Shear behaviour of non-prismatic steel reinforced concrete beams

Engineering Structures 71 (2014) 48–59

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Engineering Structures

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Shear behaviour of non-prismatic steel reinforced concrete beams

http://dx.doi.org/10.1016/j.engstruct.2014.04.0160141-0296/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +44 (0) 1225 385 096.E-mail address: [email protected] (J.J. Orr).

John J. Orr ⇑, Timothy J. Ibell, Antony P. Darby, Mark EverndenDepartment of Architecture and Civil Engineering, University of Bath, Bath, BA2 7AY, United Kingdom

a r t i c l e i n f o

Article history:Received 24 May 2013Revised 11 December 2013Accepted 7 April 2014

Keywords:ShearShear reinforcementStructural behaviourNon-prismatic beamsOptimisation

a b s t r a c t

Large reductions in embodied carbon can be achieved through the optimisation of concrete structures.Such structures tend to vary in depth along their length, creating new challenges for shear design. Toaddress this challenge, nineteen tests on non-prismatic steel reinforced concrete beams designed usingthree different approaches were undertaken at the University of Bath. The results show that the assump-tions of some design codes can result in unconservative shear design for non-prismatic sections.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Non-prismatic concrete beams can provide steel and concretesavings when used to replace equivalent strength prismatic ele-ments. In a variable section reinforced concrete beam a portionof the shear force may theoretically be carried by a suitablyinclined top or bottom flange, yet such beams have been foundto fail prematurely, suggesting codified methods are unable toaccount for the varying section shapes found in optimised struc-tures. Given that optimised structures tend to be non-prismatic,understanding these failures and providing appropriate guidancefor their design is hugely important.

2. Tapered beams

2.1. Shear behaviour

The derivation of shear stresses through equilibrium consider-ations of a homogenous uncracked and isotropic beam is relativelystraightforward, but the behaviour of a reinforced concrete sectionis more complex. In a reinforced section, cracks will form when theprincipal tensile strain exceeds the tensile capacity of concrete andthese diagonal cracks typically propagate from the tension face ofthe member towards the neutral axis.

There are conventionally considered to be six contributing fac-tors by which a reinforced concrete beam can carry shear, Fig. 1.

When present, shear reinforcement carries stress over cracks asthey open under loading and confines the section. Although aggre-gate interlock is estimated to carry significant shear force in theuncracked section [1], as cracks open the capacity to transfer stres-ses via aggregate interlock is minimal [2]. Dowel action by longitu-dinal reinforcement is contentious, with Kotsovos [2] showing it tobe extremely limited in the prismatic section.

The behaviour of prismatic and tapered concrete sections inshear is compared in Fig. 1. In sections that taper towards theirsupports, the interaction of the diagonal cracks with the path ofthe compression force at the supports is assumed to be critical.

Inclined compression or tension forces can theoretically affectthe shear resistance of the section. It is suggested [3–7] that forsections whose depth increases in the direction of increasingmoment an effective shear force for design be given by Eq. (1),which is valid for members with shear reinforcement:

V 0Ed ¼ VEd � Vccd � Vtd ð1Þ

where V 0Ed is the reduced design shear force; VEd is the shear force onthe cross section; Vccd is the vertical component of force in theinclined compression chord and Vtd is the vertical component ofthe inclined tension chord. MEd is the moment on the cross section.

Provided suitable limits on stress in the web compression strutare not exceeded, the sum of Vccd and Vtd (Eq. (1)) could theoreti-cally be made equal to the applied shear force, negating require-ments for transverse steel. For a beam with yielding tensionreinforcement of constant area (As) and without normal force thiscould be achieved by placing the bar at an effective depth that isproportional to the bending moment at each point along the beam

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Nomenclature

Asw area of transverse reinforcement (mm2)avx Max/Vax (mm) in the CFP methodbw web width (mm)dx an increment of lengthfyk characteristic yield strength of steel reinforcementMax applied bending moment (N mm) on a section in the

CFP methodMcx is the moment corresponding to shear failure (N mm) in

the CFP methodMEd,i the applied moment at position iMf the flexural capacity (N mm) in the CFP methods transverse reinforcement spacing (mm)V0Ed the reduced design shear force, Eq. (1)Vax applied shear force (N) on a section in the CFP method

Vcx shear force at failure in the CFP methodVEd,i the applied shear force at position iVf shear force corresponding to flexural failure in the CFP

methodVtd vertical component of force in the barVtd,i the vertical component of force in the bar at position ix (subscript) denotes a given cross-section at a distance x

mm from the support in the CFP methodzi the lever arm between tension and compression forces

at position iqw ratio of the area of tension steel to the web area of con-

crete to the effective depth in the CFP methodqw Asw/s/bw (reported in %)

J.J. Orr et al. / Engineering Structures 71 (2014) 48–59 49

(Eq. (2)). Such an approach would make the vertical component offorce in the bar (Vtd) theoretically equal to the applied shear force,Eq. (3).

zi ¼MEd;i

Asfykð2Þ

VEd;i ¼dMEd;i

dx¼ Vtd;i ð3Þ

where zi is the lever arm between tension and compression forces atposition i; MEd,i is the applied moment at position i; As is the con-stant area of longitudinal reinforcement and fyk is its characteristicyield strength; VEd,i is the applied shear force at position i; dx is anincrement of length and Vtd,i is the vertical component of force inthe bar at position i.

