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Based on a n a l y s i s of the frame described above, d i f f e r e n t i a l settlement may have induced an additional load of 39 kips i n Colimm E3. I t must be pointed out that t h i s computation of the e f f e c t of settlement i s only a rough e s t i - mate. The time-dependent nature of settlement and l a c k of s p e c i f i c know- ledge concerning the support i n t e n s i t y and d i s t r i b u t i o n of the s o i l support preclude a precise estimate of the induced reaction. However, t h i s a n a l y s i s shows that the d i f f e r e n t i a l settlement observed coiild place additional load on Column E3. Shear strength woxild c e r t a i n l y be influenced by such an additional column load. SHEAR STRENGTH OF SLAB Shear f a i l u r e occurred i n a l l three t e s t s . Only i n Test I I I was there evidence of flexuraJ. d i s t r e s s ; reinforcement i n the positive moment region began to y i e l d before punching. None of the meas\jred reinforcement s t r a i n s approached y i e l d i n Tests I and I I when the f i r s t shear f a i l u r e occiirred. Maxlmm applied load was s u b s t a n t i a l l y below the f l e x u r a l capacity of the slab i n each t e s t . I n Test I , the punching f a i l u r e occurred a t Colvimn Ck. This column was surrounded by the four loaded panels. The symmetry of loading and geometry of structure resulted in an e s s e n t i a l l y concentric column load. Consequently, there was l i t t l e or no moment t r a n s f e r between the slab and the column. A somewhat d i f f e r e n t shear condition was produced i n Test I I . Here, the slab f a i l e d i n shear at the edge Gol\mins. These edge coliamns were located at the inte r s e c t i o n of two loaded panels. Since the columns had very low f l e x u r a l s t i f f n e s s compared to that of the waffle slab, there was l i t t l e moment tr a n s f e r between the slab and the edge columns. The columns acted primarily as v e r t i c a l props for the slab cantilevered out from column Line D. A t h i r d type of shear condition existed i n Test I I I . I n t h i s t e s t , the i n i t i a l pionching f a i l u r e occurred at Column C5. Only one panel was loaded, and some moment was therefore transferred from the slab to the four support- ing columns. Since only two faces of the columns joined the loaded area, i t would be expected that v e r t i c a l load was transmitted to the columns p r i n c i p a l l y through these two column faces. l-3h
Test loads on Columns Ck, E3, E4, and C5 at the time of f a i l u r e are l i s t e d i n Table IX. These loads were determined as described below. Although Columns Bk and D5 also punched through, t h e i r f a i l u r e loads are not included in Table IX. F a i l u r e a t Dk took place a f t e r adja- cent Columns Ck and Ek had punched through. S i m i l a r l y f a i l u r e occurred at D5 a f t e r punching had occurred at Column C5. In both cases the structure was damaged extensively when the secondary punching occurred. For these conditions, forces acting at Bk and D5 at failvire would be impossible to assess. Column loads, designated as V^est' include both dead load and applied l i v e load. For each t e s t , the average dead load i n t e n s i t y l i s t e d i n Table IV was used. Also included was weight of loading e'quipment. This amounted to 6, 7, and 10 psf for Tests I , I I , and I I I respect- i v e l y . The nominal dead load car r i e d by a column was defined as that within the rectangular area bounded by the centerlines of panels surrounding the column. ' S i m i l a r l y , nominal applied l i v e load transferred to a column was de- fined as that within the dead load contributory area. The col\amn reac- tion due to l i v e load was determined by multiplying the nominal column load by the r a t i o s of computed to nominal l i s t e d i n Table V I I I . These mult i p l i c a t i o n r a t i o s were determined by the frame a n a l y s i s as described i n an e a r l i e r section of t h i s paper. The r a t i o for Column Ek was assumed to be the same as that for Column E3. Only the raaaimum and minimum multiplication r a t i o s were recorded for Colimin Ck. Added load of 39 kips from settlement was included i n the t e s t load for Col\min E3. Shear strength based on material properties was computed for each punch- ing location by two methods. One method was that developed by Moe. (19) The second procedure was that given by the ACI (3l8-63)(20) and described i n the Commentary on the ACI Code (SP-IO). Ì -ÌÌ ^ 1-35
