Sandwich Theory - Linear Sandwich Theory - Bending of A Sandwich Beam With Thin Facesheets - Relation Between Bending and Shear Deflections

Relation Between Bending and Shear Deflections

A relation can be obtained between the bending and shear deflections by using the continuity of tractions between the core and the facesheets. If we equate the tractions directly we get

$n_x~\sigma_{xx}^{\mathrm{face}} = n_z~\sigma_{zx}^{\mathrm{core}}$

At both the facesheet-core interfaces but at the top of the core and at the bottom of the core . Therefore, traction continuity at leads to

$2fh~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2} - (2h+f)~C_{55}^{\mathrm{core}}~\cfrac{\mathrm{d} w_s}{\mathrm{d} x} = 4h^2~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2}$

The above relation is rarely used because of the presence of second derivatives of the shear deflection. Instead it is assumed that

$n_z~\sigma_{zx}^{\mathrm{core}} = \cfrac{\mathrm{d} N_{xx}^{\mathrm{face}}}{\mathrm{d}x}$

which implies that

$\cfrac{\mathrm{d} w_s}{\mathrm{d} x} = -2fh~\left(\cfrac{C_{11}^{\mathrm{face}}}{C_{55}^{\mathrm{core}}}\right)~\cfrac{\mathrm{d}^3 w_b}{\mathrm{d} x^3}$

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