moment-curvature behaviors of concrete beams singly

Research ArticleMoment-Curvature Behaviors of Concrete Beams SinglyReinforced by Steel-FRP Composite Bars

Zeyang Sun12 Yang Yang23 Wenlong Yan12 Gang Wu1 and Xiaoyuan He2

1Southeast University Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of EducationNanjing 210096 China2School of Civil Engineering Southeast University Nanjing 210096 China3College of Civil Science and Engineering Yangzhou University Jiangsu 225127 China

Correspondence should be addressed to Gang Wu gwuseueducn

Received 6 October 2016 Revised 4 December 2016 Accepted 26 December 2016 Published 15 February 2017

Academic Editor John Mander

Copyright copy 2017 Zeyang Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A steel-fiber-reinforced polymer (FRP) composite bar (SFCB) is a kind of rebar with inner steel bar wrapped by FRP which canachieve a better anticorrosion performance than that of ordinary steel bar The high ultimate strength of FRP can also providea significant increase in load bearing capacity Based on the adequate simulation of the load-displacement behaviors of concretebeams reinforced by SFCBs a parametric analysis of the moment-curvature behaviors of concrete beams that are singly reinforcedby SFCB was conductedThe critical reinforcement ratio for differentiating the beamrsquos failure mode was presented and the conceptof the maximum possible peak curvature (MPPC) was proposed After the ultimate curvature reached MPPC it decreased with anincrease in the postyield stiffness ratio (119903sf ) and the theoretical calculationmethod about the curvatures before and after theMPPCwas derivedThe influence of the reinforcement ratio effective depth and FRP ultimate strain on the ultimate point was studied bythe dimensionlessmoment and curvature By calculating the envelope area under themoment-curvature curve the energy ductilityindex can obtain a balance between the bearing capacity and the deformation ability This paper can provide a reference for thedesign of concrete beams that are reinforced by SFCB or hybrid steel barFRP bar

1 Introduction

The corrosion of steel bars in concrete structures will demol-ish the bonding performance and lead to the cracks of con-crete due to the volume expansion of corrosion products [1]Fiber-reinforced polymer (FRP) is a type of composite withsuperior anticorrosion performance in concrete structures[2 3] the successive research and application of glass FRP(GFRP) for the repair of bridges and parking lots promptedthe Canadian government to reexamine this type of materialto extend the service life of concrete structures Experimentaland theoretical studies on concrete beams that are reinforcedby GFRP were conducted by Benmokrane et al (1996) [4]and the results of these studies indicated that the calculationmethod of the bearing capacity and the displacement of theAmerican Concrete Institute (ACI) code [5] can be employedfor a GFRP beam with minor modifications For concretebridges or marine structures in Canada a GFRP-reinforced

structure can maintain superior performance (Mufti et al2007) [6] the adhesions between GFRP and concrete wereundamaged in 5sim8 years and the alkaline substance did notpenetrate the GFRP reinforcement The Canadian HighwayBridge Design Code [7] has enabled designers to use GFRPbars as a primary means of reinforcement in concrete struc-tures but for a prestressed concrete structure the GFRP barrsquosservice strength is limited to 25 of its ultimate strength dueto its elastic property (brittle failure) and according to ACI4404R-04 [8] GFRPs are not allowed for prestressing appli-cation because of their weak creep-rupture characteristics

To improve the performance of an FRP-reinforced con-crete structure a hybrid FRP bar with different ultimatestrains was proposed [9] to achieve a certain degree of ducti-lity but this type of ductility was realized by the partial rup-ture of FRP with a low elongation rate which will exhibit sig-nificant strength degradation during cyclic loading Becausea steel bar has a large elongation rate (approximately 15)

HindawiAdvances in Civil EngineeringVolume 2017 Article ID 1309629 14 pageshttpsdoiorg10115520171309629

2 Advances in Civil Engineering

a better performance and a more acceptable cost can beachieved by combining steel and FRP Two combinationsexist (1) a concrete beam hybrid reinforced by steel bars andFRP bars (2) beams reinforced by steel-FRP composite barsTwelve concrete beams that were reinforced by a steel bar aGFRP bar or a steelGFRP bar were tested by Lau and Pam(2010) [10] the results indicated that the ductility of hybridreinforced beams was better than the ductility of a pure FRP-reinforced beam the minimum reinforcement ratio for anFRP-reinforced beam could be reduced by 25 according tothe ACI 4401R-06 [11] A study of twelve concrete beamswhich were reinforced by steel bar and GFRP bar was con-ducted by Safan (2013) [12] and the results showed that thefailure modes of concrete beams were presented as concretecrushing after the yielding of the steel bar on the tensile sideand a GFRP bar can maintain the flexural bearing capacity ofa concrete beam with a relatively small reinforcement ratioA ductility index was proposed by Pang et al (2015) [13]to evaluate the load-displacement relationship of a hybridreinforced concrete beam and the corresponding methodof differentiation of the failure mode was suggested butthis method of differentiation was slightly complicated Anexperimental study on six hybrid reinforced concrete beamswas conducted by El Refai et al (2015) [14] and the resultsindicated that the deflection of the hybrid reinforcementbeam with the higher reinforcement ratio can be well pre-dicted by current design codes the crackwidthwas calculatedby ACI-4401R-06 [11] by modifying the bonding coefficientbetween the FRP bar and concrete

Deterioration failure can occur in concrete beams that arereinforced by hybrid steel barsFRP bars when a steel bar issubjected to corrosion A better anticorrosion performancecan be achieved by a steel-FRP composite bar (SFCB) whichis a type of rebar with an inner steel bar and an outerFRP Sixteen concrete beams reinforced by hybrid rebar weretested by Nanni et al (1994) [15] and the theoretical analysisof the load-displacement curves revealed that the calculateddisplacement was slightly smaller than that of the test curveswhen the slip between the FRP and concrete was disregardedand the load-carrying capacity of the hybrid reinforced beamscan be calculated using traditional RC theory Concretebeams that are reinforced by longitudinal steel-GFRP com-posite bars and GFRP stirrups were tested by Saikia et al(2005) [16] the composite bar was composed of an inner steelbar with a diameter of 6mm that is helically wounded byGFRP with a thickness of 2mm Due to an underdevelopedmanufacturing technology a slip between the inner steel barand the outer GFRP occurred A factory-produced SFCB wasachieved by Wu et al by modifying the pultrusion process ofFRP bars [17] and its characteristics include the following(1) the initial elastic modulus and the postyield modulus canbe designed by adjusting the steelFRP ratio (2) a durabilityof SFCB that is equivalent to the durability of FRP bar canbe achieved (Figure 1(a) [18]) (3) the strength of the innersteel bar can be effectively employed and the high ultimatestrength of FRP can be employed as a reservation (4) theperformancecost ratio of a concrete structure reinforced bysteel-FRP composite bars can be optimized with consideringthe long-term performance and bearing capacity The typical

Longitudinal wrapped Inner steel barouter FRP

(a) Steel-FRP composite bars

0 5000 10000 15000 20000 25000 30000 350000

20

40

60

80

Load

(kN

)

Rupture of basalt fiber ofsteel-BFRP composite bar

Rupture of carbon fiber ofsteel-CFRP composite bar

Yielding of SFCBrsquos inner steel bar

Strain (120583120576)

(b) Load-strain curves

Figure 1 Factory production of SFCB and itsmechanical properties

load-strain curves of steel-basalt FRP composite bars andsteel-carbon FRP composite bars are illustrated in Figure 1(b)and the stress-strain model of SFCB has been comprehen-sively investigated [19] It was demonstrated that there was noslip between the inner steel and outer FRP during the loadingprocess An experimental study of a concrete beam that isreinforced with SFCBs steel bars and pure FRP bars understatic loading was conducted by the authorrsquos group [20] andresearch on the effect of blast load [21] is to be conductedIn this paper a parametric analysis of the moment-curvaturebehavior is conducted based on the simulation of the load-displacement curve of an SFCB-reinforced concrete beamthe corresponding curvature ductility index is also discussedThe research results of this paper can provide a reference forthe design of hybrid reinforced concrete beams

2 Simulation of a Concrete BeamReinforced by SFCB

21 Material Parameters Experimental studies of six con-crete beams were conducted by Sun et al (2012) [20] andthe SFCB beams exhibit a stable postyield stiffness after theyielding of SFCBrsquos inner steel bar and concrete crushed afterthe rupture of SFCBrsquos outer FRP The conventional RC beamhad the largest ductility whereas the ultimate load of the RCbeamwas approximately 31 of the ultimate load of the SFCBbeam To calculate the moment-curvature behavior of the

