Design of Simply Supported Deep Beams Using Strut-and-Tie Models

Transcription

1 ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no Design of Simply Supporte Deep Beams Using Strut-an-Tie Moels by Aolfo B. Matamoros an Kuok Hong Wong A proceure to calculate the amount of reinforcement an the strength of eep beams base on strut-an-tie moels is presente. The propose esign equations were calibrate using experimental results from 175 simply supporte beams foun in the literature with a maximum shear span-to-epth ratio of 3. The strength reuction coefficient for concrete in the main strut was foun to ecrease with the angle of inclination of the strut, resulting in Lower values than those state in Appenix A of the 2002 eition of the AC/ 318 Builing Coe for beams with shear span-to-epth ratios greater than 1. Keywors: beam; shear; strut; tie. INTRODUCTION The use of strut-an-tie moels to etermine the amount an istribution of reinforcement in concrete members entails a eparture from the traitional approach to esign. While engineers must abie by a basic set of guielines in selecting the configuration of a strut-an-tie moel, they are allowe to choose any moel eeme suitable for the particular problem. The actual choice of strut-an-tie moel is, in effect, intene to represent the path followe by internal forces within the structural element. The flexibility affore by the metho allows for the evelopment of multiple solutions for the same problem. Consequently, soun engineering jugment an esign experience are funamental to achieve a safe an optimal solution. Because several factors affect the amount an istribution of reinforcement in structural elements, unerstaning their behavior is essential for the selection of strut-an-tie moels that lea to reasonable esigns. The implications of using ifferent strut-an-tie moels for the esign of eep beams are analyze in reference to experimental results foun in the literature. transferre irectly from the loa point to the support. The fraction of the total shear resistance associate with each of the two mechanisms epens on several parameters, incluing the amount an istribution of reinforcement in the beam, compressive strength of the concrete, an the shear span-to-epth ratio. Web reinforcement in eep beams generally consists of vertically an horizontally istribute reinforcement that is place in various configurations. A number of ifferent strut-an-tie moels can be use for esign, epening on the reinforcement configuration that is aopte an the shear span-to-epth ratio of the beam. Several moels for the esign of eep beams can be foun in the literature (Rogowsky an MacGregor 1983; CEB-FlP 1990; Foster an Gilbert 1998). STRUT-AND-TIE MODELS USED TO CALCULATE STRENGTH OF DEEP BEAMS Results from four ifferent strut-an-tie moels representing basic loa transfer mechanisms in eep beams were compare with experimental results from eep beams subjecte to point loas. The analysis presente exclues beams in which the moe of failure was ue to yieling of the main tension reinforcement. The first moel (Fig. I (b)) is the simplest because it neglects the contribution of web reinforcement to the strength of the beam. It consists of a irect strut between the loa point an the support. A static analysis shows that the relationship between the applie loa an the compressive force in the strut is given by F strut = _f_ sine ( l) RESEARCH SIGNIFICANCE A set of equations to estimate the contribution of plain concrete to the total shear strength an the require amount of web reinforcement is evelope base on simple strut-an-tie moels representing the main loa transfer mechanisms of eep beams. The strength reuction factor for the main strut is calculate an compare with values specifie in the ACI (ACI Committee ) an AASHTO ( 1998) Coes. A simple expression for the strength reuction factor as a function of the angle of inclination of the strut is propose. where Fis the force applie at the loa point. Two other simple mechanisms that account for the contri?ution. of reinforcement to the total shear strength were mvstigate. I the.first moel, shear is carrie by a truss in wh1h the.vertical tie (Element 4 in Fig. I (c)) represents the vertcal r1forcement. From statics, the force carrie by the vertical tie is equal to the force applie at the loa point. MECHANISMS OF SHEAR RESISTANCE IN DEEP BEAMS Two ifferent mechanisms are commonly recognize (Aoyama 1993) as the principal mechanisms of shear resistance in eep beams with web reinforcement: 1) a truss mechanism, which is the main shear resistance mechanism in slener beams; an 2) an arch mechanism in which the loa is The thir loa transfer mechanism accounts for the forces carrie by the horizontal reinforcement. In this case, shear is 704 Ftrussl =F (2) AC! Structural Joumf!l. V. 100, No. 6, November-December ti.:f No receive April. 9, 2002, an reviewe uner In stitute publication pohcies.. Copynght. 2003, Amenan.Cocrete Institute. All rights reserve, incluing e aki of op1s unl.ess perm1ss1on 1s obtaine from the copyright proprietors. uss10n rnclurng author's.closun:, if any, will he publishe in the Scptcmhcr0erti1:ent cto r 2 AC! Structural Journal if the iscussion is receive by May I

