Development of nonlinear transfer matrix method for...

Scientia Iranica A (2015) 22(3), 639{649

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Development of nonlinear transfer matrix method forinelastic analyses of beams

M. Kwona, J. Kima;�, H. Seoa and S. Limkatanyub

a. Department of Civil Engineering, ERI, Gyeongsang National University, South Korea.b. Department of Civil Engineering, Faculty of Engineering, Prince of Songkla University, Thailand.

Received 6 August 2013; received in revised form 10 June 2014; accepted 20 October 2014

KEYWORDSMaterial nonlinear;Sti�ness matrix;TMM (TransferMatrix Method);Nonlinear analysis;Beam model.

Abstract. The objective of this study is to develop a material-nonlinear-analysisalgorithm based on the Transfer Matrix Method (TMM). This newly developed algorithmcan be used to perform nonlinear analyses of continuous beam systems. The nonlineartransfer matrix is derived from the general frame sti�ness matrix, and the Gauss-Lobattointegration scheme is employed for numerical integration. In the TMM, the systemequation has a constant number of system unknowns, regardless of the total degree-of-freedom number in the structure and the system response (either linear or nonlinear).As a result, TMM can be used e�ciently, both for linear and nonlinear structuralanalyses. In this study, a secant nonlinear algorithm, required in nonlinear TMM, isemployed, due to its good compromise between the convergence rate and numericalstability. To verify the accuracy and e�ciency of the developed TMM, four numericalexamples are selected and analyzed. The analysis results are compared with thoseobtained by the highly accurate exibility-based frame model, in terms of global and localresponses.

c 2015 Sharif University of Technology. All rights reserved.

1. Introduction

Regarded as the most powerful numerical tool, the�nite element method has been widely used by re-searchers to analyze complex structural systems. How-ever, its use by practicing engineers is still limited,due to the considerable experience required by usersto construct a suitable �nite element mesh, interpretanalysis results, and implement a numerical model.Furthermore, computational costs required in analyz-ing complex structural systems are usually high, evenwith recent drastic advances in computer technology.This lies in the fact that numerous Degrees Of Free-dom (DOFs) are required to discretize a complex

*. Corresponding author. Tel.: +82 55 7590538;Fax: +82 55 7721799E-mail address: [email protected] (J. Kim)

structural system, thus, resulting in a large sti�nessmatrix.

In order to remedy this problem, several tech-niques have been developed to reduce the size ofthe associated sti�ness matrix in the standard �niteelement method. These include, for example: thestatic condensation technique [1], the sub-structuringtechnique [2], etc. Alternatively, Cheung [3] proposedthe so-called \Finite Strip Method (FSM)" as a variantof the standard �nite element method to analyzea structure with lesser system unknowns. Anotherstructural-analysis method requiring a smaller systemmatrix was systematized by Pestel and Leckie [4],which is known as the Transfer Matrix Method (TMM).TMM is of particular interest in this study since thesize of the transfer matrix is usually much smaller thanthat of a structural sti�ness matrix and is constant,regardless of the total degree-of-freedom number in

640 M. Kwon et al./Scientia Iranica, Transactions A: Civil Engineering 22 (2015) 639{649

the structure and the system response (either linearor nonlinear). This feature is desirable, especially inthe framework of nonlinear structural analysis in whichhigh computational cost could become an issue. Thetransfer matrix is derived based on the general solutionof the governing di�erential equation [5]. For eachelement, there are two steps needed to be performed.First, the right-end variables are related to the left-end variables through the transfer matrix. Then, theright-end variables of the current element are used asthe boundary conditions for the left-end variables ofthe next element. These two steps are performed ina successive manner until all system unknowns arecomputed [6,7]. Since unknown variables at each nodalpoint can simply be determined using the transfermatrix, the analysis process required in the TMM hasto deal only with the transfer matrix of a constant size,regardless of the element number used to discretizethe system. This feature renders TMM attractivecompared to the standard �nite element method. Oneof the earliest applications of TMM in structuralanalysis was conducted by Holzer [8] to investigate thetorsional vibration problem of crankshafts. Hetenyi [9]also used TMM to study the problem of beams onelastic foundations. During the seventies and eighties,the TMM was successfully applied to the vibrationproblems of plates by several researchers, but waslimited to only linear elastic systems [10-12]. An earlyextension of TMM to nonlinear structural analysiswas the large de ection analysis of plates [13-15]. Tothe authors' knowledge, few researchers have employedTMM to perform inelastic structural analyses. Akintiloand Syngellakis [16] performed inelastic analyses ofreinforced concrete coupled shear walls using TMM.Rosignoli [17] employed TMM to analyze launchedbridges. Pfei�er [18] applied TMM to nonlinearanalyses of planar reinforced and prestressed concreteframes, including axial elongation e�ects. Starossek etal. [19] combined TMM with the displacement methodto perform material and geometrical nonlinear analysesof planar reinforced concrete frames.