Advances in Civil Engineering 3

Strain

Stress

120576cu fcu

120582E0Et

ft

120576c0 f998400c

E0 =2f998400

c120576c0

Figure 2 Stress-strain behavior of the concrete model

concrete beam that is reinforced by SFCB the comparisonanalysis between the experimental results of the SFCB beamand the calculated curves are conducted Concrete02 inOpenSees (OS) [22] is adopted for concrete stress-strainbehavior (Figure 2)The peak stress and strain of concrete canbe expressed by (1) and (2) [23] and the relationship betweenthe crush strain and the volumetric percentage of stirrups isshown in (3)

119891c0 = 1198701198911015840c = (1 + 120588sv119891yh1198911015840c )1198911015840c (1)

120576c0 = 0002119870 = 0002(1 + 120588sv119891yh1198911015840c ) (2)

120576cu = 0004 + 120588sv119891yh [MPa]300 (3)

where 1198911015840c and 120576c0 are the peak stress and peak strainrespectively 120576cu is the ultimate compressive strain of concrete120588sv is the volumetric percentage of the stirrups and 119891yh is theyield strength of the stirrups

The load-strain relationship of a SFCB is a trilinearmodel(Figure 1(b)) where the rupture of SFCBrsquos outer FRP isdefined as the failure point As a result the trilinearmodel canbe simplified in two stages (see (4)) The first stage is beforeyielding (119864I) and the second stage is between yielding andFRP rupture (119864II)

119891sf = 119864I120576sf (0 le 120576sf le 120576sfy)119891sfy + 119864II (120576sf minus 120576sfy) (120576sfy le 120576sf le 120576sfu) (4)

where119891sf and 120576sf are the stress of SFCB and the strain of SFCBrespectively 119864I is the elastic modulus before yielding 119864II isthe postyield modulus of SFCB 119891sfy and 120576sfy are the yieldstress of SFCB and the yield strain of SFCB respectively and119891sfu and 120576sfu are the ultimate stress and the ultimate strain ofSFCB at the point of FRP rupture respectively

The mechanical behavior of SFCBs in OS is realized byestablished separate fibers of steel and FRP For steel fiber thekey points are the yield point the hardening point the hard-ening slope and the ultimate point which can be describedby theChang andMander (1994)model [24] For FRP a linearelastic element is defined to represent the elastic behaviorThepostyield stiffness ratio 119903sf of an SFCB can be defined by (5)and the corresponding equivalent longitudinal reinforcementratio (120588esf ) with regard to conventional steel reinforced con-crete beam is defined by (6)

119903sf = 119864f119860 f(119864s119860 s + 119864f119860 f) =119864f119860 f119864sf119860 sf

(5)

120588esf = 119864f119860 f119903sf119864s119860g (6)

where 119864f and119860 f are the elastic modulus of outer FRP and thecross-sectional area of the outer FRP respectively 119864s and 119860 sare the elastic modulus of the inner steel bar and the cross-sectional area of the inner steel bar respectively 119864sf and 119860 sfare the elastic modulus of SFCB and the cross-sectional areaof SFCB respectively and 119860g is the total cross-sectional areaof the concrete beam

22 The Comparison between the Tested and the CalculatedResults for Concrete Beams The specimen details are pre-sented in Figure 3 the total length of the beam is 2000mmthe cross section was 220mm times 300mm and the shear-span ratio was 3 The diameter of the top bars was 12mmdiameter the diameter of the stirrupswas 8mmdiameter andthe average tested compressive strength of six concrete cubes(150 times 150 times 150mm) was 488MPa [20]

The longitudinal reinforcements of the selected concretebeams were steel bar and S10B51 and the correspondingmechanical properties are listed in Table 1 The notationldquoS10B51rdquo indicates that the SFCB is composed of an innersteel bar with a diameter of 10mm that is longitudinallywrapped by 51 bundles of 4000-tex basalt fibers where ldquotexrdquois the weight (g) of one fiber bundle per kilometerThe elasticmodulus of a steel bar is approximately 200GPa and that ofa basalt fiber is nearly 90GPa As a result the elastic modulusof an SFCB is smaller than that of a steel bar [19]

Five groups of dial gauges were evenly arranged on thefront side of the mid-span and five strain gauges were alsoinstalled on the opposite side with the same height (Fig-ure 4(a)) to verify the plane section assumption The averagestrain of B-S10B51 that wasmeasured by the dial gauges is pre-sented in Figure 4(b) the horizontal axis represents micros-train and the vertical axis represents the distance from thebeamrsquos bottom surface The average strain of B-S10B51 alongthe section height can satisfy the plane section assumptionand the neutral axis was substantially unchanged beforecracking and increased rapidly after cracking Although thecomposite bar S10B51 had stable postyield stiffness the neu-tral axis rapidly increases after the yielding of the inner steelbar to resist the increased moment


Table 1 Mechanical properties of the selected reinforcements

Reinforcement type 119889 (mm) 119864I (GPa) 119864II (GPa) 119891y (MPa) 119891u (MPa) Elongation rate ()Steel bar 12 204 415 580 145S10B51 18 9613 2980 19227 5484 26

600 600 600100 1002000

Dialindicator

JackLoad sensor

Allocation Dialindicator

Dialindicator

220B-S10B51

1206018801206018150

300

Figure 3 Specimen design

(a) Measurement of average strain at mid-span

0

50

100

150

200

250

300

minus5000 0 5000 10000 15000 20000 25000 30000

Sect

ion

heig

ht (m

m)

B-S10B51

Strain (120583120576)

3t

6t9t

1518t

2612t

2685t

(b) The strain distribution of B-S10B51

Figure 4 Verification of the plane section assumption of B-S10B51

With verification of the plain section assumption theaverage curvatures of the SFCB-reinforced concrete beam canbe calculated by

120601 = 120576c + 120576sfℎ0 (7)

where 120576c (absolute value) is the concrete compressive strainand ℎ0 is the effective height of the beamrsquos cross section Asa result MPPC 120601peak max of a hybrid reinforced beam canbe determined by the FRP rupture strain 120576FRP the sectioneffective height ℎ0 and the ultimate compressive strain ofconcrete 120576cu

The tested stress-strain curve of the composite bar S10B51and the corresponding calculated curve by OS are shownin Figure 5(a) the calculated curve was consistent withthe tested value As shown in Figure 5(b) the comparisonbetween the calculated results and the tested results for aconcrete beam reinforced by SFCB is presented in which theyield strength and the ultimate strength of S10B51 were takenas 80 of the tested values The calculated bearing capacitycorresponded with the test results whereas the correspond-ing ultimate mid-span displacement is approximately 81 ofthe tested values which is caused by the large slip between theSFCB and the concrete at mid-span (Figure 6)


0 10000 20000 300000

250

500

750St

ress

(MPa

)

TestedCalculated

Strain (120583120576)

EII

EI

120576sfy 120576sfu

(a) Mechanical properties

0

50

100

150

200

250

300

0 20 40 60 80

Beam yielding

Mid-span displacement (mm)

Load

(kN

)

Rupture of SFCBrsquosouter FRP

TestedCalculated

(b) SFCB-reinforced concrete beam

Figure 5 Comparison between tested results and calculated results of SFCB-reinforced concrete beam

SFCBrsquos outer FRP ruptured

Figure 6 Failure mode of B-S10B51

23 Critical Reinforcement Ratio The calculation methodof the RC beam can also be employed for beams that arereinforced by SFCB or hybrid FRP barsteel bar The failuremode of SFCB-reinforced concrete beam can be influencedby the yield strain of SFCB FRP rupture strain and concretecrush strain The different ultimate state (strain distribution)of hybrid reinforced concrete beam is shown in Figure 7(a)where the tensile capacity of cracked concrete is assumed tobe zero The failure mode can be divided into three casesaccording to the dominated parameters 120576sfu or 120576cu When theequivalent reinforcement ratio 120588esf exceeds 120588eb I the failuremode (Mode I) is concrete crushed before the tensile rein-forcement reached yield strain When the ultimate state isconcrete crushed after SFCBrsquos inner steel bar yielded (withoutFRP rupture) the reinforcement ratio ranges between 120588eb Iand 120588eb II (Mode II)The third failuremode is the SFCBrsquos outerFRP rupture after the inner steel yielded without concretecrushing (Mode III) and the corresponding 120588esf is smallerthan 120588eb IIWhen the failuremode includes concrete crushingthe compression stress of concrete can be simplified as arectangular block (Figure 7(b)) the average stress intensityis expressed as 12057211198911015840c and the compression height of concreteblock can be represented as 1205731119909c

According to the static equilibrium of axial force and theplain section assumption the critical reinforcement ratios

(120588eb I and 120588eb II) of SFCB-reinforced concrete beam can becalculated by equations (8) and (9) [25]

120588eb I = 120572112057311198911015840c119864s

1120576sfy (1 + 120576sfy120576cu) (8)

