Seismic-Resistant Connections of Edge Columns with Prestressed Slabs

Transcription

1 ACI STRUCTURAL JOURNAL Title no. 102-S32 TECHNICAL PAPER Seismic-Resistant Connections of Edge Columns with Prestressed Slabs by Mark Ritchie and Amin Ghali This paper reviews the procedure developed by Megally and Ghali for the seismic design of nonprestressed concrete slab-column connections and investigates the validity of the same procedure for prestressed slabs. A series of full-size specimens was tested, representing the connection of a prestressed slab, with post-tensioned unbonded strands, with an edge column. The amounts of the prestressed and the nonprestressed reinforcements were varied, while the flexural strengths were unchanged. The columns were subjected to an axial load of constant magnitude and unbalanced moment reversals, representing the effect of an earthquake, of increasing amplitude up to failure. The specimens were provided with the same amount and arrangement of stud shear reinforcement (SSR). It is concluded that the reviewed design procedure, intended for nonprestressed slabs, applies also to prestressed slabs having an average prestress of 0.4 to 1.1 MPa. The shear reinforcement, when necessary, can be designed according to ACI 421.1R-99. Keywords: seismic; shear; slabs; studs. INTRODUCTION The research on the seismic behaviour of prestressed slabcolumn connections has been limited. 1,2 The present paper summarizes the results of an experimental program conducted on connections of prestressed slabs with edge columns and proposes design recommendations. ACI requires that the factored nominal shear strength φv n exceed the maximum factored shear stress v u at the inner critical section at d/2 from the column face and at the outer critical section at d/2 from the outer edge of the shear-reinforced zone. At the inner critical section, v n = v c + v s (1/2) f c in MPa (6 f c in psi); at the outer critical section, v n = v c = (1/6) f c in MPa (2 f c in psi); where v c and v s are the nominal shear strengths provided by the concrete and shear reinforcement, respectively. Without shear reinforcement, the nominal shear strength v n = v c = (1/3) f c in MPa (4 f c in psi) at d/2 from the column face. Smaller values are permitted at columns having a relatively large ratio of long side to short side of column and large b o /d where b o is the perimeter of the critical section and d is the effective depth of the section. For seismic design, ACI permits slabs without shear reinforcement when V u 0.4φV c, where V c = v c b o d, with v c = (1/3) f c ; V u is the factored shear force caused by gravity load. For prestressed slabs, satisfying certain conditions, the code gives V c = (β p f c + 0.3f pc )b o d + V p, where β p is the smaller of 0.29 or (α s d/b o ); α s = 2.5 for exterior columns; f pc is the average effective compressive stress in concrete, in two directions, at centroid of section; and V p is the vertical component of prestressing forces crossing the critical section. In the shear reinforced zone, the shear strengths provided by the shear reinforcement and by the concrete are given (according to ACI and ACI 421.1R-99 4 ) by v s = A v f yv b o s where A v is the area of shear reinforcement within its spacing s, f yv is the specified yield strength of shear reinforcement, and v c = (1/6) f c in MPa (2 f c in psi). When stud shear reinforcements (SSR) are used, ACI 421.1R-99 4 recommends v c = (1/4) f c in MPa (3 f c in psi) and v n = v c + v s (2/3) f c in MPa (8 f c in psi); but for seismic design, the values of v c are to be reduced by 50% to account for the loss of strength provided by concrete caused by unbalanced moment reversals. Megally and Ghali 5 gave recommendations (for consideration by ACI Committee 421) for the seismic design of slab-column connections in structures having shear walls, or other bracing systems, that limit the interstory drift ratio, DR u to (permitted by IBC 6 ); DR u is the maximum interstory drift, including inelastic deformation, divided by the story height. The recommendations gave a design procedure to calculate the unbalanced moment M u, and adopted the equations of ACI 421.1R-99 4 for design of shear reinforcement and suggested an increase of the upper limit of v n at d/2 from the column by 25% when the shear stress caused by V u alone did not exceed (1/3) f c in MPa (that is, when the shear stress was caused mainly by M u rather than by V u ). Based on the experiments presented as follows, the recommendations of Megally and Ghali, 5 intended for nonprestressed slabs, will be considered for the seismic design of the connections of prestressed slabs with edge columns. The steps of design are