Derivatives of eigenvalues and eigenvectors in non-self-adjoint systems.

Abstract

U 1 = J r |_e f j [D] {s} dV; and D = matrix relating stress to elastic strain, e = total strain, s 1 = pseudo initial strain.The integrals are over the undeformed (or initial) body and can readily be evaluated. Expanding Eq. (1) yields an expression of the form dV (2)Neglecting plasticity, temperature, and other similar effects, the total strain is the sum of the elastic and initial strains, or e^e' + s 1 (3) Treating the initial strain as a constant during the variation of the total strain results in the variation of the elastic strain being identical to the variation of the total strain -i.e., 6e =(4)Combining Eqs. (3) and (4) with Eq.(2) yields the equilibrium condition 6W=l r \e, e \[JDi\{dB e }dV (5) or For the beam shown in Fig. 2, the total strain may be written in terms of the displacements as e = (du/dx) -z(d 2 w { /dx 2 ) + ^\\du/dx) 2 + (dvtf/dx) 2~\ (7) where the total normal displacement, w f , is the sum of the initial displacement (due to forming the curved surface) and the deformation due to the applied loads-i.e., w r = w 0 + w. Similarly, the initial strain is given by e' --z(d 2 w Q /dx 2 ) + $(dw 0 /dx) 2 (8) Combining Eqs. (7) and (8) with Eq. (3) yields, upon neglecting second-order terms in u and w, the elastic strain as e*' = (du/dx)-:(d 2 w/dx 2 ) + (dw/dx)(dw 0 /dx) (9) It should be noted that for the case of a curved beam the method of initial strains gives rise to a formulation equivalent to shallow shell theory.In addition to its use herein with the curved beam element, the method of initial strains should prove equally enticing for use with other elements. For example, the same logic is readily adaptable for the plane stress and three-dimensional applications suggested in Fig. 1. Again, the integrals would be evaluated over the undeformed body and could therefore be exactly determined in terms of area or volume coordinates.Two of the important questions which must be answered in the application of the method of initial strains are: 1) Is rigid body motion satisfactorily represented? 2) What happens when the combined deflections are sufficient to yield large strains?In order for the proposed technique to be readily acceptable it is indeed necessary that the answer to the first question be in the affirmative. Adequate representation of rigid body motion must be maintained for reasonably sized elements. The equivalence of the curved beam formulation to shallow shell theory indicates the possibility of encountering difficulties with rigid body representation. For shallow shell theory the rigid body displacements are readily allowed but some straining exists during rigid body rotations. In answering the second question, it will be necessary to employ many of the fundamental concepts of continuum mechanics. It is the development and ensuing application of the large strain equations which may uncover the limits of the applicability of the initial strain technique. In the initial evaluations of the proposed method, however, it will be possible to use the more elementary (small strain theory...

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