Composite Beams

 *Example:  The original composite beam shown on the left below is formed from three different materials with moduli of elasticity, E1, E2, and E3.  The bending moment in the beam is to be 100,000  lb in and  E1 = 30x106 psi,  E2 = 15x106 psi, and E3 = 10x106 psi.  Find the maximum bending stress in each material of the composite beam. 1.       Select material (3) to be the “base material”.  Therefore  E1 / E3  = 3  and E2 / E3  = 1.5.  Note:  Any one material can be selected as the “base material”.   2.      Form the “transformed section” using only the “base material” by multiplying the width of section (2) by 1.5 and the width of section 1 by 3.  The width of the base material (3) is unchanged.  The above figure on the right shows the “transformed” section.   3.      Next calculate the moment of inertia of the entire “transformed” section about its neutral axis, y.     Iyy = 2[ (1/12) b1h13 + b1h1(h1/2 + h2 + h3/2)2 + (1/12) b2h23 + b2h2(h2/2 + h3/2)2  ] + (1/12) b3h33   where        b1 = 18 in, h1 = 1 in,  b2 = 9 in, h2 = 2 in,  b3 = 6 in, h3 = 3 in        Iyy  =  2[ (1/12)(18)(13) + (18)(42) + (1/12)(9)(23) + (18)(5/2)2 ]  +  (1/12)6(33)     So for the “transformed” section    Iyy  =  829.5 in4     4.      Now the bending stress in terms of material (3) can be calculated using  σ  =  Mc/Iyy .