Composite Strut and Tie Model for Reinforced Concrete Deep Beams

Transcription

1 Journl of Advned onrete Tehnology Vol. 7, No., 97-9, Februry 9 / opyright 9 Jpn onrete Institute 97 Sientifi pper omposite Strut nd Tie Model for Reinfored onrete Deep Bems Kil-Hee Kim, Woo-Bum Kim, Jin-Mn Kim nd Sng-Woo Kim Reeived Otober 8, epted Jnury 9 Abstrt Monotoni sher loding tests were onduted on three hlf-sled reinfored onrete deep bems with sher spn-todepth rtios of.5 to.75. The obtined test results were investigted in detil bsed on the experimentl mesurements nd finite element nlysis. From these investigtions, new mro model for deep bems ws estblished. This model is omposed of two rooked min struts formed between both bem end setions nd brnhed-off sub struts. The ompressive fore introdued to min struts blnes the flexurl ompression nd the externl sher fore. The bond stress of the longitudinl reinforement nd the tensile fore of the stirrup produe the digonl ompression in the sub strut. Theoretilly predited sher strengths of tested deep bems showed good greement with experimentlly observed sher strengths, where the effetive strength of onrete ws ssumed to be 75% of the ylinder strength.. Introdution Deep bems re widely used s perimeter bems of reinfored onrete (R) frmes for funtionl nd rhiteturl resons. In multi-story wll strutures, deep bems re lso used for oupling wlls onstruted side by side. Beuse the bems hve smll spn-to-depth rtios, brittle filure in sher my be observed under seismi lterl lods for some of them. Exmples of suh filure due to erthqukes hve been reported in the literture (Kim et l. ; Puly 98; Puly nd Binney 974). To void suh filure, suffiient mount of trnsverse reinforement should be provided to indue dutile flexurl filure prior to sher filure. An lterntive method is the rrngement of digonl reinforement in bem, whih ws proposed by Prk nd Puly (975). When deep bem is subjeted to ombined bending nd sher with nti-symmetri moment distribution, the truss mehnism is hrdly developed due to the overlp of tension shift regions of longitudinl reinforement. Tht is, the min resisting mehnism is developed by single strut formed between both end setions of the member. Thus the finl sher filure my our due to rushing of digonlly ompressed onrete struts. Bsed on this bsi mehnism, severl sher design equtions for deep bems hve been proposed (AI 8; AIJ 99, 999; EB-FIP 995; Nielsen 984). However the role of the trnsverse reinforement is not neessrily ler. Assistnt Professor, Deprtment of Arhiteturl Engineering, Kongju Ntionl University, South Kore. E-mil:kimkh@kongju..kr Professor, Deprtment of Arhiteturl Engineering, Kongju Ntionl University, South Kore. Reserh Assistnt Professor, Deprtment of Arhiteturl Engineering, Kongju Ntionl University, South Kore. This pper presents new mro model to predit the sher strength of deep bems with nti-symmetri moment distribution. Sher loding tests nd nonliner finite element (FE) nlysis for deep bems were onduted to derive the mehnil sher resisting model. The proposed model is omposed of two rooked min struts nd brnhed off sub struts, where the ontribution of the trnsverse reinforement nd the bond tion of the longitudinl reinforement re tken into ount.. Experimentl progrm Three speimens with vrious sher spn-to-depth rtios nd mounts of trnsverse reinforement, s indited in Tble, were st nd tested to verify the ury of the proposed model. The detils of typil speimen S re shown in Fig.. Eh speimen hs entrl test region with mm setionl dimension nd loding stubs t both ends of the speimen. The ler spns of S-.5-5, S nd S were, 45 nd 45 mm, respetively. The speimens were designed to fil in sher before yielding of the flexurl reinforement. All the speimens ontined four deformed brs with nominl dimeter of 6 mm nd yield strength of 795 MP s top nd bottom reinforements. Deformed nd welded losed stirrups were used in this experiment. The trnsverse steel brs hd the yield strength of 5 MP nd the nominl dimeter of 6 mm. The sping of stirrups of S-.5-5, S nd S ws 5, 5 nd 75 mm, respetively. Norml weight onrete ws used in the onstrution of the test speimens. The verge ompressive strength of onrete t the time of the bem test ws.6 MP. Figure illustrtes the loding equipment nd the test setup of the speimens. This loding method ws developed by Jirs et l. (978), where the rottion of the top stub of the speimen is restrited by two vertil jks nd results in the nti-symmetri moment distribution of the entrl bem test region. In this system, the

