Research Article An Analytical Step-by-Step Procedure...

Hindawi Publishing CorporationAdvances in Civil EngineeringVolume 2013 Article ID 275657 13 pageshttpdxdoiorg1011552013275657

Research ArticleAn Analytical Step-by-Step Procedure to Derive the FlexuralResponse of RC Sections in Compression

Piero Colajanni1 Marinella Fossetti2 and Maurizio Papia3

1 Dipartimento di Ingegneria Civile Universita di Messina Contrada Di Dio 98166 Messina Italy2 Facolta di Ingegneria Architettura e Scienze Motorie Universita Kore di Enna Cittadella Universitaria 94100 Enna Italy3 Dipartimento di Ingegneria Civile Ambientale Aerospaziale dei Materiali Universita di Palermo Viale delle Scienze90128 Palermo Italy

Correspondence should be addressed to Maurizio Papia mauriziopapiaunipait

Received 6 March 2013 Revised 5 August 2013 Accepted 8 August 2013

Academic Editor Andreas Kappos

Copyright copy 2013 Piero Colajanni et alThis 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

This paper proposes an analysis procedure able to determine the flexural response of rectangular symmetrically reinforcedconcrete sections subjected to axial load and uniaxial bending With respect to the usual numerical approaches based on the fibredecompositionmethod this procedure is based on the use of analytical expressions of the contributions to the equilibrium given bythe longitudinal reinforcement and the concrete region in compression which depend on the neutral axis depth and the curvatureat each analysis step The formulation is developed in dimensionless terms after a preliminary definition of the geometrical andmechanical parameters involved so that the results are valid for classes of RC sectionsThe constitutive laws of thematerials includeconfinement effect on the concrete and postyielding behaviour of the steel reinforcement which can be assumed to be softeningbehaviour for buckled reinforcing bars The strength and curvature domains at the first yielding of the reinforcement in tensionand at the ultimate state are derived in the form of analytical curves depending on the compression level therefore the role of asingle parameter on the shape of these curves can easily be deduced The procedure is validated by comparing some results withthose numerically obtained by other authors

1 Introduction

The performance of reinforced concrete frames under severeearthquakes largely depends on the ability of the beam andcolumn sections to undergo large inelastic deformationsEspecially this ability plays a decisive role in existing build-ings the safety level of which can be estimated by employingnonlinear analysis tools (like pushover) needing a carefulinput in terms of strength domains and moment-curvaturerelationships of the critical sections

The evaluation of the moment-curvature response ofcritical sections of RC members is a complex issue mainlybecause of the interaction of various parameters constitutivelaws of materials in the elastic and plastic ranges membergeometry buckling phenomena in reinforcing steel bars andloading conditions

In order to include in a computer software the cross-sec-tion strength domain andmoment-curvature relationships ofreinforced concrete members two different approaches are

usually followed the use of strength domains [1ndash3] andmoment-curvature relationships of a usually bilinear or tri-linear prefixed shape [4ndash6] in which a degrading stiffnessmodel reproduces the effect of yielding and damage of mate-rials the layered section approach based on the fibre decom-position method [7ndash9]

The layered section approach gives a realistic and almostcomplete description of the behaviour of the critical regionsof a RC framed structure also allowing to include the bondslip effects in the moment-rotation response However theapplication of this approach to large structures with manydegrees of freedom has some limitations because numericalintegrations and a prediction of the state of materials must becarried out considering the entire layers of each region wherea plastic hinge can occur thus a high amount of informationis required to characterize each section involved and a largenumber of numerical operations is needed to reach an accept-able level of error This results in heavy computational effortsand convergence problems for nonlinear structural analysis

2 Advances in Civil Engineering

Consequently the use of moment-curvature relationships isstill the more widespread and efficient approach

The definition of moment-curvature relationships of RCsections has been a point of research interest for many years[10 11] Many analytical and numerical techniques havebeen proposed including several phenomena that affect theresponse of reinforced concrete elements

Historically moment-curvature relationships with soft-ening branch were first introduced by Wood [12] Mo [13]suggested a classical approach to reproduce the moment-curvature relationship including the softening branch basedon an FE analysis of the elastic-plastic buckling of the longi-tudinal reinforcement An alternative approachwas proposedin [14] where by using a simplified model complex nonlin-ear geometric effects were embedded in the nonlinear mate-rial behaviour of the cross-section Chandrasekaran et al [15]proposed an analytical relationship providing in explicit formthe moment-curvature response of a section by consideringnonlinear constitutive laws of materials chosen in accord-ancewith the European technical codes which can be appliedby the use of a spreadsheet

Commercial structural analysis software programs nowprovide the analyst with the option of conducting moment-curvature analyses at critical sections Although well-verifiedequations have been used in the software developments somequestions may remain (i) for a given project how can thecomputational analysis results be easily verified outside thesoftware (ii) how can general considerations on the role ofdifferent parameters characterizing the section and reinforce-ment geometry be derived

To answer to these questions a dimensionless formula-tion is required able to fully characterize the nonlinearbehaviour of an RC cross-section subjected to an assignedloading condition

In a recent paper [16] the behaviour of RC rectangularcross-sections under axial load and biaxial bending is exam-ined in dimensionless terms This paper confirms that thevalues of the ultimate bending moment and curvature in anydirection can be efficiently related to the values correspond-ing to two separate conditions of uniaxial bending

In this context the present paper proposes an analyticalmodel for the evaluation of the moment-curvature relation-ship of rectangular RC sections subjected to axial load anduniaxial bending by using an incremental strain techniquewhich could be an alternative to the classical fibre decomposi-tionmethodThemodel is based on a cross-sectional analysissatisfying strain compatibility and equilibrium conditionsand can be utilized assuming any constitutive law for theconfined concreteThemodel is able to take into account sev-eral mechanical and geometrical parameters such as sectionaspect ratio longitudinal reinforcement amount and distri-bution confinement effect on the concrete core reinforcingsteel hardening andor softening postbuckling behaviourThe formulation is carried out in dimensionless form and itis able to stress the role of the aforementioned parameters indetermining the shape of the moment-curvature relationshipand the ultimate values of the bending moment and curva-ture

The ability to analyze this role for classes of RC sectionsis the main contribution of the formulation proposed On

the other hand the following limits are related to the basicassumptions the model is applicable only to rectangularsections with symmetrical distribution of the longitudinalreinforcement no cyclic behaviour is considered Moreoverthe use of the proposedmodel to define themoment-rotationresponse of the critical region towhich the section consideredideally belongs requires a suitable modeling of the potentialldquoplastic hingesrdquo which is not discussed here

2 Analysis Model and Geometric Parameters

The analysis model adopted is an updated version of thatalready utilized in [17] where analytical expressions of theultimate moment and the corresponding compression levelwere provided for a class of sections more restricted than theone considered in the present paper

The compressed concrete rectangular section is assumedto be reinforced by longitudinal steel bars symmetricallylocated with respect to the two principal axes of inertia of thesection so that the geometrical and mechanical barycentresare coincident

The longitudinal reinforcing bars are distributed as fol-lows

(i) four bars of the same diameter are located at the cor-ners giving total sectional area 119860

119904119888

(ii) 119899119887bars of the same diameter having total area119860

119904119887 are

located on the upper and lower parts of the sectionbetween the corner bars in the direction parallel tothe width

(iii) 119899ℎbars of the same diameter having total area 119860

119904ℎ

are located on the right and on the left between thecorner bars in the direction parallel to the height ofthe section

Figure 1 shows the section geometry the reinforcing barlocation and the analysis model adopted The confined con-crete core of width 119887 and height ℎ is measured inside the stir-rup perimeter nevertheless in some applications in agree-ment with some technical codes the resisting section of theconcrete core is assumed to be defined by the axis line of theperimeter stirrups

The symbols 119888V and 119888119900in Figure 1(a) indicate the concrete

cover depths that is the distances of the horizontal and ver-tical edges of the core from the external perimeter of the sec-tion

These quantities had not been defined in the originalmodel [17] because at the ultimate state the contribution ofthe concrete cover to the bearing capacity of the section isnegligible

The mean distance of the barycentres of the sectionalarea of the upper and lower reinforcing bars from the nearesthorizontal side of the core of length 119887 is denoted as ℎ

1

By assuming that the bending moment lies in the bary-centric vertical plane of the section the upper and lower rein-forcing bars provide a primary contribution to the rotationalequilibrium of the section while the intermediate bars dis-tributedwith constant pitch119901

ℓbetween the corner bars in the

direction parallel to the height of the section give a minor

Advances in Civil Engineering 3

Gh

p1200012

b

Ash

h1

d1

As1

G

co co

Asb

c

c

p120001

Asc2

As2

d1

(a) (b)Figure 1 Section geometry and analysis model

contribution Consequently the upper and lower reinforcingbars and the intermediate reinforcing bars will be conven-tionally denoted as ldquoprincipal reinforcementrdquo and ldquosecondaryreinforcementrdquo respectively

With reference to the symbols in Figure 1(b) the follow-ing relationships must be considered

1198601199041

=119860119904119888

2+ 119860119904119887 119860

1199042= 119860119904ℎ

119901ℓ=

ℎ minus 2ℎ1

119899ℎ+ 1

1198891= 119899ℎ119901ℓ

(1)

Figure 1(b) shows that in the analysis model the secondaryreinforcement is assumed to be uniformly distributed alongthe segment 119889

1defined as shown in Figure 1(a)

Since high curvature values are involved (ultimate limitstate) unconfined concrete spalling reduces the resisting sec-tion to that of the confined concrete core all the geometricaland mechanical parameters involved in the formulation pro-posed here are referred to this reduced section

Therefore the axial load 119873 and bending moment 119872 willbe normalized to the values

119873119900= 119887ℎ119891cco 119872

119900= 119887ℎ2

119891cco (2)

where119891cco is the cylindrical strength of the confined concreteThe strains of the confined concrete and the dimension-

less curvature120593ℎwill be normalizedwith respect to the strain120576cco corresponding to119891cco and this normalization is indicatedby using the superscription () the same notation is adoptedfor the normalization of the strains of the unconfined con-crete with respect to the strain 120576co corresponding to the cyl-indrical strength 119891co The distances normalized to the heightℎ of the concrete core are denoted by using the superscription(minus

)For clarityrsquos sake the geometrical parameters thatmust be

considered assigned are denoted using Greek letters There-fore with reference to the symbols in Figure 1 the followinggeometrical parameters are defined here

120575V =119888V

ℎ 120575

119900=

119888119900

119887

120582 =ℎ1

ℎ 120572 =

1198891

ℎ=

119899ℎ

119899ℎ+ 1

(1 minus 2120582)

(3)

Stre

ss

Strain

ConfinedUnconfined

fcco

fco

120576co 120576cco 120576ccu

Figure 2 Constitutive laws of concrete

3 Constitutive Laws

31 Concrete Differently from what was necessary for theanalysis at the ultimate state [17] to derive the entiremoment-curvature curve of the class of RC sections under considera-tion the constitutive law of the concrete must be defined forboth the concrete core and the unconfined cover

Figure 2 shows typical shapes of constitutive laws of con-crete and the symbols adopted for the characteristic valuesof strains and stresses Several analytical relations expressingthese laws and confirmed by experimental investigations areavailable in the literature

The formulation proposed in the present work allows theuse of any law however the applications were carried outusing the expressions proposed in [18] which also prove tobe valid for eccentric compression and fibrous andor high-strength concrete [19ndash22]

In normalized form these expressions are written as

119904119888(120576) = (2120576 minus 120576

2

)120573

0 le 120576 le 1 (4)

119904119888(120576) = 1 + 120578 (120576 minus 1) 1 le 120576 le 120583 (5)

where 119904119888is the current stress value normalized to the cylindri-

cal strength and 120576 the strain normalized to the correspondingstrain (120576co or 120576cco for unconfined or confined concrete resp)The exponent 120573 le 1 governs the shape of the ascendingbranch of the constitutive lawThe parameter 120578 rules the neg-ative slope of the linear postpeak branch and is obtained bynormalizing the value of the softening modulus 119864

119888soft in thedimensional plane 119891

119888minus120576119888with respect to the secant modulus

expressed by the ratio between the cylindrical strength andthe corresponding strain

The symbol 120583 indicates the normalized value of the ulti-mate concrete strain and is an index of the available ldquoductilityrdquoof the material

Equations (4) and (5) are valid both for unconfined andconfined concrete upon calibration of the parameters 120573 120578and 120583


All quantities referring to the unconfined concrete arehere denoted by the subscript ( )

119888 the ascending branch is

modelled by assuming the value 120573119888= 1 so that for 120576co = 0002

(4) expresses the parabolic law first proposed in [23] and sub-sequently adopted by several researchers Considering thatthe softening branch is very steep for unconfined concreteone sets 120578

119888= minus08 so that (5) provides 119891

119888= 0 for 120576

119888= 120576cu =

00045 (120583119888= 225)

For confined concrete the characteristic quantities ofwhich are denoted by the subscript ( )cc the parameters 120573cc e120578cc depend on the effective confinement pressure and can becalculated by the procedure proposed in [18] The availableductility 120583cc = 120576ccu120576cco is obtained by imposing a conven-tional limit value of reduction to the postpeak strength withrespect to the peak value or by assuming the ultimate strain120576ccu in agreement with expressions validated experimentally[24 25]

