Nonlinear Analysis for the Crack Control of SMA Smart

Research ArticleNonlinear Analysis for the Crack Control of SMA SmartConcrete Beam Based on a Bidirectional B-Spline QR Method

Yan Li1 Tian-jing Mo1 Shuang-bei Li 12 Qing-guo Liang3 Jian-ping Li4 and Rong Qin1

1College of Civil Engineering and Architecture Guangxi University Nanning 530004 China2The Key Laboratory of Disaster Prevention and Structural Safety of the Education Ministry Guangxi UniversityNanning 530004 China3Electric Power Research Institute of Guangxi Power Grid Co Ltd Nanning 530023 China4Department of Geotechnical Engineering Tongji University Shanghai 200092 China

Correspondence should be addressed to Shuang-bei Li lsbwh90163com

Received 1 August 2017 Revised 10 November 2017 Accepted 16 November 2017 Published 11 January 2018

Academic Editor Fabrizio Greco

Copyright copy 2018 Yan Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A bidirectional B-splineQRmethod (BB-sQRM) for the study on the crack control of the reinforced concrete (RC) beam embeddedwith shape memory alloy (SMA) wires is presented In the proposed method the discretization is performed with a set of splinenodes in two directions of the plane model and structural displacement fields are constructed by the linear combination of theproducts of cubic B-spline interpolation functions To derive the elastoplastic stiffness equation of the RC beam an explicit form isutilized to express the elastoplastic constitutive law of concrete materials The proposed model is compared with the ANSYS modelin several numerical examples The results not only show that the solutions given by the BB-sQRM are very close to those givenby the finite element method (FEM) but also prove the high efficiency and low computational cost of the BB-sQRM Meanwhilethe five parameters such as depth-span ratio thickness of concrete cover reinforcement ratio prestrain and eccentricity of SMAwires are investigated to learn their effects on the crack control The results show that depth-span ratio of RC beams and prestrainand eccentricity of SMA wires have a significant influence on the control performance of beam cracks

1 Introduction

Cracking is one of the main nonlinear characteristics of rein-forced concrete (RC) structures Once tiny cracks are createdthey may expand and lead to accelerated corrosion of steelbars which reduce structural reliability and durability Andthe large cracks that exceed the limit may cause the structuralfailure [1] Therefore the durability and carrying capacityof structures can be effectively improved by controlling thedevelopment of cracks With the continuous progress ofmaterial science smart concrete provides a new and effectivemethod to solve this problem From the 1980s till now manyspecial functions of smart concrete are derived includingself-test self-adjustment self-cleaning and self-healing [2ndash5] The concrete added with new intelligent materials caneffectively control crack expansion and even cause cracks toclose so as to improve the service life of the concrete

Shape Memory Alloys (SMAs) a unique smart materialhas been researched and appliedmore andmore in intelligentstructures and essential functional components [6ndash9] Thereare two major characteristics simultaneously with the SMAsbecause of the reversibility of its phase transition One is theshape memory effect (SME) In the martensite state applyingexternal force to produce deformation on the SMAs thedeformation will be recovered gradually when the SMAs areheated up to the transformation temperature of the austenite[10] Depending on the feature above the SMAs can beused to produce SMA smart concrete which can improvethe mechanical behaviors of concrete members repair andcontrol the deformation of the structure and extend itsservice life

At present research of the SMA smart concrete is primar-ily concentrated on experiments Numerous results show thatconcrete structures embeddedwith SMAwires are effective in

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6092479 16 pageshttpsdoiorg10115520186092479

2 Mathematical Problems in Engineering

deformation and crack control Choirsquos team [11] buried SMAwires after cold drawing in the tensile zone of the reinforcedmortar beam and carried out a three-point bend test And Liet al [12] buried SMA wires in the reinforced concrete beamBoth of their results showed that the cracks could be closedwith a great restoring force generated by heating the built-in SMA wires Considering the possible complex situationsin practical applications some scholars have studied therelated parameters that affect the driving effect of SMA wiresand the controlling effect of structural deformation [13 14]Many test results show that the quantity diameter prestrainexcitationmode and volume of the SMAwires have a certaininfluence on the deformation recovery of SMA concretebeam In addition to the built-in method SMA wires alsocan be arranged outside the structure which has the effectof controlling deformation as well [15ndash17] However due tothe complexity of mechanical behaviors of RC structuresthere is less numerical analysis than experiment of theSMA concrete structure Most of them are modeled andanalyzed by software of the finite element method (FEM)such as ANSYS ABAQUS and SeismoStruct Based on thesuperelastic constitutive law of the SMAs Abdulridha etal [18] established a finite element (FE) model to simulatethe bond interface between SMA wires and concrete with abond slip element and then obtained the load-displacementrelationship of the beam Khaloorsquos team [19] used the ANSYSto establish a FE model of a cantilever RC beam embeddedwith SMA wires The analysis results showed that with theincrease of the ratio of SMA reinforcement the shear forceand displacement hysteresis curve area of concrete beamsdecrease the residual deformation of beams and the cross-sectional stiffness decrease aswell Alam et al [20] establishedthe FE model of SMA beam-column joints and analyzed itsnonlinear behaviors by the SeismoStruct software Chen andAndrawes [21] carried on the static analysis for the concretecolumn with SMA wires lateral restraint by the ABQUSsoftware When using the FEM to analyze the effect of theSMAs on the structure it is crucial for the treatment of theSMA recovery forceThere are somemethods mainly includ-ing pseudo temperature load method [22] negative thermalexpansion coefficient method [23] equivalent eccentric forcemethod [24] and equivalent reinforcement stress method[25 26]

There is no doubt that the FEM is the most widely usednumerical method so far in which the whole structure isdiscretized with meshes Due to the dependence of meshgeneration there are some drawbacks such as complicatedsolution extensive calculation and being time consuming forthe FEM inevitably [27] while the meshless method canovercome these limitations when analyzing regular struc-tures In the meshless method a set of points is used todiscretize the solution region and construct approximationfunctions which can eliminate meshes completely or partlyand need no initiation and repetition of meshes Thus thecalculation accuracy of the meshless method is ensured andthe calculation difficulty is reduced In the literature [28ndash30]the mechanical analysis of various structures is performedwith different meshless methods and a series of effectiveconclusions are obtained Thereinto the spline function

method which is semianalytical and semidiscrete utilizesspline functions to construct displacement interpolationfunctions and has a widespread applicability and feasibility indifferent structural analysis problems Liu et al [31] used B-spline finite pointmethod (B-sFPM) to analyze the free trans-verse vibration of axially functionally graded tapered Euler-Bernoulli beams Compared with the FEM the B-sFPM hasthe advantages of high efficiency and low computationalcomplexity Based on the B-sFPM Li et al [32] proposedthe bidirectional B-spline finite point method (BB-sFPM)namely the spline meshless method for the parameter iden-tification of piezoelectric laminated plates This methoddiscretized the structure along two directions with splinenodes which is more accurate and effective in calculation

Combining the advantages of the sFPM and the FEMProfessor Qin [33] put forward the unidirectional B-splineQR method (B-sQRM) named after him which has beensuccessfully implemented in the dynamic static and stabilityanalysis of various structures The B-sQRM is different fromthe sFPM In theB-sQRM one of the directions of the integralstructure is discretized by uniformly scattered spline nodesand the other is discretized by displacement shape functionswhich can reflect the deformation regularity of the structureAnd the structural displacement function is constituted by alinear combination of the product of displacement functionsand cubic B-spline functions which are compact and ofhigh order Moreover by means of element interpolationfunctions element potential energy functions and elementstiffness matrices of the FEM the displacement of elementnodes is expressed by the integral displacement of the B-sQRMThen the stiffness equation of the entire structure canbe established by the total potential energy function andthe principle of minimum potential energy Although the B-spline QRmethod depends on the discrete mesh and elementaccuracy of the FEM the number of unknowns of the B-sQRM is only related to the number of spline nodes and theseries of orthogonal polynomials also it has nothing to dowith the number of elements and the total number of nodedisplacements Therefore the B-sQRM is of simplicity andconvenience in calculation Based on the above methods thebidirectional B-spline QR method (BB-sQRM) is proposedin this paper The principle of the BB-sQRM is the sameas the B-sQRM but the discretization is performed withspline nodes in both two directions of the structure It hasthe characteristics of meshless method and can improvethe calculation accuracy while the calculation efficiency isguaranteed

In view of fact that the numerical analysis for nonlinearbehaviors of SMA smart concrete beams is less and theemployment of the FEM in modeling and analysis is moretherefore in order to enrich the theoretical system of SMAconcrete structures and to increase the diversity of methodsthe BB-sQRM is adopted in this paper to perform a nonlinearanalysis of the SMA smart concrete beam which ensuresthe accuracy of the results while the computational scale isreduced and the efficiency of numerical analysis improvedBased on the classical incremental elastoplastic theory ofthe RC the stress-strain relations of elements of concreteand concrete with steel bars under different stress conditions

Mathematical Problems in Engineering 3

0

1

2

1 2 3 4 5 6

M

N y

x

middot middot middot

Figure 1 Bidirectional spline discretization of plane structure

are given By means of the element stiffness matrix of theFEM the recovery force of SMA wires is equivalent to theeccentric load of spline node by the QR transformation andthe elastoplastic stiffness equation of the RC beam is deducedThen based on the iteration method of incremental initialstress the calculation format for nonlinear analysis of the BB-sQRM is establishedThenonlinear behavior such as crackingof the SMA concrete beam can be analyzed by controllingthe temperature of SMA wires Meanwhile several param-eters such as depth-span ratio thickness of concrete coverreinforcement ratio prestrain and eccentricity of SMAwireswill be discussed to learn their influence on the capability ofcrack control which can provide supplementary and furtheranalytical means for test methods

2 Principles of the BB-sQRM

In the BB-sQRM the two directions of the integral structureare discretized by uniformly scattered spline nodes andthe displacement field is constructed by the cubic B-splinefunctions corresponding to the boundary conditions Themost critical step in the BB-sQRM is the QR transformationpresented in Section 22 Based on element stiffness matricesthe load vector and transformed displacement vectors ofelement nodes the conversion strategy of total potentialenergy functions of elements are presented Thus the totalpotential energy function of the entire structure can beobtained directly by adding potential energy functions ofelements after the QR transformation Also the integrationof the total stiffness matrix and the total load vector can beavailable from the sum of element items in the BB-sQRMwhich need not expand before the superposition as the FEMItmeans that the solution process is greatly simplified and thecalculation ismore efficient and accurateMoreover the com-puter program would be easier to implement When usingthe BB-sQRM to analyze nonlinear problems it just needsto change the expression of element stiffness according todifferent nonlinear factors and then adjust element values incorresponding positions of the stiffnessmatrix aftermaterialsenter the plastic state

21 Construction of Integral Displacement Function of Struc-ture As can be seen from Figure 1 the plane structure isdiscretized along 119909 and 119910 directions The discrete element isa rectangle element as shown in Figure 2

1

u1

2

u2

3

u3

4

u4

Figure 2 Rectangular plane element

The effect of rotation angle is ignored The structuraldisplacement field which is constructed by the linear combi-nation of the product of B-spline functions can be expressedas follows

119906 = 119872sum119895=0

119873sum119894=0

119906119892119894119895120601119894 (119909) 120595119895 (119910) = [120595] otimes [120601] 119906119892

V = 119872sum119895=0

119873sum119894=0

V119892119894119895120601119894 (119909) 120595119895 (119910) = [120595] otimes [120601] V119892 (1)

where 119873 and119872 are the maximum numbers of spline nodesin directions of 119909 and 119910 respectively 119906119892 and V119892 are splinegeneralized displacement arrays 120601119894(119909) and120595119895(119910) are cubic B-spline interpolation basis functions of 119909 and 119910 directions

Then (1) can be written in a matrix form such as

119881 = [119873] 120575 (2)

where 119881 = 119906 V119879 is the displacement array [119873] is theshape function matrix and 120575 is the generalized displace-ment array of structural spline nodes they can be expressedby

[119873] = diag [120601 (119909) otimes 120595 (119910) 120601 (119909) otimes 120595 (119910)] [120601] = [1206010 1206011 1206012 sdot sdot sdot 120601119873minus1 120601119873] [120595] = [1205950 1205951 1205952 sdot sdot sdot 120595119872minus1 120595119872]

