Solutions for certain rectangular slabs continuous over flexible ...

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ILLINOIS ENGINEERING EXPERIMENT STATION 5. Solution for the Infinitely Long Slab Supporting a Concentrated Load.-The slab and loading are shown in Fig. 2 where the edges FIG. 2 x = 0 and x = a are assumed to be simply supported and where the slab is assumed to extend sufficiently far in the directions of positive and negative y that the deflection of the slab practically vanishes before additional supports or edges are reached. The deflection of the slab is then given by the equation* where Pa 2 1 wo = - (1 + aly - vI) e "EY'V sin au sin ax 2r'N w77-- n 3 a Nfidait defined a function 0o by means of the equation P 1 00 = NV 2 wo = --- - e - 'Y -i sin au sin ax 7r 1,2,3,.. n (10) and derivedt an expression for 0o in finite form, namely, P P0 = -- 47r Bo loge-- Ao *See, for example, A. Nadai, Die clastischen Platten, 1925, p. 85; or H. M. Westergaard, Computation of Stresses in Bridge Slabs Due to Wheel Loads, Public Roads, V. 11, No. 1, March, 1930, p. 6. Equation (9) has been adjusted for the origin of co6rdinates shown in Fig. 2 and the absolute value of (y - v) has been introduced to extend the region of applicability of the equation to the entire slab. See Appendix A for notes on the differentiation with respect to y of functions involving ]y - v\. tA. NAdai, Die elastischen Platten, 1925, p. 87. (See previous footnote.) IA. Naidai, Die elastischen Platten, 1925, p. 89.
SOLUTIONS FOR CERTAIN RECTANGULAR SLABS where Ao cosh r (y-v) o 7(x ± u) = cosh - cos (12) Bo J a a By differentiation of (10) and (11) with respect to x and y one obtains 8co 0- -P P Z e - .ly cI sin au cos ax Ox a , - P ir(x- T1 u) 1 (x +u) = - - sin - - sin 4a Bo a Ao a (13) O4)o P sgn (y - v) -0 s e - " - • sin au si n a x 9y a 2,,a... P ( 1 1 7r (y - v) _-- -- - )sinh --- 4a Bo Ao a (14)* Nadait showed that the curvatures of the slab may be expressed through the function 4o and its derivatives by means of the equations 2N daWo 900 = 40o - (y - v) ay 2 ay (15) aOxy Ox *See Appendix A for a discussion of the change of sign represented by the function sgn (y - namely, sgn (y - v) = algebraic sign of (y - v) =- for y > v v), tA. NAdai, Die elastischen Platten, 1925, p. 86.
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