2 98 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 vertil movement of the top stub is free, tht is, no dditionl vertil lod is pplied to the speimen subjeted to lterl sher fore by two prllel horizontl jks. During the test, the rottion ngle (drift) of the speimen ws mesured by two eletroni displement trnsduers nd reorded by omputer ontrolled dt quisition system. Wire-type strin guges were tthed on the surfe of the steel brs to mesure the strin of longitudinl nd trnsverse reinforements. Redings from the strin guges were used to estimte the tensile nd bond of the reinforing brs. Speimens onrete (f, MP) Tble Detils of test speimens. Pith (mm) Sher reinforement σ wy (MP) ρ w σ wy (MP) Longitudinl reinforement S D6 S f y = 795 MP S Note: S-.5-5 mens S (speimen),.5 (sher spn-to-depth rtio), nd 5 (sping of sher reinforement, mm) Wire Strin Guge Fig. Detils of speimen (S-.75-5). Loding Frme Prllelogrmi Apprtus Jk JAK A Jk JAK B Speimen Pin PIN Lod ell Jk JAK Bering Floor Fig. Test setup of speimen.

3 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, Experimentl results. Lod-displement response nd rk pttern All test speimens filed in sher due to rushing of digonlly ompressed onrete. Tble indites the experimentlly observed mximum strength of eh speimen. From the omprison between S nd S , the ontribution of the sher reinforement seems to be smll. The sher strengths predited by AI 8-8 ode (AI 8) nd AIJ (Arhiteturl Institute of Jpn) ode (AIJ 99, 999) re lso indited in Tble. It n be seen from the tble tht ll the ode equtions give very onservtive sher strengths for test deep bems. More relible design equtions for the sher strength of deep bems with nti-symmetri moment distribution should be estblished. With respet to the ontribution of the sher reinforement, Fig. shows the experimentlly observed sher fore versus drift ngle responses of test speimens. While ll speimens showed lmost the sme responses up to pek lods, the dutility of the speimen inresed s the mount of trnsverse reinforement grew higher in the post-pek behvior. rk propgtion of eh speimen for three loding stges is indited in Fig. 4. After initil development of digonl rk, the rk width inresed with inreses in sher fore up to the mximum lod. Finlly, eh speimen rehed the mximum pity showing the rushing of digonlly ompressed onrete ner the bem end setions.. Strin of reinforement Figure 5 shows the vrition of the strin distribution of reinforing brs. The x-xis of the figure represents the lotion of strin guges long bem length. In the figure, the thik solid line orresponds to the strin t the mximum pity. After digonl rking, stirrup strin inresed rpidly nd most stirrups yielded t the mximum pity. With respet to the strin of the longitudinl reinforement, tension shift ws observed fter digonl rking, nd then the strin ws mintined in tension. This mens tht the bem tion disppered over the entire bem length. It n be lso seen from Fig. 5 tht the longitudinl reinforement did not Sher fore (kn) Drift ngle (%) yield in ny of the loding stges s intended in the design of speimens. The verge bond stress long bem length ws omputed from the experimentlly obtined strin of the longitudinl reinforing brs. The omputed verge bond re indited in Fig. 6 with the drift ngle. The bond were very smll nd rnged from.5 to. MP though no splitting bond filure of the longitudinl reinforement ws observed. This my be due to the speifi mehnil behvior of deep bems. 4. Anlytil investigtion S-.5-5 S S Fig. Sher fore versus drift ngle reltionships of test speimens. To investigte the sher resisting mehnism of