given in Appendix A. RESEARCH SIGNIFICANCE This research develops a design procedure for seismic resistant slab-column connections. The code provisions of ACI and recommendations of ACI 421.1R-99 4 are supplemented to produce an integral guide for the seismic design of edge slab-column connections in prestressed slabs. EXPERIMENTAL PROGRAM Five full-scale post-tensioned slab-column connections were tested. The plan and cross-sectional properties of the test ACI Structural Journal, V. 102, No. 2, March-April MS No received May 12, 2004, and reviewed under Institute publication policies. Copyright 2005, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author s closure, if any, will be published in the January-February 2006 ACI Structural Journal if the discussion is received by September 1, (1) 314 ACI Structural Journal/March-April 2005

2 Mark Ritchie is a MSc student in the Department of Civil Engineering at the University of Calgary, Alberta, Canada. He received his BSc from the University of Calgary in His research interests include prestressed and seismic design of reinforced concrete structures. ACI member Amin Ghali is an emeritus professor of civil engineering at the University of Calgary. He is a member of ACI Committees 373, Circular Concrete Structures Prestressed with Circumferential Tendons; and 435, Deflection of Concrete Building Structures; and Joint ACI-ASCE Committees 343, Concrete Bridge Design; and 421, Design of Reinforced Concrete Slabs. His research interests include analysis and design of concrete structures. specimens are shown in Fig. 1. The specimens represented the edge column connections in a 150 mm-thick slab having 6 x 6 m 2 square panels. Of the five specimens, four were prestressed edge slab-column connections and the fifth specimen contained no prestressing and was considered the control slab. The prototype slab had a gravity service live load of 2.4 kpa, superimposed dead load of 1.3 kpa, and self-weight of 3.6 kpa. The prototype was designed such that nine 13 mm (cross sectional area = 99 mm 2 ; effective prestressing force = 110 kn (25,000 lb)) post-tensioned tendons balanced approximately 85% of the service dead load in interior spans and 65% of the service dead load in exterior spans. The tendons were banded in the x-direction and uniformly distributed in the y-direction. In the test specimens, the number of tendons was varied between nine and zero while maintaining approximately the same flexural strength in the x-direction. Stud shear reinforcement was positioned on lines perpendicular to three faces of the column. The greatest and the smallest prestress considered in this research were produced by nine and three tendons per panel of 6.0 m width of the prototype slab. The effective prestress produced by nine or by three tendons averaged 1.1 or 0.4 MPa, respectively, over a cross-sectional area equal to 6.0 x 0.15 m 2. The specimens were placed in the test frame shown in Fig. 2. The specimens were held in a vertical position on a steel support. Three edges of the slab were simply supported by neoprene pads; the fourth edge was free (Fig. 1). A 1000 kncapacity horizontal actuator (A) applied the shear force while the two 250 kn vertical actuators (B and C) produced the unbalanced moment. The column was sufficiently reinforced to prevent its failure. Prestressing reinforcement The number of tendons in the specimens varied (Table 1); nonprestressed reinforcement * was provided to maintain the same negative flexural strengths. The name of each specimen indicates the number of prestressed cables in the x-direction (running perpendicular to the free edge); for example, EC9C indicates edge column with nine prestressed strands. The reinforcement layout in Specimen EC9C is shown in Fig. 3(a) and 4. Specimen EC0C contained no prestressing reinforcement and was considered the control slab; it was provided with top * The nonprestressed bars used in the slabs are 10M and 15M; cross-sectional areas 100 and 200 mm 2, respectively; f y = 466 and 483 MPa (67.5 and 70.0 ksi), respectively. Fig. 1 Dimensions of test specimens and positive sign convention. Table 1 Number of prestressing tendons and punching shear strength of test specimens compared with source equations (1) (2) (3) (4) (5) (6) (7) (8) (9) Number of prestressed tendons Shear stress at Point A (Fig. 12); (V u ) max and negative (M u ) max ( v Test ACI ACI Megally u ) max ( v u ) A specimen y-direction x-direction f c, MPa * R-99 4 and Ghali 5 f c f c EC0C EC3C EC5C EC7C EC9C * MPa = 145 psi. v n f c ACI Structural Journal/March-April

