal planform, we derive analogies between the solutions of the nonlocal Kirchhoff theory and its local counterpart. First, we extend the Marcus decomposition method for local Kirchhoff plates, where we show that, analogous to the local case considered by Marcus, the moment sum and the nonlocal deflection are both governed by Poisson boundary value problems, which correspond to auxiliary (local) membrane problems. In the present context, it eventually follows that there is a nonvanishing term responsible for a correction of the deflection due to the nonlocal effect, while the moments and shear forces of the local and the nonlocal plate coincide. The nonlocal correction of the deflection turns out to be governed by a membrane-type boundary value problem again. From this fact, it follows immediately that the correction for the deflection due to the nonlocal effect can be derived alternatively from both the local deflection and the local moment sum, which represents a substantial simplification of the nonlocal computations. For the sake of demonstration, we consider examples with closed form solutions. We first consider (as limiting case) infinite plates under the action of single forces, where the singular behavior of Green’s function of the nonlocal Kirchhoff theory can be clarified. Then we discuss the bending of nonlocal plate strips for comparison sake, as well as equilateral triangular plates. To our best knowledge, no results for bending of nonlocal triangular plates have been presented in the literature so far. The present paper has been strongly influenced by a former contribution on analogies between simply supported polygonal Kirchhoff plates rigid in shear and shear-deformable plate solutions. This paper was co-authored together with our teacher, the late Professor Franz Ziegler, to whom we dedicate the present paper. In light of this latter contribution, and extending an idea promoted by Professor Franz Ziegler and his co-workers in the framework of various other fields, we finally show in the Appendix that the deflections of simply supported nonlocal Kirchhoff plates can be considered as deflections of a local “background” Kirchhoff plate under the action of additional fictitious eigenstrain loadings, such as thermal moments.

Boley's method is utilized in order to show that the elementary Bernoulli–Euler beam theory can be enhanced such that exact solutions of the plane-stress theory of linear elasticity are obtained for force loaded rectangular beams. An equivalent enhancement is derived for the elementary Timoshenko theory of beams. The enhancement terms act analogous to thermal loadings; they follow from the force loading of the rectangular beam in an explicit form. The resulting boundary value problem of fourth order can be efficiently solved by means of symbolic computer codes. As an illustrative example, a redundant beam is studied, which is simply supported at one end, and which is clamped at the other end. Outcomes for three alternative clamped end boundary conditions are compared.

Variable-mass conditions can occur in a variety of practical problems of engineering. Investigations on problems of this type have been figuring as a particular research field of mechanics and applied mathematics. The fundamental issue is that the basic equations of classical mechanics were originally formulated for the case of an invariant mass contained in a material volume. Therefore, appropriate formulations are required when dealing with variable-mass problems. The scope of the present article is devoted to arbitrarily moving control volumes formulated within the framework of Ritz's method, that is, to non-material volumes in the sense discussed by Irschik and Holl [10]. We aim at demonstrating a generalized version of Noether's theorem such that it can be grounded on the generalized Hamilton's principle for a non-material volume in the form derived by Casetta and Pesce [17]. This will consistently allow the consideration of conservation laws, written from a Noetherian approach, in this particular context of non-material volumes. To test the proposed formulation, the problem of a rotating drum uncoiling a strip will be addressed.

The setting of a looped drive belt on two equal pulleys is considered. The belt is modelled as a Cosserat rod, and a geometrically nonlinear model with account for tension and transverse shear is applied. The pulleys are considered as rigid bodies, and the belt–pulley contact is assumed to be frictionless. The problem has two axes of symmetry; therefore, the boundary value problem for the system of ordinary differential equations is formulated and solved for a quarter of the belt. The considered part consists of two segments, which are the free segment without the loading and the contact segment with the full frictionless contact. The introduction of a dimensionless material coordinate at both segments leads to a ninth-order system of ordinary differential equations. The boundary value problem for this system is solved numerically by the shooting method and finite difference method. As a result, the belt shape including the rotation angle, forces, moments, and the contact pressure are determined. The contact pressure increases near the end point of the contact area; however, no concentrated contact force occurs.

