Question

Using the following example to come up with a formula for displacement for a sphere instead of a cone.

I am having a hard time coming up with an expression for A(y) (circle in red). Can someone please show with steps how to come up with an expression for A(y) so that I can solve for displacement similar to the provided example. Thank you

-0.3 m A member is made of a material that has a specific weight of y = 6 kN/m and modulus of elasticity of 9 GPa. If it is in the form of a cone having the dimensions shown in Fig. 4-8a, determine how far its end is displaced due to gravity when it is suspended in the vertical position. SOLUTION Internal Force. The internal axial force varies along the member, since it is dependent on the weight Wy) of a segment of the member below any section, Fig. 4-85. Hence, to calculate the displacement, we must use Eq. 4-1. At the section located a distance y from the cone's free end, the radius x of the cone as a function of y is determined by proportion; i.e., 0.3 m x = 0.17 3 m The volume of a cone having a base of radius x and height y is (0.01) VXP = y = 0.01047y (x) = {12-22 3 m xatges ? VE TW 1 (a) Since W = 1 force at the section becomes V T(1+ry2) (4) N(y) 4 +12F, = N(y) = 6(10)(0.01047y) = 62.83y3 Displacement. The area of the cross section is also a function of position y, Fig. 4 85. We have Wy) A(y) = x = 0.03142 y2 d' = 8 T (10-ye) dh Applying Eq. 4-1 between the limits of y = 0 and y = 3 m yields (6) S SN) (62.83y) dy (0.03142y2) 9(10) c (AG)= ??? Fig. 4-8 Ay) E = 222.2(10) of vay da Yur = 1(10-) m = 1 um NOTE: This is indeed a very small amount. Aly) E Ans.

Answer 1

Date Page 25;no A ordo LYOSO X dado a sole O 2 Area element A= π (rsino volume element du = A (thickness) = Al&do) as shown in figure = A r² sin²ordo dv - AB sine do. To generate the entire sphere vary from sto n. 8- from o tor' Haloever due to symmetry of these problem we com estress 82o in terms of x,y,z, coso = 2 2 2 Zarroso. - Very?+22 /d2= sinodo . sino - (132) 2) I sin ?o=1-60520 Kothamaletament at height 2 Al Almeye dva nr²sing (rsino do) = -Ag² - 22 de using ③ ignoring - sign, dra ar? 11-2² dz Similarly A = A r²sino - nr21 - 22 ). I have expressed A as A (2), and y as V (2). You can use t for 2 it you wish so. Hope this helps Pus you need to consider the ealm of this volume element and then follow on similar shown in example of from @ A - 82 82 cone live us