Journal of the Korea Concrete Institute Vol. 23, No. 2, pp. 159~168, April, 211 GGGGG DOI 1.4334/JKCI.211.23.2.159 -» ü w» q w ³ 1) *Á½ y 1) Á½ z 1) Á y 2) 1) w w 2) w w Direct Punching Shear Strength Model for Interior Slab-Column Connections and Column Footings with Shear Reinforcement Kyoung-Kyu Choi, 1) * Sug-Hwan Kim, 1) Dong-Hoon Kim, 1)G and Hong-Gun Park 2) 2) 1) School of Architecture, Soongsil University, Seoul 156-743, Korea Dept. of Architecture, Seoul National University, Seoul 151-742, Korea ABSTRACT In the present study, an improved design method was developed for the punching shear strength of interior slabcolumn connections and column footings with and without shear reinforcement. In the evaluation of the punching shear strength, the possible failure mechanisms of the connections and column footings were considered. The considered failures modes were inclined tensile cracking of concrete, yielding of shear re-bars, and concrete crushing of compression zone/strut. The punching shear applied to the concrete critical section was assumed to be resisted mainly by the compression zone. The punching shear strength of the concrete compression zone was evaluated based on the material failure criteria of the concrete subjected to the compressive normal stress and shear stress. For verification of the proposed design method, its prediction was compared with the existing test results. The result showed that the proposed method predicted the strengths of the test specimens better than the current design methods of the KCI code for both the shear reinforced and unreinforced cases. Keywords : slab-column connections, concrete slabs, column footings, direct punching shear, shear reinforcement 1. q» e œ» œ ƒ š, ü w œ k l. ù q r e w -» w w w. 1) -» w x w j» w w w œ (Fig. 1). ù -» w w ƒw w w q f w š. Ruiz 3) Muttoni w q eƒ p w» w gj p p q (Fig. 2). *Corresponding author E-mail : kkchoi@ssu.ac.kr Received August 27, 21, Revised November 11, 21, Accepted November 22, 21 21X by Korea Concrete Institute -» w q f w» w w x w. Beutel 4) Hegger x k œ w 5) w. Pralong and Nielsen k wwe we w, Johansen 6) w f ³ w š w w w w. w -» w q f w. ù w w w š w». 7) KCI ACI 318, 8,9) Euro code2 xw 1)»» x k w x wš (Fig. 3). ù xw» x s j» w š, Fig. 4 ùkù KCI( ACI 318)» -» w w» x y w w x w d. 11,12) wr, KCI ACI 318 xw»» 8,9)» q w, -» w w s 159
Fig. 1 Various shear reinforcement methods used in slabcolumn connections 2) x w j» w š 13) (Fig. 3). Hegger et al.»» q w» x, ACI 318»» q ƒ ƒw sƒw» q ̃ É sƒw š šw. w w» w w š. p, ³ 14) gj p q» w { w š w -» w w. w w 14) k, w -» w Fig. 2 Possible failure modes at slab-column connections with shear reinforcement 3) Fig. 3 Existing design methods for direct punching shear Fig. 4 Strength predictions for existing test specimens by KCI code 16 w gj pwz 23«2y (211)
