HW6 Solutions

HW6 Solutions - Mechanics of Aircraft structures C.T. Sun...

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Unformatted text preview: Mechanics of Aircraft structures C.T. Sun 3.1 Show that there is no warping in a bar of circular cross-section. Solution: (a) Saint-Venant assumed that as the shaft twists the plane cross-sections are warped but the projections on the x-y plane rotate as a rigid body, then, zy u = zx v = (3.1.1) ) , ( y x w = where ) , ( y x is some function of x and y, called warping function, and is the angle of twist per unit length of the shaft and is assumed to be very small. (b) From the displacement field above, it is easy to obtain that = = = = xy zz yy xx So from the stress-strain relationship, we have = = = = xy zz yy xx Therefore the equilibrium equations reduce to = + y x yz xz This equation is identically satisfied if the stresses are derived from a stress function ) , ( y x , so that y xz = , x yz = (3.1.2) (c) From the displacement field and stress-strain relationship, we can obtain y x w z u x w xz = + = (3.1.3) x y w z v y w yz + = + = (3.1.4) So it forms the compatibility equation 2 = y x xz yz , or in terms of Prandtl stress function G y x 2 2 2 2 2 = + (3.1.5) (d) Boundary conditions, = ds d , or . const = But for a solid sections with a single contour boundary, this constant can be chosen to be zero. Then we have the boundary condition = on the lateral surface of the bar. (e) For a bar with circular cross-section, assume the Prandtl stress function as 3.1.1 Mechanics of Aircraft structures C.T. Sun ) 1 ( 2 2 2 2 + = a y a x C which satisfies the boundary conditions stated above. Substitute into (3.1.5), we obtain G a C 2 2 1 = Then ) ( 2 2 2 2 a y x G + = Using (3.1.2), we have y y G xz = = 1 , and x x G yz = = 1 Comparing with (3.1.3) and (3.1.4), we have y y x w xz = = => = x w . Thus, ) ( y f w = x x y w yz = + = => = y w , Thus, ) ( x g w = Hence we conclude . This means that the cross-section remains plane after torsion. In other words, there is no warping. const w = Therefore can be verified, and it successfully expresses the statement. ) , ( = y x w--- ANS 3.1.2 Mechanics of Aircraft structures C.T. Sun 3.2 Show that the Prandtl stress function for bars of circular solid sections is also valid for bars of hollow circular sections as shown in Fig. 3.34. Find the torsion constant in terms of the inner radius and outer radius , and compare with the torsion constant obtained using (3.59) for thin-walled sections. What is the condition on the wall thickness for the approximate to be within 1 percent of the exact ?...
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HW6 Solutions - Mechanics of Aircraft structures C.T. Sun...

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