Nonlinear finite element analysis of curved beams

Mixed curved-beam finite elements are developed for the geometrically nonlinear analysis of deep arches. The analytical formulation is based on a form of the nonlinear deep-arch theory with the effects of transverse shear deformation and bending-extensional coupling included. The fundamental unknowns consist of the six internal forces and generalized displacements of the arch. The generalized stiffness matrix is obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Numerical studies are presented to demonstrate the high accuracy of the solutions obtained by the mixed models and to show that their performance is considerably less sensitive to variations in the arch geometry than that of the displacement models.     

On the work theorems for finite and incremental elastic deformations with discontinuous fields: A unified treatment of different versions

Starting from the statical and kinematical relations in three different versions characterized by the use of Piola's, Kirchhoff's or Biot's (Jaumann's) stresses respectively and the corresponding deformation quantities a unified abstract formulation of the basic equations is given. Because of certain properties of the statical and geometrical operators various material independent work theorems follow including the (generalized and modified) principles of virtual displacements and virtual forces. For a hyperelastic body under conservative loading these are transformed into generalized variational principles and strengthened to complementary extremum principles. Finally the abstract formulation is applied to the incremental equations. The admissible functions are allowed to have relaxed continuity properties. Therefore the presented work theorems may be used as a basis for a consistent finite element approximation according to the mixed version or to the classical displacement formulation.     

Least square methods in structural analysis

A new approach to structural analysis is presented. The method uses equilibrium and deformation geometry field relations directly without resorting to an energy formulation. A matrix vector of errors in the field relations is minimized first with respect to the internal quantities, e.g., the moments. Subsequently the error is minimized with respect to the external forces and displacements. In the latter formulation, the internal quantities have been eliminated by back substitution. The least-square method for minimizing functional forms is the precedent for this work. The method is tested on beam and plate problems. The development of the pertinent matrices is interesting and depends on the trial functions used to represent the internal quantities.     

Nonlinear shell analysis via mixed isoparametric elements

Mixed isoparametric elements are presented for the geometrically nonlinear analysis of laminated composite shells. The analytical formulation is based on a form of the nonlinear shallow shell theory with the effects of shear deformation, material anisotropy and bending-extensional coupling included. The fundamental unknowns consist of the 13 stress resultants and generalized displacements of the shell. The generalized stiffness matrix is obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Both triangular and quadrilateral elements are considered. The accuracy of the mixed isoparametric elements developed is demonstrated by means of numerical examples and their advantages over commonly-used displacement elements are discussed. Also, computational procedures are presented for the efficient evaluation of the elemental matrices and for overcoming the difficulties associated with the large, sparse system of equations of the mixed models thus making them competitive with displacement models.     

Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study

In this study, the effect of the centrifugal forces on the eigenvalue solution obtained using two different nonlinear finite element formulations is examined. Both formulations can correctly describe arbitrary rigid body displacements and can be used in the large deformation analysis. The first formulation is based on the geometrically exact beam theory, which assumes that the cross section does not deform in its own plane and remains plane after deformation. The second formulation, the absolute nodal coordinate formulation (ANCF), relaxes this assumption and introduces modes that couple the deformation of the cross section and the axial and bending deformations. In the absolute nodal coordinate formulation, four different models are developed; a beam model based on a general continuum mechanics approach, a beam model based on an elastic line approach, a beam model based on an elastic line approach combined with the Hellinger–Reissner principle, and a plate model based on a general continuum mechanics approach. The use of the general continuum mechanics approach leads to a model that includes the ANCF coupled deformation modes. Because of these modes, the continuum mechanics model differs from the models based on the elastic line approach. In both the geometrically exact beam and the absolute nodal coordinate formulations, the centrifugal forces are formulated in terms of the element nodal coordinates. The effect of the centrifugal forces on the flap and lag modes of the rotating beam is examined, and the results obtained using the two formulations are compared for different values of the beam angular velocity. The numerical comparative study presented in this investigation shows that when the effect of some ANCF coupled deformation modes is neglected, the eigenvalue solutions obtained using the geometrically exact beam and the absolute nodal coordinate formulations are in a good agreement. The results also show that as the effect of the centrifugal forces, which tend to increase the beam stiffness, increases, the effect of the ANCF coupled deformation modes on the computed eigenvalues becomes less significant. It is shown in this paper that when the effect of the Poisson ration is neglected, the eigenvalue solution obtained using the absolute nodal coordinate formulation based on a general continuum mechanics approach is in a good agreement with the solution obtained using the geometrically exact beam model.     

