Tham khảo tài liệu 'modelling of mechanical system volume 2 episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 98 Structural elements Figure . Warping function of rectangular cross-sections that which governs the transverse displacement of a stretched membrane. More generally the problem can be solved numerically by using the finite element method. As a general result the following inequality holds Jt J which means that the stiffness due to the elastic deformation of the cross-sections is less with warping than without. The equality holds for a circular cross-section only. The values of the torsion constant for various cross-sectional shapes may be found in several books see for instance BLE 79 . A few results are quoted below Ellipse Jt na3b3 a2 b2 Square Jt 0 1406 a4 Rectangle Jt Ya3b3 a2 b2 a b 2 4 8 with Y 1 3 Straight beam models Newtonian approach 99 Hollow rectangle thickness ha and hb Jt 2hahb a - ha 2 b - hb 2 aha bhb ha hl Open thin walled cross-sections Jt Ph3 3 P length h thickness Closed thin walled cross-sections Jt 4Sh p P perimeter h thickness S area of the closed part A few numerical applications of these formulae are useful to illustrate the importance of warping. Taking a square cross-section as a first example the ratio Jt J is found to be X 6 . For a rectangular section of aspect ratio a b 2 J 5 6 b4 and Jt X 5 6 b4 or Jt J . Finally if the aspect ratio of the rectangle is very large a b 1 it is found that Jt ab3 3 so Jt J 4 a b 1 2 3 1. To conclude on the subject it may be added that if warping is prevented by clamping both ends local axial stresses are generated which are especially important in the case of open thin walled cross-sections. The reader interested in such aspects of the problem may be referred for instance to TIM 51 PIL02 . Pure bending mode of deformation As a preliminary we must emphasize that in contrast to a frequently used definition according to which pure bending of a beam means that no shear forces arise by pure bending mode of deformation we mean here that the beam is
