analysis of arbitrary composite sections in biaxial bending and axial load - vassilis k....

Computers and Structures 98-99 (2012) 33–54

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Computers and Structures

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Analysis of arbitrary composite sections in biaxial bending and axial load

Vassilis K. Papanikolaou ⇑Laboratory of Reinforced Concrete and Masonry Structures, Civil Engineering Department, Aristotle University of Thessaloniki, P.O. Box 482, Thessaloniki 54124, Greece

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 November 2011Accepted 13 February 2012Available online 11 March 2012

Keywords:Reinforced concreteComposite sectionsBiaxial bendingStress integrationUltimate strengthMoment–curvature

0045-7949/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.compstruc.2012.02.004

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A new methodology is presented for the ultimate strength and moment–curvature analysis of arbitrarycomposite sections under biaxial bending and axial load. The definition of section geometry and materialproperties can be unconditionally complex, based on an object-oriented implementation. Stress integra-tion is performed using a Green path integral, with an adaptive strain-mapped Gaussian sampling. Deriv-ative-free solution strategies for the calculation of incremental and ultimate response are applied. Resultsare presented in the form of moment–curvature curves, ultimate strength interaction curves and 3D fail-ure surfaces. The performance of the methodology is demonstrated through various case studies, compar-isons and benchmarks.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Reinforced concrete and composite structural elements, widelyused in buildings and bridges, are generally subjected to a com-bined action of biaxial bending and axial load. This is a result oftheir section geometry and material composition, their positionand orientation in the structure and, most importantly, the natureof external loading. Columns at corners or under two-way slabs,bridge piers and composite decks, subjected to wind or earthquakeexcitation, are representative cases. Assessing the adequacy ofthese sections at ultimate limit state (usually via interaction dia-grams) or providing information on their inelastic response gradu-ally up to failure (in the form of moment–curvature curves), is acomputationally intense task, mainly due to material nonlineari-ties and geometrical complexities, particularly for composite sec-tions. Therefore, addressing this problem requires efficientsolution algorithms, characterized not only by the requisite robust-ness but also by execution speed, since section analysis is often arepetitive task [1] within the broader computational frameworksof nonlinear structural analysis, design, assessment and retrofit.

The problem of arbitrary section analysis has received attentionin the literature since the early 1960’s (e.g. [2–4]), however thisattention has been intensified during the last decade along withthe advent of inexpensive personal computers. Various analytical,numerical and mixed methodologies for analyzing sections of vary-ing complexity in terms of geometry and material compositionhave been suggested, with varying degrees of reported efficiency

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in terms of convergence stability and speed. Since the literatureis extensive, a proper categorization is attempted hereinafter, inorder to assess whether the research field under discussion is openfor further novelties and improvements. In this direction, five keycharacteristics of section analysis methodologies have been identi-fied, namely (a) the type of arbitrary section addressed, i.e.whether it is reinforced concrete (R/C) or generally composite,(b) the form of material constitutive laws, e.g. polynomial or arbi-trary functions, (c) whether section subdivisions are imposed priorto stress integration, (d) the stress integration scheme and (e) thesolution strategies for applying force equilibrium conditions.

A non-exhaustive yet representative directory of previous stud-ies on the subject is summarized in Table 1, providing specificinformation on the aforementioned five key topics for each study.A critical review on this categorized literature summary leads tothe following remarks:

� A significant number of suggested algorithms are limited to R/Csections, consisting of a single concrete surface and a group ofindividual fibers for reinforcement bars (i.e. two distinct mate-rials). In order to bypass such limitations, the first prerequisiteof the present methodology will be the unconditional sectioncomplexity, i.e. unlimited number of section components (sur-faces, fiber groups and – newly introduced – lines), eachassigned to a different material constitutive law.� Most studies impose restrictions on material stress–strain con-

stitutive laws, usually in the form of specific Code directives(e.g. [5,6]), or more elaborate piecewise polynomial laws (e.g.[7,8]). For this reason, the second prerequisite of the presentmethodology will be the use of fully arbitrary material

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Table 1Features of selected previous studies on the analysis of arbitrary R/C and composite sections.

