Evaluation of Load Transfer and Strut Strength of Deep Beam

678 ACI Structural Journal/September-October 2009

ACI Structural Journal, V. 106, No. 5, September-October 2009.MS No. S-2008-205 received June 24, 2008, and reviewed under Institute publication

policies. Copyright © 2009, American Concrete Institute. All rights reserved, including themaking of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the July-August 2010 ACI Structural Journal if the discussion is received by March 1, 2010.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

This paper presents laboratory results of 12 deep beams in whichthe longitudinal reinforcement was anchored into the supportusing short straight bar anchorages. The shortest anchoragelengths provided were below 50% of those required by ACI 318-08Chapter 12 provisions. Four different specimen groups wereconstructed using three different shear span-depth ratios (a/d) andtwo longitudinal bar sizes. Most of the beams failed by strutcrushing after yielding of the main longitudinal reinforcement atmidspan. Only those specimens with the shortest anchorage lengthin each group developed concrete splitting failures along theanchorage region. The effect of a/d and anchorage length on strutstrength and load transfer mechanism observed in the tests ispresented and discussed. Test results indicate that a significantportion of the applied shear force may be transferred through trussaction even in beams with low a/d. In addition to well-known variablessuch as a/d and transverse reinforcement content, short anchoragelength affected the load transfer characteristics of the deep beamstested in this research.

Keywords: anchorage length; deep beams; strength; strut-and-tie models.

INTRODUCTIONDesign of deep beams using strut-and-tie models involves

the selection of an adequate model that captures the anticipatedbehavior with reasonable accuracy. Model selection is typicallybased on knowledge about the load transfer mechanismspresent in deep beams. It is now commonly accepted thatdeep beams transfer shear through one of two basic loadtransfer mechanisms identified as either a tied-arch mechanismor truss mechanism (Fig. 1). The two most important factorsaffecting the type of load transfer mechanism are the shearspan-depth ratio (a/d) of the beams and the amount of trans-verse reinforcement. There seems to be general agreement thatas a/d increases, a higher fraction of load is transferredthrough truss action (Fig. 1(b)). Conversely, beams withlower a/d transfer a larger portion of the load through tied-arch action (Fig. 1(a)). Beams with intermediate a/d transferload through a combination of these two mechanisms, butthere is disagreement on how to estimate the fraction transferredby each assumed model. Similarly, larger amounts of transversereinforcement lead to a higher fraction of load transferred bytruss action. A minimum amount of web reinforcement (verticaland horizontal) is required to prevent splitting of diagonal strutsforming between load and supports of deep beams.

Load transfer through tied-arch action has importantimplications in design. As can be observed by equilibrium ofthe simplified model shown in Fig. 1(a) depicting a tied archmodel, the horizontal tensile force in the tie at the bottom ofthe model is required to maintain its maximum valuethroughout the span of the beam. This assumed behaviorplaces a high anchorage demand at nodes located above thesupports, because if one designs the horizontal tie to yield atultimate load, then yielding has to be maintained at the face

of each node over each support. Anchoring these horizontalbars, which often have large diameter and are placed inmultiple layers to satisfy the tie force requirement, becomesvery difficult and results in reinforcing bar congestion nearthe ends of the beam. To anchor the bottom tie according tocode procedures (for example, ACI 318), hooked bars or tee-headed reinforcement are almost always used.

The anchorage condition over the support of deep beamsmight benefit from the presence of transverse confiningstresses that generate from the reaction and diagonal strutthat meet at the end node. The beneficial effect of transversenormal stresses on the anchorage condition of deep beamswas examined and discussed in detail by Roy and Breña(2008). The test results indicated that beams with straight baranchorages of only approximately 50% of the lengthrequired by provisions in Chapter 12 of ACI 318-08 (ACICommittee 318 2008) were able to develop peak loadscomparable with beams where longer anchorages wereprovided. In some of these beams with a/d as low as 1.0 or1.5, limited or no yielding was measured in longitudinalreinforcement at the face of the support node indicating thatload transfer by tied-arch action was not predominant evenin beams with this low a/d. Furthermore, reinforcing baryielding at midspan was observed in many of these beams andmaintained to load levels that caused crushing of the main strutsforming after cracking of the beams. These results indicatethat an alternate load path was developed in the beams withshort anchorages (truss action), which allowed developmentof higher load without placing the high anchorage demand atthe support node that is generated from tied-arch action.

The purpose of this paper is to examine and discuss thebehavior of struts and nodes in these deep beams fabricated

Title no. 106-S63

Evaluation of Load Transfer and Strut Strength of Deep Beams with Short Longitudinal Bar Anchoragesby Sergio F. Breña and Nathan C. Roy

Fig. 1—Strut-and-tie models consistent with load transfermechanisms in deep beams.

679ACI Structural Journal/September-October 2009

using short longitudinal bar anchorages. The strength ofstruts is investigated based on the results of the tests. A criticalevaluation of the apparent load transfer mechanisms in this typeof beams was conducted to provide insight into the effect ofshort bar anchorages on how these beams transfer shear.

RESEARCH SIGNIFICANCEDesign of deep beams is commonly done using strut-and-

tie models following Appendix A in ACI 318-08. Engineersare given freedom to select the strut-and-tie model for designwith little guidance as to the merits of each model. Thechoice of model is largely based on knowledge of loadtransfer mechanisms for beams with different a/d. Deepbeams with a/d < 1.0 are commonly assumed to transfer thetotal load through tied-arch action, whereas the most widelyaccepted load transfer mechanism in slender beams with a/d ≥4.0 is based on truss action. Loads in deep beams with inter-mediate a/d (between 1.0 and 4.0) are transferred through acombination of these two mechanisms. This paper willprovide data about load transfer mechanisms in deep beamswith short longitudinal bar anchorages. The informationprovided in this paper will assist in implementation of strut-and-tie models and an improved understanding of the meritsof different models when used in design.