2 AC! member Aolfo B. Matamoros is an assistant professor in the Department of Civil, Environmental, an Architectural Engineering, University of Kansas, Lawrence, Kans. He receive his MS an PhD from the University of Illinois at Urbana, Ill. He is Secretary of AC/ Committee 408, Bon an Development of Reinforcement, an is a member of AC/ Committees 341, Earthquake Resistant Concrete Briges; 439, Steel Reinforcement; an Joint ACl-ASCE Committee 445, Shear an Torsion. Kuok Hong Wong is a structural engineer at APEX Engineers, Downers Grove, Ill. He receive his bachelor's an master's of science egrees at the University of Kansas. transferre by the truss mechanism shown in Fig. l(), in which the horizontal member (Element 9) represents horizontal web reinforcement. Assuming that the angle of inclination of the strut is approximately equal to the arctangent of the shear spanto-epth ratio, the force in the horizontal tie is approximately >l b (a) F (3) where ai is the shear span-to-epth ratio. Figure 2 shows the variation of the force in the compressive strut, an horizontal an vertical truss elements with respect to the shear span-to-epth ratio calculate using Eq. (1) to (3). These forces correspon to the elastic forces in the respective elements, assuming that the total shear force is carrie by each of the three mechanisms. Figure 2 shows that, accoring to Eq. (1), the force carrie by the strut in the irect strut moel (Fig. l(b)) increases as the shear span-to-epth ratio increases. Similarly, Eq. (3) implies that the force in the horizontal element of the horizontal truss moel (Element 9 in Fig. l()) increases with ai. Experimental observations show that both the force carrie by the arch resistance mechanism (e Pavia an Siess 1965; Aoyama 1993) an the effect of the horizontal reinforcement (Rogowsky, MacGregor, an Ong 1986; Smith an Vantsiotis 1982; Warwick an Foster 1993) ecrease with the shear span-to-epth ratio. Figure 2 illustrates that the results from simple truss moels may not be consistent with experimentally observe behavior an may lea to reinforcement configurations that are not appropriate for the particular stress fiel. The CEB-FIP moel coe (CEB-FIP 1990) aresses this problem by suggesting that the total shear be istribute between ifferent strut-an-tie moels an by proviing guiance about how to istribute the total force, epening on the shear span-to-epth ratio. Element forces were calculate also using a fourth statically ineterminate strut-an-tie moel that was a combination of the three moels shown in Fig. 1. The support conitions were efine so that the overall truss woul be statically eterminate (Fig. 3). The use of the stiffness metho was necessary because the truss moel is statically ineterminate internally. The stiffness coefficients were calculate base on the assumption that the moulus of elasticity an cross section area were constant an equal for all members. This assumption consierably simplifie the moeling an analysis of the strut-an-tie moel. The shear span-to-epth ratio was assume equivalent to the cotangent of 0, the angle of inclination of the strut. Figure 2 shows the fraction of the shear loa acting on the vertical an the horizontal members of the statically ineterminate truss, for shear span-to-epth ratios ranging from 0 to 3. As inicate by Fig. 2, the magnitue of both forces increase as the ai ratio increase. Figure 2 shows that the interaction between mechanisms in the statically ineterminate truss moel resulte in a significant reuction in the magnitue of the forces in the ties. (c) () _a ---;.! Fig. 1- (a) Dimensions of noal zane; (b) compression strut mechanism; (c) vertical truss mechanism; an () horizantal truss mechanism..., 4.0 -r:::==========-=---=-=-=-=--=-=-=: , a 2.5 -= "'!-< 'C l.5 g --Fstrut - - Ftrussl Ftruss t-- Ftruss I, Ineterminate Moel 0.5,, =::=:::=== ai Fig. 2-Fraction of total shear loa carrie by truss elements in irect strut, horizantal truss, vertical truss, an statistically ineterminate truss moels. Fig. 3-Combine strut-an-tie moel. Fi Fi 705