Authors Section type Material constitutive lawfor surfaces

Section subdivision Stress integration Solution strategies

Charalampakis and Koumousis [8] Composite Piecewise up to cubicpolynomials

Curvilinear trapezoids Closed-form per trapezoid Derivative-free (Brent)

Chen et al. [16] Composite Parabolic-linear No subdivisions Closed-form Derivative-free(regula-falsi)

Chiorean [18] Composite Parabolic-linear withsoftening

No subdivisions Green/Gauss-Lobatto withadaptive bisection

With derivatives(arc-length)

Rosati et al. [5] Composite Parabolic-linear One polygon per r � epart

Closed-form per polygon With derivatives (NR)

Sfakianakis [15] Composite Parabolic-linear withsoftening

No subdivisions Fiber integration Incremental search

Sousa and Muniz [7] Composite Piecewise up to cubicpolynomials

One polygon per r � epart

Closed-form per polygon –

Brøndum–Nielsen [25], Dundar andSahin [22], Yen [21]

Reinforcedconcrete

Constant (rectangular) No subdivisions Closed-form With derivatives as finitedifferences (NR)

Bonet et al. [1] Reinforcedconcrete

Piecewise non-polynomial (arbitrary)

Polygons (thick layers)per r � e part

2D Gauss or Green/Gauss perpolygon

–

Fafitis [17] Reinforcedconcrete

Parabolic-linear No subdivisions Green/Gauss –

Pallarés et al. [6] Reinforcedconcrete

Non-polynomial – linear(EC2)

No subdivisions Closed-form With derivatives (NR)

Penelis [3], De Vivo and Rosati [26],Alfano et al. [27]

Reinforcedconcrete

Parabolic-linear One polygon per r � epart

Closed-form per polygon With derivatives (NR)

Rodriguez and Ochoa [20] Reinforcedconcrete

Parabolic-linear withsoftening

Trapezoids Closed-form per trapezoid With derivatives (NR)

Werner [4] Reinforcedconcrete

Parabolic-linear One polygon per r � epart

Closed-form per polygon Nested iterations

Yau et al. [23] Reinforcedconcrete

Constant (rectangular) No subdivisions Closed-form Derivative-free(regula-falsi)

Suggested methodology Composite Piecewise non-polynomial (arbitrary)

No subdivisions Green/Gauss with adaptivestrain-mapping

Derivative-free (Brent)

34 V.K. Papanikolaou / Computers and Structures 98-99 (2012) 33–54

constitutive laws in piecewise form, as reported only in Bonetet al. [1]. This enables the analysis of sections with materialsof non-polynomial stress–strain relationships (e.g. [9] forhigh-strength concrete and nonlinear analysis, [10] for confinedconcrete and [11,12] for plain concrete).� The most critical element that affects both the accuracy and

speed of a section analysis algorithm is the stress integrationscheme. There are three major paths that can be followed,namely (a) fiber integration (e.g. [13–15]), (b) analytical inte-gration using closed-form functions (e.g. [8,16]) and (c) numer-ical integration, usually in a form of Gaussian sampling on aGreen path integral (e.g. [17,18]). The first approach is bothapproximate and slow, requiring an increased fiber mesh den-sity and a proportional number of arithmetic operations toreach an acceptable level of accuracy. Therefore, this methodis now obsolete for ultimate strength section analysis and isused only for non-cylindrical stress fields, e.g. in cases of cyclicloading and/or load path dependency [1]. As far as the secondapproach is concerned, its main advantage is that it yields exactand quick results (especially for lower order functions), it ishowever totally restricted to a specific stress–strain expression,which is in conflict with the aforementioned second prerequi-site. Consequently, the sole path to provide a generalized solu-tion for arbitrary material constitutive laws is theimplementation of a suitable numerical integration scheme.However, it has been reported that, in certain cases, numericalintegration may be more expensive than analytical methods,while low order numerical integration may yield unacceptablylarge errors [1,19] because the required order of numerical inte-gration is not known a priori [8]. These issues will be confirmedand eventually addressed in the present methodology, byapplying an improved adaptive strain-mapped Gaussian sam-pling on a Green path integral, demonstrating fast execution

with customizable accuracy, which expresses the third prere-quisite of the present method.� A closely related issue to the aforementioned stress integration

efficiency is whether section subdivisions are required in theemployed integration scheme. There are many studies (e.g.[1,7,8,20]) where this geometric manipulation is imposed priorto stress integration, naturally leading to a reduction in execu-tion speed. For this reason, the suggested scheme will be formu-lated without need for any subdivisions (e.g. [6,18]).� The majority of solution strategies presented in the literature,

for applying force equilibrium conditions, are based on secantor tangent schemes (e.g. Newton-Raphson), which require theprior calculation of derivative measures (e.g. section stiffness).However, the disadvantages of (a) the additional computationalcost for calculating derivatives (e.g. using finite differences[21,22]) and (b) the inherent non-convergence issues relatedto secant/tangent methods, has led some researchers to adoptsimpler and more straightforward derivative-free procedures[8,16,23], which, under certain conditions, demonstrate stableand fast performance. Following the third prerequisite, thepresent methodology will apply advanced derivative-free meth-ods [8], providing, where necessary, various techniques to guar-antee convergence.

