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1 Journal of the Korea Concrete Institute Vol. 24, No. 1, pp. 015~023, February, 2012 GGGGG 인장증강효과에기반한콘크리트구조부재의사용성능검증» 1) Á½ 2) Á½ 2) Á y 3) * 1) w m 2) û w m œw 3) w w ful vp œw Serviceability Verification Based on Tension Stiffening Effect in Structural Concrete Members Gi-Yeol Lee, 1) Min-Joong Kim, 2) Woo Kim, 2) and Hwa-Min Lee 3) * 1) Dept. of Civil Engineering, Suncheon First College, Suncheon , Korea 2) Dept. of Civil Engineering, Chonnam National University, Gwangju , Korea 3) Dept. of Computer Software Engineering, Soonchunhyang University, Asan , Korea ABSTRACT This paper is about proposal of a calculation method and development of an analytical program for predicting crack width and deflection in structural concrete members. The proposed method numerically calculate stresses in steel rebar using a parabola-rectangle stress-strain curve and a modified tension stiffening factor considering the effect of the cover thickness. Based on the study results, a calculation method to predict crack width and deflection in reinforced concrete flexural members is proposed utilizing effective tension area and idealized tension chord as well as effective moment-curvature relationship considering tension stiffening effect. The calculation method was applied to the test specimens available in literatures. The study results showed that the crack width and deflections predicted by the proposed method were closed to the experimentally measured data compared the current design code provisions. Keyword : crack width, curvature, deflection, parabola-rectangle stress-strain curve, tension stiffening effect 1. gj p { m w w, ³ s, v x w w ü w dwš w w. gj p œ» 1) w š xy w», x w ³ s w» w d x w yw w y š. w k» w CEB-FIP Model Code ) (MC 90) EUROCODE 2 3) (EC 2) w ³ s z w w.» ³ w z 4) v Ì y ³ w sƒw wš. *Corresponding author leehm@sch.ac.kr Received August 17, 2011, Revised November 20, 2011, Accepted November 23, by Korea Concrete Institute z j» w w ƒ. MC 90 EC 2 gj p» - 5)(KCI) w gj p - x x ƒ w k w. ù ww k ³ w w» ƒ. 6,7) gj p - x w EC 2 w» yw w w. š, w ³ s wš, MC 90 EC 2 ³ w z w w. gj p w» w ƒ» ³ w ³ s w w. w x w x swwš» w. w w w z y sƒw» w 4) w v Ì y w. š yw w» w gj p - x s - ƒx(parabola- 15

2 rectangule) š ( w p-r š ) w. w Ì p-r š w w ³ s wš, ww v w. l ƒ» w ³ sƒwš, z w k y w. 2.» ³ 2.1 ³ s w ³ s Fig. 1(a) gj p x k. ³ s Fig. 1(b) ¼ (transfer length) l t ü Fig. 1(c) w ³ l ¼ w w x ε sx gj p x x ƒ ³ w. ( ) dx l t w 2 ε sx s MC 90, EC 2 KCI» (1)» w Fig. 1(d) s³ x w ³ s ³ w., ³ s³ x ε sm gj p s³ x m w x š w s ¼ l ³ s r, max (1) w ³ s w k w k w ³ w., ε sm m s s r max ( ) (2) w gj p s³ x 4) w MC 90 s³ x ( z ) w tx. ε sm m f so ( 1 + nρ β ef )f ct E t s E s ρ ef», f so ³ (MPa), f ct g j p s³ (MPa), E s k (MPa), n k. β t p w, KCI MC 90 ³ y» w 0.6,»w w 0.38 w. EC 2 Part I(general rules and rules for building) w xk»w 0.6,»w 0.4 w, Part II(concrete bridges) w xk 0.4 w. (3) Fig. 2(a)-(c) ³ w { Fig. 2(d) x (tension chord) yw w w. ¾ w { x w» w Fig. 2(d) sww x w Fig. 3 z¾ d ef t w z A c, ef w w. z ρ ef g j p z w w. (2) (3) ρ ef A s A c, ef A s d ef b (4)», A z ü (mm 2 s ). ³ ³ y» KCI, MC Fig. 1 Distribution of strain and bond stress in tension members (a) tension members (b) crack formation stage (c) bond stress at crack formation stage (d) stabilized cracking stage Fig. 2 Cracked RC flexural member and idealized chord model 16 w gj pwz 24«1y (2012)