However, utilising a longitudinal bar to provide vertical forcecapacity close to the supports in a simply supported beam requiresthe bar to be fully anchored at its ends and yielded along its entirelength. Furthermore, for a structure subject to an envelope of loadsthe longitudinal reinforcement position will be determined by themaximum moment on each section. It is feasible that the maxi-mum moment and shear forces on a section will not originate fromthe same load case. In such a situation, a bar placed for momentcapacity will then be incorrectly inclined to provide the desiredvertical force, and thus additional transverse reinforcement willbe required.

Prismatic section

(e) & (f)

(a) Compression concrete(b) Tension concrete(c) Flexural reinforcement

(d) Shear reinforcement(e) Aggregate interlock(f) Dowel action

Tapered section

(a)

(b)

(c)(d)

(e) & (f)

(c)(d)

(b)

(a)

Fig. 1. Contributing factors to shear resistance in tapered and prismatic sections.

2.2. Design methods

2.2.1. Truss modelACI 318 [8] and BS EN 1992-1-1 [4] allow the shear capacity of a

tapered section with transverse reinforcement to include theeffects of inclined tension and compression forces, Eq. (1). Bothcodes are based on the truss analogy, the premise of which [9,10]is that cracked concrete in the web resists shear by a diagonal uni-axial compressive stress in a concrete strut, pushing the flangesapart and causing tension in the stirrups that are then responsiblefor holding the section together. With a compression strut angle of45�, the model consistently underestimates shear strength. To cor-rect this ACI 318 [8] adds a ‘concrete contribution’, while BS EN1992-1-1 [4] assumes that once cracked the concrete provides nocontribution to shear capacity and instead allows a flatter strutangle (down to 22�, subject to stress limits in the diagonal concretestrut) to be chosen, with both approaches replicating experimentalobservations.

The additional tensile force, DF, arising from the normal stresscomponents of the inclined web compression struts of the trussmodel [4,8] must be included when calculating the force in theinclined chords to prevent over-estimation of the contribution ofan inclined chord to shear capacity.

2.2.2. Compressive force path methodThe compressive force path (CFP) method premises that the

behaviour of a reinforced concrete beam can be simplified intothree elements – a concrete frame, a steel tie of flexural reinforce-ment and a zone of concrete cantilevering teeth which formbetween successive cracks in the concrete section [11], Fig. 2.The uncracked compression zone is proposed to sustain boththe compressive flexural force and the entire shear force. Since

Cracked concrete teeth Reinforcement

Compressive force path

lever arm, z

aC + ΔCC

T + ΔTT

VVV.a = ΔT.z

P

Fig. 2. The CFP method [11].

40.720.2

166.

7

θ1 = 26ºθ2 = 24ºθ3 = 19º

100

80 140 280

500

θ1

θ2

θ3

α1

α2

α3

Fig. 4. Generic strut and tie model for a tapered section.

50 J.J. Orr et al. / Engineering Structures 71 (2014) 48–59

concrete fails in tension, those regions of the path where tensilestresses may develop will be critical [11].

Experimental data for beams subject to point loads tested byKotsovos [12] suggests that shear stirrups do not have to be pro-vided throughout the shear span in order to allow full flexuralcapacity to be reached. Since beams without continuous transversetension reinforcement cannot conform to the truss analogy, theimplication is that ‘‘truss behaviour of a section is not a necessarycondition for a beam to attain flexural capacity once shear capacityis exceeded’’ [12].

CFP-designed indeterminate frames were compared to ACI 318[13] by Salek et al. [14]. It was found that the CFP-designed framesprovided greater ductility through their positioning of transversereinforcement to carry tensile forces arising from changes in thedirection of the compressive force path. Jelic [15] also found thatCFP-designed reinforced concrete beams could provide greaterductility with less transverse reinforcement than an equivalenttruss-analogy based design.

Whilst the literature suggests that the CFP method provides anaccurate representation of the behaviour of reinforced concretebeams, the design method is based on empirical equations [16–18] and non-prismatic elements were not included in their originalderivation. There is therefore no guarantee that the method isindeed suitable for non-prismatic beams.

2.2.3. Strut and tie modelThe assumptions implicit in Eqs. (2) and (3) (which were

applied in the design of the ‘EC2’ beam series described later in thispaper) require tension and compression forces to be arranged asshown in Fig. 3(a). Such a model relies on full anchorage of thereinforcement and exact alignment of forces at the support. Suchan alignment is impossible to prove and unlikely to occur withadditional moments generated if it is not achieved (Fig. 3(b)).

