# assn2 - 3 x 2 x 3(18 3 ±or the cantilever beam loaded with...

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16.21 - Techniques of structural analysis and design Homework assignment # 2 Handed out: 2/18/05 Due: 2/25/05 February 17, 2005 1. Determine whether the following stress ±elds are possible in a structural member free of body forces: (a) (not for grade) σ 11 = 3 x 1 + 6 x 2 (1) σ 12 = 4 x 1 + 3 x 2 (2) σ 22 = 5 x 1 + 4 x 2 (3) (b) 2 σ 11 = c 1 x 1 + c 2 x 2 + c 3 x 1 x 2 + c 4 x 1 (4) 2 x 2 c 1 x 2 c 4 x 2 (5) σ 12 = c 2 3 2 σ 22 = c 4 x 1 + c 1 x 2 (6) 1

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(c) σ 11 = 2 x 2 1 2 x 1 x 2 + 6 x 3 σ 12 = x 1 x 2 + x 2 2 (7) (8) σ 13 = x 1 x 3 (9) σ 22 = 3 x 2 2 (10) σ 23 = 5 x 2 x 3 (11) σ 33 = 2( x 1 2 x 2 ) x 3 (12) 2. Given the following state of stress, determine the body forces for which the stress Feld describes a state of equilibrium: 2 σ 11 = 4 x 1 (13) 3 σ 12 = x 1 + x 2 + 2 x 1 x 2 (14) 2 2 σ 13 = 4 x 1 + 2 x 2 7 x 3 (15) 2 2 σ 22 = 3 x 2 2 x 3 (16) σ 23 = 4 x 1 x 2 x 3 (17) σ 33 = (2 x 1
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Unformatted text preview: 3 x 2 ) x 3 (18) 3. ±or the cantilever beam loaded with a point load at the free end (see Fgure), the bending moment M 3 about the x 3-axis is given by M 3 = − F x 1 . The bending stress σ 11 is given by: M 3 x 2 σ 11 = I 3 where I 3 is the moment of inertia of the cross section about the x 3-axis. Use the two-dimensional equilibrium equations in diﬀerential form to determine the stress Felds: σ 22 and σ 12 . 2 4. For the state of stress of question 3 , determine the stress vector and its normal and shear components at the point ( L, h, 0) on the plane of normal: (a) (1 , , 0) (b) √ 1 3 (1 , 1 , − 1) Determine the principal stresses and principal directions of stress at this point. 3...
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assn2 - 3 x 2 x 3(18 3 ±or the cantilever beam loaded with...

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