(3)

120575 = 1205751198790 1205751198791 sdot sdot sdot 120575119879119872119879 120575119895 = 1205750119895119879 1205751119895119879 sdot sdot sdot 120575119873119895119879119879 120575119894119895 = 119906119892119894119895 V119892119894119895119879

(4)

where 119894 = 0 1 119873 119895 = 0 1 119872 Therefore the basicunknown 120575 of the BB-sQRM is a vector with order of 2(119872+1)(119873 + 1)

Based on the displacement parameter method [33] thedisplacement of spline nodes is taken as the basic unknown


The basis function 120601119894(119909) is constructed by cubic B-splinefunctions corresponding to boundary conditions which isexpressed by

120601119894 (119909) = 103 1205933 (119909 minus 1199090ℎ minus 119894) minus 431205933 (

119909 minus 1199090ℎ minus 119894 + 12)minus 431205933 (

119909 minus 1199090ℎ minus 119894 minus 12)+ 161205933 (

119909 minus 1199090ℎ minus 119894 + 1)+ 161205933 (

119909 minus 1199090ℎ minus 119894 minus 1)

(5)

where 1205933(119909) is the cubic B-spline function which can be seenin [30 31 33]

If the spline interpolation basis function120601119894(119909) satisfies thefollowing relation the boundary problemcan be solved easilyand the computation is greatly simplified

120601119894 (119909119896) = 1 119894 = 1198960 119894 = 119896 (6)

where 119909119896 denotes the spline nodes of the cubic B-splinefunction 1205933(119909) 119896 is the number of spline nodes Therefrom119906119892 and V119892 obtained by (1) are the real displacements ofthe structure Similarly the spline basis function 120595119895(119910) canbe constructed

22 QR Transformation The transformation of the BB-sQRM which is to establish the conversion relation betweenthe displacement vector of element nodes and the generalizeddisplacement vector of structure spline nodes is the key stepof BB-sQRM

Using the element stiffness matrix of the FEM and therectangular element as shown in Figure 2 the displacementvector of element nodes in the overall coordinate system is

119881119890 = [1199061 V1 1199062 V2 1199063 V3 1199064 V4]119879 (7)

According to (1) and (7) the following is presented

119881119890 = [[119873]1 [119873]2 [119873]3 [119873]4]119879 [119906119892

V119892]= [119873]119890 120575

(8)

where [119873]119890 is the element shape function matrix [119873]119898 (119898 =1 2 3 4) is the shape function matrix of four nodes of theelement respectively which can be obtained by substitutingthe whole coordinate (119909119898 119910119898) (119898 = 1 2 3 4) of elementnodes into (3)

After the QR transformation the node displacement vec-tor dimension of each element will be expanded to be equalto the node displacement vector dimension of the wholestructure that is the sixth-order node displacement vector119881119890 of elements is to be transformed into a 2(119872+1)(119873+1)-order vector 120575 and it will provide great convenience forsubsequent calculation

3 BB-sQRM Elastoplastic StiffnessEquation of the RC Beam

31 The Elastoplastic Increment Constitutive Model of Con-crete The elastoplastic increment constitutive model of con-crete is established based on the classical theory of plasticitywhich can describe the stress-strain state of the structurein various complex loading processes And the model hasbeen extensively used in the FEM of concrete materialsElastoplastic matrices of concrete beam elements in differentstress states are given below

(i) The element elastoplastic matrix of concrete in biaxialcompressive stress state is

[119863]1198901199011 = [119863] minus [119863]1199011 = [119863] ([119868] minus [119876]1199011) (9)

where [119863]1198901199011 and [119863]1199011 are the concrete elastoplastic matrixand the concrete plastic matrix in the biaxial compressivestress state respectively and [119863] is the elastic matrix ofconcrete which is expressed by

[119863] = 1198641 minus 1205832[[[[[

1 120583 0120583 1 00 0 1 minus 120583

2

]]]]] (10)

where 119864 is the elastic modulus 120583 is the Poisson ratio [119876]1199011 isexpressed by

[119876]1199011 = [120597119865120597 [120590]] [120597119865120597 [120590]]119879 [119863]119860 + [120597119865120597 [120590]]119879 [119863] [120597119865120597 [120590]] (11)

where 119860 is the hardening parameter 119865 is the yield functionelement stress [120590] = [120590119909 120590119910 120591119909119910]

Based on the Chen-Chen yield criterion with 3 parame-ters [34] using the associated flow law [119876]1199011 is expressed byan explicit formulation [35]

[119876]1199011

= 120572[[[[

119886 (119886 + 119887120583) 119886 (119887 + 119886120583) 18119886120591119909119910 (1 minus 120583)119887 (119886 + 119887120583) 119887 (119887 + 119886120583) 18119887120591119909119910 (1 minus 120583)

36120591119909119910 (119886 + 119887120583) 36120591119909119910 (119887 + 119886120583) 6481205912119909119910 (1 minus 120583)]]]]

(12)

in which

120572 = 1198649119860 (1 minus 1205832) minus 2119864ℎ

119886 = 1198600 + 7120590119909 minus 120590119910119887 = 1198600 minus 120590119909 + 7120590119910ℎ = (11986002 + 61198600120590119909 + 61198600120590119910) (120583 + 1)


y

x

x(1)y(2)

Figure 3 Diagram of cracking in the concrete element

+ (25 minus 7120583) (1205901199092 + 1205901199102) + 3241205911199091199102 (1 minus 120583)+ 2120590119909120590119910 (25120583 minus 7)

1198600 = 1198912119887119888 minus 11989121198882119891119887119888 minus 119891119888

119860 = 119864119864119905119864 minus 119864119905 (13)

where 1198600 is a constant 119891119888 is the uniaxial compressivestrength 119891119887119888 is the biaxial compressive strength 119864119905 is thetangent modulus 119864 is the elastic modulus and 120583 is thePoisson ratio

Substituting (10) and (13) into (9) the plasticmatrix [119863]1199011and the elastoplastic matrix [119863]1198901199011 in the biaxial compressivestress state are obtained With the proposed explicit expres-sion the computer programming could be more convenient

(ii) Under the states of tension-compression stress andbiaxial tension stress the concrete may enter nonlinearitybecause of cracking It is assumed that the crack is parallelto the 1199101015840 axis when the concrete cracks And the direction ofthe main tensile stress 1205901 is parallel to the 1199091015840 axis As shownin Figure 3 1205901 and 119909-axis are with 120572 degree From 119909-axisto 1205901 counterclockwise direction is positive and clockwisedirection is negative

The elastoplastic matrix of concrete element in the localcoordinate system is expressed as [119863lowast]1198901199012

[119863lowast]1198901199012 = [119863lowast] minus [119863lowast]1199012 = [119863lowast] ([119868] minus [119876]1199012) (14)

in which

[119876]1199012 = [[[

0 0 0V 1 00 0 1 minus 120573

]]] (15)

where 120573 is the residual shear coefficient whose value is 05 inthe ordinary RC beam

The element elastoplastic matrix [119863]1198901199012 of the two stressstates in the whole coordinate system is obtained by thetransformation

[119863]1198901199012 = [119879]119879 [119863lowast]119890119901 [119879] (16)

where [119879] is the coordinate transformation matrix which isexpressed by

[119879] = [[[[

1198882 1199042 119888 times 1199041199042 1198882 minus119888 times 119904

minus2119888 times 119904 2119888 times 119904 1198882 minus 1199042]]]] (17)

where 119888 = cos 120574 119904 = sin 120574 120574 is the angle between the 1199091015840 axisof the local coordinate and the 119909-axis of the whole coordinatein Figure 3

Also the plastic matrix of concrete element of the twostress states is obtained by

[119863]1199012 = [119879]119879 [119863lowast]1199012 [119879] (18)

in which

[119863lowast]1199012 = [119863lowast] [119876]1199012 (19)

where [119863lowast]1199012 is the plastic matrix in the local coordinatesystem and [119863lowast] is the same as (11)

Substituting (10) (14) (15) and (17) into (16) the elasto-plasticmatrix [119863]1198901199012 is obtained And the plasticmatrix [119863]1199012is obtained by substituting (10) (17) and (19) into (18)

32 Additional Stress and Release Stress The plastic defor-mation of the concrete element subjected to the biaxialcompression could create the additional stress 1205900 which isobtained by

1205900 = [119863]1199011 120576 (20)

where 120576 is the strain of the concrete elementThe crack of the concrete element subjected to the biaxial

tension and tension-compression could create the releasestress 120590119877 which is obtained by

120590119877 = [[119868] minus 119901 (120574) 119902 (120574)119879]

120590119909120590119910120591119909119910 (21)

in which

119901 (120574) = [sin2120574 cos2120574 sin 120574 cos 120574]119879 119902 (120574) = [sin2120574 cos2120574 2 sin 120574 cos 120574]119879 (22)

where 120574 is the angle between the1205901 axis of the local coordinateand the 119909-axis of the whole coordinate as shown in Figure 3


y

x

A

B

b1

b2l2

a1

a2l1

u2 = 1

ub

1 2

3

4

Figure 4 Quadrilateral reinforced concrete composite element

33 Elastoplastic Stiffness Equation of Reinforced ConcreteBeam Based on BB-sQRM There are mainly three kinds ofFE models which can be chosen to analyze RC structuresintegrated separated and combined [36] In this paper thecombined model is adopted The plain concrete element andthe reinforced concrete element are created after the splinediscretization The contribution of concrete and steel bars tothe stiffness should be considered simultaneously when thestiffness matrix element is established

The element stiffness matrix of the plain concrete [119870119888]119890can be expressed by

[119870119888]119890 = int+1

minus1int+1minus1[119861]119879119890 [119863] [119861]119879119890 |119869| 119905 119889120585 119889120578 (23)

where [119861]119890 is the element strain matrix [119863] is the elasticmatrix of concrete |119869| is the Jacobian determinant 120585 and 120578are of the local coordinate

The element with steel bars is simulated by a four-nodecomposite element as shown in Figure 4 The element stiff-ness matrix of the steel bar [119870119904]119890 which is the contributionmatrix of the steel bar to the whole quadrilateral element isformed by the element stiffness matrix of the single steel bar[119870119904]119890

[119870119904]119890 = [119877]119879 [119870119904]119890 [119877] (24)

where [119877] is a coordinate transformationmatrix expressed by

[119877] =[[[[[[[[[[[

11988611198971 0 0 0 0 0 11988621198971 00 11988611198971 0 0 0 0 0 119886211989710 0 11988711198972 0 11988721198972 0 0 00 0 0 11988711198972 0 11988721198972 0 0

]]]]]]]]]]]

(25)

where 1198861 1198862 1198871 1198872 1198971 and 1198972 are the lengths shown in Figure 4It is assumed that the length of a steel bar element is 119897

and the angle between the steel bar and 119909-axis is 120579 1198881015840 = cos 120579

1199041015840 = sin 120579Then the element stiffness matrix of the single steelbar is obtained by

[119870119904]119890 = 119864119860119897[[[[[[[

1198881015840211988810158401199041015840minus11988810158402minus11988810158401199041015840

1198881015840119904101584011990410158402minus11988810158401199041015840minus11990410158402

minus11988810158402minus119888101584011990410158401198881015840211988810158401199041015840

minus11988810158401199041015840minus119904101584021198881015840119904101584011990410158402

]]]]]]] (26)

The stiffness matrix of reinforced concrete compositeelement [119870119904119888]119890 can be obtained by the superposition of theelement stiffness matrix of steel bars [119870119904]119890 and the elementstiffness matrix of the plain concrete [119870119888]119890

[119870119904119888]119890 = [119870119888]119890 + [119870119904]119890 (27)

Then the general potential energy functions of thereinforced concrete composite element and the plain concreteelement are expressed by the following

Π119890119896 = 12 119881119904119888119879119890 [119870119904119888]119890 119881119904119888119890minus 119881119904119888119879119890 (119891119904119888119890 + 119891119901119904119888119890 + 119891119877119904119888119890)

Π119890119896 = 12 119881119888119879119890 [119870119888]119890 119881119888119890minus 119881119888119879119890 (119891119888119890 + 119891119901119888 119890 + 119891119877119888 119890)

(28)

where 119891119904119888119890 and 119891119888119890 are the node force vectors of elements119891119901119904119888119890 and 119891119901119888 119890 are the additional node force vectors ofelements 119891119877119904119888119890 and 119891119877119888 119890 are the node force vectors ofcracking elements

