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ACI Structural Journal/July-August 2006 577

ACI Structural Journal, V. 103, No. 4, July-August 2006.MS No. 05-104 received May 6, 2005, and reviewed under Institute publication policies.

Copyright 2006, American Concrete Institute. All rights reserved, including the makingof copies unless permission is obtained from the copyright proprietors. Pertinent discussionincluding authors closure, if any, will be published in the May-June 2007 ACIStructural Journalif the discussion is received by January 1, 2007.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Results from an experimental investigation aimed at evaluating theadequacy of the strength factors for concrete struts in strut-and-tiemodels given in Appendix A of the 2002 ACI Building Code are

presented. The main design variables considered were: theangle between primary strut-and-tie axes, amount of reinforcementcrossing the strut, and concrete strength. A total of 12 deep beamswere tested, eight with normal strength concrete and four withhigh-strength concrete. The ratio between experimentallyobtained failure loads and the strengths predicted using the strutstrength factors given in Appendix A of the ACI Code rangedbetween 1.00 and 1.22, and between 0.91 and 1.02 for normal andhigh-strength concrete beams, respectively. Inconsistencies were

found in the provisions for minimum reinforcement crossing a strutin Sections A.3.3 and A.3.3.1 when applied to the test specimens,with the former leading to substantially larger reinforcementratios. The use of a strut strength factor s= 0.60 in high-strength

concrete bottle-shaped struts without web reinforcement led tostrength predictions approximately 10% higher than the experi-mental failure loads. The limited test results suggest that, as aminimum, an effective reinforcement ratio of 0.01, calculatedaccording to ACI Code, Section A.3.3.1, should be provided inhigh-strength concrete members when a strength factor s= 0.60

is used. Additional test data, however, are required before a definiterecommendation can be made in this regard.

Keywords: beam; cracking; shear strength.

INTRODUCTIONThe use of strut-and-tie models for the design of reinforced

concrete members was incorporated in the 2002 edition ofthe ACI Building Code (herein referred to as the ACI Code)as Appendix A. Some of the provisions for the design ofconcrete members using the strut-and-tie method, however,are still the subject of discussion and validation, such as thatrelated to the strength factors for concrete struts, which areassumed to be dependent on strut cracking condition andreinforcement provided, but independent of concretestrength and strut angle. Discussions on the adequacy ofcurrent strength factors for concrete struts have been triggeredby the significant discrepancies that exist between theproposed values in the ACI Code and those suggested byseveral researchers.1-6 In some cases, the latter ones aresubstantially lower than those given in the ACI Code, especiallyfor higher strength concrete. This paper presents results froman investigation aimed at evaluating the adequacy of thestrength factors given in Appendix A of the ACI Code for thedetermination of strength of concrete struts in strut-and-tiemodels. In particular, the ultimate capacity of diagonalcompression struts in deep concrete beams with eitherminimum or no transverse reinforcement was experimentallyevaluated. For this purpose, several concrete beams withvarious shear span-to-depth ratios, concrete strengths, andreinforcement layouts were tested.

RESEARCH SIGNIFICANCEThe adequacy of the strength factors for concrete struts

given in Appendix A of the 2002 ACI Code was evaluated.Strut strengths determined from Appendix A of the ACI Codewere compared with experimental data obtained from testsof several deep reinforced concrete beams. The datagenerated in this study also provides valuable information toACI Committees 445 and 318-E for evaluating currentdesign specifications for concrete members using thestrut-and-tie method.

STRENGTH FACTORS FOR CONCRETE STRUTSA conservative estimation of the compressive strength of

struts in strut-and-tie models is crucial to ensure adequatestructural safety and the ability to design deep beams suchthat yielding of longitudinal reinforcement occurs prior tofailure of concrete struts. If either no yielding or only limitedyielding occurs prior to failure of a concrete strut, a suddenfailure could take place with the drastic consequences thathave been observed in past shear failures.1During the lasttwo decades, several researchers have conducted studiesaimed at evaluating the strength of concrete struts for use instrut-and-tie models.1-6Generally, the approach has beento lump the effect of strut stress and strain conditions,reinforcement details, concrete strength, and uncertainties in

the truss model into a single factor, commonly referred to asthe efficiency factor or, more recently, as the strength reductionfactor v1.0.4,6,7Therefore, the strength of a concrete strutis expressed as

fc= vfc (1)

wherefc is the concrete compressive strength. To make theapplication of strut-and-tie models practical, single valuesfor the efficiency factor have been proposed, which arebased on the cracking conditions expected in the struts, thatis, struts with cracks parallel to or skewed with respect to thestrut axis, but are independent of concrete strength. Proposedvalues for the efficiency factor typically range from 0.3 forheavily cracked struts to 0.85 for struts in which the concreteis subjected to uniaxial or biaxial compression.4Appendix Aof the ACI Code8adopted this format in Article A.3.2, wherethe efficiency factor is expressed as 0.85s. Values of srange between 0.40 and 1.0, depending on the strut geometry,reinforcement provided, and stress conditions in themember. Thus, the specified values for the efficiency factor

