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1 G Journal of the Korea Concrete Institute Vol. 20, No. 3, pp. 345~356, June, 2008 p -k w gj p k k 1) Á y 2) * 1) ƒm w w 2) w w Direct Inelastic Design of Reinforced Concrete Members Using Strut-and-Tie Model Tae-Sung Eom 1) and Hong-Gun Park 2) * 1) Dept. of Architecture, Catholic University of Daegu,G Gyeongsan , Korea Y) Dept. of Architecture, Seoul National University, Seoul , Korea ABSTRACT In the previous study, direct inelastic strut-and-tie model (DISTM) was developed to perform inelastic design of reinforced concrete members by using linear analysis for their secant stiffness. In the present study, for convenience in design practice, the DISTM was further simplified so that inelastic design of reinforced concrete members can be performed by a run of linear analysis, without using iterative calculations. In the simplified direct inelastic strut-and-tie model (S-DISTM), a reinforced concrete member is idealized with compression strut of concrete and tension tie of reinforcing bars. For the strut and tie elements, elastic stiffness or secant stiffness is used according to the design strategy intended by engineer. To define the failure criteria of the strut and tie elements, concrete crushing and reinforcing bar fracture were considered. The proposed method was applied to inelastic design of various reinforced concrete members including deep beam, coupling beam, and shear wall. The design results were compared with the properties and the deformation capacities of the test specimens. Keywords : strut-and-tie model, secant stiffness, inelastic design, deformability, reinforced concrete 1. sƒ w gj p p -k (strut-and-tie model) ü y š. m p -k 1-3) 4) gj p ü gj p p k p yw, sx l ƒ p k ü w ww. x w ù x w š, w ù kü gj p r w ww. m p -k sx š, y w» w k x w š w w.. 1) p -k w, ƒ p k w j» w» w k x w š w w. ù x r w k x š w *Corresponding author Received October 29, 2007, Revised January 8, 2008, Accepted January 15, by Korea Concrete Institute. p -k w x v w, ƒ xk p -k ü w w yw. 5) 2), {w z k x ü w k x y sƒ w. p gj p q q q» gj p p k k x ƒ, k x j» ƒ. 6-8) Kim 9), Yoon k x š w 10) p -k w. w š w» p -k yw gj p k ƒ ƒ w. ù p -k w w w» x w w w w wš. y 11) r w k k p -k (direct inelastic strut-and-tie model) w.» w xw w š ƒ w 345

2 w xw k w wš x sƒw., š w w w w v w. w 11) w k p -k w, r w w y k p -k (simplified direct inelastic strut-and-tie model, S-DISTM) w. ƒ w w w xw gj p p k k wš, x sƒw. w p -k w š x sƒƒ w. 2. k p -k Fig. 1 (a) gj p ¾ (deep beam) w p -k., ƒ p k y w» w k x w š w w. Fig. 1 (b). ƒ p k w k ù w (secant stiffness) k s w w Fig. 1 Design concept of direct inelastic strut-and-tie model w V u w xw ww, w ƒ p k (performance point, Fig. 1 (b) P, P S, P T ), k x w. w ƒ p k w w gj p w. w ƒ m w gj p p k x w m xw w w, Fig. 1 (b) w w xw y w k x w ùkù (Fig. 1 (c)).» xw w, w w xw mw k ww š. w k p -k xw w k ww r w w. ù wš w ƒ gj p p k k x» ³ w ù ƒ w x ù w., p gj p w. k ƒ w.» w w ù. w p k k x x ù, w w ù j» w w w w. 3. w ƒ» w ww» w p k j» ƒ w w. e ƒ ƒ ƒ w. w w š w k ww» w, w» ƒ w v w. k x w, w q š w w k w w. Fig. 2 ù j ¾, (coupling beam), (shear wall) gj p k w p -k. ùkù ƒ ¼ w, ƒ w, ƒ w p (compression strut, Fig. 2 ) k (tension tie) w. ƒ w k ƒ p 346 w gj pwz 20«3y (2008)