120588eb II = 120572112057311198911015840c119864s

1[(120576sfu minus 120576sfy) 119903sf + 120576sfy] (1 + 120576sfu120576cu) (9)

where1205721 and1205731 are coefficients for the equivalent stress blockof concrete in compression

3 Parametric Study

31 Typical Moment-Curvature Curves of the SFCB BeamAccording to the design code of AASHTO [26] the selectedparameters are as follows the width of the beam section119887 = 200mm the aspect ratio ℎ119887 = 2sim4 120588esf = 03sim121198911015840c = 30sim90MPa the rupture strain of SFCBrsquos outer FRP120576sf = 0015sim0025 which represent carbon fiber (0015) tobasalt fiber (0025) and the postyield stiffness ratio of SFCB119903sf = 0001sim095 A total of 2880 moment-curvature analyseswere performed in this paper typical moment-curvaturecurves are illustrated in Figure 8 by varying 119903sf With anincrease in 119903sf the yield point remained stable the postyieldstiffness increased and the peak curvature (120601peak) increased


Rupture ofSFCBrsquos outer

FRP Mode IMode IIMode III

120576sfu lt 120576sf

120576sfyu120576sfy

120576cu

120588esf = 120588e

b_I

120588esf = 120588e

b_II

(a) Strain distribution of different failure modes

C

Tsf = fsfuAsf

h0

1205731x

c

1205721fc

ycMu

120588esf = 120588e

b_II

(b) Equivalent stress block of internal relationships at120601peak max

Figure 7 Schematic strain distribution and equivalent concrete stress block

000 005 010 015 0200

20

40

60

80

Cracking


Concrete crushing

Yieldpoint

= 090 120576sfu = 0015 f998400c = 30MPa h = 200mm

Residual bearing capacity decreased with

rsf = 07

rsf = 03

rsf = 01

120601peak_max

Mom

ent (

kNmiddotm

)

Curvature 120601 (radm)

the increase of rsf120588e

sf

Figure 8 Moment-curvature analyses of SFCB beams

and then decreased Before the ultimate curvature reached120601peak max the increase of 120601peak is caused by a decrease in theconcrete compression depth and the failuremode is concretecrushing As a result the ultimate curvature is dominatedby the tensile strain of SFCB in this failure mode After therupture of SFCBrsquos outer FRP the moment decreased to thelevel in which only the inner steel bar worked and the innersteel bar decreased with an increase in the 119903sf 32 Yield Point The yieldmoment increases with an increasein 120588esf whereas the yield curvature slightly decreasedWith anincrease in the concrete compressive strength both the yieldmoment and the yield curvature increased A semiempiricalequation was proposed by Aycardi et al [27] with considera-tion of the effect of yield strain and effective depth as shownin

120601y = 17 119891y119864sℎ0 = 17120576sfyℎ0 (10)

Because the yield strain and the column depth are fixedparameters the key parameter is the concrete strain whentensile reinforcement yielded The fitted yield curvature ispresented in (11) with consideration of the reinforcementratio and concrete strength

120601y regressℎ0 = 120576sfy + 00396120588esf + 000035 (11)

The comparison between the fitted results and the yieldcurvature by OS is shown in Figure 9(a) Aycardirsquos empiricalequation overestimated the yield curvature the proposed (11)is consistent with the results of OS and the coefficient ofdetermination (1198772) of the fitted equation is 0997

The fitted yield moment can be obtained based on thefitted yield curvature and the sectional force balance asshown in

119872y regress

119864s119887ℎ20 = 000157120588esf + 6233119864 minus 07 (12)

where119872y regress is the fittedmoment Figure 9(b) presents thecomparison between the fitted results and the OS values 1198772is 0998

33 Ultimate Point The ultimate point of a conventional RCstructure was defined by 80 or 85 of the peak load capac-ity For SFCB-reinforced concrete structures the rupture ofSFCBrsquos outer FRP will cause a significant decrease in the loadcapacity As a result the rupture of SFCBrsquos outer FRP wasdefined as the ultimate point of SFCB-reinforced concretebeam in this paper

331The Influence of the Reinforcement Ratio Before the sec-tion curvature reached 120601peak max both the ultimate momentand curvature increase with an increase in 120588esf (Figure 10(a))With an increase in 119903sf the section ultimate curvature with alarger 120588esf increases faster than the section ultimate curvaturewith a smaller 120588esf When the section ultimate curvature


0000 0005 0010 00150000

0005

0010

0015

0020Fi

tted

resu

ltsO

S re

sults

Yield curvature (radm)

R2 = 0987

120601y = 17fy

Esh0= 17

120576sfy

h0

(a) Curvature

0 100 200 300 400 500 600

0

100

200

300

400

500

Fitte

d re

sults

OS

resu

lts

Yield moment (kNmiddotm)

R2 = 0998

(b) Moment

Figure 9 Fitted results of yield curvature and moment

reached120601peak max the larger120588esf is the smaller the correspond-ing 119903sf is As illustrated in Figure 10(b) when the 120588esf increasedfrom 03 to 09 120601peak max decreased from 065 radm to015 radm (reduced by 77) After the curvature reached120601peak max with an increase in 119903sf the section moment contin-ued to increase while the curvature significantly decreasedwhich was caused by a decrease in the tensile strain of SFCBat the ultimate point

The dimensionless ultimate moment and curvature areshown in Figures 10(c) and 10(d) Before reaching the criticalpoint (120601peak max) the slopes of the dimensionless momentsalmost overlap When 119903sf was fixed as 01 the ultimate curva-tures were approximately ten times the yield curvature whenother parameters changed Before the ultimate curvaturereached 120601peak max with an increase in 119903sf both the slopeand the dimensionless value of the section moment withlarger 120588esf decreasedThe trend of the dimensionless curvature(Figure 10(d)) was similar to the trend of the originalcurvature (Figure 10(b)) As a result only the dimensionlesscurvature was discussed when changing other parameters

332 The Influence of SFCBrsquos Rupture Strain The rupturestrain of SFCBrsquos outer FRP has no effect on the yield curvatureor moment The dimensionless moments and curvatures ofthe SFCB beam section by changing the rupture strain ofSFCB were presented in Figure 11 the failure modes areMode II and Mode III When failure Mode II occurred theultimate point was determined by the concrete crush strainAs a result the ultimatemoment and curvature with differentSFCB rupture strains were equivalent In Mode III the largerthe SFCBrsquos rupture strain was the larger the ultimatemomentand curvature were With an increase in 119903sf the ultimatecurvature increased at a faster rate than the rate of increasein the ultimate moment which was caused by the nonlinearproperty of concrete stress-strain in the compression zone

333 The Influence of Effective Depth When the effectivedepth (ℎ0) increases and other parameters remain unchan-ged both the ultimate moment and the ultimate curvature

will increase with an increase in ℎ0 However 119903sf (approx-imately to 025) does not change when the ultimate pointreached critical point 120601peak max The dimensionless momentand curvature are also listed in Figure 12 and the dimension-less curves almost overlap

34 Curvature beforeMPPC The failure is determined by therupture strain of SFCBrsquos outer FRP in failure Mode III andthe concrete in the compression zone kept undamaged Basedon the force equilibrium and elastic assumption the concretestrain can be obtained as shown in

120576c = 120601uℎ0 minus 120576sfu = 1205741 + radic12057421 + 41205741120576sfu2 (13)

where 1205741 can be expressed as

1205741 = 2119864esf119864c 120588

esf [120576y + 119903sf (120576sfu minus 120576y)] (14)

The comparison between the calculated ultimate cur-vature and the corresponding OS curvature is shown inFigure 13

The calculated curvatures correspond with the corre-sponding OS results when 119903sf is relatively small The calcu-lated curvatures became smaller than the calculated curva-tures in OS when 119903sf is large (Figure 13(a)) The error wascaused by the elastic assumption of compressive concretewhich enlarges the contribution of compressive concreteand therefore the calculated ultimate curvature was under-estimated when 119903sf is relatively large The compressive strainof concretewill increasewith an increase in the reinforcementratio (Figure 13(b)) an increase in SFCBrsquos rupture strain(Figure 13(c)) or a decrease in 1198911015840c (Figure 13(d)) which willproduce a larger error between the calculated values and OSvalues the maximum error is approximately 30