3 15M bars in two directions (Fig. 5). One prestressing strand, running in the x direction, replaced approximately M bars. Figure 3(b), (c), and (d) depict the tendons and the top nonprestressed steel in Specimens EC7C, EC5C, and EC3C, respectively. As mentioned previously, the test specimens were representing edge column-slab connection in a prototype slab having 6.0 m panels in x- and y-directions. The test specimen s plan dimensions were 1.35 and 1.90 m in the x- and y-directions, respectively. The banded tendons, running in the x-direction, produced in the specimens, as well as in the vicinity of edge columns in the prototype slab, greater prestress than the average. The prestress in the prototype slab would approach the average values, 1.1 to 0.4 MPa only at sections far from the free edge. In the y-direction, the prototype slab had distributed tendons. To represent nine tendons per 6.0 m panel, the corresponding test specimen, EC9C, should have 9(1.35/6.0) = 2.03 tendons. The number of tendons provided in the y-direction was two in each of EC9C, EC7C, and Fig. 2 Test frame. EC5C, and one in EC3C. With these numbers and the locations of the tendons shown in Fig. 3, the prestress in the y-direction was not equal to the average values mentioned previously. Shear reinforcement The specimens were provided with more shear reinforcement than the minimum required to resist gravity loads applied to the prototype structure. Each shear stud rail consisted of eight studs (9.5 mm diameter; f yv = 462 MPa [67.0 ksi]) equally spaced at 55 mm; the distance between the column faces and the first line of peripheral studs was 35 mm. The overall height of the stud rails was 115 mm and the diameter of the anchor heads was 30 mm. The rail was 5 mm thick and 25 mm wide. The location of the stud rails of Specimen EC9C is shown in Fig. 6; this is typical for all five specimens. Flexural reinforcement The specimens were provided with sufficient flexural reinforcement to ensure that punching failure would occur prior to flexural failure. The top nonprestressed flexural reinforcement, perpendicular and parallel to the free edge, consisted of 15M bars. All prestressed specimens had approximately the same flexural capacity as the nonprestressed specimen, EC0C (Fig. 5). The flexural capacity was assessed using the yield-line theory; refer to Fig. 7 and Column 5 of Table 2. In applying the yield line theory, appropriate flexural strength values are assigned to the nonprestressed reinforcement and the prestressed tendons. All test specimens had the same uniform bottom flexural reinforcement in two orthogonal directions (Fig. 6). For both the top and bottom reinforcements, the cover was 20 mm. This resulted in effective depths of 122 and 106 mm for the two steel directions. All top and bottom flexural reinforcements running in the x-direction were bent at 180 degrees to provide anchorage near the edges of the slab (Fig. 4(a)). All reinforcement running in the y-direction had adequate development length without the need of bends. Splitting reinforcement Two studs were provided at each anchorage of the prestressing tendons to control horizontal splitting cracks that can occur through the slab thickness. An example of the location of the splitting reinforcement is shown in Fig. 6 for Fig. 3 Prestressed cables and top nonprestressed reinforcement: (a) Specimen EC9C; (b) Specimen EC7C; (c) Specimen EC5C; and (d) Specimen EC3C. 316 ACI Structural Journal/March-April 2005

4 Fig. 4 Typical cross section: (a) perpendicular to free edge; and (b) parallel to free edge. Fig. 5 Top nonprestressed reinforcement, Specimen EC0C. Specimen EC9C. The same type of stud as the SSR was used. At anchorages located at the free edge, no studs were provided, specifically to reinforce against splitting, in addition to the studs used as shear reinforcement. Loading Stage 1 This testing stage replicated the application of gravity loads prior to an earthquake occurrence. The shearing force V u and the unbalanced moment M O were applied simultaneously in load control mode such that (M O /V u ) = 0.3 m, where M O is the bending moment at the centroid of the column. The eccentricity M O /V u was determined by frame analysis of the prototype structure. The combinations of V u and M O were applied in five equal increments to reach Fig. 6 Shear and bottom nonprestressed reinforcement, typical for all specimens. Location and number of splitting reinforcement shown apply only to Specimen EC9C. V u = 110 kn (25,000 lb) and M u = 33 knm (290 kip-in.). The force V u = 110 kn was sustained during Loading Stage 2. Loading Stage 2 Loading Stage 2 simulated the seismic action on the prototype structure when subjected to an earthquake. While the force (V u ) max = 110 kn was sustained, the two column ends were displaced a distance of /2 in opposite directions (Fig. 1). The amplitude of the displacements was increased in increments of 1 mm. For each increment, three reversal cycles were performed. After reaching the maximum unbalanced moment (M O ) max, only one cycle was performed for each displacement increment. The increments of the imposed cyclic displacements continued until 20% of the ACI Structural Journal/March-April