In order to satisfy the aforementioned three prerequisites of thesuggested methodology, namely (a) unrestrained section complex-ity (b) arbitrary material constitutive laws and (c) fastest possibleexecution with customizable accuracy, a new feature set is sug-gested in the last row of Table 1. In the subsequent chapters, themathematical formulation of the methodology will be deployed,together with validation tests, case studies, extended comparisonswith previous studies and benchmarks. The ultimate goal of thepresent study is to suggest a new robust and fast procedure for

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V.K. Papanikolaou / Computers and Structures 98-99 (2012) 33–54 35

the ultimate and moment–curvature analysis of arbitrary compos-ite sections under biaxial bending and axial load, which may pro-vide in turn, a solid basis for further developments in the fieldsof structural design, assessment and optimization [24].

2. Mathematical formulation

2.1. Geometry and material definitions

Following the first prerequisite of the present methodology, thesection under consideration can be consisted of an unlimited num-ber of individual section components, namely (a) surfaces(S1,S2, . . .,Sn), describing surface-distributed materials such as con-crete, masonry or structural steel, (b) multi-segment lines(L1,L2, . . .,Ln), usually simulating linearly distributed reinforcementor fiber-reinforced polymer (FRP) strips and (c) fibergroups(FG1,FG2, . . ., FGn), i.e. dimensionless fibers representing individual

Fig. 1. Composition of arbitrary composite section.

Fig. 2. Definition of individu

reinforcement bars or tendons (Fig. 1). The above three componenttypes can contribute either positively or negatively to the sectionresponse (i.e. the corresponding internal actions will be added orsubtracted respectively), allowing the efficient embedment ofvoids or the use of multi-nested materials. This feature not onlyprovides an explicit description of section openings, withoutresorting to complicated ‘fictitious cuts’ [3,4,17,28], but also themeans to avoid double-contribution errors, i.e. by subtracting theconcrete material under structural steel or reinforcement fibers[17], similarly to the ‘background material’ concept [8].

Surfaces are defined as a counterclockwise series of vertex coor-dinates (xj,yj), together with the included angle value of the suc-ceeding circular arc segment xj (ignored for straight segments).It is also noted that a negative angle value leads to the subtractionof the corresponding circular sector from the surface. Therefore:

Si ! ½ðx1; y1;x1Þ; ðx2; y2;x2Þ . . . ðxn; yn;xnÞ� ð1Þ

Lines are defined as a series of vertex coordinates (multi-seg-ment), including the contribution area of the succeeding segment(Aj):

Li ! ½ðx1; y1;A1Þ; ðx2; y2;A2Þ . . . ðxn�1; yn�1;An�1Þ; ðxn; ynÞ� ð2Þ

Similarly, fibergroups are defined as a set of fiber coordinatesand corresponding contribution areas:

FGi ! ½ðx1; y1;A1Þ; ðx2; y2;A2Þ . . . ðxn; yn;AnÞ� ð3Þ

Fig. 2 depicts the definition of the three aforementioned sectioncomponent types.

After the definition of individual section components, an adap-tive linearization process is performed on each surface component(Si) which contains circular arc segments, transforming them toclosed polygons, in order to maintain compatibility with the Greenpath integral transformation concept [17,28], discussed later. Thisalgorithm is an improved version of the constant divisions ap-proach by Kwan and Liauw [28], outlined as follows: all arcs ofequal length are bisected in consecutive steps (sweeps), until thesurface area of the running sweep differs from the previous one,less than a requested tolerance:

jðsAtot � s�1AtotÞ=s�1Atotj < ATOL ð4Þ

where Atot is the total surface area, s is running sweep and ATOL isthe tolerance value. Suggested values of ATOL may be in the rangeof 1–2%. In the most unfavorable case of a single circle, this leadsto a 64-segment polygon for ATOL = 1% (default), with an area accu-racy of 0.16%. For ATOL = 2%, it leads to a 32-segment polygon, with

al section components.