To overcome this problem, an alternative strut and tie modelwas developed, Fig. 4, in which the flexural tension reinforcementdoes not provide full shear capacity and is not required to yield inall locations. The first tension tie is then required to carry themajority of the shear force. In the model, the concrete strut capac-ity is limited to the concrete compressive strength and the ties arelimited by the capacity of the yielding steel at each position. Inaddition a maximum link spacing of 75% of the effective depth isapplied along the beam length, following recommendations forstrut and tie modelling [4]. Areas required for anchorage anddevelopment of both longitudinal and transverse reinforcementdefine the overall section geometry.

‘EC2’

P (kN)

Position of ‘zero’ moment capacity is offset from support.

Support cannot carry horizontal force

(a)

(b)

Fig. 3. Equivalent strut and tie model assumed for EC2 model (a); potential formoment generation at the supports (b).

The proposed model does not require that all the steel is yield-ing. Instead, plastic behaviour (which is desired for a ductileresponse) is provided by ensuring that the steel yields in certainpositions only. In the proposed methodology, this yielding is tooccur at the position of maximum moment and beam depth, andaway from the support location. It is proposed that this mechanismwill provide ductility to the beam response. At the failure load, it islikely that reinforcement at the support will still be in its elasticrange.

2.3. Test data

Limited experimental data is available to describe the behaviourof tapered steel reinforced concrete beams in shear. Debaiky and El-Neima [5] compared prismatic and tapered beams in a series of 33tests. Crack number and location were found to depend on the incli-nation of the beam haunch. Compared to a prismatic section, thetotal number of cracks was found to increase for negativelyhaunched beams and to decrease for positively haunched beams(Fig. 5). For positive haunches, the position of the major shear crackmoved closer to the support, as would be expected based on anassumption that cracking occurs at the weakest section.

T-beam specimens tested later by El-Neima [19] showed similarresults, with both positive and negative haunch sections showinglower shear capacities than prismatic sections. Further tests[5,19] showed reductions in shear capacity for haunched beams,suggesting variable section beams could negatively affect the shearcapacity of the section.

Haunched beams carrying hogging in continuous beams couldbenefit from shear capacity provided by the inclined compressionzone. Macleod and Houmsi [6] and later Rombach and Nghiep [7]found that the additive method described by Eq. (1) could resultin unconservative capacity predictions for sections with inclinedcompression zones (without shear reinforcement). It was recom-mended [7] that only 50% of the value of Vccd should be included(if it is a favourable effect), Eq. (4):

V 0Ed ¼ VEd � 0:5Vccd ð4Þ

The limited available experimental data suggests that additivemethods for the shear capacity of tapered beams can lead tounconservative results. This highlights the fact that the underlyingbehaviour is not yet fully understood. This behaviour, of crucialimportance for non-prismatic beams, is assessed in this paperthrough a series of beam tests.

‘positivehaunch’

‘negativehaunch’

Fig. 5. Positive and negative haunch definition.


3. Design

To determine the most appropriate method for the design oftapered beams in shear, 19 tests on 11 beam specimens wereundertaken to consider:

1. Design using the variable angle truss model (‘EC2’).2. Design using the compressive force path method (‘CFP’).3. Design using a proposed strut and tie model (‘STM’).

3.1. Methodology

Tapered beams with shear spans of 150, 300, 500 and 1000 mmwere designed using the EC2 and CFP methods; only a 500 mmshear span was considered in the STM approach due to time limi-tations. All beams had a total span during testing of 2000 mm.Beams with shear spans of 6500 mm were cast with two taperedends and tested twice, as shown in Fig. 6. The beams are namedaccording to shear span and design methodology, with EC2 andCFP beams predicted failure modes being denoted by V (predictedshear failure) or M (predicted flexural failure).

All beams had a constant breadth of 110 mm and were rein-forced on their tension face with two 10 mm diameter high yieldU-bars to provide full anchorage at the support zone. Where linksare used, these are 3 mm diameter high yield plain bars as closedlinks. The links are anchored around two 3 mm plain bars usedas top steel (which provides a negligible contribution to flexuralcapacity). The concrete design strength was 40 MPa. Cover to thelongitudinal tensile reinforcement was 20 mm. A summary of thebeam nomenclature and design loads is provided in Table 1. Therelationship between the test load, P, and the design shear force,VEd, has been arranged such that for Beams 2–4 a comparable shearforce is imposed on the tapered end.

3.2. Beam design

3.2.1. EC2 modelThe EC2 beams series were designed according to the method

described by Eqs. (2) and (3) and uses the vertical component of

Setup ‘M’:

Setup ‘V’:

av

av

2000mm

2000mm

P

P

2000 + av

Fig. 6. Test loading arrangements.

Table 1Summary of test loads and maximum shear force.

Test Designs Testload, P

Shearspan, av

Design shear forceat the tapered end, VEd (kN)

1a EC2/CFP 32 1000 162 EC2/CFP/STM 36 500 273 EC2/CFP 31.8 300 274 EC2/CFP 29.2 150 27

a Beam 1 was symmetrical, with point load applied at midspan of the 2000 mmlong beam.

force in the yielding longitudinal steel as an effective shear link.Iteration is carried out to find the geometry that satisfies equilib-rium and provides the full flexural and shear requirements at eachpoint along the beam. Anchorage of the bar is provided by the useof U-bars. A check was also made to ensure that the steel was ableto yield under the applied loads. Such an approach, which impliesthat flexural and shear failures are equally likely, was denoted ‘V’(predicted shear failure).