In this study, an e�cient numerical algorithmis developed, with TMM incorporated, for materialnonlinear analyses of frame structures. The transfermatrix is derived from the displacement-based elementsti�ness matrix. The nonlinear algorithm implementedis based on the secant iterative scheme, which ismore stable than the widely used Newton-Raphsoniterative scheme. This is especially true when thematerial is subjected to yielding and then softeningstates or behaves elastic perfectly plastic. To verify theaccuracy and e�ciency of the developed TMM model,four numerical examples are investigated. Analysisresults obtained using the TMM are compared withthose obtained using the exibility-based �bre framemodel [20] in terms of global and local responses.

Figure 1. Transfer matrix method diagram [21].

2. Derivation of transfer matrix

2.1. General frame element sti�ness equationThree sets of governing equations used to constructthe frame element sti�ness equation are compatibility,section constitutive, and equilibrium relations. The so-called \Tonti's diagram" of Figure 1 is used to conve-niently represent these governing equations [21]. Thevirtual displacement principle is employed to representthe integral statement of the element equilibrium, thus,resulting in the element sti�ness equation. In theelement sti�ness equation, the element sti�ness matrixserves as the mapping operator between the elementnodal displacements and element nodal forces. Detailsof the derivation of the frame element sti�ness equationcan be found in any �nite element textbook [22].

Figure 2 shows a 2-node planar frame elementcon�guration. For each node, there are three dis-placement DOFs and their work-conjugate forces. The

Figure 2. Element force and deformation.

M. Kwon et al./Scientia Iranica, Transactions A: Civil Engineering 22 (2015) 639{649 641

frame element sti�ness equation is written as:

P = KU: (1)

Following the notation of Figure 2, the element nodaldisplacements are:

U =�UijUjT ; (2)

where Ui = fui vi �igT and Uj = fuj vj �jgT arearrays containing the displacements at nodes i and j,respectively. Their work-conjugate nodal forces aregrouped in the element force vector, P = fPijPjgT .The element sti�ness matrix can be expressed as:

K =ZL

BT (x)k(x)B(x)dx; (3)

where B(x) is the strain-displacement matrix and k(x)is the frame sectional sti�ness matrix which collects theaxial rigidity, EA(x), and exural rigidity, IE(x).

2.2. Partitioned form of the element sti�nessmatrix

In the TMM, the element sti�ness equation is modi�ed,such that nodal forces and displacement quantitiesat each node are grouped in the same vector andare related together through the transfer matrix [23].This could be accomplished by partitioning the strain-displacement matrix, B(x), as:

B(x) =�Bii(x) Bij(x)Bji(x) Bjj(x)

�; (4)

where:

Bii(x) =�dNui(x)dx

0 0�;

Bij(x) =�dNuj(x)dx

0 0�;

Bji(x) =�

0d2Nvi(x)dx2

d2N�i(x)dx2

�;

Bjj(x) =�

0d2Nvj(x)dx2

d2N�j(x)dx2

�; (5)

in which Nui(x), Nuj(x), Nvi(x), Nvj(x), N�i(x) andN�j(x) are polynomial interpolation functions for a 2-node planar frame element and are given in any �niteelement textbook [22].