Fig. 3. Adaptive linearization procedure.


an area accuracy of 0.64%. In addition, it has been reported in Chara-lampakis and Koumousis [8] that ultimate response discrepanciesbetween exact circle integration and equivalent polygonal repre-sentation are negligible for more than 48 vertices. Fig. 3 shows aworked example of the adaptive linearization process. The advan-tage of this improved linearization method is that smaller arc seg-ments (e.g. rounded edges of steel sections) introduce feweradditional vertices to the corresponding section surface, thusimproving the efficiency of the integration procedure.

The second prerequisite of the present method allows fully arbi-trary constitutive laws for the assigned materials. For this reason,the material definition is consisted of a series of arbitrary stress–strain (r � e) function parts (f 1, f 2, . . ., f n) in a piecewise form(Fig. 4a). These function parts may preferably be polynomials, forexact Gaussian integration; however they can be of any other form,

Fig. 4. (a) Generalized material definition, ap

(e.g. exponential [12]), which will be handled by adaptive sam-pling. Moreover, three distinct strain threshold values should bedeclared for each material, which will be used in a later stage forthe direct definition of the ultimate strain profile:

� ecu is the strain corresponding to the ultimate compressivestrength.� eco is the strain corresponding to the compressive strength

under pure compression.� etu is the strain corresponding to the ultimate tensile strength.

Examples of specific material definitions are shown in Fig. 4band 4c for concrete (parabolic-linear) and steel (bilinear) respec-tively, according to EN1992 [9]. It is possible that any of these cri-teria can be ignored when defining the ultimate profile (e.g. forconcrete tensile strength).

Throughout the present formulation, various geometrical andloading parameters of the considered section are defined (Fig. 5).Resultant internal actions (N,MX,MY) are calculated with respectto the reference coordinate system XCY, where C is the section ori-gin, that can be either the geometrical, elastic or plastic center [29].It has been previously reported that the plastic center yields betterconvergence of the solution process, especially for axial loadsapproaching the compressive capacity (because it is contained inthe isoload contour [16]). Thus, it will be adopted throughout thisstudy, and will be further improved. The exact location of the neu-tral axis is determined by its orientation (h) and depth (d), whilethe current strain profile by the curvature (u) and the strain-at-origin (eo = u�d) [8]. The neutral axis depth and the strain profileare always defined on the rotated (by angle h) coordinate systemxCy, while a positive curvature is assumed to induce compressionat the top section fiber. The strain profile is considered linear, fol-lowing the Bernoulli–Euler assumption, i.e. plane sections remainplane and perpendicular to the axis of the element. Finally, mo-ment vectors MX and MY are pointing to the positive direction ofX and Y axes respectively, and the inclination of their resultant vec-tor is denoted by angle a. It is pointed out that the orientation an-gle of the neutral axis (h) and the inclination angle of the resultant

plications on (b) concrete and (c) steel.

Fig. 5. Definition of section loading parameters.


moment (a) are generally not coincident (h – a), except for veryspecial cases. It will be shown later that this discrepancy is the rootof various divergence problems, which will be treated accordingly.

The geometry and material definitions described in this chapterare efficiently structured in computer code, using a pure object-ori-ented approach (Fig. 6). Initially, two basic classes are introduced:(a) Material, including material properties and respective stresscalculation methods and (b) Points, including a set of coordinates(xj,yj) and geometry manipulation methods (e.g. rotate, pan etc.).Subsequently, three descendant classes from ancestor class Pointsare formed, one for each section component type, namely Surface,Line and FiberGroup. Each class includes additional properties(e.g. included angles or contributing areas), an instance of theMaterial class (object) as the assigned material and a method toperform stress integration. Moreover, the Surface class includesthe method to perform adaptive linearization.

The fundamental Section class contains an unlimited number ofsection component objects (Surface, Line, FiberGroup), togetherwith methods to define centroids, determine the ultimate limitstate and summate the internal actions from the individually inte-grated component contributions. Finally, a descendant of Sectionclass, named SectionSolver, is adding the necessary methods to ap-ply force equilibrium conditions and calculate the incremental andultimate section response (solution strategies). The advantage ofthis object-oriented approach (as opposed to traditional proceduralprogramming) is that any section under consideration can be sim-ply created as a SectionSolver object, encapsulating all the neces-sary properties and numerical procedures (methods) to performsection analysis (analysis engine). These procedures will be de-scribed in detail in the subsequent paragraphs.

Fig. 6. Object-oriented formulation of the analysis engine.