In the present formulation the factors relating the cylin-drical strength and the corresponding strain of the confinedconcrete to the ones of the originally unconfined concrete aredenoted as

1198961=

119891cco119891co

1198962=

120576cco120576co

(6)

Moreover the following integral functions are defined

1198781120576

= int

120576

0

119904119888(120576) 119889120576 119878

2120576= int

120576

0

119904119888(120576) 120576 119889120576 (7)

Equations (4) and (5) show that these integrals can beexpressed in an exact analytical form only if 120573 = 1 (uncon-fined concrete) Therefore numerical analyses were carriedout in order to derive approximate analytical expressions forthe confined concrete to be utilized in practical applications

As a result it can be shown that in the field 04 le 120573 le 1which includes all real cases the following expressions implya maximum error of 8 with respect to the values calculatedby numerical integration for 120576 ge 01

1198781120576

= (1205762

minus1

31205763

)

1199031

1198782120576

= (2

31205763

minus1

41205764

)

1199032

01 le 120576 le 1

(8)

where

1199031= 076 + 048 (120573 minus 05) 119903

2= 087 + 026 (120573 minus 05)

(9)

On the other hand the analytical form expressed by (5) showsthat (7) can be exactly integrated in the field of the normalizedpostpeak strains

1198781120576

= 11987811

+ (120576 minus 1) +1

2120578(120576 minus 1)

2

1198782120576

= 11987821

+1

2(1205762

minus 1) +1

6120578(120576 minus 1)

2

(1 + 2120576)

1 le 120576 le 120583

(10)

These expressions also provide approximate values for con-fined concrete because of the approximation by which

0

Stre

ss

0

Strain

Compression

fsy Atan Esc

Atan Es02fsy

minus120576su minus120576sy

120576sy 120576lowasts

Tension

Atan Esh minusfsy

Figure 3 Constitutive law of longitudinal steel reinforcement

the addends 11987811

and 11987821

are affected while for 120573 = 1 (uncon-fined concrete) (9) give 119903

1= 1199032

= 1 and both (8) and (10)express the integral quantities (7) in the exact analytical form

32 Longitudinal Steel Reinforcement Figure 3 shows thebilinear simplified constitutive law adopted for the longitu-dinal steel bars in tension and compression and the symbolsdenoting the characteristic quantities involved

With respect to the analyses made in [17] the restrictiveassumption that the postyielding modulus in compressionmust be the same as in tension is removed

Under tension the possible hardening behaviour isdefined by themean hardeningmodulus119864

119904ℎ under compres-

sion the slope of the postyielding branch can bemodified (thisoption is shown in the figure) so that it can becomenegative ifthe transverse reinforcement is not able to prevent buckling ofthe longitudinal bars In this case the slope of the postyield-ing branch is governed by the modulus 119864

119904119888 which can be

determined by using the model proposed in [26] assumingthat the ultimate strain corresponds to a reduction of 80 ofthe yielding stress 119891

119904119910 This ultimate strain in compression is

denoted as 120576lowast119904in Figure 3

The simple equations that analytically express the lawsof the elastic and postyielding branches in Figure 3 are hereomitted for brevityrsquos sake They will be introduced in thecourse of the formulation

The dimensionless parameters characterizing the slopesof the postyielding branches and the ultimate strain in anideal normalized stress-strain diagram are denoted as

120578119904ℎ

=119864119904ℎ

119864119904

120578119904119888

=119864119904119888

119864119904

120583119903=

120576119904119906

120576119904119910

(11)

where 119864119904is the Young modulus and 120576

119904119910 120576119904119906

are the yield-ing and ultimate strains respectively For compressed steelbars subject to buckling 120576

119904119906is understood to be replaced by

120576lowast

119904in (11)A further parameter relating the characteristic strain

values of steel and concrete is

120577119904119888

=

120576119904119910

120576cco (12)


Finally the amount of principal and secondary longitudinalreinforcement is related to the section of the concrete core bymeans of the mechanical ratios of reinforcement

1205961

= 2

1198601199041119891119904119910

119887ℎ119891cco 120596

2= 2

1198601199042119891119904119910

119887ℎ119891cco (13)

4 Equilibrium of Section

Denoting as119873 and119872 the axial load and the bendingmomentacting on the section and as

119899 =119873

119873119900

=119873

119887ℎ119891cco 119898 =

119872

119872119900

=119872

119887ℎ2119891cco (14)

these quantities normalized with respect to the ones definedby (2) the equilibrium of the section in dimensionless formis expressed by the following equations

119899 = 1198991199041

+ 1198991199042

+ 119899119888 (15)

119898 = 1198981199041

+ 1198981199042

+ 119898119888 (16)

At the secondmember of these equations there are clearlyindicated and ordered contributions offered by the principalreinforcement secondary reinforcement and concrete

These contributions are analytically expressed in the fol-lowing sections by assuming the classical hypothesis that thesection remains plane and neglecting the tensile concretestrength

41 Contribution of Principal Reinforcement In the analyticalformulation shown in this section the following parametersare involved the geometrical parameter 120582 defined in (1) themechanical parameters characterizing the constitutive law ofthe steel reinforcement 120578

119904119888 120578119904ℎ and 120577

119904119888 defined by (11) and

(12) and the mechanical ratio of reinforcement 1205961expressed

by the first of (13)Figure 4 shows the lateral view of the RC member con-

sidered and a generic state of strain and stressBy using the symbols shown in the figure the upper and

lower reinforcement bars are subjected respectively to thestrains

120576119904119886

= 120593 (119909119888minus ℎ1) = 120593ℎ [(119909

119888minus

1

2) + 1205821]

120576119904119887

= 120593 (119909119888minus ℎ + ℎ

1) = 120593ℎ [(119909

119888minus

1

2) minus 1205821]

(17)

where

119909119888=

119909119888

ℎ(18)

is the neutral axis depth normalized to the height ℎ of thesection of concrete core and

1205821=

1 minus 2120582

2 (19)

The corresponding stresses denoted as 119891119904119886

and 119891119904119887

inFigure 3(c) can be expressed in the forms

119891119904119886

= 119864119904120576119904119910

+ 119864119904119888(120576119904119886

minus 120576119904119910) (20)

119891119904119887

= minus119864119904120576119904119910

+ 119864119904ℎ

(120576119904119887

+ 120576119904119910) (21)

G

120576sah

2

120576sb

minus120576sy

120576syh minus 2h1

2

h1xc

fsa

fsb

(a) (b) (c)

120593

Figure 4 Strain and stress states of principal reinforcement ofsection

It must be observed that (20) expresses the linear elasticbehaviour of the upper reinforcement if in the field of strainsminus120576119904119910

le 120576119904119886

le 120576119904119910one sets 119864

119904119888= 119864119904 analogously (21) expresses

the elastic response of the lower reinforcement if for minus120576119904119910

le

120576119904119887

le 120576119904119910one sets 119864

119904ℎ= 119864119904

The axial load and the bendingmoment that the principalreinforcement can bear with respect to the barycentre of thesection are expressed by

1198731199041

= 1198601199041(119891119904119886

+ 119891119904119887)

1198721199041

= 1198601199041(119891119904119886

minus 119891119904119887) (

ℎ

2minus ℎ1)

(22)

By substituting (17) into (20) and (21) and by introducingthe resulting expressions of 119891

119904119886and 119891

119904119887into (22) in normal-

ized form one obtains

1198991199041

=1205961

2[

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

+ 120578119904119888)

+(1 minus120593

120577119904119888

1205821) (120578119904ℎ

minus 120578119904119888)]

(23)

1198981199041

=1205961

21205821[2 minus

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

minus 120578119904119888)

minus(1 minus120593

120577119904119888

1205821) (120578119904ℎ

+ 120578119904119888)]

(24)

where

120593 =120593ℎ

120576cco(25)

is the dimensionless curvature normalized with respectto 120576cco

On the basis of what was observed commenting on (20)and (21) (23) and (24) have a general validity under the con-dition that one sets

120578119904119888

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 120582) le 1

120578119904ℎ

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 1 + 120582) le 1

(26)


G

(b) (c)(a)

120576s1

120576s2

a1

a2

Δf2

Δf1

fs2

fs1

fe2

fe1

minus120576sy

120576sy

hd1

h1

h minus 2h12

h

2

xc

120593

2120576sy

120593

Figure 5 Strains and stresses of uniformly distributed secondary reinforcement

42 Contribution of Secondary Reinforcement With respectto the formulation shown in the previous section instead of120582 and 120596

1the parameters 120572 and 120596

2are now involved which

are expressed in (3) and (13) respectivelyFigure 5 shows the symbols used to define a generic strain

state of the section and the corresponding stress state Fromthis figure one can deduce that the distances 119886

1and 1198862can be

expressed by

1198861=

1198891

2minus

ℎ

2+ 119909119888minus

120576119904119910

120593 119886

2=

1198891

2+

ℎ

2minus 119909119888minus

120576119904119910

120593

(27)

By normalizing these distances to the height ℎ and remem-bering the expression of the normalized dimensionless cur-vature (25) one obtains

1198861= (

120572

2minus

120577119904119888

120593) + (119909

119888minus

1

2)

1198862= (

120572

2minus

120577119904119888

120593) minus (119909

119888minus

1

2)

(28)

The actual stress at the upper ldquofibrerdquo of the uniformly distrib-uted secondary reinforcement denoted as 119891

1199041in Figure 5(c)

can be expressed by separately evaluating the stress 1198911198901that

one would have in the case of infinitely elastic behaviour andthe difference Δ119891

1due to the actual postyielding behaviour

These quantities are expressed respectively by

1198911198901

= 119864119904120593(1198861+

120576119904119910

120593) = 119891

119904119910

120593

120577119904119888

(1198861+

120577119904119888

120593) (29)

Δ1198911= (119864119904minus 119864119904119888) 1205931198861= 119891119904119910

120593

120577119904119888

(1 minus 120578119904119888) 1198861 (30)

Analogously at the lower fibre of the secondary reinforce-ment one has

1198911198902

= minus119864119904120593(1198862+

120576119904119910

120593) = minus119891

119904119910

120593

120577119904119888

(1198862+

120577119904119888

120593) (31)

Δ1198912= minus (119864

119904minus 119864119904ℎ) 1205931198862= minus119891119904119910

120593

120577119904119888

(1 minus 120578119904ℎ) 1198862 (32)

Then the axial load and bending moment with respect to thebarycentre of the section that the secondary reinforcementcan bear can be calculated by the following expressions

1198731199042

=21198601199042

1198891

ℎ

2[1198911198901

(1198861+

120577119904119888

120593) + 119891

1198902(1198862+

120577119904119888

120593)

minus (Δ11989111198861+ Δ11989121198862) ]

1198721199042

=21198601199042

1198891

ℎ2

2[(1198911198901

minus 1198911198902)1205722

6minus Δ11989111198861(120572

2minus

1198861

3)

+Δ11989121198862(120572

2minus

1198862

3)]

(33)

Equations (28) (29) and (31) show that

1198911198901

(1198861+

120577119904119888

120593) + 1198911198902

(1198862+

120577119904119888

120593) = 2119891

119904119910

120593

120577119904119888

(119909119888minus

1

2) 120572

1198911198901

minus 1198911198902

= 119891119904119910

120593

120577119904119888

120572

(34)

Therefore substituting (34) into (33) in normalized form oneobtains

1198991199042

=1198731199042

119873119900

= 1205962

120593

120577119904119888

(119909119888minus

1

2) minus

1

2120572

times [(1 minus 120578119904119888) 1198862

1minus (1 minus 120578

119904ℎ) 1198862

2]

(35)

1198981199042

=1198721199042

119872119900

=1205962

2120572

120593

120577119904119888

[1205723


119904119888) (

120572

2minus

1198861

3) 1198862

1

minus (1 minus 120578119904ℎ) (

120572

2minus

1198862

3) 1198862

2]

(36)


h

c

c

co cob

x3

xc

Confinedconcrete

1205761

1205762

1205763

1205764

(a)

h

c

c

co cob

xc

1205761

1205762

1205763 = 1205764 = 0x3 = 0

Confinedconcrete

(b)

Figure 6 Strain state of concrete section (a) uncracked section (b) cracked section

It must be observed that if one sets 119864119904119888

= 119864119904in (30) andor

119864119904ℎ

= 119864119904in (32) (33) expresses the contributions to the axial

load and bendingmoment in the case inwhich the upper fibreandor the lower fibre of the secondary reinforcement remainin the elastic field As a consequence (35) and (36) are validfor any value of 119909

119888e 120593 under the condition that one sets

120578119904119888=1 if minus 120576

119904119910le 1205761199041

le 120576119904119910

997904rArr minus1le120593

120577119904119888

(119909119888minus

1 minus 120572

2)le1

120578119904ℎ=1 if minus 120576

119904119910le 1205761199042

le 120576119904119910


120577119904119888

(119909119888minus

1 + 120572

2)le1

(37)

43 Contribution of Concrete Figure 6 shows the strain statesof the section corresponding to the cases of uncracked andcracked sections By using the symbols in the figure the con-tributions to the equilibrium provided by the concrete coreare expressed by

119873cc = 119887int

119909119888

1199093

119891119888(119909) 119889119909 =

119887ℎ119891cco120593

int

1205762

1205763

119904119888(120576) 119889120576

119872cc = 119887int

119909119888

1199093

119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus

ℎ

2)