5 Table 2 Test values of unbalanced moments, drift ratios, and shear stresses* (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) Maximum shear stress at d/2 from column (v u ) max / f c Test (M O ) max, Positive M (M specimen f c, MPa u ) max, (M O ) yield line, o Negative M o knm knm knm DR y DR u DR u80 µ Side BC Point A Side BC Point A EC0C 27.6 EC3C 25.8 EC5C 25.6 EC7C 29.4 EC9C * Edge slab-column connection subjected to (V u ) max = 110 kn combined with cyclic moment transfer. 1 MPa = 145 psi. 1 knm = kip-in. Value of (M O ) max and drift ratios for this specimen are smaller than what would have been reached without malfunction of loading equipment. malfunctioning of the loading equipment (very high V u was inadvertently introduced). Consequently, for this specimen, the values of the drift ratio after reaching (M O ) max were not recorded. Fig. 7 Yield line pattern used to calculate values of (M O ) yield line that would produce flexural failure. Fig. 8 Definition of DR y, DR u, and DR u80. unbalanced moment capacity was lost (Fig. 8). The positive directions of V u, M O, deflection, and DR are indicated in Fig. 1. TEST RESULTS It should be mentioned that Specimen EC7C failed prematurely before completing Loading Stage 2, due to Ultimate loads and mode of failure All slabs failed by punching shear inside the shearreinforced zone. Strain measurements on selected shear reinforcing studs indicated that the yielding of studs occurred in each of the five tested specimens. The recorded yield was on studs located within the first three peripheral rows of studs adjacent to the faces of the columns. Before failure, cracks widened considerably; large deflections (40 to 44 mm at ultimate) were recorded around the column slab connection. The maximum positive and negative values of the unbalanced moment (M O ) max are listed in Table 2. The values of (V u ) max /V c vary between 0.54 and 0.58, where V c = b o d f c /3, with f c in MPa. The value of V c is the nominal shear strength of the connection in the absence of unbalanced moment and shear reinforcement (ACI ). The values of V u at O, the center of the column, combined with M O are statically equivalent to V u at the centroid of the critical section combined with M u = M O + V u x O ; where x O = m, the x-coordinate of O (Fig. 9). For each specimen, two values of (M u ) max are given in Table 2, corresponding to the positive and the negative (M O ) max. Splitting of the slab through the thickness did not occur at any of the anchors of the tendons. Hysteresis loops The large drift ratios recorded in tests (Column 6 to 8, Table 2) indicate the ability of the tested slab-column connections to undergo plastic displacement without punching failure. An example of the hysteresis loops of DR versus M O is shown in Fig. 10 for specimen EC0C. Envelopes of the loops for all specimens are shown in Fig. 11. Columns 6 to 8 of Table 2 list the values of DR y, DR u, and DR u80 (defined in Fig. 8). DISCUSSION OF TEST RESULTS Flexural strength As expected, the values of (M O ) max recorded in the tests were smaller than the calculated (M O ) yield line, where (M O ) yield line is the unbalanced moment that can produce 318 ACI Structural Journal/March-April 2005

6 Fig. 9 Shear-critical section at d/2 from column face. Fig. 11 Envelopes of drift ratio-unbalanced moment hysteresis loops. Fig. 10 Hysteresis loops of drift ratio versus unbalanced moment at column centroid, Specimen EC0C. flexural failure when combined with (V u ) max. The yield line pattern shown in Fig. 7 was used, with the distances x 1, x 2, and x 3 that minimize M O (for example, for EC5C, x 1, x 2, and x 3 are 0.58, 0.2, and 0.25 m, respectively, differing slightly for the other prestressed specimens). It can be seen in Columns 3 and 5 of Table 2, that the absolute values of (M O ) yield line are greater than the absolute values of (M O ) max recorded in the tests, confirming that the failures were by punching, before the flexural strength could be reached. In the yield line analysis, the flexural strength provided by individual strands and reinforcing bars are taken considering their respective depths at their intersection with the yield lines and f ps or f y for the prestressed and the nonprestressed reinforcement, respectively, where