For the Kelvin state as considered in Example 15.4, explicitly justify the displacement and stress results given
Question:
For the Kelvin state as considered in Example 15.4, explicitly justify the displacement and stress results given in relations (15.2.8) and (15.2.10).
Data from example 15.4
Equation 15.2.8
Equation 15.2.10
Transcribed Image Text:
Consider next the combined Kelvin problem with unit loads aa (a = 1, 2, 3) acting along each of three coordinate directions, as shown in Fig. 15.6. This singular state is denoted by S"(x) and is given by the potentials p = 0, y = -C- R 1 8 (1-v) The displacements and stresses corresponding to this case become +(3-4v) ai] = C Xaxi 2R R UR = TR$ = C(1-2v) of= C [3x a Xixj R R As a special case of this state, consider aa= [0, 0, 1], which would be state S*(x) with potentials - Jijxa) - -+ (1-2v)(daixj + dajxi dijxa, sino where C = 0==4=0, = (15.2.9) and in spherical coordinates (R0,0) (see Fig. 1.6) yields the following displacements and stresses 2C(1-v) cosp C(3 - 4v) sind R 2 R OR=-2C(2-v)- 9 coso R = up R TRO= Te = 0 de=0=C(12v)- (15.2.7) Ug = 0 coso R (15.2.8) (15.2.10)
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KelvinStatewith Unit Load 0 y C where C R 2 x y 41v y C2 401v C34v C 2R 34v 8 2v8 12v x 10 10 C R C ...View the full answer
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Related Book For
Elasticity Theory Applications And Numerics
ISBN: 9780128159873
4th Edition
Authors: Martin H. Sadd Ph.D.
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