Taking the same geometry as the ‘V’ series beams, EC2 beamswith predicted flexural failures (denoted ‘M’) disregarded thepotential shear contribution of inclined longitudinal steel and usedonly yielding transverse reinforcement designed according to BSEN 1992-1-1 [4] (Eq. (6.8)) to provide shear capacity. The strutinclination was typically 21.8�, and VRd,max was not a governing cri-teria. A concrete compressive strength of 40 MPa, and steel yieldstrength of 500 MPa, where used in the design and all partial safetyfactors were set to 1.0. The four beam elements, designed to theloads and shear spans given in Table 1 are shown in Fig. 7.

3.2.2. Compressive force path (CFP)The CFP method was applied by considering the two potential

failure modes for the section: ductile flexure or brittle flexure-shear. An empirical equation for the moment corresponding toshear failure (Mcx) is used for design and given by Eq. (5)[16,17,20].

Mcx ¼ 0:875avxd 0:342b1 þ 0:3Mf

d2

ffiffiffiffiffiffiffiz

avx

r� � ffiffiffiffiffiffiffiffiffiffiffiffiffi16:66pwfy

4

sð5Þ

Vcx ¼Mcx

avxð6Þ

Where the subscript x denotes a given cross-section at a distance xmm from the support; Mcx is the moment corresponding to shearfailure (N mm); Mf is the flexural capacity (N mm); avx is the ratioMax/Vax (mm); Max and Vax are the applied bending moment(N mm) and shear force (N) on the section; z is the lever arm(mm); d is the effective depth (mm); qw is the ratio of the area oftension steel to the web area of concrete to effective depth; fyk isthe characteristic strength of the tension steel (N/mm2); bw is theweb width (mm); and Vcx is the shear force at failure.

The design method was originally applied to prismatic beams.For variable section members, the beam is divided into sections,and at each section the required effective depth for flexural capac-ity according to the plane sections theory is determined. Each sec-tion is then assessed against Eq. (5) to determine a value of Mcx forthe section, with a corresponding shear capacity given by Eq. (6).The effective depth at each point along the length of the beam isiterated until either Vf or Vcx exceeds the design shear at each point.

A graph of the applied moment-shear envelope (Va, Ma), themoment and shear forces corresponding to flexural failure (Mf,Vf) and the moment and shear forces corresponding to shear failure(Mcx, Vcx) are then plotted (shown in Fig. 8 for a beam subject tomultiple point loads).

From this two types of failure mode can be designed for. Thefirst, shear failure (denoted ‘V’), requires only that the shear capac-ity at each point on the beam be equal to the applied load enve-lope. In the support zones Vcx therefore governs the failure mode.

The second mode is to ensure flexural failure in the entire beam(denoted ‘M’). This is achieved by adding shear capacity to all posi-tions on the beam at which Vcx 6 Vf until flexural failure is predictedto occur before shear failure. In the end zones, considerable trans-verse reinforcement was required to achieve this, Eq. (7).

As ¼Vf � Vcx

fykð7Þ

52

50

50220

20

500500

300 300

19860

129

150 150

82

50

50

50

10

10

10

Test Setup ‘V’

Test Setup ‘M’

2000

320

320

75150

225100

150100

80

10080

60

60

ø10

ø10mm

ø3mm ø3mm

ø10mm

ø3mm ø3mm

ø10mm

ø3mm ø3mm

110mm

110mm

110mm

110

220

198

129

82

Section at load point:

320

ø3 ø10

ø3 ø10

ø3 ø10

ø10Beam 1_EC2

Beam 2_EC2

Beam 3_EC2

Beam 4_EC2

Fig. 7. EC2 designed beams 1–4.

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35 40 45

Mom

ent (

kNm

)

Shear (kN)

ExampleLoading

5kN5kN 5kN5kN5kNVcx : Mcx

Vf : Mf

Flexural Failure, Vf < Vcx

Shear Failure, Vcx < Vf

Load envelope (Va, Ma)

Capacity for shear failure(Vcx, Mcx)

Capacity for flexural failure(Vf, Mf)

500mm = = == =

Fig. 8. Failure modes and interaction diagram for example beam subject to symmetric loading.


Where Vf is the shear force corresponding to flexural failure; Vcx isgiven by Eq. (6) and fyk is the characteristic steel yield strength.

Following the proposed method four beams were designed withone end predicted to fail in shear (‘V’) and one end in flexure (‘M’).The resulting beam layouts are shown in Fig. 9. The curved end ofthe beam arises as a result of the design method and was con-structed using bespoke formwork.