=119887ℎ2

119891cco (1

1205932int

1205762

1205763

119904119888(120576) 120576 119889120576minus

119909119888minus 05

120593int

1205762

1205763

119904119888(120576) 119889120576)

(38)

where

1199093=

0 for 119909119888le ℎ

119909119888minus ℎ for 119909

119888gt ℎ

(39)

and consequently

1205762= 120593119909119888

1205763=

0 for 119909119888le 1

120593 (119909119888minus 1) for 119909

119888gt 1

(40)

Considering (7) in the normalized form one obtains

119899cc =119873cc119873119900

=1

120593(11987811205762

minus 11987811205763

) (41)

119898cc =119872cc119872119900

=1

1205932(11987821205762

minus 11987821205763

) minus 119899cc (119909119888 minus1

2) (42)

The contribution of the surrounding unconfined concrete(concrete cover) can be calculated by subtracting the contri-bution of the confined region to that of the whole section

Considering that the strains involved in the functions 1198781120576

and 1198782120576

must be normalized with respect to the strain 120576co(unconfined concrete) by following the same procedure asthat leading to (41) and (42) and by introducing the con-finement efficiency factors defined by (6) for the whole sec-tion one obtains

119899un1 =1 + 2120575

119900

1198961

1

1198962120593

(11987811205761015840

1

minus 11987811205761015840

4

)

119898un1 =1 + 2120575

119900

1198961

1

(1198962120593)2(11987821205761015840

1

minus 11987821205761015840

4

) minus 119899un1 (119909119888 minus1

2)

(43)where

1205761015840

1= 1198962120593 (119909119888+ 120575V) le 120583

119888

1205761015840

4=

0 for 119909119888le (1 + 120575V)

1198962120593 [119909119888minus (1 + 120575V)] for 119909

119888gt (1 + 120575V)

(44)

The contribution to be subtracted from the above quantitiesare calculated bymodifying (41) and (42) in order to considerthe different constitutive law of the unconfined concrete andthe different base of normalization of strain and stress It caneasily be shown that one obtains

119899un2 =1

11989611198962120593

(11987811205761015840

2

minus 11987811205761015840

3

) (45)

119898un2 =1

1198961(1198962120593)2(11987821205761015840

2

minus 11987821205761015840

3

) minus 119899un2 (119909119888 minus1

2) (46)


where

1205761015840

2= 1198962120593119909119888le 120583119888

1205761015840

3=

0 for 119909119888le 1

1198962120593 (119909119888minus 1) for 119909

119888gt 1

(47)

On the basis of what was said above the effective contributionof the concrete section in compression in a generic loadingstate is expressed by

119899119888= 119899cc + 119899un1 minus 119899un2 (48)

119898119888= 119898cc + 119898un1 minus 119898un2 (49)

If the confinement effect is negligible (6) gives 1198961= 1198962= 1

By comparing (40) (41) and (42) with (47) (45) and (46)respectively it can easily be observed that 119899cc = 119899un2 and119898cc = 119898un2 Therefore in this case the contributions of thewhole section of reacting concrete given by (48) and (49) arereduced to the values 119899cc = 119899un1 and 119898cc = 119898un1 and areexpressed by (45) and (46) for 119896

1= 1198962= 1

5 Analysis Procedure

The flexural response of a class of RC sections characterizedby assigned values of the geometrical andmechanical param-eters defined above is obtained by imposing the equilibriumcondition expressed by (15) and (16)

For each configuration the contributions to the equi-librium provided by the principal reinforcement ((23) and(24) under the conditions expressed by (26)) the secondaryreinforcement ((28) (35) and (36) under the conditionsexpressed by (37)) and the compressed region of concrete((48) and (49) considering (41)ndash(47)) only depend on thevariables 119909

119888and 120593 which are independent of or related to one

another according to the type of analysis required (see below)In all cases it must be assumed that the proposed formulationis applied for 120593 gt 0

51 Field of Application The field of validity of the procedureproposed here is limited by the fact that the constitutive law ofthe steel reinforcement does not include unloading branchesstarting from a point of the post yielding branches in com-pression or in tension Therefore for any loading step theupper principal reinforcement and the upper fibre of the dis-tributed secondary reinforcement both potentially subject tocompression cannot be in the postyielding field in tensionthe lower principal reinforcement and the lower fibre of thedistributed secondary reinforcement both potentially subjectto tension cannot be in the postyielding field in compression

Because of these assumptions the field of values of 119909119888

consistent with a given value of 120593 is limited by the initial andend values provided by (26) or (37)

More precisely in the absence of secondary reinforce-ment (120596

2= 0) (26) gives

119909119888min = 120582 minus

120577119904119888

120593 119909

119888max = (1 minus 120582) +120577119904119888

120593 (50)

while if the secondary reinforcement is present (37) leads tomore restrictive limitations

119909119888min =

1 minus 120572

2minus

120577119904119888

120593 119909

119888max =1 + 120572

2+

120577119904119888

120593 (51)

These limitations also occur in a classical numericalapproach based on the strip decomposition of the section

However it must be observed that (50) or (51) allows oneto construct the moment-curvature curve for a large field ofvalues of assigned level of compression including the moreusual cases occurring in practical applications

52 Moment-Curvature Curves The more usual applicationof the analytical expressions derived in the previous sectionsis to derive by a step-by-step procedure the moment-curva-ture119898-120593 curve of a class of RC sections for a given value of 119899

To this purpose the procedure is applied by the followingsteps

(i) assigning the lowest value of curvature 120593 that onewants to consider increasing values of 119909

119888are consid-

ered from the minimum to the maximum expressedby (50) or (51)

(ii) for each value of the couple 120593 119909119888the corresponding

value of 1198991199041 1198991199042 and 119899

119888are calculated and their sum

increasing for each increase in 119909119888

(iii) when the assigned value of 119899 is reached with anacceptable tolerance (ie (15) is verified with verygood approximation) the assigned values of 120593 and 119899and the value of 119909

119888that was found make it possible

to calculate the corresponding values of11989811990411198981199042 and

119898119888expressed by (24) (36) and (49) and the sum

of these contributions of moment is the ordinate ofthe point having the abscissa 120593 belonging to the119898-120593curve sought

(iv) considering an increased value of 120593 the procedure isrepeated to obtain a subsequent point of themoment-curvature curve

The procedure stops when the maximum available ulti-mate confined concrete strain or the maximum available ten-sile strain of the lower reinforcement is exceeded These twooccurrences imply respectively that 120593119909

119888gt 120583cc or 120593(119909119888 minus 1 +

120582) lt minus120577119904119888120583119903 where 120583cc = 120576ccu120576cco is the available ductility of

the confined concrete defined in Section 31 and 120583119903is the steel

ductility defined in (11)If the stress-strain law of the steel reinforcement also

exhibits a softening postyielding branch corresponding to apostbuckling behaviour up to the strain value 120576

∙

119904(Figure 3)

a further limit at which the procedure must be stoppedderives from the condition that 120593(119909

119888minus 120582) gt 120577

119904119888120583119903119888 in

which 120583119903119888

= 120576lowast

119904120576119904119910 Beyond this condition the residual stress

of the compressed reinforcement should be assumed to beconstant and equal to 02119891

119904119910 but this branch of the stress-

strain law is not considered in the proposed model as statedin Section 32


53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level

The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898

119906-119899 120593119906-119899 and119898

119910-119899

120593119910-119899 domains which give useful information on the flexural

strength and curvature ductility of an RC section subjected toan assigned compression level

The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909

119888and 120593 being related to one

anotherIn order to construct the 119898

119906-119899 and 120593

119906-119899 curves two dif-

ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is

120593119906max =

120583cc + 120577119904119888120583119903

1 minus 120582 (52)

The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed

120593119906(119909119888minus 1 + 120582) = minus 120577

119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus

120577119904119888120583119903

120593119906

(53)

Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909

119888 and (15) and (16)

by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898

119906

Once 120593119906max is reached the second branch of the 119898

119906-119899

120593119906-119899 curves must be determined considering that for further

decreasing values of 120593119906 the collapse is produced by the

achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is

120593119906119909119888= 120583cc 997904rArr 119909

119888=

120583cc120593119906

(54)

By using the second of these equations (15) and (16) againlink 119899 and119898

119906to the only variable 120593

119906

It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙

119904in Figure 3 This strain

value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593

119906max = 120577119904119888(120583119903119888

+

120583119903)(1minus2120582) where 120583

119903119888= 120576lowast

119904120576119904119910 and the second branch of the

ultimate strength domains must be determined by assumingthat for values of 120593

119906that decrease from this maximum the

relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577

119904119888120583119903119888120593

The 119898119910-119899 120593119910-119899 curves can be simply determined by

assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576

119904119910 This condition in dimensionless terms gives

120593119910(119909119888minus 1 + 120582) = minus 120577

119904119888997904rArr 119909119888= (1 minus 120582) minus

120577119904119888

120593119910

(55)

By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593

119910 The 119898

119910-119899 120593119910-119899 curves can be con-

structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576

∙

119904in the principal

reinforcement in compression if this reinforcement is subjectto buckling

Considering the field of validity of the procedure pro-posed the minimum value of 120593

119906that can be considered to

construct the119898119906-119899 120593119906-119899 domains is obtained by introducing

the second of (53) into the first of (50) or (51) so that

120593119906min = 120577

119904119888

120583119903minus 1

1 minus 2120582 (56)

in the absence of the secondary reinforcement and

120593119906min = 120577

119904119888

120583119903minus 1

(1 + 120572) 2 minus 120582 (57)

if the secondary reinforcement is also presentObviously the 119898

119910-119899 120593119910-119899 domains can be constructed

starting from any value 120593119910gt 0

In relation to the field of applications (RC columns) thefirst values of 120593

119906and 120593

119910that are to be actually considered are

the ones first producing 119899 ge 0

6 Applications

The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers

Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical

10 Advances in Civil EngineeringN

orm

aliz

ed st

ress

1

00 1 2 3 4 5

Saatcioglu and Razvi (1992)Mander et al (1988)

Normalized strain

Figure 7 Calibration of constitutive law of confined concrete

strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891

119904119910= 420Nmm2 uniformly distributed along the

perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891

ℓ119890= 011119891

1015840

co where 1198911015840

co =

085119891co is the reduced strength value prescribed by the ACI318 code [28]

The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors

Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045

and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered

The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]

Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573

119888= 1 120578

119888= minus08

and 120576co = 0002On the basis of the above data the square cross-section of

the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to

400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)

Zahn et al (1989)Present model

Figure 8 Comparison of results119872-120593 curve for given119873

23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899

119887= 119899ℎ= 2 and the distance

of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ

1=

162 = 8mmThe principal reinforcement consists of four upper and

four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864

119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888

= 0Therefore the dimensionless parameters involved in the

formulation proposed take on the following values 120575V = 120575119900=

23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ

= 0Figure 8 compares the 119872-120593 curve obtained in [27] with

that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887

2

times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here

Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling

The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors

The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load

As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by


the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect

The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873

119900expressed by (2) The procedure

is applied by determining the value of 119909119888from (15) for the

assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected

The specimen considered had a square section of side 119861 =

210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891

119904119910=

517Nmm2 and average hardening modulus 119864119904ℎ

= 0013119864119904

The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm

The unconfined concrete had cylindrical strength 119891co =

3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups

On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =

000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575

119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ

= 0013Figure 9 shows that the procedure proposed provides

results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models

It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work

Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576

119904119906= 0066 (120583

119903= 27) The

results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level

The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this

100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)

ExperimentalModelPresent model

Saatcioglu et al (1995)

Curvature (1mm times 10minus4)

Figure 9 Comparison of theoretical and experimental results

point the 120593119906-119873 curve proceeds by a very brief stretch (not

very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)

Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings

7 Conclusions

A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study

At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section

Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain

For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to


100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)

Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement

the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept

The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage

Acknowledgment

Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated

References

[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001

[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004

[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006

[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966

[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970

[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987

[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973

[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006

[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964

[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986

[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968

[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992

[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002

[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011


[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012

[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013

[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992

[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995

[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006

[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010

[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012

[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951

[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982

[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988

[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002

[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989

[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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 athttpwwwhindawicom

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



Consequently the use of moment-curvature relationships isstill the more widespread and efficient approach

The definition of moment-curvature relationships of RCsections has been a point of research interest for many years[10 11] Many analytical and numerical techniques havebeen proposed including several phenomena that affect theresponse of reinforced concrete elements

Historically moment-curvature relationships with soft-ening branch were first introduced by Wood [12] Mo [13]suggested a classical approach to reproduce the moment-curvature relationship including the softening branch basedon an FE analysis of the elastic-plastic buckling of the longi-tudinal reinforcement An alternative approachwas proposedin [14] where by using a simplified model complex nonlin-ear geometric effects were embedded in the nonlinear mate-rial behaviour of the cross-section Chandrasekaran et al [15]proposed an analytical relationship providing in explicit formthe moment-curvature response of a section by consideringnonlinear constitutive laws of materials chosen in accord-ancewith the European technical codes which can be appliedby the use of a spreadsheet