yw sƒw t w. w w w ³ q, gj p / l q w q f š w. w -» w» q,» x m w z w. 2. gj p -» w q w { x w w x j { ³ 17) w. Kotsovos, 15) Zararis, 16) Jelic w, j { z gj p ù z ƒ j w gj p ƒ w {w (Tureyen et al., 18) Kotsovos 15) ). w ƒ k ³, 19,2) ³ 14) w š y, 21) w x» w. gj p yw w» w, gj p w w š w. w w gj p q» Rankine(Chen ) q 22)» w. q» gj p w q ƒ w ƒ w. ƒ w f ck w q ƒ w, f' t w q ƒ w (Fig. 5). q» l ( x ) w. 19-21) v nc ( z) f ck ( f ck ( z) ) v nt + ( z) f t ( f t ( z) ) (1a) (1b)» f t w w 12). (1) gj p s s xk w. Fig. 5 Principal stress failure criteria of concrete subjected to shear-compression x, gj p ¾, z l. gj p x (αε o ) w (1) { š ( { x) yw. q { w w, w d w w. ³ w {w. w x ¼ z w s³ w ( (1)) mw w. v c v nc ( z)dz/ w r w» w, ³ w z w s³ ( ) w (1), (3) w ³ s³ w. for α 1 for α > 1 d or v nt ( z)dz/d v cc v nc ( z)dz/d f ck ( f ck ) /d (3) (4a) ( z) f ck 2 ----- αz αz 2 -----» αε o gj p x, ε o (.2) gj p w w x, α gj p x w w (2) /α v cc v nc ( z)dz/d f ck ( f ck ) /d/α for α 1 v ct v nt ( z)dz/d f t ( f t ) /d + (4b) (4c) -» ü w» q w 161
for α > 1 /α v ct v nt ( z)dz/d f t ( f t + ) /d/α (4d) (4), gj p ƒ d x gj p w w x j (α >1), ƒ w w {w w. α >1 (4b) (d) y ( z / α) { š ƒ w. (2) w s³ w. for α 1 for α >1 3. (5a) (5b) -» w» w q. -» w { x w { ƒ š q x ¼ ƒ ƒ š ƒ». xw»» l ³ (q ) w š ƒ wš» d /2 j e x w. ù -» w e w q f w w ƒ q. 11) ( z)dz/ ( α α 2 /3)f ck /α ( z)dz/ ( 2/3)f ck Fig. 2(a) -» gj p ³ z w w ƒ q. -» w x gj p wì w {w. wr e Á ( ) gj p q, gj p w w { (Fig. 2(b), Fig. 6). -» w gj p p q (Fig. 2(c)). q f -» w ƒƒ w. 3.1 e -» w ƒ ü q (Fig. 6(a)), w gj p (V c )» (V s ) w. 8,9,1,23) V n V c + V s V s A v f yt gj p ƒ w q w. gj p ( (4c)) š w (8). V c v ct d f t f t + ( ) x ¼, w q ƒ w 45 ³ š w» s z Ì w x s c 1 + d( c 2 + d) w (Fig. 3, KCI 9) ). ü w w x ¼ 2c 1 + 2c 2 +4d. (4) ùkù, gj p { x ƒw yw. w 14) r w» w, ³ w q w x αε o.196(α 1, Kinnunen (6) (7) (8) Fig. 6 Failure modes and the critical shear sections inside/outside shear reinforced zone around slab-column connections 162 w gj pwz 23«2y (211)
and Nylander ) w 24) (2/3)f ck w. w w š w g j p z f (1 / 3)f t w. 14) t wr» š j» z 25) š w» 4 w, j» z λ s [ 2/d ] w. w x 26) x z (ACI 318-8 ) š w» 8) w, Manterola x w x x λ bo [ 3/ /d w.,, j» z x 1] f t w w (V c ) (9) w. V c λ s λ bo f t f t 2/3 ( + ( )f ck ) Á q ƒ w gj p w -» w. ù q x s x w. Fig. 6(b) e, Á req x. 9) d 4 2 d max + -- 2 + + 2c 1 2c 2 (9) (1)» d max» ƒ ¾, (6) V s. 3.2 w 27) Beutel w ƒ -» w gj p p q. xw KCI ACI 318-8» 8) w q š w V max.5 f ck d ( l.67 f ck d ) wš. gj p q f w (4b) w gj p q w. V max v cc d (11) v cc (4a) 28) (4b) w w. Shehata w x αε o.35(α 1.75) w -» w ƒ gj p w q. (11) v cc αε o.35 ƒ w. Table 1 Specimen properties of slab-column connections without shear reinforcement, and strength predictions Investigators 11) No. of specimens c 1 (c 2 ) (1) f ck (2) (MPa) d ρ t (%) Hallgren and Kinnunen 7 25 84.1-18.8 194-22.3-1.2 1.12-1.33.54-.97 Tomaszewicz 13 1-2 64.3-119. 88-275 1.5-2.6 1.25-1.59 1.26-1.65 Ramdane, Regan et al. 15 15 32.9-11.6 98-12.6-1.3 1.6-1.5.77-1.55 Marzouk and Hussein 15 15-3 42.-8. 