Solutions for certain rectangular slabs continuous over flexible ...

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ILLINOIS ENGINEERING EXPERIMENT STATION we is the deflection of an infinitely long slab due to a concentrated load at the point x = u, y = -v. The part of the deflection w' is given by the equation Pa 2 1 w/ = l--- -- [1 + a (y + v)] e - " (+') sin au sin ax. (64) 2rN 21,2,..- n A particular problem is solved, then, by taking the difference between two solutions. Westergaard's numerical values and curves, mentioned previously, may be used in each of these solutions. 13. The Finite Edge Fixed.-When the finite edge, coinciding with I FIG. 15 the x axis, is fixed as indicated in Fig. 15, one may write the expression for the deflection of the slab as the sum of three parts: w = wo + w' + 2w, (65) where wo, wo and wi are given by Equations (9), (64) and (21) respectively. In (21) one may replace ly\ by y since the slab no longer exists for y < 0. Since the moments, shears and reactions are obtainable in finite form for each part of (65), they are obtainable for the entire solution. The results obtained in previous sections may be applied to this solution. It is of interest to observe that the part of the deflection produced by the fixing of the edge, obtained as the difference between (65) and (63), namely, Pay 1 wf = 2 (wi ± W 1 ) = -__N _ - ay e - *(v+) sin au sin ax, (66)
SOLUTIONS FOR CERTAIN RECTANGULAR SLABS may be represented in finite form. Substitution of nir/a for a in (66), and comparison with (10) and (11), shows that cosh (v + y) 7r (x + u) cosh -- - cos Pvy a a w f = - 47rgN loge i (v+y) a ir (x - u) (67) cosh - cos a a One has the interesting result, therefore, that (67) expresses in finite form the deflection produced by the fixed-end moment and the corresponding edge reactions. Nhdai* showed in like manner that the deflection of the infinitely long slab due to a concentrated moment applied at a point on the x axis, and having the same direction as M,, may be represented by an expression similar to (67). The fixed-end moment is twice as great as that given by Equations (32) and (36) for the corresponding positions of load. In other words, continuity of the slab across a rigid beam in this particular example produces a moment My in the slab over the beam equal to 50 per cent of the fixed-end moment. Newmarkt generalizes further on this representation of a continuous slab as being "50 per cent fixed" under certain conditions. 14. The Finite Edge Free.-When the finite edge is free, as shown FiG. 16 in Fig. 16, the boundary conditions on the free edge are *A. NAdai, Die elastischen Platten, 1925, p. 163. tN. M. Newmark, "A Distribution Procedure for the Analysis of Slabs Continuous Over Flexible Beams," Bulletin 304, Engineering Experiment Station, University of Illinois, 1938, Chapter IX.
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