SP6-1. (a) A very long, prismatic elastic shaft of circular cross section is embedded in an...

 

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SP6-1. (a) A very long, prismatic elastic shaft of circular cross section is embedded in an elastic medium. When a length dx of the shaft rotates an amount 8, the medium applies torque kedx to the length dx. Torque To is applied to the end of shaft at x = 0. Obtain expressions for rotation = (x) and torque T = T(x) in terms of To, k (foundation modulus), G (shear modulus), J (polar second moment of area), and x. (b) Revise part (a) so that the medium resists axial displacement rather than rotation, with force kudx for axial displacement u of a length dx. The bar resists stretching rather than twisting, and the loading at x = 0 is axial force Po rather than torque To. Obtain expressions for axial displacement u = u(x) and axial force P = P(x) in terms of Po, k, E (Young's modulus), A (cross-sectional area), and x. SP6-1. (a) A very long, prismatic elastic shaft of circular cross section is embedded in an elastic medium. When a length dx of the shaft rotates an amount 8, the medium applies torque kedx to the length dx. Torque To is applied to the end of shaft at x = 0. Obtain expressions for rotation = (x) and torque T = T(x) in terms of To, k (foundation modulus), G (shear modulus), J (polar second moment of area), and x. (b) Revise part (a) so that the medium resists axial displacement rather than rotation, with force kudx for axial displacement u of a length dx. The bar resists stretching rather than twisting, and the loading at x = 0 is axial force Po rather than torque To. Obtain expressions for axial displacement u = u(x) and axial force P = P(x) in terms of Po, k, E (Young's modulus), A (cross-sectional area), and x.

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