PSI - Issue 71

Ishaan Tapas et al. / Procedia Structural Integrity 71 (2025) 477–483

479

K 2

Correction term due to taper

2. Illustrations In this section, we present the formulation of the Hamiltonian for the tapered cantilever beam in brief. A schematic illustration of a tapered beam is shown in Figure 1. The global coordinate system O (x, y, z) is attached at the root of the beam. In this work, the taper is characterized by the varying cross-sectional area along the x-axis. The x-axes of these coordinate systems are aligned with the centroidal axis of the beam. The y and z axes are oriented along the principal axes of the cross-section of the beam. In this work, the taper is assumed to be linear along the length of the beam.

Fig. 1. Schematic of a tapered beam.

Let u(x) and w(x) be the axial and transverse displacement variables of any point on the cross-section in the global coordinate system. The Lagrangian is developed such that the beam cross-section is symmetric in two planes and the bending is uncoupled in two perpendicular planes. The circumferential strain equation for the beam in the transverse direction is given by the equation: = = − ( 2 2 ) (1) It can be shown that the shear stresses are identically zero. The strain energy is given by, = 1 2 ∫ 0 ∫ 2 = 12 ∫ 0 ∫ 2 ( 2 2 ) 2 (2) where, L is the length of the beam, E is the Young’s modulus of the material and A is the cross -section of the beam, and ∫ 2 = , where Izz is the area moments of inertia about the z-axis. Note, as the cross-sectional area is varying along the length, Izz is a function of the length coordinate x. Thus, the potential energy of the tapered beam is given by, = 1 2 ∫ 0 ( 2 2 ) 2 (3) Similarly, the kinetic energy of the tapered beam is given by, = 2 2 ∫ 0 ( ) 2 (4)