Commercial structural analysis software programs nowprovide the analyst with the option of conducting moment-curvature analyses at critical sections Although well-verifiedequations have been used in the software developments somequestions may remain (i) for a given project how can thecomputational analysis results be easily verified outside thesoftware (ii) how can general considerations on the role ofdifferent parameters characterizing the section and reinforce-ment geometry be derived

To answer to these questions a dimensionless formula-tion is required able to fully characterize the nonlinearbehaviour of an RC cross-section subjected to an assignedloading condition

In a recent paper [16] the behaviour of RC rectangularcross-sections under axial load and biaxial bending is exam-ined in dimensionless terms This paper confirms that thevalues of the ultimate bending moment and curvature in anydirection can be efficiently related to the values correspond-ing to two separate conditions of uniaxial bending

In this context the present paper proposes an analyticalmodel for the evaluation of the moment-curvature relation-ship of rectangular RC sections subjected to axial load anduniaxial bending by using an incremental strain techniquewhich could be an alternative to the classical fibre decomposi-tionmethodThemodel is based on a cross-sectional analysissatisfying strain compatibility and equilibrium conditionsand can be utilized assuming any constitutive law for theconfined concreteThemodel is able to take into account sev-eral mechanical and geometrical parameters such as sectionaspect ratio longitudinal reinforcement amount and distri-bution confinement effect on the concrete core reinforcingsteel hardening andor softening postbuckling behaviourThe formulation is carried out in dimensionless form and itis able to stress the role of the aforementioned parameters indetermining the shape of the moment-curvature relationshipand the ultimate values of the bending moment and curva-ture

The ability to analyze this role for classes of RC sectionsis the main contribution of the formulation proposed On

the other hand the following limits are related to the basicassumptions the model is applicable only to rectangularsections with symmetrical distribution of the longitudinalreinforcement no cyclic behaviour is considered Moreoverthe use of the proposedmodel to define themoment-rotationresponse of the critical region towhich the section consideredideally belongs requires a suitable modeling of the potentialldquoplastic hingesrdquo which is not discussed here

2 Analysis Model and Geometric Parameters

The analysis model adopted is an updated version of thatalready utilized in [17] where analytical expressions of theultimate moment and the corresponding compression levelwere provided for a class of sections more restricted than theone considered in the present paper

The compressed concrete rectangular section is assumedto be reinforced by longitudinal steel bars symmetricallylocated with respect to the two principal axes of inertia of thesection so that the geometrical and mechanical barycentresare coincident

The longitudinal reinforcing bars are distributed as fol-lows

(i) four bars of the same diameter are located at the cor-ners giving total sectional area 119860

119904119888

(ii) 119899119887bars of the same diameter having total area119860

119904119887 are

located on the upper and lower parts of the sectionbetween the corner bars in the direction parallel tothe width

(iii) 119899ℎbars of the same diameter having total area 119860

119904ℎ

are located on the right and on the left between thecorner bars in the direction parallel to the height ofthe section

Figure 1 shows the section geometry the reinforcing barlocation and the analysis model adopted The confined con-crete core of width 119887 and height ℎ is measured inside the stir-rup perimeter nevertheless in some applications in agree-ment with some technical codes the resisting section of theconcrete core is assumed to be defined by the axis line of theperimeter stirrups

The symbols 119888V and 119888119900in Figure 1(a) indicate the concrete

cover depths that is the distances of the horizontal and ver-tical edges of the core from the external perimeter of the sec-tion

These quantities had not been defined in the originalmodel [17] because at the ultimate state the contribution ofthe concrete cover to the bearing capacity of the section isnegligible

The mean distance of the barycentres of the sectionalarea of the upper and lower reinforcing bars from the nearesthorizontal side of the core of length 119887 is denoted as ℎ

1

By assuming that the bending moment lies in the bary-centric vertical plane of the section the upper and lower rein-forcing bars provide a primary contribution to the rotationalequilibrium of the section while the intermediate bars dis-tributedwith constant pitch119901

ℓbetween the corner bars in the

direction parallel to the height of the section give a minor


Gh

p1200012

b

Ash

h1

d1

As1

G

co co

Asb

c

c

p120001

Asc2

As2

d1




1198601199041

=119860119904119888

2+ 119860119904119887 119860

1199042= 119860119904ℎ

119901ℓ=

ℎ minus 2ℎ1

119899ℎ+ 1

1198891= 119899ℎ119901ℓ

(1)





119873119900= 119887ℎ119891cco 119872

119900= 119887ℎ2

119891cco (2)





120575V =119888V

ℎ 120575

119900=

119888119900

119887

120582 =ℎ1

ℎ 120572 =

1198891

ℎ=

119899ℎ

119899ℎ+ 1

(1 minus 2120582)

(3)

Stre

ss

Strain

ConfinedUnconfined

fcco

fco

120576co 120576cco 120576ccu


3 Constitutive Laws





119904119888(120576) = (2120576 minus 120576

2

)120573

0 le 120576 le 1 (4)

119904119888(120576) = 1 + 120578 (120576 minus 1) 1 le 120576 le 120583 (5)














119888= 0 for 120576

119888= 120576cu =

00045 (120583119888= 225)



1198961=

119891cco119891co

1198962=

120576cco120576co

(6)


1198781120576

= int

120576

0

119904119888(120576) 119889120576 119878

2120576= int

120576

0

119904119888(120576) 120576 119889120576 (7)



1198781120576

= (1205762

minus1

31205763

)

1199031

1198782120576

= (2

31205763

minus1

41205764

)

1199032

01 le 120576 le 1

(8)

where

1199031= 076 + 048 (120573 minus 05) 119903

2= 087 + 026 (120573 minus 05)

(9)


1198781120576

= 11987811

+ (120576 minus 1) +1

2120578(120576 minus 1)

2

1198782120576

= 11987821

+1

2(1205762

minus 1) +1

6120578(120576 minus 1)

2

(1 + 2120576)

1 le 120576 le 120583

(10)


0

Stre

ss

0

Strain

Compression

fsy Atan Esc

Atan Es02fsy



Tension

Atan Esh minusfsy



and 11987821


1= 1199032













120578119904ℎ

=119864119904ℎ

119864119904

120578119904119888

=119864119904119888

119864119904

120583119903=

120576119904119906

120576119904119910

(11)


119904119910 120576119904119906



120576lowast



120577119904119888

=

120576119904119910

120576cco (12)



1205961

= 2

1198601199041119891119904119910

119887ℎ119891cco 120596

2= 2

1198601199042119891119904119910

119887ℎ119891cco (13)



119899 =119873

119873119900

=119873

119887ℎ119891cco 119898 =

119872

119872119900

=119872

119887ℎ2119891cco (14)


119899 = 1198991199041

+ 1198991199042

+ 119899119888 (15)

119898 = 1198981199041

+ 1198981199042

+ 119898119888 (16)




119904119888 120578119904ℎ and 120577

119904119888 defined by (11) and





120576119904119886

= 120593 (119909119888minus ℎ1) = 120593ℎ [(119909

119888minus

1

2) + 1205821]

120576119904119887

= 120593 (119909119888minus ℎ + ℎ

1) = 120593ℎ [(119909

119888minus

1

2) minus 1205821]

(17)

where

119909119888=

119909119888

ℎ(18)


1205821=

1 minus 2120582

2 (19)


and 119891119904119887


119891119904119886

= 119864119904120576119904119910

+ 119864119904119888(120576119904119886

minus 120576119904119910) (20)

119891119904119887

= minus119864119904120576119904119910

+ 119864119904ℎ

(120576119904119887

+ 120576119904119910) (21)

G

120576sah

2

120576sb

minus120576sy

120576syh minus 2h1

2

h1xc

fsa

fsb

(a) (b) (c)

120593



le 120576119904119886

le 120576119904119910one sets 119864



le

120576119904119887

le 120576119904119910one sets 119864

119904ℎ= 119864119904


1198731199041

= 1198601199041(119891119904119886

+ 119891119904119887)

1198721199041

= 1198601199041(119891119904119886

minus 119891119904119887) (

ℎ

2minus ℎ1)

(22)


119904119886and 119891

119904119887into (22) in normal-


1198991199041

=1205961

2[

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

+ 120578119904119888)

+(1 minus120593

120577119904119888

1205821) (120578119904ℎ

minus 120578119904119888)]

(23)

1198981199041

=1205961

21205821[2 minus

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

minus 120578119904119888)

minus(1 minus120593

120577119904119888

1205821) (120578119904ℎ

+ 120578119904119888)]

(24)

where

120593 =120593ℎ

120576cco(25)



120578119904119888

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 120582) le 1

120578119904ℎ

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 1 + 120582) le 1

(26)


G

(b) (c)(a)

120576s1

120576s2

a1

a2

Δf2

Δf1

fs2

fs1

fe2

fe1

minus120576sy

120576sy

hd1

h1

h minus 2h12

h

2

xc

120593

2120576sy

120593







1and 1198862can be

expressed by

1198861=

1198891

2minus

ℎ

2+ 119909119888minus

120576119904119910

120593 119886

2=

1198891

2+

ℎ


120576119904119910

120593

(27)


1198861= (

120572

2minus

120577119904119888

120593) + (119909

119888minus

1

2)

1198862= (

120572

2minus

120577119904119888

120593) minus (119909

119888minus

1

2)

(28)







1198911198901

= 119864119904120593(1198861+

120576119904119910

120593) = 119891

119904119910

120593

120577119904119888

(1198861+

120577119904119888

120593) (29)

Δ1198911= (119864119904minus 119864119904119888) 1205931198861= 119891119904119910

120593

120577119904119888

(1 minus 120578119904119888) 1198861 (30)


1198911198902

= minus119864119904120593(1198862+

120576119904119910

120593) = minus119891

119904119910

120593

120577119904119888

(1198862+

120577119904119888

120593) (31)

Δ1198912= minus (119864

119904minus 119864119904ℎ) 1205931198862= minus119891119904119910

120593

120577119904119888

(1 minus 120578119904ℎ) 1198862 (32)


1198731199042

=21198601199042

1198891

ℎ

2[1198911198901

(1198861+

120577119904119888

120593) + 119891

1198902(1198862+

120577119904119888

120593)

minus (Δ11989111198861+ Δ11989121198862) ]

1198721199042

=21198601199042

1198891

ℎ2

2[(1198911198901

minus 1198911198902)1205722

6minus Δ11989111198861(120572

2minus

1198861

3)

+Δ11989121198862(120572

2minus

1198862

3)]

(33)


1198911198901

(1198861+

120577119904119888

120593) + 1198911198902

(1198862+

120577119904119888

120593) = 2119891

119904119910

120593

120577119904119888

(119909119888minus

1

2) 120572

1198911198901

minus 1198911198902

= 119891119904119910

120593

120577119904119888

120572

(34)


1198991199042

=1198731199042

119873119900

= 1205962

120593

120577119904119888

(119909119888minus

1

2) minus

1

2120572

times [(1 minus 120578119904119888) 1198862


119904ℎ) 1198862

2]

(35)

1198981199042

=1198721199042

119872119900

=1205962

2120572

120593

120577119904119888

[1205723


119904119888) (

120572

2minus

1198861

3) 1198862

1

minus (1 minus 120578119904ℎ) (

120572

2minus

1198862

3) 1198862

2]

(36)


h

c

c

co cob

x3

xc

Confinedconcrete

1205761

1205762

1205763

1205764

(a)

h

c

c

co cob

xc

1205761

1205762

1205763 = 1205764 = 0x3 = 0

Confinedconcrete

(b)



= 119864119904in (30) andor

119864119904ℎ




120578119904119888=1 if minus 120576

119904119910le 1205761199041

le 120576119904119910


120577119904119888

(119909119888minus

1 minus 120572

2)le1

120578119904ℎ=1 if minus 120576

119904119910le 1205761199042

le 120576119904119910


120577119904119888

(119909119888minus

1 + 120572

2)le1

(37)


119873cc = 119887int

119909119888

1199093

119891119888(119909) 119889119909 =

119887ℎ119891cco120593

int

1205762

1205763

119904119888(120576) 119889120576

119872cc = 119887int

119909119888

1199093

119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus

ℎ

2)

=119887ℎ2

119891cco (1

1205932int

1205762

1205763

119904119888(120576) 120576 119889120576minus

119909119888minus 05

120593int

1205762

1205763

119904119888(120576) 119889120576)

(38)

where

1199093=

0 for 119909119888le ℎ

119909119888minus ℎ for 119909

119888gt ℎ

(39)

and consequently

1205762= 120593119909119888

1205763=

0 for 119909119888le 1

120593 (119909119888minus 1) for 119909

119888gt 1

(40)


119899cc =119873cc119873119900

=1

120593(11987811205762

minus 11987811205763

) (41)

119898cc =119872cc119872119900

=1

1205932(11987821205762

minus 11987821205763

) minus 119899cc (119909119888 minus1

2) (42)



and 1198782120576


119899un1 =1 + 2120575

119900

1198961

1

1198962120593

(11987811205761015840

1

minus 11987811205761015840

4

)

119898un1 =1 + 2120575

119900

1198961

1

(1198962120593)2(11987821205761015840

1

minus 11987821205761015840

4

) minus 119899un1 (119909119888 minus1

2)

(43)where

1205761015840

1= 1198962120593 (119909119888+ 120575V) le 120583

119888

1205761015840

4=

0 for 119909119888le (1 + 120575V)