f ps is the prestressing tendon stress at ultimate and f y is the measured yield strength of nonprestressed reinforcement. The value of f ps was taken equal to 1204 MPa (174.7 ksi) using an equation of ACI Only the yield line values with M O in the positive direction are given in Table 2. Similar yield line analysis with M O reversed confirmed that the failure in the tests was by punching, not by flexure. Displacement ductility For seismic design, Pan and Moehle 7 consider that the slabcolumn connections have adequate ductility when µ 1.2; where µ is displacement ductility, defined as DR µ u = DR y where DR u and DR y are identified graphically in Fig. 8. The values of µ for the test specimens are given in Column 9 of (2) Fig. 12 Shear-critical section at d/2 from outermost peripheral line of SSR (typical for all test specimens). Table 2. According to Park and Paulay 8 µ values, defined in a similar way, without prestressing, range between 3 and 5 for seismic-resistant reinforced concrete framed structures. For all the tested specimens, µ values are within or exceed the values suggested by Pan and Moehle 7 and Park and Paulay. 8 Shear stresses At an edge column, ACI 421.1R-99 4 calculates the assumed linear stress distribution over the perimeter of the critical sections shown in Fig. 9 and 12 by the equations v u = V u γ vy M uy x b o d I y γ vy 1 1 = l x where b o is the length of the perimeter of the shear critical section; x is the coordinate, with respect to the centroidal principal axes, of the point at which v u is calculated; l x and l y l y (3) (4) ACI Structural Journal/March-April

7 ACI : v n = v s + v c -- f (5) 2 c where v c = (1/6) f c (6) ACI 421.1R : v n = v s + v c -- f (7) 3 c where v c = (1/8) f c (8) Fig. 13 Shear stress at Point A of critical section (Fig. 9) during Loading Stage 2, Specimen EC0C (nonprestressed slab). Megally and Ghali 5 5 : v n = v s + v c -- f (9) 6 c where v c = (1/8) f c (10) Fig. 14 Shear stress at Point A of critical section (Fig. 9) during Loading Stage 2, Specimen EC9C (slab with highest amount of prestressing). are the projection of critical section on the centroidal principal x and y axes; and I y is the second moment of area about the y axis. Figure 9 and 12 show the positive directions of x and y axes, V u and M uy when (l x /l y ) < 0, γ vy = 0. It should be mentioned that ACI uses a parameter J instead of I in Eq. (3) and gives equations for J and γ vy that are applicable for a critical section in the shape of a closed rectangle. For a critical section of general shape, ACI 421.1R-99 4 replaces J by I, as in Eq. (3) and gives Eq. (4) for γ vy for edge columns. Shear strength at critical section at d /2 from column face With unbalanced moment reversals, the maximum shear stress (v u ) max occurs at Side BC or at Points A and D of the critical section (Fig. 9). Table 2 gives values of (v u ) max / f c at Side BC and at Point A (or D). The values of (v u ) max are calculated by Eq. (3) with the test values (V u ) max and (M O ) max. It can be seen that the greatest shear stress occurs at the outer edge, Point A (or D) due to (V u ) max combined with negative (M O ) max (Table 2, Column 13). For seismic design of shear reinforcement, ACI , 3 ACI 421.1R-99, 4 and Megally and Ghali 5 express the nominal shear strength v n at the critical section d/2 from the column face as (in MPa) The equation for v s, the nominal shear stress provided by shear reinforcement, is common for all three sources and is given by Eq. (1). In all the equations containing f c when the stress is in psi, replace f c by 12 f c. Note that ACI does not give special equations for the seismic design of stud shear reinforcement in slabs. On the other hand, the equations given by ACI 421.1R-99 4 and by Megally and Ghali 5 provide provisions for the seismic design of stud shear reinforcement; they recommend a relatively low value of v c to account for the reduction in shear strength of cracked concrete caused by cyclic loading. Regardless of the amount of shear reinforcement, each of the above three sources sets an upper limit for v n ; the highest upper limit is (5/6) f c in MPa (10 f c in psi). It can be seen from Table 2, Column 13 that the upper limit (5/6) f c is exceeded in all specimens at Point A of the critical section at d/2 from the column face. The nominal shear strength provided by SSR in each specimen is calculated by Eq. (1), with A v = 426 mm 2, s = 55 mm, f yv = 462 MPa, and b o = 978 mm v s = A v f yv b o s = 3.66 MPa The nominal shear strength predicted by Eq. (5), (7), and (9) is the lesser of (v c + v s ) and the upper limit specified for each equation; the result of this calculation is