By Prof. Dr. A. N. Guz (auth.)

At the current time balance conception of deformable structures has been built right into a manifold box inside stable mechanics with equipment, options and techniques of its personal. we will be able to infrequently identify a department of or civil engineering the place the result of the steadiness concept haven't came across their program. This vast improvement including engineering functions are mirrored in a flurry of papers showing in periodicals in addition to in a lots of monographs, textbooks and reference books. In so doing, vast majority of researchers, con cerned with the issues of functional curiosity, have handled the lack of balance within the thin-walled structural components. attempting to simplify resolution of the issues, they've got used - and one-dimensional theories according to a variety of auxiliary hypotheses. This task contributed much to the preferential improvement of the soundness idea of thin-walled constructions and corporation of this idea right into a department of good mechanics with its personal updated tools and traits, yet left 3-dimensional linearised concept of deformable our bodies balance (TL TDBS), tools of fixing and ideas of the three-d balance difficulties themselves virtually with no cognizance. It needs to be emphasized that through 3 dimensional theories and difficulties during this booklet are intended these theories and difficulties which don't draw two-dimensional plate and shell and one-dimensional rod theories.

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**Sample text**

In the small deformations theory the change in the metric tensor is neglected as well. Let us discuss now incompressibility condition under the simplifications of the small deformations theory. 185) Let us consider now how contravariant components of "tracking" loads simplify in the small deformations theory. 186) it is unimportant with respect to what state, deformed or undeformed, the load intensity is calculated. 184) are the basic relations of the small deformations theory built up on simplification accepted in this subsection.

5 Simplifications for small rotation angles 39 denotes elongation factor in direction of the infinitesimal element. If after deformation the infinitesimal element is located out of plane x3 = const , let oN· denote its projection onto the plane x 3 = const ; all the above considerations remain valid in this case as well. Thus, the rotation angle of an infinitesimal element around an axis to which it was perpendicular before deformation is said to be an angle between this element before deformation and a projection of the element after deformation onto the plane orthogonal to the axis of rotation.

R ;;f. n r' = __1_ dgnn. _ dg "" ()(}' ' I 2 r~~ = r· = _1_ ag lim mn 2g •• I •. ,. __a_l 2 ()(}m ng... I. 110) should be taken into account. 1). 116) All other relations in cylindrical co-ordinates can be obtained in a similar way. It is worth noting that in Cartesian co-ordinates diagonal elements of the metric tensor are identically unity and all non-diagonal elements are identically zero. Besides that, all covariant derivatives reduce to the familiar partial derivatives with respect to appropriate co-ordinate.