3 Fig. 2 Strut-and-tie model for shear-dominated RC members x x w k k (inelastic tension tie, Fig. 2 ) k w k k (elastic tension tie, Fig. 2 ƒ ) w w. q w» w w. 1) gj p p q ¼ w { w w. 2) { w z w w. w w f ù ƒ ƒ k w w. jš» w wù, ƒ w w f j» w f ü. 3.1 ¼ w q { w w, { { p w w. ¼ w gj p p k k k w { w ƒ w. 5) ¼ w d k d gj p p. { w ¼ w k T w. T = M u ( jd )» T = k w, M u = { (1) p -k w gj p k 347

4 p (= V u l s ), V u =, l s = g j p ¼, ( jd )= pq ¼. Fig. 2 (a)~(c) ùkù { p w ww ¼ w k w k k w. k x sƒwš p -k w ƒ š w w, ¼ w k k w k s w. ³ gj p š w, k k (4) f. ù 14) w j w e, r ³ gj p ƒ z š w. ¾ (Fig. 2 (a)) (Fig. 2 (c)) p -k ¼ w p j» { M u w j» w. ƒ (equivalent stress block) w A c w. k s T = = tu M u ( jd)ε u l t (2) A c = M u ( jd)f ce (5)» tu = k k x, ε u = ¼ w k k x, l t = k ¼. k k x ε u gj p q q» w w. ù w k x w, xk, w w w w», p. e x š w ε u = 0.002~0.04 ƒ w w, 4 ƒ gj p p q ƒ w ε u k. ü w x, xƒ k x» w. θ pu ( jd) ε u = ε y l t» θ tu = xƒ, ε y = w x. θ pu w ù, FEMA ), 356» ü 13) e x šw w. { p (¾ (Fig. 2 (a)), (Fig. 2 (b), (c)) e ¼ w k k w. k k k w w {w z x { p w ù { k k». k k k k e w. k e T = = ty M u ( jd)ε y l t» ty = k w x. (3) (4)» f ce =gj p z (= 0.85f ck ), f ck = gj p. gj p k ùkü. ù x w gj p x j»ƒ x, k x gj p p x yƒ e w. r w ¼ w gj p p k k e k w. k e E c A c = = l e E c M u l e ( jd)f ce» E c =gj p k f ck ƒ 30 MPa j 3300 f ck (MPa), f ck ƒ 30 MPa w 4700 f ck (MPa) w. 1,3) l c = p ¼. Fig. 2 ùkù gj p ¼ w A' s. e ƒ ewš w š w p w. ù gj p» q ƒ ù, w š w š (6) w w ƒ j. (1)~(6) (jd) gj p ƒ b w ùký. M u a ( jd) = d -- = d ( jd)f ce b (6) (7a)» a = ƒ ¾, b =, d = z¾. (7a) (jd) w 2 w, l ¾ w p -k (jd). 348 w gj pwz 20«3y (2008)