7-125.4-2.1 1.11-1.66.7-1.68 Lovrovich and McLean 4 1 39.3 83 1.7.85-1.34 1.9-1.72 Tolf 8 125-25 22.9-28.6 98-2.4-.8 1.8-1.35.82-1.47 Regan 23 54-2 9.5-42.6 64-2.8-2.4.92-1.4 1.8-1.66 Swamy and Ali 2 15 37.4-4.1 1.6-.7 1.11-1.21.94-1.9 Marti et al. Pralong et al. 2 3 26.2-34.6 143-171 1.2-1.5 1.24-1.32 1.23-1.38 Schaefers 2 21 23.1-23.3 113-17.6-.8 1.32-1.57 1.18-1.3 Ladner et al. Schaeidt et al., Ladner 6 1-5 27.9-33.5 8-24 1.2-1.8 1.17-1.57 1.38-1.75 Corley and Hawkins 2 23-254 44.4 111 1.-1.5.91-.92.86-.93 Moe 14 152-35 2.5-35.2 114 1.1-2.6.87-1.51 1.13-1.64 Kinnunen and Nylander 12 15-3 24.6-31.4 117-128.5-2.1 1.7-1.34.78-1.63 Elstner and Hognestad 24 254-356 1.2-4.4 114-121.5-6.9.86-1.55.58-2.8 Park and Choi 5 25-64 26.4-28.6 9-13 1.-2. 1.2-1.48 1.12-1.49 Teng et al. 5 2 33.-4.2 12 1.1-1.2 1.5-1.36 1.5-1.38 Bernaert and Puech 2 23 14.-41.4 12-124 1.-1.9.87-1.51.87-1.92 Manterola 12 1-45 24.2-39.7 17.5-1.4.9-1.32.65-1.45 Yitzhaki 16 119-3 9.8-21.6 78-82.5-8.5 1.9-1.96 1.21-1.91 Mean 1.27 1.34 COV.152.232 (1) Specimen 4 of Park and Choi, and OC13, OC13-1.6, OC13-.63, and OC 15 of Teng et al., had rectangular column-sections. The others had square or circular column-sections. For the circular column-sections, c 1 c 2 ( π /2)D was used, where D diameter of circular column-section. (2) f ck concrete compressive cylinder strength (.8 f c,cube ). (3) Strength-predictions by KCI (or ACI 318-8) ------------- V pred. ----------- V ACI. (3) -» ü w» q w 163
(6) -» w V n V max w. 4. w w w -» w»» q w» 11) x w. CEB-FIP» š» x13,29-38) -» w w 27 x -» w w 187 x,»» q w 81 x ƒ. Tables 1, 2, 3 ƒƒ -» w, Table 2 Specimen properties of slab-column connections with shear reinforcement, and strength predictions Investigators 11,29-36) No. of specimens c 1 (c 2 ) f' ck (MPa) d ρ t (%) ρ v (%) or A v f vy (MN) (1) Failure mode (2) Graf 6 2-3 14.4-16.4 27-47.54-2.34.366-.1184 1.26-1.82.87-1.46 O Keefe 2 15 25.-26.3 11 3.1.87.96-1. 1.71-1.75 C Elstner and Hognestad 9 25 13.8-44.8 11 1.44-11.4.73-.513.99-1.93.92-1.5 O Moe 1 2 23. 1 2.14.72 1.54 1.3 O Andersson 2 15-3 2.8-28.4 12-13.72-12.63.34-.37.81-1.63.83-1.64 I/O/C Franz 13 21 19.8-26.9 13.39-3.34.2-.123.74-1.21.99-1.81 I/O/C Narasiham 1 31 33.4 14 2.92.242 1.7 1.17 O Petcu and Stanculescu 9 2 2.4-32.1 11-21 1.9-4.7.74-.325.9-1.61.79-1.17 O Marti et al. 1 3 34. 15.79.4 1.16 1.81 I Sunquist 4 4-25 2.6-26.8 17 4.37-7.97.227-.256 1.-1.31.93-1.12 O/C Seible et al. 3 31 33.6 12 1.33-2.42.56-.12 1.12-1.24.99-1.14 O Swamy and Ali 2 15 31.7 1 3.17-4.16.69-.86 1.13-1.21 1.19-1.27 O Pral et al., Muller et al. 2 3 3.-32.9 15 2.62-4.96.12-.24.86-1.38.84-1.41 O Hallgren and Kinnunen 2 25 85.-92.4 2 6.76-8.48.455-.569 1.17-1.22.78-.94 O Broms 6 25 14.6-23.7 15 3.9-4.73.113-.245.86-1.88.89-2.6 O/C Lovrovich and Mc Lean 5 1 38.9 8.95-34.45.19-.698 1.22-2.56 1.56-3.11 O Regan 1 6 35.4 26 1.15.181 1.2 1.3 O Kinnunen et al. 2 8 26.3-26.7 67.94-.95.814 1.55 1.23 O Tolf et al. 8 13-25 2.4-22.3 1-2 1.25-2.65.16-.126.94-1.17.79-1.34 O/C Chana and Desai 12 3-4 22.7-32. 19-21.39-11.45.4-.943 1.18-2.8 1.12-2.86 I/O Yamada et al. 11 3 16.-19.5 17.47-4.21.42-.398 1.15-2.4 1.8-2.54 I/O Chana 3 3 29.