From Park and Gamble Book.

Transcription

1 From Park and Gamble Book. 1

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6 Behavior of Two-Way Slabs Figure 1. Two-Way Slab on Simple Edge Supports: (a) Bending of Center Strips of Slab and (b) Grid Model of Two-Way Slab. Figure 1(a) 5wl 5wl w l a a b b a b = = 4 EI EI wb la where: w a = w b = Share of load w carried in short direction Share of load w carried in short direction The larger share of load is carried by short span. This is approximate because the actual behavior of a slab is more complex than that of two intersecting strip (Figure 1.a). Figure 1(b) Consider the intersections of s and l. We can see that at the intersection for the exterior edge of strip l both edges are the same elevation; the strip is twisted. This twisting results in torsional stresses and torsional moments that are seen to be most pronounced near corners. Therefore, the total load on the slab is carried not only by bending moments in two directions but also by the twisting moments. For this reason bending moments in elastic slabs are smaller than would be 6

7 computed for sets of unconnected strips loaded by w a and w b. For example, for square slab with simply supported edges ( wa = wb = w), the maximum moment in each strip would be ( w/) l 8 = 0.065wl The exact theory of elastic plates show that the maximum bending moment in such a square slab is only wl. Therefore, in this case the twisting moments relieve the bending moments by about 5 percent. Figure 3. Moments and in a Uniformly Loaded Slab with Simple Supports on Four Sides. 7

8 Figure 4. Simply Supported Slab Maximum Moment Contour. 8

9 Figure 5. Deflection 9

10 Figure 6. Moment Diagram 10

11 The twisting moments are usually of consequence only at exterior comers of a two-way slab system, where they tend to crack the slab at the bottom along the panel diagonal, and at the top perpendicular to the panel diagonal. Special reinforcement should be provided at exterior corners in both the bottom and top of the slab, for a distance in each direction from the comer equal to one-fifth the longer span of the comer panel, as shown in Figure 7. The reinforcement at the top of the slab should be parallel to the diagonal from the corner, while that at the bottom should be perpendicular to the diagonal. Alternatively, either layer of steel may be placed in two bands parallel to the sides of the slab. The positive and negative reinforcement, in any case, should be of a size and spacing equivalent to that required for the maximum positive moment (per foot of width) in the panel, according to ACI Code Figure 7. Special Reinforcement at Exterior Coners of a Beam-Supported Two-Way Slab. 11

12 Definitions (ACI 13.): Slab Panel is assumed to be rectangular, with sides of l 1 and l measured center to center of supporting columns: l 1 = is always the span being considered l = is the transverse span l n = is the clear span face to face of supporting columns ACI Definition of column strip. Column strip is a design strip with a width on each side of column centerline equal to 0.5l or 0.5l 1 whichever is less. Column strip includes beams if any. ACIT 13.. Definition of middle strip Middle strip is a design strip bounded by two column strips. Figure 8. Layout of Typical Interior Panel. 1

13 C 1 in Figure 8 above is size of equivalent rectangular column, capital, or bracket measured in the direction of the span for which moments are being determined. 13

14 Loads on the Equivalent Frame L1 L L 1 1 L 1 L L L L L L n M l M r Figure 9. 14

15 M pos wl u M l M l M r M r M r + M l M l M r wl u M pos M r M ( ) 0 u n + l 1 M = wl L 8 Figure

16 Figure 11. Moment Variation in Column-Supported Two-Way Slabs: (a) Critical moment Sections; (b) Moment Variation along a Span; (c) Moment Variation Across the Width of Critical Section (From Nilson s Book). 16

17 Figure 11.a shows a flat plate floor supported by columns at A, B, C, and D. Figure 11.b show moment diagram for the direction span l 1. In this direction, the slab may be considered as a broad, flat beam of width l. Accordingly, the load per foot of span is wl. In any span of a continuous beam, the sum of the midspan positive moment and the average of the negative moments at adjacent supports is equal to the midspan positive moment of a corresponding simply supported beam. In terns of the slab, this requirement of statics may be written 1 1 ab cd ef 8 ( M + M ) + M = wl l 1 A similar requirement exists in the perpendicular direction, leading to the relation: 1 1 ac bd gh 8 ( M + M ) + M = wll 1 17

18 Example Determine the static moment is each panel for the example slab given below. Example Slab Floor, Showing division into Panels for Static Moment Computations. The static moment is calculated for each panel for the example slab as follows: w = 1.w + 1.6w = 1.(94)+1.6(70) = 4.8 psf u D L For Panel I (ACI ): 1 1 M 0 = wll u n = (0.48)(0)(18) = 18 ft kips= 8 8 For Panel IV (See ACI ) 1 1 M 0 = wll u n = (0.48)(10.67)(18.33) = 101 ft kips