Thermal postbuckling of thin-walled composite stiffeners

A study is made of the thermal postbuckling response of composite stiffeners subjected to prescribed edge displacement and a temperature rise. The flanges and web of the stiffeners are modeled by using two-dimensional plate finite elements. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the plate. A reduction method is used in conjunction with mixed finite element models for determining the postbuckling response of the stiffeners. Sensitivity derivatives are evaluated and used to study the effects of variations in the different lamination and material parameters of the stiffeners on their postbuckling response characteristics. Numerical studies are presented for anisotropic stiffeners with Zee and channel sections.     

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.     

A mixed shear flexible finite element for the analysis of laminated plates

A mixed shear flexible finite element based on the Hencky-Mindlin type shear deformation theory of laminated plates is presented and their behavior in bending is investigated. The element consists of three displacements, two rotations, and three moments as the generalized degrees of freedom per node. The numerical convergence and accuracy characteristics of the element are investigated by comparing the finite element solutions with the exact solutions. The present study shows that reduced-order integration of the stiffness coefficients due to shear is necessary to obtain accurate results for thin plates.     

Tangible user interfaces for physically-based deformation: design principles and first prototype

We present design principles for conceiving tangible user interfaces for the interactive physically-based deformation of 3D models. Based on these design principles, we developed a first prototype using a passive tangible user interface that embodies the 3D model. By associating an arbitrary reference material with the user interface, we convert the displacements of the user interface into forces required by physically-based deformation models. These forces are then applied to the 3D model made out of any material via a physical deformation model. In this way, we compensate for the absence of direct haptic feedback, which allows us to use a force-driven physically-based deformation model. A user study on simple deformations of various metal beams shows that our prototype is usable for deformation with the user interface embodying the virtual beam. Our first results validate our design principles, plus they have a high educational value for mechanical engineering lectures.     

Bent-beam electrothermal actuators-Part I: Single beam and cascadeddevices

This paper describes electrothermal microactuators that generate rectilinear displacements and forces by leveraging deformations caused by localized thermal stresses. In one manifestation, an electric current is passed through a V-shaped beam anchored at both ends, and thermal expansion caused by joule heating pushes the apex outward. Analytical and finite element models of device performance are presented along with measured results of devices fabricated using electroplated Ni and p⁺⁺ Si as structural materials. A maskless process extension for incorporating thermal and electrical isolation is described. Nickel devices with 410-μm-long, 6-μm-wide, and 3-μm-thick beams demonstrate 10 μm static displacements at 79 mW input power; silicon devices with 800-μm-long, 13.9-μm-wide, and 3.7-μm-thick beams demonstrate 5 μm displacement at 180 mW input power. Cascaded silicon devices using three beams of similar dimensions offer comparable displacement with 50-60% savings in power consumption. The peak output forces generated are estimated to be in the range from 1 to 10 mN for the single beam devices and from 0.1 to 1 mN for the cascaded devices. Measured bandwidths are ≈700 Hz for both. The typical drive voltages used are ⩽12 V, permitting the use of standard electronic interfaces that are generally inadequate for electrostatic actuators     

Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams

Most existing formulations for structural elements such as beams, plates and shells do not allow for the use of general nonlinear constitutive models in a straightforward manner. Furthermore, such structural element models, due to the nature of the generalized coordinates used, do not capture some Poisson modes such as the ones that couple the deformation of the cross section of the structural element and stretch and bending. In this paper, beam models that employ general nonlinear constitutive equations are presented using finite elements based on the nonlinear absolute nodal coordinate formulation. This formulation relaxes the assumptions of the Euler–Bernoulli and Timoshenko beam theories, and allows for the use of general nonlinear constitutive models. The finite elements based on the absolute nodal coordinate formulation also allow for the rotation as well as the deformation of the cross section, thereby capturing Poisson modes which can not be captured using other beam models. In this investigation, three different nonlinear constitutive models based on the hyper-elasticity theory are considered. These three models are based on the Neo–Hookean constitutive law for compressible materials, the Neo–Hookean constitutive law for incompressible materials, and the Mooney–Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. These results show that the use of nonlinear constitutive models can significantly enhance the performance and improve the computational efficiency of the finite element models based on the absolute nodal coordinate formulation. The results also show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.     

Design of a planar multibody dynamic system with ANCF beam elements based on an element-wise stiffness evaluation procedure

The absolute nodal coordinate formulation (ANCF) has been widely applied for large deformation analysis in flexible multibody dynamics. Although the formulation led to stable solutions for time integration under large rotations and deformations, excessive time consumption was recorded. The nonlinear relationship between the deformation and the internal force accounted for repeated adjustment to the force equilibrium state as the structure deformed. In this research, an equivalent model of the ANCF beam structure was constructed. The stiffness evaluation method was applied in an element-wise manner. In this model, the irrelevant parts were separated from those that relate to the displacements and design parameters enabling efficient updates of internal forces to achieve force equilibrium. Therefore, by using this model, optimization problems, in which displacements as well as design parameters keep changing can be efficiently approached. To verify the proposed method, two examples of optimization problems related to a free-falling pendulum and a slider-crank mechanism are demonstrated.     

Shear-Bending-Torsion Interaction in Structural Concrete Members: A Nonlinear Coupled Sectional Approach

Inclined cracks that take place in reinforced concrete elements due to tangential internal forces, such as shear and torsion, produce a non-isotropic response on the structure in the post-cracked regime and up to failure, also known as crack-induced-anisotropy. The result is that all six internal forces acting in a cross-section are generally coupled. A generalized beam formulation for the nonlinear coupled analysis of non-isotropic elements under six internal forces is presented. The theory is based on a cross-section analysis approach with both warping and distortion capabilities, which were proved necessary to correctly handle the problem with frame element analysis. In this paper, the non-linear mechanical aspects of cracked concrete structures under tangential forces are summarized. A state of the art review of beam formulations for the non-linear analysis of concrete structures is presented, and the approaches followed to account for the interaction of shear and torsion forces are discussed. After presenting the proposed formulation, its capabilities are shown by means of an application example of a cross section under coupled bending-shear and torsion, finally main conclusions are drawn.     

Sensitivity analysis for the dynamic response of viscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric response of viscoplastic shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. Spatial discretization is performed using finite elements, with discontinuous stress resultants across element interfaces. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) with an implicit constitutive update. The sensitivity coefficients are evaluated using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading, and a circular plate subjected to impulsive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of these quantities at selected times.     

Theoretical research on naturally curved and twisted beams under complicated loads

This paper presents an analytical method for the study of naturally curved and twisted beams under complicated loads, with special attention devoted to the solving process of governing equations which take into account the effects of torsion-related warping as well as transverse shear deformations. The solutions derived in this paper can be used for the analysis of the beams, including the calculation of various internal forces, stresses, strains and displacements. These governing equations, in special cases, can be readily solved and yield the solutions to the problem. A generalized warping coordinate for a curved coplanar beam subjected to the action of vertical distributed loads is given for verification.     

