hw21_Solutions

# Gde 2l eiw x f 0 stp x 3 boundary conditions w0

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Unformatted text preview: axial stress along the bottom, s max (x ) = ˘ M 0b È Lˆ 3Ê 1 ˆ Ê Í1 - stpÁ x - ˜ + Á Á a + 1 3 ˜(x - L )˙ ˜ 2I Î 2¯ 8Ë Ë ¯ ˚ 17. By inspection, y * = y, z * = z ﬁ (0,0 ) By symmetry, I *z = 0 y * I zz = Â Ei I zz + y 2 A i = 13.33 in 4 E0 [ ( )] * GDE: E 0 I zz v' ' ' ' (x ) = -10 BC: v(0 ) = v' (0 ) = v' ' ( ) = v' ' ' ( ) = 0 100 100 * E 0 I zz v' ' ' (x ) = -10 x + C1 * E 0 I zz v' ' (x ) = -5 x 2 + C1 x + C 2 C 5 * E 0 I zz v' (x ) = - x 3 + 1 x 2 + C 2 x + C 3 3 2 C C 5 * E 0 I zz v(x ) = - x 4 + 1 x 3 + 2 x 2 + C 3 x + C 4 12 6 2 ﬁ C1 = 1000, C 2 = -50000, C 3 = C 4 = 0 Therefore, we have * E 0 I zz v(x ) = - 5 4 1000 3 50000 2 x+ xx 12 6 2 Deflection: v(x ) = 1 Ê - 5 4 500 3 ˆ x+ x - 25000 x 2 ˜ *Á 3 E 0 I zz Ë 12 ¯ Moment: * E 0 I zz v' ' (x ) = -5 x 2 + 1000 x - 50000 Stress: s xx ÈNc Ê M cI* - M cI* z yy y yz Í - yÁ * Á I* I* - I* 2 ÍA yz Ë yy zz Î Mc ˆ EÊ Á - y *z ˜ = E0 Á I zz ˜ Ë ¯ E = E0 () ˆ Ê M cI* - M cI* z yz ˜ - z Á y zz ˜ Á I* I* - I* 2 yz ¯ Ë yy zz () ˆ˘ ˜˙ ˜˙ ¯˚ At point A, v( 00 ) = -0.9377 in 1 s xx = 0 since M zc = 0 at x = 100 At point C, v(0 ) = 0 by the first boundary condition lb È (- 50000 )˘ s xx = 1Í- 2 ˙ = 7502 in 2 13.33 ˚ Î 18a) Boundary conditions: w(0 ) = w' (0 ) = w(L ) = w' ' (L ) = 0 GDE: (EIw' ')' ' = f z (x ) = q0 Ê1 - x ˆ ﬁ EIw' ' ' ' (x ) = q0 Ê1 - x ˆ Á ˜ Á ˜ Ë L¯ Ë L¯ b) q0 x 2 EIw' ' ' = q 0 x + C1 2L q x2 q x3 EIw' ' = 0 - 0 + C1 x + C 2 2 6L 3 q0 x q 0 x 4 C1 2 EIw' = + x + C 2 x + C3 6 24 L 2 q x4 q x5 C C EIw = 0 - 0 + 1 x 3 + 2 x 2 + C 3 x + C 4 24 120 L 6 2 ﬁ C1 = - 2q 0 L q L2 , C 2 = 0 , C3 = C 4 = 0 5 15 q 0 x 4 q 0 x 5 q 0 Lx 3 q 0 L2 x 2 EIw(x ) = + 24 120 L 15 30 Nondimensionalizing, EIw(x ) - 1 Ê x ˆ 1 Êxˆ 1 Êxˆ 1 Êxˆ = Á ˜+ Á ˜- Á ˜+ Á ˜ 4 120 Ë L ¯ 24 Ë L ¯ 15 Ë L ¯ 30 Ë L ¯ q0 L 5 4 3 2 c) M y (x ) = EIw' ' (x ) M y (x ) = q 0 L2 Ê x ˆ q L2 Ê x ˆ 2q L2 Ê x ˆ q L2 - 0 Á ˜ - 0 Á ˜+ 0 Á˜ 2 ËL¯ 6 ËL¯ 5 Ë L ¯ 15 2 3 Nondimensionalizing, M y (x ) q 0 L2 -1Ê x ˆ 1 Ê x ˆ 2Ê x ˆ 1 = Á ˜ + Á ˜ - Á ˜+ 6 ËL¯ 2Ë L¯ 5ËL¯ 5 3 2 19a) Ê 3x ˆ A(x ) = a(x )b = abÁ1 ˜ Ë 4L ¯ ba 3 (x ) a 3 b Ê 3 x ˆ I yy (x ) = = Á1 ˜ 12 12 Ë 4 L ¯ 3 The applied load will have two effects - extension: force p0b - bending: moment –p0ba/8 No distributed load: the only loading is applied at x = L which shows up in the boundary conditions. GDE: (EA(x )u ' (x ))' = - f x = 0 (EI yy (x )w' ' (x ))' ' = f z = 0 Boundary conditions: u (0 ) = w(0 ) = w' (0 ) = ( yy w' ')L = 0 EI ' EA(L )u ' (L ) = p 0 b (EI yy w' ') = - p 0 L ab 8 Note that...
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## This note was uploaded on 01/18/2010 for the course AE 322 taught by Professor Lambros during the Spring '04 term at University of Illinois at Urbana–Champaign.

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