. A treatise on the mathematical theory of elasticity . oint on a boundary, the term of (27) whichcontains Q can be one-valued indepen-dently of any adjustment of v!, v. Itis merely necessary to fix the meaningof e. In Fig. 17, OX is the initialline, drawn into the plate, and the angleXOT=a. Then 0 may be taken to liein the interval We may seek the stress-system thatwould correspond with (27) if u and vwere put equal to zero. We should find^ 2(\ + /x) ^= X + 2fi r* Pig. 17. 2{X + ^) xf _2(\ + f,) x>y ^y- x+2f. ^^ ^- x + 2/. ^^ ^^^) In polar coordinates the same stress-system is expressed by
. A treatise on the mathematical theory of elasticity . oint on a boundary, the term of (27) whichcontains Q can be one-valued indepen-dently of any adjustment of v!, v. Itis merely necessary to fix the meaningof e. In Fig. 17, OX is the initialline, drawn into the plate, and the angleXOT=a. Then 0 may be taken to liein the interval We may seek the stress-system thatwould correspond with (27) if u and vwere put equal to zero. We should find^ 2(\ + /x) ^= X + 2fi r* Pig. 17. 2{X + ^) xf _2(\ + f,) x>y ^y- x+2f. ^^ ^- x + 2/. ^^ ^^^) In polar coordinates the same stress-system is expressed by the equationsp^2JX^)^c_os^ ^ = 0, Te^O (31) This distribution of stress is described by Michell* as a simple radialdistribution. Such a distribution about a point cannot exist if the pointis within the body. When the origin is a point on the boundary, the state ofstress expressed by (31) is that due to a single force at the point. Wecalculate the resultant traction across a semicircle with its centre at theorigin. The a;-component of the resultant is. or it is X-I-/X\ + 2fi. — I rr. cos 6. rdO, J —IT+a. IT. The y-component of the resultant is -/ rr . sin 0 .rd6. — TT+a London Math. Soc. Proc, vol. 32 (1900), p. 35. 148-151] AT A POINT OF A PLATE 209 or it is zero. Thus the resultant applied force acts along the initial line andits amount is ?7rA(X +fj,)/(X + 2fi); the sense is that of the continuation ofthe initial line outwards from the body when A is positive. This result gives us the solution of the problem of a plate with a straightboundary, to which force is applied at one point in a given direction. Takingthat direction as initial line, and F as the amount of the force, the stress-system is expressed by the equations ;^=__i^£^!_ ,^ = 0, ^=0, (32) and these qiiantities are of course averages taken through the thickness ofthe plate. 150. Case of a straight boundary. In the particular case where the boundary is the axis of x, the axis of y penetrates intothe plate, a
Size: 1541px × 1622px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1920, bookpublishercambr, bookyear1920