ACI 318-08 STRUT AND NODESTRENGTH EQUATIONS

Design provisions for strut-and-tie models are containedin Appendix A of ACI 318-08. The strength of elementsdesigned using strut-and-tie models, according to thesedesign provisions, is governed by the strength of the weakestelement of the strut-and-tie model (strut, tie, or node). Forcesacting in struts, ties, or nodes must be equated to the resistanceof each element to solve for the load carrying capacity of thesystem. The focus in this paper will be those modelsgoverned by either strut or node strength. It will be assumedthat tie strength does not govern the capacity of the chosenstrut-and-tie model. The nominal strength of struts in a strut-and-tie model built according to ACI 318-08 is determined from

Fns = fceAcs (1)

fce = 0.85βs fc′ (2)

where Acs is the minimum cross sectional area (strut width wtimes element thickness b) at one of the two ends of the strut;fce is equal to the effective concrete strut strength; and βs isan effective concrete strength factor dependent on strut geometryand transverse strain conditions. The reduction in uniaxialconcrete compressive strength, when subjected to off-axistensile strains, is well known. In ACI 318-08, this decreasein strut strength is captured through the use of βs. The factorβs equals 1.0 for prismatic struts that have a constant cross-

sectional area throughout their length. For bottle-shapedstruts, βs equals 0.75 if the strut is crossed by the minimumamount of reinforcement specified in ACI 318-08 sectionA.3.3, or 0.60λ (λ is a lightweight aggregate concrete factor)if the reinforcement does not comply with ACI 318-08,Section A.3.3, or no reinforcement is present.

Similarly, the nominal strength of nodal zones (Fnn) inACI 318-08 is calculated using

Fnn = fceAnz (3)

where Anz represents the area of the node face perpendicularto the applied force on the node that results in the higheststress on the nodal zone. In the case of nodes, fce is calculatedusing Eq. (2) with βn instead of βs. The factor βn depends onthe type of node, and it equals 1.0 for nodal zones with onlycompression forces acting on them, 0.80 in nodal zoneswhere one tie is anchored, or 0.60 for nodal zones anchoringtwo or more ties. These nominal strengths computed by Eq. (1)and (3) must be multiplied by a strength reduction factor of0.75 to calculate the design strength according to ACI 318.

COMMONLY ACCEPTED FORCE TRANSFER MECHANISMS FOR DEEP BEAMS

Rather than relying on empirical procedures as was donein the past, strut-and-tie models are now used extensively inthe design of deep beams. These models are commonlyconstructed to approximately follow the force transfer pathanticipated in deep beams. The two most commonlyaccepted strut-and-tie models used in deep beam design aretied arch or truss models depending on the a/d of the beam.A tied arch model is predominantly used in design of shortbeams (a/d equal to 1.0 or less), with the assumption thatload is transferred directly from load point into supportthrough the formation of concrete struts (direct strut mechanism).The horizontal component of each strut at the support is set inequilibrium by a horizontal tie extending the full length ofthe beam (Fig. 1(a)). The tie force in this model is constantthroughout the span, so the longitudinal reinforcementforming the tie has to be anchored at the face of the node overeach support to develop the yield stress fy. The criticalsection for development of reinforcement in ACI 318-08 isdefined as the section where the centroid of the reinforcementleaves the extended nodal zone.

For longer deep beams (a/d of 4.0 or above), load transferoccurs indirectly from load point into support through two ormore struts that form between diagonal cracks within theshear span of the beam (Fig. 1(b)). The vertical componentof struts that reach the bottom longitudinal reinforcementwithin the shear span is set in equilibrium by vertical tieforces generated in stirrups that enclose the longitudinalreinforcement and are well anchored in the compressionzone of the beams. The bottom longitudinal reinforcement isagain required to equilibrate the horizontal component ofstrut forces at the bottom nodes, and compression stressesnear the top of the beam equilibrate horizontal strut forcecomponents at top nodes. The force transfer mechanism inthis type of beams closely resembles forces in a truss, withtop and bottom chords carrying compressive and tensileforces, respectively, diagonal web members carryingcompressive forces, and vertical web members carryingtensile forces.

In beams of intermediate length (a/d between 1.0 and 4.0),a combination of force transfer mechanisms is commonly

Sergio F. Breña, FACI, is an Associate Professor in the Department of Civil andEnvironmental Engineering at the University of Massachusetts Amherst, Amherst,MA. He is Secretary of ACI Committee 369, Seismic Repair and Rehabilitation, and amember of the ACI Publications Committee; ACI Committees 374, Performance-Based Seismic Design of Concrete Buildings, and 440, Fiber Reinforced PolymerReinforcement; Joint ACI-ASCE Committee 445, Shear and Torsion; and E803,Faculty Network Coordinating Committee.

Nathan C. Roy is a Structural Engineer with LeMessurier Consultants, Cambridge,MA. He received his BS and MS in civil engineering from the University of MassachusettsAmherst in 2004 and 2006, respectively.


assumed. Guidance on the fraction of force transferred byeither truss or tied arch action is sparse. Federation Internationalde La Precontrainte Recommendations (FIP 1999) indicate thatthe force transfer in beams with a/d between 0.5 and 2.0occurs through a combination of truss and tied arch action.Force transfer in beams transitions from entirely tied archaction to entirely truss action as a/d increases from 0.5 to 2.0using the following relationship

(4)

where F represents the total force being transferred, F1 is theforce transferred through truss action, and z is the internallever arm (distance between top and bottom chords of themodel). Using strain gauge data obtained from instrumentedstirrups in tests of deep beams with a/z = 1.49, Uribe andAlcocer (2001) found that the direct strut mechanismcontributed slightly more than implied by the FIP (1999)recommendations although the implied failure mode wasconsistent with test observations.

The models mentioned above only account for the contributionof horizontal reinforcement near the bottom of the beam.Vertical web reinforcement is only explicitly included intruss models but not in tied arch models. Horizontal webreinforcement is not included in either model and its effect isonly considered when determining the strength of struts.