In the present paper, a formulation of Lagrange’s equations, written in the framework of the Lagrange (material) description of Continuum Mechanics, is provided for open systems. An open system is constituted by (the portion of) a continuous material body that is enclosed by a nonmaterial surface. Such a surface, through which a flow of mass takes place, is also denoted as a control surface, and the corresponding volume is called a control volume. In order to apply Lagrange’s equations of analytical mechanics, the motion and the deformation of the body is modeled in the framework of the Ritz approximation technique by means of a finite number of generalized coordinates. However, since mass may not be conserved in an open system due to the flow of mass through the control surface, the original form of Lagrange’s equations must be accomplished by proper flux terms to be considered at the control surface. In order to derive this extended form, a local version of Lagrange’s equations is derived first, using a proper mathematical manipulation of the local relation of balance of linear momentum written in the Lagrange description of Continuum Mechanics. This local form is integrated over the volume that instantaneously is enclosed by the image of the control surface in a properly chosen reference configuration. In the integrated form, the Truesdell–Toupin method of fictitious particles and generalized Reynolds transport theorems are utilized in order to exchange the integrals and the partial derivatives with respect to time and generalized coordinates and velocities. This yields the desired form of Lagrange’s equations for open systems, written in the Lagrange description of Continuum Mechanics. Illustrative examples demonstrate the consistence of this novel form with the Euler (spatial) version of Lagrange’s equations for open systems, which was derived earlier by the present authors.

In the present paper a concept of static and dynamic shape control for slender piezoelastic beams is addressed. The notion shape control means to find a weighting sequence for the piezoelectric control units, such that force-induced lateral deflections are completely annihilated along the beam axis. In our case this spatial control intensity for the piezoelectric transducers is realized with so-called resistive electrodes: by prescribing only the maximum and minimum voltage level, any desirable voltage distribution along the beam axis may be obtained. For the derivation of the shape control method an extended version of the Bernoulli–Euler beam theory is used, which unifies the kinematics of a slender piezoelectric beam and the Telegrapher’s equations for moderately conductive electrodes. The outcome is an actuator equation for the displacement, which is fourth order in space and second order in time, and a sensor equation, which is second order in space and first order in time (diffusion equation for the voltage). The developed shape control method is verified by several numerical examples in the static case as well as in the dynamic frequency regime for a clamped-free beam. We show that the presented method is perfectly suited for static loads, being able to cancel force-induced structural deformations along the beam axis, whereas the deflections in the higher frequency regime can be nearly canceled out. The method is efficient, as long as a non-dimensional constant, involving the resistance, the capacitance per unit length, the length of the electrodes and the excitation frequency, is small.

SUMMARY Structural control has been comprehensively studied over the world as a multidisciplinary research field. The present work is motivated by an attempt to give a common frame to the recent research and applications of structural control technology in civil engineering across Europe. They include novel passive dampers, functional materials and semi-active dampers, active control systems, and their performance investigations. Design methods for the vibrations reduction of buildings, bridges, and wind turbines are discussed with reference to case studies. Control algorithms and dimension reduction techniques are also studied. Adaptation strategies and techniques based on the potential offered by piezoelectricity are reviewed. Copyright © 2014 John Wiley & Sons, Ltd. Copyright © 2014 John Wiley & Sons, Ltd.