In accordance with Eq. (4), the element sti�nessmatrix of Eq. (3) can be partitioned as:

K =�Kii KijKji Kjj

�; (6)

where each sub-matrix can be expressed as:

Kii =ZL

Bii(x)TEA(x)Bii(x)dx

+ZL

Bji(x)TEI(x)Bji(x)dx;

Kij =ZL

Bii(x)TEA(x)Bij(x)dx

+ZL

Bji(x)TEI(x)Bjj(x)dx:

Kji = KTij ;

Kjj =ZL

Bij(x)TEA(x)Bij(x)dx

+ZL

Bjj(x)TEI(x)Bjj(x)dx: (7)

2.3. Derivation of transfer matrixBased on the partitioned form of the element sti�nessmatrix of Eq. (6), the element sti�ness equation ofEq. (1) can be partitioned as:�

Pi

Pj

�=�Kii KijKji Kjj

��Ui

Uj

�; (8)

where Pi = fFix Fiy MigT and Pj = fFjx Fjy MjgTare arrays containing the forces at nodes i and j,respectively.

The core idea of the TMM is to modify thesti�ness equation of Eq. (8), such that the forces anddisplacements at node j are expressed in terms offorces and displacements at node i through the transfermatrix. Therefore, the transfer matrix equation can beexpressed in the following form:

Vj = TMVi; (9)

where Vi = fui vi �i Fix Fiy MigT and Vj =fuj vj �j Fjx Fjy MjgT are arrays containing thedisplacements and forces at nodes i and j, respectively,and TM is the transfer matrix, de�ned as:

TM =� �K�1

ij Kii K�1ij

Kji �KjjK�1ij Kii �KjjK�1

ij

�: (10)

2.4. Numerical integration schemeIn this study, the numerical integration required inthe TMM relies on the Gauss-Lobatto integrationscheme, which includes extreme integration points atthe element ends, as shown in Figure 3. This numericalintegration scheme is preferable to the conventionalGauss integration scheme, since it allows sampling ofthe element-end response, which, most likely, becomesinelastic [20]. Thus, each sub-matrix of the element


Figure 3. Gauss-Lobatto integration points.

sti�ness can be written in the Gauss-Lobatto numericalintegration form as:

Kii =nIPXn=1

Bii(�n)TEA(�n)Bii(�n):J:wn

+nIPXn=1

Bji(�n)TEI(�n)Bji(�n):J:wn;

Kij =nIPXn=1

Bii(�n)TEA(�n)Bij(�n):J:wn

+nIPXn=1

Bji(�n)TEI(�n)Bjj(�n):J:wn;

Kji = KTij ;

Kjj =nIPXn=1

Bij(�n)TEA(�n)Bij(�n):J:wn

+nIPXn=1

Bjj(�n)TEI(�n)Bjj(�n):J:wn; (11)

where J is the element Jacobian; nIP is the integrationpoint number; �n is the integration-point position inthe natural coordinate; and wn is the integration-pointweight.

3. TMM (Transfer Matrix Method)

In TMM, displacement and force quantities at eachnode are computed by successively multiplying thetransfer matrix of each element, enforcing the nodalcompatibility with imposed boundary conditions, andsatisfying the nodal equilibrium with external loads.

3.1. Boundary conditionThe boundary conditions commonly imposed on abeam system can be classi�ed as �xed, hinged, rollerand free conditions. The boundary condition matrix,An, reaction-force vector, BCn, and prescribed dis-placement vector, Rn, associated with the boundarycondition at node n are given as [24]:

Fixed:

An =

241 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

35 ;BCn =

�0 0 0 bHn bPn bMn

T ;Rn =

�run rvn r�n

T: (12)

Hinged:

An =�1 0 0 0 0 00 1 0 0 0 0

�;

BCn =�

0 0 0 bHn bPn 0T ;

Rn =�run rvn

T : (13)

Roller:

An =�0 1 0 0 0 0

�;

BCn =�

0 0 0 0 bPn 0T ;

Rn = frvngT : (14)

Free:

An =�0 0 0 0 0 0

�;

BCn =�

0 0 0 0 0 0T ;

Rn = f0gT ; (15)

where variables b and r represent the unknown reactionforce and the prescribed displacement, respectively.