35 The Maximum Possible Curvature (MPPC) A total of 98sets of ultimate points were observed around the maximum


00 02 04 06 08 10

50

100

150

200

250

300

350

Failure mode changed toMode II

Mom

ent (

kNmiddotm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =

(a) 119903sfndash moment

00 02 04 06 08 10004

005

006

007

008

009Failure mode changed to Mode II

Curv

atur

e (ra

dm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =

(b) 119903sfndash curvature

00 02 04 06 08 101

2

3

4

5

6

7

8

Dim

ensio

nles

s mom

ent

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =

(c) 119903sfndash dimensionless moment

00 02 04 06 08 10

6

7

8

9

10

11

12

13

14

15D

imen

sionl

ess c

urva

ture

Failure modechanged to

Mode II

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =

(d) 119903sfndash dimensionless curvature

Figure 10 Influence of the reinforcement ratio

possible peak curvature The comparison between the calcu-lated 120601peak max using (7) and the corresponding curvature inOS is shown in Figure 14(a) where the former was generallylarger than the latterThemaximum error was approximately20 which was primarily caused by the error in concretecompressive strain The fitted 120601peak max is listed in (15) andthe corresponding comparison between the fitted value andthe OS results is presented in Figure 14(a) which indicatesthat (15) can yield a better prediction of 120601peak max

120601peak max = 120576sfu + 0985120576cu minus 00045ℎ0 (15)

When the section curvature reached 120601peak max the cor-responding reinforcement ratio 120588esf is 120588eb II (Figure 7(a)) The

calculated 120588eb II by (9) was larger than 120588eb II in OS the maxi-mum error was approximately 40 (Figure 14(b)) Equation(16) is the fitted critical reinforcement ratio (120588eb II regress) by theregression in concrete compressive strain

120588eb II regress

= 1198911015840c119864s (1 + 120576sfu120576cu) (120576ssfu minus 120576sfy)times 1[1379 (120576ssfy (120576ssfu minus 120576sfy)) + 1278119903sf minus 0003]

(16)


00 02 04 06 08 101

2

3

4

5D

imen

sionl

ess m

omen

t

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =

(a) 119903sfndash dimensionless moment

00 02 04 06 08 10

6

7

8

9

10

11

12

13

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =

(b) 119903sfndash dimensionless curvature

Figure 11 Influence of SFCBrsquos rupture strain

00 02 04 06 08 100

100

200

300

400

500

600

700

Dim

ensio

nles

s mom

ent

0

1

2

3

4

5

rsf

f998400c = 30MPa

120576sfu = 0025

Dimensionless momenth0 = 02mh0 = 04mh0 = 06m

Mom

ent (

kNmiddotm

)

Moment (kNmiddotm h0 = 02m)Moment (kNmiddotm h0 = 04m)Moment (kNmiddotm h0 = 06m)

06120588esf =

(a) 119903sfndashmoment

00 02 04 06 08 10 12

005

010

015

020

025

Curv

atur

e (ra

dm

)

0

2

4

6

8

10

12

14

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

120576sfu = 0025

Dimensionless curvatureh0 = 02mh0 = 04mh0 = 06m

Curvature (radm)h0 = 02mh0 = 04mh0 = 06m

06120588esf =

(b) 119903sfndashcurvature

Figure 12 Influence of effective depth

The comparison between 120588eb II regress and the reinforce-ment ratio in OS is illustrated in Figure 14(b) The errorwas plusmn10 percent which indicates that (16) can be used topredict the failure mode of a designed concrete beam that isreinforced by SFCB

36 Curvature after MPPC After the ultimate curvaturereached120601peak max the ultimate curvaturewill decreasewith anincrease in 119903sf and the corresponding failuremode is concretecrushing (Mode III) Based on the force equilibrium and the

simplified compression block the tensile strain of SFCB canbe calculated by

120576sf = minus1205742 + radic12057422 minus 8119903sf12057432119903sf (17)

where 1205742 and 1205743 can be expressed by (18) and (19) respec-tively

1205742 = 120576y (1 minus 119903sf) + 120576cu119903sf (18)


000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =

(b) Different reinforcement ratio

00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =

(c) Different SFCBrsquos rupture strain

00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c

Figure 13 Comparison between the calculated curvatures and the corresponding OS results in failure Mode III

1205743 = 120576cu120576y (1 minus 119903sf) minus 119891c120576cu119864esf120588esf (19)

The comparison between the calculated curvatures by(17) and the OS values is presented in Figure 15 Whenthe curvature ductility is relatively large the calculatedcurvature was similar to the OS results When the curvatureductility is relatively small the calculated curvature may beapproximately 135 times the calculated curvature of the OSvalues and 120601calculated120601peak max OS is approximately a powerfunction of the curvature ductility (Figure 15)

4 Discussion of Curvature Ductility

Many indexes were proposed for the performance evaluationof a conventional RC structure ductility is a main index butthe ductility substantially varied because the yield curvature(displacement) that was obtained using different methodssignificantly varied [28 29] The yield point can be deter-mined by the graphingmethod and the equal energymethodas shown in Figure 16 For a SFCB beam the yield curvature isdetermined by the yield of the inner steel bar and the ultimatecurvature is defined by the rupture of SFCBrsquos outer FRP or thecrushing of concrete in the compression zone


002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Fitted resultsOS results

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)

120601peak_max by (7) over OpenSess result

(a) Comparison between the calculated 120601peak max and the correspondingOS results

002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)

by (9) over OpenSess result


120588eb_II

(b) Comparison of the critical reinforcement ratio 120588esf

Figure 14 Maximum possible peak curvature and critical reinforcement ratio

5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y

Figure 15 Comparison between the calculated curvatures and theOS values in Mode II

The differences among different ductility factors arecalculated using (20) to (22) where 1205831 is the ratio betweenthe yield curvature and ultimate curvature (see (20)) 1205832 isa ductility index that considers the effect of the ultimatemoment (see (21)) and 1205833 is the ratio of the total envelopearea and the area before yielding (see (22))

1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)

The development trend of 1205831 1205832 and 1205833 is presentedin Figure 17(a) when the failure mode was SFCBrsquos outer

FRP rupture the three ductility indexes increased withan increase in 119903sf and 1205833 has the largest absolute valueWhen the failure mode was concrete crushing the threeductility indexes decreased with an increase in 119903sf the largestductility was observed at the point of theMPPC Figure 17(b)presents the development trend of ldquodimensionlessrdquo ductility(120583120583119903sf=005) compared with the slopes before MPPC thedecreasing slope of ldquodimensionlessrdquo 1205831 was the largest andthe decreasing slope of ldquodimensionlessrdquo 1205832 was comparativelyflat For ldquodimensionlessrdquo 1205833 both the ultimate moment andthe ultimate curvature were considered and the decreasingslope of ldquodimensionlessrdquo 1205833 ranged between ldquodimensionlessrdquo1205831 and ldquodimensionlessrdquo 1205832

The influence of 120588esf 120576sfu 1198911015840c and ℎ0 on 1205833 is shown inFigure 18 When the failure mode is the rupture of SFCBrsquosouter FRP (Mode III)120588esf 1198911015840c and ℎ0 have aminimal influenceon 1205833 By changing the rupture strain of SFCB 120576sfu the slopeof 1205833 increases with an increase in 119903sf which is caused by theincreased ultimate moment and ultimate curvature Whenthe failure mode was concrete crushing after the yielding ofSFCBrsquos inner steel (Mode II) the ductility varies within arelatively small range 1205833 decreases with an increase in 119903sf and1198911015840c and 120576sfu had minimal influence on the softening branchThe ldquodimensionlessrdquo 1205833 (120574120583 = 12058331205833 119903sf=005) was not affectedby changes in ℎ0 (Figure 18(d)) and 1205833 was approximately 43times 1205831 when the MPPC was attained

5 Conclusions

Based on the simulation of the test results of a concrete beamreinforced by SFCB the parametric analysis of moment-curvature behavior of singly reinforced concrete beams wasconducted according to AASHTO design code The mainconclusions are as follows

(1) The failure modes of a concrete beam reinforced bySFCB including concrete crushing before the tensilereinforcement reached yield strain (Mode I) concrete


0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y

(a) Different methods to determine yield point and ultimatepoint

0

A

B

120601y 120601u

Mu

My

Ay

Aplastic

Atot = Ay + Aplastic

(b) Ductility defined by energy

Figure 16 Different methods for defining ductility

00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =

(a) Differences among 1205831 1205832 and 1205833

00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05

(b) ldquoDimensionlessrdquo ductility

Figure 17 Comparison of 1205831 1205832 and 1205833

crushed after SFCBrsquos inner steel bar yielded (withoutouter FRP fiber ruptured Mode II) and the SFCBrsquosouter FRP ruptured after the inner steel yieldedwithout concrete crushing (Mode III) In addition theultimate curvature increases with an increase in 119903sfin failure Mode III whereas the ultimate curvaturedecreases with an increase in 119903sf in failure Mode II

(2) When the rupture of SFCBrsquos outer FRP accompa-nied the crushing of concrete in the compressionzone the section reached the MPPC and the corre-sponding parameters were critical for the ductilityof a concrete beam Before reaching the MPPC thecalculated ultimate curvature was consistent withthe OpenSees (OS) results when 119903sf was relativelysmall and the ultimate curvature was underestimatedwith an increase in 119903sf After reaching the MPPC

the calculated curvature corresponds with the OSresults when the curvature ductility is large and thecurvature will be overestimated when the curvatureductility is comparatively small