3 + -Propose Cc, Eq (4) Experimental Cc, Eq (.J (a) 1.5 (.J Experimental Cwv, Eq. 22 -Propose Cwv, Eq IIi (b) 1.5 ' t I Experimental Cwh, Eq. 24 -Propose Cwh, Eq. 25.a I ai (c) Fig. 4-Propose coefficients for three loa-transfer mechanisms. PROPOSED MODEL To aress the inconsistencies between the observe behavior of eep beams an the magnitue of the internal forces calculate using elastic truss moels, correction factors were evelope base on experimental results. These factors inicate the fraction of the total force carrie by each of the three loa-transfer mechanisms as a function of the shear span-to-epth ratio. Each loa-transfer mechanism is represente with a simple strut-an-tie moel so esign forces in ties an struts coul be easily calculate using statics. The propose methoology further simplifie calculations by ientifying key elements in each mechanism an establishing factors irectly relating the nominal strength of those elements to the strength of the beam. STRENGTH OF STRUT IN DEEP BEAMS WITHOUT WEB REINFORCEMENT In eep beams without web reinforcement, the shear force is resiste primarily by a strut that forms between the loa point an the support. For beams in which bearing a? anchorage failures at the noes are prevente, shear capacity is governe by the compressive capacity of the strut. The mean stress in the strut epens on its imensions, which are efine by the engineer. To conuct a consistent evaluation, an given that some flexibility exists in efining the with of the strut at the loa point, a simple iealization of the shape of strut was aopte (Fig. l(b)). The strut was assume to have a uniform with wst efine by the geometry of the noe at the support (Fig. l(a)), which is base on a moel use by Collins an Mitchell ( 1991) 706 where lb is the with of base plate, an ha is twice the i stance between the centroi of the main reinforcement an the bottom of the beam. Because the expression aopte to calculate the with of the strut is base on the geometry of the noe at the support, there are limits to its appl icabi lity. The efinition of the with of the strut that was use woul not be applicable in cases in which the main tension reinforcement is not properly anchore or istribute throughout the tie. The moe of failure of the beams use in the analysis was escribe by the respective authors as shear failure, so estimates of strength base on Eq. (4) are not applicable to failures ue to yieling of the main tension reinforcement or to cases in which the moe of failure is ue to beari ng at the noal zones. The angle e was approximate accoring to the followi ng relationship tane ""' - 1 ai a = - (5) Analysis of the test ata consiere in this stuy showe that the average ratio of the arc tangent of the /a ratio to the angle of the strut at failure was 5 with a coefficient of variation of 4%. For the ieal conition, in which the strut is subjecte to a uniaxial compressive stress fiel, the strength of the strut can be calculate as the prouct of the area of the strut an the compressive strength of concrete. For beams with a rectangular cross section, the nominal uniaxial strength of the strut S strut can be calculate base on Eq. (4) as where b is the thickness of the web. A coefficient Cc was obtaine relating the nominal uniaxial strength of the strut Sstrut an the force F applie at the support. Calculations were carrie out using failure loas of beams without web reinforcement (a full list of the specimens an their respective references is given by Wong [2001]). This coeffi cient may be interprete as an inicator of the relative contribution of the arch mechanism to the shear strength of eep beams. Accoring to Eq. (6), Values of Cc, obtaine by evaluating Fas the failure loa of each specimen in Eq. (7), are shown in Fig. 4 with respect to the shear span-to-epth ratio. The following expression is propose as a lower boun for the uniaxial strength parameter Cc in terms of the shear span-to-epth ratio e Cc = 0I 3 < - o.85 sm a (8) The propose expression for Cc inicates that the contri bution of the arch resistance mechanism ecreases as the ai ratio increases, which is consistent with the observe behavior of eep an slener beams (e Pavia an Siess 1965; Smith an Vantsiotis 1982). An upper limit of 0.85 was set

4 for Cc as the ai ratio approaches zero because that conition resembles an unreinforce concrete block subjecte to uniaxial compression. COMPARISON OF PROPOSED METHOD AND OTHER PROCEDURES TO ESTIMATE STRUT STRENGTH An approach that is commonly use for esign (AASHTO 1998; ACI Committee ; CEB-FIP 1990) is to calculate the strength of the strut Sstrut as the prouct of a concrete strength coefficient for the strut v an the compressive strength of the concrete (Alshegeir an Ramirez 1992). The strength coefficient for the strut accounts for the reuction in compressive strength that occurs because concrete is subjecte to transverse tensile stresses an for ifferences between the simple stress fiel of the iealize moel an the complex stress fiel that exists in the structural member. Several proceures have been propose for calculating the magnitue of this coefficient; a summary of these expressions is presente by Alshegeir an Ramirez (1992). Expressions for v propose by Warwick an Foster (1993), the 2002 eition of the ACI 318 Coe (ACI Committee ), an the AASHTO LRFD Brige