By P. R. Lancaster, D. Mitchell
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Extra info for Advanced Solid Mechanics: Theory, worked examples and problems
LSa the stresses at a point are shown in both rectangular and polar coordinates. lSb. 16). 11 (a) Fig. 15 T~--------------~a•L-----------~--~ Fig. 16 28 To obtain the value of ar note that 6 is the angle measured anticlockwise from the x-axis to the normal to the plane ac. On the Mohr circle this corresponds to measuring 26 anticlockwise from the datum ab. Note that points in the Mohr circle correspond to planes in the stress diagram. Thus ar + a6 --:2,--- + R cos (26 - a) where R is the radius of ar + a6 ----2--- cry and + ~1ohr circle.
The additional terms in the expressions for a and a , not found by the simple theory of bending, allow for the X y normal loading on the top surface of the bar, which produces compressive stresses (a ) and also compensation to correct for the y 43 condition of zero bending moment at the ends of the plate (o ). 1. - c2y2 v = - 2EI [12 3 + ~ 3 + v (L- c2y2} 6 5 Now for compatibility of strain w 2c 2x ( 2 x 3) = 2EI [2xy2 - -5-+ R, X - 3 aul(Y) ay av 1 (x) + vx(y 2 - c2) + vxy2] + - - - + _a_x_ which is true for all values of x andy.
10b are, as they stand, independent but as with stresses, they must be compatible with each other so that the strain distribution throughout the plate is continuous. 18. 10c is an identity and the variables can be separated following the usual mathematical techniques in the solution of partial differential equations. 10c that a constant -Pa 2/2GI occurs. 10b). It is usual in such instances to let A+ B = -Pa2/2GI. This does not increase the constants of integration, which decide the boundary conditions because if A is known, B can be found.