3.2. Transfer matrixThe schematic notion of the transfer matrix methodat each loading stage is conveyed in Figure 4. At ageneric node, n, the right nodal variable vector, VR

n ,is computed as the summation of the load vector, Ln,and boundary condition matrix, BCn.

VRn = Ln + BCn; (16)

where the load vector, Ln = f0 0 0 LHn LPn LMn gT ,collects the horizontal force, LHn , vertical force, LPn ,and moment, LMn , acting at the node. The left nodalvariable vector, VL

n+1, of node n+ 1 is then computedfrom the right nodal variable vector, VR

n , of node nthrough the following transfer matrix relation:

VLn+1 = TMnVR

n : (17)

The compatibility between nodal and prescribed dis-


Figure 4. Nodal and element vectors.

placements is satis�ed through the following matrixrelation:

An+1VLn+1 = Rn+1: (18)

At the end node, the following relation is used to satisfythe equilibrium condition:

AendVRn = Rend; (19)

where:

Aend =

240 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

35 ;Rend = f0 0 0gT : (20)

4. Material nonlinearity

In nonlinear structural analysis, the Newton-Raphsoniteration method has been widely used since it yieldsa quadratic convergence characteristic. However, itcould become numerically unstable when the tangentialsti�ness of a structure approaches or equals zero. Thisis in the case when the post-yielding sti�ness of theforce-deformation relation approaches or equals zero.Several researchers have employed di�erent approachesto avoid this problem. For example, Manoharan andDasgupta [25] employed the modi�ed Newton-Raphsoniteration method to perform nonlinear consolidationanalysis of elastic-perfectly plastic soil. Vecchio [26]used the secant-sti�ness iteration method to performinelastic analysis of RC membrane structures. The prosand cons of various iteration methods are thoroughlydiscussed in Yang and Kou [27].

In this study, the secant-sti�ness iteration methodis adopted to solve the nonlinear responses of a beamsystem, due to its good compromise between conver-gence rate and numerical stability. The schematicrepresentation of the secant-sti�ness iteration methodis shown in Figure 5. Sectional responses (axialand exure) are assumed to be bilinear, as shown inFigure 5. For each section at an integration point, thesecant axial sti�ness and the exural sti�ness (EAsecantand EIsecant) are determined. The partitioned elementsti�ness matrix of Eq. (6) and element transfer matrixof Eq. (10) are computed based on these sectional se-cant properties. The residual work concept is employedas a convergence criterion [27].

Figure 5. Solutions strategy.

5. Numerical examples

Four numerical examples are used to verify the accu-racy and demonstrate the e�ciency of the proposedmaterial nonlinear transfer matrix method. Corre-lation studies are performed by comparing the ob-tained numerical results with those obtained withthe exibility-based �bre frame element [20]. This exibility-based �bre frame element has been widelyused in the research community and was implementedinto the Open System for Earthquake EngineeringSimulation Platform [28]. In all examples, the load-control marching scheme is used for solution procedure,and �ve Gauss-Lobatto integration points are employedto perform the required numerical integration.

5.1. Cantilever beamA cantilever beam of length L = 2 m is subjectedto an applied load, P , at its free end, as shownin Figure 6. The beam has a square cross sectionwith dimension of b = h = 50 mm. For themechanical and strength properties of the beam, aninitial elastic modulus of 200 GPa, a yield strength

Figure 6. Cantilever system.


of 400 MPa, and a strain-hardening ratio of 0.01are assumed. Four proposed TMM elements areused to represent the cantilever beam, as shown inFigure 6. To ease the global and local responsecomparisons, four exibility-based frame elements areemployed to discretize this cantilever beam, eventhough one exibility-based element would be su�cientto yield accurate results. For the exibility-basedmodel, 15 �bers are used to discretize the beamsection, while, for the proposed TMM model, exactintegration is employed to derive the beam sectionalresponse.

Figures 7 and 8 compare the tip load-displacementand load-rotation curves obtained with the two models.Clearly, global responses obtained with the two modelsare in good agreement, thus, con�rming the accuracyof the proposed TMM model. To investigate localresponses of both models, sectional moment-curvatureresponses, sampled at the �rst integration point of eachelement, are compared in Figure 9. The result clearly

Figure 7. Load vs. displacement for node 5 in thecantilever system in Figure 6.

