1.3 FUNDAMENTAL FORMULAS
In the development of formulas for deflection, stresses, and buckling of sandwich components, mathematical expressions for bending, extensional, and shear stiffness often appear as do parameters involving these stiffnesses. It is convenient to present the fundamental stiffness formulas at the outset. Here also are discussed the effects of facing and core stiffness on sandwich bending stiffness so that the degree of approximation implied by simplified formulas neglecting facing and core stiffness is known.
1.3.1 Sandwich Bending Stiffness
A structural sandwich under forces normal to its facings has a bending stiffness, per unit width, given by the formula
where E' is the effective modulus of elasticity of facing; E'c is effective core elastic modulus in the appropriate L or W direction*--not the Ô direction--(see chapter 2 for details); ë is one minus the product of two Poisson's ratios (ë = 1 - ìab ìba ); t is facing thickness; 1 and 2 are subscripts denoting facing 1 and 2; t is core thickness; and h is distance between facing centroids. (See sketch of figure 1-2 for notation.) For many combinations of facing materials it will be found advantageous to choose thicknesses such that E1t1 = E2t2.
For sandwich with facings of the same material and thickness, formula (1:1) reduces to
which can also be written as
The second term of formula (l:la) incorporating facing stiffness and core stiffness is neglected for most sandwich. The effect of this second term in increasing basic sandwich stiffness is obtained from values of Ê (formula l:lb) shown graphically in figure 1-3.
If the sandwich has thin facings on a core of negligible bending stiffness, as is usually the case, and after assuming ë1 = ë2 = ë, the bending stiffness is given by the formula:
1.3.2 Sandwich Extensional Stiffness
The extensional stiffness of a sandwich, stretched or compressed by force in its plane, is given by the formula
1.3.3 Sandwich Shear Stiffness
A sandwich that has fairly thin facings on a thick core has a transverse shear stiffness per unit width given approximately by the formula
where tc is the core thickness and Gc is the core shear modulus associated with the distortion of the TL or TW plane (see chapter 2 for details).
*For honeycomb cores the elastic moduli (not shear moduli) parallel and perpendicular to the core ribbon (L or W direction) are essentially zero.