119891119901119896 119890 = int119890 [119861]119879119890 1205900 119889Ω119891119877119896 119890 = int119890 [119861]119879119890 120590119877 119889Ω

(29)

where the corner ldquo119896rdquo represents the composite element (119904119888) orthe plain concrete element (119888) and [119861]119890 is the element strainmatrix 1205900 and 120590119877 are obtained by (20) and (21)

After theQR transformation the general potential energyfunction of element which is expressed by the generalizeddisplacement vector of spline nodes can be presented

Π119890 = 12 120575119879 [119866]119890 120575 minus 120575119879 (119865119890 + 119865119901119890 + 119865119877119890) (30)

in which

[119870]119890 = [119873]119879119890 [119870119904119888]119890 [119873]119890 + [119873]119879119890 [119870119888]119890 0 [119873]119890 119865119890 = [119873]119879119890 119891119904119888119890 + [119873]119879119890 119891119888119890 119865119901119890 = [119873]119879119890 119891119901119904119888119890 + [119873]119879119890 119891119901119888 119890 119865119877119890= [119873]119879119890 119891119877119904119888119890 + [119873]119879119890 119891119877119888 119890

(31)

where [119870]119890 is the element general stiffness matrix 119865119890 isthe element load vector 119865119901119890 is the element additional load


vector created by the plastic deformation 119865119877119890 is the loadvector transformed by release stress of the cracking elements

If the concrete beam is divided into 119899 elements its generalpotential energy function can be expressed by

Π = 12 120575119879 [119870] 120575 minus 120575119879 (119865 + 119865119901 + 119865119877) (32)

in which

[119870] = 119899sum119890=1

[119866]119890

[119865] = 119899sum119890=1

119865119890

[119865119901] = 119899sum119890=1

119865119901119890

[119865119877] = 119899sum119890=1

119865119877119890

(33)

So it is quite easy and convenient to establish thegeneral stiffness matrix and the general load vector with thesuperposition of element stiffness matrices and element loadvectors by using the BB-sQRM According to the principleofminimumpotential energy the elastoplastic stiffness equa-tion of the RC beam can be obtained by (32)

[119870] 120575 = 119865 + 119865119901 + 119865119877 (34)

Then the displacement vector and the stress vector canbe obtained by

120575 = [119870]minus1 (119865 + 119865119901 + 119865119877) 120590 = [119863]119890119901 [119861] [119873]119890 120575 (35)

where the elastoplastic matrix [119863]119890119901 is obtained by (9) or (16)34The Solution Process of the BB-sQRMElastoplastic StiffnessEquation In this paper the incremental iteration methodfor initial stress is adopted to solve the elastoplastic stiffnessequation Here are the steps

(1) Divide the external load into 119873 stages The initialdisplacement 1205750 = 0 and the initial stress 1205900 = 0

(2) Apply the first stage load 1198651 which usually takes 10to 30 of the predicted limit load Make the displacementincrement norm of the first iteration zeroThen the structureis in an elastic state at present According to the elasticstiffness matrix [1198701] calculate the increment of the structuredisplacement

1198891205751 = [1198701]minus1 1198651 (36)

(3) According to the geometric equation calculate theincrement of the element strain 1198891205761 and according to theelastic stress-strain relationship the increment of the elementstress 1198891205901 can be obtained by

1198891205901 = [119863] [119861]119890 [119873]119890 1198891205751 (37)

Then the element stress components120590119909120590119910 and 120591119909119910 underfirst stage load can be derived Also the main stresses 1205901 and1205902 under this stage load can be obtained

(4) Determine the stress state of each element on the basisof 1205901 and 1205902

If the element is under the state of biaxial compressionstress calculate the element elastoplastic matrix [119863]1198901199011 andthe plastic matrix [119863]1199011 by (9) (10) and (12) and calculatethe additional node force vector 119891119901119896 119890 by (20) and (29)

If the elements are subjected to the tension-compressionstress or the biaxial tension stress calculate the elementelastoplastic matrix [119863]1198901199012 and the plastic matrix [119863]1199012 by(14)ndash(19) and calculate the node force vectors of crackingelements 119891119877119896 119890 by (21) (22) and (29)

Then transform the obtained matrices and vectors aboveinto the correspondingmatrices and vectors of the BB-sQRM

(5) After processing all the elements by step (4) the loadincrement array under current iteration can be obtained by(29) (31) and (33)

1198891198651 = 119889119865119901 + 119889119865119877 (38)

Then calculate the new general stiffness matrix [119870] andnew displacement increment

(6) Determine whether the iteration convergesIf the norm of the displacement increment satisfies the

criterion of iteration convergence the iterative computationof the load on the current stage is stopped and then enters thenext step

10038171003817100381710038171003817119889120575119895110038171003817100381710038171003817 lt 10minus8 (39)

where the corner ldquo119895rdquo represents the number of iterations afterincreasing each load increment

If the condition above is not satisfied repeat steps (4)ndash(6)on the basis of the stress increment obtained by the newdisplacement increment

(7) With the element state after finishing the first loadcalculation as the initial state apply the second stage loadCalculate the displacement increment of the second load

1198891205752 = [1198702]minus1 1198652 (40)

Then the stress increment can be obtained Repeat steps(3)ndash(6) until the second load iteration finishes(8) Apply the load progressively to number 119873 load and

complete the solution

4 Crack Control Effect Analysis ofSMA Smart Concrete Beam

41 The Stiffness Equation of the SMA Concrete Beam Basedon BB-sQRM The introduction of the SMA driving force isone of the key steps while establishing the crack equationof the SMA concrete beam In this paper the SMA recoveryforce is simulated by the equivalent eccentric force methodThat is the SMA wires are not processed into elements in thebidirection spline discretization of the structure Instead therecovery force of SMA wires is equivalent to the eccentric


Table 1 The recovery stress 120590119903 in a certain temperature range of SMA wires with different prestrains (MPa)

Temperature (∘C) Stress of 4 prestrain Stress of 6 prestrain Stress of 8 prestrain20 0 0 030 2714 2731 274840 7412 9543 1364750 62341 39423 2748160 165834 197854 11257270 218125 275371 15840780 241534 309025 33941890 281413 365412 394857100 296417 379142 501637110 325426 395427 537486120 327013 399652 595213

force which is introduced in Section 42 The spline nodeload 119891SMA based onBB-sQRM is transformed by the drivingforce of SMA wires

119891SMA = [119873]119894 119879SMA

0 (41)

where the SMA driving force 119879SMA which is produced by theelectric excitation can be obtained by the equivalent eccentricforce method

Adding 119891SMA into (28) the general potential energyfunction of the concrete composite element with SMA wirescan be obtained Then repeat the calculations from (24)to (34) According to the QR transformation and elementdiscretization the stiffness equation of the SMA concretebeam is derived by the principle of minimum potentialenergy and the BB-sQRM

[119870] 120575 = 119865 + 119865119901 + 119865119877 + 119865SMA (42)

in which

119865SMA =119899sum119890=1

[119873]119879119890 119879SMA

0 (43)

When cracks of concrete beam are produced under theload stimulate the SMA wires which are buried in the beamwith electric currentsThen the phase transition of SMAwiresoccurs and the recovery stress is produced And the drivingforce vector 119865SMA is formed which can control cracksdevelopment Therefore in order to obtain the elastoplasticstiffness equation of the SMA smart concrete beam thedriving force of SMAwires should be added to the last step inSection 34That is ldquoapply the load progressively to the no119873load and apply the driving force of SMAwires then completethe solutionrdquoThus the displacement of the structure and themaximum stress and strain of the steel bar elements can beobtained by the proposed model and the maximum crackwidth of the concrete beam can be calculated

The maximum crack width of the concrete beam isobtained by the following

120596max = 120572cr120593120576max119871 (44)where 119871 is the length of the steel bar elements 120572cr and 120593 aredetermined by [37]

2

10

050

100150200250300350400450500550600650

Reco

very

stre

ss (M

Pa)

40 60 80 100 120 140 160 180 200 220 24020Temperature (∘C)

4 6

8 9

Figure 5 The recovery stress-temperature curves of SMA withdifferent prestrains

42 Test and Simulation Method of SMA Recovery Stress Inorder to learn the driving characteristics of SMA wires in theelectric excitation the recovery stress test of SMA wires wascarried out [25] SMAmaterials used were Ni-Ti wires wherethe Ni accounted for 4424 and Ti accounted for 5576The original length was 300mm and diameter was 2mmThemaximumrecovery stresswas 680MP and themaximumrecovery strain was 8 The constant electric current wasdetermined to be adopted in the next test recovery stress-temperature curves of SMAwires in different prestrains from2 to 10 were measured The curves are shown in Figure 5The recovery stress values in a certain temperature range ofSMA wires with 4 6 and 8 prestrains are shown inTable 1

As shown in Figure 6 and Table 1 when the prestrainis identical with the rise of temperature the austenitetransformation occurs and the recovery stress of SMA wires


Neutral axisT３－T３－ SMAs ℎs

Figure 6 The equivalent eccentric force of SMA wires

SMAsℎs

500

200

5empty4

L

q = 22 kNm

Steel bars

Figure 7 The simply supported reinforced concrete beam

increases The stress tends to be stable at 120∘C After heatingup to 200∘C SMA wires soften and lose shape memoryeffect gradually When the prestrain of SMA wires is notgreater than 8 the maximum recovery stress of SMA wiresincreases with the increasement of the prestrains Converselyafter more than 8 the maximum recovery stress of SMAwires decreases with the rise of prestrains

The recovery stress of SMA wires can be processed basedon a set of related test data The methods for simulating therecovery force of SMA wires are mainly divided into twocategories One relates to thermodynamic analysis whichneeds to define the elastic modulus at different temperaturesduring the application of temperature such as the pseudotemperature loading method and negative thermal strainmethodThe other is applying the recovery force as the exter-nal load on the structure such as the equivalent eccentricforce method and the equivalent steel bar stress method Inthis paper the equivalent eccentric force method is adoptedThis method which does not involve the complex thermalbehavior in the analysis process is more convenient to realizein model establishment and computer programming

Based on the equivalent eccentric force method therecovery force of SMA wires is equivalent to a pair of hori-zontal forces which are applied to both ends of the concretebeam as shown in Figure 7 According to the arrangementof longitudinal steel bars the SMA wires are embedded andanchored in the tension zone at the bottom of the concretebeamWhen SMAwires are heated the phase transformationand the contraction of SMA wires occur Because of theconstraint of the anchorages the contraction of SMA wires islimited And the eccentric driving force is formed and appliedat the bottom of the concrete beam Therefore the activecontrol of the structure deformation and the crack control canbe realized

The equivalent horizontal force 119879SMA and the bendingmoment119872SMA of SMA wires can be obtained by

119879SMA = 119899120590119903119860SMA119872SMA = 119879SMAℎ119904 (45)

Figure 8 FE model of the SMA reinforced concrete beam

where 119899 is the number of SMA wires 120590119903 is the recovery stressof SMA wires in Table 1 119860SMA is the cross-sectional area ofsingle SMA wire ℎ119904 is the distance from SMA wires to theneutral axis of the concrete beam that is the eccentricity ofSMA wires In ANSYS the recovery force of SMA wires isequivalent to the horizontal eccentric force which is appliedon the concrete beam model as the external load Thereforethe finite element (FE) simulation of the concrete crackcontrol with SMA wires can be realized

5 Result and Discussion

In this paper the crack control effect of a simply supportedRC beam embedded with SMA wires is considered Throughthe numerical example the correctness of the proposed BB-sQRM can be verified Also a discussion for the influence ofseveral parameters on crack control effect is presented in theexampleThe basic RC beammodel is shown in Figure 8Thecross section is 119887timesℎ = 200mm times 500mm and the calculatedspan is expressed in 119871 The uniformly distributed load is 119902 =22 kNm and the C25 concrete is adopted The tension zoneis embedded with three HRB400 steel bars and five SMAwires where the diameter is 4mm The relevant parametersare 119864119862 = 255 times 104Nmm2 119864119904 = 2 times 105Nmm2 119891119888119896 =167Nmm2 and 119891119905119896 = 178Nmm2 The recovery stress of


Table 2 Designation and parameters of SMA reinforced concrete beams

DesignationBeam

dimension(mm)

