A Hamiltonian State Space Approach for 3D Analysis of Circular Cantilevers

Abstract

A 3D exact analysis of extension, torsion and bending of a cantilever of a circular cross section is studied with emphasis on the fixed-end effect. Through Hamiltonian variational formulation, the basic equations of elasticity in cylindrical coordinates and the boundary conditions of the problem are formulated into the state space setting in which the state vector comprises the displacement vector and the conjugate stress vector as the dual variables. Upon delineating the Hamiltonian characteristics of the system, 3D solutions for transversely isotropic circular cantilevers subjected to an axial force, a torque, terminal couples and transverse forces are determined, thereby, the fixed-end effects and applicability of the solutions of generalized plane strains and the elementary theory of bending of beams are evaluated.

Exact Analysis of Curved Beams and Arches with Arbitrary End Conditions: A Hamiltonian State Space Approach

“…The first and the second eigenvalues are zero and ±i are repeated eigenvalues. The non-zero eigenvalues may be used to estimate the stress disturbance due to the end conditions [7,8]. As the dimension b/a decreases, the eigenvalues for both orthotropic materials and isotropic materials increase, which implies the end effects diminish more rapidly.…”

“…Following the same line as the derivation for problems of circular cylinders [7,8], it can be shown that the symplectic orthogonality for the problem are …”

“…which shows that, when θ is taken to be the distinguished coordinate that plays the role of the time variable, the conjugate variable dual with u = [u r , u θ ] T is p = [σ rθ , σ θθ ] T , not [rσ rθ , rσ θθ ] T as in the case of r or z being the distinguished coordinate [7,8].…”

“…Recently, we have developed a Hamiltonian state space formalism for anisotropic elasticity [5][6][7][8]. A salient feature of the formalism is that it brings out the inherent Hamiltonian character of the system, which provide an essential basis for a solution approach by using eigenfunction expansion.…”

State space formulation of magneto-electro-elasticity in Hamiltonian system and applications

“…Moreover, the state matrix of the Hamiltonian state equations was symplectic and its eigenvalues have physical meaning. By expansion of the eigen-functions of the state matrix, Ding et al [40] and Tarn and co-workers [41][42][43][44][45] found some new solutions which were difficult or even impossible to be obtained by the original state space method. On the other hand, the state space equations in Hamiltonian system had the corresponding variational principle and it was convenient to develop approximate method, for example, finite element method, to solve the state equation [46][47][48][49].…”