Arbitrarily curved and twisted space beams
Hunter, William Francis
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A derivation of the equations which govern the deformation of an arbitrarily curved and twisted space beam is presented. These equations differ from those of the classical theory in that extensional effects are included. Other departures from the previous theory are that the strain - displacement relations are derived and that the expressions for the stress resultants are developed from the strain - displacement relations instead of assuming that the resultants are proportional to changes in the curvatures. It is shown that the torsional stress resultant obtained by the classical approach is basically incorrect except when the cross-section is circular. Using a vector approach the exact expressions for the curvature components of a deformed space beam are developed. Because inextension of the beam is not assumed an additional term appears in each of the linearized curvature expressions. These expressions are utilized in the derivation of the strain - displacement relations. The normal and shearing physical components of the strain tensor are given. These relations are not restricted to beams whose cross-sectional dimensions are very small compared to the radius of curvature. Next, a development of the stress resultants is presented. Effects arising from the initial twist of the beam are obtained which are not reflected in the classical theory. Finally, the six equilibrium equations are derived using a vector approach. The governing equations are given in the form of twelve first-order differential equations. A numerical algorithm is given for obtaining the natural vibration characteristics and example problems are presented.
- Doctoral Dissertations