Title no. 103-S61

Strength of Struts in Deep Concrete Members Designed

Using Strut-and-Tie Method

by Carlos G. Quintero-Febres, Gustavo Parra-Montesinos, and James K. Wight


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ACI Structural Journal/July-August 2006578

in the ACI Code range between 0.34 and 0.85. As might beexpected, these discrete values of the efficiency factor aresomewhat arbitrary and primarily based on experimentalevidence. Background information about provisions forconcrete struts given in Appendix A of the ACI Code can befound elsewhere.7

Other more sophisticated expressions for estimating thestrength of concrete struts include the effect of concretecompressive strength,3,5in which the value of the efficiencyfactor decreases as the concrete strength is increased. Onesuch expression is given in Eq. (2)5

(2)

In Eq. (2), the efficiency factor is represented by the termsin parenthesis. Evaluating this expression forfc = 28 and70 MPa (4000 and 10,000 psi), values for the efficiencyfactor of 0.56 and 0.35 are obtained, respectively. Ramirezand Breen3proposed an efficiency factor for concrete strutsin beams of 2.5/fc MPa. In this case, is equal to 0.47 and0.30 for 28 and 70 MPa concrete, respectively.

Extensive research has also been conducted at the Universityof Toronto by Vecchio and Collins9,10 to evaluate thecompression behavior of cracked concrete. Expressions foreffective concrete strength developed by Vecchio andCollins are based on the strain conditions in the concrete, andthus their application to strut-and-tie models requires anaccurate estimation of the state of strain in the strut. Twoexpressions they developed are

(3)

and

(4)

fc 0.7fc

200---------

fc MPa( )=

fc1

0.85 0.2712----

--------------------------------

fc fc=

c1

0.80 0.341o----+

--------------------------------

fc fc=

where 1and 2are the principal tension and compressionstrains, respectively, and o is the concrete strain at peakcompressive strength. In Eq. (3), the principal strain ratio 1/2needs to be known in order to determine the effectiveconcrete strength in the strut. Equation (4) represents asimpler version, in which only the principal tension strainneeds to be determined. If Eq. (4) is evaluated for principaltensile strains of 0.002 and 0.006 assuming o= 0.002, theefficiency factor would be equal to 0.88 and 0.55, respec-tively. More recently, Vecchio11proposed a reduction in the

degree of concrete softening predicted by Eq. (3) and (4) toaccount for the fact that some of the softening, which wasattributed to the concrete in previous tests, was actually dueto crack slip and not to transverse tensile strains. Thus, forcases where crack reorientation does not occur, highervalues of the efficiency factor would be obtained comparedto those determined from Eq. (3) and (4).

A simple expression for the concrete efficiency factor as afunction of the angle of inclination of the strut with respectto the tie direction was recently proposed by Matamoros andWong,12with the values of the efficiency factor decreasingas the angle of inclination of the strut decreases. That modelgave lower strength estimates compared with the ACI Codefor the range of strut angles considered in their study

(between 30 and 60 degrees).From the previous discussion, it is clear that, depending on

the model used, significant differences can be obtained in thevalues of the efficiency factor used for concrete struts.Values as low as 0.3 have been proposed in the past, whichis less than the lowest factor specified in the ACI Code.Therefore, an evaluation of the adequacy of the strengthfactors for concrete struts given in Appendix A of the ACI Code,in particular those corresponding to struts crossed by no or lowamounts of transverse reinforcement, was performed.

EXPERIMENTAL INVESTIGATIONThe experimental program reported herein was carried out

at the Structural Engineering Laboratory of the University of

Michigan. It comprised the testing to failure of 12 reinforcedconcrete deep beam specimens designed with various shearspan-to-depth ratios, concrete strengths, and reinforcementlayouts. Detailed information about this testing program canbe found elsewhere.13All beams were 2440 mm (8 ft) long,460 mm (18 in.) deep, and 150 mm (6 in.) wide, except forthe high-strength concrete specimens, in which the beam webwidth in the test region was reduced to 100 mm (4 in.) toforce a diagonal compression strut failure prior to yielding ofthe longitudinal reinforcement. The clear-span-to-total-depth ratio for all beams was less than 4.0, which classifiesthem as deep beams, according to the ACI Code. The beamswere loaded at a single point on the top face close to one ofthe supports in order to develop a single concrete strut actingapproximately at a predetermined angle. Figure 1 shows theload and support configuration for the beams.

Specimen design parametersThree main design parameters were considered in this

study, primary strut angle with respect to the memberlongitudinal (tie) axis, concrete compressive strength, andamount of web reinforcement crossing the primary strut.Shallow and steep strut angles were considered, withvalues of 25 to 29degrees, and 35 to 44degrees, respectively,based on a strut-and-tie model constructed following theprovisions in Appendix A of the 2002 ACI Code. The

ACI member Carlos G. Quintero-Febresis a Professor of Civil Engineering at theUniversidad de los Andes, Mrida, Venezuela. He received his degree in civil engineering

from the Universidad de los Andes, Venezuela; his MS from the University of Iowa,

Iowa City, Iowa; and his PhD from the University of Michigan, Ann Arbor, Mich. He

is a recipient of the ACI Structural Research Award in 2003. His research interests

include the earthquake-resistant design of reinforced concrete structures and

structural reliability.