5 ( jd) = 2M d d 2 u f ce b (7b) ¾, (l s / h, h = ¾, Fig. 2)ƒ 1.0 w ¼ w gj p p k w w. ƒ 1.0 w ¼ w w x». 15), ƒ p w, ¼ w Áw. ƒ ¼ w k w (Fig. 2 (b)). ƒ j { p d k k, ù ¼ w k k ƒƒ w. k k w k k k ƒ ƒ (2) (4) w. (jd) (Fig. 2(b)). ( jd)=d c' (7c)» c' = gj p v Ì. ƒ 1.0 j e ¼ w. w w xw ¼ w ƒ, gj p p w p -k w w. 3.2 ƒ w ƒ w w k w. gj p w ³ ƒ gj p p w» q ƒ. w ƒ w gj p p j» w ƒ w k x x w w. ƒ w k k w. k e T v = = ty sv u dε y l t» T v = k ƒ w (= sv u / d), s = ƒ w k k (Fig. 2). Fig. 2 (a) ùkù ƒ ƒ gj p p w. w ƒ w k w j»ƒ (8) T v. ƒ w k k (8) w (8) Fig. 3 Modeling of diagonal concrete strut w, w k yƒ j w e. j x p ü»w w gj p w w p -k w w w. 1) 3.3 ƒ w Fig. 3 ƒ w gj p p. ùkù ƒ w gj p p p e xkƒ w sw x, ƒx Õ xk. ƒ gj p p Paulay, 15) Kim, 9) Park and Eom 16) s³ A c w (Fig. 3). b w ( w 1 + w 2 ) DPT θ A c = » b w =, w 1, w 2 = gj p p s, θ = ƒ w g j p p ƒ 31 o ~59 w 2). o ¼ w gj p p ƒ, ƒ w gj p p k x w, k k c w. k e E c A c b ( w + w w 1 2 ) DPT θ = = l c E c l c 4. q» (9) (10) w y k p -k. 1) gj p w w ( V u, { p M u ) w. 2) ¼ w, ƒ w, ƒ w p w wš w gj p p -k w, w w (Fig. 2 ). p -k w gj p k 349

6 3) ƒ p k k k e w k s w. e ƒ w ù, z š w k w 3 k w w. 4) w w w xw w w ƒ p k x w. 5) ƒ p k mw. ¼ w p ACI318-05, 1) CEB-FIP» gj p» 2) mw. ¼ w k ƒ w gj p p q q» w m w. p, k, w w w w, p k w ù w w. 6) k w w w. ¼ w ƒ w» gj p» 1-3) ww ³ w w. gj p,, ¾ w x 8,17-20) w w, w w gj p q ƒ gj p p, q, gj p ƒ ³ q w. gj p ƒ ³ q w w. ù q ¼ w k q ƒ w gj p p mw. Wood 8),»w gj p { q { (flexural stress index)ƒ 0.15 w š k x q x ε fr w w. gj p { (11). ρ Flexural stress index = l f y + P/A g (11)» ρ l = w ¼ w 8). Wood x ε fr =0.04 w. ƒ p - 2 w gj p p ƒ w ³ w gj p p ƒ w. 6,7) 6)ƒ Vecchio and Collins w w 1 f ck ƒ w gj p p z f ce p ƒ w x ε t w w. f ce = f ck f ck f ck 0.34ε t /ε co (12) Fig. 4 Evaluation of transverse strain of diagonal concrete strut» ε co = gj p w w x, ε t = p ƒ w x. ε t ƒ w g j p p ƒ (virtual plane element) x x w (straincompatibility) w sƒw (Fig. 4). 16) ε c + = ε 1 + ε 2 ε t (13)» ε c = gj p p x, ε 1, ε 2 =ƒ ¼ w ƒ w s³ x. Fig. 4 ùkü ε 1 ε 2 ƒ x w w. ww w p w (12) (13) w w ƒ w gj p p w z f ce j ƒ p w q ƒ. ƒ w gj p p z w» w p s³ s (Fig. 3 (9)) p sww s³ x (Fig. 4) w. 1) ACI gj p m ww e ƒ š ƒ p s w w. q ù ƒ p q ƒ ù, k w ƒ j ù ( ƒ k k x ε u ) ƒ g w w. 5. 3~4 y k p -k (S-DISTM) ƒ ¾, 17), 18) 20) gj p k ww. 350 w gj pwz 20«3y (2008)

7 x {w z k x ƒ gj p q x. S-DISTM w k 3 w w., w ³ j» w ƒ w k k k w. w { w w gj p w» w { pƒ w ¼ w k k k w. p k w š w (φ =1.0). 5.1 ¾ (deep beam) ¾ Foster and Gilbert 17)ƒ xw B2.0-3 x. Fig. 5 (a) w ¾ x x, Fig. 5 (b) p -k S-DISTM w p -k. Fig. 5 (b) p -k ƒ gj p p ƒx p ƒ w x w (Fig. 4 ). ƒ w gj p p w sƒw» w, ƒ (dummy element, Fig. 5 (b) ƒ ) ƒw. ƒ Fig. 5 Design example for deep beam (B2.0-3) p -k w gj p k 351