(3) The curvature ductility with consideration of theenvelope area can reflect an increase in the ultimatemoment when 119903sf is large and the sectional curvatureductility can be justified by changing 119903sf 120576sfu 120588esf 1198911015840c and ℎ0 Before reaching the MPPC 120588esf 1198911015840c and ℎ0have a minimal influence on the ldquodimensionlessrdquo 1205833After reaching the MPPC 1205833 slightly decreased withan increase in 119903sf For the design of SFCB-reinforcedconcrete beams Mode II is preferred without SFCBrsquosouter FRP rupture and the bearing capacity and duc-tility of concrete beams are considered synthetically


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)

(a) Change reinforcement ratio

00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100

Failure mode changed to Mode II

Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =

(b) Change SFCB rupture strain

00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0

Figure 18 Comparison of 1205833 with different parameters

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors acknowledge the financial support from theNational Key Technology Support Program of China(2014BAK11B04) the National Natural Science Foundationof China (nos 51408126 and 51528802) the Natural ScienceFoundation of Jiangsu Province China (no BK20140631)and Project Funding from the Priority Academic ProgramDevelopment of Jiangsu Higher Education Institutions(CE01-2-3)

References

[1] S Dong B Zhao C Lin R Du R Hu and G X Zhang ldquoCor-rosion behavior of epoxyzinc duplex coated rebar embeddedin concrete in ocean environmentrdquo Construction and BuildingMaterials vol 28 no 1 pp 72ndash78 2012

[2] C E Bakis L C Bank V L Brown et al ldquoFiber-reinforcedpolymer composites for constructionmdashstate-of-the-art reviewrdquoJournal of Composites for Construction vol 6 no 2 pp 73ndash872002

[3] X Huang J Wang F Zhang S-S Niu and J Ding ldquoAn experi-mental investigation on the failure behavior of a notched con-crete beam strengthenedwith carbon fiber-reinforced polymerrdquoInternational Journal of Polymer Science vol 2015 Article ID729320 17 pages 2015


[4] B Benmokrane O Chaallal and R Masmoudi ldquoFlexural res-ponse of concrete beams reinforced with FRP reinforcing barsrdquoACI Structural Journal vol 93 no 1 pp 46ndash55 1996

[5] ACI Committee 318 Building Code Requirements for StructuralConcrete (ACI 318-05) and Commentary (ACI 318-89) Ameri-can Concrete Institute Detroit Mich USA 1989

[6] A A Mufti N Banthia B Benmokrane M Boulfiza and JP Newhook ldquoDurability of GFRP composite rodsrdquo ConcreteInternational vol 29 no 2 pp 37ndash42 2007

[7] Canadian Standards Association Canadian Highways BridgeDesign Code Section 16 Fiber Reinforced Structures CanadianStandards Association Ottawa Canada 2000

[8] ACI 4404R-04 Prestressing Concrete Structures with FRP Ten-dons vol 440 ACI Committee Farmington Hills Mich USA2004

[9] E E-S Etman ldquoInnovative hybrid reinforcement for flexuralmembersrdquo Journal of Composites for Construction vol 15 no 1pp 2ndash8 2011

[10] D Lau and H J Pam ldquoExperimental study of hybrid FRPreinforced concrete beamsrdquo Engineering Structures vol 32 no12 pp 3857ndash3865 2010

[11] ACI ldquoGuide for the design and construction of concrete rein-forced with FRP barsrdquo ACI 4401R-06 American Concrete Ins-titute Detroit Mich USA 2006

[12] M A Safan ldquoFlexural behavior and design of steel-GFRPreinforced concrete beamsrdquo ACI Materials Journal vol 110 no6 pp 677ndash685 2013

[13] L Pang W Qu P Zhu and J Xu ldquoDesign propositions forhybrid FRP-steel reinforced concrete beamsrdquo Journal of Com-posites for Construction ASCE vol 20 no 4 pp 1ndash9 2015

[14] A El Refai F Abed and A Al-Rahmani ldquoStructural perfor-mance and serviceability of concrete beams reinforced withhybrid (GFRP and steel) barsrdquo Construction and BuildingMaterials vol 96 pp 518ndash529 2015

[15] A Nanni M J Henneke and T Okamoto ldquoBehaviour of conc-rete beams with hybrid reinforcementrdquoConstruction and Build-ing Materials vol 8 no 2 pp 89ndash95 1994

[16] B Saikia J Thomas A Ramaswamy and K S Nanjunda RaoldquoPerformance of hybrid rebars as longitudinal reinforcement innormal strength concreterdquoMaterials and Structures vol 38 no10 pp 857ndash864 2005

[17] GWu Z SWu Y B Luo andH CWei ldquoA new reinforcementmaterial of steel fiber composite bar (SFCB) and its mechanicspropertiesrdquo in Proceedings of 9th International Symposium onlsquoFiber Reinforced Polymer (FRP) Reinforcement for ConcreteStructuresrsquo (FRPRCS rsquo09) University of Adelaide AdelaideAustralia July 2009

[18] Z-Y Sun G Wu Z-S Wu and J Zhang ldquoNonlinear behaviorand simulation of concrete columns reinforced by steel-FRPcomposite barsrdquo Journal of Bridge Engineering vol 19 no 2 pp220ndash234 2014

[19] G Wu Z-S Wu Y-B Luo Z-Y Sun and X-Q Hu ldquoMechan-ical properties of steel-frp composite bar under uniaxial andcyclic tensile loadsrdquo Journal of Materials in Civil Engineeringvol 22 no 10 Article ID 010010QMT pp 1056ndash1066 2010

[20] Z Y Sun Y Yang W H Qin S T Ren and G Wu ldquoExperi-mental study on flexural behavior of concrete beams reinforcedby steel-fiber reinforced polymer composite barsrdquo Journal ofReinforced Plastics and Composites vol 31 no 24 pp 1737ndash17452012

[21] M Andreou A Kotsoglou and S Pantazopoulou ldquoModellingblast effects on a reinforced concrete bridgerdquo Advances in CivilEngineering vol 2016 Article ID 4167329 11 pages 2016

[22] S Mazzoni F McKenne M H Scott and G L Fenves OpenSystem for Earthquake Engineering Simulation User ManualVersion 20 Pacific Earthquake Engineering Center Universityof California Berkeley Calif USA 2009 httpopenseesberke-leyeduOpenSeesmanualsusermanualindexhtml

[23] H M Yassin Mohd Nonlinear analysis of prestressed concretestructures under monotonic and cyclic [PhD thesis] Universityof California Berkeley Calif USA 1994

[24] G A Chang and J B Mander ldquoSeismic energy based fatiguedamage analysis of bridge beams part Imdashevaluation of seismiccapacityrdquo NCEER Technical Report 94-0006 NCEERStateUniversity of New York at Buffalo Buffalo NY USA 1994

[25] Z Sun W Xiao G Wu and Z Wu ldquoStudy on moment-curva-ture behavior of concrete column reinforced by steel-fiber rein-forced polymer composite barsrdquo in Proceedings of 12th Interna-tional Symposium on Fiber Reinforced Polymers for ReinforcedConcrete Structures (FRPRCS rsquo12) amp The 5th Asia-Pacific Con-ference on Fiber Reinforced Polymers in Structures (APFIS rsquo15)Nanjing China December 2015

[26] AASHTOAASTOLRFDBridge Design Specifications (SI Units)American Association of State Highway and TransportationOfficials Washington DC USA 3rd edition 2004

[27] L E Aycardi J B Mander and A M Reinhorn ldquoSeismic resis-tance of reinforced concrete frame structures designed only forgravity loads experimental performance of subassemblagesrdquoACI Materials Journal vol 91 no 5 pp 552ndash563 1994

[28] F Oudah and R El-Hacha ldquoA new ductility model of reinforcedconcrete beams strengthened using Fiber Reinforced Polymerreinforcementrdquo Composites Part B Engineering vol 43 no 8pp 3338ndash3347 2012

[29] Z Sun G Wu J Zhang Y Zeng and W Xiao ldquoExperimentalstudy on concrete columns reinforced by hybrid steel-fiberreinforced polymer (FRP) bars under horizontal cyclic loadingrdquoConstruction and Building Materials vol 130 pp 202ndash211 2017