DESCRIPTION OF LABORATORY TESTSTests reported in this paper were conducted for simply

supported deep beams subjected to a single concentratedload at midspan. Laboratory specimens consisted of 12 deepbeams divided into four different groups according to threedifferent a/d (1.0, 1.5, or 2.0) and two sizes of the mainlongitudinal bars (No. 5 or No. 6). The main variable in eachgroup of beams was the anchorage condition of longitudinalbars over one of the supports. On the test side, the longitudinalreinforcement was continued on different distances past thesupport node in the various specimens (straight baranchorage). Longitudinal reinforcement on the far side of thebeams was anchored past the support using a standard90-degree hook to preclude anchorage failures there.

Specimens are designated according to a/d and anchoragelength of straight bars on the test side of the beams. The firsttwo digits in the beam designation correspond to a/d (1.0,1.0L, 1.5, or 2.0), and the last three digits correspond to theratio of provided-to-design anchorage length on the test sideof the specimens. The design anchorage length is defined asthe length calculated using Chapter 12 of ACI 318-08. Thisratio varied in each group of beams as listed in the specimendesignation in Table 1, but all groups had beams with 75 and 50%of the anchorage length required by ACI 318-08, Chapter 12.

Specimen nominal dimensions and reinforcing patternsare shown in Fig. 2. All beams had a nominal width of 152 mm(6 in.) and total depths of 635, 457, or 356 mm (25, 18, or 14 in.)for beams with a/d of 1.0, 1.5, or 2.0, respectively. Longitudinalreinforcement consisted of No. 5 or No. 6 main bottom bars andNo. 3 top bars needed for reinforcing cage fabrication. Inspecimens in Group 1.0L, the longitudinal bar size used wasNo. 6, whereas No. 5 bars were used in the other threegroups. Vertical stirrups were fabricated using deformedwire (D4) with an actual yield stress of 605 MPa (88 ksi).Horizontal bars made from D4 wire were placed at 152 mm(6 in.) spaces in the beam web, close to each of the lateralbeam faces. Web reinforcement complied with section A.3.3in ACI 318-08.

Beams were designed using a nominal compressivestrength of concrete equal to 28 MPa (4 ksi) and a reinforcingsteel nominal yield stress fy equal to 414 MPa (60 ksi). Justbefore testing each beam, specimen dimensions were verifiedand companion concrete cylinders were tested to determine theas-built geometry of the beams and the actual strength ofconcrete. Due to formwork flexibility, the actual width of thebeams varied slightly from the nominal value of 152 mm (6 in.).The measured concrete compressive strength along with the as-built width of each specimen at the time of testing is listed inTable 1. Further details of the beam reinforcement configurationand provided anchorage length on the test side of the beams arepresented in a previous paper (Roy and Breña 2008).

Test setup and instrumentationThe span in all beams was equal to 1.22 m (48 in.).

Specimens were subjected to a single concentrated force atmidspan. Thick steel plates (25 mm [1 in.]) were placed belowthe loading point and above reaction points to avoid localized

F113--- 2a

z------ 1–⎝ ⎠⎛ ⎞F=

Table 1—As-built properties of specimens and observed failure parameters

Group a/d Specimen fc′ , MPa (psi) Beam width b, mm (in.) Peak load Pu, kN (kip) Failure mode Pu/Py* Pu/Pcr-d

†

1.0

DB1.0-1.00 33.3 (4830) 165 (6.5) 677 (152) Strut – S 1.43 2.17

DB1.0-0.75 31.7 (4600) 173 (6.8) 743 (167) Strut – H 1.55 2.10

DB1.0-0.50 30.6 (4440) 157 (6.2) 729 (164) Strut – H 1.53 2.34

DB1.0-0.32 27.0 (3915) 152 (6.0) 667 (150) Strut – S 1.43 2.14

1.0LDB1.0-0.75L 29.9 (4340) 155 (6.1) 741 (167) Strut – H 1.15 2.39

DB1.0-0.28L 29.4 (4265) 155 (6.1) 642 (144) Anchorage ‡ 2.06

1.5

DB1.5-0.75 32.7 (4745) 152 (6.0) 459 (103) Strut – S 1.49 1.72

DB1.5-0.50 34.1 (4945) 152 (6.0) 423 (95) Strut – H 1.44 1.58

DB1.5-0.38 33.8 (4900) 152 (6.0) 427 (96) Anchorage 1.45 2.40

2.0

DB2.0-0.75 34.7 (5035) 155 (6.1) 313 (70) Strut – H 1.56 2.33

DB2.0-0.50 33.0 (4790) 155 (6.1) 297 (67) Strut – H 1.40 1.68

DB2.0-0.43 35.6 (5165) 155 (6.1) 266 (60) Anchorage 1.28 1.50*Ratio of maximum load to yield load.†Ratio of maximum load to diagonal cracking load.‡Yielding not observed.Note: S is strut crushing observed on test end (straight anchorage) of specimen; and H is strut crushing observed on far end (hooked anchorage) of specimen.


crushing at the nodal zone. The beam supports consisted of apin (far end of beam) and a roller (test end of beam) in all tests.A 445 kN (100 kip) load cell was placed underneath eachsupport to measure reactions throughout the tests.

External and internal instrumentation was placed atselected locations in the specimens to relate the measuredresponse with parameters from common strut-and-tie modelsused for design of deep beams. To investigate the effect ofthe short reinforcing bar anchorage on tie stresses, reinforcingbars were instrumented internally using strain gauges. Theresults of these measurements were presented and discussedin detail in a separate paper (Roy and Breña 2008). Theinstrumentation that was used to determine the behavior ofstruts in the four specimen groups is described in detail in thissection.

During testing, linear potentiometers were attached on bothbeam faces to threaded rods embedded into the concretebefore beam casting and defined the control points to calculateaxial strains developed along struts. The rod placement wasselected to measure concrete deformation along the directionof main struts in selected strut-and-tie models of the differentspecimen groups depending on beam a/d. For SpecimenGroups 1.0 and 1.0L, the potentiometers measured axialdeformation of a direct strut forming between loading point

and support on the test side of the beams. In Groups 1.5 and2.0, two possible strut orientations were monitored by placinginstruments that followed a direct path (tied-arch model) or anindirect path (truss model) from loading point to supports.Figure 3 illustrates the potentiometer placement for the fourspecimen groups tested in this research.