2.2. Stress integration

Following the geometry and material definitions, perhaps themost demanding numerical task that greatly influences both theaccuracy and speed of the analysis, is the employed stress integra-tion scheme. The fundamental integration problem is defined asfollows: calculate the resultant internal actions of the compositesection (axial N and moments MX, MY) for a prescribed neutral axisorientation (h) and strain profile (eo,u), denoted as I(eo,u,h). Thesection origin (C) is initially set to the plastic center [16,29] andthe reference coordinate system XCY is rotated by angle h, resulting

to the rotated coordinate system xCy (see Fig. 5). Since the sectionis composed of a specific number of components, i.e. (nS) surfaces,(nL) lines and (nFG) fibergroups, the resultant actions are the sum-mation of individual component contributions as follows:

R ¼XnS

i¼1

½signðSiÞRS;i� þXnL

i¼1

½signðLiÞRL;i� þXnFG

i¼1

½signðFGiÞRFG;i� ð5Þ

where R is the resultant action (N, Mx, My, compression negative), nis the component count, indices S, L and FG correspond to surface,line and fibergroup components respectively and sign() denoteswhether the component contribution is positive (actual material)or negative (void). Following the integration process, moments Mx

and My are transformed back to the reference system XCY asfollows:

MX ¼ Mx cosð�hÞ þMy sinð�hÞMY ¼ �Mx sinð�hÞ þMy cosð�hÞ

ð6Þ

The integration algorithms for the individual component types (Si,Li, FGi) are described below.

2.2.1. Surface stress integrationFor the stress integration of individual surfaces (Si) it has been

already justified that, in order to handle arbitrary stress–strainconstitutive laws, the most appropriate solution is to apply anumerical integration scheme. In this study, the Gauss–Legendrequadrature method, originally introduced in Fafitis [17], will beadopted. Initially, the resultant actions (RS,i) are represented byarea integrals of general form:

RS;i ¼ZZ

Si

xrysrðyÞdxdy ð7Þ

where Si represents the surface under consideration, (x,y) are thesurface planar coordinates, r(y) is the stress function expressed interms of the coordinate y, and (r,s) are exponents depending onthe requested resultant action, i.e. (r,s) = (0,0) for axial NS,i, (0,1)for moment MxS,i and (1,0) for moment �MyS,i. By applying Green’stheorem, the area integral is transformed into a line integral alongthe closed boundary Li that encloses the area Si as follows:

RS;i ¼1

r þ 1

ILi

xrþ1ysrðyÞdy ¼ 1r þ 1

Xn‘i

j¼1

I‘j

xrþ1ysrðyÞdy

" #

¼ 1r þ 1

Xn‘i

j¼1

Ij ð8Þ

where the surface boundary Li is represented by a closed polygon ofn ‘i segments. Therefore, the integration problem breaks down tothe integration of each individual polygon segment ‘j, denoted asthe elementary integral Ij. The equation of this line segment maybe expressed as:

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‘j ! x ¼ ajyþ bj ð9Þ

where aj and bj is the slope and intercept of line ‘j, respectively.Introducing (9) in (8), the elementary integral is further simplifiedinto a line integral of a single-variable nonlinear function Fj(y):

Ij ¼I‘j

ðajyþ bjÞrþ1ysrðyÞdy ¼I‘j

FjðyÞdy ð10Þ

At this stage, the Gauss–Legendre integration scheme for theelementary integral Ij is applied. This scheme is theoretically yield-ing exact solutions for polynomial functions Fj(y) of order n, when aminimum number of nG ¼ n n 2þ 1 Gauss points are utilized (thesymbol ‘n’ denotes integer division). In the original study of Fafitis[17], a constant number of Gauss points nG = 3 was selected, claim-ing exact results for Code provided stress–strain laws for concrete,e.g. stress-block or parabolic-linear [9,30]. Nonetheless, it has beenobserved that, even for these piecewise-polynomial functions (e.g.parabolic-linear), the originally suggested application of Gaussianquadrature per segment ð‘jÞ, may lead to non-exact results, becauseit is highly probable that multiple function parts will be mapped onthe same polygon segment, exhibiting non-continuity in the deriv-ative and virtually resulting to a non-polynomial form. This obser-vation is clearly depicted in Fig. 7, for the case presented in theoriginal study [17] (circular points). It is therefore suggested here-in, that the sampling process should be applied per function partwithin each segment (rectangular points), in order to guarantee ex-act solutions for the widely used piecewise-polynomial stress–strain relationships. Table 2 shows a comparison between the ori-ginal and the improved integration scheme (described in detailhereinafter) for the example shown in Fig. 7, where it is demon-strated that the latter yields absolutely exact results for all consid-ered actions.