ACI member Gustavo Parra-Montesinosis an Assistant Professor of Civil Engineeringat the University of Michigan. He is Secretary of ACI Committee 335, Composite and

Hybrid Structures, and a member of ACI Committees 318-F, New Materials, Products

and Ideas (Structural Concrete Building Code), and 544, Fiber Reinforced Concrete. He isalso a member of Joint ACI-ASCE Committee 352, Joints and Connections in Monolithic

Concrete Structures. His research interests include the seismic behavior and design ofreinforced concrete, hybrid steel-concrete, and fiber-reinforced concrete structures.James K. Wight, FACI, is a Professor of Civil Engineering at t he Univers ity of

Michigan. He is Chair of ACI Committee 318, Structural Concrete Building Code,

and a member of Joint ACI-ASCE Committees 352, Joints and Connections in

Monolithic Concrete Structures, and 445, Shear and Torsion. He has received several

ACI awards, including the Delmar L. Bloem Award in 1991, the Joe W. Kelly Award in

1999, and the ACI Structural Research Award in 2003. His research interests include

the earthquake-resistant design of reinforced concrete structures.


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smallest angle between a strut and a tie is limited to 25 degreesin Section A.1 of the ACI Code. Therefore, beams testedwith shallow strut angles were at or close to the lower limit,and thus likely to depart from a deep beam behavior (that is,shear resistance through a single direct strut between theload point and support). Beams tested with steep strut anglesdid represent deep beam behavior. Normal-strength concrete(in the order of 28 MPa [4000 psi]) and high-strengthconcrete (in the order of 55 Mpa [8000 psi]) were used toevaluate the adequacy of current strength factors for a wide

range of concrete strengths. Based on these design parameters,the test specimens were grouped in three series of fourspecimens each, as follows: Series A having normal strengthconcrete and a main strut angle of approximately 28 degrees;Series B with normal strength concrete and a primary strutangle ranging between 41 and 44degrees; and Series H withhigh-strength concrete, two specimens having a strut angleof approximately 25 degrees and two with a strut angle ofeither 35 or 38degrees. Table 1 lists the primary strut angle,, for each test specimen.

Two amounts of reinforcement crossing the primary strutwere considered in this study. For Series A and B, twospecimens had no reinforcement crossing the strut, andtwo specimens had minimum reinforcement as per

Section A.3.3.1. For each strut angle considered in Series H,one specimen had no web reinforcement and one had webreinforcement crossing the primary strut. The web reinforcementprovided in the specimens of Series H, however, did notsatisfy the minimum reinforcement ratio required inSection A.3.3 of the ACI Code, as will be explained indetail later.

Design of test specimensThe test specimens were designed using the strut-and-tie

model shown in Fig. 2. In this model, a single strut isassumed between the load point and the closer support,which is located at approximately one-third or less of the

clear span away from the load. For the other portion of thebeam specimen (left portion of beam in Fig. 2) a morecomplex truss is required to avoid a strut angle less than

25 degrees. With this configuration, the strength of theprimary strut to the closer support was computed based onSection A.3 of the ACI Code. Bearing strengths at nodalzones were then checked and a prediction of the failure loadwas obtained for the selected strut-and-tie model. Calculationsfor determining the beam strength for the critical shear spanof one of the test beams in Series A is provided in theAppendix of this paper. In addition, a complete designexample for a deep beam using a strut-and-tie model similarto that shown in Fig. 2 can be found elsewhere.14

Reinforcement detailsDetails of the reinforcement for the test specimens are

given in Fig. 3 through 5. The longitudinal reinforcement for

Fig. 1Overall dimensions and load configuration of deepbeam specimens.

Fig. 2Strut-and-tie model for deep beam design.

Table 1Properties of test specimens and summary of test results

Specimen bw, mm d, mm a, mm , degree

Effectivereinforcementratio provided

Effectivereinforcementratio required Failure mode Vu, kN vu/fc, MPa VACI, kN Vu/VACI

A1 150 370 525 28.5 0.0042* 0.003(0.0096) DS 251 0.95 215 1.17

A2 150 370 525 28.5 0.0042 0.003 (0.0096) DS 237 0.90 215 1.13

A3 150 370 525 27.9 DS 221 0.84 185 1.20

A4 150 370 525 27.9 DS 196 0.74 185 1.06

B1 150 375 335 41.3 0.0038 0.003 (0.013) DC 456 1.40 425 1.07

B2 150 375 335 41.3 0.0038 0.003 (0.013) DC 426 1.31 425 1.00B3 150 375 305 43.8 DC 468 1.44 385 1.22

B4 150 375 305 43.8 DC 459 1.41 385 1.19

HA1 100 380 595 25.0 0.0056 NA (0.014) SC 265 0.64 275 0.96

HA3 100 380 545 25.0 SC 292 0.71 320 0.91

HB1 100 380 340 35.0 0.0085 NA (0.021) DC 484 1.18 475 1.02

HB3 100 380 310 37.6 DC 460 1.12 505 0.91

*Based on Eq. (5).Based on ACI 318, Section A.3.3.1.Based on ACI 318, Section A.3.3.DS = diagonal splitting; DC = diagonal compression; SC = shear compression; Vu= shear force at failure; vu= average shear stress at failure (Vu/bwd), VACI= calculated strength

based on ACI 318, Appendix A.Notes: NA = not applicable; 1 mm = 0.394 in.; 1 kN = 0.225 kips; and 1 MPa = 0.145 ksi.


