Reissner mixed variational principle is employed for establishment of the nonlinear differential and boundary conditions of dynamic equilibrium governing the flexure of beams when the effects of true ...shear stresses are included. Based on the Reissner mixed variational principle, the nonlinear size-dependent model of the Reissner nano-beam is derived in the framework of the nonlocal strain gradient elasticity theory. Furthermore, the closed form analytical solutions for the geometrically nonlinear flexural equations are derived and compared to the nonlinear flexural results of the Timoshenko size-dependent beam theory. The profound differences in the assumptions and formulations between the Timoshenko and the Reissner beam theory are also comprehensively discussed. The Reissner beam model is shown not to be a first-order shear deformation theory while comprising the influences of the true transverse shearing stress and the applied normal stress. Moreover, it is exhibited that the linear and nonlinear deflections obtained based on the Reissner beam theory are consistently lower than their Timoshenko counterparts for various gradient theories of elasticity.
The single shooting method is applied to the optimal control and combined structural and control design (co-design) of very flexible beams. The objective is to assess feasibility, advantages and ...limitations of an integrated design approach when dealing with actuated structures exhibiting large oscillations and, more generally, strongly nonlinear couplings. A gradient based approach is proposed for the large design space defined by both the optimal control and co-design problems. Numerical studies are presented for the case of a very flexible actuated pendulum with large rigid-body motion. The impact of local (B-splines) and global (discrete sines) set of basis functions is investigated for increasing levels of actuation authority, showing the importance of the time–frequency resolution of the parametrisation on the convergence properties and outcome quality of the process. Locking between control and structural disciplines around specific design points is found, thus highlighting the disadvantage of a sequential design approach. Simultaneous designing of control law and structure is seen, instead, to explore efficiently larger regions of the design space.
Plants constantly undergo external mechanical loads such as wind or touch and respond to these stimuli by acclimating their growth processes. A fascinating feature of this mechanical-induced growth ...response is that it can occur rapidly and at long distance from the initial site of stimulation, suggesting the existence of a fast signal that propagates across the whole plant. The nature and origin of the signal is still not understood, but it has been recently suggested that it could be purely mechanical and originate from the coupling between the local deformation of the tissues (bending) and the water pressure in the plant vascular system. Here, we address the physical origin of this hydromechanical coupling using a biomimetic strategy. We designed soft artificial branches perforated with longitudinal liquid-filled channels that mimic the basic features of natural stems and branches. In response to bending, a strong overpressure is generated in the channels that varies quadratically with the bending curvature. A model based on a mechanism analogous to the ovalization of hollow tubes enables us to predict quantitatively this nonlinear poroelastic response and identify the key physical parameters that control the generation of the pressure pulse. Further experiments conducted on natural tree branches reveal the same phenomenology. Once rescaled by the model prediction, both the biomimetic and natural branches fall on the same master curve, enlightening the universality of our poroelastic mechanism for the generation of hydraulic signals in plants.
In this paper, we present an original energy-preserving numerical formulation for velocity-based geometrically exact three-dimensional beams. We employ the algebra of quaternions as a suitable tool ...to express the governing equations and relate rotations with their derivatives, while the finite-element discretization is based on interpolation of velocities in a fixed frame and angular velocities in a moving frame description. The proposed time discretization of governing equations directly relates the energy conservation constraint with the time-discrete kinematic compatibility equations. We show that a suitable choice of primary unknowns together with a convenient choice of the frame of reference for quantities and equations is beneficial for the conservation of energy and enables admissible approximations in a simple manner and without any additional effort. The result of this study is simple and efficient, yet accurate and robust numerical model.
Finite deformations of planar slender beams for which shear strain can be neglected are described by the extensible- elastica model, where the strain-displacement relation is geometrically exact and ...the Biot stress–strain relation is linear. However, if the formulation is expressed in terms of displacements without rotation, the kinematics are described by a partial differential equation involving a fourth-order spatial operator, which cannot be approximated by the classical
C
0
-continuous FE method in the standard Galerkin framework. In this work, we propose the spatial approximation of such high-order PDE by means of NURBS-based isogeometric analysis (IGA) which allows the use of globally high-order continuous basis functions. The employed IGA approach possesses three advantages: first, it facilitates the encapsulation of the exact geometric representation of the beams in the spatial approximation with fewer discrete points, especially useful for curved structures; second, it allows the discretization of high-order spatial operators; and third, it provides an efficient numerical solution of the discrete problem by using a limited number of degrees of freedom since the employed standard Galerkin formulation does not require rotational degrees of freedom. Yet this approach has not been directly compared to appropriate analytical solutions. To this end, we compare and validate numerical results from FE with the closed-form solutions for a set of static beam problems, including a newly derived solution for an initially curved beam, based on the extensible-elastica theory, by estimating the convergence orders of the errors. We also highlight the advantages of this formulation with the numerical solution of three dynamic problems: the swinging of a pinned beam, the propagation of solitons (nonlinear waves) in post-buckled beams, and snap-through buckling of a pinned beam that is axially buckled before transverse loading.