Fig. 7. Improvement of the Gaussian integr

Table 2Comparison between Fafitis [17] and present integration scheme.

h = �45�, d = �9.972 in N (kip)

Exact (100 Gauss points) �1999.66Fafitis (reported values) �1997.29Fafitis (simulated) �1997.14Fafitis – error% 0.13%Present strain-mapped scheme �1999.66

A schematic view of the suggested strain-mapped Gaussian sam-pling procedure is shown in Fig. 8. Initially, for each function part f k

of the constitutive law (Fig. 4), the values of the (predefined) strainlimits elimA,k and elimB,k are mapped on the segment ‘j under consid-eration by a simple strain-to-coordinate transformation, whichleads to the corresponding coordinate limits ylimA,k and ylimB,k:

yðeÞ ¼ ðe� eoÞ=u ð11Þ

where u and eo are the current strain profile parameters. If anycoordinate limit falls outside a segment limit (yA,j or yB,j), it is re-placed by the segment limit itself. If both coordinate limits fall out-side the same segment limit, the respective function part f k is notcontributing to the integral and is subsequently ignored. Finally,Gaussian sampling is applied for all contributing function parts ofthe considered segment ‘j, resulting to the elementary integral Ij,following (10):

Ij ¼I‘j

FjðyÞdy¼:Xnf

k¼1

12ðylim B;k � ylim A;kÞ

XnGk

m¼1

½wmFjðymÞ�" #

ð12Þ

where nf is the number of stress–strain function parts, nGk is thenumber of utilized Gauss points i.e. the quadrature order, ym isthe coordinate where function Fj is sampled and wm is the corre-sponding weight. The coordinate ym for the Gauss the point numberm is derived from the corresponding Gauss index km = [�1. . .1] asfollows:

ym ¼12ðylim B;k þ ylim A;kÞ þ

km

2ðylim B;k � ylim A;kÞ ð13Þ

Finally, the function Fj is sampled at coordinate ym, using thefunction part f k, in stress–strain terms, as follows:

FjðymÞ ¼ ðajym þ bjÞrþ1ysmfkðeo �uymÞ ð14Þ

ation scheme suggested by Fafitis [17].

Mx (kip-in) My (kip-in)

�7405.90 2820.60�7411.72 2796.55�7412.03 2797.00

0.08% 0.84%�7405.90 2820.60

Fig. 8. Adaptive strain-mapped Gaussian integration.


An important advantage of the suggested strain-mapped tech-nique is that each function part (f k) can acquire a different numberof Gauss points (nGk) which, if set equal to ðnþ r þ 1þ sÞ n 2þ 1 forpolynomial parts of order n, produces exact integration resultswith an optimum reduction in computational cost, as comparedto a constant quadrature order [1,17,19]. Another advantage is thatno geometric manipulations, such as section subdivisions[1,4,5,7,8,20,26,27], are imposed prior to stress integration, whichobviously enhances the overall analysis performance. More specif-ically, it is assumed that the present strain-mapped approachyields similar results to the MTLI method suggested by Bonetet al. [1] (where the section is decomposed into thick layers corre-sponding to each function part and integrated separately), yetwithout the laborious section decomposition and the restrictionof a constant Gauss quadrature order.

There are certain cases where a function part (f k) is of non-poly-nomial form (e.g. ascending branch in high-strength concretestress–strain relationship, according to EN1992 [9]) and therefore,Gaussian integration can yield only approximate estimates. Forthis reason, an adaptive integration technique is introduced herein,which is described as follows: each function part (f k) data structureis upgraded with memory, which stores the largest strain rangeintegrated for that part Demax,k, throughout the analysis process(initially set to zero):

Demax;k ¼ Maxðjuðylim B;k � ylim A;kÞjÞ ð15Þ

During the running integration, if the strain range surpasses therespective stored value, an adaptive process is triggered, whichsequentially increases the utilized number of Gauss points, untila prescribed integration accuracy is reached:

XnGkþ1

m¼1

½wmFjðymÞ� �XnGk

m¼1

½wmFjðymÞ� !, XnGk

m¼1

½wmFjðymÞ� !��

�� < ITOL

ð16Þ

where ITOL is a tolerance value (default 5‰). The main advantage ofthe suggested adaptive integration technique is that is capable toautomatically handle fully arbitrary stress–strain functions withcustomized accuracy and optimized execution speed, followingthe third prerequisite of the present methodology.