1198962120593 [119909119888minus (1 + 120575V)] for 119909

119888gt (1 + 120575V)

(44)


119899un2 =1

11989611198962120593

(11987811205761015840

2

minus 11987811205761015840

3

) (45)

119898un2 =1

1198961(1198962120593)2(11987821205761015840

2

minus 11987821205761015840

3

) minus 119899un2 (119909119888 minus1

2) (46)


where

1205761015840

2= 1198962120593119909119888le 120583119888

1205761015840

3=

0 for 119909119888le 1

1198962120593 (119909119888minus 1) for 119909

119888gt 1

(47)


119899119888= 119899cc + 119899un1 minus 119899un2 (48)

119898119888= 119898cc + 119898un1 minus 119898un2 (49)



1= 1198962= 1










2= 0) (26) gives

119909119888min = 120582 minus

120577119904119888

120593 119909

119888max = (1 minus 120582) +120577119904119888

120593 (50)


119909119888min =

1 minus 120572

2minus

120577119904119888

120593 119909

119888max =1 + 120572

2+

120577119904119888

120593 (51)






119888are consid-



value of 1198991199041 1198991199042 and 119899










119888gt 120583cc or 120593(119909119888 minus 1 +





∙

119904(Figure 3)


119888minus 120582) gt 120577

119904119888120583119903119888 in

which 120583119903119888

= 120576lowast








119906-119899 120593119906-119899 and119898

119910-119899






119906-119899 and 120593



120593119906max =

120583cc + 120577119904119888120583119903

1 minus 120582 (52)


120593119906(119909119888minus 1 + 120582) = minus 120577

119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus

120577119904119888120583119903

120593119906

(53)


119888 and (15) and (16)


119906


119906-119899




120593119906119909119888= 120583cc 997904rArr 119909

119888=

120583cc120593119906

(54)



119906




119906max = 120577119904119888(120583119903119888

+

120583119903)(1minus2120582) where 120583

119903119888= 120576lowast





119904119888120583119903119888120593




120593119910(119909119888minus 1 + 120582) = minus 120577

119904119888997904rArr 119909119888= (1 minus 120582) minus

120577119904119888

120593119910

(55)


119910 The 119898

119910-119899 120593119910-119899 curves can be con-


∙







120593119906min = 120577

119904119888

120583119903minus 1

1 minus 2120582 (56)


120593119906min = 120577

119904119888

120583119903minus 1

(1 + 120572) 2 minus 120582 (57)





119906and 120593



6 Applications




orm

aliz

ed st

ress

1

00 1 2 3 4 5


Normalized strain





ℓ119890= 011119891

1015840

co where 1198911015840

co =







119888= 1 120578

119888= minus08



400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)






1=



119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888



23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ



2














119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Gh

p1200012

b

Ash

h1

d1

As1

G

co co

Asb

c

c

p120001

Asc2

As2

d1




1198601199041

=119860119904119888

2+ 119860119904119887 119860

1199042= 119860119904ℎ

119901ℓ=

ℎ minus 2ℎ1

119899ℎ+ 1

1198891= 119899ℎ119901ℓ

(1)





119873119900= 119887ℎ119891cco 119872

119900= 119887ℎ2

119891cco (2)





120575V =119888V

ℎ 120575

119900=

119888119900

119887

120582 =ℎ1

ℎ 120572 =

1198891

ℎ=

119899ℎ

119899ℎ+ 1

(1 minus 2120582)

(3)

Stre

ss

Strain

ConfinedUnconfined

fcco

fco

120576co 120576cco 120576ccu


3 Constitutive Laws





119904119888(120576) = (2120576 minus 120576

2

)120573

0 le 120576 le 1 (4)

119904119888(120576) = 1 + 120578 (120576 minus 1) 1 le 120576 le 120583 (5)














119888= 0 for 120576

119888= 120576cu =

00045 (120583119888= 225)



1198961=

119891cco119891co

1198962=

120576cco120576co

(6)


1198781120576

= int

120576

0

119904119888(120576) 119889120576 119878

2120576= int

120576

0

119904119888(120576) 120576 119889120576 (7)



1198781120576

= (1205762

minus1

31205763

)

1199031

1198782120576

= (2

31205763

minus1

41205764

)

1199032

01 le 120576 le 1

(8)

where

1199031= 076 + 048 (120573 minus 05) 119903

2= 087 + 026 (120573 minus 05)

(9)


1198781120576

= 11987811

+ (120576 minus 1) +1

2120578(120576 minus 1)

2

1198782120576

= 11987821

+1

2(1205762

minus 1) +1

6120578(120576 minus 1)

2

(1 + 2120576)

1 le 120576 le 120583

(10)


0

Stre

ss

0

Strain

Compression

fsy Atan Esc

Atan Es02fsy



Tension

Atan Esh minusfsy



and 11987821


1= 1199032













120578119904ℎ

=119864119904ℎ

119864119904

120578119904119888

=119864119904119888

119864119904

120583119903=

120576119904119906

120576119904119910

(11)


119904119910 120576119904119906



120576lowast



120577119904119888

=

120576119904119910

120576cco (12)



1205961

= 2

1198601199041119891119904119910

119887ℎ119891cco 120596

2= 2

1198601199042119891119904119910

119887ℎ119891cco (13)



119899 =119873

119873119900

=119873

119887ℎ119891cco 119898 =

119872

119872119900

=119872

119887ℎ2119891cco (14)


119899 = 1198991199041

+ 1198991199042

+ 119899119888 (15)

119898 = 1198981199041

+ 1198981199042

+ 119898119888 (16)




119904119888 120578119904ℎ and 120577

119904119888 defined by (11) and





120576119904119886

= 120593 (119909119888minus ℎ1) = 120593ℎ [(119909

119888minus

1

2) + 1205821]

120576119904119887

= 120593 (119909119888minus ℎ + ℎ

1) = 120593ℎ [(119909

119888minus

1

2) minus 1205821]

(17)

where

119909119888=

119909119888

ℎ(18)


1205821=

1 minus 2120582

2 (19)


and 119891119904119887


119891119904119886

= 119864119904120576119904119910

+ 119864119904119888(120576119904119886

minus 120576119904119910) (20)

119891119904119887

= minus119864119904120576119904119910

+ 119864119904ℎ

(120576119904119887

+ 120576119904119910) (21)

G

120576sah

2

120576sb

minus120576sy

120576syh minus 2h1

2

h1xc

fsa

fsb

(a) (b) (c)

120593



le 120576119904119886

le 120576119904119910one sets 119864



le

120576119904119887

le 120576119904119910one sets 119864

119904ℎ= 119864119904


1198731199041

= 1198601199041(119891119904119886

+ 119891119904119887)

1198721199041

= 1198601199041(119891119904119886

minus 119891119904119887) (

ℎ

2minus ℎ1)

(22)


119904119886and 119891

119904119887into (22) in normal-


1198991199041

=1205961

2[

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

+ 120578119904119888)

+(1 minus120593

120577119904119888

1205821) (120578119904ℎ

minus 120578119904119888)]

(23)

1198981199041

=1205961

21205821[2 minus

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

minus 120578119904119888)

minus(1 minus120593

120577119904119888

1205821) (120578119904ℎ

+ 120578119904119888)]

(24)

where

120593 =120593ℎ

120576cco(25)



120578119904119888

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 120582) le 1

120578119904ℎ

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 1 + 120582) le 1

(26)


G

(b) (c)(a)

120576s1

120576s2

a1

a2

Δf2

Δf1

fs2

fs1

fe2

fe1

minus120576sy

120576sy

hd1

h1

h minus 2h12

h

2

xc

120593

2120576sy

120593







1and 1198862can be

expressed by

1198861=

1198891

2minus

ℎ

2+ 119909119888minus

120576119904119910

120593 119886

2=

1198891

2+

ℎ


120576119904119910

120593

(27)


1198861= (

120572

2minus

120577119904119888

120593) + (119909

119888minus

1

2)

1198862= (

120572

2minus

120577119904119888

120593) minus (119909

119888minus

1

2)

(28)







1198911198901

= 119864119904120593(1198861+

120576119904119910

120593) = 119891

119904119910

120593

120577119904119888

(1198861+

120577119904119888

120593) (29)

Δ1198911= (119864119904minus 119864119904119888) 1205931198861= 119891119904119910

120593

120577119904119888

(1 minus 120578119904119888) 1198861 (30)


1198911198902

= minus119864119904120593(1198862+

120576119904119910

120593) = minus119891

119904119910

120593

120577119904119888

(1198862+

120577119904119888

120593) (31)

Δ1198912= minus (119864

119904minus 119864119904ℎ) 1205931198862= minus119891119904119910

120593

120577119904119888

(1 minus 120578119904ℎ) 1198862 (32)


1198731199042

=21198601199042

1198891

ℎ

2[1198911198901

(1198861+

120577119904119888

120593) + 119891

1198902(1198862+

120577119904119888

120593)

minus (Δ11989111198861+ Δ11989121198862) ]

1198721199042

=21198601199042

1198891

ℎ2

2[(1198911198901

minus 1198911198902)1205722

6minus Δ11989111198861(120572

2minus

1198861

3)

+Δ11989121198862(120572

2minus

1198862

3)]

(33)


1198911198901

(1198861+

120577119904119888

120593) + 1198911198902

(1198862+

120577119904119888

120593) = 2119891

119904119910

120593

120577119904119888

(119909119888minus

1

2) 120572

1198911198901

minus 1198911198902

= 119891119904119910

120593

120577119904119888

120572

(34)


1198991199042

=1198731199042

119873119900

= 1205962

120593

120577119904119888

(119909119888minus

1

2) minus

1

2120572

times [(1 minus 120578119904119888) 1198862


119904ℎ) 1198862

2]

(35)

1198981199042

=1198721199042

119872119900

=1205962

2120572

120593

120577119904119888

[1205723


119904119888) (

120572

2minus

1198861

3) 1198862

1

minus (1 minus 120578119904ℎ) (

120572

2minus

1198862

3) 1198862

2]

(36)


h

c

c

co cob

x3

xc

Confinedconcrete

1205761

1205762

1205763

1205764

(a)

h

c

c

co cob

xc

1205761

1205762

1205763 = 1205764 = 0x3 = 0

Confinedconcrete

(b)



= 119864119904in (30) andor

119864119904ℎ




120578119904119888=1 if minus 120576

119904119910le 1205761199041

le 120576119904119910


120577119904119888

(119909119888minus

1 minus 120572

2)le1

120578119904ℎ=1 if minus 120576

119904119910le 1205761199042

le 120576119904119910


120577119904119888

(119909119888minus

1 + 120572

2)le1

(37)


119873cc = 119887int

119909119888

1199093

119891119888(119909) 119889119909 =

119887ℎ119891cco120593

int

1205762

1205763

119904119888(120576) 119889120576

119872cc = 119887int

119909119888

1199093

119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus

ℎ

2)

=119887ℎ2

119891cco (1

1205932int

1205762

1205763

119904119888(120576) 120576 119889120576minus

119909119888minus 05

120593int

1205762

1205763

119904119888(120576) 119889120576)

(38)

where

1199093=

0 for 119909119888le ℎ

119909119888minus ℎ for 119909

119888gt ℎ

(39)

and consequently

1205762= 120593119909119888

1205763=

0 for 119909119888le 1

120593 (119909119888minus 1) for 119909

119888gt 1

(40)


119899cc =119873cc119873119900

=1

120593(11987811205762

minus 11987811205763

) (41)

119898cc =119872cc119872119900

=1

1205932(11987821205762

minus 11987821205763

) minus 119899cc (119909119888 minus1

2) (42)



and 1198782120576


119899un1 =1 + 2120575

119900

1198961

1

1198962120593

(11987811205761015840

1

minus 11987811205761015840

4

)

119898un1 =1 + 2120575

119900

1198961

1

(1198962120593)2(11987821205761015840

1

minus 11987821205761015840

4

) minus 119899un1 (119909119888 minus1

2)

(43)where

1205761015840

1= 1198962120593 (119909119888+ 120575V) le 120583

119888

1205761015840

4=

0 for 119909119888le (1 + 120575V)

1198962120593 [119909119888minus (1 + 120575V)] for 119909

119888gt (1 + 120575V)

(44)


119899un2 =1

11989611198962120593

(11987811205761015840

2

minus 11987811205761015840

3

) (45)

119898un2 =1

1198961(1198962120593)2(11987821205761015840

2

minus 11987821205761015840

3

) minus 119899un2 (119909119888 minus1

2) (46)


where

1205761015840

2= 1198962120593119909119888le 120583119888

1205761015840

3=

0 for 119909119888le 1

1198962120593 (119909119888minus 1) for 119909

119888gt 1

(47)


119899119888= 119899cc + 119899un1 minus 119899un2 (48)

119898119888= 119898cc + 119898un1 minus 119898un2 (49)



1= 1198962= 1










2= 0) (26) gives

119909119888min = 120582 minus

120577119904119888

120593 119909

119888max = (1 minus 120582) +120577119904119888

120593 (50)


119909119888min =

1 minus 120572

2minus

120577119904119888

120593 119909

119888max =1 + 120572

2+

120577119904119888

120593 (51)






119888are consid-



value of 1198991199041 1198991199042 and 119899










119888gt 120583cc or 120593(119909119888 minus 1 +





∙

119904(Figure 3)