Figure 8. Load vs. rotation for node 5 in the cantileversystem in Figure 6.

shows that the proposed TMM model is capable of wellrepresenting local responses.

5.2. Simply supported beamA simply supported beam of length L = 3 m issubjected to a four-point bending loading, as shownin Figure 10. The beam has a rectangular cross sectionand is 40 mm wide and 100 mm thick. For themechanical and strength properties of the beam, aninitial elastic modulus of 210 GPa, a yield strengthof 420 MPa, and a strain-hardening ratio of 0.002are assumed. Six proposed TMM elements are usedto represent the beam, as shown in Figure 10. Toexpedite the global and local response comparisons,six exibility-based frame elements are employed todiscretize this beam, even though a mesh with fewer el-ements would be su�cient to produce accurate results.Two concentrated forces are applied at nodes 3 and5. For the exibility-based model, 15 �bers are usedto discretize the beam section, while, for the proposedTMM model, exact integration is employed to derivethe beam sectional response.

The load-displacement response at node 3 andload-rotation response at node 5 obtained with the twomodels are shown in Figures 11 and 12, respectively.Clearly, there is good agreement between both modelsat the global level. The investigation into the localresponse obtained with both models is performed bycomparing the moment-curvature responses at integra-tion points along element 2. As shown in Figure 13,the proposed TMM model is capable of resemblingthe local response obtained with the exibility-basedmodel. The progressive yielding migration along theelement length is also well presented by the proposedTMM model.

5.3. Continuous beamFigure 14 shows a three-span continuous beam systemsubjected to two equal concentrated forces within itsmiddle span. The wide- ange beam section is 300 mmin height, 200 mm in width, 14 mm in ange thickness,and 9 mm in web thickness. For the mechanicaland strength properties of the beam, an initial elasticmodulus of 210 GPa, a yield strength of 420 MPa, anda strain-hardening ratio of 0.01 are assumed. Fourteenproposed TMM elements are used to represent thebeam, as shown in Figure 14. The same number of exibility-based elements is also used to discretize thiscontinuous beam system for the sake of global and localresponse comparisons. Two equal concentrated forcesare imposed at nodes 7 and 9. For the exibility-basedmodel, 5, 12, and 5 �bers are used to discretize theupper ange, web, and lower ange, respectively.

The load-displacement response at node 7 andthe load-rotation response at node 9 obtained withthe two models are compared in Figures 15 and 16,


Figure 9. Section moment vs. curvature for elements 1, 2, 3 and 4 in the cantilever system in Figure 6.

Figure 10. Simply supported beam system.

Figure 11. Load vs. displacement for node 3 in thesimply supported beam system in Figure 10.

Figure 12. Load vs. rotation for node 5 in the simplysupported beam system in Figure 10.

respectively. In general, there is good agreement be-tween the TTM model results and the results obtainedwith the exibility-based model. Small discrepanciesbetween the two models are observed in the post-yielding responses due to �ber section discretization.It is important to note that the proposed TMM modeluses exact integration to derive the beam-section re-sponse, while the exibility-based model employs �ber-section discretization to perform numerical sectionalintegration. Figure 17 compares the sectional moment-


Figure 13. Section moment vs. curvature at integration points of element 2 in the simply supported beam system inFigure 10.

Figure 14. Three-span continuous beam system.

Figure 15. Load vs. displacement for node 7 in the3-span continuous beam system in Figure 14.

Figure 16. Load vs. rotation for node 9 in the 3-spancontinuous beam system in Figure 14.


Figure 17. Section moment vs. curvature at the last integration point of elements 3, 4, 7, and 8 for the 3-span continuousbeam system in Figure 14.

curvature responses at the last integration point ofelements 3, 4, 7, and 8. Clearly, the proposed TMMmodel is capable of representing the local responseobtained with the exibility-based model.

