By A. L. Gol'Denveizer, Th. Von Kármán, H. L. Dryden
ISBN-10: 0080095615
ISBN-13: 9780080095615
Idea of Elastic skinny Shells discusses the mathematical foundations of shell idea and the approximate tools of resolution. the current quantity was once initially released in Russian in 1953, and continues to be the one textual content which formulates as thoroughly as attainable different units of uncomplicated equations and numerous approximate tools of shell research emphasizing asymptotic integration.
The booklet is equipped into 5 elements. half I provides the final formula and equations of the idea of shells, that are according to the well known speculation of the maintenance of the traditional aspect. half II is dedicated to the membrane theory--the most generally used approximate approach to research of shells that was once formulated at nearly an identical time because the extra basic bending thought. partially III equipment of research of round cylindrical shells through trigonometric sequence are thought of. half IV is basically mathematical in personality and its objective is to justify the approximate tools of shell research. partially V approximate equipment of study of shells are formulated.
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Extra info for Theory of Elastic Thin Shells. Solid and Structural Mechanics
Sample text
P, from which we obtain, finally, the two relations which give the restrictions of the vectors K and L. 4)· We then obtain where n dK . , a,fc,c, v. remain arbitrary, and are the resulting stress functions. 6) can be replaced by scalar ones in the same manner as was done in section 12. We then obtain the formulas which relate the stresses and moments to the stress functions. »_i— · ♦In this formula the divisors sin χ have been introduced, and the signs have been selected in a special way in order to give simpler forms to the formulas that follow.
Figure 10 If (α, β) are the parameters of an orthogonal system of curvilinear coordinates, then u, υ% w will coincide with the projections of the total displacement on the positively-directed tangents of the a- and β-curves and on the negatively-directed normal to the surface, respectively. Let us introduce, also, the following notation for the angles of rotation which arise in the process of deformation or displacement of the shell (Figure 11)ι γΑ is the angle through which the element directed along the vector Ma rotates towards the vector n in the plane (Jfe, n); γ2 is the angle through which the element directed along the vector 31 p rotates towards the vector n in the plane (il/p, n); Ü)J is the angle through which the element directed along the vector Ma rotates towards M^ in the tangent plane; ω2 is the angle through which the element directed along the vector M p rotates towards M* in the tangent plane· Obviously «) = o>1 + ω2 gives the change in the angle χ between the coordinate curves, while S= M *~Ml can be taken as the measure of the angle of rotation of an element of the middle surface around the normal from M$ to Ma„ The vector Ö which satisfies the following conditions will be called the vector of elastic rotation: 1) the projection of Q of the vector JV(p> is equal to γ,$ 2) the projection of Ω on JX{*} is equal to — γ2; 3) the projection of Q on the vector n is equal to δ.
7) if one sets 8 = const and a = const» The area of the curvilinear quadrangle will be dS = ABs'mxdad$. The following forces will act on the isolated element of the shells 1) R^Ada, acting on ab; 2) - [ **'* + ^ - 3) RmBd$, 4) - ( ^ + ^ Λ Χ * + ■ £ « * « )rfß·acting on M; 5) JMßsinxdadß, d ? J 0 4 + -^ <*ß)«*·. acting on cd: acting on ac: acting on the area abed. Let us set the sum of these forces equal to zero, and neglect infinitesimals of the third order. After some transformations we then obtain -yp(ARi°)-l