The torsional centre position of stocky beams with arbitrary noncircular cross-sectional shapes and with arbitrary elastic material properties

An innovative method for surmounting the constrained torsional problems of stocky beams with arbitrary noncircular cross-sectional shapes and with arbitrary elastic material properties

“…For the meaning of the symbols in the formula and the creation process of the formula, please refer to the literature [2] . when establishing the out-of-plane strain law of the cross-section under stress, the plane-section hypothesis utilized by the traditional beam theory is no longer employed, but the mathematical approximation method is adopted to develop the out-of-plane strain law of the cross-section of a beam after deformation.…”

“…The meaning of the symbols in the formula is the same as that explained in literature [2] . The so-called Ludwig Prandtl membrane is the analogy membrane used by Ludwig Prandtl in the study of the torsion of thin-walled bars with noncircular cross sections.…”

“…My years of research experience signifies that if the interpolation function represented by formula (2) is selected, the numerical solution of the mathematical model will be comparable to the analytical solution (see the verification example 2 in the literature [2] ) because formula (2) is a mathematical evolvement to Prandtl's membrane [9] utilized in the study on the maximum shearing stress of beams with noncircular thin-walled cross sections due to torsion.…”

“…A, B, C, D are the cross-section tension-compression stiffness matrix, anti-bending stiffness matrix, anti-warping stiffness matrix, anti-shear-torsion stiffness matrix, respectively. The form of matrix A, B, C, D is the same as that showed in the appendix of literature [2] .…”

“…If the mechanics behavior of the beam under the condition of pure constrain torsion is desired, the first target will become to determine the position of the torsional centre of the cross section according to the method introduced in reference [2] , and then take the connections of all the torsional centres of the cross sections as the x -axis and establish a global coordinate system. Then successively take above steps (3)-(7), the mechanics behavior of the beam under the condition of pure constraint torsion will be obtained.…”