Concrete cover(mm)

Reinforcementratios ()

Prestrain ofSMA wires ()

Eccentricdistances ofSMA wires

(mm)B1 200 times 500 times 4000 25 076 6 220B2 200 times 500 times 5000 25 076 6 220B3 200 times 500 times 6000 25 076 6 220B4 200 times 500 times 5000 45 076 6 220B5 200 times 500 times 5000 65 076 6 220B6 200 times 500 times 5000 25 046 6 220B7 200 times 500 times 5000 25 114 6 220B8 200 times 500 times 5000 25 076 4 220B9 200 times 500 times 5000 25 076 8 220B10 200 times 500 times 5000 25 076 6 110B11 200 times 500 times 5000 25 076 6 160Note B1 B2 and B3 are different in depth-span ratios B2 B4 and B5 are different in concrete cover B2 B6 and B7 are different in reinforcement ratios B2B8 and B9 are different in prestrain of SMA wires B2 B10 and B11 are different in eccentric distances of SMA wires

SMA wires comes from the test data shown in Table 1 andits simulation employs the equivalent eccentric force methodintroduced in Section 42 Based on the basicmodel shown inFigure 8 the different beammodels are designed according toseveral parameters which are listed in Table 2

51 Comparison of the Results of BB-sQRM and FEM inANSYS A comparison example of the two methods is pre-sented which is to verify the correctness of the proposedcrack control analysis model based on BB-sQRM Beam 2(B2) in Table 2 is used as the SMA reinforced concrete modelfor this numerical example In order to obtain the maxi-mum crack width of the SMA concrete beam in differenttemperatures both ANSYS and BB-sQRM need to be usedin calculating the maximum strain of steel bar elementsThe final result can be calculated by (44) Consequently theobtained maximum strain of steel bars is utilized to comparein the verification process First use the ANSYS in finiteelement analysis Both the SOLID65 elements and the LINK180 elements are adopted in modeling When dividing themesh considering the equivalent load of SMA wires need tobe applied on the node the width direction of the concretebeam is divided into 6 equal parts And onemesh is increasedalong the height direction based on 25 equal parts That isthe mesh generation of the concrete beam is namely 50 times 26times 6 in which the number of elements is 10550 the numberof nodes is 12393 and the total number of the degrees offreedom is 74313 The model of FEM is shown in Figure 9Second based on BB-sQRM a planemodel shown in Figure 1is used for analysis The spline node discretization of 50times 25 has 1250 elements 1326 nodes and 3975 degrees offreedom Meanwhile in order to discuss the effect of splinenode discretization on the precision of calculation and checkthe convergence of the BB-sQRM the maximum strain inother discrete cases has also been calculated The numericalresults are listed in Table 3Themaximum strain error of steelbars obtained by comparing the twomethods under different

ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

35 times 25 40 times 25 45 times 25 50 times 25 55 times 25 60 times 2530 times 25

Spline nodes discretization of BB-sQRM

Figure 9 Convergence efficiency of the BB-sQRM

discretization is shown in Figure 10 And the convergenceefficiency of the BB-sQRM in different discretization isillustrated in Figure 11

As shown in Table 3 and Figure 10 the more the numberof spline nodes is the closer the results between ANSYS andBB-sQRM are When the discretization is 30 times 25 35 times 25 40times 25 45 times 25 50 times 25 55 times 25 and 60 times 25 respectively themaximum error is minus668 minus484 minus271 minus192 minus106minus051 and minus013 which converges quickly As shown inFigure 11 under different SMA temperatures of 20∘C 60∘Cand 120∘C with the spline node discretization from 30 times 25 to60times 25 themaximum strain of steel bar elements is stabilizedgradually However as far as the total degrees of freedom are


BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)

Figure 10 Maximum strain of bar steel elements with respect to spline notes discretization (a) 20∘C (b) 60∘C and (c) 120∘C

Table 3 Maximum strain of steel bar elements with different discretization and temperatures (times10minus4)Method Discretization Temperature (∘C)

20 40 60 80 100 120

BB-sQRM

30 times 25 9803 9766 9046 8620 8352 827430 times 25 9979 9943 9216 8788 8517 843840 times 25 10180 10140 9412 8980 8707 862845 times 25 10260 10220 9486 9053 8779 869950 times 25 10350 10310 9570 9132 8856 877555 times 25 10410 10370 9623 9183 8905 882460 times 25 10450 10410 9660 9218 8938 8857

ANSYS 50 times 26 times 6 10460 10420 9660 9230 8950 8860


Table 4 The recovery situation of maximum crack width with different depth-span ratios

Temperature of SMA wires(∘C)

Maximum crack width (mm) Recovery rate of maximum crack width ()1205771 1205772 1205773 1205771 1205772 1205773

20 01002 02098 03500 000 000 00040 00996 02090 03491 059 039 02760 00886 01940 03315 1161 754 52980 00821 01851 03211 1810 1177 826100 00780 01795 03146 2220 1444 1011120 00768 01779 03128 2340 1522 1064Depth-span ratios (120577) 1205771 = ℎ11198711 = 1 8 1205772 = ℎ21198712 = 1 10 and 1205773 = ℎ31198713 = 1 12

The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12

Figure 11 The recovery rates of maximum crack width withdifferent depth-span ratios of the beams

concerned the computation of the BB-sQRM is much lessthan that of the FEM

The main reasons for the error between the BB-sQRMand the FEM are as follows (1) The type of elements usedin the two methods is different The plane stress elementswere utilized in the BB-sQRM while the three-dimensionalsolid elements were adopted in the ANSYS differently Thusthe degrees of freedom of nodes are distinguishing (2) Thematerial constitutive law used in the twomethods is differentThe BB-sQRM was applied to different stress zone of rein-forced concrete beam with different yield criteria and failurecriteria In the FEManalysis byANSYS the same stress-strainrelationship was assumed for all stress zones Therefore theabove-mentioned errors of the two methods are permissibleand reasonable and the model of the SMA concrete beambased on the BB-sQRM is correct and effective

52 Parameter Analysis Based on the above verificationusing the BB-sQRM elastoplastic analysis program of theSMA smart concrete beam with the crack control propertythemaximumcrackwidth can be calculated and the influence

Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm

Figure 12 The recovery rates of maximum crack width withdifferent thickness of concrete cover

on the crack control effect of different parameters can be dis-cussed In this paper depth-span ratio thickness of concretecover reinforcement ratio of concrete beams and prestrainand eccentricity of SMA wires are selected as the parametersUtilizing the recovery stress-temperature curve of SMAwiresmeasured by test the nonlinear analysis and the crack controlanalysis of the SMA concrete beam under different workingconditions are carried outThe various parameter models areshown in Table 2 The materials of the reinforced concretebeam the cross section and the load are unchanged Theexample is calculated by the BB-sQRMwith the discretizationof 50 times 25521 Effect of Depth-Span Ratios The effect of crack controlon SMA reinforced concrete beam under different depth-span ratios is considered by using the models B1 B2 and B3The height of the cross section remained and the calculatedspan of the beams is 4m 5m and 6m respectively Asshown in Table 4 and Figure 12 when the SMA is in thenormal temperature (20∘C) the maximum crack width is01002mm 02098mm and 03500mm severally After the


Table 5 The recovery situation of maximum crack width with different thickness of concrete cover



20 02098 02848 03700 000 000 00040 02090 02838 03687 039 037 03460 01940 02625 03403 754 782 80480 01851 02501 03234 1177 1219 1258100 01795 02422 03129 1444 1497 1543120 01779 02399 03098 1522 1578 1626Concrete covers (119888) 1198881 = 25mm 1198882 = 45mm and 1198883 = 65mm

Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114

Figure 13 The recovery rates of maximum crack width withdifferent reinforcement ratios

SMA wires are heat-treated into 120∘C the recovery rateof the maximum crack width reached 2340 1522 and1064 Significantly with the depth-span ratios reductionthe beam is more slender and the initial maximum crackwidth of the beam without heating of SMA wires increasesbut the recovery rate of the maximum crack width decreasessignificantly which indicates that the depth-span ratio has aconsiderable influence on the crack control effectThe reasonis that when cracks are driven to control by SMA wiresthe higher the depth-span ratio is the greater the eccentricmoment formed by the recovery force of SMA wires is andthe better the driving effect is Conversely the lower thedepth-span ratio is the greater the beam deformation is andthe weaker the driving effect of SMA wires is

522 Effect of theThickness of the Concrete Cover Dependingon the specification of the design of reinforced concrete beamthe concrete cover affects the development of the beam crackssignificantly As shown in Table 5 and Figure 13 when thethickness of the concrete cover is 25mm (in B2) 45mm(in B4) and 65mm (in B5) the maximum crack width

of the beam before heat-treating is 02098mm 02848mmand 03700mm respectively And the recovery rate of themaximum crack achieves 1522 1578 and 1626 afterheating the SMA wires to 120∘CThus with the increasing ofthe thickness of the concrete cover the recovery rate of themaximum crack width increases However the increment ofthe recovery-temperature curve is very small with the samethickness of the concrete cover because for the reinforcedconcrete beam increasing the thickness of concrete cover canmake the crack develop more fully Then the prestrain of theSMA wires increases accordingly and the recovery stress isgreater after heating the SMA wires with electric excitationSo it is useful for control crack by increasing the concretecover thickness though the effect is not evident

523 Effect of the Reinforcement Ratios in Tension Zone Inorder to check the influence on the crack control effect ofthe reinforcement ratios the models of B6 B2 and B7 areused With the same number of SMA wires embedded inthe concrete beam make the reinforcement ratios 046076 and 114 respectivelyThat is 3 steel bars of HRB400in different diameters of 14mm 18mm and 22mm areembedded in the tension zone The calculated results areshown in Table 6 and Figure 14 In the normal temperaturethe maximum crack width of the reinforced concrete beamis 03935mm 02098mm and 01275mm Heating the SMAwires to 120∘C the recovery rate of the maximum crackwidth is 1597 1522 and 1478 respectively The resultsillustrate that the larger the reinforcement ratio is the worsethe property of the crack control on the SMA concrete beamis In respect that the plastic deformation of steel bars andthe bond degrees of concrete both hinder the recovery ofcracks and deformation which make the driving effect ofSMA wires weak the hindering effect becomes obvious withthe increment of the reinforcement ratio

524 Effect of the Prestrain of SMA Wires The prestrain ofSMA wires in models B8 B2 and B9 is 4 6 and 8respectively The SMA wires are heat-treated to the tem-perature of the maximum recovery stress and the result isillustrated in Figure 15 The prestrain of the three modelsis 02098mm When the exciting temperature is 180∘C themaximum crack width of the concrete beam is 01822mm01754mm and 01595mm and the recovery is correspond-ingly 1317 1641 and 240 Obviously with the increase


Table 6 The recovery situation of maximum crack width with different reinforcement ratios



20 03935 02098 01275 000 000 00040 03918 02090 01270 043 039 03860 03611 01940 01181 823 754 73480 03448 01851 01129 1237 1177 1144100 03340 01795 01096 1513 1444 1402120 03307 01779 01087 1597 1522 1478Reinforcement ratios (120572119904)1205721199041 =046 (reinforcement diameter 14mm)1205721199042 =076 (reinforcement diameter 18mm) and1205721199043 = 114 (reinforcement diameter22mm)

Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468

Figure 14 The maximum crack widths with different prestrains ofSMA wires

of the prestrain of SMA wires the recovery rate of themaximum crack width increases greatly which explains thatthe prestrain of SMA wires has a significant influence onthe control effect of beam cracks It is because the larger theprestrain is the more the austenite transforms during thephase transition and temperature rise and the greater therecovery stress of SMA wires is the better the effect of crackcontrol is

525 Effect of the Eccentricity of SMAWires The eccentricityof SMA wires refers to the distance from SMA wires to thebeam neutral axis The SMA wires with the eccentricity of110mm 160mm and 220mm are used in the models B10B11 and B2 respectively The calculation result is shownin Figure 15 In the normal temperature of SMA wiresthe maximum crack width of the three beam models is02098mm Heating the SMA wires to 120∘C correspondingto the eccentricity from small to large the maximum crackwidth of the beam is 01857mm 01826mm and 01779mmand the recovery rate of the maximum crack width is 1147

Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm

Figure 15Themaximum crack widths with different eccentricity ofSMA wires

1298 and 1522 respectively Therefore the maximumcrack width recovery rate increases with the increment of theSMA wire eccentricity The reason is that when the recoverystress is certain the greater the eccentricity of SMA wiresis the greater the reverse moment is formed and the moresignificant the driving effect of SMA wires is the better thecrack control effect of the beam is

6 Conclusions

Based on the BB-sQRM a new model is presented in thispaper which is used in the nonlinear analysis for crackcontrol capability of the SMA concrete beamThe discretiza-tion of the BB-sQRM is performed with a set of uniformlyscattered spline nodes in both two directions of the structureThen a transformation relationship is established which isbetween the node displacement of elements and the splinenode displacement of the structure Through the conversionthe element displacement vector can be expanded to the


vector which has the same dimension as the structural dis-placement vector Consequently it is convenient to generatestiffness matrices and load vectors and the stiffness equationhas a small number of unknowns Considering the nonlin-earity of concrete materials an explicit form is proposed toexpress the elastic-plastic constitutive law of concrete whichalso simplifies the calculation and programming Throughseveral numerical examples available in this literature theconvergence and accuracy of the BB-sQRM are proved Theresults show that the solutions obtained by the proposedmodel with relatively few nodes are very close to thosegiven by the space model of ANSYS with a large numberof nodes Therefore the examples demonstrate the higherefficiency and lower computational cost of the BB-sQRMthan those of the FEM simultaneously Meanwhile it isshown by the parametric analysis results that the change inthickness of concrete cover and reinforcement ratio makesno difference to the beam crack control capability And notonly the recovery of maximum crack width but also the crackcontrol effect of the SMA concrete beam can be improvedusefully by the increase of the depth-span ratio of the beamand prestrain and eccentricity of SMA wires Thereby thedurability and carrying capacity of the RC structure can beeffectively improved

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work of this paper has been supported by the grantsawarded by the National Natural Science Foundation ofChina (Project no 11562001) the Guangxi Natural ScienceFoundation ofChina (Project no 2014GXNSFAA118020) andthe Systematic Project of Guangxi Key Laboratory of DisasterPrevention and Structural Safety (Project no 2014ZDX03)

References

[1] Y Yao S E Tung and B Glisic ldquoCrack detection and character-ization techniques An overviewrdquo Structural Control and HealthMonitoring vol 21 no 9 pp 1387ndash1413 2014

[2] C DMoen and S R Sharp ldquoBond properties between concreteand corrosion-resistant reinforcing steelsrdquo ACI Structural Jour-nal vol 113 no 2 pp 383ndash392 2016

[3] AMichel B J Pease andA Peterova ldquoPenetration of corrosionproducts and corrosion-induced cracking in reinforced cemen-titious materials Experimental investigations and numericalsimulationsrdquo Cement Concrete Composites vol 47 no SI pp75ndash86 2014

[4] M A Adam M Said A A Mahmoud and A S ShanourldquoAnalytical and experimental flexural behavior of concretebeams reinforced with glass fiber reinforced polymers barsrdquoConstruction and Building Materials vol 84 pp 354ndash366 2015

[5] B Han YWang S Dong et al ldquoSmart concretes and structuresa reviewrdquo Journal of Intelligent Material Systems and Structuresvol 26 no 11 pp 1303ndash1345 2015

[6] M M Sherif J Tanks and O E Ozbulut ldquoAcoustic emissionanalysis of cyclically loaded superelastic shape memory alloyfiber reinforced mortar beamsrdquo Cement and Concrete Researchvol 95 pp 178ndash187 2017

[7] B Silwal R J Michael and O E Ozbulut ldquoA superelasticviscous damper for enhanced seismic performance of steelmoment framesrdquo Engineering Structures vol 105 pp 152ndash1642015

[8] S M R Khalili M Botshekanan Dehkordi and E Carrera ldquoAnonlinear finite element model using a unified formulation fordynamic analysis of multilayer composite plate embedded withSMA wiresrdquo Composite Structures vol 106 pp 635ndash645 2013

[9] J M Jani M Leary A Subic and M A Gibson ldquoA review ofshape memory alloy research applications and opportunitiesrdquoMaterials and Corrosion vol 56 pp 1078ndash1113 2014

[10] E Patoor D C Lagoudas P B Entchev L C Brinson andX Gao ldquoShape memory alloys Part I General properties andmodeling of single crystalsrdquoMechanics of Materials vol 38 no5-6 pp 391ndash429 2006

[11] E Choi D J Kim J-H Hwang and W J Kim ldquoPrestressingeffect of cold-drawn short NiTi SMA fibres in steel reinforcedmortar beamsrdquo Smart Materials and Structures vol 25 no 8Article ID 085041 2016

[12] H Li Z-Q Liu and J-P Ou ldquoBehavior of a simple concretebeamdriven by shapememory alloywiresrdquo SmartMaterials andStructures vol 15 no 4 article no 017 pp 1039ndash1046 2006

[13] Z Deng Q Li and H Sun ldquoBehavior of concrete beam withembedded shape memory alloy wiresrdquo Engineering Structuresvol 28 no 12 pp 1691ndash1697 2006

[14] M K Kim D J Kim Y-S Chung and E Choi ldquoDirect tensilebehavior of shape-memory-alloy fiber-reinforced cement com-positesrdquo Construction and Building Materials vol 102 pp 462ndash470 2016

[15] L Sun D Liang Q Gao and J Zhou ldquoAnalysis on factorsaffecting the self-repair capability of SMA wire concrete beamrdquoMathematical Problems in Engineering vol 2013 Article ID138162 6 pages 2013

[16] C Czaderski M Shahverdi R Bronnimann C Leinenbachand M Motavalli ldquoFeasibility of iron-based shape memoryalloy strips for prestressed strengthening of concrete structuresrdquoConstruction and Building Materials vol 56 pp 94ndash105 2014

[17] M Shin and B Andrawes ldquoEmergency repair of severely dam-aged reinforced concrete columns using active confinementwith shapememory alloysrdquo Smart Materials and Structures vol20 no 6 Article ID 065018 2011

[18] A Abdulridha D Palermo S Foo and F J Vecchio ldquoBehaviorand modeling of superelastic shape memory alloy reinforcedconcrete beamsrdquo Engineering Structures vol 49 pp 893ndash9042013

[19] A R Khaloo I Eshghi and P P Aghl ldquoStudy of behavior ofreinforced concrete beams with smart rebars using finite ele-ment modelingrdquo International Journal of Civil Engineering vol8 no 3 pp 221ndash231 2010

[20] M S AlamM A Youssef andMNehdi ldquoAnalytical predictionof the seismic behaviour of superelastic shape memory alloyreinforced concrete elementsrdquo Engineering Structures vol 30no 12 pp 3399ndash3411 2008

[21] Q Chen and B Andrawes ldquoFinite element analysis of activelyconfined concrete using shape memory alloysrdquo Journal ofAdvanced Concrete Technology vol 12 no 12 pp 520ndash534 2014


[22] P Lu F S Cui and M J Tan ldquoA theoretical model for thebending of a laminated beam with SMA fiber embedded layerrdquoComposite Structures vol 90 no 4 pp 458ndash464 2009

[23] M-Y Wang H-L Ji J-H Qiu and C Zhang ldquoPlate stiffnessactive control with shape memory alloys and layout optimiza-tionrdquo Journal of Vibration and Shock vol 33 no 23 pp 30ndash362014

[24] H Li Z-Q Liu Z-W Li and J-P Ou ldquoStudy on damageemergency repair performance of a simple beam embeddedwith shape memory alloysrdquo Advances in Structural Engineeringvol 7 no 6 pp 495ndash501 2004

[25] S-B Li C-M Mo and Q-G Liang ldquoRecovery performance ofshape memory alloy and its application in repairing cracks ofconcrete beamsrdquo Concrete vol no 4 pp 68ndash73 2015

[26] S B Li Q G Liang and J P Li ldquoFinite Element Analysis for theCrack Self-Repairing Behavior of the Concrete Beam Embed-ded Shape Memory Alloyrdquo Applied Mechanics and Materialsvol 507 pp 455ndash459 2014

[27] V P Nguyen T Rabczuk S Bordas and M Duflot ldquoMeshlessmethods a review and computer implementation aspectsrdquoMathematics and Computers in Simulation vol 79 no 3 pp763ndash813 2008

[28] L X Peng Y-P Tao H-Q Li and G-K Mo ldquoGeometric non-linear meshless analysis of ribbed rectangular plates based onthe FSDT and the moving least-squares approximationrdquoMath-ematical Problems in Engineering vol 2014 Article ID 54870813 pages 2014

[29] L X Peng ldquoFree Vibration Analysis of Symmetrically Lami-nated Folded Plate Structures Using an Element-Free GalerkinMethodrdquo Mathematical Problems in Engineering vol 2015Article ID 124296 13 pages 2015

[30] S-B Li H Wu and C-M Mo ldquoDynamic analysis of thickthinpiezoelectric FGMplates with splinemeshless methodrdquo Journalof Vibration and Shock vol 36 no 7 pp 116ndash122 2017

[31] P Liu K Lin H Liu and R Qin ldquoFree Transverse Vibra-tion Analysis of Axially Functionally Graded Tapered Euler-Bernoulli Beams through Spline Finite Point Methodrdquo Shockand Vibration vol 2016 Article ID 5891030 23 pages 2016

[32] S-B Li L-X Huang L-J Jiang and R Qin ldquoA bidirectional B-spline finite point method for the analysis of piezoelectric lami-nated composite plates and its application inmaterial parameteridentificationrdquoComposite Structures vol 107 no 1 pp 346ndash3622014

[33] R Qin Computational Structure Mechanics Guangxi NormalUniversity Press Guilin China 1997

[34] A C T Chen and W F Chen ldquoConstitutive relations for con-creterdquo Journal of Engineering Mechanics vol 101 no EM4 pp465ndash481 1975

[35] R Q Nolinear Analysis of Engineering Structures Science PressBeijing China 2006

[36] J-Q Jiang and X-Z Lu Finite Element Analysis of ConcreteStructures Tsinghua University Press Beijing China 2013

[37] GB 50010-2010 Code for Design of Concrete Structures Gen-eral Administration of Quality Supervision Inspection andQuarantine of the Peoplesrsquos Republic of China (AQSIQ) 2010

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Submit your manuscripts atwwwhindawicom


deformation and crack control Choirsquos team [11] buried SMAwires after cold drawing in the tensile zone of the reinforcedmortar beam and carried out a three-point bend test And Liet al [12] buried SMA wires in the reinforced concrete beamBoth of their results showed that the cracks could be closedwith a great restoring force generated by heating the built-in SMA wires Considering the possible complex situationsin practical applications some scholars have studied therelated parameters that affect the driving effect of SMA wiresand the controlling effect of structural deformation [13 14]Many test results show that the quantity diameter prestrainexcitationmode and volume of the SMAwires have a certaininfluence on the deformation recovery of SMA concretebeam In addition to the built-in method SMA wires alsocan be arranged outside the structure which has the effectof controlling deformation as well [15ndash17] However due tothe complexity of mechanical behaviors of RC structuresthere is less numerical analysis than experiment of theSMA concrete structure Most of them are modeled andanalyzed by software of the finite element method (FEM)such as ANSYS ABAQUS and SeismoStruct Based on thesuperelastic constitutive law of the SMAs Abdulridha etal [18] established a finite element (FE) model to simulatethe bond interface between SMA wires and concrete with abond slip element and then obtained the load-displacementrelationship of the beam Khaloorsquos team [19] used the ANSYSto establish a FE model of a cantilever RC beam embeddedwith SMA wires The analysis results showed that with theincrease of the ratio of SMA reinforcement the shear forceand displacement hysteresis curve area of concrete beamsdecrease the residual deformation of beams and the cross-sectional stiffness decrease aswell Alam et al [20] establishedthe FE model of SMA beam-column joints and analyzed itsnonlinear behaviors by the SeismoStruct software Chen andAndrawes [21] carried on the static analysis for the concretecolumn with SMA wires lateral restraint by the ABQUSsoftware When using the FEM to analyze the effect of theSMAs on the structure it is crucial for the treatment of theSMA recovery forceThere are somemethods mainly includ-ing pseudo temperature load method [22] negative thermalexpansion coefficient method [23] equivalent eccentric forcemethod [24] and equivalent reinforcement stress method[25 26]