International Journal of

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RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



a better performance and a more acceptable cost can beachieved by combining steel and FRP Two combinationsexist (1) a concrete beam hybrid reinforced by steel bars andFRP bars (2) beams reinforced by steel-FRP composite barsTwelve concrete beams that were reinforced by a steel bar aGFRP bar or a steelGFRP bar were tested by Lau and Pam(2010) [10] the results indicated that the ductility of hybridreinforced beams was better than the ductility of a pure FRP-reinforced beam the minimum reinforcement ratio for anFRP-reinforced beam could be reduced by 25 according tothe ACI 4401R-06 [11] A study of twelve concrete beamswhich were reinforced by steel bar and GFRP bar was con-ducted by Safan (2013) [12] and the results showed that thefailure modes of concrete beams were presented as concretecrushing after the yielding of the steel bar on the tensile sideand a GFRP bar can maintain the flexural bearing capacity ofa concrete beam with a relatively small reinforcement ratioA ductility index was proposed by Pang et al (2015) [13]to evaluate the load-displacement relationship of a hybridreinforced concrete beam and the corresponding methodof differentiation of the failure mode was suggested butthis method of differentiation was slightly complicated Anexperimental study on six hybrid reinforced concrete beamswas conducted by El Refai et al (2015) [14] and the resultsindicated that the deflection of the hybrid reinforcementbeam with the higher reinforcement ratio can be well pre-dicted by current design codes the crackwidthwas calculatedby ACI-4401R-06 [11] by modifying the bonding coefficientbetween the FRP bar and concrete

Deterioration failure can occur in concrete beams that arereinforced by hybrid steel barsFRP bars when a steel bar issubjected to corrosion A better anticorrosion performancecan be achieved by a steel-FRP composite bar (SFCB) whichis a type of rebar with an inner steel bar and an outerFRP Sixteen concrete beams reinforced by hybrid rebar weretested by Nanni et al (1994) [15] and the theoretical analysisof the load-displacement curves revealed that the calculateddisplacement was slightly smaller than that of the test curveswhen the slip between the FRP and concrete was disregardedand the load-carrying capacity of the hybrid reinforced beamscan be calculated using traditional RC theory Concretebeams that are reinforced by longitudinal steel-GFRP com-posite bars and GFRP stirrups were tested by Saikia et al(2005) [16] the composite bar was composed of an inner steelbar with a diameter of 6mm that is helically wounded byGFRP with a thickness of 2mm Due to an underdevelopedmanufacturing technology a slip between the inner steel barand the outer GFRP occurred A factory-produced SFCB wasachieved by Wu et al by modifying the pultrusion process ofFRP bars [17] and its characteristics include the following(1) the initial elastic modulus and the postyield modulus canbe designed by adjusting the steelFRP ratio (2) a durabilityof SFCB that is equivalent to the durability of FRP bar canbe achieved (Figure 1(a) [18]) (3) the strength of the innersteel bar can be effectively employed and the high ultimatestrength of FRP can be employed as a reservation (4) theperformancecost ratio of a concrete structure reinforced bysteel-FRP composite bars can be optimized with consideringthe long-term performance and bearing capacity The typical

Longitudinal wrapped Inner steel barouter FRP

(a) Steel-FRP composite bars

0 5000 10000 15000 20000 25000 30000 350000

20

40

60

80

Load

(kN

)

Rupture of basalt fiber ofsteel-BFRP composite bar

Rupture of carbon fiber ofsteel-CFRP composite bar

Yielding of SFCBrsquos inner steel bar

Strain (120583120576)

(b) Load-strain curves

Figure 1 Factory production of SFCB and itsmechanical properties

load-strain curves of steel-basalt FRP composite bars andsteel-carbon FRP composite bars are illustrated in Figure 1(b)and the stress-strain model of SFCB has been comprehen-sively investigated [19] It was demonstrated that there was noslip between the inner steel and outer FRP during the loadingprocess An experimental study of a concrete beam that isreinforced with SFCBs steel bars and pure FRP bars understatic loading was conducted by the authorrsquos group [20] andresearch on the effect of blast load [21] is to be conductedIn this paper a parametric analysis of the moment-curvaturebehavior is conducted based on the simulation of the load-displacement curve of an SFCB-reinforced concrete beamthe corresponding curvature ductility index is also discussedThe research results of this paper can provide a reference forthe design of hybrid reinforced concrete beams

2 Simulation of a Concrete BeamReinforced by SFCB

21 Material Parameters Experimental studies of six con-crete beams were conducted by Sun et al (2012) [20] andthe SFCB beams exhibit a stable postyield stiffness after theyielding of SFCBrsquos inner steel bar and concrete crushed afterthe rupture of SFCBrsquos outer FRP The conventional RC beamhad the largest ductility whereas the ultimate load of the RCbeamwas approximately 31 of the ultimate load of the SFCBbeam To calculate the moment-curvature behavior of the


Strain

Stress

120576cu fcu

120582E0Et

ft

120576c0 f998400c

E0 =2f998400

c120576c0



119891c0 = 1198701198911015840c = (1 + 120588sv119891yh1198911015840c )1198911015840c (1)

120576c0 = 0002119870 = 0002(1 + 120588sv119891yh1198911015840c ) (2)

120576cu = 0004 + 120588sv119891yh [MPa]300 (3)






119903sf = 119864f119860 f(119864s119860 s + 119864f119860 f) =119864f119860 f119864sf119860 sf

(5)

120588esf = 119864f119860 f119903sf119864s119860g (6)








600 600 600100 1002000

Dialindicator

JackLoad sensor


Dialindicator

220B-S10B51

1206018801206018150

300



0

50

100

150

200

250

300

minus5000 0 5000 10000 15000 20000 25000 30000

Sect

ion

heig

ht (m

m)

B-S10B51

Strain (120583120576)

3t

6t9t

1518t

2612t

2685t




120601 = 120576c + 120576sfℎ0 (7)




0 10000 20000 300000

250

500

750St

ress

(MPa

)

TestedCalculated

Strain (120583120576)

EII

EI

120576sfy 120576sfu


0

50

100

150

200

250

300

0 20 40 60 80

Beam yielding


Load

(kN

)


TestedCalculated








120588eb I = 120572112057311198911015840c119864s

1120576sfy (1 + 120576sfy120576cu) (8)

120588eb II = 120572112057311198911015840c119864s



3 Parametric Study





120576sfu lt 120576sf

120576sfyu120576sfy

120576cu

120588esf = 120588e

b_I

120588esf = 120588e

b_II


C

Tsf = fsfuAsf

h0

1205731x

c

1205721fc

ycMu

120588esf = 120588e

b_II



000 005 010 015 0200

20

40

60

80

Cracking


Concrete crushing

Yieldpoint

= 090 120576sfu = 0015 f998400c = 30MPa h = 200mm


rsf = 07

rsf = 03

rsf = 01

120601peak_max

Mom

ent (

kNmiddotm

)



sf



120601y = 17 119891y119864sℎ0 = 17120576sfyℎ0 (10)





119872y regress

119864s119887ℎ20 = 000157120588esf + 6233119864 minus 07 (12)





0000 0005 0010 00150000

0005

0010

0015

0020Fi

tted

resu

ltsO

S re

sults


R2 = 0987

120601y = 17fy

Esh0= 17

120576sfy

h0

(a) Curvature

0 100 200 300 400 500 600

0

100

200

300

400

500

Fitte

d re

sults

OS

resu

lts


R2 = 0998

(b) Moment










1205741 = 2119864esf119864c 120588






00 02 04 06 08 10

50

100

150

200

250

300

350


Mom

ent (

kNmiddotm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10004

005

006

007

008


Curv

atur

e (ra

dm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 101

2

3

4

5

6

7

8

Dim

ensio

nles

s mom

ent

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

14

15D

imen

sionl

ess c

urva

ture


Mode II

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =







120588eb II regress


(16)


00 02 04 06 08 101

2

3

4

5D

imen

sionl

ess m

omen

t

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =



00 02 04 06 08 100

100

200

300

400

500

600

700

Dim

ensio

nles

s mom

ent

0

1

2

3

4

5

rsf

f998400c = 30MPa

120576sfu = 0025


Mom

ent (

kNmiddotm

)


06120588esf =


00 02 04 06 08 10 12

005

010

015

020

025

Curv

atur

e (ra

dm

)

0

2

4

6

8

10

12

14

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

120576sfu = 0025



06120588esf =








1205742 = 120576y (1 minus 119903sf) + 120576cu119903sf (18)


000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c







002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Strain

Stress

120576cu fcu

120582E0Et

ft

120576c0 f998400c

E0 =2f998400

c120576c0



119891c0 = 1198701198911015840c = (1 + 120588sv119891yh1198911015840c )1198911015840c (1)

120576c0 = 0002119870 = 0002(1 + 120588sv119891yh1198911015840c ) (2)

120576cu = 0004 + 120588sv119891yh [MPa]300 (3)






119903sf = 119864f119860 f(119864s119860 s + 119864f119860 f) =119864f119860 f119864sf119860 sf

(5)

120588esf = 119864f119860 f119903sf119864s119860g (6)








600 600 600100 1002000

Dialindicator

JackLoad sensor


Dialindicator

220B-S10B51

1206018801206018150

300



0

50

100

150

200

250

300

minus5000 0 5000 10000 15000 20000 25000 30000

Sect

ion

heig

ht (m

m)