presented for each specimen in Table 1 in terms of its value of f c (Columns 5 to 7). These values can be compared with the maximum shear stress (v u ) max determined from the tests (Column 8 of Table 1, or Column 13 of Table 2). The comparison indicates that each of Eq. (5), (7), and (9) gives a conservative prediction of v n. Figure 13 and 14 show hysteresis loops plotted during loading stage 2 for specimens EC0C and EC9C, respectively. The abscissas in the figures represent the drift ratio; the ordinates represent the shear stress v u / f c, and the value of v u is for Point A of the critical section at d/2 from the column face (Fig. 9), calculated using Eq. (3) with V u = 110 kn combined with M u varying as recorded during the loading 320 ACI Structural Journal/March-April 2005

8 cycles. The horizontal dashed lines in Fig. 13 and 14 represent the upper limit (v n ) upper limit according to ACI , 3 ACI 421.1R-99 4 and Megally and Ghali, 5 (1/2) f c, (2/3) f c, and (5/6) f c, respectively. From the two figures and Column 8 of Table 1, it can be seen that in seismic design for punching shear, it is safe to consider that (v n ) upper limit = (5/6) f c in MPa (10 f c in psi). Shear strength for critical section at d /2 from outermost peripheral line of SSR ACI requires that the shear-reinforced zone extends to a distance (αd) such that the shear stress v u at a critical section at d/2 from the outermost peripheral line of shear studs be less than 1 v n = v c = -- f 6 c MPa (11) ACI 421.1R-99 4 and Megally and Ghali 5 adopt the same v c value as ACI In the tested specimens, the greatest shear stress is at the free edge, Point A, when M u is negative. Table 1, Column 9, gives values of (v u ) max / f c at Point A of the critical section at d/2 from the outermost peripheral line of SSR (Fig. 12). It can be seen that the values of (v u ) max vary between 1.6 and 1.8 times the predicted value: v c = (1/6) f c in MPa (2 f c in psi). It should be possible by the use of Eq. (3) combined with Eq. (11) to predict the value of M uy, and hence, M O, that produces punching shear failure at the outer critical section (Fig. 12). Similarly, the value of M O that produces punching shear failure at the inner critical section (Fig. 9) can be predicted by Eq. (3) combined with one of Eq. (5), (7), or (9). Consider as an example Specimen EC0C and use Eq. (9) (Megally and Ghali 5 ), the predicted moment at the column centroid that would produce punching shear failure at the inner and outer critical sections would be (M O ) for failure at inner section = 36 knm ( 318 kip-in.) (M O ) for failure at outer section = 4 knm ( 35 kip-in.) The lesser of the two absolute values of M O governs. Thus, based on this analysis, one would predict that the failure would occur at the outer critical section at M O = 4 knm ( 35 kip-in.). The failure in the tests of this specimen occurred at the inner critical section. Similar calculations for the other specimens would also predict the failure at the outer critical section; in all tests, the punching failure was at the inner critical section. The reason for the false prediction is believed to be that Eq. (3) overestimates the values of v u at the outer edge due to V u combined with negative M u. This is further discussed in the following section. Calculation of maximum shear stress at critical section outside shear-reinforced zone When the earthquake produces a negative unbalanced moment, Eq. (3) gives an overestimate of the shear stress v u at points on (or in the vicinity of) the free edge of the critical section outside the shear reinforced zone (Fig. 12). No more accurate method exists to determine v u in this particular case. To solve this problem in design, it is suggested herein that when M uy in Eq. (3) is negative, to safely ignore the SSR on the rails running perpendicular to the inner column face (the Fig. 15 Alternative shear-critical section at d/2 from outermost peripheral line of SSR when M uy is negative. studs on the right-hand side of the column in Fig. 12). The outer shear critical section will become rectangular as shown in Fig. 15. The ratio (l x /l y ) will become smaller than or close to 0.2, resulting in γ vy becoming equal or close to zero (Eq. (4)). This will avoid an overestimation of v u that will lead in design to an excessively large shear-reinforced zone (refer to the example that follows). Example: Design of extent of shear-reinforced zone To show a case where the design can lead to an excessively large shear reinforced zone, consider an edge-column connection (Fig. 12) with where (according to ACI ) V u = 0.5V c (12) V c = 1 -- f, where f c is in MPa (13) 3 c b o d V c is equal to the nominal