Engineering Rock Mechanics
408 Appendix A: Stress and strain analysis from which COS' e = YZ (1 + COS 4) or cos 4 = (1 - sin2 e) - sin' e = 1 - 2 sin2 e fr ...
Stress analysis 409 positive shear stresses plot below the o-axis. The trigonometric relations used to simplify the equations r ...
41 0 Appendix A: Stress and strain analysis Calculate the rotation angle and direction from ox to q. Remember that rotations on ...
Strain analysis 41 1 Using Mohr's circle to determine stresses on u plune Follow points 1,2,3 and 5 of the method for determinin ...
41 2 Appendix A: Stress and strain analysis P moves to P’, Q moves to Q. The vector P’Q’ may have a different magnitude and dire ...
Strain analysis 41 3 that P and P' remain coincident and Q moves to Q. Hence du = - sin a-dy The total change in u (i.e. du) is ...
41 4 Appendix A: Stress and strain analysis For small rotations sin R = R, so that du=-Rdy dv = Rdx or (b) Normal strain ...
Strain analysis 41 5 Hence or Note that the tensorial shear strain is half the engineering shear strain. (d) Combined strain and ...
41 6 Appendix A: Stress and strain analysis Collecting these expressions together: Ex = -au/ax, E, = -av/ay r These are the equa ...
Strain analysis 41 7 Note that strains are dimensionless. Transformation of displacement (plane strain) Say point P moves to P' ...
41 8 Appendix A: Stress and strain analysis Now so By convention E,’ = decrease in lengthhit length in x’ direction %’ = decreas ...
Strain analysis 41 9 Now, if we replacef with u', v' and yIXy in turn: auf auf aut ax! ax aY = -cos 8 + - sin0 a a ax aY = cos ...
420 Appendix A: Stress and strain analysis au av av av av JY aY ax ax ax aU av au ax JY aY = -cos2 8 + -cos 8 sin8 - -cos 8 sin8 ...
Strain analysis 421 1 125 .sx=-=8O0Op or e,, = SOOOp E =-=zoo0 ps or err = 2OOOp 1 500 yxy=-= 9000ps or exy = 4500p. 1000 Comput ...
422 Appendix A: Stress and strain analysis The directions x’ and y’ corresponding to this value of p are known as principal dire ...
Strain analysis 423 Example. Continuing from before, with e,, = 8000 p, eyy = 2000 ps, eq = 4500 ,us: B I \ B Q at (2000,4500) ...
424 Appendix A: Stress and strain analysis It is not possible to measure shear strains in practice, so for two- dimensional stra ...
Strain analysis 425 (a) Delta rosette E~ = c + r COS 28 Q = c + r COS (28 + 120) ER = c + r cos (28 + 240). Using the double ang ...
426 Appendix A: Stress and strain analysis rearranging (4) gives (b) Rectangular rosette 0 .cP = c + r cos 28 % = c + r cos (28 ...
Strain analysis 427 In summary then: Delta rosette 1 c=- &,+EQ+€, l( 3 d3(ER -'Q) tan28 = Rectangular rosette c=- E,+&, ...
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