Coe (AASHTO 1998), which is base on the moifie compression fiel theory (Collins an Mitchell 1991), are presente an compare with the propose moel next. Assuming that the strut has a uniform with, the mean compressive stress in the strut can be calculate as the force acting on the strut ivie by its cross-sectional area. For beams with rectangular cross sections, the unit compressive stress in the strut can be calculate base on Eq. (1) an (4). (J strut = Fstrut A strut = (9) F w Sib sine F At failure, the value of the imensionless coefficient v can be calculate base on the shear strength, material, an geometric properties of beams without web reinforcement, as inicate by Eq. (10). v Fstrut sstrut (Jstrut (10) 1: F = <f: lbsino + hacoso)bsino lcu = 0.85sl: (12) where s = 0.75 for bottle-shape struts if the amount of web reinforcement provie is greater than the minimum specifie in Section A.3.3, an s = 0.60A. otherwise. For normalweight concrete, the factor A.= 1, an accoring to Eq. (12), the compressive strength coefficient is v = 0.85 s = 0.64 if mi.nimum einforcement reqmrement is met 1 Warwick an Foster (1993) investigate the sensitivity of the compressive strength coefficient to four ifferent parameters: concrete strength, shear span-to-epth ratio, vertical steel area, an horizontal steel area. They conclue that the strength of the strut was not very sensitive to the amount of reinforcement an propose the following expression in terms of the remaining two parameters: compressive strength an shear span-to-epth ratio v = : -o.n() () (11) Equation (11) can be solve for the limit case using Eq. (5). The 0.85 limit for v controls for shear span-to-epth ratios smaller than 0.38 or angles of inclination for the strut greater than 70 egrees. for al ::; 2 (14) where 1; is the compressive strength of the concrete in MPa, an ai is the shear span-to-epth ratio. The expression is applicable in the range between 20 an 100 MPa. The fourth expression presente for the compressive strength coefficient is that aopte by the AASHTO LRFD Brige Design Specifications (AASHTO 1998) to calculate the compressive strength of struts, v = (15) E1 where E1 is the principal tensile strain in the concrete, which is approximate as (16) sine o._3 _ ::; 0.85 al sino (13) 0.51 otherwise Cc Base on Eq. (8) an (10), the following expression is propose for the strut strength coefficient in eep beams v = The 2002 eition of the ACI Builing Coe (ACI Committee ) inclues a new appenix with provisions for esign using strut-an-tie moels. Accoring to Appenix A of the 2002 ACI Builing Coe, the strength of struts shoul be taken as where Es is tensile strain in the ties, an 0 is the angle between the tie an the strut. For the purpose of esign, the AASHTO LRFD Coe allows the aoption of E5 as the yiel strain of the web reinforcement an requires a minimum amount of web reinforcement to limit the size of the cracks. Noting that tano:::::: la, the compressive strength factor can be erive by substituting Eq. (16) into Eq. (15). Assuming that the yiel strain an stress of the steel are 2 an 415 MPa (60 ksi), respectively, v = (a/ ) 2 (17) 707

5 AC), Eq. 13 without Web Reinf. ACI, Eq. 13 Minimwn Web Reinf Warwick & Fosla', Eq. 14 (t'c3ksi) -Warwick& Fosla', Eq. 14 (fc-10ksi ; --AASIITO LRFD Coe, Eq Propose moel. Eq. 11 o.s strut strut r--;;rt--&: :xexperun en'.::::ta.lre=sults,eq::i:...:..: 10: --J i ""' I J 0.6 g VJ VJ '"""""'=-""-c-----' ,-----j _,_.----,- ----,.----,.---J'-- --r----f ai 2.5 a/3 STRENGTH OF DEEP BEAMS WITH WEB REINFORCEMENT Two aitional factors were erive correlating the total applie loa with the nominal strength of the ties in the two remaining simple moels (Fig. l (c) an ()). The total strength of the beam is calculate as the sum of the components attribute to each of the three basic loa transfer mechanisms, represente by the forces in each of the main tru ss elements, multiplie by a correction coefficient 3.0 Fig. 5-Comparison of strut strength factor v calculate accoring to ifferent methos. k moels is consiere (cp = 0.75), the magnitue of the strength coefficient rops to 0.48 or 0.38, epening on whether the minimum web reinforcement requirement is provie. These values are close to the minimum observe for the test results that were analyze. >I (I 8) An effective with for the ties of the two truss moel s shown in Fig. l(c) an () was efine to quantify the contribution of istribute reinforcement to the total shear strength. Figure 6 shows a typical eep beam after cracking occurs. Strains near the support an the loa point are small, reucing the effectiveness of reinforcement. Furthermore, the strut-an-tie moels in Fig. l(c) an () have the horizontal an vertical ties locate at the center of the shear span an epth of the beam. For those reasons, only the reinforcement place in the vicinity of the ties was taken into