119888minus 120582) gt 120577

119904119888120583119903119888 in

which 120583119903119888

= 120576lowast








119906-119899 120593119906-119899 and119898

119910-119899






119906-119899 and 120593



120593119906max =

120583cc + 120577119904119888120583119903

1 minus 120582 (52)


120593119906(119909119888minus 1 + 120582) = minus 120577

119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus

120577119904119888120583119903

120593119906

(53)


119888 and (15) and (16)


119906


119906-119899




120593119906119909119888= 120583cc 997904rArr 119909

119888=

120583cc120593119906

(54)



119906




119906max = 120577119904119888(120583119903119888

+

120583119903)(1minus2120582) where 120583

119903119888= 120576lowast





119904119888120583119903119888120593




120593119910(119909119888minus 1 + 120582) = minus 120577

119904119888997904rArr 119909119888= (1 minus 120582) minus

120577119904119888

120593119910

(55)


119910 The 119898

119910-119899 120593119910-119899 curves can be con-


∙







120593119906min = 120577

119904119888

120583119903minus 1

1 minus 2120582 (56)


120593119906min = 120577

119904119888

120583119903minus 1

(1 + 120572) 2 minus 120582 (57)





119906and 120593



6 Applications




orm

aliz

ed st

ress

1

00 1 2 3 4 5


Normalized strain





ℓ119890= 011119891

1015840

co where 1198911015840

co =







119888= 1 120578

119888= minus08



400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)






1=



119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888



23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ



2














119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119888= 0 for 120576

119888= 120576cu =

00045 (120583119888= 225)



1198961=

119891cco119891co

1198962=

120576cco120576co

(6)


1198781120576

= int

120576

0

119904119888(120576) 119889120576 119878

2120576= int

120576

0

119904119888(120576) 120576 119889120576 (7)



1198781120576

= (1205762

minus1

31205763

)

1199031

1198782120576

= (2

31205763

minus1

41205764

)

1199032

01 le 120576 le 1

(8)

where

1199031= 076 + 048 (120573 minus 05) 119903

2= 087 + 026 (120573 minus 05)

(9)


1198781120576

= 11987811

+ (120576 minus 1) +1

2120578(120576 minus 1)

2

1198782120576

= 11987821

+1

2(1205762

minus 1) +1

6120578(120576 minus 1)

2

(1 + 2120576)

1 le 120576 le 120583

(10)


0

Stre

ss

0

Strain

Compression

fsy Atan Esc

Atan Es02fsy



Tension

Atan Esh minusfsy



and 11987821


1= 1199032













120578119904ℎ

=119864119904ℎ

119864119904

120578119904119888

=119864119904119888

119864119904

120583119903=

120576119904119906

120576119904119910

(11)


119904119910 120576119904119906



120576lowast



120577119904119888

=

120576119904119910

120576cco (12)



1205961

= 2

1198601199041119891119904119910

119887ℎ119891cco 120596

2= 2

1198601199042119891119904119910

119887ℎ119891cco (13)



119899 =119873

119873119900

=119873

119887ℎ119891cco 119898 =

119872

119872119900

=119872

119887ℎ2119891cco (14)


119899 = 1198991199041

+ 1198991199042

+ 119899119888 (15)

119898 = 1198981199041

+ 1198981199042

+ 119898119888 (16)




119904119888 120578119904ℎ and 120577

119904119888 defined by (11) and





120576119904119886

= 120593 (119909119888minus ℎ1) = 120593ℎ [(119909

119888minus

1

2) + 1205821]

120576119904119887

= 120593 (119909119888minus ℎ + ℎ

1) = 120593ℎ [(119909

119888minus

1

2) minus 1205821]

(17)

where

119909119888=

119909119888

ℎ(18)


1205821=

1 minus 2120582

2 (19)


and 119891119904119887


119891119904119886

= 119864119904120576119904119910

+ 119864119904119888(120576119904119886

minus 120576119904119910) (20)

119891119904119887

= minus119864119904120576119904119910

+ 119864119904ℎ

(120576119904119887

+ 120576119904119910) (21)

G

120576sah

2

120576sb

minus120576sy

120576syh minus 2h1

2

h1xc

fsa

fsb

(a) (b) (c)

120593



le 120576119904119886

le 120576119904119910one sets 119864



le

120576119904119887

le 120576119904119910one sets 119864

119904ℎ= 119864119904


1198731199041

= 1198601199041(119891119904119886

+ 119891119904119887)

1198721199041

= 1198601199041(119891119904119886

minus 119891119904119887) (

ℎ

2minus ℎ1)

(22)


119904119886and 119891

119904119887into (22) in normal-


1198991199041

=1205961

2[

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

+ 120578119904119888)

+(1 minus120593

120577119904119888

1205821) (120578119904ℎ

minus 120578119904119888)]

(23)

1198981199041

=1205961

21205821[2 minus

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

minus 120578119904119888)

minus(1 minus120593

120577119904119888

1205821) (120578119904ℎ

+ 120578119904119888)]

(24)

where

120593 =120593ℎ

120576cco(25)



120578119904119888

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 120582) le 1

120578119904ℎ

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 1 + 120582) le 1

(26)


G

(b) (c)(a)

120576s1

120576s2

a1

a2

Δf2

Δf1

fs2

fs1

fe2

fe1

minus120576sy

120576sy

hd1

h1

h minus 2h12

h

2

xc

120593

2120576sy

120593







1and 1198862can be

expressed by

1198861=

1198891

2minus

ℎ

2+ 119909119888minus

120576119904119910

120593 119886

2=

1198891

2+

ℎ


120576119904119910

120593

(27)


1198861= (

120572

2minus

120577119904119888

120593) + (119909

119888minus

1

2)

1198862= (

120572

2minus

120577119904119888

120593) minus (119909

119888minus

1

2)

(28)







1198911198901

= 119864119904120593(1198861+

120576119904119910

120593) = 119891

119904119910

120593

120577119904119888

(1198861+

120577119904119888

120593) (29)

Δ1198911= (119864119904minus 119864119904119888) 1205931198861= 119891119904119910

120593

120577119904119888

(1 minus 120578119904119888) 1198861 (30)


1198911198902

= minus119864119904120593(1198862+

120576119904119910

120593) = minus119891

119904119910

120593

120577119904119888

(1198862+

120577119904119888

120593) (31)

Δ1198912= minus (119864

119904minus 119864119904ℎ) 1205931198862= minus119891119904119910

120593

120577119904119888

(1 minus 120578119904ℎ) 1198862 (32)


1198731199042

=21198601199042

1198891

ℎ

2[1198911198901

(1198861+

120577119904119888

120593) + 119891

1198902(1198862+

120577119904119888

120593)

minus (Δ11989111198861+ Δ11989121198862) ]

1198721199042

=21198601199042

1198891

ℎ2

2[(1198911198901

minus 1198911198902)1205722

6minus Δ11989111198861(120572

2minus

1198861

3)

+Δ11989121198862(120572

2minus

1198862

3)]

(33)


1198911198901

(1198861+

120577119904119888

120593) + 1198911198902

(1198862+

120577119904119888

120593) = 2119891

119904119910

120593

120577119904119888

(119909119888minus

1

2) 120572

1198911198901

minus 1198911198902

= 119891119904119910

120593

120577119904119888

120572

(34)


1198991199042

=1198731199042

119873119900

= 1205962

120593

120577119904119888

(119909119888minus

1

2) minus

1

2120572

times [(1 minus 120578119904119888) 1198862


119904ℎ) 1198862

2]

(35)

1198981199042

=1198721199042

119872119900

=1205962

2120572

120593

120577119904119888

[1205723


119904119888) (

120572

2minus

1198861

3) 1198862

1

minus (1 minus 120578119904ℎ) (

120572

2minus

1198862

3) 1198862

2]

(36)


h

c

c

co cob

x3

xc

Confinedconcrete

1205761

1205762

1205763

1205764

(a)

h

c

c

co cob

xc

1205761

1205762

1205763 = 1205764 = 0x3 = 0

Confinedconcrete

(b)



= 119864119904in (30) andor

119864119904ℎ




120578119904119888=1 if minus 120576

119904119910le 1205761199041

le 120576119904119910


120577119904119888

(119909119888minus

1 minus 120572

2)le1

120578119904ℎ=1 if minus 120576

119904119910le 1205761199042

le 120576119904119910


120577119904119888

(119909119888minus

1 + 120572

2)le1

(37)


119873cc = 119887int

119909119888

1199093

119891119888(119909) 119889119909 =

119887ℎ119891cco120593

int

1205762

1205763

119904119888(120576) 119889120576

119872cc = 119887int

119909119888

1199093

119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus

ℎ

2)

=119887ℎ2

119891cco (1

1205932int

1205762

1205763

119904119888(120576) 120576 119889120576minus

119909119888minus 05

120593int

1205762

1205763

119904119888(120576) 119889120576)

(38)

where

1199093=

0 for 119909119888le ℎ

119909119888minus ℎ for 119909

119888gt ℎ

(39)

and consequently

1205762= 120593119909119888

1205763=

0 for 119909119888le 1

120593 (119909119888minus 1) for 119909

119888gt 1

(40)


119899cc =119873cc119873119900

=1

120593(11987811205762

minus 11987811205763

) (41)

119898cc =119872cc119872119900

=1

1205932(11987821205762

minus 11987821205763

) minus 119899cc (119909119888 minus1

2) (42)



and 1198782120576


119899un1 =1 + 2120575

119900

1198961

1

1198962120593

(11987811205761015840

1

minus 11987811205761015840

4

)

119898un1 =1 + 2120575

119900

1198961

1

(1198962120593)2(11987821205761015840

1

minus 11987821205761015840

4

) minus 119899un1 (119909119888 minus1

2)

(43)where

1205761015840

1= 1198962120593 (119909119888+ 120575V) le 120583

119888

1205761015840

4=

0 for 119909119888le (1 + 120575V)

1198962120593 [119909119888minus (1 + 120575V)] for 119909

119888gt (1 + 120575V)

(44)


119899un2 =1

11989611198962120593

(11987811205761015840

2

minus 11987811205761015840

3

) (45)

119898un2 =1

1198961(1198962120593)2(11987821205761015840

2

minus 11987821205761015840

3

) minus 119899un2 (119909119888 minus1

2) (46)


where

1205761015840

2= 1198962120593119909119888le 120583119888

1205761015840

3=

0 for 119909119888le 1

1198962120593 (119909119888minus 1) for 119909

119888gt 1

(47)


119899119888= 119899cc + 119899un1 minus 119899un2 (48)

119898119888= 119898cc + 119898un1 minus 119898un2 (49)



1= 1198962= 1










2= 0) (26) gives

119909119888min = 120582 minus

120577119904119888

120593 119909

119888max = (1 minus 120582) +120577119904119888

120593 (50)


119909119888min =

1 minus 120572

2minus

120577119904119888

120593 119909

119888max =1 + 120572

2+

120577119904119888

120593 (51)






119888are consid-



value of 1198991199041 1198991199042 and 119899










119888gt 120583cc or 120593(119909119888 minus 1 +





∙

119904(Figure 3)


119888minus 120582) gt 120577

119904119888120583119903119888 in

which 120583119903119888

= 120576lowast








119906-119899 120593119906-119899 and119898

119910-119899






119906-119899 and 120593



120593119906max =

120583cc + 120577119904119888120583119903

1 minus 120582 (52)


120593119906(119909119888minus 1 + 120582) = minus 120577

119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus

120577119904119888120583119903

120593119906

(53)


119888 and (15) and (16)


119906


119906-119899




120593119906119909119888= 120583cc 997904rArr 119909

119888=

120583cc120593119906

(54)



119906




119906max = 120577119904119888(120583119903119888

+

120583119903)(1minus2120582) where 120583

119903119888= 120576lowast





119904119888120583119903119888120593




120593119910(119909119888minus 1 + 120582) = minus 120577

119904119888997904rArr 119909119888= (1 minus 120582) minus

120577119904119888

120593119910

(55)


119910 The 119898

119910-119899 120593119910-119899 curves can be con-


∙







120593119906min = 120577

119904119888

120583119903minus 1

1 minus 2120582 (56)


120593119906min = 120577

119904119888

120583119903minus 1

(1 + 120572) 2 minus 120582 (57)





119906and 120593



6 Applications




orm

aliz

ed st

ress

1

00 1 2 3 4 5


Normalized strain





ℓ119890= 011119891

1015840

co where 1198911015840

co =







119888= 1 120578

119888= minus08



400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)






1=



119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888



23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ



2














119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1205961

= 2

1198601199041119891119904119910

119887ℎ119891cco 120596

2= 2

1198601199042119891119904119910

119887ℎ119891cco (13)



119899 =119873

119873119900

=119873

119887ℎ119891cco 119898 =

119872

119872119900

=119872

119887ℎ2119891cco (14)


119899 = 1198991199041

+ 1198991199042

+ 119899119888 (15)

119898 = 1198981199041

+ 1198981199042

+ 119898119888 (16)




119904119888 120578119904ℎ and 120577

119904119888 defined by (11) and





120576119904119886

= 120593 (119909119888minus ℎ1) = 120593ℎ [(119909

119888minus

1

2) + 1205821]