3.2.3. Strut and tie model (STM)The strut and tie model is shown in Fig. 4 and is differentiated

from the EC2 model as it does not require the steel to be yieldingeverywhere within the tapered zone (the steel should be yieldingat the position of maximum moment to provide plastic behaviour).Design is initiated by calculating the required depth at the positionof maximum moment that will ensure the predefined area of inter-nal flexural steel reinforcement steel has yielded in this location.The actual, rather than characteristic, yield strength of the steelshould be used. This provides the overall geometry of the beam,with the force in each section limited by concrete crushing andsteel yielding (each element is not required to be at maximumstress). Tension paths are required to form a straight line betweennodes. At each node, equilibrium considerations provide the forcesin each tie and thus the area of transverse reinforcement required.

The model geometry is iterated to provide a reinforcement lay-out with the required shear and flexural capacities, resulting in thebeam shown in Fig. 10. At the support, the forces must still line upquite precisely, so one end of the beam was cast around a steelangle anchored into the concrete by a 10 mm diameter reinforcingbar. This is the only difference between the two ends of the taperedSTM beams (denoted ‘1’ without the end angle and ‘2’ with the endangle).

4. Testing

A steel mould was used to create each beam profile (Fig. 7,Figs. 9 and 10). Longitudinal and transverse reinforcement wascut and bent to shape in the laboratory using prepared templatesfor each beam. The beams were cast in sets of four, demouldedafter 3 days and cured in laboratory conditions prior to testing.

The test set up for each beam is shown in Fig. 11. This setup hasbeen carefully chosen to allow each specimen to be tested twice.The first test has only a minor effect on the prismatic zone, andno effect on the tapered zone, of the second test. Steel strains inthe flexural reinforcement were monitored at three positions inthe tapered section with strain gauges. High definition video andstill images were recorded for post-processing using digital imagecorrelation to determine strains on the concrete surface.

Beam 1_CFP

Beam 2_CFP

Beam 3_CFP

Beam 4_CFP

22020

10040

1000

ø10mm

110

220


ø3mm ø3mm

70

80

214ø10mm

110

214ø3mm ø3mm

500 500

70200 205

100 140 16025010

200

10

320

10

100050

50

ø10mm

110

181ø3mm ø3mm

70200

155 25180120

50200

300 300

181320

5015010

120 18050

200

150 150

158ø10mm

110

158ø3mm ø3mm

10

10ø3 ø10

ø3 ø10

ø3 ø10

ø3 ø10

Test Setup ‘V’

Test Setup ‘M’

Fig. 9. CFP designed beams 1–4.

500ø10mm

110

208


ø3mm ø3mm

5001525100 100

ø3 ø1011050

12015

6030

165110110 50

12015

6030

165

Test Setup ‘2’

Test Setup ‘1’

208

10

Fig. 10. STM design beam 2.

RollersupportPin

support

av

P

2000-avm

50mm

50mm

50mmHydraulic jack andDisplacement transducer

Spot patternfor DIC camera

Straingauge

270

0.2av0.34av

Fig. 11. Test layout and monitoring.

Table 3Recorded 28 day concrete cube compressive strength.

Age Average compressive strength (MPa) Standard deviation (MPa)

28 days 47.9 3.8


4.1. Materials

All specimens were cast from the same concrete mix design(Table 2). Average cube strengths at 28 days are given in Table 3.High yield deformed bar was used as longitudinal reinforcement(characteristic yield strength, fyk = 500 MPa, average yield strengthaccording to tests fy = 562 MPa), undeformed high yield bar wasused as transverse reinforcement. The aggregate (see Table 2)was provided in accordance with the requirements of BS EN12620 [21].

Table 2Concrete mix design per m3.

Cement (CEM IIbv) 4–8 mm aggregate 0–5 mm aggregate Water

450 kg 1055 kg 705 kg 190 kg

4.2. Test results

A summary of all the tests is provided in Table 4. Crack patternsat failure for all beams are provided in Fig. 12. Load–displacementplots (normalised for design maximum load) are shown in Figs. 13–15. Beam 4_CFP_V results are omitted due to test problems. Thetransverse reinforcement ratio (qw = Asw/s/bw) within the shearspan for each beam is given.

4.3. Discussion

The results of nineteen beam tests undertaken on eleven spec-imens are presented in Fig. 13. On average, beams designed usingthe ‘CFP’ method exceeded their design load by 16%, thosedesigned using the ‘STM’ method by 10% and the ‘EC2’ beams werefound to be unconservative, falling short of the design load by 34%on average.

The results suggest that the EC2 design method can lead tounconservative designs for tapered beams in shear. Both the CFPand STM models provide consistent, conservative, results. Withinthe STM test series, tests with and without external steel platesanchored into the section with a horizontal bar showed no differ-ence in global behaviour.

The tests suggest that simplified optimisation methods may notnecessarily be appropriate for elements of complex geometry andsome existing design guidance may be inappropriate for the sheardesign of tapered beams.