3 Fig. 4 Strain distribution and deflection curve Fig EC 2-Part II (5a), EC 2-Part I (5b) w ρ ef d b s r, max d, 3.4c e 1 e b s r max Effective tension area, (a) beams dominantly subjected to flexure (b) slabs subjected to flexure (c) tension members ρ ef (5a) (5b)», d b (mm), c v Ì(mm), e 1 p w x 0.8, x 1.6 š, e 2 x s { 0.5, 1.0. JSCE Standard specifications for concrete structures Structural Performance Verification 8) (JSCE 02)» ³ s d w {³ s(flexural crack width) ³ w. f so [ ( )] E s w 1.1 j 1 j 2 j 3 4c c s d b sd», j 1 t x 1.0, x 1.3 š, j 2 (7a) gj p, j 3 (7b) ed(layer) w. w, c v Ì (mm), c s (mm), sd j v w x , š gj p ³ w. 15 j f ck + 20 (6) (7a) ( + ) 5 n l 2 j n l + 8 z» w gj p (7b)», f ck» (MPa), n l ed. xk 9) Gergly-Lutz x ³ s w m w ³ s dw w, rw x ¾ š. 2.2 { gj p w p-š w, š ¼ ƒ y w w. š ³ j»ƒ yw ³ p M cr w ƒ p M a w M a < M cr ³ M a M cr ³ w w. Fig. 4(a) ³ š ( 1 r ) I k w w w yw ƒ w. Fig. 4(b) ³ ³ x x j ƒ ³ x y ³ p w w. ³ y š ( 1 r ) II g j p z š w x w w. l ³ š r 1 -- r I II ε so d c 1 : uncracked section (8a) ε so ε ts d c 2 d : cracked section (8b)», d z¾ (mm), c 1 c 2 ³ ¾ (mm), ε so x, ε ts z w x. (8) š w Fig. 4(c) ƒ y dθ w š, ƒ w w. 17

4 xd θ x 1 -- r, { p ³ w ³ w. w š w w w. w MC 90 EC 2 zš ³ w r e 1 r ( ) e ξ 1 -- r II dx (9) w rw w + ( 1 ξ 1 ) -- r I (10)» ξ z sww ³ w s. ξ 1 β f scr k f so (11)», β s³ x w w xk MC 90»w 0.8,»w w 0.5, EC 2»w 1.0,»w 0.5. k MC 90 EC (10) w zš w. ηl r e (12)», η w xk { p xk 10), L z ¼ (mm). KCI JSCE 02 (10) zš 11) Branson w z 2 p w w. ³ 2 p, I crack ³ 2 p. α KCI 3.0, JSCE (13) w z 2 pƒ w. M ηl a E c I e (14)» E c gj p k. (13) w ³ p l { w, p-š w š rw w. w r KCI JSCE 02 m wš, x w z 2 p p, w w» w ù ƒ w w. 12) 3. š w w ³ s z wì w w w š w.»» w w gj p - x w { w - x k w w x - x wš. { w w w gj p w ƒ w w. w MC 90 EC 2 gj p - x w Fig. 5 s - ƒx - x š (parabolarectangle stress-strain curve, p-r š ) w w. I e α M cr M a M cr I uncrack I crack M a (13)», M a p, M cr ³ p, I uncrack α f c f cd for 0 2 f c n ε c 2 f cd for 2 u2 (15a) (15b) Fig. 5 Parabola-rectangle stress-strain curve for concrete 18 w gj pwz 24«1y (2012)