TABLE IX SHEAR STRENGTH OF SLAB Location Shear Strength, kips Test Column Reaction Test No. Frame Analysis t e s t by Moe's Equation ^Moe by ACI's Equation ACI by ACI Beam Equation Beam t e s t Moe t e s t "ACI t e s t Beam Column C4 I C 4a 1130 1150 1420 1420 970 â 0.81 0.81 1.16 1.19 Column E3 IIA 3 E 390 440 800 870 930 460 0.49 0.51 0.42 0.47 0.85 0.96 Column E4 I I B 3 E 390 460 710 770 800 430 0.55 0.60 0.49 0.57 0.91 1.07 Column C5 I I I C 5 700 730 1090 1120 1180 0.64 0.65 0.59 0.62
Moe developed h i s equation from a large number of t e s t s on slabs. Additional data was evaluated using r e s u l t s from other investigations. From a t h e o r e t i c a l examination of shear strength i n slabs an equation was derived with constants determined from t e s t data. The equation takes into accovint column s i z e , slab thickness, and t e n s i l e strength of concrete. For a concentrically loaded colimin, shear strength i s given by the expression: V^ = hdyl7[i5(l - 0.075 7 < a ) ] / [ l + 5.25(bdyi7/V^^^J] ( l ) where b = periphery of c r i t i c a l section d = e f f e c t i v e depth f^ = compressive strength of concrete r = length of side of square column or equivalent for rectangular column Vâ, = shear force a t f l e x u r a l f a i l u r e f l e x V = ultimate shear force for concentric load u When both moment and shear are transferred at a slab-column j o i n t , shear strength i s given by: V = V^ - M/r (2) where M = moment transferred to the coliamn V = ultimate shear force for combined moment and shear In Eq. 1, the c r i t i c a l section b xs the perimeter of the column. For rectangular columns, the length r i s equal to the quantity (x^ + y ^ ) / ( x + y ) , where x and y are the dimensions of the column. Sheax strengths computed by Eqs. 1 and 2 are l i s t e d i n Table IX. Parameters used i n the calculations are given in Table X. Moments used i n Eq. 2 are those determined from the frame a n a l y s i s . For Column C^i, no moment was transferred. I n Colimins E3 and Ei*-, moment transferred was only i n the direction of column Lines 3 sxid k, respectively. Moment i n Column C5 was 1- 37
TABLE X SHEAR EQUATION PARAMETERS Moe 's Equation ACI Moe's and ACI Equation Location Test No. Column Size, i n . X i n . b, i n . r , i n . V f l e x , kips Equation, b, i n . d, i n . p s i Unbalanced Moment, k-in. Colimin C4 I 26 X 26 104 26.0 2190 2240 179 18.8 72.4 â Column E3 I I 12 X 32 56 26.6* 620 740 113 28.3 72.4 530 Column E4 I I 12 X 24 48 20.0* 620 740 102 27.1 72.4 630 Column C5 I I I 26 X 26 104 26.0 l l 4 0 1200 109 21.5 72.4 4510 * Equivalent value of r for a rectangular column was taken as x^ + y^/x + y, where x and y are the column dimensions.
applied from two directions, across Line C and across Line 5- The moment used i n calculations was that across Line C, the greater of the two. ^ f l e x ^ term in Eq. 1 to indicate the condition of the slab r e s u l t i n g from bending moments. The term accounts for cracking, depth of neutral a x i s , compressive s t r e s s e s , and other states of damage. In these calcu- l a t i o n s , Vâ was taken as the column reaction found by the frame an a l y s i s with an applied load equal to the f l e x u r a l capacity determined by y i e l d l i n e a n a l y s i s . Loads determined by y i e l d l i n e a n a l y sis are shown i n Fig. 56. In the 1963 ACI Code, a method for computing shear strength of slabs i s contained i n Section 1707 of Chapter 17. Provisions for t r a n s f e r of moments are specified i n Chapter 9, Section 920 of the Code. Additional information i s included m the Code Commentary. (21) For v e r t i c a l loads only, ultimate shear strength i s given by: = 4bdv/I7 (3) Notation i n t h i s equation i s s i m i l a r to that for Eq. 1. In Eq. 3, however, the c r i t i c a l section, b, i s defined as the perimeter located at d/2 from the column faces. When moment i s transferred, the c r i t i c a l section b i s d i f f e r e n t from that given above. I n the direction of moment, the c r i t i c a l section remains equal to the column width plus d /2 each side of the column. The section peirpendi- cular to the direction of moment extends I . 5 times the slab depth to each side of the column. The Code Commentary ^^^^ s p e c i f i e s that moment transferred by torsion i s that part of the unbalanced moment exceeding the f l e x u r a l capacity of the c r i t i c a l sections. Unbalanced moments at Columns E3, E4 , and C5 were i n each case l e s s than f l e x u r a l capacities at the c r i t i c a l section. Therefore, no part of the unbalanced moments added to the v e r t i c a l shear s t r e s s e s . 1-39