consieration, an the effective withs for the vertical an horizontal ties were efine as al3 an l3, respectively. The contribution of the reinforcement outsie these two bans was neglecte. Accoring to the propose moel, the nominal strength of each truss element is expresse as ( 19) a (20) Fig. 6-Effective area for reinforcement. Figure 5 shows a comparison between ACI 318 (ACI Committee ), the Warwick an Foster (1993) moel, the AASHTO LRFD Coe (AASHTO 1998) equation, an the propose Eq. (11). The propose equation was more conservative in the range of strut angles between 25 an 65 egrees (these values are presente as a reference because the ACI Coe limits the angle between struts an ties to values greater than 25 egrees). The more conservative nature of the propose equation can be attribute to test results with ai of approximately 1, which ha strength coefficients as low as 0.35 (Fig. 5). Similar coefficients were observe in specimens with vertical web reinforcement an steep strut angles (low ai), in which the strength of the specimen was primarily controlle by the strength of the strut. All methos presente, with the exception of Appenix A of the ACI Coe (ACI Committee ), inicate that the compressive strength factor ecreases as the angle of inclination of the strut ecreases. As Fig. 5 shows, the coefficients of the ACI Coe provie reasonable estimates of strength for angles of inclination for the strut as low as 30 egrees. If the strength reuction factor specifie by the ACI Coe for strut-an-tie 708 (21) where f; is the compressive strength of concrete; b is the thickness of the beam; Pwv an Pwh are the vertical an horizontal reinforcement ratios, respectively; Arv an A th are the areas of the vertical an horizontal ties falling within the effective withs, respectively; anfyv anfyh are the yiel strengths of the vertical an horizontal reinforcement, respectively. Accoring to Eq. (18), the contribution of the strut is the prouct of the strength parameter Cc an the nominal strength of the strut given by Eq. (19). The value of Cc propose in Eq. (8) was aopte, an the remaining experimental results were use to calibrate the two remaining coefficients. The ata bank consiste of a total of 175 reinforce concrete eep beams with various reinforcement configurations. A etaile listing of the experimental results that were use in the analysis an their respective references is presente by Wong (2001). Concrete strength range from 14 to 73 MPa ( to 10.6 ksi) an the ai ratio range from 0.35 to The next step consiste of eveloping expressions for the coefficients Cwv an Cwh in terms of ai.

6 Table 1-Mean ratio of experimental to estimate strength Unreinforce Type of istribute reinforcement Number of specimens 45 Average 1.60 Stanar eviation 0.44 Maximum 2.29 Minimum 0.77 Coefficient of variation 0.28 Vertical Horizontal an Overall, AIJ Horizontal vertical Overall guieline Vrest!Vcalc11late l STRENGTH OF SPECIMENS WITH VERTICAL REINFORCEMENT Test results for a subset of beams having only vertical web reinforcement were analyze to calibrate the coefficient Cwv It was assume that in these specimens the shear force was carrie by the combine action of the strut an the truss mechanism that inclues a vertical tie (Fig. 2(c)). Substituting Srh equal to zero in Eq. (18) an solving for the coefficient Cwv, (22) where Sstrut an Srv are given by Eq. (19) an (20), an Vis the measure strength of the beam. Values of Cwv calculate from the subset of beams without horizontal web reinforcement are shown Fig. 4. Given the high egree of scatter in the ata, particularly for low values of ai, an the tren exhlbite by Ftrussl in Fig. 2 (Eq. (2)), a constant value of unity was aopte as a lower boun for Cwv (23) SHEAR STRENGTH ACCORDING TO JAPANESE DESIGN GUIDELINE (25) The results of the propose moel were compare with those from a esign guieline by the Architectural Institute of Japan (AU) (Aoyama 1993). Although this is not strictly a strut-an-tie approach, it was selecte for comparison because it assumes that the total shear capacity consists of components associate with a truss mechanism an an arch mechanism. The total shear strength is obtaine by aing the component associate with the arch an truss mechanisms (26) where= Pw<Jwy (1+cot 2 0);v=0.7 -a (ab in MPa); Vo<JB Pwfwy "C. V<JB/2; anfwy 25a 8. At very steep angles of inclination for the strut (ai close to zero), it is expecte that the vertical reinforcement will be subjecte to compression an still carry a fraction of the total force proportional to the amount of reinforcement provie. STRENGTH OF SPECIMENS WITH DISTRIBUTED HORIZONTAL AND VERTICAL REINFORCEMENT Similarly, the subset of beams with vertical an horizontal reinforcement were use to calibrate the parameter Cwh Equation (18) was solve for the parameter Cwh an propose expressions for Cc an Cwv (Eq. (8) an (23), respectively) were