5 Fig. 6 Distribution of stress and strain for single RC beam», 2 x, u2 ww x, n š x w, Fig. 5 t. f cd gj p (MPa), w. f cd α cc f ck γ c (16)», α cc» w š w» w z 0.85, γ c gj p w k 1.0, ww k 1.5. w 1.0 w. gj p p-r š w ƒx { w» w x s Fig. 6 ww. Fig. 6(a) { p M w w w j». C (17a) (17b)», ρ A { š, s bd γ s w k 1.0, w w k š α w w v w, s³ f cd w. Fig. 5 w p-r š w - x (15) w α d gj p x w w w. ( 6 ) α for 0 ε 12 c α for ε 3ε c2 u2 c», αf cd bc ( ( )) T A s f s γ s ρf s bd γ s (18a) (18b) gj p x ( ) txw. (17) w p q¼ z. z d βc ( 1 βk )d z» w gj p (19)», k c d z¾ w ¾. š β p-r š s w e w, l d w ¾ ¾ w. (15) w - x (18) w α w gj p x w w w. ( 8 ) β for 0 ε 46 ( 2 c ) ( 3 4 ) + 2 β for ε 2 ( 3 2 c 2 u2 ) (20a) (20b) βc w C e l ƒ, { p. M Cz αf cd y( 1 βy)bd 2 M Tz f s A s ( 1 βy)d γ s (21a) (21b) (21b) w w» w v w y { p j» w» w» w sx x w w ¾ w w. rw w w w { p j»(intensity) ùkü m w. m M f cd bd 2 (22) m (21a) w w y w w. y 1 1 4mβ α β (23) l w» w x ƒ w, (18) (20) w α β w. (23) w ¾ y w z, ƒ w wì Fig. 6(b) x s»w w. 1 y f s E s ε s y f y γ s --- (24) 19

6 Fig. 8 Crack width calculation program Fig. 7 Algorithm of steel stress calculation (24) (21b) { p w w w ƒ w x w { p w w. f s γ s M A s ( 1 βy)d (25), (24) (25) w ƒ w, w ¾ x y j w w w w. w w w w w. w w» w Fig. 7 š w w ³ s v w ³ s gj p w ³ s w w gj p p p-r š w w 4) w w., ³ s w (2) ³ MC 90 w (4a), gj p x (3) r 4) (5) w. wì w w. š y y w Fig. 8 ³ s v w. v w š w» ³ s Gergly- Lutz w ƒ w w y w. w ³ s ƒ» ³ s y y w» w Bilal, 13), š 15) ww { x 14) w Fig. 9 w. ³ s ƒ» ³ w w, x Fig. 3 (5) w z w. š» w x k w w w. Fig. 9 r, w w w ³ s x sƒw ³ y» ³ x yw dw. ù EC 2, JSCE 02 Gergly Lutz ³ s sƒwš, MC 90 x sƒw ùkû.» p»w w MC 90 EC 2-Part 1 ³ w ƒ w, ³ s w gj p s³ x (3), ³ w» q., MC 90 z, EC 2-Part 1 v Ì ƒ s ww ³ w» ³ j sƒ. 20 w gj pwz 24«1y (2012)

7 Fig. 9 Comparison of proposed crack width calculation method vs various design codes Table 1 Statistical analysis of crack width calculation results MC 90 EC 2 (Part 1) JSCE 02 Gergly & Lutz Proposed model Mean Var St. dev CV w» ³ y q w» w x w m w Table 1 w. m r,» ³ w d y ƒ yw y w. ù MC 90 EC 2-Part 1 yw y w, JSCE 02 Gergly Lutz y ƒ w ùk û. ww, gj p x p w v Ì w w z w ³ s yw w q. 4.2 w ³ s Fig. 10 Deflection calculation program w w., w (12) (11) w zš (10) w. w k 4) w (7) w. š, w w y w Fig. 10 v w. v w š w» ƒ w w y w. r 4) w w w» ³ w» w Alwis, 16) z» w gj p 21

