Elastic Eighth-Space. Elastic Quarter-Space
In what follows we deal with the study of the elastic eighth-space subjected to the action of a normal or tangential load; the results thus obtained will then be particularized for the case of the elastic quarter-space. We remember that they have been use
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Elastic Eighth-Space. Elastic Quarter-Space
In what follows we deal with the study of the elastic eighth-space subjected to the action of a normal or tangential load; the results thus obtained will then be particularized for the case of the elastic quarter-space. We remember that they have been used in Sect. 9.2 for the elastic half-space. We will take advantage of the stress functions introduced in [3, 4, 13, 16].
10.1 Elastic Eighth-Space The problem of the state of strain and stress in the interior of an elastic eighthspace has been only a few times studied. But the problem is of interest, because one can thus obtain informations concerning such a local state around a solid angle with three faces of an elastic solid; if the three plane faces are not orthogonal one each other, then the problem may be studied similarly, passing to a system of oblique Cartesian co-ordinate axes. Interesting results are known for the corresponding plane case (the elastic quarter-plane [51], the elastic wedge). A particular case of loading with an internal concentrated force has been considered by W. A. Hijab [2]. The elastic eighth-space, acted upon by a periodic load on one of its faces, may practically constitute the end beam in the case of a continuous wall-beam of infinite height, having an infinity of equal spans and identically acted upon. The results corresponding to a local loading may be as well used for the study of the local effect (corner effect) in the support zone of a wall-beam with a single span. We will consider the elastic eighth-space xi 0; i ¼ 1; 2; 3, acted upon by normal loads, using the results given in [6, 8] or by tangential loads, using the results given in [7]. We will present also a particular case of loading, which leads to results in finite form, as it has been shown in [10, 14].
P. P. Teodorescu, Treatise on Classical Elasticity, Mathematical and Analytical Techniques with Applications to Engineering, DOI: 10.1007/978-94-007-2616-1_10, Ó Springer Science+Business Media Dordrecht 2013
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Elastic Eighth-Space. Elastic Quarter-Space
10.1.1 Action of a Periodic Normal Load Let be the elastic eighth-space xi 0; i ¼ 1; 2; 3, acted upon by a normal load p3 ðx1 ; x2 Þ; periodic (in two directions) on the face x3 ¼ 0. To fix the ideas, we represent this load by means of a double Fourier series, even with respect to the variables x1 and x2 , of the form (9.2.1), (9.2.10 ).
10.1.1.1 General Considerations The elastic eighth-space xi 0; i ¼ 1; 2; 3, may be thought as a part of the elastic half-space, acted upon by the periodic load (9.2.1), (9.2.10 ), symmetric with respect to the Ox1 and Ox2 axes. The tangential stresses vanish on the faces x1 ¼ 0 and x2 ¼ 0, while the normal stresses are given by m b0 1m X X bnm þ a2n ð1 cnm x3 Þ þ 2mb2m ecnm x3 cos bm x2 ; 2 cnm n m m b0 r22 ðx1 ; 0; x3 Þ ¼ 1m X X bnm þ b2m ð1 cnm x3 Þ þ 2ma2n ecnm x3 cos an x1 ; 2 cnm n m
r11 ð0; x2 ; x3 Þ ¼
ð10:1Þ
where we took into account the formulae (9.2.11). The state of stress thus calculated c
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