The stress condition in the extended nodal zone on the testside of the beams was of interest because of the shortlongitudinal bar anchorage used in the design of the specimens.Properly anchored straight bars extending past the back face ofthe nodal zone are assumed to generate a uniform stressdistribution on the vertical node face. These stresses areessential to preserve node equilibrium in strut-and-tiemodels and avoid node failure. Strains in the nodal zone onthe test side of the specimens were measured using surfacestrain gauges bonded to the surface of the concrete in a 0-45-90degree rosette pattern. The instrument placement within theextended nodal zone is illustrated in Fig. 3(d).

Strain gauges were also bonded to reinforcing bars atmidspan and within the extended nodal zone over thesupports. These gauges were used to evaluate the developmentof yield stresses, primarily within the extended nodal zoneon the test side of the specimens as an indicator of the loadtransfer mechanism being developed in the beams. Details of

Fig. 2—Specimen geometry, reinforcement, and experimental test setup.


reinforcing bar gauge placement and a description of themeasured strains at these locations is provided by Roy andBreña (2008).

TEST RESULTSAll specimens failed at higher loads than anticipated by the

strut-and-tie models used for design. Even specimens with theshortest longitudinal bar anchorages in each group (DB1.0-0.32,DB1.0L-0.28, DB1.5-0.38, and DB2.0-0.43) were capable ofreaching a peak load approaching the maximum loadmeasured in specimens within their corresponding group andexhibited yielding of the main reinforcement and moderateductility before failure. Ultimate load was between 1.15 and1.56 times higher than the load corresponding to yielding oflongitudinal reinforcement at midspan, indicating thedevelopment of an alternate load transfer mechanism. Inspecimens where anchorage failure was observed on the testend of the beams, the ratio between ultimate and yield loadwas similar to or lower than other specimens in the samegroup (Table 1). The loads measured at flexural and diagonalcracking, yielding, and failure of the specimens are listed inthe companion paper (Roy and Breña 2008).

Observed failure modeFailure of all specimens, except those with the shortest

anchorage length, was characterized by strut crushing eitheron the test side or on the far side of the beams independentof provided anchorage length. Strut crushing was typicallyobserved at the boundary between the top end of the strut andthe nodal zone under the loading point (refer to Fig. 4). Onlyspecimens DB1.0-0.28L, DB1.5-0.38, and DB2.0-0.43exhibited an anchorage failure on the test side of the beamsprior to strut crushing. Strains and stresses in struts on thetest side of the beams were able to be examined in detail forall other specimens that failed by strut crushing. Detailedcracking maps for all specimens are presented and discussedin detail by Roy and Breña (2008).

Figure 4 illustrates typical cracking and strut crushing inthree different specimens from groups representing differenta/d ratios. Other specimens failing by strut crushing in eachgroup exhibited similar cracking and crushing characteris-tics. As shown in Fig. 4(a) and (b), the inclination of diagonalcracks and strut crushing in beams from Specimen Groups 1.0and 1.5 approximately followed the direction of a direct strutbetween the load point and one of the supports. A compar-ison of Fig. 4(c) and (d), which show the front and back faceof a specimen in Group 2.0, illustrates how the inclination ofdiagonal cracks and strut crushing roughly followed thedirection of an indirect (truss) load-transfer mechanism thatformed between loading point and support. This behavior isconsistent with the recognized influence of a/d on behaviorof deep or slender beams (MacGregor and Wight 2005).

Comparison of strains in nodal zone with strains measured in direct strut

Compressive strains were measured in all specimengroups along lines that approximately coincided with theexpected direction that struts would form if loads were transferredthrough tied-arch action. In Specimen Groups 1.5 and 2.0,compressive strains were also measured along the directionof struts in a strut-and-tie model representing transfer ofshear through truss action. Additionally, strains weremeasured within the extended nodal zone on the test side ofthe specimens using surface strain gauges bonded at an angleof 45 degrees with the horizontal in all specimens. Strainsmeasured in the struts and nodes are compared for the threespecimens with longest anchorage lengths in Group 1.0 inFig. 5. In this group, the direct strut forming between loadpoint and support was inclined at approximately 45 degreesso the readings on the two elements of the strut-and-tie tied-arch model (strut and node) could be compared directly.

Several distinct features can be observed from the load-strain plots in Fig. 5. The strain measured in differentinstruments at lower loads is approximately equal inSpecimens DB1.0-1.00 and DB1.0-0.75, but quite differentin Specimen DB1.0-0.50. Strains at lower loads, below aload approximately corresponding to first diagonal cracking,seemed to be increasing linearly. This load level is indicatedby an oval or circle in the figures. After this point, the readingsin different instruments departed significantly with strainsmeasured in Instruments L2 or L3, positioned within the strutwidth registering the largest value and the strain measuredwithin the extended nodal zone being one of the lowest. Thedifference in maximum compressive strain at peak loaddepended largely on whether crushing and diagonal cracksformed within the strut width on the test side of the beams ornot. In Specimen DB1.0-1.00, crushing was observed at the

Fig. 3—Specimen instrumentation used in struts and nodes.

ACI Structural Journal/September-October 2009 683

top of the diagonal strut on the test side and explains thelarge compressive strains registered in this specimen. Strutcrushing occurred on the far side of Specimens DB1.0-0.75and DB1.0-0.50, explaining the lower compressive strainsrecorded at failure compared with Specimen DB1.0-1.00. InSpecimen DB1.0-0.75, diagonal cracks did not form withinthe estimated width of the direct strut, whereas cracksformed at approximately the same inclination as the strut inSpecimen DB1.0-0.50. The presence of diagonal crackswithin the strut width may have attributed to a stiffnessreduction of the strut, leading to higher compressive strains.Instruments L2 and L3 in Specimen Groups 1.0 and 1.0Lwere thought to be the most representative of observed strut

Fig. 4—Selected pictures illustrating strut crushing inspecimens from different groups.

Fig. 5—Measured strain in direct strut and nodal zonestrain at 45 degrees.


behavior, so only these are used to determine the averagestrut strain in subsequent calculations.