deep bems, nonliner FE nlysis ws rried out for speimens S-.5-5 nd S-.75-5, with sher spn-todepth rtios of.5 nd.75, respetively. The FE nlysis ws performed using two-dimensionl FE nlysis progrm, VeTor, developed by Vehio nd his reserh group t the University of Toronto in nd. The nlytil model used in the VeTor ws the Modified Field Theory (MFT) (Vehio nd ollins 986) nd the onstitutive models tht were used (Kent nd Prk 97; Kupfer et l. 969; Popovis 97) re listed in Tble. The detils of eh onstitutive model nd their implementtion into the VeTor Tble Experimentl results of test speimens. Speimens onrete (f, MP) ρ w σ wy (MP) V AIJ (kn) V AI (kn) V exp (kn) Filure mode S S S V AIJ, V AI : sher fore lulted by AIJ nd AI ode, respetively; V exp : Experimentlly obtined sher fore. Sher filure (rushing)

4 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 Digonl rk Digonl rk Digonl rk Pek Pek Pek After pek After pek After pek () S-.5-5 (b) S () S Fig. 4 rk propgtion. Stirrup Hoop Strin strin (μ) (μ) PEAK Yielding Strin Digonl rk Stirrup Hoop strin Strin (μ) PEAK Yield Strin Stirrup Hoop strin Strin (μ) (μ) PEAK Yield Strin Longitudinl reinforement Rebr strin (μ) strin (μ) Digonl rk Digonl rk ler length (mm) ler length (mm) ler length (mm) PEAK Digonl rk Rebr Strin (μ) Longitudinl reinforement strin (μ) PEAK Digonl rk Rebr Strin (μ) Longitudinl reinforement strin (μ) PEAK Digonl rk ler length (mm) ler length (mm) ler length (mm) () S-.5-5 (b) S () S Fig. 5 Strin of reinforements long length of speimen.

5 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 Bond stress (MP) Drift ngle (%) S-.5-5 S S Fig. 6 Averge bond stress versus drift ngle responses of speimens. progrm hve been reported elsewhere (Wong nd Vehio ). The longitudinl nd trnsverse reinforements were modeled s truss elements nd onneted to nodl points of onrete elements by the ontt elements. A omprison of the nlytil nd experimentl results for the sher fore versus displement response of test speimens is shown in Fig. 7. In S-.5-5, the mximum lod ws rehed t step 5 in the omputtionl proess, nd in S-.75-5, t step. These predited mximum sher strengths were lmost the sme s the experimentlly observed ones. After the pek lod, however, the predited urve showed somewht premture redution of lod rrying pity ompred to the experimentl one. Figure 8 illustrtes the rk ptterns of S-.5-5 nd S t loding steps 8 nd 9, respetively. It n be seen from this figure tht the theoretilly predited rk propgtion greed well with experimentl observtion (see Fig. 4). Bsed on the omprison results, the pplied FE nlysis method in this pper n be used for Tble onstitutive models used in FE nlysis. Anlytil prmeters Models Pre-pek urve Popovis post-pek urve Modified Prk-Kent softening Vehio 99-A onrete Tension stiffening Modified Bentz Tension softening Liner/no residul Tension splitting Not onsidered onfined strength Kupfer/Rihrt Diltion Vrible-Kupfer rking riterion Mohr-oulomb (Stress) rk width hek rk limit (Agg/) onrete hysteresis Liner w/plsti offsets Steel br Steel hysteresis Sekin model Dowel tion Tssios et l. Bond onrete bond Fujii model prediting the internl stress flow of speimens. Figure 9 shows the mgnitude nd diretion of prinipl tensile nd ompressive t three loding stges, i.e. pre-pek, pek nd post-pek lods. From the figure for the pre-pek stge, it n be seen tht one min ompressive strut developed on bem digonl. At pek nd post-pek lods, however, this digonl strut seems to seprte into two min struts s the width of the digonl rk inreses. These two min struts re loted t both sides of bem digonl. During this proess, the truss tion produed by the stirrup tension nd the digonl ompression of the onrete existed outside of the min strut. However, this truss tion ws smll nd most of the sher fore ould be resisted by the digonl min strut. 5 5 Step 4 Step 5 Anlysis Test 5 5 Step Anlysis Test Fore (kn) 5 Fore (kn) Step Displement (mm) 5 5 Displement (mm) () S-.5-5 (b) S Fig. 7 omprison of nlytil nd experimentl results for sher fore versus displement reltionship of speimens.

