Vibrations of pinned-fixed heterogeneous circular beams pre-loaded by a vertical force at the crown point

Green’s functions for nonhomogeneous curved beamswith applications to vibration problems

“…Here we summarize the most important hypotheses and governing equations of the model we have established to tackle the vibratory problem. A thorough description is available, e.g., in [22,23,24]. We use a curvilinear coordinate system whose orthogonal unit vectors e ξ ; e η ; e ζ are attached to the E-weighted centerline which intersects the cross-section at the point C e .…”

“…Thus, the constitutive equation is Hooke's law in the following form: σ ξ = E(η, ζ)ε ξ . Making use of Hooke's law and the kinematic relations (1)-(2) we can set up the following equations for the axial force N and the bending moment M -as regards the details the reader is referred to [22,23]:…”

“…Here the virtual quantities are denoted by the symbol δ while f n , f t are distributed loads in the normal and tangential direction. The principle of virtual work yields two non-linear equilibrium equations [23]:…”

“…With the knowledge of the Green function matrix the solution sought can be given by the integral: (14) in which G(ϕ, γ) is the Green function matrix and ϕ, γ are angle coordinates. The Green function matrix is defined by the following four properties [1,23]:…”

Dynamic Stability of a Structurally Damped Delaminated Beam Using Higher Order Theory

“…Natural frequency and static buckling analyses of intact straight beams and plates [2] as well as circular arches [3][4][5] are a widely investigated topic. However laminated composite structures due to their heterogeneous material are able to become partially or totally damaged in the presence of fabrication defects, impact loading, or chemical corrosion [6].…”