An Explicit Stiffness Matrix for Parabolic Beam Element

Abstract

This study is devoted to strain-based formulation for a curved beam. Arches with parabolic geometry, which have a variety of applications, belong to this structural type. Dependency of the curvature radius to the arch length creates some complexities in the solution process. To analyze these complex structures, a twonode beam with six degrees of freedom is suggested by utilizing closed-form solution and the stiffness-based finite element method. Considering the effect of shear deformation, and incorporating equilibrium conditions into the finite element model, lead to the exact strains. Displacements and explicit stiffness matrix are found based on these exact strains. To validate the efficiency of the author's formulation, seven numerical tests are performed. The outcomes demonstrate that by employing only a single element, the locking-free answers can be found. KeywordsFinite element method, parabolic beam, explicit stiffness matrix, strain-based formulation, equilibrium conditions.

Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions

“…where D i 1 and D i 2 define the displacement components of an arbitrary point in the longitudinal (x i ) and thickness (z) directions, respectively, whereas d i 1 and d i 2 refer to the displacement field of the neutral axis in the longitudinal (x i ) and thickness (z) directions, respectively. For details about the geometrical definition of this system in its bottom and upper parts can be found in [1]. The relation between the strain components of an arbitrary point and the reference neutral strain field is defined as…”

“…This kind of system consists of beams with a straight or curved geometry, connected by an elastic layer, with a linear, non-linear, and plastic behavior. Despite the large amount of studies from the literature focusing on the structural behavior of single beams [1][2][3][4], some static and dynamic aspects for coupled beam systems have already been explored in the pioneering works by Seelig and Hoppmann [5] and Dublin and Friedrich [6], drawing increased attention from researchers in the recent years. These studies focused on static and dynamic behavior [7][8][9], transverse vibration [10], and forced vibration [11], using viscously damped interlayer [8,12] among others.…”

Finite Element Analysis of Curved Beam Elements Employing Trigonometric Displacement Distribution Patterns

“…Third, transverse shear deformation and rotational inertia effects are taken into account, while cross-sectional distortions and shear lag effects are neglected. Finally, the curved beam element is treated as a circular arc element with a constant radius of curvature for each individual element [16].…”

Stability of a half-sine shallow arch under sinusoidal and step loads in thermal environment

“…Such deformations cause severe changes in strains and stresses. The geometry of the arch is an influential parameter on its load-bearing capacity (Cai et al, 2012;Bateni and Eslami, 2015;Bradford et al, 2015;Rezaiee-Pajand and Rajabzadeh-Safaei, 2016). In addition, various loadings (e.g., the sinusoidal (Plaut and Johnson, 1981), concentrated (Pi et al, 2008;Chandra et al, 2012;Tsiatas and Babouskos, 2017), distributed (Moghaddasie and Stanciulescu, 2013b) and end moment loads (Chen and Liao, 2005;Chen and Lin, 2005)), geometric imperfections (Virgin et al, 2014;Zhou et al, 2015a), and boundary conditions (Pi and Bradford, 2012;Pi and Bradford, 2013;Han et al, 2016) are other important factors in the structural design.…”