Dynamic soil–structure interaction (SSI) represents an interdisciplinary subject characterized by complex geometrical and material nonlinearities affecting both the soil and the structural components ...of the system under study. In the current practice, SSI problems are often solved using the so-called
substructure method
based on decomposing the superstructure-foundation-soil system into two subsystems whose response is determined independently. The
direct method
is a more general procedure to tackle SSI problems which is in principle capable of accounting for both soil and structural nonlinearities. Despite this generality, most of the current computational platforms are specialized either for structural applications or for geotechnical applications. Whereas the formers are capable to accurately reproduce the nonlinear response of the structural elements, they are usually poor in modeling soil nonlinear behaviour especially under earthquake loading. On the other hand, the latter do the reverse: they are advanced in modeling soil response but typically rough in reproducing the behaviour of the structural components of the system. Inevitably, all structures interact with the ground and therefore the need to model material and geometric nonlinearities maybe important both in the ground and structural elements. The variation of the stiffness mismatch between soil and structural elements during ground motions is one of the most influential parameters for the assessment of the kinematic component in SSI problems. The variation of the stiffness ratio is mainly governed by material nonlinearity that can develop in either the soil or the structural parts of the system. This paper presents a nonlinear constitutive model for beam elements that is compatible with performance–based earthquake engineering principles because it is capable to simulate the cyclic degradation of mechanical properties and to assess the damage suffered by the structural system. The proposed model is based on distributed plasticity and can be easily implemented in any computational platform allowing the use of an explicit solver. The article describes the theoretical aspects of the model. Validation and application of the model to the solution of a dynamic soil–structure interaction problem where the nonlinear behaviour of geomaterials is coupled with the inelasticity of structural elements, are presented in a companion paper (Andreotti and Lai in Bull Earthq Eng
2017
).
In this article, an analytical approach, namely, multi-level residue harmonic balance is introduced and developed for the nonlinear free vibration analysis of axially loaded beams with an internal ...hinge. The main advantage of this method is that only one set of nonlinear algebraic equations is required to be solved for obtaining the zero level solution while the high accuracy of the higher level solutions can be obtained by solving a set of linear equations. The new approximate analytical solution method is developed for solving the governing differential equations. The accuracy and efficiency of the proposed method are verified by a numerical method. In the comparison, the results obtained from the proposed method well agree with those from other methods. The effects of vibration amplitude, axial force, and hinge location on the fundamental frequencies of various beam cases are investigated. The optimum and worst hinge locations are also studied.
Celotno besedilo
Dostopno za:
BFBNIB, DOBA, GIS, IJS, IZUM, KILJ, KISLJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
This paper presents a corotational formulation of a three-dimensional elasto-plastic mixed beam element that can undergo large displacements and rotations. The corotational approach applies to a ...two-noded element a coordinate system which continuously translates and rotates with the element. In this way, the rigid body motion is separated out from the deformational motion. In this paper, a mixed formulation is adopted for the derivation of the local element tangent stiffness matrix and nodal forces based on a two-field Hellinger–Reissner variational principle. The local beam kinematics is based on a low-order nonlinear strain expression using Timoshenko assumption. The warping effects are characterized by adopting Benscoter theory that describes the warping degree of freedom by an independent function. Shape functions that satisfy the nonlinear local equilibrium equations are selected for the interpolation of the stress resultants. This local element, together with the corotational framework, can be used to analyze the nonlinear buckling and postbuckling of thin-walled beams with generic cross-section. The mixed formulation solution is compared against the results obtained from a corotational displacement-based formulation having the same beam kinematics. The superiority of the mixed formulation is clearly demonstrated.
On a nonlinear theory of thin rods Luo, Albert C.J.
Communications in nonlinear science & numerical simulation,
12/2010, Letnik:
15, Številka:
12
Journal Article
Recenzirano
In this paper, a nonlinear theory for a straight rod is presented from the general theory of the three-dimensional deformable-body in the Cartesian coordinate frame. A set of nonlinear strains is ...presented, and the stretch on central curve exactly satisfies the deformation geometrical relations. The relations between the Euler angles and deformation are given from the curvatures and torsion curvatures of the central curves, which can easily explain the existing theories of rods and beams. Full nonlinear equations of motion for a nonlinear rod are developed via the vector form. Such a treatise is different from the traditional treatises of nonlinear rods, based on the Cosserat’s theory (e.g., Cosserat and Cosserat
1 in 1896) or the Kirchhoff assumptions (e.g., Kirchhoff
18 in 1859; Love
3 in 1944). This paper extends the ideas of Galerkin
4 in 1915. The nonlinear theory of thin rods can reduce to the existing theories for thin rods and beams, such ideas presented in this paper can be applied for development of the nonlinear theory for plates and shells as well.
One order parabolic approximation KZK equation is considered to be reliable inside ±15 degree relative to the KZK centerline, so KZK equation is unsuitable for studying steered beams in its classical ...configuration of having beam axis aligned with transducer central axis. An equivalent parametric array model was proposed based on the collimation characteristic of primary field, so the problem of steered beams calculation using KZK equation is converted to source condition calculation of equivalent parametric array. In this paper, a time domain solution of KZK equation for rectangular source is used to calculate steered nonlinear beams, the natural directivity beam calculated under this model is compared with classical KZK approach, and the influence of the distance between the center of parametric array and equivalent parametric array in calculating steered nonlinear field is analyzed at the same time. Finally, comparison is made between the calculated and measured steered nonlinear field which shows that the equivalent parametric array model is an effective theory model in steered nonlinear beams computation, and this research gives a helpful direction to the future study.
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