A new model for slab and beam structures––comparison with other models

In this paper an optimized model for the analysis of plates reinforced with beams is presented as compared with other models used by various researchers. The adopted model contrary to the models used previously takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system. According to this model the stiffening beams of the structure are isolated from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The adopted model describes better the actual response of the plate-beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. Four additional models neglecting the shear forces at the interfaces are presented and used for comparison reasons, while a three-dimensional elasticity model is also employed for the verification of the accuracy of the results of the examined models. The findings from this investigation, using the adopted model, which approximates better the actual response of the plate-beams system, necessitate the consideration of the inplane forces and deformations.     

Stresses and displacements for floating pan roofs

Procedures are developed and parametric curves presented for pan-type roofs floating on various liquids. The rim essentially subjects the central, liquid-supported plating to a vertical load but, at the same time, it provides radial stiffness to the plating. The effect of the trade-off between the rim's weight and stiffness is observed. An understanding of this effect is important in the design of these roofs. Commonly used rim configurations and tank sizes are considered in the presentation and, thus, the results are useful in approximating the displacements of and forces in typical floating pans.     

Study of the ligament tension and cross-section deformation using nonlinear finite element/multibody system algorithms

In this investigation, the effects of the knee-joint movements on the ligament tension and cross-section deformation are examined using large displacement nonlinear finite element/multibody system formulations. Two knee-joint models that employ different constitutive equations and significantly different deformation kinematics are developed and implemented to analyze the ligament dynamics in a computational solution procedure that integrates large displacement finite element and multibody system algorithms. The first model employs a lower fidelity large displacement cable element that does not capture the cross-section deformations and allows for using only nonlinear classical beam theory with a linear Hookean material law instead of a general continuum mechanics approach. In the second model, a higher fidelity large displacement beam model that captures more coupled deformation modes including Poisson modes as well the cross-section deformation is used. This higher fidelity model also allows for a straight forward implementation of general nonlinear constitutive models, such as Neo Hookean material laws, based on a general continuum mechanics approach. Cauchy stress tensor and Nanson’s formula are used to obtain an accurate expression for the ligament tension forces, which as shown in this investigation depend on the ligament cross section deformation. The two models are implemented in a general multibody system algorithm that allows introducing general constraint and force functions. The finite element/multibody system computational algorithm used in this investigation is based on an optimum sparse matrix structure and ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The results obtained in this investigation show that models that ignore coupled deformation modes including some Poisson modes and the cross-section deformations can lead to inaccurate prediction of the ligament forces. These simpler models, as demonstrated in this investigation, can be used to obtain only simplified expressions for the ligament tensions. A three-dimensional knee-joint model that consists of five bodies including two flexible bodies that represent the medial collateral ligament (MCL) and lateral collateral ligament (LCL) is used in the numerical comparative study presented in this paper. The large displacement procedure presented in this investigation can be applied to other types of Ligaments, Muscles, and Soft Tissues (LMST) in biomechanics applications.     

On the role of geometrically exact and second-order theories in buckling and post-buckling analysis of three-dimensional beam structures

Several geometrically nonlinear beam models are evaluated with respect to their utility in the analysis of buckling and post-buckling behavior of three-dimensional beam structures. The first two models are based on the so-called geometrically exact beam theory capable of representing finite rotations and finite displacements. The principal difference between these models concerns only the chosen parameterization of finite rotations, with the orthogonal matrix used in the first and the rotation vector used in the second one. The third beam model based on the second-order approximation of finite rotations is also discussed along with its application to constructing a consistent formulation of the linear eigenvalue problem for computing an estimate of the critical load. Exact linearized forms, which are crucial for facilitating the buckling load computation and assuring a robust performance of a Newton-method-based continuation strategy, are presented for all three beam models. An elaborate set of numerical simulations of buckling and post-buckling analysis of beam structures is given in order to illustrate the performance of each of the presented models. Finally, some conclusions are drawn.