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580 ACI Structural Journal/July-August 2006

all beam specimens was provided by four equal size barsdistributed in two layers whose centroid coincided with thetheoretical centroid of the main tie. This reinforcement waschosen such that the stress in the steel corresponding to themaximum predicted tie force for a particular strut strengthwould be at or below the yield strength of the steel fy. Thereinforcement selected to carry the maximum tie force waskept constant along the specimen length. At both beam ends,the longitudinal reinforcement was extended beyond thesupports and terminated with either standard 90- or 180-

degree hooks to ensure proper development.Transverse reinforcement over the portion of the beamaway from the primary strut consisted of U-shaped, No. 10M(area = 71 mm2) deformed bars. More stirrups than requiredfor shear strength were used to force a failure in the strutbeing tested. The amount of reinforcement used in each casewas checked using the strut-and-tie model of Fig. 2. Theextensions of the beam beyond the supports were reinforcedwith U-shaped stirrups, as per ACI Code Section 12.5, toensure proper development of reinforcement.

Transverse reinforcement crossing the primary diagonal strutwas provided through No. 6M (area = 32 mm2) horizontal bars,

and either U-shaped stirrups (Series A and B) or two single-legties (Series H) (Fig. 3 through 5). For Series A and B, which hada specified concrete strength less than 41 MPa (6000 psi), ACICode Eq. (A-4) was used to select the reinforcement

(5)

whereAsiis the total area of reinforcement at a spacing siin

a layer of reinforcement iwith bars at an angleito the axisof the strut. For Series H (concrete strength above 41 MPa[6000 psi]), a relaxation of the requirements of Section A.3was made. In this case, instead of assuming that the compressiveforce in the strut would spread at a 2:1 slope (ACI Code,Section A.3.3 and Fig. RA.1.8(b)), a shallower slope of6:1 was used to avoid excessive amounts of web rein-forcement. For all beams, a single U-shaped stirrup (Series Aand B) or a set of two single-leg ties (Series H) was placedat the location of the applied load and the reaction force.Table 1 gives the web reinforcement ratios provided in thecritical shear span, calculated according to Eq. (A-4) of theACI Code.

MaterialsThe concrete for the three series was obtained in differentbatches from a local ready mixed concrete supplier. Themaximum coarse aggregate size was 10 mm (3/8 in.). Foreach batch, 12 100 x 200 mm (4 x 8 in.) cylinders were takenfor later evaluation of concrete compressive strength. Alllongitudinal reinforcement was made of Grade 60 (nominalfy= 410 MPa (60 ksi) deformed steel. Main strut transversereinforcement consisted of smooth round bars (notcompliant with ASTM A 615M standards), while all otherbeam transverse reinforcement consisted of Grade 60 steeldeformed bars. Reinforcing steel for each series wasobtained separately from a local steel supplier. Measuredconcrete compressive strengths, obtained from at least three

cylinder tests, and steel yield strengths are given in Table 2.

Fabrication of test unitsBoth the longitudinal and transverse reinforcement were

bent by a local steel supplier. After strain gauges wereattached to the bars at the desired locations, steel reinforcementcages were assembled in the laboratory and placed into theforms. The beams were cast following common proceduresand wet-cured for seven to 10 days before being removedfrom their wood formwork. Beams from Series A and B,which had a uniform cross-section, were cast upright, whilethose from Series H, which had a reduction of the crosssection in the shear span, were cast upside down to ease thecasting process and to ensure proper consolidation of the

concrete in the forms. Once the concrete had gained sufficientstrength, the beams were placed in the testing rig.

Asi

bs i------- sin i 0.003

Fig. 4Reinforcement details for Series B.

Fig. 3Reinforcement details for Series A.

Table 2Material properties

Series

fc, MPa

(ksi)

Reinforcement yield strengthfy , MPa (ksi)

No. 6M No. 10M No. 13M No. 19M No. 22M

A22.0(3.2)

407(59.0)

441(64.0)

455(66.0)

462

(67.1)

B32.4(4.7)

545(79.1)

476(69.0)

427

(61.9)

H50.3(7.3)

586(85.0)

476(69.0)

434

(63.0)


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Test setup and instrumentationThe beams were tested in two different setups. A test bed

with a 490 kN (110 kip) actuator was used for Series A, whilefor Series B and H, a steel loading frame with a 1780 kN(400 kip) capacity was used. The same load configuration wasused in both test setups, as shown in the schematic drawingof Fig. 1. All beams were simply supported through the useof restrained and free rollers, and were loaded on the top faceat a single location. The applied load and the reaction forceswere distributed on the beam surface through 25 mm (1 in.)

thick by 150 mm (6 in.) square steel plates. High strengthgrout was used for proper seating of the steel plates on thebeams surfaces. The load was applied monotonically up tofailure in increments that corresponded to approximatelyone-tenth of the predicted failure load for each beam.