-31.1 19.84.11 1.2-1.54 1.15-1.49 O Marzouk and Jiang 3 25 68.-74. 12 4.33-12.25.157-.444.79-1.19.56-.82 O Beutel and Hegger 1 4-32 23.2-46.3 19-23.76-2.63.6-.226 1.27-1.58.96-2.19 I/O Lee et al. 5 29-4 6.3-66.3 16-23.59-2.9.8-.126 1.1-1.47 1.7-1.81 I/O Olivera et al. 9 12 6.-66.3 1-11.75-7.4.16-.13.89-1.25 1.1-1.64 I/O Sherif and Dilger 1 25 33. 114 1.4.217 3) 1.2 1.39 I Mokhtar et al. 7 25 23.-41. 121.4 1.24.119-.279 3).9-1.23 1.24-1.93 I Pilakoutas and Li 3 2 39.-42.2 139.72.536-.894 3).8-.85.93-.99 C Adetta and Polak 3 15 41. 9.88.216 3).85-1.4 1.7-1.34 I/O/C Langohr et al. 4 35 27.6 127 1.14.185-.37 3).8-1.6.84-1.13 O Seible et al. 7 35 33.6 121 1.17.26-.61 3).91-1.1.9-1.36 O Van der Voet et al. 6 25 29.-37.1 114 1.46.96-.53 3).89-1.25 1.4-2.4 I/O Broms 6 25 17.-24. 15.58-1.11.41-.8 3).9-1.71.88-2.3 C Mean 1.2 1.32 COV.238.311 (1) (2) (3) Strength-predictions by KCI (or ACI 318-8) Predicted failure mode by proposed method: I - punching failure inside shear reinforced zone ; O - punching failure outside shear reinforced zone; and C - concrete compression crushing Contribution of shear reinforcement, A v f vy (MPa). ------------- V pred. ----------- V ACI. 164 w gj pwz 23«2y (211)
Table 3 Specimen properties of column footings, and strength predictions Investigators No. of specimens c 1 (c 2 ) l 1 (l 2 ) d f ' c (MPa) f ' y (MPa) ρ t (%) Support condition ( 1 ) ---------------- V pred. ( 1 ) ------------------ ( 2 ) V pred. (1) (2) Hegger et al. 37) 5 15 9 15 17.6-3.7 548.62-1.3 Sand 1.28-1.64 1.49-2.33 Hegger Sand /Car 13 2 1,2-1,8 25-47 19.-38.1 552.87-.91 et al. 13) spring 1.18-1.96 1.7-2.26 Richart 38) 63 15-3 9 2-4 13.9-34.8 384-571.2-1.25 Car spring 1.31-2.8.89-2.5 Mean 1.62 1.47 COV.128.196 Soil pressure applied inside the area of critical section(a ) was neglected (Hegger et al. 37) ) Strength-predictions by ACI 318-8 8) q, 13.9 f ck 38.1 MPa, 2.98 L 1 /d 1.51, 25 d 47 mm,.2 ρ 1.25(percent). Figs. 4 7 xw KCI» w -» w x ƒ. KCI, s³ 1.34, COV 23.2%,, s³ 1.32, COV 31.1% r ƒ j p 2% w j sƒw., s³ 1.27, COV 15.2%,, s³ 1.2, COV 23.8%, xw ACI» x y Fig. 7 Strength predictions by proposed method for existing test specimens (slab-column connections) -» w,»» q x p, x e, š x x ƒ. x» x11,13,29-38),. -» w, 9.8 f ck 119. MPa,.46 c 1 / d 8.33, 1. c 1 / c 2 5., 64 d 275 mm,.3 ρ 8.5(percent), -» w, 13.8 f ck 92.4 MPa,.24 c 1 / d 2.52, 8.25 d 669 mm,.34 ρ 3.2(percent), ΣA v.118(m 2 ) ( x ü). w»» Fig. 8 Strength predictions for existing test specimens by KCI (or ACI 318-8) and proposed model (column footings) -» ü w» q w 165
w. ùkù w yw wš. wr x w gj p ƒ ƒw ù ƒ ƒw ƒ ƒw. ( (9)) ƒ g j p ¾ ƒ j w ƒ ƒw. 5. Fig. 9 Strength predictions by proposed method for various types of shear reinforcements w wš. w Fig. 8»» q w x d ƒ. KCI( ACI 318-8)» x d ƒ s³ 1.47, COV 19.6%., x d ƒ s³ 1.62, COV 12.8% d r ùkü. x,, l, p, w w x ƒ sw. Fig. 9 -» w š, l wš x w x d š yw w ùkû. Table 2 w w q f w d wì. Table 2 w» x š ƒ, -» w ü w w ù(i), gj pƒ ³ w q ù(o), g j p ƒ q (C) w xk q d. Fig. 1 w,,, y -» w»» q yw sƒw r w. w k, gj p { x w w y š w» w gj p q» w, x w g j p w. -» w ³ w q gj p / p q w q, w q f š w w w. w w w sww 394 -» w 81»» q w» x w.,»» -» w»» q ywš d dw. 21 ( w» ) w w» (No. 21-15547). Fig. 1 Variation of punching shear strength according to design parameters 166 w gj pwz 23«2y (211)
