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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II SOLUTIONS FOR HOMEWORK NO. Problem 1 (20 points) (a) The equilibrium equations are ∂σ 11 ∂x 1 + ∂σ 12 ∂x 2 + ∂σ 13 ∂x 3 + ρb 1 = (1) ∂σ 21 ∂x 1 + ∂σ 22 ∂x 2 + ∂σ 23 ∂x 3 + ρb 2 = (2) ∂σ 31 ∂x 1 + ∂σ 32 ∂x 2 + ∂σ 33 ∂x 3 + ρb 3 = (3) All shear stresses are zero. Furthermore, the gravitational body force loading b has one non-zero component only, in the direction of e 3 . Therefore, from the third equilibrium equation we get: ∂σ 33 ∂σ 33 ∂x 3 + ρb 3 = ⇔ ∂x 3 = − ρb 3 (4) We also know that σ 33 ( x ) = − p ( x ) and b 3 = − g . We can substitute these into Equation 4 and integrate both sides with respect to x 3 x 3 p ( x ) x 3 dp ( x ) dx 3 − ρb 3 dx 3 ⇔ − dp ( x ) = ρgx 3 ⇔ − p ( x 3 ) + p = ρgx 3 (5) − = dx 3 p p ( x 3 ) = p − ρgx 3 (6) Note that p is only a function of x 3 . (b) 1. The traction vector on a surface can be found by multiplying the stress with the unit outward normal vector on that surface. In this case, the normal vector is n l = ⎧ ⎨ ⎩ − cos θ − sin θ ⎫ ⎬ ⎭ (7) The traction vector is then 1 3 = − p ( x ) − cos θ = p ( x ) cos θ t l ⎡ ⎣ − p ( x ) − p ( x ) ⎤ ⎦ ⎧ ⎨ ⎩ − sin θ ⎫ ⎬ ⎭ ⇔ t l ⎧ ⎨ ⎩ ⎫ ⎬ (8) ⎭ p ( x ) sin θ 2. Action and reaction, forces need to balance at the interface. ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ − p ( x ) cos θ − p ( x ) sin θ (9) t l = 3. ⎤ ⎦ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ ⎡ ⎣ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 cos θ sin θ (10) t d = σ 31 σ 32 σ 33 which gives 3 linear equations involving σ ij − p cos θ = σ 11 cos θ + σ 13 sin θ (11) = σ 21 cos θ + σ 23 sin θ (12) − p sin θ = σ 31 cos θ + σ 33 sin θ (13) Problem 2 (20 points) 1 + ν ν ij = σ ij − δ ij E 1 + ν σ kk + α Δ Tδ ij (14) 3 k =1 The procedure is similar to what was discussed in class for elastic constitutive relations without thermal effects. The idea...
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This note was uploaded on 02/23/2012 for the course MECHANICAL 2.002 taught by Professor Davidparks during the Spring '04 term at MIT.

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hw3_sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT...

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