Imagine cutting a cube of material from around the

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Imagine cutting a cube of material from around the material Each face has one normal stress and two shear stresses
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ME 382 Lecture 03 10/ix/07 2 Sign convention: Positive stresses act in the positive directions on positive faces of the cube. In general need to determine σ xx , σ yy , σ zz , τ xy , τ xz , τ yz - but at least two shear stresses and at least one normal stress will obviously be zero in ME382. Shear and normal stresses at a point on a free surface are zero. B EAMS OF ARBITRARY CROSS SECTION Axial tension: σ zz = N zz A zz Beam bending: σ zz = M xx y I xx M yy x I yy Shear force: (Approximate for some sections) τ zx ( x ) = V zx Q ( x ) I xx t x ( ) ; τ zy ( y ) = V zy Q ( y ) I yy t y ( ) where Q x ( ) = x dA ( x ) * A *( x ) (or, Q x ( ) = xA * x ( ) , where x is the distance between the centroid of A zz and A *( x) ) N zz M xx V zx M yy V zy
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ME 382 Lecture 03 10/ix/07 3 T ORSION OF CYLINDERS τ z θ = T zz r J zz where J zz = π R o 4 R i 4 ( ) 2 For a thin walled tube J zz = 2 π R 3 t T HIN - WALLED PRESSURE VESSELS Cylinder Axial (longitudinal) stress: σ zz = PR 2 t Hoop (circumferential) stress: σ θθ = PR t Radial stress: σ rr 0 7. Use superposition to add up all the separate contributions to the stress components For example, the axial stress may have a component from (i) Axial load (ii)
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