Effect of anchorage length on strain developed in direct strut

The influence of anchorage length on strain developed inthe direct strut on the test side of specimens with different a/dwas examined by comparing strains developed in strutswithin specimens of the same group (Fig. 6). The largestdifferences in strut strains were observed in Group 1.0 specimens(Fig. 6(a)), with lower compressive strains developing (ingeneral) as anchorage length decreased. The only specimenin Group 1.0 that did not fit this trend was Specimen DB1.0-0.75,in which no diagonal cracks formed within the direct strut widthas previously discussed. Differences in strut strains were lesspronounced in specimens within Groups 1.5 and 2.0, asshown in Fig. 6(b) and (c), so no clear detrimental influence

of anchorage length was observed in strain development forspecimens in these groups. Also, due to the larger a/d inthese specimen groups, it is conceivable that load transferwould occur predominantly by truss action so anchoragedifferences would not affect development of direct strutstrains as much. It is interesting to note that strut strains inSpecimen DB2.0-0.43, the specimen with the shortestanchorage length of this group, became tensile as loadincreased, perhaps due to the critical diagonal crackwidening, excessively creating a vertical offset between themonitoring points for instrument L1.

EVALUATION OF STRUT STRENGTH AND EFFECTIVE STRENGTH FACTOR

A key parameter in the analysis and design of concreteelements using strut-and-tie models according to ACI 318-08 isthe definition of a strut effective strength factor used to calculatestrut strength. The ACI strut effective strength factor (βs) ismeant to capture the lower compressive strength of concretewhen subjected to off-axis tensile strains as commonlyoccurs in webs of beams. The instrumentation used tomonitor strains in struts in the specimens was used with anassumed strut-and-tie model to determine experimental valuesof strut-effective strength factors for the different specimens,considering that the majority of the specimens tested in thisresearch failed by strut crushing.

To facilitate calculations to determine the strut strengthfrom experimental results, the total load in the specimenswas assumed to be transferred through tied-arch action fromthe loading point to support in all specimens. This assumptionis not entirely correct because part of the load is also transferredthrough truss action, particularly in the more slender specimengroups. The number and inclination of struts entering the topnode adjacent to the loading point of the specimens is illustratedin Fig. 7. Only this node was considered critical, because strutcrushing was always observed at or near the top node. Asingle strut entering the top node, as opposed to various strutsrepresenting the different load transfer mechanisms, may beused to represent the resultant of all struts. This resultant wouldhave a different angle than a strut following a direct path fromload to support because of the presence of other load transfermechanisms. The slight difference in angle between the resultantstrut and a direct strut, however, is likely to affect strut strengthcalculations minimally (Fig. 7).

A tied-arch strut-and-tie model consistent with the peakmeasured load in the tests was then developed for strutstrength evaluation. The geometry of the tied-arch modelwas established so that node strength was not reached at thepeak load because no crushing was observed in the nodalregion of the beams. The procedure used followed the oneproposed by Wight and Parra-Montesinos (2003), where thestrut inclination angle and width are determined iteratively.From symmetry of the test setup, the total applied force onthe beams resulted in equal shear force transferred to eachsupport. The force in the direct strut forming between loadand support in the tied-arch model is FS = V/sinα, where α isthe angle of inclination of the strut measured from horizontal(Fig. 8). The angle α depends on the horizontal distancebetween the resultant of one-half of the applied force and thesupport (560 mm [22 in.]), and the vertical distance betweenthe top and bottom chords in the model (z). The top-nodeheight hnode-top was initially assumed equal to 50 mm (2 in.)to initiate iterations. After determining FS for the initial α,the top horizontal compressive force was calculated as

Fig. 6—Direct strut strain development in specimens withindifferent groups.


FC = FS cosα; this force was then equated to the nominalstrength of the top node given by Eq. (2) and (3) with βn = 1.0to solve for the required top-node height to avoid node crushing

(5)

The depth between top and bottom chords in the model, z,was then revised and a new strut angle α was determined toavoid nodal failure

(6)

Iterations were conducted until hnode-top was approximatelythe same in two subsequent calculation cycles. In all iterations,the height of the bottom node was assumed equal to 102 mm(4 in.), twice the distance between the bottom face of thebeams and the centroid of the bottom longitudinal reinforcement.

Once convergence was achieved and an appropriate strutinclination was found, the top and bottom widths of the strutwere determined from

wstrut = Lplatesinα + hnodecosα (7)

where the plate widths and node heights used for the top andbottom nodes in the previous equation corresponded to thoseshown in Fig. 8(a). Because the bottom plate is wider thanhalf the top plate, the top end of the strut ended up governingstrut strength in all the specimens. This is consistent with thelocation of observed concrete spalling near the top of thestruts in all specimens that failed by strut crushing (refer toFig. 4). The stress at the top end of the strut was thencalculated using

(8)

where the as-built width of the specimens, b, was used andws-top is the width of the strut determined from Eq. (7) at thetop node. Table 2 summarizes the final strut angle, top-nodeheight, and strut stress at the top end for the assumed tiedarch model.

hnode-topFC

fceb---------

FS αcos

0.85fc′ b------------------- V

0.85fc′ b αtan--------------------------------= = =

z dhnode-top

2--------------------–=

fstrutFS

ws-topb-----------------=

Effective concrete strength factorStrut stresses fstrut determined using the peak measured

shear force in the tests was equated to the ACI strut strengthequation (Eq. (2)) to estimate the strut effective strengthfactor βs for all specimens. The resulting values for βs rangedbetween the value corresponding to prismatic and bottle-shaped struts as defined in ACI 318-08, with higher averagevalues obtained in Specimen Groups 1.0 and 1.0L. A plotshowing the decrease of βs with increasing a/d is shown inFig. 9. For comparison, results from deep beams testsreported by Quintero-Febres et al. (2006) are also shown inthe figure (hollow symbols). The decreasing trend in βs, withan increase in a/d, is less pronounced in their specimens butthe plot clearly demonstrates an influence of a/d on theconcrete strut strength factor (βs). It is also of interest to notethat the value recommended for prismatic struts in ACI 318(βs = 1.00) was not reached even for deep beams with thelowest a/d. On the other hand, the value assumed for bottle-shaped struts satisfying web reinforcement requirements ofACI A.3.3 (βs = 0.75) was conservative for all ranges of a/d;that is, higher βs values were obtained in all specimens. Shortbar anchorages did not seem to influence βs values significantlyas observed in values reported in Table 2 for specimens witheven the shortest anchorage length within each group.