use to account for the contribution of the strut an the vertical reinforcement. (24) The calculate values for Cwh are shown in Fig. 4. Past research inicates that the effect of the horizontal reinforcement ecreases as ai increases (Rogowsky, MacGregor, an Ong 1986; Smith an Vantsiotis 1982; Warwick an Foster 1993). The tren is reflecte in the ata in Fig. 4. Consequently, an expression in which the coefficient Cwh was limite to zero for ai greater than unity was propose as a lower boun The value of cotcp is the largest value within the range of cotcp 2 J, Dtan0 (27) In Eq. (26) an (27), it is the istance between centrois of axial reinforcement; Pw is web reinforcement ratio; awy is the yiel strength of web reinforcement; <I> is the angle of the strut in the truss mechanism; Dis the epth of the beam; v 0 is the strength reuction factor for the strut; <Jn is the cyliner strength of concrete, in MPa; is the fraction of the total force in the strut assigne to the truss mechanism; an 0 is the angle of inclination of the arch with respect to the specimen. For simplification, the moel assumes that the angle of inclination of the struts in the truss an arch mechanism are equal. ANALYSIS OF RESULTS Figure 7 shows the ratio of measure-to-calculate strength with respect to the ai for all beams in the atabase, 709

7 2.S , l e u ] > j > O.S i i t, l l.s ai Fig. 7-Ratio of measure-to-calculate strength versus shear span-to-epth ratio accoring to propose equations j 1.50 'C 5 I,, >..f j 0 all > * Fig. 8-Ratio of measure-to-calculate strength versus shear span-to-epth ratio using All guielines (Aoyama 1993) i 1.40 ; "B 1 00 >. j 0.80 x > ai X Beams with Vertical Web Reinforcement o Beams with Vertical an Horizontal Web Reinforcement Vs b.[f, Fig. 9-Ratio of measure-to-calculate shear strength versus shear stress carrie by reinforcement accoring to propose equations. accoring to the propose moel. Table 1 summarizes the statistical results for concrete specimens with ifferent reinforcement configurations, incluing the results obtaine with the AU guieline. Table 1 inicates that for the propose moel, the scatter of results was larger in the case of specimens without web reinforcement. The coefficient of variation was significantly lower for specimens that ha any type of web reinforcement, an no test results were unerestimate for beams with both horizontal an vertical web reinforcement. The propose moel resulte in a mean ratio of measure-to-calculate strength of 1.40 an a stanar eviation of 0.31, with a coefficient of variation of I > j 0.80 > Specimens without Reinforcement X Specimens with Vertical Reinforcement o Specimens with Vertical an Horizontal Reinforcement S Ve Fig. JO- Ratio of measure-to-calculate shear strenj.:lh versus shear stress carrie by concrete accoring to propose equations. The AU guieline (Fig. 8), which ha a mean ratio of measure-to-calculate strength of 7 an a stanar eviation of 0.20, provie a better estimate of the mean strength. The lower mean obtaine with the AU guieline, which correspons to approximately one stanar eviation below the propose moel, is attribute to the fact that the propose equations were evelope to provie a lower-boun estimate of strength. A comparison between the coefficients of variation obtaine with the two approaches-0.22 for the propose moel an 0.19 for the AU guieline- inicate that the accuracy was similar for the set of ata that was investigate. The formulation use in the AIJ guieline inicates that the compressive stress acting on the strut of the truss mechani sm must be smaller than or equal to the effective strength of the concrete in the web, an any remaining capacity is taken by the arch mechanism. This establishes an effective upper limit on the amount of vertical reinforcement that can be pl ace in the web, which epens on the strength of the concrete. Accoring to the propose moel, any reuction in the strength of the strut relate to the amount of web reinforcement is reflecte by the coefficients for the horizontal an vertical truss mechanisms. Warwick an Foster (1993) conclue in their research that the amount of web reinforcement i not have a significant effect on the strength of the strut, an their propose equation for the strut strength factor v epens only on the compressive strength of concrete an the shear span-to-epth ratio of the beam. Figure 9 an l 0 show the relationship between the ratio of measure-to-calculate strength an the mean shear stress attribute to plain concrete an web reinforcement, accoring to the propose moel. In the majority of the beams that were analyze, the unit shear stresses attribute to the web reinforcement range from 0 to 8...Jf; (in units of psi). The measure strength was not unerestimate in any of the specimens in which the unit shear stresses attribute to the shear reinforcement range between 4 an 8...Jf;. Figure IO shows that in the majority of the ata the unit shear stress carrie by the concrete range between 2 an 1 O...Jf;. Figure 11 shows the ratio of measure-to-calculate strength versus the percentage of web reinforcement of the specimens in the atabase. The figure shows that a wie range was represente by the set of ata- between 0.15 an 3.5%. The amount of reinforcement calculate using Eq. (22) an (24) was compare with that require by the strut-antie moels shown in Fig. I (c) an 3, which may not represent 710