6 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 undeformed deformed Wr< mm Wr mm undeformed deformed Wr<mm Wr mm () S-.5-5 () (b) S (Step 9) Fig. 8 deformtion nd rk pttern. Wr: rk width Step 4 Step 5 Steel Steel Tension Tension Steel Tension () S-.5-5 Step Step 9 Steel Tension Steel Tension Steel Tension (b) S Fig. 9 Prinipl onrete. The bove-mentioned phenomen n be onfirmed by the sher stress nd prinipl ompressive stress distributions long setion height, s shown in Fig.. At the ritil setion of the bem, Setion A-A, the lotion of the pek stress shifted from the setion edge to the enter of the setion s the progress of loding steps. This pek shift ws observed in both bems nd my be due to the lol rushing of the onrete t the extreme ompression fiber. A similr phenomenon ws observed t Setion B-B of the S-.5-5 bem. The other speifi behvior ws the seprtion of the digonl min strut t nd fter pek lod. At Setion - of S-.5-5, nd t Setions B-B nd - of S-.75-5, the stress distributions hd two peks t nd fter pek lod. This suggests tht the min digonl strut is going to seprte into two struts due to the hnge of internl mehnism. Finl filure of the bem ourred due to rushing of the digonlly ompressed onrete, nd verge prinipl ompressive stress t pek lod ws from.7 to.8 times the ompressive strength of onrete. 5. urved dul strut model (DS-M) When deep bem with sher spn-to-depth rtio of one or less is subjeted to bending moment nd sher, uniform ompression stress field (in other word, truss mehnism) is hrdly developed due to the tension shift of the flexurl reinforement. This mens tht the min resisting mehnism for sher should be digonl strut developed between both bem ends. This hs been being pointed out experimentlly nd theoretilly in pst reserh works. Therefore design equtions for deep bems hve been developed bsed on the strut nd tie model. However, the bond tion of longitudinl reinforement nd the fore of the trnsverse reinforing br nnot be properly evluted using the strut nd tie model, beuse the model is ble to ontrol the sher strength of deep bems only by the ompressive strength of digonl strut. In this hpter, new sher resisting model inorporting dul urved struts is proposed, where the bond tion of longitudinl reinforement nd the fore of the stirrup re tken into ount. Bsed on the numeril

7 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 Lotion on vertil diretion (mm) A A B B Step 4 Step 5 Setion A-A Equivlent Sher Stress (MP) Lotion in vertil diretion (mm) Step 4 Step 5 Setion B-B Equivlent sher stress (MP) Lotion in vertil diretion (mm) Step 4 Step 5 Setion Equivlent sher stress (MP) () Sher in onrete (S-.5-5) Lotion in vertil diretion (mm) Setion A-A Step 4 Step Prinipl ompression stress (MP) Lotion in vertil diretion (mm) 5 5 Step 4 5 Step 5 Setion B-B Prinipl ompression stress (MP) (b) Prinipl (S-.5-5) Lotion in vertil diretion (mm) Setion - Step 4 Step Prinipl ompression stress (MP) Lotion in vertil diretion (mm) Step Setion A-A Step Equivlent sher stress (MP) Lotion in vertil diretion (mm) Setion B-B Step Step Equivlent sher stress (MP) () Sher in onrete (S-.75-5) Lotion in vertil diretion (mm) Step Setion - Step Equivlent sher stress (MP) Lotion in vertil diretion (mm) Setion A-A Step Step Prinipl ompression stress (MP) Lotion in vertil diretion (mm) Setion B-B Step Step Prinipl ompression stress (MP) (d) Prinipl (S-.75-5) Lotion in vertil diretion (mm) Step Setion - Step Prinipl ompression stress (MP) Fig. Distribution of sher nd prinipl. simultions nd experimentl observtions desribed in the previous hpter, new mro model to predit the sher strength of deep bems ws developed. 