L7 composites - Materials Science and Engineering

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46 Intro Composite Applns. Properties Voigt, Reuss, Hill Anistrpy. CTE Cellular Matls. Wood Wood: modulus, contd. • The second equation is more subtle and states that the elastic modulus transverse to the grain varies more rapidly - with the square of the density - than the axial modulus. The reason for this can be understood very simply in terms of the cellular structure. When wood is loaded across the grain, the cell walls bend like miniature beams. This response can be quantified by use of beam theory to arrive at the functional dependence of equation 2.
47 Intro Composite Applns. Properties Voigt, Reuss, Hill Anistrpy. CTE Cellular Matls. Wood ! Introduction to beam theory • Consider a 3-point beam with length, l: supported at either end and loaded in the center with a force, F. The most important point is that there is a neutral point in the beam, n, at which the stress is zero; above this it is compressive, and below it is tensile. The stress is proportional to distance from the neutral plane. δ max "l l 0 = (R + y)# $ R# R# but % = "l l 0 &% = y R = 8'max y l 2 = 3Fly Ewt 3 ! Moment of Inertia, I : I = y 2 y= t " dA = y= 0 wt 3 12 Moment on the beam, M : " y= t M = # ydA y= 0 Stress varies linearly with strain : # y E$ E(y /R) = = = y y E R This shows that stress varies linearly with y so #/y is a constant : M = # y " y= t y= 0 y 2 dA = # y I Thus this double equality is true : M I # E = = y R For a force, F, at the center of the beam the maximum deflection, % max is : % max = l2 8R = l2M 8EI = l2Fl /4 8EI = l3F 3Fl3 = 32EI 8Ewt 3 w t
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L7 composites - Materials Science and Engineering

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