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The free vibrations of physical systems
Published in A. P. French, Vibrations and Waves, 2017
The relation of the torsion constant to the basic elastic properties of the twisted material is less direct than the relation of a spring constant to Young's modulus for a stretched wire or rod. The essential process is called a shear deformation of the material. Suppose that a rectangular block of material is firmly glued at its base to a table, and that its top face is glued to a flat board [Fig. 3-9(a)]. Then a horizontal force Ρ applied to the board, in a direction parallel to two of the top edges of the block, causes a deformation as shown.1 Two of the side faces are changed from rectangles into parallelograms. Thus the deformation can be characterized by the angle of shear, ∝. In terms of the actual transverse displacement x of the top end of a block of height l, we have (approximately)
Effect of strain hardening on the rotation capacity of welded I-section high-strength steel beams
Published in Ships and Offshore Structures, 2023
Wei Jun Wong, Carey L. Walters
To study the behaviour of the plastic hinge localisation independently from length-dependent moment gradient and global buckling effects, the beams in this study were loaded with a uniform moment along the length (pre-buckling) and modelled such that the non-dimensional lateral-torsional-buckling slenderness (Equation (5)) was equal to 0.1. The parameter is where is the moment at first yielding in the cross-section, and is the elastic critical buckling load, for which an exact solution exists for a beam loaded subject to a uniform moment, as is given by Equation (6) (Bažant and Cedolin 2010): where L is the length of the beam; E is the Young's modulus; is the second moment of area about the minor axis; G is the shear modulus; is the warping stiffness; and J is the torsion constant.
Comparative study of analytical/semi-analytical methods for prediction of axial strength of cold-formed steel wall panels with sheathing
Published in Australian Journal of Structural Engineering, 2022
Chanchal Sonkar, Achal Kumar Mittal
Considering a column of any cross-section, U is given by Equation (2), where u, v=generalised displacements of the shear centre, and= rotation of the column sectionwhereE -steel modulus of elasticity;Ix and Iy- moments of inertia of the section in x and y directions, respectively;-product of inertia;Cw – torsion warping constant;G-shear modulus of the CFS; andJ- St. Venant torsion constant.
Animation of cycloid and spiral curves in companion with instantaneous center of rotation and radius of curvature
Published in Journal of the Chinese Institute of Engineers, 2023
Jeng-Tzong Chen, Chia-Ying Yang, Yen-Ting Chou, Chi-Ning Tsang
Zhou and Zhu (2007) discussed the relationship between natural coordinates in fluid mechanics and orthogonal curvilinear coordinates. Since orthogonal curvilinear coordinates may have some excellent mathematical properties, natural coordinates can be applied more widely if they can be transformed to the orthogonal curvilinear coordinates. Frenet formula, which describes the differential property of natural coordinates, was compared with the derivative formulae of orthogonal curvilinear coordinates to show that natural coordinates are not generally orthogonal curvilinear coordinates. The geometry of a space curve can be completely defined in terms of two parameters: the horizontal and vertical curvatures, or equivalently, the curvature and torsion in the literature of Riley and Hobson (2008). Distinction is made between the track angle and space-curve bank angle, referred to as the Frenet bank angle by Ling and Shabana (2021). In railroad vehicle systems, the track bank angle measures the track super-elevation required to define a balance speed and achieve a safe vehicle operation by Ling and Shabana (2021). Frenet formula is the governing equation of a curve in space. Two-dimensional curve is a special case. Once the radius of curvature is determined, the location of instantaneous center of rotation can be found. We may wonder what is the curve constructed by the instantaneous center of rotation. Regarding the two parameters in the Frenet formulae, the radius of curvature and the torsion constant are referred in the literature. We intend to explore the geometric meaning of the two parameters in the 2D cycloid and 3D spiral curve by way of animation. Both are demonstrated to see the trajectories of instantaneous center of rotation.