5.4. Cantilever columnA cantilever column of height H = 2:5 m is subjectedto a constant axial load, P = 270 kN, and anincrementally increasing moment, M , at its free end, asshown in Figure 18. The column section is square witha dimension of b = H = 300 mm For the mechanical

Figure 18. Column system.

and strength properties of the beam, an initial elasticmodulus of 20 GPa, a yield strength of 40 MPa, and astrain-hardening ratio of 0.01 are assumed. As shown inFigure 18, the column is discretized into �ve proposedTMM elements. One exibility-based element is usedto model this column. For the exibility-based model,15 �bers are used to discretize the column section,while, for the proposed TMM model, exact integrationis employed to derive the column sectional response.

Figure 19 shows the tip load-displacement re-sponses obtained with the two models. Clearly, there isexcellent agreement between the two models. Figure 20compares the axial-moment interaction diagrams con-

Figure 19. Load vs. displacement for node 6 in thecolumn system in Figure 18.


Figure 20. P-M interaction diagram for the columnsystem in Figure 18.

Table 1. Comparison of computational time ratio.

Case of analysis TMM Flexibility-basedCantilever beam 1.00 1.36

Simply supported beam 1.00 1.26Continuous beam 1.00 2.04Cantilever column 1.00 1.35

structed by the two models and indicates that theymatch quite perfectly.

The relative computational time of each analysisis shown in Table 1. The TMM model is a minimumof 1.26 times and a maximum of 2.04 times faster thanthe exibility-based frame model in simply supportedbeams and continuous beams, respectively.

6. Conclusion

In this study, a material-nonlinear-analysis algorithmbased on the Transfer Matrix Method (TMM) ispresented. This newly developed algorithm can beused to perform nonlinear analyses of continuous beamsystems. In the TMM, the system equation has aconstant number of system unknowns, regardless ofthe total degrees-of-freedom number in a structure andthe system response (either linear or nonlinear). Thatis, the TMM model is at least 1.26 times faster thanthe exibility-based frame model in the analyses. As aresult, TMM can be used e�ciently both for linear andnonlinear structural analyses. Four numerical studiescon�rm the accuracy and e�ciency of the proposedTMM model. In all examples, there is good agreementbetween the proposed TMM model and the exibility-based �ber element model, both at global and locallevels.

Acknowledgment

This research was supported by a grant (13SCIPA01)from the Smart Civil Infrastructure Research Program,

funded by the Ministry of Land, Infrastructure andTransport (MOLIT) of the Korean government and theKorean Agency for Infrastructure Technology Advance-ment (KAIA). Special thanks go to senior lecturer, Mr.Wiwat Sutiwipakorn, for reviewing and correcting theEnglish of this paper.

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Biographies

Minho Kwon is Professor of Structural Engineeringin the Department of Civil Engineering at GyeongsangNational University, Jinju, South Korea. His mainresearch interests include �nite element modeling forRC and steel structures, aircraft crash analysis, earth-quake resisting design, constitutive modeling, andexperimenting with RC structures under severe loads.He has conducted several research projects with thegovernment and industry, and is member of the Amer-ican Society of Civil Engineers and the Korean Societyof Civil Engineers.

Jinsup Kim received his PhD degree in StructuralEngineering from the University of Gyeongsang Na-tional University, Jinju, South Korea, in 2014. Hisresearch interests include �nite element analysis forRC and steel structures, and seismic reinforcing andretro�tting for RC and steel structures using �ber-reinforced polymer and high performance �ber rein-forced concrete.

Hyunsu Seo is a PhD candidate in Civil Engineeringfrom Gyeongsang National University, Jinju, SouthKorea. His major �eld of interest is structural engineer-ing and numerical modeling of civil structures. He isalso interested in numerical simulation of experimentalmodeling in reinforced concrete and steel frame struc-tures. He is currently involved in several national R&Dprojects.

Suchart Limkatanyu is Associate Professor in theDepartment of Civil Engineering at the Prince ofSongkla University, Thailand. He received his PhDdegree in Structural Engineering and Structural Me-chanics (SESM) from the University of Colorado,Boulder, USA. His research interests center on theseismic analysis, design and retro�tting of reinforcedconcrete structures, earthquake engineering, compu-tational mechanics, nonlinear frame analysis, nano-engineering and technology, and multi-physics sys-tems.

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