shear strength (in force units) at the inner critical section (Fig. 9) in the absence of unbalanced moment or shear reinforcement. The critical section at d/2 from the column (Fig. 9) has the properties b o = 8.57d, x O = 0.75d, x A = 1.85d, γ vy = 0.35, and I y = 6.87d 4. Combined with V u = 0.5V c = (1/6) f c b o d, assume that the edge column in Fig. 9 is subjected to unbalanced moment (same value as its upper limit) 1 4I M O = -- f 6 c y b γ vy x o dx O A (14) This is the value of the unbalanced moment that, combined with V u = 0.5V c, produces shear stress in the inner critical section at the outer edge equal to (5/6) f c, with f c in MPa. This is the upper limit for v n according to Megally and Ghali. 5 Equation (12) and (14) give for the critical section in Fig. 9: V u = 1.43 f d 2 and M O = 5.99 d 3 c f c. Consider the critical sections shown in Fig. 12 and 15. The dashed line in Fig. 16 represents α y versus the shear stress (v u ) A (calculated by Eq. (3)) at Point A of the critical section in Fig. 12. The point of intersection of the graph (at α y = 6.9) with the horizontal line at coordinate (1/6) f c gives the extent of the outer critical section. The intersection of the solid line in Fig. 16 with the horizontal line at stress = (1/6) f c, at α y = ACI Structural Journal/March-April

9 Fig. 16 Maximum shear stress outside shear-reinforced zone at point on free edge. Example column dimensions c x = c y = 250 mm. 5.5, indicates the extent of shear reinforced zone when the outer critical section has the shape of three sides of a rectangle, as shown in Fig. 15. CONCLUSIONS On the basis of tests and the analysis, the following conclusions can be made concerning seismic design for punching shear of prestressed slab-edge column connections: 1. The design recommendations of Megally and Ghali, 5 intended for nonprestressed slabs, can also be used for prestressed slabs, having effective prestressing equal to 0.4 to 1.1 MPa. In practice, the effective prestress is within or close to this range. The design steps are given in Appendix A; 2. The experiments confirm Megally and Ghali s 5 recommendations that for seismic design, the upper limit of the nominal shear strength v n at the critical section at d/2 from the column can be taken equal to (5/6) in MPa (10 f c in psi); 3. With negative unbalanced moment M u the maximum shear stress v u is at the outer critical section at the free edge. The use of this stress in design can lead to an excessively large shear-reinforced zone. This problem can be avoided by ignoring the shear reinforcement positioned on lines perpendicular to the free edge in calculating v u. With positive M u, no shear reinforcement should be ignored in determining v u ; 4. Effective prestressing of 0.4 to 1.1 MPa in edge slabcolumn connections do not adversely affect the ductility or the maximum interstory drift ratio that they can undergo without punching failure; and 5. Headed studs prevented splitting through the slab thickness at the anchorages of the strands. Two studs of nominal strength 28 kn were sufficient to prevent the splitting due to a single strand with a jacking force of 110 kn. ACKNOWLEDGMENTS The stud shear reinforcement used in the tests was donated by Decon, USA and Canada. Dywidag Systems International, Vancouver, Canada, donated the prestressing strands, ducts, and anchors. Canada s Natural Sciences and Engineering Research Council financially supported the research. NOTATION A v = cross-sectional area of shear reinforcement on one peripheral line, m 2 b o = length of perimeter of shear-critical section, m c x, c y = column dimensions in the x- and y-directions, respectively, m DR u = maximum interstory drift, including inelastic deformation, divided by l c d = average of distances from extreme compression fiber to centroid of tension reinforcements running in two orthogonal directions, m f c = specified concrete strength, Pa f pc = average effective prestress in two directions at section centroid, Pa f yv = specified yield strength of shear reinforcement, Pa I y = second moment of area of assumed critical section about y-axis, m 4 l c = story height, m l x, l y = projections of shear-critical section on its principal x- and y-axes, respectively, m M O = unbalanced moment transferred between slab and column, Nm M u = unbalanced moment transferred between slab and column at centroid of shear-critical section, Nm s = spacing between peripheral lines of shear reinforcement, m V c = nominal shear strength provided by concrete, N V u = factored shear force at section, N v c = nominal shear strength (in stress units) provided by concrete, Pa v n = nominal shear strength (in stress units), Pa v s = nominal