Load cells, displacement transducers, and electricalresistance strain gauges were used to monitor the appliedloads, deflections and deformations, and strains in the steelreinforcement, respectively. Load cells provided readingsfor the loads imposed by the actuator and hydraulic jack usedin the beam tests. A displacement transducer was placedunder the beam at the load point to record the imposeddisplacements. An arrangement of three pairs of displacementtransducers, oriented vertically, horizontally, and at 45 degrees,

was used in each beam to monitor average strains in the criticalshear span. Cracks were also marked at selected load points

up to approximately 75% of the predicted peak load andnumbered according to the loading stage.

ANALYSIS OF TEST RESULTSThe behavior of the test specimens was evaluated in terms

of load versus deflection and load versus shear span distortionresponses, cracking pattern, failure mode, and strain historyat selected reinforcement locations.

Overall response

All beam specimens exhibited a nearly linear response up tofailure, as shown by the load versus deflection curves ofFig. 6 through 8. Even though there was a larger region of lowstiffness for the specimens in Series A (due to support seating),it is clear that the specimens in Series A (shallow strut angle)experienced larger load-point deflections than those in Series B(steep strut angle) at peak load. Also, from Fig. 6 it is seen thatthe specimens of Series A with web reinforcement had a largerstrength than those without web reinforcement, while no majordifferences in strength were observed for the beams in Series Band H (Fig. 7 and 8, respectively).

Average shear distortions in the critical shear span (mainstrut) were computed based on the readings obtained from anarrangement of linear potentiometers, as explained earlier.

The load versus shear distortion response was characterized

Fig. 5Reinforcement details for Series H.

Fig. 6Load versus displacement response (Series A).

Fig. 7Load versus displacement response (Series B).


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by a linear uncracked response up to first diagonal cracking(loads in the order of 25% of the peak load), followed by anearly linear cracked-elastic response up to peak load, whichcorresponded roughly to an average shear distortion of 0.004to 0.005 rad for all test specimens. Once the peak load wasattained, a rapid decrease in the applied load with increasingshear distortions was observed. Providing minimum webreinforcement crossing the strut did not seem to noticeablyaffect the deformation capacity of the struts.

Cracking patterns and failure modesShear force at failure Vu and failure modes for the testspecimens are listed in Table 1. Figure 9 through 11 showtypical crack patterns at failure for the specimens in Series A,B, and H, respectively. In general, a few flexural cracksformed first, which remained at a narrow width throughoutthe tests. Diagonal cracks then formed at approximately 25%of the peak load, as mentioned earlier, defining the directionof the main concrete strut. Failure for all test specimens wasbrittle and the failure modes were identified as follows:

1) Diagonal-splitting failure, in which diagonal cracks thatformed initially at mid-depth of the beam in the direction ofthe main strut propagated to the outside edge of the loadingplate and the inside edge of the bearing plate at the support

(Fig. 9). This type of failure was typical for Series A, regardlessof the amount of web reinforcement provided;

2) Strut crushing failure at beam mid-depth following theformation of several diagonal cracks (Fig. 10). This failuremode was typical for Series B and Specimens HB-1 and HB-3from Series H; and

3) Shear-compression failure near the loading point (Fig. 11)after formation of one or two main diagonal cracks, which haddeveloped at mid-depth of the beam and propagated towardthe outside edge of the loading plate. Specimens HA1 and

HA3 (shallow strut angle with and without web reinforcement)had this type of shear-compression failure.

Strains in longitudinal tension (tie) reinforcementThe strain profile in the main longitudinal reinforcement

was very similar for all specimens. The formation of a tieaction was confirmed by the nearly uniform strain distribution inthe longitudinal reinforcement over the shear span. Thestrains measured in the lower reinforcement layer,however, were slightly larger than those measured in theupper reinforcement layer. A significant reduction inlongitudinal reinforcement strains between the load pointand the support away from the failure region was observed,following a trend similar to that predicted by the strut-and-

Fig. 8Load versus displacement response (Series H).

Fig. 9Cracking pattern at failure for Specimen A1.

Fig. 10Cracking pattern at failure for Specimen B3.

Fig. 11Cracking pattern at failure for Specimens HA1and HA3.


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tie model shown in Fig. 2 for the left side of the beam. Thelongitudinal reinforcement in all specimens of Series A andH remained elastic throughout the tests, while one or twostrain gauges had just reached the yield point when a strutfailure occurred in the specimens of Series B. No signs ofanchorage problems were observed in any of the tests.

Strains in longitudinal compression reinforcementTypically, two strain gauges were attached to the compression

reinforcement, one just under the load point and the other at

the outer face of the theoretical upper compression node(right side inclined face in Fig. 2). Strains recorded at thesetwo locations were well below yielding for all specimens,with the strains under the load point being larger. In somecases, one or both of the strain gauges recorded tensilestrains at later stages of the test when the top reinforcementwas crossed by diagonal cracks (Fig. 9).