8 p ƒ w x w» w, w e. x w w w w { p ƒƒ V u = 700 kn, M u = 578 knm. gj p f ck =78MPa š, { w ƒƒ f y = 440, 590 MPa w. p -k w sx ƒ ƒ w, Fig. 5 (c) ùkù. S-DISTM w p -k k 3 4 ww. ƒ p q w e ƒ w x ε t ƒ w k ¼ w k w. w ƒ p q ƒ w» w, ƒ w k k ƒ w. ¼ w k k x x w. x l j w ¾ j» w». k k k x ε u =0.01 ƒ w w. ù m q ƒ k x ε u = w. ƒ w gj p p (Fig. 5 (b) ) w q (12) (13) w mw. S-DISTM w p -k k Fig. 5 (c) (d) ùkþ. w p -k w ¾ wì ùkþ. p -k p f (truss mechanism) ƒ w k (Fig. 5 (b)) j 700 kn w, ρ v =2.3% e w ùkû. p -k e f (arch mechanism) p f w w, ƒ f w j» x w (displacement compatibility) sx (force equilibrium) w. S-DISTM p k ƒ w e (ρ v = 0.83%) j w. Fig. 5 (e) S-DISTM p -k k k ƒ w gj p p - x. k k x ε u =0.01 ƒ w, gj p p z ƒ f ce = 53.3 MPa p w 55.3 MPa q ƒ. k x ε u = w, p w q ƒ w. w x p -k w. Fig. 5 (a) (d) ¾ S-DISTM w. S-DISTM w p -k x w, p -k ƒ w w. wr S-DISTM x d., ¼ w ρ l = 2.69% ƒ w ρ v = 0.83%ƒ x ¼ w ρ l = 2.15% ƒ w ρ v = 0.67% j. x ³ ¼ w 6D6 (f y =590MPa) š w š (Fig. 5 (a)), w wù ywš q w e f w sƒw». Fig. 5 (f) x S-DISTM w - ùkü. ε u = w ùkþ. ε u = w w 6.9 mm x w x 8.8 mm d sƒ. ε u =0.01 w S-DISTM k 9.9 mm ƒ w p q ƒ d, x 8.8 mm q ƒ x ew. 5.2 (coupling beam) (coupled wall) w j k x w z. p, w yw k x s ƒƒ. 18)ƒ Galano and Vignoli xw w P01 x»w P02 x. x p w. Fig. 6 (a) x x,, j», Fig. 6 (b) x S-DISTM w p -k ƒ ƒ ùkþ. ƒ 0.75 P01 ¼ w w x, 15) Fig. 6 (b) ùkù ¼ w k w. w., Fig. 6 (b) ùk ù d ƒ ¼ w ƒ w x z x { pƒ 0 w. gj p w ƒƒ f ck =48.9MPa, f y =567MPa w. w { p ƒƒ V u =224kN, M u =67.2kNm x w P01 x w w w. x ¼ w k (Fig. 6 (b)) w w 352 w gj pwz 20«3y (2008)

9 Fig. 6 Design example for coupling beam (P01 and P02) 13) w. FEMA 356 x w xƒ j» θ pu = 0.025rad, (3) w x xƒ w k k x ε u =0.04 w. w xƒ w θ pu =0.04rad w k x ε u =0.063 w S-DISTM k ww. Fig. 6 (c) S-DISTM w ƒ p k ü ƒ k x j» w š. Fig. 6 (d) S-DISTM. Fig. 6 (c) (d) ùkù k k k x w { y, d k x 17.6 mm 26.8 mm j ƒ. Fig. 6 (e) k k ƒ gj p p - x ùkþ. k k k x ƒƒ ε u = 0.04 (θ pu = rad) (θ pu =0.04rad) w ƒ gj p p (8.1 MPa) (12) (13) w gj p z f ce. ƒ gj p p w q xƒ θ pu = rad w ùkû. w, Fig. 6 (a) ùkù P01 x S-DISTM w. S-DISTM ƒ w w, ¼ w (ρ l = 2.64%) x (ρ l = 2.10%) j. S-DISTM e w š w š, w z ƒ w, x w ¼ w š w p -k w gj p k 353