Notation -» w q x ¼ gj p ¾ f' t w w gj p v cc gj p ³ s³ v ct gj p ³ s³ v nc gj p ƒ e v nt gj p ƒ e V c gj p V max -» w V s» z l ƒ e α gj p x w w x αε o gj p x ε o gj p w w x λ s j» z λ bo x x gj p w j» w s³ š x 1. MacGregor, J. G. and Wight, J. K., Reinforced Concrete: Mechanics and Design, Prentice Hall, NJ, 25, 1132 pp. 2., x qp l w, 23, pp. 285~286. 3. Ruiz, M. F. and Muttoni, A., Applications of Critical Shear Crack Theory to Punching of Reinforced Concrete Slabs with Transverse Reinforcement. ACI Struct. J., 16~S46, 29, pp. 485~494. 4. Beutel, R. and Hegger, J., Punching Shear Resistance of Shear Reinforced Flat Slabs, Arbeitsgemeinshaft industrieller Forschungsvereinigungen Otto von Guericke e. V., Research Programm Nr.1644-N, DBV 185, 1998. 5. Pralong, J., Poinçonnement Symétrique des Plachers-Dalles, IBK-Bericht Nr. 131, Insitut für Baustatik und Konstruktion ETH Zürish, 1982. 6. Johansen, K. W., Yield-Line Theory, Cement and Concrete Ass., London, 1962, 181 pp. 7. ³, y, v v p-» w, gj pwz, 16«, 2y, 24, pp. 163~174. 8. ACI Committee 318, Building Code Requirements for Structural Concrete (ACI 318-8) and Commentary (ACI 318R-8), USA, 28, 473 pp. 9. w gj pwz, gj p» w, 27, 524 pp. 1. EC 2, Design of Concrete Structures Part I: General Rules and Rules for Buildings, European Committee for Standardization Brussels, 22, 225 pp. 11. FIP 12, Punching of Structural Concrete Slabs, CEB-FIP Task Group, Lausanne, Switzerland, 21, 314 pp. 12. Park, H. and Choi, K., Improved Strength Model for Interior Flat Plate-Column Connections Subject to Unbalanced Moment, ASCE J. Structural Engr., Vol. 132, No. 5, 26, pp. 694~74. 13. Hegger, J., Ricker, M., and Sherif, A. G., Punching Strength of Reinforced Concrete Footings, ACI Structural Journal, Vol. 16, No. 5, 29, pp. 76~715. 14. ³, y, q -» w w, gj pwz, 22«, 3y, 21, pp. 345~356. 15. Kotsovos, M. D. and Pavlovic, M. N., Ultimate Limit-State Design of Concrete Structures, a New Approach, Thomas Telford, London, 1998, 28 pp. 16. Zararis, P. D. and Papadakis, G. C., Diagonal Shear Failure and Size Effect in RC Beams without Web Reinforcement, J. Struct. Engrg., ASCE, Vol. 127, No. 7, 21, pp. 733~742. 17. Jelic, I., Pavlovic, M. N., and Kotsovos, M. D., A Study of Dowel Action in Reinforced Concrete Beams, Magazine of Concrete Research, Vol. 51, No. 2, 1999, pp. 131~141. 18. Tureyen, A. K. and Frosch, R. J., Concrete Shear Strength, Another Perspective, ACI Struct. J., Vol. 1, No. 5, 23, pp. 69~615. 19. Choi, K., Park, H., and Wight, J. K., Unified Shear Strength Model for Reinforced Concrete Beams-Part I: Development, ACI Struct. J., Vol. 14, No. 2, 27, pp. 142~152. 