EXPERIMENTAL EVALUATION OF SHEAR TRANSFER MECHANISM

As mentioned previously, two major force transfer pathsfrom load point application to support (direct or indirectpath) commonly occur in deep beams with different a/d. The

Fig. 7—Replacement of forces at top node with singlediagonal resultant.

Fig. 8—Model and node geometry used for strut strengthevaluation.


direct path is often represented using a tied-arch strut-and-tiemodel and the indirect path is often related to a truss model.An experimental evaluation of the fraction of the total sheartransferred through both load mechanisms was onlyconducted for specimens in Groups 1.5 and 2.0 using theaverage strain data measured during the tests. Control pointsin these specimen groups were set following the approximatedirection of tied-arch (direct) and truss-model struts (indirect)forming between loading point and support, as shown inFig. 3(b) and (c). Instrumentation in Specimen Groups 1.0 and1.0L was only placed following the direction of a direct strut asassumed in a tied-arch model (Fig. 3(a)) so no informationcould be obtained regarding the shear transferred by trussaction, in these groups, if any. This instrumentation, incombination with assumed strut-and-tie models, permitted anestimation of the fraction of shear transferred from load point tosupport through each mechanism.

Figure 10 shows average strains in struts measured duringthe tests for specimens in Groups 1.5 and 2.0. Averagestrains were measured in struts that followed a directionapproximately parallel to that predicted from a truss loadtransfer model (refer to sketches in Fig. 10). In a truss modelwith the two struts inclined at the same angle as shown, theforces in diagonal struts are theoretically equal. The testresults showed, however, that strains measured along the twostruts developing in this hypothetical load-path model weredifferent. This phenomenon may be due to the presence ofdiagonal cracks crossing the interior strut where InstrumentsL3 were placed, resulting in lower measured compressivestrains compared with those measured in Instrument L2.Because these observed differences were larger in specimensin Group 2.0, only the measurements taken in the exteriorstrut along L2 were used to estimate stresses at the peak loadmeasured in the tests in Specimen Group 2.0. The averagestrain values measured in Instruments L2 and L3 at peak loadwere used to estimate strut stresses in Specimen Group 1.5.

The uniaxial stress-strain model proposed by Hognestad(1951) was used to compute stresses from measured averagestrains. To capture the stress-strain response of the concretestruts to the point of peak stress, the ascending portion in themodel as described in Eq. (9) was used

(9)fc kfc′2εc

εco

--------εc

εco

-------⎝ ⎠⎛ ⎞ 2

–=

where εco = 2fc′ /Ec represents the strain at peak stress (fc′ )and the modulus of concrete Ec was equal to 4730 MPa(57,000 psi). Because Eq. (9) was used to estimate stressesfrom strains measured experimentally, k was equal to 1.0instead of the typically-assumed value of 0.85. Only thestress at maximum measured strain was of interest forestimation of the fraction of load transferred throughtruss action.

Strut stresses determined from strain measurements takenon Instruments L2 and L3 (Fig. 10) were used to calculatethe fraction of shear transferred through truss action (Vtruss).Figure 11 illustrates the assumed truss model used, where thetop node geometry under the point load was assumed, asillustrated in Fig. 11(b). Forces acting at the top node weredetermined using Eq. (10) through (12) for an assumed valueof shear transferred through truss action (Vtruss)

(10)

(11)

FC(L) = FC(R) – Fscosγ (12)

where the shear span a equals 610 mm (24 in.), and the anglethat the strut makes with the horizontal, γ, was determined

fc′fc′

FS-trussVtruss

γsin-------------=

FC R( )

Vtrussa

z----------------

Vtrussa

d hnode-top R( )–-----------------------------------= =

Table 2—Model parameters for strut strength evaluation

SpecimenApplied shear V,

kN (kip)Beam width b,

mm (in.) Strut angle α, degTop node height

hnode-top , mm (in.)Strut width (top end)

ws-top , mm (in.) fstrut, MPa (ksi) βs

DB1.0-1.00 338 (76) 165 (6.5) 44.3 74.3 (2.9) 124 (4.89) 23.6 (3.43) 0.84

DB1.0-0.75 371 (83.5) 173 (6.8) 44.0 82.6 (3.3) 130 (5.12) 23.8 (3.45) 0.88

DB1.0-0.50 365 (82) 157 (6.2) 43.8 92.9 (3.7) 137 (5.41) 24.4 (3.53) 0.94

DB1.0-0.32 334 (75) 152 (6.0) 43.6 100.3 (3.9) 143 (5.62) 22.3 (3.23) 0.97

DB1.0-0.75L 371 (83.5) 155 (6.1) 43.6 99.1 (3.9) 142 (5.59) 24.5 (3.55) 0.97

DB1.0-0.28L 320 (72) 155 (6.1) 44.0 85.8 (3.4) 132 (5.21) 22.5 (3.27) 0.90

DB1.5-0.75 229 (51.5) 152 (6.0) 32.9 83.5 (3.3) 125 (4.93) 22.1 (3.2) 0.79

DB1.5-0.50 211 (47.5) 152 (6.0) 33.3 72.9 (2.9) 117 (4.59) 21.6 (3.14) 0.75

DB1.5-0.38 214 (48) 152 (6.0) 33.3 74.4 (2.9) 118 (4.64) 21.7 (3.14) 0.75

DB2.0-0.75 156 (35) 155 (6.1) 25.5 71.6 (2.8) 108 (4.26) 21.6 (3.13) 0.73

DB2.0-0.50 149 (33.5) 155 (6.1) 25.5 72.1 (2.8) 109 (4.28) 20.6 (2.98) 0.73

DB2.0-0.43 133 (30) 155 (6.1) 26.0 58.3 (2.3) 97 (3.82) 20.2 (2.94) 0.67

Fig. 9—Variation of strut factor as a function of shear span-depth ratio.