5. Modeling nd bsi ssumptions When deep bem is subjeted to reverse symmetri moment under onstnt sher (e.g. oupling bem), the sher resisting mehnism is generlly expressed by digonl diret strut nd supplementl fn shped mehnism (AIJ 999). However, when the theory of plstiity is pplied for the modeling, the stress ondition of the onrete ner the pivot of fn is not ler nd hrdly modeled beuse of the hnging width of eh fn bone. In this study, the sher resisting mehnism is modeled using the lower bound of the theory of plstiity. The feture of this modeling is tht two urved digonl min struts re introdued s the min sher resisting mehnism in onjuntion with supplementl sub struts. This urved dul strut model ws devised from the experimentl observtions nd the theoretil nlysis using the FE method. The new model is indited in Fig. for two bems with different sher spn-to-depth rtios. All of the min

8 4 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 nd sub struts re subjeted to n xil ompressive stress,. Two min struts re hnging their ngle long bem length due to the externl ompression given by the sub struts. At n intersetion point between the min nd sub struts, tringulr hydro pressure stress field is ssumed. The xil ompressive fores of sub struts blne the horizontl bond fores of the longitudinl reinforement nd the vertil tension fores of the stirrup t the intersetion points of the sub strut nd the longitudinl reinforement. The bsi ssumptions used in the proposed model re s follow: () At the mximum sher resistne, ll of the sher reinforement is to be yielded. This omes from the experimentlly observed strin distributions of stirrups indited in Fig. 5. (b) Bsed on the experimentl results desribed in Setion., the distribution of the bond stress of longitudinl reinforement is ssumed to hve tringulr shpe s reognized from Fig.. () The resultnt vetor of the fore of the sub-strut loted in the region in tension indites point T. Eqution () shows the vetor eqution, nd the ngles of the other two sub-struts n be omputed from Eq. () bsed on the ssumptions of () nd (b) bove. d tnθ = () tn θ = (/ ) tnθ, tn = (/ 5) tn θ θ () where d is the distne between the enters of the strut nd the bottom re-br; is third of the ler spn of bem; nd θ is the ngle between the substrut nd the longitudinl xis. (d) As shown in Fig., the rim of rooked min strut A in the A region is bent by the influene of the sub-struts, nd thus it is ssumed tht the rooked Min Strut strts from the enter of the min re-br. (e) The width of rooked min strut A is set below one tenth of the effetive depth of the member. This imittion is bsed on the result of the FE nlysis for the reson tht the region trversing the digonl sher rk beomes quite lrge if the width of strut A is too lrge, s n be seen in Fig.. 5. omputtion of bsi vlues for nlysis First of ll, the ngles of the sub-struts re determined bsed on the experimentl nd nlytil results. Figure shows tht the sher resistne mehnism is governed by geometri ondition. Bsed on the ssumption tht the stress of eh strut is the sme, the sher fore n be obtined by omputing the width of the strut tht stisfies the geometri onstrint. Figure n be drwn in