8 Fig. 11 Comparison of proposed deflection calculation method vs various design codes Table 2 Statistical analysis of deflection calcualtion results MC 90 EC 2 JSCE 02 KCI Proposed model Mean Var St. dev CV Shah 17) Tan 18) ww x w Fig. 11 w. x gj p f ck,»ww, w xk - z¾ a dƒ x w. Fig. 11 r, w dw w w w w 1.5 M a M cr 3 19)» ³ w x yw dwš y w. w w 3M cr z w d y y š. MC 90, EC 2, KCI JSCE 02» x sƒw ùkû. xw» w xk w w ƒ û sƒ» q. wì zš j» w (11) s ξƒ v Ì w w w wš» q. w ƒ» ³ y» w x w m wš Table 2 w. m r, w EC 2» ³ w d y ƒ yw y w. ù MC 90 y w y w, z 2 p w KCI» w w y w. JSCE 02» w y yw, ƒ j ùkû. ww, ³ s w g j p x p w v Ì w w z w yw w q. 5. gj p w ³ s yw w w. w gj p x p w š wš, w v Ì w w z w., gj p s³ x w zš s ƒƒ w. w Ì, gj p p š w w» ³ w ³ s w v w. v w ³ s w x w,» ³ d y ƒ w ùkû ( w» ) w» w ( y : ).. 22 w gj pwz 24«1y (2012)

9 š x 1. Kelvin, F. and Peter, H. B., Tension Stiffening and Cracking of High-Strength Reinforced Concrete Tension Members, ACI Structural Journal, Vol. 101, No. 4, 2004, pp. 447~ CEB-FIP, CEB-FIP Model Code 1990, Comite Euro-International Du Beton, Paris, 1991, pp. 247~ European Committee for Standardization, Eurocode 2- Design of Concrete Structures, European Committee for Standardization, Brussels, 2002, pp. 124~ », ½, ½, y, v Ì š w gj p z, gj pwz, 23«, 6y, 2011, pp. 791~ mw, gj p», w gj pwz, 2007, pp. 340~ , ½, gj p - x» w gj p, wm wz, 30 «, 4Ay, 2010, pp. 383~ , y, ½, gj p ³ s w z w, gj pwz, 22«, 6y, 2010, pp. 761~ JSCE, Standard Specifications for Concrete Structures- 2002, Structural Performance Verification, Japan Society of Civil Engineers, Tokyo, 2005, pp. 107~ Gergely, P. and Lutz, L. A., Maximum Crack Width in Renforced Concrete Flexural Members, Causes, Mechanism, and Control of Cracking in Concrete ACI Special Publication SP-20, ACI, Michigan, 1968, pp. 87~ Eibl, J., Concrete Structures Euro-Design Handbook, Ernst & Sohn, 1994, pp. 244~ Branson, D. E., Deformation of Concrete Structures, McGraw-Hill, Newyork, 1977, 546 pp. 12., gj p p š w { p-š w, gj pwz, 3«, 4y, 1991, pp. 97~ Bilal, S. H. and Mohamad, H. H., Effect of Fiber Reinforcement on Bond Strength of Tension Lap Splices in High-Strength Concrete, ACI Structural Journal, Vol. 98, No. 5, 2001, pp. 638~ , y, gj p { ³ s ³, wm wz, 5«, 4y, 1985, pp. 103~ š, ³, p š w gj p { ³ s, wm wz, 22«, 4-Ay, 2002, pp. 825~ Alwis, W. A. M., Trilinear Moment-Curvature Relationship for Reinforced Concrete Beams, ACI Structural Journal, Vol. 87, No. 3, 1990, pp. 276~ Shah, S. P. and Shrikrishna, M. K., Response of Reinforced Concrete Beams at High Strain Rates, ACI Structural Journal, Vol. 95, No. 6, 1998, pp. 705~ Tan, K. H., Paramasivam, P., and Tan, K. C., Instantaneous and Long-Term Deflections of Steel Fiber Reinforced Concrete Beams, ACI Structural Journal, Vol. 91, No. 4, 1994, pp. 384~ ½, ½, y,,, gj p, y», 2007, pp. 260~261. gj p w ³ s w wš w v w. w gj p p s - ƒx - x š w v Ì w w w š w e w. wì ³ s w v w z z ³ w { x yw w. š w zš w. ³ s w ww x w, xw» ³ x yw dw ùkû. w : ³ s, š,, s - ƒx - x š, z z» w gj p 23