120576119904119887

= 120593 (119909119888minus ℎ + ℎ

1) = 120593ℎ [(119909

119888minus

1

2) minus 1205821]

(17)

where

119909119888=

119909119888

ℎ(18)


1205821=

1 minus 2120582

2 (19)


and 119891119904119887


119891119904119886

= 119864119904120576119904119910

+ 119864119904119888(120576119904119886

minus 120576119904119910) (20)

119891119904119887

= minus119864119904120576119904119910

+ 119864119904ℎ

(120576119904119887

+ 120576119904119910) (21)

G

120576sah

2

120576sb

minus120576sy

120576syh minus 2h1

2

h1xc

fsa

fsb

(a) (b) (c)

120593



le 120576119904119886

le 120576119904119910one sets 119864



le

120576119904119887

le 120576119904119910one sets 119864

119904ℎ= 119864119904


1198731199041

= 1198601199041(119891119904119886

+ 119891119904119887)

1198721199041

= 1198601199041(119891119904119886

minus 119891119904119887) (

ℎ

2minus ℎ1)

(22)


119904119886and 119891

119904119887into (22) in normal-


1198991199041

=1205961

2[

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

+ 120578119904119888)

+(1 minus120593

120577119904119888

1205821) (120578119904ℎ

minus 120578119904119888)]

(23)

1198981199041

=1205961

21205821[2 minus

120593

120577119904119888

(119909119888minus

1

2) (120578119904ℎ

minus 120578119904119888)

minus(1 minus120593

120577119904119888

1205821) (120578119904ℎ

+ 120578119904119888)]

(24)

where

120593 =120593ℎ

120576cco(25)



120578119904119888

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 120582) le 1

120578119904ℎ

= 1 if minus 1 le120593

120577119904119888

(119909119888minus 1 + 120582) le 1

(26)


G

(b) (c)(a)

120576s1

120576s2

a1

a2

Δf2

Δf1

fs2

fs1

fe2

fe1

minus120576sy

120576sy

hd1

h1

h minus 2h12

h

2

xc

120593

2120576sy

120593







1and 1198862can be

expressed by

1198861=

1198891

2minus

ℎ

2+ 119909119888minus

120576119904119910

120593 119886

2=

1198891

2+

ℎ


120576119904119910

120593

(27)


1198861= (

120572

2minus

120577119904119888

120593) + (119909

119888minus

1

2)

1198862= (

120572

2minus

120577119904119888

120593) minus (119909

119888minus

1

2)

(28)







1198911198901

= 119864119904120593(1198861+

120576119904119910

120593) = 119891

119904119910

120593

120577119904119888

(1198861+

120577119904119888

120593) (29)

Δ1198911= (119864119904minus 119864119904119888) 1205931198861= 119891119904119910

120593

120577119904119888

(1 minus 120578119904119888) 1198861 (30)


1198911198902

= minus119864119904120593(1198862+

120576119904119910

120593) = minus119891

119904119910

120593

120577119904119888

(1198862+

120577119904119888

120593) (31)

Δ1198912= minus (119864

119904minus 119864119904ℎ) 1205931198862= minus119891119904119910

120593

120577119904119888

(1 minus 120578119904ℎ) 1198862 (32)


1198731199042

=21198601199042

1198891

ℎ

2[1198911198901

(1198861+

120577119904119888

120593) + 119891

1198902(1198862+

120577119904119888

120593)

minus (Δ11989111198861+ Δ11989121198862) ]

1198721199042

=21198601199042

1198891

ℎ2

2[(1198911198901

minus 1198911198902)1205722

6minus Δ11989111198861(120572

2minus

1198861

3)

+Δ11989121198862(120572

2minus

1198862

3)]

(33)


1198911198901

(1198861+

120577119904119888

120593) + 1198911198902

(1198862+

120577119904119888

120593) = 2119891

119904119910

120593

120577119904119888

(119909119888minus

1

2) 120572

1198911198901

minus 1198911198902

= 119891119904119910

120593

120577119904119888

120572

(34)


1198991199042

=1198731199042

119873119900

= 1205962

120593

120577119904119888

(119909119888minus

1

2) minus

1

2120572

times [(1 minus 120578119904119888) 1198862


119904ℎ) 1198862

2]

(35)

1198981199042

=1198721199042

119872119900

=1205962

2120572

120593

120577119904119888

[1205723


119904119888) (

120572

2minus

1198861

3) 1198862

1

minus (1 minus 120578119904ℎ) (

120572

2minus

1198862

3) 1198862

2]

(36)


h

c

c

co cob

x3

xc

Confinedconcrete

1205761

1205762

1205763

1205764

(a)

h

c

c

co cob

xc

1205761

1205762

1205763 = 1205764 = 0x3 = 0

Confinedconcrete

(b)



= 119864119904in (30) andor

119864119904ℎ




120578119904119888=1 if minus 120576

119904119910le 1205761199041

le 120576119904119910


120577119904119888

(119909119888minus

1 minus 120572

2)le1

120578119904ℎ=1 if minus 120576

119904119910le 1205761199042

le 120576119904119910


120577119904119888

(119909119888minus

1 + 120572

2)le1

(37)


119873cc = 119887int

119909119888

1199093

119891119888(119909) 119889119909 =

119887ℎ119891cco120593

int

1205762

1205763

119904119888(120576) 119889120576

119872cc = 119887int

119909119888

1199093

119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus

ℎ

2)

=119887ℎ2

119891cco (1

1205932int

1205762

1205763

119904119888(120576) 120576 119889120576minus

119909119888minus 05

120593int

1205762

1205763

119904119888(120576) 119889120576)

(38)

where

1199093=

0 for 119909119888le ℎ

119909119888minus ℎ for 119909

119888gt ℎ

(39)

and consequently

1205762= 120593119909119888

1205763=

0 for 119909119888le 1

120593 (119909119888minus 1) for 119909

119888gt 1

(40)


119899cc =119873cc119873119900

=1

120593(11987811205762

minus 11987811205763

) (41)

119898cc =119872cc119872119900

=1

1205932(11987821205762

minus 11987821205763

) minus 119899cc (119909119888 minus1

2) (42)



and 1198782120576


119899un1 =1 + 2120575

119900

1198961

1

1198962120593

(11987811205761015840

1

minus 11987811205761015840

4

)

119898un1 =1 + 2120575

119900

1198961

1

(1198962120593)2(11987821205761015840

1

minus 11987821205761015840

4

) minus 119899un1 (119909119888 minus1

2)

(43)where

1205761015840

1= 1198962120593 (119909119888+ 120575V) le 120583

119888

1205761015840

4=

0 for 119909119888le (1 + 120575V)

1198962120593 [119909119888minus (1 + 120575V)] for 119909

119888gt (1 + 120575V)

(44)


119899un2 =1

11989611198962120593

(11987811205761015840

2

minus 11987811205761015840

3

) (45)

119898un2 =1

1198961(1198962120593)2(11987821205761015840

2

minus 11987821205761015840

3

) minus 119899un2 (119909119888 minus1

2) (46)


where

1205761015840

2= 1198962120593119909119888le 120583119888

1205761015840

3=

0 for 119909119888le 1

1198962120593 (119909119888minus 1) for 119909

119888gt 1

(47)


119899119888= 119899cc + 119899un1 minus 119899un2 (48)

119898119888= 119898cc + 119898un1 minus 119898un2 (49)



1= 1198962= 1










2= 0) (26) gives

119909119888min = 120582 minus

120577119904119888

120593 119909

119888max = (1 minus 120582) +120577119904119888

120593 (50)


119909119888min =

1 minus 120572

2minus

120577119904119888

120593 119909

119888max =1 + 120572

2+

120577119904119888

120593 (51)






119888are consid-



value of 1198991199041 1198991199042 and 119899










119888gt 120583cc or 120593(119909119888 minus 1 +





∙

119904(Figure 3)


119888minus 120582) gt 120577

119904119888120583119903119888 in

which 120583119903119888

= 120576lowast








119906-119899 120593119906-119899 and119898

119910-119899






119906-119899 and 120593



120593119906max =

120583cc + 120577119904119888120583119903

1 minus 120582 (52)


120593119906(119909119888minus 1 + 120582) = minus 120577

119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus

120577119904119888120583119903

120593119906

(53)


119888 and (15) and (16)


119906


119906-119899




120593119906119909119888= 120583cc 997904rArr 119909

119888=

120583cc120593119906

(54)



119906




119906max = 120577119904119888(120583119903119888

+

120583119903)(1minus2120582) where 120583

119903119888= 120576lowast





119904119888120583119903119888120593




120593119910(119909119888minus 1 + 120582) = minus 120577

119904119888997904rArr 119909119888= (1 minus 120582) minus

120577119904119888

120593119910

(55)


119910 The 119898

119910-119899 120593119910-119899 curves can be con-


∙







120593119906min = 120577

119904119888

120583119903minus 1

1 minus 2120582 (56)


120593119906min = 120577

119904119888

120583119903minus 1

(1 + 120572) 2 minus 120582 (57)





119906and 120593



6 Applications




orm

aliz

ed st

ress

1

00 1 2 3 4 5


Normalized strain





ℓ119890= 011119891

1015840

co where 1198911015840

co =







119888= 1 120578

119888= minus08



400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)






1=



119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888



23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ



2














119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










G

(b) (c)(a)

120576s1

120576s2

a1

a2

Δf2

Δf1

fs2

fs1

fe2

fe1

minus120576sy

120576sy

hd1

h1

h minus 2h12

h

2

xc

120593

2120576sy

120593







1and 1198862can be

expressed by

1198861=

1198891

2minus

ℎ

2+ 119909119888minus

120576119904119910

120593 119886

2=

1198891

2+

ℎ


120576119904119910

120593

(27)


1198861= (

120572

2minus

120577119904119888

120593) + (119909

119888minus

1

2)

1198862= (

120572

2minus

120577119904119888

120593) minus (119909

119888minus

1

2)

(28)







1198911198901

= 119864119904120593(1198861+

120576119904119910

120593) = 119891

119904119910

120593

120577119904119888

(1198861+

120577119904119888

120593) (29)

Δ1198911= (119864119904minus 119864119904119888) 1205931198861= 119891119904119910

120593

120577119904119888

(1 minus 120578119904119888) 1198861 (30)


1198911198902

= minus119864119904120593(1198862+

120576119904119910

120593) = minus119891

119904119910

120593

120577119904119888

(1198862+

120577119904119888

120593) (31)

Δ1198912= minus (119864

119904minus 119864119904ℎ) 1205931198862= minus119891119904119910

120593

120577119904119888

(1 minus 120578119904ℎ) 1198862 (32)


1198731199042

=21198601199042

1198891

ℎ

2[1198911198901

(1198861+

120577119904119888

120593) + 119891

1198902(1198862+

120577119904119888

120593)

minus (Δ11989111198861+ Δ11989121198862) ]

1198721199042

=21198601199042

1198891

ℎ2

2[(1198911198901

minus 1198911198902)1205722

6minus Δ11989111198861(120572

2minus

1198861

3)

+Δ11989121198862(120572

2minus

1198862

3)]

(33)


1198911198901

(1198861+

120577119904119888

120593) + 1198911198902

(1198862+

120577119904119888

120593) = 2119891

119904119910

120593

120577119904119888

(119909119888minus

1

2) 120572

1198911198901

minus 1198911198902

= 119891119904119910

120593

120577119904119888

120572

(34)


1198991199042

=1198731199042

119873119900

= 1205962

120593

120577119904119888

(119909119888minus

1

2) minus

1

2120572

times [(1 minus 120578119904119888) 1198862


119904ℎ) 1198862

2]

(35)

1198981199042

=1198721199042

119872119900

=1205962

2120572

120593

120577119904119888

[1205723


119904119888) (

120572

2minus

1198861

3) 1198862

1

minus (1 minus 120578119904ℎ) (

120572

2minus

1198862

3) 1198862

2]

(36)


h

c

c

co cob

x3

xc

Confinedconcrete

1205761

1205762

1205763

1205764

(a)

h

c

c

co cob

xc

1205761

1205762

1205763 = 1205764 = 0x3 = 0

Confinedconcrete

(b)



= 119864119904in (30) andor

119864119904ℎ




120578119904119888=1 if minus 120576

119904119910le 1205761199041

le 120576119904119910


120577119904119888

(119909119888minus

1 minus 120572

2)le1

120578119904ℎ=1 if minus 120576

119904119910le 1205761199042

le 120576119904119910


120577119904119888

(119909119888minus

1 + 120572

2)le1

(37)


119873cc = 119887int

119909119888

1199093

119891119888(119909) 119889119909 =

119887ℎ119891cco120593

int

1205762

1205763

119904119888(120576) 119889120576

119872cc = 119887int

119909119888

1199093

119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus

ℎ

2)

=119887ℎ2

119891cco (1

1205932int

1205762

1205763

119904119888(120576) 120576 119889120576minus

119909119888minus 05

120593int

1205762

1205763

119904119888(120576) 119889120576)

(38)

where

1199093=

0 for 119909119888le ℎ

119909119888minus ℎ for 119909

119888gt ℎ

(39)

and consequently

1205762= 120593119909119888

1205763=

0 for 119909119888le 1

120593 (119909119888minus 1) for 119909

119888gt 1

(40)


119899cc =119873cc119873119900

=1

120593(11987811205762

minus 11987811205763

) (41)

