Tham khảo tài liệu 'finite element analysis - thermomechanics of solids part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 14 Torsion and Buckling TORSION OF PRISMATIC BARS Figure illustrates a member experiencing torsion. The member in this case is cylindrical with length L and radius r0. The base is fixed and a torque is applied at the top surface which causes the member to twist. The twist at height z is 6 z and at height L it is 60. Ordinarily in the finite-element problems so far considered the displacement is the basic unknown. It is approximated by an interpolation model from which an approximation for the strain tensor is obtained. Then an approximation for the stress tensor is obtained using the stress-strain relations. The nodal displacements are solved by an equilibrium principle in the form of the Principle of Virtual Work. In the current problem an alternative path is followed in which stresses or more precisely a stress potential is the unknown. The strains are determined from the stresses. However for arbitrary stresses satisfying equilibrium the strain field may not be compatible. The compatibility condition see Chapter 4 is enforced furnishing FIGURE Twist of a prismatic rod. 181 2003 by CRC CRC Press LLC 182 Finite Element Analysis Thermomechanics of Solids a partial differential equation known as the Poisson Equation. A variational argument is applied to furnish a finite-element expression for the torsional constant of the section. For the member before twist consider points X and Y at angle Ị and at radial position r. Clearly X r cosộ and Y r sin Ộ. Twist induces a rotation through angle ớ z but it does not affect the radial position. Now x r cos 0 ớ y r sin 0 ớ . Use of double-angle formulae furnishes the displacements and restriction to small angles ớ furnishes to first order u -Yớ v Xớ. It is also assumed that torsion does not increase the length of the member which is attained by requiring that axial displacement w only depends on X and Y The quantity w X Y is called the warping function. It is readily verified that all strains vanish except
