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# lec23 - 2.001 MECHANICS AND MATERIALS I Lecture#25 Prof...

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± 2.001 - MECHANICS AND MATERIALS I Lecture 12/6/2006 Prof. Carol Livermore Recall from last time: Deformation Strain (Compatibility) Equilibrium Stress (Constitutive) Moment-Deformation Moment of Inertia Stress-Loading Deﬂections/Angle of Twist Example: What is ϕ (3 L )? M t M t Beam Bending 1 = k = ρd x ± xx = ρ y M z = A σ xx ydA Ey σ xx = ρ M z = EI ± ρ I = A y 2 dA σ xx = M z y I = 1 = M z dx ρ EI = T, 0 x 2 L =0 , 0 x 3 L Shaft Torsion dx = r dϕ ± θx ± 2 dx M t = A σ θx rdA = Gr σ θθ dx M t = GJ ± dx J = r 2 dA A σ θx = M t r J = M t dx GJ 1 #25

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±² ³´ M t = GJ dx J = π R 4 1 = J 1 , 0 x L 2 J = π R 2 4 = J 2 ,L x 3 L 2 T = , 0 x L dx GJ 1 T = x 2 L dx GJ 2 =0 , 2 L x 3 L dx So: T ϕ ( x )= x + c 1 , 0 x L GJ 1 T ϕ ( x x + c 2 x 2 L GJ 2 ϕ ( x c 3 , 2 L x 3 L Boundary Conditions: ϕ (0) = 0 c 1 ϕ ( L ) is continuous TL = = 1 1 GJ 1 GJ 2 + c 2 c 2 GJ 1 J 2 ϕ (2 L ) is continuous.
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lec23 - 2.001 MECHANICS AND MATERIALS I Lecture#25 Prof...

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