more detil s Fig. (). The fore per unit length of strut A nd its oordinte n be expressed in Eqs. () nd (4), respetively. STA = σ b x () y = (tn β ) x+ d (4) where σ is the stress of onrete on the strut; b is the width of the ross setion; x is the width of strut A; β is the ngle between the strut A nd the longitudinl xis; x nd y re the x- nd y-xes in the x-y oordinte system, respetively. Similrly, the fore of strut E nd its oordinte n be obtined s STE = σ b s (5) y = (tn θ ) x ( / ) tnθ ( = L/) (6) where s is the width of strut E; θ is n ngle between the strut E nd the longitudinl xis; nd L is the ler spn of the member. The oordintes for the nodl point II of struts A nd E n be given by the following Eqs. (7) nd (8) bsed on Eqs. (4) nd (6). x d + ( /)tnθ = = tn β tnθ α (7) y = (tn θ ) α ( / ) tnθ (8) The B region of Fig. is illustrted in detil in Fig. (b). The fore per unit length nd the oordinte of strut B n be given by Eqs. (9) nd (), respetively. STB = σ b x (9) y = (tn φ ) x+ α(tnφ tn θ ) ( / ) tnθ () tnφ x sin β + s sinθ = x os β + s osθ = ( sin + sin ) + ( os + os ) () x x β s θ x β s θ () where x is the width of strut B nd φ is the ngle between the strut B nd the longitudinl xis. Sub-strut F n be expressed s the following equtions. STF = σ b s () y = (tn θ ) x ( / ) tnθ (4) where s is the width of strut F nd θ is the ngle between the strut F nd the longitudinl xis. The oordintes for the nodl point III of struts B nd F n be given by the following Eqs. (5) nd (6) bsed on Eqs. () nd (4). x ( / ) tn θ ( / ) tn θ α(tnφ tn θ ) γ (tnθ tn φ ) (5) = =

9 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 5 D region region B region A region D region / Ⅰ Strut E (sub) Strut F (sub) Strut H Strut G (sub) Strut A (min) Ⅱ Ⅴ Strut B (min) Ⅳ Strut D (min) Ⅲ Strut (min) Strut (min) Ⅲ Ⅳ Strut D (min) M V V M Strut B (min) Ⅴ Strut G (sub) Ⅱ Strut F (sub) Strut A (min) θ θ Strut E (sub) Ⅰ Strut H Bond Fore τ b Sher Reinforement Fore () S-.5-5 ρ w σ wy rooked Min Strut Sub-Strut M V rooked Min Strut V θ Sub-Strut θ M σ s b θ ρ σ w wy b ΣΨ τ b (b) S Fig. rooked dul strut model.

10 6 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 y = (tn θ ) γ ( / ) tnθ (6) Figure () depits the region of Fig. in detil. The fore per unit length nd the oordinte of strut n be given by Eqs. (7) nd (8), respetively. ST = σ b x (7) y = (tn φ ) x+ γ(tn φ tn θ ) ( / ) tnθ (8) tnφ x sin β + s sinθ + s sinθ = x os β + s osθ + s osθ (9) ( xsin β + ssinθ + ssinθ) = + ( xos β + sosθ + sosθ) x () where x is the width of strut, nd φ is the ngle between the strut nd the longitudinl xis. Sub-strut G n be expressed s the following equtions. STG = σ b s () y = (tn θ ) x (5 / ) tnθ () where s is the width of strut G nd θ is the ngle between the strut G nd the longitudinl xis. x Strut B (min) point Ⅲ Strut (min) Strut B (min) Strut A (min) x point Ⅱ φ θ β d φ φ point Ⅱ Strut E (sub) s point Ⅰ x θ x Sher Reinforement Fore ρ w σ wy b ρ w σ wy Strut F (sub) s τ b Bond Fore (Στ b ψ) Sher Reinforement Fore ρ w σ wy b ρ w σ wy τ b () A region Bond Fore (Στ b ψ) (b) B region x 4 φ φ Strut D (min) Strut (min) point Ⅰ Strut A (min) point Ⅳ x Strut H β x x os β x 4 /(osφ ) x 4 os φ Strut G (sub) θ s point Ⅲ φ point Ⅴ Strut D (min) x 4 sin φ x 4 x sinβ tnφ point Ⅳ x sin β Sher Reinforement Fore ρ w σ wy b ρ w σ wy Bond Fore (Στ b ψ) () region τ b (d) D region Fig. Detils of rooked dul strut model.

