Tham khảo tài liệu 'principles of engineering mechanics (2nd edition) episode 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 236 Introduction to continuum mechanics Ut Uv Us therefore t7s ut-uv 1171 1 V _2 _2x 2 Iff à 1 í 2p 2 l ơ lơ 2 or2Ơ-3 T3Ơ-1 Now E 2G 1 v and K - 3 1 2v so 2G 14-p 3 1 20 and strain which occur at 45 to the bar axis. The strain energy based on this calculation is i7 -i k1 2 2 - a12 ẢG oG The difference is due to the fact that in this situation the element is still being distorted because the transverse stress is still zero even though the other two normal stresses are equal to each other. Example A thin-walled cylindrical pressure vessel shown in Fig. contains a gas at pressure p. The diameter of the shell is m and the thickness of Substituting these values into the previous equation Figure - ơì ơ2 Ơ2ƠỊ Ơ3Ơ1 Ậ r O - Ơ2 2 r2 - Ơ3 2 0-3 - O 2 2G o the steel is 6 mm. Given that the yield stress of steel is 300 MN m2 find the maximum allowable pressure based on the stresses on the curved surfaces remote from the ends. Example The von Mises-Hencky theory of failure for ductile materials suggests that under the action of multi-axial stresses failure will occur when the maximum shear strain energy is equal to that which occurs when failure occurs in a simple tensile test. Show that this leads to the formula o-l - Ơ2 2 r2 - Ơ3 2 0-3 - O 2 2ay2 where Oy the yield stress. Solution From example the expression for shear strain energy is K0-1 z 2 T - O3 2 3 - O 2 In a simple tensile test ơ2 0-3 0 so r7 1 r 1 7 Gs 2o- Ơ- where O ơy IZCi oG therefore crj - r2 2 cr2 - Ơ3 2 ơ-3 - O 2 2 ry2 It is interesting to note that the shear strain energy in the simple tensile test is greater than that calculated from the maximum shear stress Solution The material is under the action of a longitudinal stress and a hoop stress as shown in Fig. . We shall consider the equilibrium of a unit length of the cylinder. Resolving in the y direction 7i_2fl7t rHt Figure p2r 2tơH 0 where O-H is known as the hoop stress therefore _pr ƠHt Considering the equilibrium