Strains in web or strut reinforcementAs shown in Fig. 3 through 5, minimum web reinforcement,

calculated according to Appendix A of the ACI Code, wasprovided in Specimens A1 and A2 (shallow strut angle), andB1 and B2 (steep strut angle), but less than the minimumspecified web reinforcement was placed in Specimens HA1(shallow strut angle) and HB1 (steep strut angle). Strainswere recorded at various locations along the main strut onthe vertical stirrup legs and on the horizontal reinforcingbars. Strain measurements and visual observations indicatedthat the web reinforcement was effective in controlling crackopening. Figure 12 shows the measured strain response forvertical and horizontal web reinforcement for Specimen A1.The strain response for the web reinforcement of all testspecimens was similar and can be summarized asfollows: 1) strains in the horizontal reinforcement werelower than those in the vertical reinforcement; 2) for thespecimens with a shallow angle strut, the level of strains inthe vertical stirrups was higher than that for specimens witha steep angle strut; and 3) for specimens with web reinforcementin Series A and B, the vertical reinforcement was close to orhad just exceeded the yield point at the moment of failure,while for Specimens HA1 and HB1, the strains measured atfailure corresponded to a stress of approximately 0.7fyand0.5fy, respectively. It should be mentioned that smooth bars,as opposed to deformed bars, were used as web reinforcement inthe critical shear span. Strain measurements, however,indicated good bond between the bars and the concrete andthus, no major differences in behavior were expected ifdeformed reinforcement had been provided.

PREDICTED VERSUS EXPERIMENTALSTRUT STRENGTHS

The main objective of this research project was toexperimentally evaluate the adequacy of the expressions inSection A.3 of the ACI Code for the strength of concretestruts in strut-and-tie models. The nominal compressivestrength of a strut without reinforcement in the direction ofthe strut axis is taken as (ACI Code Eq. (A-2))

Fns=fcuAc (6)

whereAcis the smaller cross-sectional area at either end ofthe strut, andfcuis the effective compressive strength of theconcrete in a strut, given by Eq. (7) (Code Eq. (A-3))

fcu= 0.85sfc (7)

where

s = 0.75 for a bottle-shaped strut with reinforcementsatisfying Section A.3.3 (A.3.2.2(a)); and

s = 0.60for a bottle-shaped strut without reinforcementsatisfying Section A.3.3 (A.3.2.2(b)), = 1 fornormal weight concrete.

The design of the test specimens was carried out using thestrut-and-tie model shown in Fig. 2 and following theprovisions in Appendix A of the ACI Code. The procedurefollowed to calculate the strength in the critical span ofSpecimen A1 is demonstrated in the Appendix of this paper. Itshould be mentioned that strength factors used for concretestruts (s in the ACI Code) are intended to account notonly for the effect of strut stress and strain conditions andreinforcement details, but also for uncertainties in the trussmodel,7 as discussed earlier. Therefore, any strut failuremode was considered for evaluation of the adequacy of theACI strut strength factors.

Table 1 and Fig. 13 show a comparison between thecalculated and experimental failure loads. It is worthmentioning that in order to properly evaluate the adequacy of

Fig. 12Strains in web reinforcement crossing primarydiagonal strut (Specimen A1).

Fig. 13Experimental versus calculated strengths.


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the provisions in Appendix A of the ACI Code, the predictedstrengths reported in Table 1 correspond to the maximumvalues obtained after considering various admissible strutangles. Thus, the reported values represent the admissiblesolution that led to either the least conservative or moreunconservative prediction. As can be seen in Fig. 13 andTable 1, the calculated strengths were lower than or equal tothe experimental results for the specimens in Series A and Bwith normal strength concrete. For the high-strengthconcrete beams, however, a strut strength factor s= 0.60

was found to be inadequate for use in struts not crossed byweb reinforcement, as indicated by the failure loads inSpecimens HA3 and HB3 (appear as a single point in Fig. 13),which were approximately 10% lower than the predictedstrengths. For the case of Specimens HA1 and HB1, whichhad effective web reinforcement ratios of 0.0056 and 0.0084,respectively, strengths calculated according to ACI CodeSection A.3.3.1 and using a strength factor s= 0.60 led topredicted values nearly equal to the failure loads. Thus, theselimited test results suggest that, as a minimum, an effectivereinforcement ratio of 0.01, calculated according to ACICode Section A.3.3.1, should be provided in high-strengthconcrete members when a strength factor s= 0.60 is used.Additional test data, however, are required before a definite

recommendation can be made in this regard.Peak average shear stress demands for each specimen

are also given in Table 1. Regardless of the amount of webreinforcement provided, test specimens with shallow and steepstruts sustained peak average shear stress demands exceeding0.6 and 1.1fc (MPa) (7.2 and 13.2fc [psi]), respectively. Forthe specimens subjected to average shear stresses greaterthan the limit given for deep beams in Chapter 11 of the ACICode (0.83fc [MPa]), however, the calculated strut andnode dimensions were believed to be excessive.

EVALUATION OF MINIMUM STRUTREINFORCEMENT REQUIREMENTS

IN APPENDIX A OF ACI CODE

Minimum strut reinforcement requirements are defined inSection A.3.3 of Appendix A in the ACI Code. In thatsection, sufficient steel reinforcement is required to resist atension force perpendicular to the strut axis, which iscalculated by assuming that the strut compression forcespreads out at a slope of 2:1 (longitudinal:transversedirection, refer to ACI Code Fig. RA.1.8(b)). On the otherhand, Section A.3.3.1 specifies that this requirement can beassumed to be satisfied for members with a concretecompressive strengthfc not greater than 41 MPa (6000 psi),if sufficient web reinforcement is provided such that Eq. (5)(Eq. (A-4) in ACI Code) is satisfied.