10 ». Fig. 6 (f) S-DISTM x w P01 ( w) w - x. ù kù FEMA 356 xƒ (0.025rad) w k k x ε u =0.04 ƒ w x w q x j e w 17.6 mm k x d. FEMA 356 xƒ d». k x ε u = ƒ w ƒ w x P01 x 30 mm 26.8 mm k x. p Fig. 6 (f) ùkù ƒ g j p p w (8.1 MPa) k w z (f ce =8.7MPa)ƒ, x 30 mm q w w w w x q ew. Fig. 6 (f)»w x P02 w - x wì, P02 ƒ w x P01 x 30 mm 16 mm» w wƒ. w»w gj p ùkù x - x w 21). Lee and Watanabe»w gj p x w w x (elongation), w gj p q ƒ» šw.»w x d sƒw» w ƒ gj p p z d sƒw w, ƒ gj p p z w e k x d w v ƒ š. 5.3 (shear wall) S-DISTM w 20) Sittipunt et al. xw»w x W1 ( = 1.45) w k ww. x {w z w. Fig. 7 (a) x W1 x,, j», W1» x xk. ƒ» 6D16 2D12 ¼ w e (ρ l = 1.28%, f y = 473 MPa), s ƒƒ (f y =450MPa) e. x k w p -k Fig. 7 (b) ùkþ. w w w ƒ w. gj p f ce = 36.6 MPa w. x w Fig. 7 Design example for shear wall (W1) 354 w gj pwz 20«3y (2008)

11 w w V u = 491 kn. k (Fig. 6 (b)) 13) w w k x, FEMA 356 x xƒ 0.01rad w k x ε u = (3) l w. W1 { ƒ 0.15 j š 8)ƒ w ¼ w k x Wood w q x ε fr (= 0.04) d, { q. Fig. 7 (c) S-DISTM k w ƒ p k š, Fig. 7 (d) w ew. gj p p q w mw q ƒ ùkû, Fig. 7 (c) (d) zw. w, S-DISTM x w. Fig. 7 (a) (d) ƒƒ x e S-DISTM. S-DISTM w x e w ù, S-DISTM w ¼ w ƒ w ƒƒ 17%, 14%. S-DISTM ³ w e w w». S-DISTM v w ù,». Fig. 7 (f) W1 x w»w x S- DISTM w w -. ù kù»w w x W1 g j p q w z x 37 mm w w w. k x ε u = (θ pu = 0.01rad) ƒ w S-DISTM w k x 31 mm x d d. 6. p -k w xw k x w y k p -k (Simplified direct inelastic strut-and-tie model, S-DISTM) w. S-DISTM gj p k gj p p, k k k w, ƒ k w x w. ƒ k ww ¾,, ƒ p w p ƒ w w w w. w ƒ q sƒw» w gj p, q q» š w. ¾,, {w z k x w w, x w., k k x, w w k k x w w. y k p -k w. 1) p -k w xw w r w k ww. 2) k x w wš k xw q mw, ü sƒ. 3) ƒ w, q, gj p p w. wz y w» w w ƒ p q w» w ƒ v w. š x 1. American Concrete Institute (ACI), Building Code Requirements for Structural Concrete and Commentary, ACI , ACI 318R-05, Farmington Hills, Mich., Comite Euro-International du Beton/Federation Internationale de la Precontrainte (CEB-FIP), CEB-FIP Model Code 1990: Design Code, Thomas Telford, London, 1993, 437 pp. 3. w wz,» (KBC), m, 2005, 597 pp. 4. Schlaich, J., Schafer, K., and Jennewein, M., Toward a Consistent Design of Structural Concrete, PCI Journal, Vol. 32, No. 3, 1987, pp. 74~ American Concrete Institute, Examples for the Design of Structural Concrete with Strut-and-Tie Models, SP-208, Farmington Hills, 2002, 242 pp. 6. Vecchio, F. and Collins, M. P., The Modified Compression- Field Theory for Reinforced Concrete elements Subjected to Shear, ACI Structural Journal, Vol. 83, No. 2, 1986, pp. 219~ Hsu, T. T. C., Toward a Unified Nomenclature for Reinforced Concrete Theory, Journal of Structural Engineering, ASCE, Vol. 122, No. 3, 1996, pp. 275~ Wood, S. L.. Minimum Tensile Reinforcement Requirements in Walls, ACI Structural Journal, Vol. 86, No. 4, 1990, pp. 582~ Kim, J., Seismic Evaluation of Shear-Critical Reinforced Concrete Columns and Their Connections, Ph.D. Dissertation, Univ. of New York at Buffalo, 1996, 391 pp. 10. Yun, Y., Nonlinear Strut-Tie Model Approach for Structural Concrete, ACI Structural Journal, Vol. 97, No. 4, 2000, pp. 581~ y, ½ š, k, w w k p -k, gj pwz, 17«, 2y, p -k w gj p k 355

