Solutions for certain rectangular slabs continuous over flexible ...

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68 ILLINOIS ENGINEERING EXPERIMENT STATION 26. Concentrated Loads Over the Edge Stringers; Loads Symmetrical With Respect to the Center Stringer.-The slab is shown in Fig. 28. & P Beam Ezl , FIG. 28 From previous results the slab deflection for b >y >0 is found to be Pa 2 k 4k W = E. - f ( H cosh ay+sinh ay -ay cosh ay WIN 27- n' nrHi (162) 4 |1 + n4H- f9 ay sinh ay sin au sin ax nvHi J where f2 and f/ are given in Appendix B, and where 1 k = 4 (163) 4 -- +f1 The deflection of the center beam is 1 4Pa 3 k 2 z 1 = W = i- E. f2 sin au sin ax (164) J-0o rE 1 1i 1 .. n 4 and the deflection of each of the outer beams is 1 2Pa 3 1 Z2= w= W kf 6 sin au sin ax z 2 - 7r4E l 1 ,2,.. 165) 2Pa 3 1 E1 2 2 = - 2zoE 4 k 2 f6 sin au sin ax 7r E2 12.I n 4 A El
SOLUTIONS FOR CERTAIN RECTANGULAR SLABS where z 0 is the simple beam deflection given by (115), and where f6 is given in Appendix B. 27. Concentrated Loads Giving a Slab Deflection Which Is Antisymmetrical With Respect to the Longitudinal Center Line.-The neglect of the effect of twisting of the beams and the fact that there is no deflection of the longitudinal center line permit the corresponding solutions given in Chapter V, Sections 20 and 21 to be used also when the slab is continuous over a longitudinal center stringer. These anti-symmetrical solutions may be combined with the corresponding symmetrical solutions to obtain the effect of a single concentrated load at any point on the slab. 28. Load Uniformly Distributed Over the Entire Slab.*-Section 22 gives the solution for the uniformly loaded slab with edge stiffeners only, namely 4pa 4 wi = --- 1 - (1 - b cosh ay + ca ay sinh ay) sin ax (166) 7T'iV ,-- n where b 3 and ca are given in Appendix B. To this a second solution, 4pa 4 k, 2 (f cosh ay - sinh ay 7r N .-. n 6 (167) + ay cosh ay - f 3 ay sinh ay) sin ax, similar to (109), may be added to correct for the effect of the center / L zoao p A's" Urn'" Aree / b/ FiG. 29 *This problem has been treated previously in an approximate manner. See Hjalmar Granholm, "'Om Lastf6rdelning Mellan Balkar Med SArskild HAnsyn Till Farbaneplattans Inverkan," Betong, 1936, No. 4, pp. 139-219, Stockholm. Granholm takes account of the interaction between the beams and slab and also includes the effect of torsional rigidity of the beams in his analyses, but considers the slab to be replaced by a series of thin strips which act independently of each other.
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H ILL I N 0 I S UNIVERSITY OF ILLIN
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