There is no doubt that the FEM is the most widely usednumerical method so far in which the whole structure isdiscretized with meshes Due to the dependence of meshgeneration there are some drawbacks such as complicatedsolution extensive calculation and being time consuming forthe FEM inevitably [27] while the meshless method canovercome these limitations when analyzing regular struc-tures In the meshless method a set of points is used todiscretize the solution region and construct approximationfunctions which can eliminate meshes completely or partlyand need no initiation and repetition of meshes Thus thecalculation accuracy of the meshless method is ensured andthe calculation difficulty is reduced In the literature [28ndash30]the mechanical analysis of various structures is performedwith different meshless methods and a series of effectiveconclusions are obtained Thereinto the spline function

method which is semianalytical and semidiscrete utilizesspline functions to construct displacement interpolationfunctions and has a widespread applicability and feasibility indifferent structural analysis problems Liu et al [31] used B-spline finite pointmethod (B-sFPM) to analyze the free trans-verse vibration of axially functionally graded tapered Euler-Bernoulli beams Compared with the FEM the B-sFPM hasthe advantages of high efficiency and low computationalcomplexity Based on the B-sFPM Li et al [32] proposedthe bidirectional B-spline finite point method (BB-sFPM)namely the spline meshless method for the parameter iden-tification of piezoelectric laminated plates This methoddiscretized the structure along two directions with splinenodes which is more accurate and effective in calculation

Combining the advantages of the sFPM and the FEMProfessor Qin [33] put forward the unidirectional B-splineQR method (B-sQRM) named after him which has beensuccessfully implemented in the dynamic static and stabilityanalysis of various structures The B-sQRM is different fromthe sFPM In theB-sQRM one of the directions of the integralstructure is discretized by uniformly scattered spline nodesand the other is discretized by displacement shape functionswhich can reflect the deformation regularity of the structureAnd the structural displacement function is constituted by alinear combination of the product of displacement functionsand cubic B-spline functions which are compact and ofhigh order Moreover by means of element interpolationfunctions element potential energy functions and elementstiffness matrices of the FEM the displacement of elementnodes is expressed by the integral displacement of the B-sQRMThen the stiffness equation of the entire structure canbe established by the total potential energy function andthe principle of minimum potential energy Although the B-spline QRmethod depends on the discrete mesh and elementaccuracy of the FEM the number of unknowns of the B-sQRM is only related to the number of spline nodes and theseries of orthogonal polynomials also it has nothing to dowith the number of elements and the total number of nodedisplacements Therefore the B-sQRM is of simplicity andconvenience in calculation Based on the above methods thebidirectional B-spline QR method (BB-sQRM) is proposedin this paper The principle of the BB-sQRM is the sameas the B-sQRM but the discretization is performed withspline nodes in both two directions of the structure It hasthe characteristics of meshless method and can improvethe calculation accuracy while the calculation efficiency isguaranteed

In view of fact that the numerical analysis for nonlinearbehaviors of SMA smart concrete beams is less and theemployment of the FEM in modeling and analysis is moretherefore in order to enrich the theoretical system of SMAconcrete structures and to increase the diversity of methodsthe BB-sQRM is adopted in this paper to perform a nonlinearanalysis of the SMA smart concrete beam which ensuresthe accuracy of the results while the computational scale isreduced and the efficiency of numerical analysis improvedBased on the classical incremental elastoplastic theory ofthe RC the stress-strain relations of elements of concreteand concrete with steel bars under different stress conditions


0

1

2

1 2 3 4 5 6

M

N y

x







1

u1

2

u2

3

u3

4

u4



119906 = 119872sum119895=0

119873sum119894=0

119906119892119894119895120601119894 (119909) 120595119895 (119910) = [120595] otimes [120601] 119906119892

V = 119872sum119895=0

119873sum119894=0

V119892119894119895120601119894 (119909) 120595119895 (119910) = [120595] otimes [120601] V119892 (1)



119881 = [119873] 120575 (2)



(3)


(4)








119909 minus 1199090ℎ minus 119894 + 1)+ 161205933 (


(5)



120601119894 (119909119896) = 1 119894 = 1198960 119894 = 119896 (6)




119881119890 = [1199061 V1 1199062 V2 1199063 V3 1199064 V4]119879 (7)


119881119890 = [[119873]1 [119873]2 [119873]3 [119873]4]119879 [119906119892

V119892]= [119873]119890 120575

(8)






[119863]1198901199011 = [119863] minus [119863]1199011 = [119863] ([119868] minus [119876]1199011) (9)


[119863] = 1198641 minus 1205832[[[[[

1 120583 0120583 1 00 0 1 minus 120583

2

]]]]] (10)


[119876]1199011 = [120597119865120597 [120590]] [120597119865120597 [120590]]119879 [119863]119860 + [120597119865120597 [120590]]119879 [119863] [120597119865120597 [120590]] (11)



[119876]1199011

= 120572[[[[

119886 (119886 + 119887120583) 119886 (119887 + 119886120583) 18119886120591119909119910 (1 minus 120583)119887 (119886 + 119887120583) 119887 (119887 + 119886120583) 18119887120591119909119910 (1 minus 120583)

36120591119909119910 (119886 + 119887120583) 36120591119909119910 (119887 + 119886120583) 6481205912119909119910 (1 minus 120583)]]]]

(12)

in which

120572 = 1198649119860 (1 minus 1205832) minus 2119864ℎ

119886 = 1198600 + 7120590119909 minus 120590119910119887 = 1198600 minus 120590119909 + 7120590119910ℎ = (11986002 + 61198600120590119909 + 61198600120590119910) (120583 + 1)


y

x

x(1)y(2)



1198600 = 1198912119887119888 minus 11989121198882119891119887119888 minus 119891119888

119860 = 119864119864119905119864 minus 119864119905 (13)






in which

[119876]1199012 = [[[

0 0 0V 1 00 0 1 minus 120573

]]] (15)



[119863]1198901199012 = [119879]119879 [119863lowast]119890119901 [119879] (16)


[119879] = [[[[





[119863]1199012 = [119879]119879 [119863lowast]1199012 [119879] (18)

in which

[119863lowast]1199012 = [119863lowast] [119876]1199012 (19)




1205900 = [119863]1199011 120576 (20)



120590119877 = [[119868] minus 119901 (120574) 119902 (120574)119879]

120590119909120590119910120591119909119910 (21)

in which




y

x

A

B

b1

b2l2

a1

a2l1

u2 = 1

ub

1 2

3

4




[119870119888]119890 = int+1

minus1int+1minus1[119861]119879119890 [119863] [119861]119879119890 |119869| 119905 119889120585 119889120578 (23)



[119870119904]119890 = [119877]119879 [119870119904]119890 [119877] (24)


[119877] =[[[[[[[[[[[

11988611198971 0 0 0 0 0 11988621198971 00 11988611198971 0 0 0 0 0 119886211989710 0 11988711198972 0 11988721198972 0 0 00 0 0 11988711198972 0 11988721198972 0 0

]]]]]]]]]]]

(25)




[119870119904]119890 = 119864119860119897[[[[[[[

1198881015840211988810158401199041015840minus11988810158402minus11988810158401199041015840

1198881015840119904101584011990410158402minus11988810158401199041015840minus11990410158402

minus11988810158402minus119888101584011990410158401198881015840211988810158401199041015840

minus11988810158401199041015840minus119904101584021198881015840119904101584011990410158402

]]]]]]] (26)


[119870119904119888]119890 = [119870119888]119890 + [119870119904]119890 (27)


Π119890119896 = 12 119881119904119888119879119890 [119870119904119888]119890 119881119904119888119890minus 119881119904119888119879119890 (119891119904119888119890 + 119891119901119904119888119890 + 119891119877119904119888119890)

Π119890119896 = 12 119881119888119879119890 [119870119888]119890 119881119888119890minus 119881119888119879119890 (119891119888119890 + 119891119901119888 119890 + 119891119877119888 119890)

(28)


119891119901119896 119890 = int119890 [119861]119879119890 1205900 119889Ω119891119877119896 119890 = int119890 [119861]119879119890 120590119877 119889Ω

(29)



Π119890 = 12 120575119879 [119866]119890 120575 minus 120575119879 (119865119890 + 119865119901119890 + 119865119877119890) (30)

in which

[119870]119890 = [119873]119879119890 [119870119904119888]119890 [119873]119890 + [119873]119879119890 [119870119888]119890 0 [119873]119890 119865119890 = [119873]119879119890 119891119904119888119890 + [119873]119879119890 119891119888119890 119865119901119890 = [119873]119879119890 119891119901119904119888119890 + [119873]119879119890 119891119901119888 119890 119865119877119890= [119873]119879119890 119891119877119904119888119890 + [119873]119879119890 119891119877119888 119890

(31)





Π = 12 120575119879 [119870] 120575 minus 120575119879 (119865 + 119865119901 + 119865119877) (32)

in which

[119870] = 119899sum119890=1

[119866]119890

[119865] = 119899sum119890=1

119865119890

[119865119901] = 119899sum119890=1

119865119901119890

[119865119877] = 119899sum119890=1

119865119877119890

(33)


[119870] 120575 = 119865 + 119865119901 + 119865119877 (34)


120575 = [119870]minus1 (119865 + 119865119901 + 119865119877) 120590 = [119863]119890119901 [119861] [119873]119890 120575 (35)




1198891205751 = [1198701]minus1 1198651 (36)


1198891205901 = [119863] [119861]119890 [119873]119890 1198891205751 (37)







1198891198651 = 119889119865119901 + 119889119865119877 (38)




10038171003817100381710038171003817119889120575119895110038171003817100381710038171003817 lt 10minus8 (39)




1198891205752 = [1198702]minus1 1198652 (40)









119891SMA = [119873]119894 119879SMA

0 (41)



[119870] 120575 = 119865 + 119865119901 + 119865119877 + 119865SMA (42)

in which

119865SMA =119899sum119890=1

[119873]119879119890 119879SMA

0 (43)




2

10

050

100150200250300350400450500550600650

Reco

very

stre

ss (M

Pa)

40 60 80 100 120 140 160 180 200 220 24020Temperature (∘C)

4 6

8 9







SMAsℎs

500

200

5empty4

L

q = 22 kNm

Steel bars






119879SMA = 119899120590119903119860SMA119872SMA = 119879SMAℎ119904 (45)







DesignationBeam

dimension(mm)

Concrete cover(mm)







ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()







BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









0

1

2

1 2 3 4 5 6

M

N y

x







1

u1

2

u2

3

u3

4

u4



119906 = 119872sum119895=0

119873sum119894=0

119906119892119894119895120601119894 (119909) 120595119895 (119910) = [120595] otimes [120601] 119906119892

V = 119872sum119895=0

119873sum119894=0

V119892119894119895120601119894 (119909) 120595119895 (119910) = [120595] otimes [120601] V119892 (1)



119881 = [119873] 120575 (2)



(3)


(4)








119909 minus 1199090ℎ minus 119894 + 1)+ 161205933 (


(5)



120601119894 (119909119896) = 1 119894 = 1198960 119894 = 119896 (6)




119881119890 = [1199061 V1 1199062 V2 1199063 V3 1199064 V4]119879 (7)


119881119890 = [[119873]1 [119873]2 [119873]3 [119873]4]119879 [119906119892

V119892]= [119873]119890 120575

(8)






[119863]1198901199011 = [119863] minus [119863]1199011 = [119863] ([119868] minus [119876]1199011) (9)


[119863] = 1198641 minus 1205832[[[[[

1 120583 0120583 1 00 0 1 minus 120583

2

]]]]] (10)


[119876]1199011 = [120597119865120597 [120590]] [120597119865120597 [120590]]119879 [119863]119860 + [120597119865120597 [120590]]119879 [119863] [120597119865120597 [120590]] (11)



[119876]1199011

= 120572[[[[

119886 (119886 + 119887120583) 119886 (119887 + 119886120583) 18119886120591119909119910 (1 minus 120583)119887 (119886 + 119887120583) 119887 (119887 + 119886120583) 18119887120591119909119910 (1 minus 120583)

36120591119909119910 (119886 + 119887120583) 36120591119909119910 (119887 + 119886120583) 6481205912119909119910 (1 minus 120583)]]]]

