On the Derivation of Exact Solutions of a Tapered Cantilever Timoshenko Beam

Abstract

A tapered beam is a beam that has a linearly varying cross section. This paper presents an analytical derivation of the solutions to bending of a symmetric tapered cantilever Timoshenko beam subjected to a bending moment and a concentrated force at the free end and a uniformly-distributed load along the beam. The governing differential equations of the Timoshenko beam of a variable cross section are firstly derived from the principle of minimum potential energy. The differential equations are then solved to obtain the exact deflections and rotations along the beam. Formulas for computing the beam deflections and rotations at the free end are presented. Examples of application are given for the cases of a relatively slender beam and a deep beam. The present solutions can be useful for practical applications as well as for evaluating the accuracy of a numerical method.

Locking-free Kriging-based Timoshenko Beam Elements using an Improved Implementation of the Discrete Shear Gap Technique

“…At the right end, the beam is subjected to a prescribed deflection wL and a prescribed rotation θL (see Fig 1(b) ), wL = θL = 0 if the right end is clamped. The weak form of the governing equations for the beam static deformation is given as [12] ∫ 𝛿𝜃, 𝑥 𝐸𝐼 𝜃, 𝑥 𝑑𝑥 + 𝛿𝑤(0)𝑃 0 + 𝛿𝜃(0)𝑀 0 ∀ 𝛿𝑤, 𝛿𝜃 ∈ 𝕍 = {𝑣|𝑣 ∈ ℍ 1 (0, 𝐿), 𝑣(𝐿) = 0}…”

Improved numerical integration for locking treatment in the Peridynamic Timoshenko beam model

“…With respect to the displacement field u of a beam with length L 0 exposed to a distributed axial load f x , a distributed transverse load f z and an inertia force ρu¨, the term of minimum potential energy (Wong et al ., 2019) is described as:where Π int denotes the internal strain energy, Π ext represents the energy of external forces and δ signifies the variation operator concerning the state variables (u = { u , ϕ , w }).…”

“…With respect to the displacement field u of a beam with length L 0 exposed to a distributed axial load f x , a distributed transverse load f z and an inertia force ρ€ u, the term of minimum potential energy (Wong et al, 2019) is described as:…”

Structural analysis of non-prismatic beams: Critical issues, accurate stress recovery, and analytical definition of the Finite Element (FE) stiffness matrix

“…Both the above mentioned models are derived for prismatic bodies i.e., for beams with constant axial, shear, and bending stiffnesses (proportional to cross-section area and second moment of area) [24]. Non-prismatic beam model is obtained just substituting the constant stiffness coefficients of prismatic beam ODEs with functions accounting for the variations of cross-section area and second moment of area [65,52,78,69,72]. This apparently trivial modification of the beam model however leads to ODEs for which an explicit analytical solution is difficult to compute.…”