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1. Stiffness basics 35
when 0.6 < R/w < 8 – Young and Budynas [4], whereas for 8 <R/w < 10, the
eccentricity can be approximated to:
where is the moment of inertia about the centroidal axis and A is the cross-
sectional area, as indicated by Young and Budynas [4].
Figure 1.21 Cross-secion geometry and eccentricity position for a thick curved beam
5.3.1.1 In-Plane Compliances
The circular segment of Fig. 1.20 undergoes deformations within its own
plane under the action of and which are located at its free end 1.
Unlike the case of the straight beam, where the action of for instance, is
strictly axial, all three loads combine in the present case to produce axial,
shearing and bending deformations. The strain energy U contains
contributions from the bending moment, the normal force, and the shearing
force, in the form:
where is a generic angle ranging between 0 and The displacements at
point 1 are calculated by means of Castigliano’s displacement theorem as:
