PSI - Issue 24

Francesco De Crescenzo et al. / Procedia Structural Integrity 24 (2019) 28–39 Francesco De Creascenzo and Pietro Salvini / Structural Integrity Procedia 00 (2019) 000 – 000

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It is convenient to assign a value to the helix angle and then to compute the corresponding radius and load. By this way, it is possible to table loads against deflection and to compute the tangent characteristic of the coil using central difference = 0 [ cos 1 2 (sin 0 cos 0 tan + 2 cos 2 0 ) ( 2 + tan 2 )] (25) = c o s ( sin cos − sin 0 cos 0 0 ) (26) = − 2 0 0 cos 0 (sin − sin 0 ) (27) Finally, tangent stiffness of the spring is given by: ℎ = (28) 3.4. Coil shearing characteristics In the 2D discrete model, shearing of coils is lumped in two springs for each coil. In Fig.3a this is obtained with a full spring k s that lumps two-quarters of coil and with two half-springs 2k s , that lump a quarter of coil each. The simplest model of coil shearing is the one used by Haringx in his work on spring buckling (Fig. 3b). The coil is projected on a plane perpendicular to helical axis, resulting in an open ring. One end of the ring is clumped and a radial force is applied to the other. A bending moment ′ is here also applied to the free end, in order to hold it against rotation. The bending moment along the wire and deformation energy are then: = ′ + sin (29) = 1 2 ∫ 2 2 0 = 1 2 ∫ 2 2 0 (30) where it is evident that the force is not inducing any rotation of the free end so that ′ = 0 . The radial displacement is found applying Castigliano's theorem: = = 3 (32) and the integration gives: = ′2 + 2 3 2 (31)

Shearing stiffness of the coil is then: = = 3

(33)