119898cc =119872cc119872119900

=1

1205932(11987821205762

minus 11987821205763

) minus 119899cc (119909119888 minus1

2) (42)



and 1198782120576


119899un1 =1 + 2120575

119900

1198961

1

1198962120593

(11987811205761015840

1

minus 11987811205761015840

4

)

119898un1 =1 + 2120575

119900

1198961

1

(1198962120593)2(11987821205761015840

1

minus 11987821205761015840

4

) minus 119899un1 (119909119888 minus1

2)

(43)where

1205761015840

1= 1198962120593 (119909119888+ 120575V) le 120583

119888

1205761015840

4=

0 for 119909119888le (1 + 120575V)

1198962120593 [119909119888minus (1 + 120575V)] for 119909

119888gt (1 + 120575V)

(44)


119899un2 =1

11989611198962120593

(11987811205761015840

2

minus 11987811205761015840

3

) (45)

119898un2 =1

1198961(1198962120593)2(11987821205761015840

2

minus 11987821205761015840

3

) minus 119899un2 (119909119888 minus1

2) (46)


where

1205761015840

2= 1198962120593119909119888le 120583119888

1205761015840

3=

0 for 119909119888le 1

1198962120593 (119909119888minus 1) for 119909

119888gt 1

(47)


119899119888= 119899cc + 119899un1 minus 119899un2 (48)

119898119888= 119898cc + 119898un1 minus 119898un2 (49)



1= 1198962= 1










2= 0) (26) gives

119909119888min = 120582 minus

120577119904119888

120593 119909

119888max = (1 minus 120582) +120577119904119888

120593 (50)


119909119888min =

1 minus 120572

2minus

120577119904119888

120593 119909

119888max =1 + 120572

2+

120577119904119888

120593 (51)






119888are consid-



value of 1198991199041 1198991199042 and 119899










119888gt 120583cc or 120593(119909119888 minus 1 +





∙

119904(Figure 3)


119888minus 120582) gt 120577

119904119888120583119903119888 in

which 120583119903119888

= 120576lowast








119906-119899 120593119906-119899 and119898

119910-119899






119906-119899 and 120593



120593119906max =

120583cc + 120577119904119888120583119903

1 minus 120582 (52)


120593119906(119909119888minus 1 + 120582) = minus 120577

119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus

120577119904119888120583119903

120593119906

(53)


119888 and (15) and (16)


119906


119906-119899




120593119906119909119888= 120583cc 997904rArr 119909

119888=

120583cc120593119906

(54)



119906




119906max = 120577119904119888(120583119903119888

+

120583119903)(1minus2120582) where 120583

119903119888= 120576lowast





119904119888120583119903119888120593




120593119910(119909119888minus 1 + 120582) = minus 120577

119904119888997904rArr 119909119888= (1 minus 120582) minus

120577119904119888

120593119910

(55)


119910 The 119898

119910-119899 120593119910-119899 curves can be con-


∙







120593119906min = 120577

119904119888

120583119903minus 1

1 minus 2120582 (56)


120593119906min = 120577

119904119888

120583119903minus 1

(1 + 120572) 2 minus 120582 (57)





119906and 120593



6 Applications




orm

aliz

ed st

ress

1

00 1 2 3 4 5


Normalized strain





ℓ119890= 011119891

1015840

co where 1198911015840

co =







119888= 1 120578

119888= minus08



400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)






1=



119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888



23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ



2














119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










h

c

c

co cob

x3

xc

Confinedconcrete

1205761

1205762

1205763

1205764

(a)

h

c

c

co cob

xc

1205761

1205762

1205763 = 1205764 = 0x3 = 0

Confinedconcrete

(b)



= 119864119904in (30) andor

119864119904ℎ




120578119904119888=1 if minus 120576

119904119910le 1205761199041

le 120576119904119910


120577119904119888

(119909119888minus

1 minus 120572

2)le1

120578119904ℎ=1 if minus 120576

119904119910le 1205761199042

le 120576119904119910


120577119904119888

(119909119888minus

1 + 120572

2)le1

(37)


119873cc = 119887int

119909119888

1199093

119891119888(119909) 119889119909 =

119887ℎ119891cco120593

int

1205762

1205763

119904119888(120576) 119889120576

119872cc = 119887int

119909119888

1199093

119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus

ℎ

2)

=119887ℎ2

119891cco (1

1205932int

1205762

1205763

119904119888(120576) 120576 119889120576minus

119909119888minus 05

120593int

1205762

1205763

119904119888(120576) 119889120576)

(38)

where

1199093=

0 for 119909119888le ℎ

119909119888minus ℎ for 119909

119888gt ℎ

(39)

and consequently

1205762= 120593119909119888

1205763=

0 for 119909119888le 1

120593 (119909119888minus 1) for 119909

119888gt 1

(40)


119899cc =119873cc119873119900

=1

120593(11987811205762

minus 11987811205763

) (41)

119898cc =119872cc119872119900

=1

1205932(11987821205762

minus 11987821205763

) minus 119899cc (119909119888 minus1

2) (42)



and 1198782120576


119899un1 =1 + 2120575

119900

1198961

1

1198962120593

(11987811205761015840

1

minus 11987811205761015840

4

)

119898un1 =1 + 2120575

119900

1198961

1

(1198962120593)2(11987821205761015840

1

minus 11987821205761015840

4

) minus 119899un1 (119909119888 minus1

2)

(43)where

1205761015840

1= 1198962120593 (119909119888+ 120575V) le 120583

119888

1205761015840

4=

0 for 119909119888le (1 + 120575V)

1198962120593 [119909119888minus (1 + 120575V)] for 119909

119888gt (1 + 120575V)

(44)


119899un2 =1

11989611198962120593

(11987811205761015840

2

minus 11987811205761015840

3

) (45)

119898un2 =1

1198961(1198962120593)2(11987821205761015840

2

minus 11987821205761015840

3

) minus 119899un2 (119909119888 minus1

2) (46)


where

1205761015840

2= 1198962120593119909119888le 120583119888

1205761015840

3=

0 for 119909119888le 1

1198962120593 (119909119888minus 1) for 119909

119888gt 1

(47)


119899119888= 119899cc + 119899un1 minus 119899un2 (48)

119898119888= 119898cc + 119898un1 minus 119898un2 (49)



1= 1198962= 1










2= 0) (26) gives

119909119888min = 120582 minus

120577119904119888

120593 119909

119888max = (1 minus 120582) +120577119904119888

120593 (50)


119909119888min =

1 minus 120572

2minus

120577119904119888

120593 119909

119888max =1 + 120572

2+

120577119904119888

120593 (51)






119888are consid-



value of 1198991199041 1198991199042 and 119899










119888gt 120583cc or 120593(119909119888 minus 1 +





∙

119904(Figure 3)


119888minus 120582) gt 120577

119904119888120583119903119888 in

which 120583119903119888

= 120576lowast








119906-119899 120593119906-119899 and119898

119910-119899






119906-119899 and 120593



120593119906max =

120583cc + 120577119904119888120583119903

1 minus 120582 (52)


120593119906(119909119888minus 1 + 120582) = minus 120577

119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus

120577119904119888120583119903

120593119906

(53)


119888 and (15) and (16)


119906


119906-119899




120593119906119909119888= 120583cc 997904rArr 119909

119888=

120583cc120593119906

(54)



119906




119906max = 120577119904119888(120583119903119888

+

120583119903)(1minus2120582) where 120583

119903119888= 120576lowast





119904119888120583119903119888120593




120593119910(119909119888minus 1 + 120582) = minus 120577

119904119888997904rArr 119909119888= (1 minus 120582) minus

120577119904119888

120593119910

(55)


119910 The 119898

119910-119899 120593119910-119899 curves can be con-


∙







120593119906min = 120577

119904119888

120583119903minus 1

1 minus 2120582 (56)


120593119906min = 120577

119904119888

120583119903minus 1

(1 + 120572) 2 minus 120582 (57)





119906and 120593



6 Applications




orm

aliz

ed st

ress

1

00 1 2 3 4 5


Normalized strain





ℓ119890= 011119891

1015840

co where 1198911015840

co =







119888= 1 120578

119888= minus08



400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)






1=



119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888



23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ



2














119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where

1205761015840

2= 1198962120593119909119888le 120583119888

1205761015840

3=

0 for 119909119888le 1

1198962120593 (119909119888minus 1) for 119909

119888gt 1

(47)


119899119888= 119899cc + 119899un1 minus 119899un2 (48)

119898119888= 119898cc + 119898un1 minus 119898un2 (49)



1= 1198962= 1










2= 0) (26) gives

119909119888min = 120582 minus

120577119904119888

120593 119909

119888max = (1 minus 120582) +120577119904119888

120593 (50)


119909119888min =

1 minus 120572

2minus

120577119904119888

120593 119909

119888max =1 + 120572

2+

120577119904119888

120593 (51)






119888are consid-



value of 1198991199041 1198991199042 and 119899










119888gt 120583cc or 120593(119909119888 minus 1 +





∙

119904(Figure 3)


119888minus 120582) gt 120577

119904119888120583119903119888 in

which 120583119903119888

= 120576lowast








119906-119899 120593119906-119899 and119898

119910-119899






119906-119899 and 120593



120593119906max =

120583cc + 120577119904119888120583119903

1 minus 120582 (52)


120593119906(119909119888minus 1 + 120582) = minus 120577

119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus

120577119904119888120583119903

120593119906

(53)


119888 and (15) and (16)


119906


119906-119899




120593119906119909119888= 120583cc 997904rArr 119909

119888=

120583cc120593119906

(54)



119906




119906max = 120577119904119888(120583119903119888

+

120583119903)(1minus2120582) where 120583

119903119888= 120576lowast





119904119888120583119903119888120593




120593119910(119909119888minus 1 + 120582) = minus 120577

119904119888997904rArr 119909119888= (1 minus 120582) minus

120577119904119888

120593119910

(55)


119910 The 119898

119910-119899 120593119910-119899 curves can be con-


∙







120593119906min = 120577

119904119888

120583119903minus 1

1 minus 2120582 (56)


120593119906min = 120577

119904119888

120583119903minus 1

(1 + 120572) 2 minus 120582 (57)





119906and 120593



6 Applications




orm

aliz

ed st

ress

1

00 1 2 3 4 5


Normalized strain





ℓ119890= 011119891

1015840

co where 1198911015840

co =







119888= 1 120578

119888= minus08



400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)






1=



119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888



23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ



2














119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119906-119899 120593119906-119899 and119898

119910-119899






119906-119899 and 120593



120593119906max =

120583cc + 120577119904119888120583119903

1 minus 120582 (52)


120593119906(119909119888minus 1 + 120582) = minus 120577

119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus

120577119904119888120583119903

120593119906

(53)


119888 and (15) and (16)


119906


119906-119899




120593119906119909119888= 120583cc 997904rArr 119909

119888=

120583cc120593119906

(54)



119906




119906max = 120577119904119888(120583119903119888

+

120583119903)(1minus2120582) where 120583

119903119888= 120576lowast





119904119888120583119903119888120593




120593119910(119909119888minus 1 + 120582) = minus 120577

119904119888997904rArr 119909119888= (1 minus 120582) minus

120577119904119888

120593119910

(55)


119910 The 119898

119910-119899 120593119910-119899 curves can be con-


∙







120593119906min = 120577

119904119888

120583119903minus 1

1 minus 2120582 (56)


120593119906min = 120577

119904119888

120583119903minus 1

(1 + 120572) 2 minus 120582 (57)





119906and 120593



6 Applications




orm

aliz

ed st

ress

1

00 1 2 3 4 5


Normalized strain





ℓ119890= 011119891

1015840

co where 1198911015840

co =







119888= 1 120578

119888= minus08



400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)






1=



119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888



23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ



2














119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










orm

aliz

ed st

ress

1

00 1 2 3 4 5


Normalized strain





ℓ119890= 011119891

1015840

co where 1198911015840

co =







119888= 1 120578

119888= minus08



400

300

200

100

00 001 002 003 004 005

Mom

ent (

kNm

)

Curvature (1m)






1=



119904= 210GPa 120576

119904119910= 119891119904119910119864119904=

0002) and 119864119904ℎ

= 119864119904119888



23354 = 0065 120582 = 8354 = 00226 120572 = 06365 120577119904119888

=

000200073 = 0274 1198961

= 4630 = 153 1198962

= 00073

0002 = 365 1205961= 0117 120596

2= 12059612 and 120578

119904119888= 120578119904ℎ



2














119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















119904119910=


= 0013119864119904






119900= 120575V = 0087 120582 = 00492 120572 = 0601 120577

119904119888= 04724

1198961= 1347 119896

2= 2735 120596

1= 0280 120596

2= 12059612 and 120578

119904119888=

120578119904ℎ





119904119906= 0066 (120583

119903= 27) The



100

80

60

40

20

00 1 2 3 4

Mom

ent (

kNm

)








7 Conclusions






100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100

80

60

40

20

00 300 600 900 1200 1500

Mom

ent (

kNm

)

Axial load (kN)

Mu

My

My without cover

(a)

8

6

4

2

00 300 600 900 1200 1500

Axial load (kN)

120593u120593y

Curv

atur

e(1

mm

times10

minus4)

(b)




Acknowledgment


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









Research Article An Analytical Step-by-Step Procedure...

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