5. Digital image correlation

Digital image correlation (DIC) was used to monitor strain dis-tributions in the concrete. DIC employs a static camera to take pho-tos of a unique and random pattern, painted on the side of eachbeam, which is then processed to determine displacement and

Table 4Test failure modes and ultimate capacity.

qw (%) PIV collected (Yes/No) Design maximum load, P (kN) [A] Maximum load achieved in test (kN) [B] Failure mode [B]/[A]

1_EC2_V 0.00 N 32.0 19.0 Shear 0.592_EC2_V 0.00 Y 36.0 28.2 Shear 0.782_EC2_M 0.28 Y 36.0 32.1 Shear 0.893_EC2_V 0.00 N 31.8 17.1 Shear 0.543_EC2_M 0.47 Y 31.8 18.8 Shear 0.594_EC2_V 0.00 N 29.2 26.1 Shear 0.894_EC2_M 0.69 N 29.2 9.6 Anchorage 0.33

(Fig. 13) EC2 Average [B]/[A]: 0.661_CFP_V 0.06 N 32.0 29.6 Shear 0.932_CFP_V 0.18 Y 36.0 46.9 Flexure 1.302_CFP_M 0.41 Y 36.0 48.6 Flexure 1.353_CFP_V 0.26 Y 31.8 43.8 Shear 1.383_CFP_M 0.56 Y 31.8 31.5 – 0.994_CFP_M 0.77 Y 29.2 28.7 – 0.98

(Fig. 14) CFP Average [B]/[A]: 1.162_STM_1 (i) 0.31 Y 36.0 41.8 Shear/flexure 1.162_STM_2 (i) 0.31 Y 36.0 41.5 Shear/flexure 1.152_STM_1 (ii) 0.31 Y 36.0 38.7 Flexure 1.082_STM_2 (ii) 0.31 Y 36.0 37.9 Flexure 1.052_STM_1 (iii) 0.31 Y 36.0 37.4 Flexure 1.042_STM_2 (iii) 0.31 Y 36.0 40.6 Flexure 1.13

(Fig. 15) STM Average [B]/[A]: 1.10

Beam 3_CFP_V/M

Beam 3_EC2_V/M

Beam 2_EC2

Beam 2_CFP

CrackingFailure location CrushingFlexural Reinforcement Transverse Reinforcement

Beam 1_CFP_V

Beam 1_EC2_V

Beam 2_STM(ii)

Beam 2_STM(iii)

1

1

1

V

V

2

2

2

M

M

Beam 2_STM(i)

V M

V M

V M

Beam 4_CFP_V/M

V M

Beam 4_EC2_V/M

Fig. 12. Summary of crack patterns at failure for all beams tested.


strain distributions. The freeware software MatchID [22] was usedto carry out the analysis.

5.1. Results

DIC data was collected for each test as described in Table 4.Fig. 13 shows that the design methods give contrasting behaviourunder loading. The CFP beams displayed ductility in combinationsof flexure and flexure–shear failure. By contrast, all the EC2 beamsfailed in a less desirable brittle manner. The STM model providesductility through a design method that does not rely on assump-tions of steel yielding or on empirical equations for shear behaviour.

In the EC2 test series, elements with transverse reinforcement(denoted ‘M’) were found to behave almost identically to thosewithout transverse reinforcement (denoted ‘V’), Table 4. It was pro-posed in the design model for ‘V’ type beams that the inclined steelreinforcement would both carry the entire shear force and act as

flexural reinforcement. This was not achieved, as the beams failedto reach their design loads, but more importantly there was no sig-nificant difference in failure load between the ‘V’ and ‘M’ typebeams. This suggests that the transverse reinforcement, which isgenerally placed in concrete sections to increase their shear capac-ity, did not achieve this.

By contrast, the effective placement of transverse reinforcementforms a central design criterion for the CFP method, with verticalsteel provided only where it is needed in positions where tensionforces are developed [11].

5.2. Analysis

Differences between the design methods are highlighted inFig. 16, where principal strains e1 and e2 for Beam 2_EC2, Beam2_CFP_M and Beam 2_STM_2(i) are presented at their respectivepeak loads. Also shown in Fig. 16 is an overlay of reinforcement

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.04010 20 30 50 60 70

Deflection at load point (mm)

Appl

ied

load

/ D

esig

n lo

ad4_V

1_EC2_V

2_EC2_V

2_EC2_M

3_EC2_M

3_EC2_V

4_EC2_M

4_EC2_V

2_V2_M

3_M1_V

4_M

Fig. 13. Normalised load–displacement results for EC2 beam series.

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.04010 20 30 50 60 70

1_CFP_V

2_CFP_M

2_CFP_V

3_CFP_M

4_CFP_M

3_CFP_V


Appl

ied

load

/ D

esig

n lo

ad

1_V

2_M

2_V

3_V

3_M4_M

Fig. 14. Normalised load–displacement results for CFP beam series.

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.04010 20 30 50 60 70

2_STM_1(i)

2_STM_2(i)2_STM_1(ii)

2_STM_2(ii)2_STM_1(iii)

2_STM_2(iii)


Appl

ied

load

/ D

esig

n lo

ad

2_1(i)

2_1(iii)

2_2(ii)

Fig. 15. Normalised load–displacement results for STM beam series.


locations and crack patterns at failure. The plots illustrate how thedistribution of strains differs by design method. In the EC2 beam,which failed at an applied load 14 kN less than that for the CFPbeam, a strain concentration in the support zone can been seen,

while much lower strains in the ‘body’ of the tapered beam areevident.