shear strength (in stress units) provided by shear reinforcement, Pa α = ratio of distance between column face and shear-critical section outside shear-reinforced zone to slab effective depth = interstory drift, m γ v = fraction of unbalanced moment transferred by eccentricity of shear stresses µ = displacement ductility REFERENCES 1. Martinez-Cruzado, J. A.; Qaisrani, A. N.; and Moehle, J. P., Post- Tensioned Flat Plate Slab-Column Connections Subjected to Earthquake Loading, 5th U.S. National Conference on Earthquake Engineering, Chicago, Ill., 1994, V. 2, pp Hawkins, N. M., Lateral Load Resistance of Unbonded Post-Tensioned Flat Plate Construction, PCI Journal, V. 26, No. 1, Jan. 1981, pp ACI Committee 318, Building Code Requirements for Structural Concrete (ACI ) and Commentary (318R-02), American Concrete Institute, Farmington Hills, Mich., 2002, 443 pp. 4. ACI Committee 421, Shear Reinforcement for Slabs (ACI 421.1R-99), American Concrete Institute, Farmington Hills, Mich., 1999, 15 pp. 5. Megally, S., and Ghali, A., Seismic Behavior of Edge Slab-Column Connections with Stud Shear Reinforcement, ACI Structural Journal, V. 97, No. 1, Jan.-Feb. 2000, pp International Code Council, International Building Code (IBC-2003), Ill., 2003, 660 pp. 7. Pan, A., and Moehle, L. P., Lateral Displacement Ductility of Reinforced Concrete Flat-Slabs, ACI Structural Journal, V. 86, No. 3, May-June 1989, pp Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley and Sons, Inc., New York, 1975, 769 pp. APPENDIX A STEPS FOR DESIGN Consider a structure provided by shear walls or other bracing systems such that the drift ratio in the x-direction, including plastic deformation, DR u It is required to design seismic resistant slab-column connections at an edge parallel to the y-axis. The steps of design (based on Megally and Ghali 5 ) are summarized below. The values of f c, f yv, V u, V c, DR u, and d are given. Steps 1 and 2 calculate the value of M u to be used in the design and Steps 3 to 6 design the shear reinforcement, if required (refer to flow chart in Fig. A1). Step 1 Calculate the maximum elastic interstory drift e e = DR u l C c (A1) where l c is the story height and C is a dimensionless factor specified by IBC(2003), Section , 6 representing the inherent inelastic deformability of the primary resisting system and the occupancy importance of the structure (for example, with shearwalls, 2.7 C 6.7). Step 2 Calculate the unbalanced moment M uy by linear elastic analysis of an equivalent frame, as specified by 322 ACI Structural Journal/March-April 2005

10 Fig. A1 Design steps for punching shear. ACI , 3 Section 13.7, subjected to prescribed interstory drifts e. The value of M uy need not exceed the upper limit M u M ( ) pr limit (A2) where M pr is the probable flexural strength of the inner side of the critical section in Fig. 9, having a width equal to l y. For an edge column, the empirical coefficient α m is (A3) where γ vy is the fraction of moment transferred by vertical shear stresses in the slab, and ρ is the ratio of tensile flexural reinforcement passing through the inner side l y of the shearcritical section. The tensile flexural reinforcement is at the top or at the bottom when M u is positive or negative, respectively. In calculating the probable flexural strengths, use 1.25 times the specified yield strength of the nonprestressed flexural reinforcement and a strength reduction factor equal to Step 3 Calculate the maximum shear stress v u at d/2 from the column face (Eq. (3)). Verify that v u (5/6) f c ; if this is not satisfied, the structural members should be changed to reduce v u (for example, increase column sides, α m l α m = 0.55 γ y vy ρ 40 l x or reduce drift by stiffening the bracing system). If v u φv n, go to Step 6 where v n is the nominal shear strength without shear reinforcement. Step 4 When DR u and (V u /φv c ) 0.4 or when DR u and (V u /φv c ) ( DR u ) the connection does not require shear reinforcement and the design is complete. If either of these conditions are not satisfied, go to Step 5. Step 5 Provide the minimum amount of shear reinforcement such that the steel contribution to the nominal shear strength satisfies the inequality A v s = v f yv f MPa (A4) b o s 4 c A v s = v f yv 3 f psi (A4) b o s c Step 6 Provide shear reinforcement according to ACI 421.1R-99 2 such that v u φv n, where the nominal shear strength v n = v s + v c, where v c and v s are the shear strength provided by the concrete and shear reinforcement, respectively (Eq. (10) and (1)); extend the shear-reinforced zone such that v u in the outer critical section does no exceed (1/6) f c in MPa (2 f c in psi). ACI Structural Journal/March-April