11 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 7 The oordintes for the nodl point IV of struts nd G n be given by the following Eqs. () nd (4) bsed on Eqs. (8) nd (). x γ(tnφ tn θ ) + (5 /)tn θ ( /)tnθ λ (tnφ tn θ ) () = = y = (tn θ ) λ (5 / ) tnθ (4) The fore per unit length nd the oordinte of strut D n be given by Eqs. (5) nd (6), respetively. STD = σ b x (5) 4 y = (tn φ ) x+ λ(tnφ tn θ ) (5 / ) tnθ (6) tnφ x sin β + s sinθ + s sinθ + s sinθ = x os β + s osθ + s osθ + s osθ ( xsin β + ssinθ + ssinθ + ssinθ) 4 = + ( xos β + sosθ+ sosθ + sosθ) x (7) (8) where x is the width of strut D nd φ 4 is the ngle between the strut D nd the longitudinl xis. The oordintes for the point V of strut D in Fig. (d) n be given by the following equtions. x = ( ) (9) y = jt d η () [ x os ( x ) /( sin ) x sin tn x /( os )] η = β + φ + β φ β 4 θ of the three sub-struts tht re formed by the sum of the resultnt fores of the bond stress of the min re-br nd the tensile fore of sher reinforement; () ssuming the ompressive stress ( ) of the strut nd the ngle β, whih is formed by min strut A nd the longitudinl xis; nd () lulting the width of the rooked min strut, whih stisfies the geometri onstrints illustrted in Fig.. 6. Anlysis results nd disussion Experiments nd FE nlysis with the sher-spn rtio s the vrible of interest were rried out to propose new sher resistne model for the rooked min- nd sub-struts nd n nlytil method to predit the sher fore, s illustrted in Fig. () nd Fig. (b) for the sher-spn rtios of.5 nd.75, respetively. From the figures, it n be seen tht the min strut A inlines (is rooked) more s the sher-spn rtio gets shorter, nd the min strut exhibits the tendeny to bend little more due to the sub strut under the influene of the bond stress of the min re-br, whih gets lrger s the sherspn rtio beomes greter. Additionlly, the hrteristi mehnism of the proposed model in this study is tht the sher fore is START Speify geometri nd mteril properties Given θ, θ, θ Assume σ (< σ <f ) where j is the distne between the enters of the t min re-brs. Substituting Eqs. (9) nd () for Eq. (6), Eq. () is obtined. Then, the vlue of the onrete stress in the strut is used to solve the eqution to determine the width of eh strut. j d η + (5 / ) tn θ tn φ λ(tnφ tn θ ) = t () Assume β (β = ) lulte s, s, s lulte tnφ, tnφ, tnφ lulte x, x, x, x 4 lulte hnge β hnge σ In ddition, the width of strut H in the longitudinl xis n be expressed s in the following eqution. V i = σ b = x sin β + x osφ () 4 Thus, the desired sher fore of the member is given by the following eqution. V = σ b = σ b( x sinβ + x os φ ) () 4 As hs been shown so fr, the sher fore n be omputed by solving geometri problem. The flow of the nlysis is shown in Fig.. It involves severl proesses, nmely () determining the ngles θ, θ, nd No β < 9 Yes No V = mx? Yes Output V END Fig. lultion proedure.

12 8 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 trnsmitted through by-pssing route to void the digonl sher rks, whih onnet both ends of the bem, s the width of the digonl rks inreses fter the ourrene of the digonl sher rk. This llows the rther free use of the effetive ftor of onrete ompressive strength inside the strut. Shown in Fig. 4 obtined from nlytil simultions is the reltionship between the sher-spn rtio nd the effetive strength of onrete inside the strut. There is good greement between the model predition nd the test results of the two ses. Additionlly, the rooked min-strut nd sub-strut mehnism, bsed on the experimentl nd FE nlytil results, re most onsistent with eh other when the effetive ompressive strength of onrete is set to.75 times the ylinder strength. Additionlly, when the effetive ftor of onrete ompressive strength of Fig. 4 is set to.75, the rooked min-strut nd sub-strut mehnism works only for the se of the member with sher-spn rtio below.75. When the geometri behvior for the member with sher-spn rtio below.75 is determined by the nlysis results so fr disussed, the sher resistne of the member n be predited with muh improved ury by the nlytil omputing method proposed nd explined in this study. However, to inrese the preision of the mro model proposed in this study, it is neessry to determine the bond behvior of the min re-brs in more detil. Thus, dditionl studies in this re re expeted in ner future. 