2. Choi, K., Reda Taha, M. M., Park, H., and Maji, A. K., Punching Shear Strength of Interior Concrete Slab-Column Connections Reinforced with Steel Fibers, Cement and Concrete Composites, Vol. 29, No. 5, 27, pp. 49~42. 21. Park, H., Choi, K., and Wight, J. K., Strain-Based Shear Strength Model for Slender Beams without Web Reinforcement, ACI Struct. J., Vol. 13, No. 6, 26, pp. 783~793. 22. Chen W. F., Plasticity in Reinforced Concrete, NewYork, McGraw-Hill, 1982, pp. 24~25. 23. CSA A23.3-M4 Technical Committee, Design of Concrete Structures, Canadian Standards Associations, Toronto, Ontario, 24, 258 pp. 24. Kinnunen, S. and Nylander, H., Punching of Concrete Slabs without Shear Reinforcement, Transactions No. 158, Royal Institute of Technology, Stockholm, 196, 112 pp. 25. Bažant, Z. P. and Cao, Z., Size Effect in Punching Shear Failure of Slabs, ACI Struct. J., Vol. 84, No. 1, 1987, pp. 44~53. 26. Manterola, M., Poinçonnement de Dalles Sans Armature D effort Trenchant, Comité Européen du Béton (Hrsg.), Dalles, Structures Planes, CEB-Bull, Paris, D Information 1966, 58 pp. 27. Beutel, R., Punching of Flat Slabs with Shear Reinforcement at Inner Columns, Rheinisch-Westfälischen Technischen Hochschule Aachen, Aachen, Germany, 22, 267 pp. -» ü w» q w 167
28. Shehata, I. A. E. M., Theory of Punching in r. c. Slabs, Ph.D, Thesis, Polytechnic of Central London, 1985. 29. Sherif, A. G. and Dilger, W. H., Tests of Full-Scale Continuous Reinforced Concrete Flat Slabs, ACI Struct. J., Vol. 97, No. 3, 2, pp. 455~467. 3. Mokhtar, A., Ghali, A., and Dilger, W. H., Stud Shear Reinforcement for Flat Concrete Plates, ACI J., Vol. 82, No. 5, 1985, pp. 676~683. 31. Pilakoutas, K. and Li, X., Alternative Shear Reinforcement for Reinforced Concrete Flat Slabs, ASCE, Vol. 129, No. 9, 23, pp. 1164~1172. 32. Adetta, B. and Polak, M. A., Retrofit of Slab Column Interior Connections Using Shear Boltss, ACI Struct. J., Vol. 12, No. 2, 25, pp. 268~274. 33. Langohr, P. H., Ghali, A., and Dilger, W. H., Special Shear Reinforcement for Concrete Flat Plate, ACI Journal, Vol. 73, No. 3, 1976, pp. 141~146. 34. Seible, F., Ghali, A., and Dilger, W. H., Preassembled Shear Reinforcing Units for Flat Plates, ACI Journal, Vol. 77, No. 1, 198, pp. 28~35. 35. Vam der Voet, A. F., Dilger, W. H., and Ghali, A., Concrete Flat Plates with Well-Anchored Shear Reinforcement Elements, Canadian Journal of Civil Engineering, Vol. 9, No. 1, 1982, pp. 17~114. 36. Broms, C. E., Shear Reinforcement For Deflection Ductility of Flat Plates, ACI Struct. J., Vol. 87, No. 6, 199, pp. 696~75. 37. Hegger, J., Sherif, A. G., and Ricker, M., Experimental Investigations on Punching Behavior of Reinforced Concrete Foooting, ACI Structural Journal, Vol. 13, No. 4, 26, pp. 64~612. 38. Richart, F. E., Reinforced Concrete Wall and Column Footings Part 1, J. of ACI, Vol. 2, No. 2 1948, pp. 97~127. w / -» ü w» q w w. -» w» q w q f ( ³ q, w, gj p / p q ) š w w. gj p x w gj p w š ƒ w, gj p w w gj p q» w w. x mw w., w xw KCI» w ƒ š x. w : -» w, gj p,» q,, 168 w gj pwz 23«2y (211)