12 2005, pp. 201~ American Society of Civil Engineers (ASCE), NEHRP Guidelines for the Seismic Rehabilitation of Buildings, FEMA 273, Federal Emergency Management Council, Washington, DC, 1997, 369 pp. 13. American Society of Civil Engineers (ASCE), Prestandard and Commentary for the Seismic Rehabilitation of Buildings, FEMA 356, Federal Emergency Management Council, Washington, DC, 2000, 429 pp. 14. Eom, T., Design-Oriented Inelastic Analysis for Earthquake Design of Reinforced Concrete Structures, PhD dissertation, Seoul National University, Seoul, February 2006, 326 pp. 15. Paulay, T., Coupling Beams of Reinforced Concrete Shear Walls, Journal of the Structural Division, ASCE, Vol. 97, No. 3, 1971, pp. 843~ Park, H. G. and Eom, T. S., Nonlinear Analysis of Reinforced Concrete Members using Truss Model, Journal of Structural Engineering, ASCE, Vol. 133, No. 10, 2007, pp. 1351~ Foster, S. J. and Gilbert, R. I., Experimental Studies on High-Strength Concrete Deep Beams, ACI Structural Journal, Vol. 95, No. 4, 1998, pp. 382~ Galano, L. and Vignoli, A., Seismic Behavior of Short Coupling Beams with Different Reinforcing-bar Layouts, ACI Structural Journal, Vol. 97, No. 6, 2000, pp. 876~ Oesterle, R. G. et al., Earthquake-resistant Structural Walls. Tests of Isolated Walls, Report to the National Science Foundation, Construction Technology Laboratories, Portland Cement Association, Skokie, Ill., 1976, 315pp. 20. Sittipunt, C., Wood, L. S., Lukkunaprasit, P., and Pattararattanakul, P., Cyclic Behavior of Reinforced Concrete Structural Walls with Diagonal Web Reinforcement, ACI Structural Journal, Vol. 98, No. 4, 2001, pp. 554~ Lee, J. and Watanabe, F., Shear Deterioration of Reinforced Concrete Beams Subjected to Reversed Cyclic Loading, ACI Structural Journal, Vol. 100, No. 4, 2003, pp. 480~489. w w w xw ww r w k w k p -k.» k p -k w, w w w xw gj p k ww y k p -k (simplified direct inelastic strut-and-tie model, w S-DISTM) w. S-DISTM gj p gj p p k w. p k k w x w. p k q» w» w gj p q q š w. S-DISTM w ¾,, w gj p k ww š, k, x» x w. w : p -k, w, k, x, gj p 356 w gj pwz 20«3y (2008)

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