2.2.2. Line integrationThe possibility to include and directly integrate linearly distrib-

uted materials in a composite section is introduced for the firsttime in the present study. This feature is practical for simulatingclosely spaced reinforcement (without resorting to a dense arrayof bar fibers), linear reinforcement distributions of unknown barspacing and FRP strips [31]. It can also reduce modeling time andpotential errors, as the exact choice of rebar diameter and spacingusually emanates from Code regulations rather than strengthrequirements. The adaptive strain-mapped integration method de-scribed in the previous paragraph can be directly adapted to linecomponents (Li), following a few necessary modifications. Theresultant actions (RLi), contributing to the total section response(Eq. (5)) are calculated using the following line integral, similarlyto (8):

RL;i ¼I

Li

xrystðyÞrðyÞdy ¼Xn‘ij¼1

tj

I‘j

xrysrðyÞdy

" #¼Xn‘i

j¼1

Ij ð17Þ

where the multi-segment line Li is represented by a series of n‘i

consecutive segments of thickness tj. Therefore, the integrationproblem breaks down again to the integration of the elementaryintegral Ij, using Gaussian quadrature (Fig. 8), similarly to Eqs.(10) and (12):

Ij ¼ tj

I‘j

ðajyþ bjÞrysrðyÞdy ¼ tj

I‘j

FjðyÞdy ð18Þ

Ij ¼ tj

I‘j

FjðyÞdy¼:Xnf

k¼1

Ak

2

XnGk

m¼1

½wmFjðymÞ�" #

ð19Þ

where Ak is the contributing area of the function part f k of segment‘j:

Ak ¼ Ajylim B;k � ylim A;k

yB;j � yA;j

�� ð20Þ

and Aj is the area of segment ‘j, as defined in (2). All other formula-tions are identical to the surface integration formulation. However,contrary to surface integration, horizontal segments are now con-tributing to the total integral and they are specially treated asfollows:


Nj ¼ AjrðyA;jÞMx;j ¼ AjyA;jrðyA;jÞ

My;j ¼ AjxA;j þ xB;j

2rðyA;jÞ

ð21Þ

For the exact integration of polynomial functions, the direct lineintegration needs even fewer Gauss points than surface integra-tion, specifically ðnþ r þ sÞ n 2þ 1 for polynomial parts of ordern. This renders this approach superior in terms of execution speedthan the equivalent fiber discretization of linearly distributedmaterials, which may require notably more arithmetic operationsto reach an equivalent level of accuracy.

2.2.3. Fibergroup integrationThe integration procedure for each fibergroup component (FGi)

is straightforward and performed analytically as follows:

RFG;i ¼XnFGi

j¼1

½Ajxrj y

sjrðeo �uyjÞ� ð22Þ

where nFGi is the number of group fibers, Aj is the fiber contributingarea and (xj,yj) are the fiber coordinates, as defined in (3).

2.3. Multicriteria ultimate limit states

Following the implementation of the stress integration schemethat calculates the resultant internal forces for a given neutral axisorientation (h) and strain profile (eo,u), the next step is to establisha method to derive the ultimate strain profile (uu,eou) for a givennatural axis depth (d) and subsequently integrate the ultimatestrain profile, as a function of the natural axis depth and orienta-tion: I(eou,uu,h) ? I(d,h). The resultant actions emanating fromthis strain profile define the locus of points in the triaxial stressspace that correspond to the conventional (or nominal) ultimatestrength of the section [8]. It has to be noted though, that the con-ventional ultimate strength may slightly differ in certain casesfrom the physical one, which corresponds to the top of the respec-tive moment–curvature diagram [15]. However, in the presentstudy, the former definition will be employed, since it is compati-ble with Code requirements (e.g. [9,30]) and its derivation is deter-ministic, leading to fast and accurate solutions, as opposed to thelaborious iterative calculation of moment–curvature maxima[8,15]. This derivation will be the basis for the construction of ulti-

Fig. 9. Ultimate strain profile for in

mate strength interaction curves and 3D failure surfaces, usingspecial solution strategies presented in the subsequent chapter.