B-S10B51

Strain (120583120576)

3t

6t9t

1518t

2612t

2685t




120601 = 120576c + 120576sfℎ0 (7)




0 10000 20000 300000

250

500

750St

ress

(MPa

)

TestedCalculated

Strain (120583120576)

EII

EI

120576sfy 120576sfu


0

50

100

150

200

250

300

0 20 40 60 80

Beam yielding


Load

(kN

)


TestedCalculated








120588eb I = 120572112057311198911015840c119864s

1120576sfy (1 + 120576sfy120576cu) (8)

120588eb II = 120572112057311198911015840c119864s



3 Parametric Study





120576sfu lt 120576sf

120576sfyu120576sfy

120576cu

120588esf = 120588e

b_I

120588esf = 120588e

b_II


C

Tsf = fsfuAsf

h0

1205731x

c

1205721fc

ycMu

120588esf = 120588e

b_II



000 005 010 015 0200

20

40

60

80

Cracking


Concrete crushing

Yieldpoint

= 090 120576sfu = 0015 f998400c = 30MPa h = 200mm


rsf = 07

rsf = 03

rsf = 01

120601peak_max

Mom

ent (

kNmiddotm

)



sf



120601y = 17 119891y119864sℎ0 = 17120576sfyℎ0 (10)





119872y regress

119864s119887ℎ20 = 000157120588esf + 6233119864 minus 07 (12)





0000 0005 0010 00150000

0005

0010

0015

0020Fi

tted

resu

ltsO

S re

sults


R2 = 0987

120601y = 17fy

Esh0= 17

120576sfy

h0

(a) Curvature

0 100 200 300 400 500 600

0

100

200

300

400

500

Fitte

d re

sults

OS

resu

lts


R2 = 0998

(b) Moment










1205741 = 2119864esf119864c 120588






00 02 04 06 08 10

50

100

150

200

250

300

350


Mom

ent (

kNmiddotm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10004

005

006

007

008


Curv

atur

e (ra

dm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 101

2

3

4

5

6

7

8

Dim

ensio

nles

s mom

ent

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

14

15D

imen

sionl

ess c

urva

ture


Mode II

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =







120588eb II regress


(16)


00 02 04 06 08 101

2

3

4

5D

imen

sionl

ess m

omen

t

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =



00 02 04 06 08 100

100

200

300

400

500

600

700

Dim

ensio

nles

s mom

ent

0

1

2

3

4

5

rsf

f998400c = 30MPa

120576sfu = 0025


Mom

ent (

kNmiddotm

)


06120588esf =


00 02 04 06 08 10 12

005

010

015

020

025

Curv

atur

e (ra

dm

)

0

2

4

6

8

10

12

14

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

120576sfu = 0025



06120588esf =








1205742 = 120576y (1 minus 119903sf) + 120576cu119903sf (18)


000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c







002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












600 600 600100 1002000

Dialindicator

JackLoad sensor


Dialindicator

220B-S10B51

1206018801206018150

300



0

50

100

150

200

250

300

minus5000 0 5000 10000 15000 20000 25000 30000

Sect

ion

heig

ht (m

m)

B-S10B51

Strain (120583120576)

3t

6t9t

1518t

2612t

2685t




120601 = 120576c + 120576sfℎ0 (7)




0 10000 20000 300000

250

500

750St

ress

(MPa

)

TestedCalculated

Strain (120583120576)

EII

EI

120576sfy 120576sfu


0

50

100

150

200

250

300

0 20 40 60 80

Beam yielding


Load

(kN

)


TestedCalculated








120588eb I = 120572112057311198911015840c119864s

1120576sfy (1 + 120576sfy120576cu) (8)

120588eb II = 120572112057311198911015840c119864s



3 Parametric Study





120576sfu lt 120576sf

120576sfyu120576sfy

120576cu

120588esf = 120588e

b_I

120588esf = 120588e

b_II


C

Tsf = fsfuAsf

h0

1205731x

c

1205721fc

ycMu

120588esf = 120588e

b_II



000 005 010 015 0200

20

40

60

80

Cracking


Concrete crushing

Yieldpoint

= 090 120576sfu = 0015 f998400c = 30MPa h = 200mm


rsf = 07

rsf = 03

rsf = 01

120601peak_max

Mom

ent (

kNmiddotm

)



sf



120601y = 17 119891y119864sℎ0 = 17120576sfyℎ0 (10)





119872y regress

119864s119887ℎ20 = 000157120588esf + 6233119864 minus 07 (12)





0000 0005 0010 00150000

0005

0010

0015

0020Fi

tted

resu

ltsO

S re

sults


R2 = 0987

120601y = 17fy

Esh0= 17

120576sfy

h0

(a) Curvature

0 100 200 300 400 500 600

0

100

200

300

400

500

Fitte

d re

sults

OS

resu

lts


R2 = 0998

(b) Moment










1205741 = 2119864esf119864c 120588






00 02 04 06 08 10

50

100

150

200

250

300

350


Mom

ent (

kNmiddotm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10004

005

006

007

008


Curv

atur

e (ra

dm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 101

2

3

4

5

6

7

8

Dim

ensio

nles

s mom

ent

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

14

15D

imen

sionl

ess c

urva

ture


Mode II

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =







120588eb II regress


(16)


00 02 04 06 08 101

2

3

4

5D

imen

sionl

ess m

omen

t

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =



00 02 04 06 08 100

100

200

300

400

500

600

700

Dim

ensio

nles

s mom

ent

0

1

2

3

4

5

rsf

f998400c = 30MPa

120576sfu = 0025


Mom

ent (

kNmiddotm

)


06120588esf =


00 02 04 06 08 10 12

005

010

015

020

025

Curv

atur

e (ra

dm

)

0

2

4

6

8

10

12

14

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

120576sfu = 0025



06120588esf =








1205742 = 120576y (1 minus 119903sf) + 120576cu119903sf (18)


000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c







002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 10000 20000 300000

250

500

750St

ress

(MPa

)

TestedCalculated

Strain (120583120576)

EII

EI

120576sfy 120576sfu


0

50

100

150

200

250

300

0 20 40 60 80

Beam yielding


Load

(kN

)


TestedCalculated








120588eb I = 120572112057311198911015840c119864s

1120576sfy (1 + 120576sfy120576cu) (8)

120588eb II = 120572112057311198911015840c119864s



3 Parametric Study





120576sfu lt 120576sf

120576sfyu120576sfy

120576cu

120588esf = 120588e

b_I

120588esf = 120588e

b_II


C

Tsf = fsfuAsf

h0

1205731x

c

1205721fc

ycMu

120588esf = 120588e

b_II



000 005 010 015 0200

20

40

60

80

Cracking


Concrete crushing

Yieldpoint

= 090 120576sfu = 0015 f998400c = 30MPa h = 200mm


rsf = 07

rsf = 03

rsf = 01

120601peak_max

Mom

ent (

kNmiddotm

)



sf



120601y = 17 119891y119864sℎ0 = 17120576sfyℎ0 (10)





119872y regress

119864s119887ℎ20 = 000157120588esf + 6233119864 minus 07 (12)





0000 0005 0010 00150000

0005

0010

0015

0020Fi

tted

resu

ltsO

S re

sults


R2 = 0987

120601y = 17fy

Esh0= 17

120576sfy

h0

(a) Curvature

0 100 200 300 400 500 600

0

100

200

300

400

500

Fitte

d re

sults

OS

resu

lts


R2 = 0998

(b) Moment










1205741 = 2119864esf119864c 120588






00 02 04 06 08 10

50

100

150

200

250

300

350


Mom

ent (

kNmiddotm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10004

005

006

007

008


Curv

atur

e (ra

dm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 101

2

3

4

5

6

7

8

Dim

ensio

nles

s mom

ent

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

14

15D

imen

sionl

ess c

urva

ture


Mode II

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =







120588eb II regress


(16)


00 02 04 06 08 101

2

3

4

5D

imen

sionl

ess m

omen

t

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =



00 02 04 06 08 100

100

200

300

400

500

600

700

Dim

ensio

nles

s mom

ent

0

1

2

3

4

5

rsf

f998400c = 30MPa

120576sfu = 0025


Mom

ent (

kNmiddotm

)


06120588esf =


00 02 04 06 08 10 12

005

010

015

020

025

Curv

atur

e (ra

dm

)

0

2

4

6

8

10

12

14

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

120576sfu = 0025



06120588esf =








1205742 = 120576y (1 minus 119903sf) + 120576cu119903sf (18)