8 the best choice in some instances, but are acceptable moels accoring to the ACI Coe (ACI Committee ). The ratio of shear force in the vertical tie accoring to each of the two strut-an-tie moels to the force calculate with Eq. (22) is shown Fig. 11, for the subset of specimens with only vertical web reinforcement. The amount of vertical reinforcement require by the combine strut-an-tie moel (Fig. 3) was similar to that require by the propose equations for ai greater than 1.5 (Fig. 12). In specimens that ha span-toepth ratios less than 1.5, the combine moel resulte in a smaller amount of vertical reinforcement. A similar comparison with the vertical truss moel (Fig. 1(c)) shows that it require approximately twice the amount of reinforcement than the propose equations. In the case of horizontal web reinforcement, the comparison was limite to only four test specimens because the propose equations suggest neglecting the effect of the horizontal reinforcement when ai is greater than 1. A comparison of the amount of reinforcement require by the two strut-antie moels an the propose showe that the amount of reinforcement require by the combine moel is slightly less but comparable with that require by the propose equations. The simple strut-an-tie moel (Fig. I ()) also require approximately 50% more reinforcement than the propose equations in this case, although it is important to point out that this particular truss moel neglecte the contribution of the vertical reinforcement. SUMMARY AND CONCLUSIONS This investigation focuse on eveloping a relationship between the strength of eep beams an the forces in the main strut an ties of moels representing the loa carrie by plain concrete, vertical, an horizontal web reinforcement. Accoring to the propose moel, the total shear strength is given by V = if; bw 51 + Atvfyv+ 3(1- al )A 1hfyh (28) where tan0 :::: 1/(a/), an the with wst of the strut is calculate accoring to Eq. (4). The following limits apply to Eq. (28): the term 0.3/(al) has an upper limit of 0.85sin0, an the term (1 - ai) has a lower limit of 0. The ratio of calculate force to nominal strength of the strut, esignate concrete strength coefficient v, was compare with the new strut-an-tie provisions of the ACI Coe (ACI Committee ), equations propose by Warwick an Foster (1993), an AASHTO LRFD Brige Coe (AASHTO 1998). Safety was the primary concern consiere in the formulation of the propose equation, which resulte in a conservative estimate of the strength in beams with strut inclination angles ranging between 30 an 60 egrees (shear span-to-epth ratios ranging between approximately 0.5 an 1.5). Results inicate that the strut factor v ecrease as the angle of inclination of the strut ecrease. Estimates of shear strength accoring to the propose moel were compare with estimates accoring to a guieline by the AIJ (Aoyama 1993). The accuracy of the AIJ guieline for the entire group of specimens was comparable to that of the propose equations (coefficient of variation of 0.19 for the AU guieline compare with 0.22 for the propose equations). The propose equations provie safer 0.40 X Beams with Vertical Web Reinforcement 0.20 o Beams with Vertical an Horizontal Web Reinforcement ==:r===r===;:::===;:::====;='-=-t===-_j P., +P Fig. 11-Ratio of measure-to-calculate shear strength versus web reinforcement ratio for all specimens consiere r.========== , +Vertical Truss Moel 0 Statically Inctemlinate Truss Moel _,, _,..,., i: :: ,.._-----.rlll'>it----l , oel o o.so B g 0 uo ai Fig. 12-Comparison of calculate amount of vertical reinforcement. estimates of strength for beams with shear span-to-epth ratios of less than 1. A comparison of the amount of reinforcement require by four ifferent strut-an-tie moels showe that the amount of reinforcement require by a statically ineterminate moel (Fig. 3) was closest to the propose equations, inicating that the assumption of equal moulus of elasticity an crosssectional area for all members was aequate for esign purposes. The two simple strut-an-tie moels analyze in this paper (Fig. 1(c) an ()) were foun to require approximately ouble the amount of reinforcement require by the propose equations (Wong 2001 ), which inicates that, although their use leas to safe esigns, the results are very conservative. ACKNOWLEDGMENTS The authors woul like to acknowlege Steve McCabe, Davi Darwin, an JoAnn Browning of the Department of Civil Engineering at the University of Kansas for their valuable comments an contributions for the evelopment of this work. A strut A1h Arv a b Cc Cwh Cwv D NOTATION cross-sectional area of compressive strut area of horizontal tie area of vertical tie shear span, istance between center of loa to center of reaction with of beam correction factor for calculate force in strut correction factor for calculate force in horizontal tie correction factor for force calculate in vertical tie total epth of beam effective epth of beam 71 1

