Plane Strain: The Direct Determination of Stress . y Fig 3. JC The stresses are yy xy- P iraih—a) fe(tan ^—^tan^-—-j + ^^tMtan ^^—tan \ y y I V y y ah tan~^ -—- — tan^ ? -f by log — ? ;/—i \ y y / (a?—a)2-f3/ //rj»—. f) I* —I- Qj + ^w/ log ^ ^ ^^^ ^ r^* + 3/ Traih^a) fe/tan ^ 2/ tan 1.::—_) -I- . (34) 2/ —a&ftan 1 Jj~~ C tan y ^y iraQy—oC) 1) i tan ^ ic ^ tan^ .1 a X y a tan ^ -^ — tan v y As these results are of some practical importance, the stresses could beexpressed more simply by writing 6 =^ tain~^xjy, #i = tan*^(;«—a)/^/, and02 = tan-^ (^-~^)/3/j where 6, 6i, and 02 are measured fr
Plane Strain: The Direct Determination of Stress . y Fig 3. JC The stresses are yy xy- P iraih—a) fe(tan ^—^tan^-—-j + ^^tMtan ^^—tan \ y y I V y y ah tan~^ -—- — tan^ ? -f by log — ? ;/—i \ y y / (a?—a)2-f3/ //rj»—. f) I* —I- Qj + ^w/ log ^ ^ ^^^ ^ r^* + 3/ Traih^a) fe/tan ^ 2/ tan 1.::—_) -I- . (34) 2/ —a&ftan 1 Jj~~ C tan y ^y iraQy—oC) 1) i tan ^ ic ^ tan^ .1 a X y a tan ^ -^ — tan v y As these results are of some practical importance, the stresses could beexpressed more simply by writing 6 =^ tain~^xjy, #i = tan*^(;«—a)/^/, and02 = tan-^ (^-~^)/3/j where 6, 6i, and 02 are measured from the planes ^ = 0^,X = (Xj and X :=^b respectively, and are positive on the side on which x increasesand negative on the other side. Plane Strain: the Direct Determination of Stress. 121. Fig ^. The stresses at the point Q, for example, could be written thus:- P 7ra(&—a) yy -hx{e+ei)-^ax{e + d2)-ab {62-61)i-hx(6+6,) + ax(6 + 62)-ah(62-6i)] ?^ L ^ (35) (17) The following example is of considerable theoretical interest. Consideran infinitely long slab of thickness 27r, the axis of ^ being in the centralplane of the slab, as shown in fig. 5. 4 « I « I « i 1 4 -a- r y.,, t „r 0 I. f,.. .f Fig 5. Mr. S. D. Carothers. If we take t , • t _/»/ .N 1. 1 COS i^ , -f 1 sin e// ^4-^i|r =:/(^ + ^^) == tan l--_--}-^|log—^ : ; then the scheme of stresses tLitRj yy xy . 3(f) ,1 ^ (36) gives tension over the central half of the edge 3/ =? 0 of intensity tt andpressure of equal intensity over the remainder. There are certain shearingstresses on the edge 2/ = 0, but these are in equilibrium and may be consideredas having a purely local effect. The normal pressures and tensions on the edge ^ = 0 are in equilibrium,and it will be a matter of great interest to asc
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