000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c







002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120576sfu lt 120576sf

120576sfyu120576sfy

120576cu

120588esf = 120588e

b_I

120588esf = 120588e

b_II


C

Tsf = fsfuAsf

h0

1205731x

c

1205721fc

ycMu

120588esf = 120588e

b_II



000 005 010 015 0200

20

40

60

80

Cracking


Concrete crushing

Yieldpoint

= 090 120576sfu = 0015 f998400c = 30MPa h = 200mm


rsf = 07

rsf = 03

rsf = 01

120601peak_max

Mom

ent (

kNmiddotm

)



sf



120601y = 17 119891y119864sℎ0 = 17120576sfyℎ0 (10)





119872y regress

119864s119887ℎ20 = 000157120588esf + 6233119864 minus 07 (12)





0000 0005 0010 00150000

0005

0010

0015

0020Fi

tted

resu

ltsO

S re

sults


R2 = 0987

120601y = 17fy

Esh0= 17

120576sfy

h0

(a) Curvature

0 100 200 300 400 500 600

0

100

200

300

400

500

Fitte

d re

sults

OS

resu

lts


R2 = 0998

(b) Moment










1205741 = 2119864esf119864c 120588






00 02 04 06 08 10

50

100

150

200

250

300

350


Mom

ent (

kNmiddotm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10004

005

006

007

008


Curv

atur

e (ra

dm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 101

2

3

4

5

6

7

8

Dim

ensio

nles

s mom

ent

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

14

15D

imen

sionl

ess c

urva

ture


Mode II

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =







120588eb II regress


(16)


00 02 04 06 08 101

2

3

4

5D

imen

sionl

ess m

omen

t

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =



00 02 04 06 08 100

100

200

300

400

500

600

700

Dim

ensio

nles

s mom

ent

0

1

2

3

4

5

rsf

f998400c = 30MPa

120576sfu = 0025


Mom

ent (

kNmiddotm

)


06120588esf =


00 02 04 06 08 10 12

005

010

015

020

025

Curv

atur

e (ra

dm

)

0

2

4

6

8

10

12

14

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

120576sfu = 0025



06120588esf =








1205742 = 120576y (1 minus 119903sf) + 120576cu119903sf (18)


000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c







002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0000 0005 0010 00150000

0005

0010

0015

0020Fi

tted

resu

ltsO

S re

sults


R2 = 0987

120601y = 17fy

Esh0= 17

120576sfy

h0

(a) Curvature

0 100 200 300 400 500 600

0

100

200

300

400

500

Fitte

d re

sults

OS

resu

lts


R2 = 0998

(b) Moment










1205741 = 2119864esf119864c 120588






00 02 04 06 08 10

50

100

150

200

250

300

350


Mom

ent (

kNmiddotm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10004

005

006

007

008


Curv

atur

e (ra

dm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 101

2

3

4

5

6

7

8

Dim

ensio

nles

s mom

ent

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

14

15D

imen

sionl

ess c

urva

ture


Mode II

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =







120588eb II regress


(16)


00 02 04 06 08 101

2

3

4

5D

imen

sionl

ess m

omen

t

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =



00 02 04 06 08 100

100

200

300

400

500

600

700

Dim

ensio

nles

s mom

ent

0

1

2

3

4

5

rsf

f998400c = 30MPa

120576sfu = 0025


Mom

ent (

kNmiddotm

)


06120588esf =


00 02 04 06 08 10 12

005

010

015

020

025

Curv

atur

e (ra

dm

)

0

2

4

6

8

10

12

14

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

120576sfu = 0025



06120588esf =








1205742 = 120576y (1 minus 119903sf) + 120576cu119903sf (18)


000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c







002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00 02 04 06 08 10

50

100

150

200

250

300

350


Mom

ent (

kNmiddotm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10004

005

006

007

008


Curv

atur

e (ra

dm

)

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 101

2

3

4

5

6

7

8

Dim

ensio

nles

s mom

ent

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

14

15D

imen

sionl

ess c

urva

ture


Mode II

rsf

f998400c = 30MPa

120576sfu = 0025

h0 = 04m

03120588esf =

06120588esf =

09120588esf =







120588eb II regress


(16)


00 02 04 06 08 101

2

3

4

5D

imen

sionl

ess m

omen

t

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =



00 02 04 06 08 100

100

200

300

400

500

600

700

Dim

ensio

nles

s mom

ent

0

1

2

3

4

5

rsf

f998400c = 30MPa

120576sfu = 0025


Mom

ent (

kNmiddotm

)


06120588esf =


00 02 04 06 08 10 12

005

010

015

020

025

Curv

atur

e (ra

dm

)

0

2

4

6

8

10

12

14

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

120576sfu = 0025



06120588esf =








1205742 = 120576y (1 minus 119903sf) + 120576cu119903sf (18)


000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c







002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00 02 04 06 08 101

2

3

4

5D

imen

sionl

ess m

omen

t

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =


00 02 04 06 08 10

6

7

8

9

10

11

12

13

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

06120588esf =



00 02 04 06 08 100

100

200

300

400

500

600

700

Dim

ensio

nles

s mom

ent

0

1

2

3

4

5

rsf

f998400c = 30MPa

120576sfu = 0025


Mom

ent (

kNmiddotm

)


06120588esf =


00 02 04 06 08 10 12

005

010

015

020

025

Curv

atur

e (ra

dm

)

0

2

4

6

8

10

12

14

Dim

ensio

nles

s cur

vatu

re

rsf

f998400c = 30MPa

120576sfu = 0025



06120588esf =








1205742 = 120576y (1 minus 119903sf) + 120576cu119903sf (18)


000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c







002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










000 005 010 015 020

070

075

080

085

090

095

100

105

120601Eq120601

OS

120601OS

rsf increased

(a) All data

00 02 04 06075

080

085

090

095

100

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

120601Eq120601

OS

03120588esf =

06120588esf =

09120588esf =


00 02 04 06 08

070

075

080

085

090

095

100

rsf

f998400c = 30MPa

h0 = 04m

120576sfu = 0015

120576sfu = 002

120576sfu = 0025

120601Eq120601

OS

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

080

085

090

095

100

105

rsf

120576sfu = 0025

h0 = 04m

f998400c = 30MPa

f998400c = 60MPa

f998400c = 90MPa

120601Eq120601

OS

06120588esf =

(d) Different 1198911015840c







002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










002 004 006 008 010 012 014 016 018 02006

08

10

12

14


Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



002 004 006 008 010 012 014 016 018 02006

08

10

12

14

Calc

ulat

ed re

sults

OS

resu

lts

Curvature (radm)



120588eb_II



5 10 1508

10

12

14

120601120601

max

_OS

120601max_OS120601y



1205831 = 120601u120601y (20)

1205832 = 119872u120601u119872y120601y (21)

1205833 = 119864tot119864y =120601u120601y minus

119872u119872y+ 119872u120601u119872y120601y (22)




5 Conclusions




0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

Mu

120601y 120601u120601

09Mu08Mu

M

1205831 =120601u120601y


0

A

B

120601y 120601u

Mu

My

Ay

Aplastic




00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

Duc

tility

rsf

120576sfu = 002

f998400c = 30MPa


120583112058321205833

06120588esf =


00 01 02 03 04 05 06 07 08 09 10

10

15

20

25

30

35


rsf

120576sfu = 002

f998400c = 30MPa

120583112058321205833

06120588esf =

120583120583

r sf=05








00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Mode II

rsf

120576sfu = 0025

f998400c = 30MPa

h0 = 04m

(1205833)

Duc

tility

1205831 (120588esf = 06)

1205833 (120588esf = 03)

1205833 (120588esf = 06)

1205833 (120588esf = 09)


00 01 02 03 04 05 06 07 08 09 100

20

40

60

80

100


Duc

tility

rsf

h0 = 04mf998400

c = 30MPa

1205833 (120576sfu = 0015)

1205833 (120576sfu = 002)

1205831 (120576sfu = 002)

1205833 (120576sfu = 0025)

06120588esf =


00 01 02 03 04 05 06 07 08 09 100

50

100

150

Duc

tility

rsf

120576sfu = 0025

h0 = 04m

1205831 (f998400c = 30MPa)

1205833 (f998400c = 30MPa)

1205833 (f998400c = 60MPa)

1205833 (f998400c = 90MPa)


06120588esf =

(c) Change 1198911015840c

00 01 02 03 04 05 06 07 08 09 1000

05

10

15

20

25

30

35

40

Left axis

Left axis

Right axis

Right axis

0

20

40

60

rsf

120576sfu = 0025

f998400c = 30MPa

1205833 (h0 = 06m)

1205831 (h0 = 06m)

1205833 (h0 = 04m)

1205833 (h0 = 06m)

1205833 (h0 = 08m)

(1205831)

Duc

tility

06120588esf =

(h0 = 06m)12058311205831_Duc

tility

Dr s

f=05

rsf=05

(d) Change ℎ0


Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









moment-curvature behaviors of concrete beams singly

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