(12)

in which

120572 = 1198649119860 (1 minus 1205832) minus 2119864ℎ

119886 = 1198600 + 7120590119909 minus 120590119910119887 = 1198600 minus 120590119909 + 7120590119910ℎ = (11986002 + 61198600120590119909 + 61198600120590119910) (120583 + 1)


y

x

x(1)y(2)



1198600 = 1198912119887119888 minus 11989121198882119891119887119888 minus 119891119888

119860 = 119864119864119905119864 minus 119864119905 (13)






in which

[119876]1199012 = [[[

0 0 0V 1 00 0 1 minus 120573

]]] (15)



[119863]1198901199012 = [119879]119879 [119863lowast]119890119901 [119879] (16)


[119879] = [[[[





[119863]1199012 = [119879]119879 [119863lowast]1199012 [119879] (18)

in which

[119863lowast]1199012 = [119863lowast] [119876]1199012 (19)




1205900 = [119863]1199011 120576 (20)



120590119877 = [[119868] minus 119901 (120574) 119902 (120574)119879]

120590119909120590119910120591119909119910 (21)

in which




y

x

A

B

b1

b2l2

a1

a2l1

u2 = 1

ub

1 2

3

4




[119870119888]119890 = int+1

minus1int+1minus1[119861]119879119890 [119863] [119861]119879119890 |119869| 119905 119889120585 119889120578 (23)



[119870119904]119890 = [119877]119879 [119870119904]119890 [119877] (24)


[119877] =[[[[[[[[[[[

11988611198971 0 0 0 0 0 11988621198971 00 11988611198971 0 0 0 0 0 119886211989710 0 11988711198972 0 11988721198972 0 0 00 0 0 11988711198972 0 11988721198972 0 0

]]]]]]]]]]]

(25)




[119870119904]119890 = 119864119860119897[[[[[[[

1198881015840211988810158401199041015840minus11988810158402minus11988810158401199041015840

1198881015840119904101584011990410158402minus11988810158401199041015840minus11990410158402

minus11988810158402minus119888101584011990410158401198881015840211988810158401199041015840

minus11988810158401199041015840minus119904101584021198881015840119904101584011990410158402

]]]]]]] (26)


[119870119904119888]119890 = [119870119888]119890 + [119870119904]119890 (27)


Π119890119896 = 12 119881119904119888119879119890 [119870119904119888]119890 119881119904119888119890minus 119881119904119888119879119890 (119891119904119888119890 + 119891119901119904119888119890 + 119891119877119904119888119890)

Π119890119896 = 12 119881119888119879119890 [119870119888]119890 119881119888119890minus 119881119888119879119890 (119891119888119890 + 119891119901119888 119890 + 119891119877119888 119890)

(28)


119891119901119896 119890 = int119890 [119861]119879119890 1205900 119889Ω119891119877119896 119890 = int119890 [119861]119879119890 120590119877 119889Ω

(29)



Π119890 = 12 120575119879 [119866]119890 120575 minus 120575119879 (119865119890 + 119865119901119890 + 119865119877119890) (30)

in which

[119870]119890 = [119873]119879119890 [119870119904119888]119890 [119873]119890 + [119873]119879119890 [119870119888]119890 0 [119873]119890 119865119890 = [119873]119879119890 119891119904119888119890 + [119873]119879119890 119891119888119890 119865119901119890 = [119873]119879119890 119891119901119904119888119890 + [119873]119879119890 119891119901119888 119890 119865119877119890= [119873]119879119890 119891119877119904119888119890 + [119873]119879119890 119891119877119888 119890

(31)





Π = 12 120575119879 [119870] 120575 minus 120575119879 (119865 + 119865119901 + 119865119877) (32)

in which

[119870] = 119899sum119890=1

[119866]119890

[119865] = 119899sum119890=1

119865119890

[119865119901] = 119899sum119890=1

119865119901119890

[119865119877] = 119899sum119890=1

119865119877119890

(33)


[119870] 120575 = 119865 + 119865119901 + 119865119877 (34)


120575 = [119870]minus1 (119865 + 119865119901 + 119865119877) 120590 = [119863]119890119901 [119861] [119873]119890 120575 (35)




1198891205751 = [1198701]minus1 1198651 (36)


1198891205901 = [119863] [119861]119890 [119873]119890 1198891205751 (37)







1198891198651 = 119889119865119901 + 119889119865119877 (38)




10038171003817100381710038171003817119889120575119895110038171003817100381710038171003817 lt 10minus8 (39)




1198891205752 = [1198702]minus1 1198652 (40)









119891SMA = [119873]119894 119879SMA

0 (41)



[119870] 120575 = 119865 + 119865119901 + 119865119877 + 119865SMA (42)

in which

119865SMA =119899sum119890=1

[119873]119879119890 119879SMA

0 (43)




2

10

050

100150200250300350400450500550600650

Reco

very

stre

ss (M

Pa)

40 60 80 100 120 140 160 180 200 220 24020Temperature (∘C)

4 6

8 9







SMAsℎs

500

200

5empty4

L

q = 22 kNm

Steel bars






119879SMA = 119899120590119903119860SMA119872SMA = 119879SMAℎ119904 (45)







DesignationBeam

dimension(mm)

Concrete cover(mm)







ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()







BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018













119909 minus 1199090ℎ minus 119894 + 1)+ 161205933 (


(5)



120601119894 (119909119896) = 1 119894 = 1198960 119894 = 119896 (6)




119881119890 = [1199061 V1 1199062 V2 1199063 V3 1199064 V4]119879 (7)


119881119890 = [[119873]1 [119873]2 [119873]3 [119873]4]119879 [119906119892

V119892]= [119873]119890 120575

(8)






[119863]1198901199011 = [119863] minus [119863]1199011 = [119863] ([119868] minus [119876]1199011) (9)


[119863] = 1198641 minus 1205832[[[[[

1 120583 0120583 1 00 0 1 minus 120583

2

]]]]] (10)


[119876]1199011 = [120597119865120597 [120590]] [120597119865120597 [120590]]119879 [119863]119860 + [120597119865120597 [120590]]119879 [119863] [120597119865120597 [120590]] (11)



[119876]1199011

= 120572[[[[

119886 (119886 + 119887120583) 119886 (119887 + 119886120583) 18119886120591119909119910 (1 minus 120583)119887 (119886 + 119887120583) 119887 (119887 + 119886120583) 18119887120591119909119910 (1 minus 120583)

36120591119909119910 (119886 + 119887120583) 36120591119909119910 (119887 + 119886120583) 6481205912119909119910 (1 minus 120583)]]]]

(12)

in which

120572 = 1198649119860 (1 minus 1205832) minus 2119864ℎ

119886 = 1198600 + 7120590119909 minus 120590119910119887 = 1198600 minus 120590119909 + 7120590119910ℎ = (11986002 + 61198600120590119909 + 61198600120590119910) (120583 + 1)


y

x

x(1)y(2)



1198600 = 1198912119887119888 minus 11989121198882119891119887119888 minus 119891119888

119860 = 119864119864119905119864 minus 119864119905 (13)






in which

[119876]1199012 = [[[

0 0 0V 1 00 0 1 minus 120573

]]] (15)



[119863]1198901199012 = [119879]119879 [119863lowast]119890119901 [119879] (16)


[119879] = [[[[





[119863]1199012 = [119879]119879 [119863lowast]1199012 [119879] (18)

in which

[119863lowast]1199012 = [119863lowast] [119876]1199012 (19)




1205900 = [119863]1199011 120576 (20)



120590119877 = [[119868] minus 119901 (120574) 119902 (120574)119879]

120590119909120590119910120591119909119910 (21)

in which




y

x

A

B

b1

b2l2

a1

a2l1

u2 = 1

ub

1 2

3

4




[119870119888]119890 = int+1

minus1int+1minus1[119861]119879119890 [119863] [119861]119879119890 |119869| 119905 119889120585 119889120578 (23)



[119870119904]119890 = [119877]119879 [119870119904]119890 [119877] (24)


[119877] =[[[[[[[[[[[

11988611198971 0 0 0 0 0 11988621198971 00 11988611198971 0 0 0 0 0 119886211989710 0 11988711198972 0 11988721198972 0 0 00 0 0 11988711198972 0 11988721198972 0 0

]]]]]]]]]]]

(25)




[119870119904]119890 = 119864119860119897[[[[[[[

1198881015840211988810158401199041015840minus11988810158402minus11988810158401199041015840

1198881015840119904101584011990410158402minus11988810158401199041015840minus11990410158402

minus11988810158402minus119888101584011990410158401198881015840211988810158401199041015840

minus11988810158401199041015840minus119904101584021198881015840119904101584011990410158402

]]]]]]] (26)


[119870119904119888]119890 = [119870119888]119890 + [119870119904]119890 (27)


Π119890119896 = 12 119881119904119888119879119890 [119870119904119888]119890 119881119904119888119890minus 119881119904119888119879119890 (119891119904119888119890 + 119891119901119904119888119890 + 119891119877119904119888119890)

Π119890119896 = 12 119881119888119879119890 [119870119888]119890 119881119888119890minus 119881119888119879119890 (119891119888119890 + 119891119901119888 119890 + 119891119877119888 119890)

(28)


119891119901119896 119890 = int119890 [119861]119879119890 1205900 119889Ω119891119877119896 119890 = int119890 [119861]119879119890 120590119877 119889Ω

(29)



Π119890 = 12 120575119879 [119866]119890 120575 minus 120575119879 (119865119890 + 119865119901119890 + 119865119877119890) (30)

in which

[119870]119890 = [119873]119879119890 [119870119904119888]119890 [119873]119890 + [119873]119879119890 [119870119888]119890 0 [119873]119890 119865119890 = [119873]119879119890 119891119904119888119890 + [119873]119879119890 119891119888119890 119865119901119890 = [119873]119879119890 119891119901119904119888119890 + [119873]119879119890 119891119901119888 119890 119865119877119890= [119873]119879119890 119891119877119904119888119890 + [119873]119879119890 119891119877119888 119890

(31)





Π = 12 120575119879 [119870] 120575 minus 120575119879 (119865 + 119865119901 + 119865119877) (32)

in which

[119870] = 119899sum119890=1

[119866]119890

[119865] = 119899sum119890=1

119865119890

[119865119901] = 119899sum119890=1

119865119901119890

[119865119877] = 119899sum119890=1

119865119877119890

(33)


[119870] 120575 = 119865 + 119865119901 + 119865119877 (34)


120575 = [119870]minus1 (119865 + 119865119901 + 119865119877) 120590 = [119863]119890119901 [119861] [119873]119890 120575 (35)




1198891205751 = [1198701]minus1 1198651 (36)


1198891205901 = [119863] [119861]119890 [119873]119890 1198891205751 (37)







1198891198651 = 119889119865119901 + 119889119865119877 (38)




10038171003817100381710038171003817119889120575119895110038171003817100381710038171003817 lt 10minus8 (39)




1198891205752 = [1198702]minus1 1198652 (40)









119891SMA = [119873]119894 119879SMA

0 (41)



[119870] 120575 = 119865 + 119865119901 + 119865119877 + 119865SMA (42)

in which

119865SMA =119899sum119890=1

[119873]119879119890 119879SMA

0 (43)




2

10

050

100150200250300350400450500550600650

Reco

very

stre

ss (M

Pa)

40 60 80 100 120 140 160 180 200 220 24020Temperature (∘C)

4 6

8 9







SMAsℎs

500

200

5empty4

L

q = 22 kNm

Steel bars






119879SMA = 119899120590119903119860SMA119872SMA = 119879SMAℎ119904 (45)







DesignationBeam

dimension(mm)

Concrete cover(mm)







ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()







BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









y

x

x(1)y(2)



1198600 = 1198912119887119888 minus 11989121198882119891119887119888 minus 119891119888

119860 = 119864119864119905119864 minus 119864119905 (13)






in which

[119876]1199012 = [[[

0 0 0V 1 00 0 1 minus 120573

]]] (15)



[119863]1198901199012 = [119879]119879 [119863lowast]119890119901 [119879] (16)


[119879] = [[[[





[119863]1199012 = [119879]119879 [119863lowast]1199012 [119879] (18)

in which

[119863lowast]1199012 = [119863lowast] [119876]1199012 (19)




1205900 = [119863]1199011 120576 (20)



120590119877 = [[119868] minus 119901 (120574) 119902 (120574)119879]