The drive belt set on two pulleys is considered as a nonlinear elastic rod deforming in plane. The modern equations of the nonlinear theory of rods are used. The static frictionless contact problem for the rod is derived. The nonlinear boundary value problems for the ordinary differential equations are solved by the finite differences method and by the shooting method by means of computer mathematics. The belt shape and the stresses are determined in the nonlinear formulation which delivers the contact reaction and the contact area. The developed method allows performing calculations for any set of geometrical and stiffness parameters. (VLID)4844522

In the present contribution, the possibility of controlling dynamic stresses in force-loaded bodies by means of actuating eigenstrain fields is addressed. The action of eigenstrains, such as thermal or piezoelectric actuating strains, is subsequently gathered under the notion of actuating stresses. Our study is performed in the framework of the theory of small incremental dynamic deformations superimposed upon a state of possibly large static pre-deformation of a hyperelastic body. Particularly, we present a solution for the general problem of producing certain incremental stress trajectories by means of specifically tailored actuation stresses that are superimposed onto the force-loaded body. This we shortly call the stress tracking problem. The problem of suppressing incremental stresses is contained as a special case. Subsequently, particular emphasis is given to the systematic derivation of necessary and sufficient conditions that must be satisfied in order to solve the stress tracking problem. Necessary conditions are presented that must be satisfied by the intermediate configuration and by the desired incremental stress field that shall be tracked, and sufficient conditions are derived that must be satisfied by the incremental actuating stresses. As an illustrative example, our three-dimensional formulation is eventually applied to the one-dimensional dynamic case of a straight homogeneous rod with a support excitation at one end and a single point-mass at the other end.

This paper is concerned with the discussion of a general methodology to design continuously distributed strain sensors; the latter are also known as spatial filters. The methodology is based on the application of the principle of virtual work. Hence, solutions for the design problem can be computed from static auxiliary problems; yet, they are valid for actual dynamic problems as well. Moreover, the use of the principle of virtual work enables a simple and straightforward translation of the results to a structural level. Different types of spatial filters, such as modal filters, displacement filters and compatibility filters are presented and their use for structural and health monitoring is discussed. Some illustrative examples for each type of spatial filter are presented, in which piezoelectric sensor networks are used to put the filters into practice; in each example experimental results validate the presented design methodology.

The static and dynamic version of the body force analogy of linear thermoelasticity is extended with respect to the following problem: In a dynamic setting, we present a distribution of imaginary forces, such that the isothermal displacements due to the latter are equal to the difference between the thermal displacements due to a prescribed thermal actuation stress and a certain prescribed displacement field.

A beam resting on spatially fixed supports may slide relatively to these as soon as external forces are applied. Consequently, the length of the portion of the reference configuration, which is currently located between the supports, depends on the loading and therefore is not known in advance. In the present paper, the problem of a slender beam under a uniformly distributed force is investigated, which is clamped at one side but may slide through another clamping device in axial direction at the opposite side. In combination with a suitable coordinate transformation by which the numerical treatment is simplified, a finite element approach is utilized to determine the equilibrium shape for the maximum critical load that can be imposed on the beam. In the course of this, the influence of the extensibility of the beam axis is studied. A theory based on Reissner’s geometrically exact relations for the plane deformation of beams is adapted such that it allows constitutive relations on stress–strain level to be integrated consistently. In addition to the classical equations of the extensible elastica, a constitutive model derived from the St. Venant–Kirchhoff material of non-linear continuum mechanics is studied. The results obtained in this survey are finally compared to those from the linear beam theory, which turns out to be incapable of describing the problem under consideration in a satisfactory manner.

This article deals with the non-linear modelling of beams that are bent, sheared and stretched by external forces and moments. In the following, we restrict to plane-deformations and static conditions. Our task is to present a continuum mechanics-based interpretation of the celebrated large displacement finite deformation structural mechanics theory, which was presented by Eric Reissner [On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. The latter formulation was restricted to the notions of structural mechanics and thus did not use the notions of stress and strain, which are fundamental for continuum mechanics. Thus, the common continuum mechanics-based constitutive modelling at the stress–strain level cannot be utilized in connection with Reissner's original theory. Instead, Reissner suggested that constitutive relations between certain generalized strains (bending, shear and axial force strains) and generalized static entities (bending moment...