9 F Fstrut Ftrussl Ftruss2 f: fcu it Ll 1 Es <!> /... e v Vo Pw Pwh Pwv <JB (Jstrut <Jwy applie shear force calculate force in strut calculate force in vertical tie calculate force in horizontal tie compressive strength of concrete strength of concrete in struts accoring to Appenix A of 2002 ACI Coe yiel stress for horizontal web reinforcement in beam yiel stress for vertical web reinforcement in beam twice istance from bottom of beam to centroi of main tensile reinforcement istance between centrois for top an bottom tensile reinforcement length of support stiffness matrix for statically ineterminate truss moel nominal strength of struts nominal strength of horizontal tie nominal strength of vertical tie calculate shear strength of beam accoring to propose moel calculate component of total strength attribute to arch action accoring to AU guieline calculate component of total strength attribute to truss action accoring to AIJ guieline calculate total shear strength accoring to AIJ guieline with of strut fraction of total force in strut attribute to truss mechanism factor to account for effect of cracking an confining reinforcement on effective compressive strength of concrete in strut accoring to Appenix A of ACI isplacement vector for statically ineterminate truss moel principal tensile strain in concrete tensile strain in web reinforcement angle of main strut with respect to horizontal plane correction factor relate to unit weight of concrete angle between strut an horizontal plane effective compressive strength factor for concrete in strut effective compressive strength factor for concrete in strut reinforcement ratio for vertical web reinforcement reinforcement ratio for horizontal web reinforcement reinforcement ratio for vertical web reinforcement compressive strength of concrete, in MPa (cyliner strength) mean compressive stress acting on strut tensile strength of web reinforcement REFERENCES AASHTO, 1998, "AASHTO LRFD Brige Design Specifications," American Association of State Highway an Transportation Officials, Washington, D.C. ACI Committee 318, 2002, "Builing Coe Requirements for Structural Concrete (ACI ) an Commentary (3 I 8R-02)," American Concrete Institute, Farmington Hills, Mich., 443 pp. AJshegeir, A., an Ramirez, J. A., 1992, "Strut-Tie Approach in Pretensione Beams," AC/ Structural Journal, V. 89, No. 3, May-June, pp Aoyama, H., 1993, "Design Philosophy for Shear in Earthquake Resistance in Japan," Earthquake Resistance of Reinforce Concrete Structures-A Volume Honoring Hiroyuki Aoyama, University of Tokyo Press, Tokyo, Japan, pp CEB-FIP, J 990, "CEB-FIP Moel Coe for Concrete Structures," Comite Euro-International u Seton/Feeration Internationale e la Precontrainte, Paris. Collins, M., an Mitchell, D., 1991, Prestresse Concrete Structures, Prentice Hall, Upper Sale River, N.J., 766 pp. e Paiva, H. A. R., an Siess, C. P., 1965, "Strength an Behavior of Deep Beams in Shear," Journal of Structural Engineering, ASCE, Y. 91, No. ST5. Foster, S. J., an Gilbert, R. I., 1998, "Experimental Stuies on High- Strength Concrete Deep Beams," AC/ Structural Journal, V. 95, No. 4, July-Aug., pp Ricketts, D. R., an MacGregor, J. G., 1985, "Ultimate Behaviour of Continuous Deep Reinforce Concrete Beams," MS thesis, Department of Civil Engineering, University of Alberta, Canaa, 90 pp. Rogowsky, D. M., an MacGregor, J. G., 1983, "Shear Strength of Deep Reinforce Concrete Continuous Beams," Structural Engineering Report 110, Department of Civil Engineering, University of Alberta, Canaa, 178 pp. Rogowsky, D. M.; MacGregor, J. G.; an Ong, S. Y., 1986, "Tests of Reinforce Concrete Deep Beams." ACI JOURNAL, Proceeings V. 83, No. 4, July-Aug., pp Smith, K. N., an Vantsiotis, A. S., 1982, "Shear Strength of Deep Beams," ACI JOURNAL, Proceeings V. 79, No. 3, May-June, pp Warwick, W. B., an Foster, S. J., 1993, "Investigation Into the Efficiency Factor Use in Non-Flexural Reinforce Concrete Member Design," UNICN Report No. R-320, School of Civil Engineering, University of New South Wales, Kensington, 81 pp. Wong, K. H., 2001, "Design of Reinforce Concrete Beams Using Strutan-Tie Moels," MS thesis, Department of Civil an Environmental Engineering, University of Kansas, Lawrence, Kans., July, 68 pp. 712