Nonuniform elastic torsion (Q1329497)

An approximate analysis of linearly elastic homogeneous isotropic prismatic bar of doubly symmetric solid cross-section twisted through a small angle by externally applied couples which may vary, continuously or discontinuously, along the length of the bar, is presented. The displacement components of the beam are postulated in the form: \(u_ 1=w(x_ 1,x_ 2,x_ 3)\), \(u_ 2=-x_ 3t(x_ 1)\), \(u_ 3=x_ 2 t(x_ 1)\), where \((x_ 2,x_ 3)\) is a point of an end section of the beam, \(x_ 1\) is the centroidal axis \((x_ 1\geq 0)\); the twist angle \(t=t(x_ 1)\) and the warping displacement \(w=w(x_ 1,x_ 2,x_ 3)\) are to be found from the associated approximate field equations subject to suitable torsional boundary conditions (one of the approximations is that the stress components \(S_{22}\) and \(S_{33}\) vanish throughout the beam). Also, a torsion problem is discussed in which \(t=t(x_ 1)\) is prescribed and a torque \(T=T(x_ 1)\) and warping function \(w=w(x_ 1,x_ 2,x_ 3)\) are to be found. A series solution to these problems is proposed, and numerical calculations particularly for bars with doubly symmetric cross-sections are given.