By Howard G. Allen

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14) is not. 3. 9). 15b) and 2 aP B22 = — . 27b): w2 = — ~ If i - . 15a), the secondary deformation can be determined: H>2 = —If- {Cxoc\ sinh ocjX-h C 2af cosh a x x — C 3af sin oc2x — C^l cos oc2x 4- C e } . ) FIG. 3. Conditions at a "pinned" end, showing three hinges. Six boundary conditions are needed for the determination o f the constants C\— C%. For example, in a pin-ended strut in which the ends o f the faces are also free to rotate independently ( F i g . 3), (i) χ = 0, Wi = 0; (iv) χ = L, w'x' = 0; (ii) χ = 0, w[' = 0; (iii) χ = L, wL = 0, ( ν ) χ = 0, (vi) w>2 = 0; Λ: = 0, w'2 = 0.

When the faces are thicker (as with asbestos cement or plywood for example) and the core is weak the effect may be significant. It may also be necessary to take the effect into account when performing tests on sandwich beams to determine the shear modulus o f the core. T o discuss the nature o f this interaction between the bending stiffness o f the faces and the shear stiffness o f the core, consider first a sandwich with a core which is rigid in shear (G = « » ) and with a load qi per unit length.

In this state the shear stress τ is uniform across the thickness o f the core (c) and diminishes linearly to zero across the thickness of each face (Fig. 4c). 20) = -bdt+EIfW^'. Also, 9x = -Q'i> Qi = M'l9 f Mx = -Dw[ . 21) ( 2 1 9 23 SANDWICH BEAMS A s a result o f the shear stress τ , the core undergoes a shear strain γ = τ / G , which corresponds to an additional transverse beam deflection w2. The faces must share this extra deflection and, in order to do so, they must be subjected to additional distributed loads q2, shear forces Q2 and bending moments M 2 such that q2 = - Ô 2 ; Ö 2 = K; M2 = -EIfw'2'.

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