assuming an initial depth of the top node (hnode-top(R)) ofapproximately 75 mm (3 in.). With known values of theseforces acting at the top node, the node dimensions wereadjusted so that nodal failure would not occur to be consistentwith the observed behavior during the tests. The revisedheight of the right face of the node (hnode-top(R)), the heightof the left face of the node (hnode-top(L)), and strut width(wS-truss) were determined using

(13)

(14)

(15)

(16)

Notice that the node heights calculated in Eq. (13) and (14)represent the minimum heights required to avoid nodal zonecrushing failure at a stress equal to 0.85fc′ . With the nodegeometry defined, a revised value for the diagonal strut forceFS-truss was estimated using the strut stresses determinedduring the tests (fS-truss(test)) applying Eq. (17), which thenpermitted evaluation of the shear force transferred into thesupport by truss action (Vtruss) through the use of

FS-truss = fS-truss(test)wS-trussb (17)

hnode-top R( )

FC R( )

fceb------------

FS-truss γcos

0.85fc′ b-----------------------------

Vtruss

0.85fc′ b γtan------------------------------= = =

hnode-top L( )

FC L( )

fceb------------

FC R( ) FS-truss γcos–

0.85fc′ b------------------------------------------------= =

Δhnode-top hnode-top R( ) hnode-top L( )–=

wS-trussLplate-top

2--------------------- γ Δhnode-top γcos+sin=

Fig. 10—Average strains measured in struts of strut-and-tietruss model: (a) Group 1.5 exterior strut; (b) Group 1.5interior strut; (c) Group 2.0 exterior strut; and (d) Group 2.0interior strut.

Fig. 11—Truss model for load transfer: (a) geometry ofmodel; and (b) top node details.


Vtruss = FSsinγ (18)

With these revised values of FS-truss and Vtruss, Eq. (11)through (18) were applied again to adjust the top node geometryand forces until convergence was achieved in two subsequentcycles within reasonable tolerance. The effective depth z wascalculated in each iteration using the most current value forhnode-top(R). Table 3 summarizes the diagonal strut force,FS-truss, corresponding to the measured strut stress and thecorresponding strut width determined from the proceduredescribed above.

The fraction of shear force transferred by truss action issummarized in Table 4 for each of the specimens in Groups1.5 and 2.0. It can be observed that this fraction depends ona/d, as would be expected. For specimens where anchoragefailures were not observed, the fraction of total shear transferredthrough truss action was approximately 0.50 and 0.80 forGroups 1.5 and 2.0, respectively. Interestingly, specimens withthe shortest anchorage length in these two groups, in whichanchorage failure was observed during the tests, seemed totransfer the entire applied shear by truss action. The differencein observed load transfer mechanism of specimens withinadequate anchorage length can be attributed to theinability to develop high stresses at the bottom node over thesupport, as required by a tied-arch model. For comparison,the value considered transferred by truss action included inthe “FIP Recommendations 1999” (Federation Internationalde La Precontrainte 1999) (Eq. (4)) is listed in the lastcolumn of Table 4. The values determined in these tests areconsistently lower than those computed using the FIPrecommendations by approximately 40% for specimens inGroup 1.5 and 20% for specimens in Group 2.0. ACI 318-08does not give guidance as to specific strut-and-tie modelsto use for design of deep beams as indicated earlier inthis paper.

According to the assumed load transfer models, the differencein total shear not transferred by truss action must be transferredby tied-arch action (VTA = Vtest – Vtruss). A tied arch modelsimilar to that illustrated in Fig. 8 was used in combinationwith strut stresses experimentally determined by using readingsfrom Instrument L1 (Fig. 3(b)) to assess whether the procedureto estimate Vtruss previously described gave reasonableresults. The assessment was based on comparing the strutstress determined from the assumed tied-arch model withstrut stresses determined experimentally. For this comparison,the fraction of shear force transferred by tied-arch action wascalculated as the difference between the total applied shearforce and the fraction of shear transferred by truss action aspreviously calculated. As previously discussed, the total

shear force was assumed to be transferred by truss action inspecimens with the shortest anchorage lengths in each group.

Table 3 lists relevant parameters of the diagonal strut inthe tied-arch model for each specimen in Groups 1.5 and 2.0.The value of these parameters was obtained, assuming thatonly a fraction of the total shear corresponding to VTA wasapplied to the model. The assumed angle of inclination of thediagonal strut in the model (α) was calculated assuming anapproximate node height of 75 mm (3 in.). The diagonal strutforce FS-TA required for equilibrium was calculated usingEq. (19), from which the minimum node height to avoidnodal crushing was estimated using Eq. (20). The diagonalstrut width at the face of the top node was estimated usingEq. (21).

(19)

(20)

(21)

With these properties and forces determined, a diagonalstrut stress could be determined analytically (fS-TA) andcompared with the strut stress experimentally estimated( fS-TA(test)). These values and their ratio are listed in the lastthree columns of Table 3. As shown in the table, strut stressvalues determined analytically with an assumed tied-archmodel compare favorably with values experimentallydetermined. The maximum difference observed wasapproximately 12% in specimens in Group 2.0, givingconfidence in the procedure and models adopted for the

FS TA–VTA

αsin-----------

Vtest Vtruss–

αsin------------------------------= =

hnode-TAFS TA–

0.85fc′ b-------------------=

wS TA–Lplate-top

2--------------------- α hnode-TA αcos+sin=

Table 3—Geometry of strut-and-tie models for load transfer evaluation

Specimen

Truss model Tied arch model

fS-TA(test), MPa

Truss strut angle γ, deg

fS-truss(test), MPa ws-truss, mm FS-truss, kN

Tied arch strut angle α, deg FS-TA, kN ws-TA, mm fS-TA, MPa

DB1.5-0.75 52.7 9.8 93 139 32.9 219 92 15.7 14.5 1.08

DB1.5-0.50 52.7 9.0 91 125 33.3 203 88 15.2 16.8 0.90

DB1.5-0.38 52.7 17.8 105 284 33.3 * * * 15.3 *

DB2.0-0.75 43.7 12.9 91 181 25.5 69 56 8.0 9.1 0.88

DB2.0-0.50 43.7 11.6 90 162 25.5 85 60 9.2 10.4 0.88

DB2.0-0.43 43.7 16.8 99 258 26.0 * * * 5.9 **Specimen where all load was transferred through truss action.Note: 1 MPa = 145 psi; 1 mm = 0.0394 in.; 1 kN = 0.2248 kip.