The requirements in Sections A.3.3 and A.3.3.1, whenapplied to the test specimens, were found to be neitherconsistent nor equivalent. Specimens in Series A and B, withnormal-strength concrete, were designed to satisfy therequirements of Section A.3.3.1. If, instead, Section A.3.3were to be satisfied, a much larger amount of transversereinforcement would have been required. Table 1 shows theamount of web reinforcement required by Appendix A of theACI Code and the amount provided in the test specimens. Ascan be seen, the provisions in Section A.3.3 required morethan three times the minimum reinforcement specified inSection A.3.3.1 for the specimens in Series A and B.

For the specimens of Series H, ACI Code, Section A.3.3,required reinforcement ratios of 1.4 and 2.1%, respectively,

leading to impractical and heavily congested designs. InSeries H, for which Section A.3.3.1 is not applicable becauseof the use of high strength concrete, a 6:1 slope was assumedfor the spreading of the strut force instead of the 2:1 slopespecified in Section A.3.3. As discussed in the previoussection, unconservative strength predictions wereobtained for the high-strength concrete beams withoutweb reinforcement, while relatively good agreementbetween predicted and experimental results was obtained inSpecimens HA1 and HB3, which had effective reinforcement

ratios of approximately 40% of those required by Section A.3.3.Clearly, additional experimental information needs to begenerated to draw definite conclusions with regard to theminimum web reinforcement required in high-strengthconcrete members designed using strut-and-tie models.Based on the very limited results presented herein and untilfurther data become available, a minimum effective webreinforcement ratio of 0.01, as defined by Eq. (5) (ACI CodeSection A.3.3.1), is recommended for use in high-strength concrete members designed with a strut strengthfactor s= 0.60.

The limited scope and budget for this experimentalprogram did not permit a thorough investigation of whatreasonable percentage of web reinforcement would be

required to justify the use of a strut strength factor, s= 0.75,for high-strength concrete struts. As mentioned previously,the compliance with ACI Code Section A.3.3 may lead toexcessive amounts of reinforcement that could result inimpractical designs.

SUMMARY AND CONCLUSIONSThe adequacy of the strength factors for concrete struts

given in Appendix A of the 2002 ACI Code was evaluatedbased on the results from the tests of 12 reinforced concretedeep beams. The main design variables considered weremain strut angle with respect to the member longitudinal (tie)axis, amount of web reinforcement crossing the strut, andconcrete strength. The following conclusions were drawn

from the results of this investigation:1. The strut strength factors given in Appendix A of the

2002 ACI Code were found to be adequate for use in normal-strength concrete bottle-shaped struts crossed by either noreinforcement (s = 0.60) or crossed by the minimumtransverse reinforcement specified in ACI Code SectionA.3.3.1 (s= 0.75);

2. The provisions for required minimum transversereinforcement in normal-strength concrete members givenin Appendix A of the 2002 ACI Building Code should bereevaluated. Substantial differences in the required amountof reinforcement were obtained when applying the provisions inSections A.3.3 and A.3.3.1 to the test beams, with the formerleading to substantially larger reinforcement ratios; and

3. Results from the tests of four high-strength concrete beamssuggest that, as a minimum, an effective web reinforcementratio of 0.01, calculated according to ACI Code SectionA.3.3.1, should be provided in high-strength concretemembers when a strength factor s= 0.60 is used. Additionaltest data, however, are required before a definite recommen-dation can be made in this regard.

ACKNOWLEDGMENTSThe work presented in this paper was carried out at the Structural Engineering

Laboratory, University of Michigan, Ann Arbor, Mich. Financial supportwas provided by the ACI Concrete Research Council and the University ofMichigan. The first author would like to thank the Universidad de los Andes,


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Mrida, Venezuela, for its support during his sabbatical leave. Thanks arealso extended to University of Michigan student A. L. Libbrecht, and toL. B. Fargier-Gabaldn and B. Afsin Canbolat for their valuable help during theexperimental phase of this project. The conclusions and recommendationscontained in this paper are those of the authors and do not necessarily representthe opinion of the sponsors.

REFERENCES1. MacGregor, J. G.,Reinforced Concrete, Mechanics and Design, 2nd

Edition, Prentice Hall, 1988, 848 pp.

2. Schlaich, J.; Schfer, K.; and Jennewein, M., Toward a ConsistentDesign of Structural Concrete, PCI Journal, V. 32, No. 3, 1987, pp. 74-150.

3. Ramirez, J. A., and Breen, J. E., Evaluation of a Modified Truss-Model Approach for Beams in Shear, ACI Structural Journal, V. 88,No. 5, Sept.-Oct. 1991, pp. 562-571.

4. Yun, Y. M., and Ramirez, J. A., Strength of Struts and Nodes in Strut-Tie Model,Journal of Structural Engineering, ASCE, V. 122, No. 1, 1996,pp. 20-29.

5. Nielsen, M. P., Limit Analysis and Concrete Plasticity, CRC PressLLC, 1999, 936 pp.

6. Rogowsky, D. M., and MacGregor, J. G., Design of ReinforcedConcrete Beams, Concrete International, V. 8, No. 8, Aug. 1986, pp. 49-58.

