PSI - Issue 28

Christos F. Markides et al. / Procedia Structural Integrity 28 (2020) 710–719 Christos F. Markides and Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2019) 000–000

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The idea of adopting this alternative configuration of the uniform disc to describe the flattened one lies in the fact that a point force, applied at an internal point z k of the uniform disc, is understood as the resultant vector due to a stress distribution applied to the boundary of a deleted circular hole on the uniform disc’s cross-section, with z k as its center, where, as the hole shrinks onto z k , stresses increase beyond all bounds, maintaining the particular magnitude of P (Muskhelishvili 1963). In this context, letting n tend to infinity is equivalent to introducing an infinite number of con secutive infinitesimal wholes (subjected to point forces of the same, here, magnitude P ), located along z 1 z 2 and z 3 z 4 , thus cutting the uniform disc along these segments z 1 z 2 and z 3 z 4 , transforming it (by justifiably omitting the shaded segments of the uniform disc (Fig.2b)) into the flattened one (Fig.2c). The above described problem is here confronted as a first fundamental problem of plane linear elasticity and, under the assumptions of homogeneity and isotropy, Muskhelishvili’s (1963) complex potentials method is adopted, providing closed-form full-field expressions for the stress field components. Then, according to the previous reasoning, these stresses will approximate those of the flattened disc in question. It is to be mentioned that the solution of this particular problem for the uniform disc, results actually from the super position of the solutions of two problems: (a) that of the uniform disc under vertical forces of magnitude P at the circum ferential points z 1 , z 2 , z 3 , z 4 , and (b) that of the same disc under vertical forces of magnitude P at the internal points , k k z z (1≤ k ≤ n ). However, for brevity, the two problems are presented here simultaneously, reaching directly the overall solution. Regarding the degree of approximation, it is to be mentioned that the solution of the particular problem of the uniform disc approaches better and better that of the flattened disc as the number of internal points of application of the point forces becomes larger and, also, as the length of the chords z 1 z 2 and z 3 z 4 becomes smaller, taking into consideration that even if the former condition is satisfied, the shaded sectors of the uniform disc under and below the chords z 1 z 2 and z 3 z 4 will be never completely stress free. 2.2 The complex potentials The cross-section of the uniform disc is considered lying in the z = x + iy = re i θ complex plane, with its center as the origin of the Cartesian reference; R denotes the disc’s radius. Vertical forces (0, –P ), per unit thickness, act at the points z 1 = Re i Θ , z 2 = – Re – i Θ , and z k , while opposite forces (0, P ) act at the points z 3 = – Re i Θ , z 4 = Re – i Θ , and k z (Fig.3a). Regarding, for example, the internal points z k of application of the point forces (0, –P ) on the chord z 1 z 2 , a number n of them is considered, so that the total number of points of application of the point forces (0, –P ), i.e., of z k , along with z 1 , z 2 , will be n +2. Then, the magnitude of P , per unit thickness, will be:

frame P

(1)

P





 2 n t 

y

y

P frame

P frame

 2 frame P n t 

P=const.

.

P

const







z 2

z 1

z 2

z 1

k z

R

θ

n +2 total amount of points

z=re i θ

Θ

r

x

x

Ο

Ο

L

k z

z 4 n -intenral points

n +1 portions

z 3

z 3

z 4

(a) (b) Fig. 3. (a) The problem of the uniform disc; (b) The partition of the chord z 1 z 2 and the magnitude of P . P frame P frame