By contrast, strains in the CFP beam are at their greatest con-centration beneath the load point, at the position of maximum

-6.7x10-4-6.0x10-3 -3.3x10-3 2.0x10-3 4.7x10-3 7.3x10-3 1.0x10-2

P = 32kN

Beam 2_EC2_M at peak load

x

y

Beam 2_STM_2(i) at peak load

Principal Strain, ε2


P = 46kN

Beam 2_CFP_M at peak load



Contour Scale



P = 41.5kN

Crack pattern(over Principal Strain, ε1)



Reinforcement plotted in white

Fig. 16. Strain plot comparison between EC2, STM and CFP beams at their respective peak loads.


moment. Moving from this position to the support location, threebands of inclined zones of higher strain can then be seen in thee2 plot, with the magnitude of the maximum strain at the supportbeing lower than that in the EC2 Beam. These zones can be imag-ined to demarcate the concrete ‘teeth’ described by the CFP model.This is further reflected by the much deeper cracking that was ableto occur before failure in the CFP beam.

In the EC2 beam, cracks at the support dominate the failuremode. A line drawn over the top of the maximum crack depthfor the 2_EC2_M and 2_CFP_M beams shows how the EC2 designis governed by the end zone, suggesting a shear critical designmethod. In the CFP designed beam, the neutral axis (denoted bythe tip of the cracks) is almost horizontal in the tapered sectionof the beam, suggesting a greater utilisation at its peak load.

P = 32kN

Beam 2_EC2_M at 32kN

x

y



Beam 2_CFP_M

Pri

Pri

-6.7x10-4-6.0x10-3 -3.3x10-3 2.Contour Scale

Fig. 17. Strain plot comparison between EC2, CFP and

Fig. 17 compares principal strains for the three design methodsat the failure load of the EC2 beam (32kN), further demonstratingthe changes described above. Beams 3_EC2 and 3_CFP were foundto display the same general behaviour as those of the 2_EC2 and2_CFP. In Beam 3_EC2_M, a peak of principal strain e2 was seenat the supports just prior to failure (which occurred at just overhalf of the design load). This contrasts to the behaviour of Beam3_CFP, Fig. 18, and provides additional weight to the analysis andcomparisons between the EC2 and CFP models described above.

It is thus shown that by controlling the compression path theCFP method is better able to facilitate a conservative design predic-tion whose failure mode is also predicted. DIC analysis presented inthis section has shown how the behaviour of the EC2 and CFPbeams differ, and whilst data was not collected for all the EC2

P = 32kN

Beam 2_STM_2(i) at 32kN at 32kN

ncipal Strain, ε2

ncipal Strain, ε1



0x10-3 4.7x10-3 7.3x10-3 1.0x10-2

STM beams at the failure load of Beam 2_EC2_M.

ε1: 3_CFP_M (P= 32kN)

P = 32kNP = 18kN

ε1: 3_EC2_M (P = 18kN)

Fig. 18. Comparison of principal strains in Beams 3_EC2_M and 3_CFP_M at their respective failure loads.

39kN66kN84kN

Values in design model

100mm170mm

270mm

72kN

102kN

Recorded testvalues

68kN

Fig. 19. Forces derived from strain gauge readings compared to STM design model.

8 8 6

8kN

141612

1412

1614

1614

141210

1612

16kN

10128

8

221814

241612

26221412

1614

1614

141210

1630 30

22

10 1128

261

Failure (32kN)

12 12

1210

1212

12kN

10 8 8

1

3222

1614 38

322216

12

2826

2418

1412

381214

1824 28

26

34 3410

Failur

2826

2418

1412

381214

1824 28

26

10

8

32kN

_MTS_2M_2CE_2

12kN

6 121012

16kN

14 1210

14

8kN

6

Fig. 20. Crack propagation comp


beams, their failure modes recorded during testing are consistentwith the behaviour seen in Beam 2_EC2 and described above.

The crack and strain distributions seen in the STM beam seriesdemonstrate the advantages of the design method. By movingstrain concentrations away from the support, where their influencein the EC2 beam has been seen to cause premature shear failures,the STM beams fail in a ductile manner as shown in Fig. 13.

The STM beams have a deeper support section than thosedesigned using the EC2 model. This results from anchorage anddevelopment length requirements for transverse reinforcement,coupled with the model stipulation for straight lines between

1216 16

1614

16kN

44

1242

44 26

3814

4228 32

1616

2028

44

44kN

44

1242

44 26

3814

4228 32

1616

2028

44

Failure (48kN)

12

12kN

2822 16

12 10

32 2622

16 3412

e (34kN)

12

26

14

28 32

1616

2028

2024

14

1618

1618

3030

1814

32kN28

22 1612 10

32 2622

16 3412

2

M_PFC_2)i(2

101212

1612 10

1612

8kN

arisons for series 2 Beams.


tension nodes. The minimum STM depth was almost identical tothat for the CFP method. In the STM design model, all the forcesin the beam during loading are predicted. During testing, strainsin the longitudinal tension steel were recorded. It is thus possibleto compare the predicted and actual load distribution in the beam,as shown in Fig. 19 at the design load (36 kN).

