By Howard G. Allen
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Comparison o f (ii) and ( v i ) leads to the conclusion that C2 and C4 are zero, as a consequence o f which (i) gives a zero value for C5. This leaves equations ( i i i ) and ( i v ) for the determination o f C± and C 3 : C ] sinh a i L + C3 sin *%L — 0, Cia? sinh olXL — C 3 a | sin a 2 L = 0. One solution is C± = C3 = 0, which implies that the strut is straight and unbuckled. Another is C i = 0, C 3 ^ 0, α 2 £ = ηπ, where η is an integer. This last condition gives o|L2 = nW = ^ [ - ^ ± ^ + 4 ^ ] .
This suggests that one way of eliminating errors in tests of this kind would be to glue rigid inserts into the overhanging parts, in place o f the core. Some o f the simplicity o f the method is then lost, however, SANDWICH BEAMS 41 because tests can no longer be performed on unmodified sample beams. If G -»» c», then a -*· «>, ßs 0 and the fractional error vanishes. In this case the sandwich acts as a single composite beam without shear deformation. If G 0, then α 0, /? 3-* —aL 6 and the fractional error becomes (7/7^— 1).
L aL WL „ T „ B± n —1 cosh — + —1 s i n h — + — - + C 3 L + C 4 = - 0 -2 + JS4. a 2 a 2 16 a (xi) vv2 continuous at Β aL ^ . aL W C i c o s h - y + C 2 sinh — + y = Βλ. ( x i i ) w[' continuous at Β . öZ/ aL I WL _ \ „ Λ C i s i n h — + C 2 c o s h - ^ - + l — - + 2 C 3 U = B2 + 2B3a. A. 31). The total deflection, w, may therefore be expressed as a function of χ in the region AB. 24b) may be used to express the core shear stress r as a function o f χ in the same region. Furthermore, the maximum direct stress in the faces at any section is: c + 2t t 29 SANDWICH BEAMS D o u b l e differentiation o f e q u a t i o n s ( 2 .