tailieunhanh - Applied Structural Mechanics Fundamentals of Elasticity Part 6

Tham khảo tài liệu 'applied structural mechanics fundamentals of elasticity part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 136 8 Disks By substituting 12 a b c into 5a b c one obtains the stresses 13a 13b 13c For presenting the results the stresses are normalized as F b-a t. This delivers the following stress distributions where the abbreviation p b a is used Fig. B-8 Circumferential and radial stresses in the quarter - circle disk Exercise B-8-5 137 ơw ơ a-b t F r 2 3a 0 sin p Try F r 2 a Y 1 . 2 a i p - 1 P ---2-. ----sTI--cos f p2 - 1 1 p2 In p i or o are obtained at p and V has its rTmax Wmx 2 The maximum stresses Ơ maximum at jp 0. Fig. B-8 presents the distribution of the normal stresses over the cross section for different ratios of the dimensionless radii p. If p increases approaches a linear distribution corresponding to that of a straight beam. Exercise B-8-5 A semi - infinite disk y 0 - co X co thickness t is subjected to a concentrated moment M at the origin 0 as shown in Fig. B-9a. a Determine the stress function for this load case by using a force couple as shown in Fig. B-9b and by applying the stress function radiating stress state Í F ff t r ip cos 9 for a concentrated force F as formulated in ET2 . b Which stresses occur in the semi-infinite disk Solution a In order to determine the stress function o we substitute the prescribed moment by a force couple as shown in Fig- B-9b. Then the stress function for F in 0j can be written as - Q x - y . 138 8 Disks If we superpose the above stress function by the corresponding stress function for F in 0 . X y we obtain - x-E JiO y . 1 According to the rules of the differential calculus the partial derivative is defined as a ạ . 2 ùx x ỵ The difference in 1 can by applying 2 be written in the form 1 .X- 3a After transformation into polar coordinates ò ồr r ò òy the difference reads I E ._ Blli p rcos V - r 3b Tn ET2 the stress function of a single load on a semi-plane is derived as F r ffi cos qj . KÍ The partial derivatives of 4 F F . . V t-r cos V - cpsin p yield by substitution into 3b the following stress .