120590119909120590119910120591119909119910 (21)

in which




y

x

A

B

b1

b2l2

a1

a2l1

u2 = 1

ub

1 2

3

4




[119870119888]119890 = int+1

minus1int+1minus1[119861]119879119890 [119863] [119861]119879119890 |119869| 119905 119889120585 119889120578 (23)



[119870119904]119890 = [119877]119879 [119870119904]119890 [119877] (24)


[119877] =[[[[[[[[[[[

11988611198971 0 0 0 0 0 11988621198971 00 11988611198971 0 0 0 0 0 119886211989710 0 11988711198972 0 11988721198972 0 0 00 0 0 11988711198972 0 11988721198972 0 0

]]]]]]]]]]]

(25)




[119870119904]119890 = 119864119860119897[[[[[[[

1198881015840211988810158401199041015840minus11988810158402minus11988810158401199041015840

1198881015840119904101584011990410158402minus11988810158401199041015840minus11990410158402

minus11988810158402minus119888101584011990410158401198881015840211988810158401199041015840

minus11988810158401199041015840minus119904101584021198881015840119904101584011990410158402

]]]]]]] (26)


[119870119904119888]119890 = [119870119888]119890 + [119870119904]119890 (27)


Π119890119896 = 12 119881119904119888119879119890 [119870119904119888]119890 119881119904119888119890minus 119881119904119888119879119890 (119891119904119888119890 + 119891119901119904119888119890 + 119891119877119904119888119890)

Π119890119896 = 12 119881119888119879119890 [119870119888]119890 119881119888119890minus 119881119888119879119890 (119891119888119890 + 119891119901119888 119890 + 119891119877119888 119890)

(28)


119891119901119896 119890 = int119890 [119861]119879119890 1205900 119889Ω119891119877119896 119890 = int119890 [119861]119879119890 120590119877 119889Ω

(29)



Π119890 = 12 120575119879 [119866]119890 120575 minus 120575119879 (119865119890 + 119865119901119890 + 119865119877119890) (30)

in which

[119870]119890 = [119873]119879119890 [119870119904119888]119890 [119873]119890 + [119873]119879119890 [119870119888]119890 0 [119873]119890 119865119890 = [119873]119879119890 119891119904119888119890 + [119873]119879119890 119891119888119890 119865119901119890 = [119873]119879119890 119891119901119904119888119890 + [119873]119879119890 119891119901119888 119890 119865119877119890= [119873]119879119890 119891119877119904119888119890 + [119873]119879119890 119891119877119888 119890

(31)





Π = 12 120575119879 [119870] 120575 minus 120575119879 (119865 + 119865119901 + 119865119877) (32)

in which

[119870] = 119899sum119890=1

[119866]119890

[119865] = 119899sum119890=1

119865119890

[119865119901] = 119899sum119890=1

119865119901119890

[119865119877] = 119899sum119890=1

119865119877119890

(33)


[119870] 120575 = 119865 + 119865119901 + 119865119877 (34)


120575 = [119870]minus1 (119865 + 119865119901 + 119865119877) 120590 = [119863]119890119901 [119861] [119873]119890 120575 (35)




1198891205751 = [1198701]minus1 1198651 (36)


1198891205901 = [119863] [119861]119890 [119873]119890 1198891205751 (37)







1198891198651 = 119889119865119901 + 119889119865119877 (38)




10038171003817100381710038171003817119889120575119895110038171003817100381710038171003817 lt 10minus8 (39)




1198891205752 = [1198702]minus1 1198652 (40)









119891SMA = [119873]119894 119879SMA

0 (41)



[119870] 120575 = 119865 + 119865119901 + 119865119877 + 119865SMA (42)

in which

119865SMA =119899sum119890=1

[119873]119879119890 119879SMA

0 (43)




2

10

050

100150200250300350400450500550600650

Reco

very

stre

ss (M

Pa)

40 60 80 100 120 140 160 180 200 220 24020Temperature (∘C)

4 6

8 9







SMAsℎs

500

200

5empty4

L

q = 22 kNm

Steel bars






119879SMA = 119899120590119903119860SMA119872SMA = 119879SMAℎ119904 (45)







DesignationBeam

dimension(mm)

Concrete cover(mm)







ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()







BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









y

x

A

B

b1

b2l2

a1

a2l1

u2 = 1

ub

1 2

3

4




[119870119888]119890 = int+1

minus1int+1minus1[119861]119879119890 [119863] [119861]119879119890 |119869| 119905 119889120585 119889120578 (23)



[119870119904]119890 = [119877]119879 [119870119904]119890 [119877] (24)


[119877] =[[[[[[[[[[[

11988611198971 0 0 0 0 0 11988621198971 00 11988611198971 0 0 0 0 0 119886211989710 0 11988711198972 0 11988721198972 0 0 00 0 0 11988711198972 0 11988721198972 0 0

]]]]]]]]]]]

(25)




[119870119904]119890 = 119864119860119897[[[[[[[

1198881015840211988810158401199041015840minus11988810158402minus11988810158401199041015840

1198881015840119904101584011990410158402minus11988810158401199041015840minus11990410158402

minus11988810158402minus119888101584011990410158401198881015840211988810158401199041015840

minus11988810158401199041015840minus119904101584021198881015840119904101584011990410158402

]]]]]]] (26)


[119870119904119888]119890 = [119870119888]119890 + [119870119904]119890 (27)


Π119890119896 = 12 119881119904119888119879119890 [119870119904119888]119890 119881119904119888119890minus 119881119904119888119879119890 (119891119904119888119890 + 119891119901119904119888119890 + 119891119877119904119888119890)

Π119890119896 = 12 119881119888119879119890 [119870119888]119890 119881119888119890minus 119881119888119879119890 (119891119888119890 + 119891119901119888 119890 + 119891119877119888 119890)

(28)


119891119901119896 119890 = int119890 [119861]119879119890 1205900 119889Ω119891119877119896 119890 = int119890 [119861]119879119890 120590119877 119889Ω

(29)



Π119890 = 12 120575119879 [119866]119890 120575 minus 120575119879 (119865119890 + 119865119901119890 + 119865119877119890) (30)

in which

[119870]119890 = [119873]119879119890 [119870119904119888]119890 [119873]119890 + [119873]119879119890 [119870119888]119890 0 [119873]119890 119865119890 = [119873]119879119890 119891119904119888119890 + [119873]119879119890 119891119888119890 119865119901119890 = [119873]119879119890 119891119901119904119888119890 + [119873]119879119890 119891119901119888 119890 119865119877119890= [119873]119879119890 119891119877119904119888119890 + [119873]119879119890 119891119877119888 119890

(31)





Π = 12 120575119879 [119870] 120575 minus 120575119879 (119865 + 119865119901 + 119865119877) (32)

in which

[119870] = 119899sum119890=1

[119866]119890

[119865] = 119899sum119890=1

119865119890

[119865119901] = 119899sum119890=1

119865119901119890

[119865119877] = 119899sum119890=1

119865119877119890

(33)


[119870] 120575 = 119865 + 119865119901 + 119865119877 (34)


120575 = [119870]minus1 (119865 + 119865119901 + 119865119877) 120590 = [119863]119890119901 [119861] [119873]119890 120575 (35)




1198891205751 = [1198701]minus1 1198651 (36)


1198891205901 = [119863] [119861]119890 [119873]119890 1198891205751 (37)







1198891198651 = 119889119865119901 + 119889119865119877 (38)




10038171003817100381710038171003817119889120575119895110038171003817100381710038171003817 lt 10minus8 (39)




1198891205752 = [1198702]minus1 1198652 (40)









119891SMA = [119873]119894 119879SMA

0 (41)



[119870] 120575 = 119865 + 119865119901 + 119865119877 + 119865SMA (42)

in which

119865SMA =119899sum119890=1

[119873]119879119890 119879SMA

0 (43)




2

10

050

100150200250300350400450500550600650

Reco

very

stre

ss (M

Pa)

40 60 80 100 120 140 160 180 200 220 24020Temperature (∘C)

4 6

8 9







SMAsℎs

500

200

5empty4

L

q = 22 kNm

Steel bars






119879SMA = 119899120590119903119860SMA119872SMA = 119879SMAℎ119904 (45)







DesignationBeam

dimension(mm)

Concrete cover(mm)







ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()







BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018











Π = 12 120575119879 [119870] 120575 minus 120575119879 (119865 + 119865119901 + 119865119877) (32)

in which

[119870] = 119899sum119890=1

[119866]119890

[119865] = 119899sum119890=1

119865119890

[119865119901] = 119899sum119890=1

119865119901119890

[119865119877] = 119899sum119890=1

119865119877119890

(33)


[119870] 120575 = 119865 + 119865119901 + 119865119877 (34)


120575 = [119870]minus1 (119865 + 119865119901 + 119865119877) 120590 = [119863]119890119901 [119861] [119873]119890 120575 (35)




1198891205751 = [1198701]minus1 1198651 (36)


1198891205901 = [119863] [119861]119890 [119873]119890 1198891205751 (37)







1198891198651 = 119889119865119901 + 119889119865119877 (38)




10038171003817100381710038171003817119889120575119895110038171003817100381710038171003817 lt 10minus8 (39)




1198891205752 = [1198702]minus1 1198652 (40)









119891SMA = [119873]119894 119879SMA

0 (41)



[119870] 120575 = 119865 + 119865119901 + 119865119877 + 119865SMA (42)

in which

119865SMA =119899sum119890=1

[119873]119879119890 119879SMA

0 (43)




2

10

050

100150200250300350400450500550600650

Reco

very

stre

ss (M

Pa)

40 60 80 100 120 140 160 180 200 220 24020Temperature (∘C)

4 6

8 9







SMAsℎs

500

200

5empty4

L

q = 22 kNm

Steel bars






119879SMA = 119899120590119903119860SMA119872SMA = 119879SMAℎ119904 (45)







DesignationBeam

dimension(mm)

Concrete cover(mm)







ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()







BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018












119891SMA = [119873]119894 119879SMA

0 (41)



[119870] 120575 = 119865 + 119865119901 + 119865119877 + 119865SMA (42)

in which

119865SMA =119899sum119890=1

[119873]119879119890 119879SMA

0 (43)




2

10

050

100150200250300350400450500550600650

Reco

very

stre

ss (M

Pa)

40 60 80 100 120 140 160 180 200 220 24020Temperature (∘C)

4 6

8 9







SMAsℎs

500

200

5empty4

L

q = 22 kNm

Steel bars






119879SMA = 119899120590119903119860SMA119872SMA = 119879SMAℎ119904 (45)







DesignationBeam

dimension(mm)

Concrete cover(mm)







ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()







BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018











SMAsℎs

500

200

5empty4

L

q = 22 kNm

Steel bars






119879SMA = 119899120590119903119860SMA119872SMA = 119879SMAℎ119904 (45)







DesignationBeam

dimension(mm)

Concrete cover(mm)







ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()







BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










DesignationBeam

dimension(mm)

Concrete cover(mm)







ANSYS

BB-sQRM

20∘C 60∘C100∘C 100∘C

minus6

minus4

minus2

0

2

4

6

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()







BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









BB-sQRMANSYS



96

98

100

102

104

106

108

110

112M

axim

um st

rain

of b

ar el

emen

ts (times

10minus4)

(a)

BB-sQRMANSYS



88

90

92

94

96

98

100

102

104

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(b)

BB-sQRMANSYS



80

82

84

86

88

90

92

94

Max

imum

stra

in o

f bar

elem

ents

(times10

minus4)

(c)



20 40 60 80 100 120

BB-sQRM








The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018













The r

ecov

ery

rate

of c

rack

wid

th (

)

Temperature (∘C)

24

20

16

12

8

4

020 40 60 80 100 120

1 81 101 12





Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

65mm45mm25mm








Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018













Temperature (∘C)20 40 60 80 100 120

0

4

8

12

16

20

24

The r

ecov

ery

rate

of c

rack

wid

th (

)

046076114












Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018













Max

imum

crac

k w

idth

(mm

)

Temperature (∘C)

021

020

019

018

017

016

20 40 60 80 100 120 140 160 180

468




Temperature (∘C)20 40 60 80 100 120

017

018

019

020

021

022

Max

imum

crac

k w

idth

(mm

)

110 mm160mm220mm



6 Conclusions






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018












Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018
































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Nonlinear Analysis for the Crack Control of SMA Smart