The presence of a high vertical strains a small distance from thesupport in the STM beam was correlated to the design layout usingthe DIC data. The contribution of the first vertical link in the STMdesign is critical as the force in this link is approximately equalto the full reaction. It is further shown in Fig. 19 that the forces pre-dicted in the model correlate well to the recorded steel strains.

The STM model provides advantages to the beam design bymoving areas of high tension away from the support zone. Whilstthe STM and CFP beams both failed in a ductile manner, it may besurmised from these plots that the STM beam was closer to a brit-tle failure than the CFP beam. This suggests that CFP the method isan appropriate and conservative design approach.

6. Discussion

A comparison between the overall profiles of the end zone ofthe three beam types is provided by Fig. 7, Figs. 9 and 10. Whilstthe shallow support zone of the EC2 beam might be expected toyield the lowest capacities, it is the manner in which the STMand CFP models are able to adjust and control the behaviour ofthe beams that is most relevant about these results. This is demon-strated by comparing reinforcement ratios for the test specimens(Table 4). For example, Beam 2_CFP_V has less transverse rein-forcement than Beam 2_EC2_M (0.18% versus 0.28%), but achievesa higher failure load (46.9 kN versus 32.1 kN). The failure loads ofBeam 2_CFP_V and Beam 2_STM_1 are similar (46.9 kN versus41.8 kN), but the beams have quite different transverse reinforce-ment ratios (0.18% versus 0.31%). It is only by pushing the EC2model to its feasible limit that the deficiencies in such an approachare revealed.

6.1. Crack progressions

The progression of cracking is summarised for all Beam 2 vari-ants in Fig. 20. The EC2 beams were found to crack initially at theend zone (regions of high strain seen in the DIC results) before lim-ited crack progression towards the point of load application. In theSTM model, cracking begins in the centre of the tapered zonebefore spreading towards the support and the point of load appli-cation. In the CFP model, cracking begins beneath the load pointand spreads towards the support before failure.

These patterns of crack progression are seen in all the beamstested and highlight the shear critical nature of the EC2 design.By cracking first in the end zone, it is clear that shear is the dom-inant condition at the ultimate load, and thus a brittle failure iscreated.

The EC2 model is, in itself, a strut and tie model that assumesthe steel yields so that plastic behaviour is possible and stressesmay be redistributed. This approach forms the basis of the BS EN1992-1-1 [4]. The STM model, in which the steel is not assumedto be yielding in all positions, is therefore based on lower boundplasticity theory but is applied to a situation where the behaviouris not plastic. The success of the STM model is found in providing aconservative design method for tapered beams that do not fail inshear. Ductile behaviour is ensured by controlling the yieldingposition such that it occurs away from the support (in the beamstested in this paper it occurs primarily beneath the loading point).This provides the element with the ductile behaviour that is

desired in the design process, something that may not be guaran-teed by the EC2 method.

In this way, the STM and EC2 models are opposite in theirapproach. Whereas the STM model becomes a self-fulfilling proph-ecy in which the section can be designed to be ductile and so it isductile, the EC2 model can become a vicious circle in which theductile failure that is desired cannot be achieved because the steelis not able to yield prior to failure of the critical section. This fun-damental result demonstrates the power of the STM method overthe EC2 approach described in this paper.

7. Conclusions

This paper has shown that the STM and CFP methods providereliable design processes that result in an advantageous internalstrain distribution, while the EC2 method does not marry withthe experimental data. The importance of the compression pathin shear behaviour is demonstrated by the CFP and STM designmethods. In both, the change in direction of the compression pathgives rise to the critical link position.

The test results (Fig. 13) show that the EC2 model is shear crit-ical and unconservative, while the STM is less shear dominant thanthe EC2 model, and the CFP model provides perhaps the idealbehaviour (limited cracking and lower strains in the end zone,leading to predictable and ductile failures).

Results from both the CFP and STM beam design methods, sup-ported by DIC data, suggest that they are better able to model thebehaviour of tapered beams in shear. In light of the results pre-sented for the EC2 beam series, it is recommended that taperedbeams do not rely entirely on the contribution of their flexuralsteel to provide shear capacity. This means that the value of thevertical component of the steel force (Vtd) should not comprisethe full value of the shear resistance of the section until furtherguidance can be determined. The DIC analysis shows that the ten-sion zones seen in the support zone of EC2 beams is moved in boththe STM and CFP beams to a distribution across the taper that sat-isfies the design model.

Only simply-supported beams have been considered in thispaper, and many more challenges and opportunities will arisethrough the use of non-prismatic beams in frame elements, or incast in situ construction where the support conditions allowmoments to be carried. This introduces additional challenges forthe optimisation of concrete structures.

References

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Shear behaviour of non-prismatic steel reinforced concrete beams