7. MacGregor, J. G., Derivation of Strut-and-Tie Models for the 2002ACI Code, Examples for the Design of Structural Concrete with Strut-and-Tie Models, SP-208, K.-H. Reineck, ed., American Concrete Institute,Farmington Hills, Mich., 2002, pp. 7-40.

8. ACI Committee 318, Building Code Requirements for StructuralConcrete (ACI 318-02) and Commentary (318R-02), American Concrete

Institute, Farmington Hills, Mich., 2002, 443 pp.9. Vecchio, F., and Collins, M. P., The Modified Compression FieldTheory for Reinforced Concrete Elements Subjected to Shear, ACIJOURNAL, ProceedingsV. 83, No. 2, Feb. 1986, pp. 219-231.

10. Vecchio, F. J., and Collins, M.P., Compression Response of CrackedReinforced Concrete, Journal of Structural Engineering, ASCE, V. 119,No. 12, 1993, pp. 3590-3610.

11. Vecchio, F. J., Disturbed Stress Field Model for ReinforcedConcrete: Formulation,Journal of Structural Engineering, ASCE, V. 126,No. 9, 2000, pp. 1070-1077.

12. Matamoros, A., and Wong, K. H., Design of Simply SupportedDeep Beams Using Strut-and-Tie Models,ACI Structural Journal, V. 100,No. 6, Nov.-Dec. 2003, pp. 704-712.

13. Quintero-Febres, C.; Parra-Montesinos, G.; and Wight, J. K.,Evaluation of Strength Factors for Concrete Struts in Deep ConcreteMembers,Report No. UMCEE 05-04, Department of Civil and EnvironmentalEngineering, University of Michigan, Ann Arbor, Mich., 2005, 78 pp.

14. Wight, J. K., and Parra-Montesinos, G., Use of Strut and Tie Modelfor Deep Beam Design as Per ACI 318 Code, Concrete International,V. 25, No. 5, May 2003, pp. 63-70.

APPENDIXFigure A-1 shows the strut-and-tie model for the critical

span of Specimen A1. The selection of the strut angle andthus, the depth of the top node hn, was made such as tomaximize the predicted shear strength. This was done withthe purpose of determining the most critical admissible solutionfor the strut-and-tie model considered, according toAppendix A of the ACI Code. Other admissible solutionswill be below this maximum prediction and, thus, theywould represent more conservative predictions.

From Fig. A-1, it can be seen that the applied load at thetop of the beam, P, was divided into two forces of magnitudeequal to the beam shears at the left and right beam portions,0.35P and 0.65P, respectively, acting over a proportionalplate width (refer to Reference 14 for detailed informationabout this procedure). The critical solution was found for astrut angle = 28.5 degrees. For this angle, hn= 195 mm.The forces acting at the nodes and struts can be expressed interms of the applied load Pas follows

Fstrut= = 1.36P (A-1)0.65P

sin --------------

Ftie= Fhorizontal at nodes= Fstructcos= 1.20P (A-2)

Check of top node and adjacent inclined strutStrut width and strength at intersection with top node, (ws)t

and (Fns)t, respectively

(ws)t= (0.65 150 mm)sin+ hncos= 125 mm (A-3)

(Fns)t= 0.85sfc (ws)tbw (A-4)

(Fns)t= 0.85 0.75 22 MPa (215 mm)(150 mm) = 450 kN

Equation (A-4) leads to an applied load P= (Fns)t/1.36 =330 kN.

Strength of vertical and horizontal faces of upper rightnode, (Fnn)t-vand (Fnn)t-h, respectively

(A-5)

(Fnn)t-v= 0.85 1.0 22 MPa(0.65 150 mm)(150 mm) = 275 kN

(A-6)

(Fnn)t-h= 0.85 1.0 22 MPa(195 mm)(150 mm) = 545 kN

Equation (A-5) leads to P= (Fnn)t-v/0.65 = 420 kN, whileEq. (A-6) leads to P= (Fnn)t-h/1.20 = 455 kN.

Check of bottom node and adjacent inclined strutStrut width and strength at intersection with bottom node,

(ws)band (Fns)b, respectively

(ws)b= (150 mm)sin+ (180 mm)cos= 230 mm (A-7)

(A-8)

(Fns)b= 0.85 0.75 22 MPa(230 mm)(150 mm) = 485 kN

Equation (A-8) leads to P= 355 kN.

Fnn( )t-v 0.85nfc 0.65 150 mm( ) bw=

Fnn( )t-h 0.85nfchn bw=

Fns( )b 0.85sfc ws( )b bw=

Fig. A-1Strut-and-tie model in critical span of Specimen A1.


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ACI Structural Journal/July-August 2006586

Strength of vertical and horizontal faces of bottom node,

(Fnn)b-vand (Fnn)b-h, respectively

(Fnn)b-v= (A-9)

(Fnn)b-v= 0.85 0.80 22 MPa(150 mm)(150 mm) = 335 kN

(Fnn)b-h = (A-10)

405 kN < 715 kN

Equation (A-9) and (A-10) lead to P= 515 kN and P=340 kN, respectively.

The strength of the beam is then governed by the upper strutend, which leads to a maximum applied load of 330 kN and acorresponding shear in the critical span of 215 kN.

0.85nfc 150mm( ) bw

0.85 nfchti ebw Asfy

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