Solutions for certain rectangular slabs continuous over flexible ...

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ILLINOIS ENGINEERING EXPERIMENT STATION where (2ab - av) sinh av - av sinh (2ab - av) c4 = (208) (sinh 2ab - 2ab) (cosh 2ab - 1) Equation (207) is valid over the entire loaded panel. It also represents the total deflection in the unloaded panel provided that x is taken positive to the left. One has, therefore, the deflection of the slab, for x > 0, W = Wo + Wcor where w 0 is the deflection of a rectangular slab, b by a, simply supported on all edges and loaded as in the right-hand panel of Fig. 40, and w,or is given by Equation (207). When the ratio b/a is small, that is when (b/a) < 0.4, one may take wo as approximately equal to the deflection of the infinitely long slab of span b. Then Pb 2 1 nry \ "ry nrV mrX Wo = - + - ) e sin - sin - .(209) 2r-N ... n 3 b b b The bending moment M. in the slab over the interior beam is M = -N-- = -N--] x=0 ax X0=0 aX2 (210) P ( -- 1) sinh av -sinh (2ab-av) Pv _v = - cos ay. a 75.. sinh 2ab - 2ab 38. The Infinitely Long Slab Simply Supported at the Edges, Continuous Over a Central Line Support, and Carrying a Concentrated Load.-In the preceding solution, by allowing the supports at y = ± a/2 to recede sufficiently far from the load, one may determine approximately the deflection of an infinitely long slab having a
SOLUTIONS FOR CERTAIN RECTANGULAR SLABS FiG. 41 central interior line support as shown in Fig. 41. The approximation is quite good if one lets a = 27b in Equations (207) and (208) and confines the application of the results to a region 0 < y < 2b. For convenience, one may use dimensionless variables: x, = x/(2b), 1 yj = y/(2b), (211) vi = v/(2b). Then in the loaded strip, for 0 < y < 2b, the deflection of the slab is where W = wo + woor (212) Pb 2 1 nry "* nrv n W = - -1 + - e b sin - sin -, (213) 27r'N ... n b b b 2Pb 2 d 4 Wcor= -- [(1-xi) sinh nx-x sinh (n-nxi)] cos ny 1 (214) W rN ,3- n and (1 - vi) sinh nvi - vi sinh (n - nvi) d4 = (215) (sinh n - n) (cosh n - 1) The deflection of the unloaded panel, when 2b > y > given by (214) provided that x is taken positive to the left. - 2b, is
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H ILL I N 0 I S UNIVERSITY OF ILLIN
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3000-5-38-14400 pEssF II
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CONTENTS (CONTINUED) V. THE RECTANG
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CONTENTS (CONCLUDED) PAGE 36. Recta
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