The present paper is concerned with vibrations of an elastic bridge loaded by a moving elastic beam of a finite length, which is an extension of the authors\' previous study where the second beam was modeled as a semi-infinite beam. The second beam, which represents a train, moves with a constant speed along the bridge and is assumed to be connected to the bridge by the limiting case of a rigid interface such that the deflections of the bridge and the train are forced to be equal. The elastic stiffness and the mass of the train are taken into account. The differential equations are developed according to the Bernoulli-Euler theory and formulated in a non-dimensional form. A solution strategy is developed for the flexural vibrations, bending moments and shear forces in the bridge by means of symbolic computation. When the train travels across the bridge, concentrated forces and moments are found to take place at the front and back side of the train.

An extension of Mohr’s analogy to bending of shear-deformable beams with eigenstrain-type actuation, such as a piezoelectric actuation, is presented first. Various refined shear-deformable beam theories are included by means of a theory-dependent parameter. The one-dimensional version of Reissner’s sixth-order plate theory is exemplarily addressed. The Bernoulli–Euler theory of beams rigid in shear, as well as the shear-deformable theory of Timoshenko, are included as special cases. Afterwards, the extended Mohr analogy is applied to the bending of smart beams with piezoelectric patch actuators. The following special problem of static shape control is solved: Seek a placement of single patch actuators, such that the displacement and the cross-sectional rotation vanish at some pre-selected locations of the beam, despite the beam is loaded by external forces. Using Mohr’s analogy, it is shown that the auxiliary loading of the adjoint beam must form a self-equilibrated system of loading in order to achieve the latter goal. The high potential of the proposed actuator placement is demonstrated for the case of a cantilever beam with a single force acting at the tip. Placements of single actuators are represented such that the tip displacement and the tip cross-sectional rotation vanish. The outcomes of shear-deformable theories are compared to the Bernoulli–Euler theory and to a Finite Element computation using piezoelectrically coupled elements.

The mechanical response of complex elastic structures that are assembled of substructures is significantly influenced by joints such as bolted joints, spot-welded seams, adhesive-glued joints, and others. In this respect, computational techniques, which are based on the direct finite element method or on classical modal reduction procedures, unfortunately showan inefficient balance between computation time andaccuracy. In the present paper, a novel reduction method for the physical (nodal) joint interface degrees of freedom is presented, which we call joint interface modes. For the computation of the joint interface modes, Newton’s third law (principle of equivalence of forces) across the joint is explicitly accounted for the mode generation. This leads to a dimension of the generalized joint interface degrees of freedom in the reduced system, which is a factor of 2 or more smaller than in conventional reduction methods, which do not consider Newton’s third law. Two different approaches for the computation of the joint interface modes are presented. Numerical studies with bolted joints of different complexities are performed using a simple but representative constitutive joint model. It is demonstrated that the new joint-interface-mode formulation leads to both excellent accuracy and high computational efficiency.

In finite element methods that are based on position and slope coordinates, a representation of axial and bending deformation by means of an elastic line approach has become popular. Such beam and plate formulations based on the so-called absolute nodal coordinate formulation have not yet been verified sufficiently enough with respect to analytical results or classical nonlinear rod theories. Examining the existing planar absolute nodal coordinate element, which uses a curvature proportional bending strain expression, it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and, even more serious, the normal force is incorrect. A correction based on the classical ideas of the extensible elastica and geometrically exact theories is applied and a consistent strain energy and bending moment relations are derived. The strain energy of the solid finite element formulation of the absolute nodal coordinate beam is based on the St. Venant–Kirchhoff material: therefore, the strain energy is derived for the latter case and compared to classical nonlinear rod theories. The error in the original absolute nodal coordinate formulation is documented by numerical examples. The numerical example of a large deformation cantilever beam shows that the normal force is incorrect when using the previous approach, while a perfect agreement between the absolute nodal coordinate formulation and the extensible elastica can be gained when applying the proposed modifications. The numerical examples show a very good agreement of reference analytical and numerical solutions with the solutions of the proposed beam formulation for the case of large deformation pre-curved static and dynamic problems, including buckling and eigenvalue analysis. The resulting beam formulation does not employ rotational degrees of freedom and therefore has advantages compared to classical beam elements regarding energy-momentum conservation.