fS TA test( )–

fS TA–

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

Table 4—Fraction of shear transferred by individual load transfer mechanisms

SpecimenVtest, kN

Vtruss, kN Vtruss/Vtot VTA, kN VTA/Vtot

Vtruss/Vtot (FIP)

DB1.5-0.75 229 110 0.48 119 0.52 0.79

DB1.5-0.50 211 100 0.47 112 0.53 0.77

DB1.5-0.38 214 226 1.06 * * 0.78

DB2.0-0.75 156 125 0.81 30 0.19 1.00

DB2.0-0.50 149 112 0.75 36 0.25 1.00

DB2.0-0.43 133 178 1.34 * * 1.00

Note: 1 kN = 0.2248 kip.


experimental evaluation of shear force transfer. The fractionof shear force transferred by tied-arch action is listed inTable 4 for the two groups of specimens.

CONCLUSIONSTests of deep beams having three different a/d, where the

longitudinal reinforcement was anchored using shorterlengths than required by the ACI 318-08 Code, were reportedin this paper. Two parameters that are relevant to the developmentof strut-and-tie models for this class of deep beams were criticallyexamined in the paper (effective concrete strength factor βs andthe force transfer mechanism). The transverse reinforcementcontent in all beams was kept constant in order to isolate theeffects of a/d and anchorage conditions. The effects of lateralconfining stresses along the straight anchored bars werereported in a previous paper (Roy and Breña 2008). Thefollowing conclusions can be drawn from the test resultspresented in this paper, which are strictly applicable only forthe range of variables included in the present study:

1. Longitudinal bars in the beams reported in this paperwere anchored at nodes over a support generating transversestresses along the bar within the nodal region. These transversestresses prevented bars from pulling out from the nodal zoneand promoted stress redistribution within the beams after baryielding. Even beams with short longitudinal bar anchorageswere capable of redistributing load into a load path otherthan directly from load to support, which allowed developmentof higher loads leading to strut crushing at failure of mostof the specimens.

2. Because of the transverse stresses developed within thesupport node, deep beams with short longitudinal baranchorages were able to transfer a fraction of the total loadfrom loading point to support through truss action. Thisphenomenon is not expected to occur in beams supportedthrough indirect supports where no clamping stresses aredeveloped on longitudinal bars anchored in the nodal region.

3. The fraction of load transfer by truss action in deepbeams is significant, even for relatively low a/d. In thesetests, the fraction of load transferred by truss action in beamswith a/d = 1.5 and a/d = 2.0 was approximately 50% and80% of the total shear, respectively. These percentageswould likely be different in beams with other amounts oftransverse reinforcement.

4. The value of the effective concrete strength factor (βs)used in ACI 318-08 for bottle-shaped struts (βs = 0.75) was

conservative for the beams tested in this research. βs seems todecrease with increasing a/d, however, which would beconsistent with the presence of higher longitudinal strainsdeveloping in the web. Further research seems warranted inthis area.

5. Although short bar anchorages should not be encouraged indesign to ensure ductility of beams prior to failure, the testsreported in this paper show that even with short anchoragesyielding of the longitudinal reinforcement occurred atmidspan, and, depending on the load transfer mechanismthat is mobilized, could spread all the way into the supportnode. The high fraction of applied shear carried through trussaction in beams with a/d greater than 1.0 and shortanchorages allowed bottom bars to develop stresses lowerthan yield at the support node and still be able to carry theapplied load safely. The tensile force demand at the supportnode decreased when truss action developed in the beams.

ACKNOWLEDGMENTSGraduate studies for the second author were supported through a Brack

Structural Engineering Graduate Student Fellowship at the University ofMassachusetts Amherst. The authors would like to express their mostsincere gratitude to R. Brack for establishing the graduate fellowship. Inkind material donations received for specimen fabrication from Barker Steeland Nucor Steel Connecticut are deeply appreciated.

REFERENCESACI Committee 318, 2008, “Building Code Requirements for Structural

Concrete (ACI 318-08) and Commentary,” American Concrete Institute,Farmington Hills, MI, 465 pp.

Federation Internationale de la Precontrainte, Commision 3 Recommenda-tions, 1999, “FIP Practical Design of Structural Concrete,” FIP Symposium,London, UK, 113 pp.

Hognestad, E., 1951, “A Study of Combined Bending and Axial Load inReinforced Concrete Members,” Bulletin Series No. 399, University of IllinoisEngineering Experiment Station, Urbana, IL, V. 49, No. 22, 128 pp.

MacGregor, J. G., and Wight, J. K., 2005, Reinforced Concrete:Mechanics and Design, fourth edition, Prentice Hall, NJ, 1132 pp.

Quintero-Febres, C. G.; Parra-Montesinos, G.; and Wight, J. K., 2006,“Strength of Struts in Deep Concrete Members Designed Using Strut-and-TieMethod,” ACI Structural Journal, V. 103, No. 4, July-Aug., pp. 577-586.

Roy, N. C., and Breña, S. F., 2008, “Behavior of Deep Beams with ShortLongitudinal Bar Anchorages,” ACI Structural Journal, V. 105, No. 4,July-Aug., pp. 460-470.

Uribe, C. M., and Alcocer, S. M., 2001, “Behavior of Deep BeamsDesigned using Strut-and-Tie Models,” Centro Nacional de Prevención deDesastres, México, D.F., 247 pp. (in Spanish)

Wight, J. K., and Parra-Montesinos, G., 2003, “Strut and Tie Model forDeep Beam Design,” Concrete International, V. 25, No. 5, May, pp. 63-70.

Evaluation of Load Transfer and Strut Strength of Deep Beam

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