7. onlusions An experimentl investigtion ws rried out for the R bems with min vrible of the sher-spn rtio. The test speimen ws subjeted to monotoni ntisymmetri moment. An nlytil method to ompute the sher fore of the deep bems nd new mro model for the sher resistne mehnism were proposed bsed on the results of the experiments nd numeril nlysis. From the experimentl nd nlytil investigtions, the following onlusions n be dedued: () FE nlysis ws rried out on the test speimens with sher-spn rtios of.5 nd.75 to delinete the stress flow nd the filure proess, nd the numeril results showed good orreltion with the experimentl results. () An nlytil method for omputing the sher fore nd new mro model, whih onsidered the bond tion of the min re-br nd used the geometri property of the rooked dul strut model, ws proposed. () The nlytil results reveled tht the proposed sher resistne tion ws observed t sher-spn rtio of less thn.75. In ddition, when the effetive ftor of onrete ompressive strength ws.75, the results of the nlysis nd experiment were the most onsistent with eh other. Nottions The following symbols re used in this pper: = third of the ler spn of member b = width of bem setion = length of strut H in the longitudinl xis ST = fore per unit length of strut d = distne between the enters of the strut nd bottom re-br f = ylinder ompressive strength of onrete j t = distne between enters of min re-brs L = ler spn of member s i = width of sub-strut V = sher fore of member W r = rk width x, y = x- nd y-xes in the x-y oordinte system, respetively x i = width of min strut β = ngle between min strut A nd longitudinl xis φ = ngle between min strut B~D nd longitudinl xis θ = ngle between sub-strut nd longitudinl xis = sher reinforement rtio ρ w 6 6 V n (kn) 8 S-.5-5 (test) V n (kn) 8 S (test) 4 Sher Spn Rtio = Effetiveness ftor ν ( / f ) 4 Sher Spn Rtio = Effetiveness ftor ν ( / f ) () S-.5-5 (b) S Fig. 4 Sher fore versus effetiveness ftor of onrete reltionships.

13 K.-H. Kim, W.-B. Kim, J.-M. Kim nd S.-W. Kim / Journl of Advned onrete Tehnology Vol. 7, No., 97-9, 9 9 σ wy = stress of onrete on the strut = yield strength of sher reinforement Referenes Amerin onrete Institute, (8). Building ode requirements for struturl onrete (AI 8M-8) nd ommentry. Frmington Hills, MI: Amerin onrete Institute. Arhiteturl Institute of Jpn, (99). Design guidelines for erthquke resistnt reinfored onrete buildings bsed on ultimte strength onept. Tokyo: Arhiteturl Institute of Jpn, 4-5. Arhiteturl Institute of Jpn, (999). Design guidelines for erthquke resistnt reinfored onrete buildings bsed on inelsti displement onept. Tokyo: Arhiteturl Institute of Jpn, 8-6. EB-FIP, (995). Model ode for onrete strutures. rd ed. EB-FIP Interntionl Reommendtions, -8. Jirs, J. O., Mruym, K. nd Rmirez, H. (978). Development of loding system nd initil tests short olumns under bidiretionl loding. ESRL Report No Kent, D.. nd Prk, R. (97). Flexurl members with onfined onrete. Journl of the Struturl Division, ASE, 97(ST7), Kim, K.-H., Yoshid, A., Sto, Y. nd Fujii, S. (). Effet of sher spn rtio nd reinforing rrngement on sher nd bond pities of R bem. Proeedings of the Jpn onrete Institute,, 7-. (in Jpnese) Kupfer, H., Hilsdorf, H. K. nd Rüsh, H. (969). Behvior of onrete under Bixil Stress. AI Journl, 66(8), Nielsen, M. P. (984). Limit nlysis nd onrete plstiity. Prentie Hll. Prk, R. nd Puly, T. (975). Reinfored onrete strutures. New York: John Wiley & Sons, Puly, T. (98). Simulted seismi loding of spndrel bems. Journl of the Struturl Division, ASE, 97(ST9), Puly, T. nd Binney, J. R. (974). Digonlly reinfored oupling bems of sher wlls. AI Speil Publition, 4, Popovis, S. (97). A review of stress-strin reltionships for onrete. AI Journl, 67(), Vehio, F. J. nd ollins, M. P. (986). The modified ompression field theory for reinfored onrete elements subjeted to sher. AI Journl, 8(), 9-. Wong, P. S. nd Vehio, F. J. (). VeTor nd FormWorks user s mnul. Tehnil Report, Dept. of ivil Engineering, Univ. of Toronto, Toronto, nd,