Since the section consists of an unlimited number of compo-nents and different materials, multiple failure criteria should be ta-ken into account in order to determine the global ultimate strainprofile that corresponds to the conventional ultimate strength ofthe section. This is a necessary improvement of the existing litera-ture, where either a single concrete [16] or a dual concrete/steel[18] strain criterion is taken into account for composite sections.Moreover, the pure compression criterion imposed by EN1992§6.1(5) [9] and other Codes (referred hereinafter as ‘C-restriction’,after the pertinent figure 6.1 of EN1992) is generally neglected, ex-cept in Rosati et al. [5].

According to the present approach, for a given natural axisdepth (d), the ultimate strain profile (uu,i, eou,i) is initially definedfor each section component (Si,Li,FGi) (always on the rotated coor-dinate system xCy), using the three material strain threshold val-ues (ultimate strength criteria) already declared in the materialdefinition (ecu,eco,etu), as follows (Fig. 9):

uu;i ¼ umin ¼ Min

ecu;i=ðymax;i � dÞ if d < ymax;i

eco;i=ðyo;i � dÞ if d < ymin

etu;i=ðymin;i � dÞ if d > ymin;i

8><>: ð23Þ

yo;i ¼ ymin þ ðymax;i � yminÞecu;i

eco;ið24Þ

eou;i ¼ uu;id ð25Þ

where ymax,i and ymin,i are the component coordinate limits and ymin

is the whole section coordinate minimum. The first row in Eq. (23)corresponds to the ultimate compressive strength, the second to theultimate compressive strength under pure compression (C-restric-tion) and the third to the ultimate tensile strength. The derived va-lue of uu,i essentially corresponds to the curvature value attained onthe first triggering of any of the aforementioned failure criteria. It isalso observed in Fig. 9, that the C-restriction criterion is applicableonly when the neutral axis falls below the section (d < ymin), i.e. thewhole section is under compression. It is pointed out that the C-restriction as described in EN1992 [9] refers to the contribution ofconcrete in a reinforced concrete section, i.e. an effectively compos-ite section. It is deemed that this restriction should still hold forconcrete that participates in fully arbitrary composite sections.Either way, the application of the C-restriction is fully optional

dividual section components.


and can be easily disabled. However, it has been observed (see nextchapters) that it leads to stable solutions even for the uncommoncases of high axial compression.

Following the calculation of all component ultimate curvaturevalues (uu,i), the resulting ultimate curvature for the whole sectionis simply derived as the globally minimum value, as follows:

uu ¼Minðuu;iÞ 8 ðSi; Li; FGiÞ ð26Þeou ¼ uud ð27Þ

The above definition of the global ultimate curvature value impliesthat the section is considered to have reached its ultimate limitstate when any of its components first reaches its own ultimatestate, as defined by its corresponding material criteria (Eq. (23)).It is also noted that each section component may be optionally ex-cluded from the process of ultimate limit state definition, uponengineering judgment. The advantage of this approach is that itcan automatically handle sections of any complexity. Figs. 10 and11 show schematically the derivation of the multicriteria ultimatestrain profile for composite sections under prevailing flexureand pure compression (where C-restriction is also triggered),respectively.

Fig. 11. Ultimate strain profile of compos

Fig. 10. Ultimate strain profile of compos

2.4. Solution strategies

The last component of the present methodology is the imple-mentation of robust and rapid solution strategies for applyingequilibrium conditions between external and internal actions andsubsequently derive the requisite moment–curvature curves, ulti-mate strength interaction curves and 3D failure surfaces. Followingthe prerequisite of speed optimization, derivative-free procedureswill be adopted herein, avoiding the additional computational costof derivative calculation and any inherent non-convergence issuesrelated to secant/tangent methods.

2.4.1. Interaction curves and failure surfacesThe ultimate failure surface is defined as the locus of resultant

action points (N,MX,MY) in stress space, calculated from the stressintegration procedure applied on the ultimate strain profile, for aprescribed position of the neutral axis: I(d,h). Furthermore, planarinteraction curves are usually expressed either as (a) MX �MY con-tours for a constant external axial force N or (b) M � N contours fora constant inclination of the resultant moment vector a.

ite section under pure compression.

ite section under prevailing flexure.


For the construction of the 3D failure surface, there are threedifferent approaches of increasing computational cost, namely:

� The d � h method where both neutral axis depth (d) and orien-tation (h) values are incremented, and resultant points arederived from direct integration of the corresponding ultimatestrain profile.� The N � h method where the natural axis depth d is iteratively

calculated for incrementing values of axial load (N) and naturalaxis orientation (h), producing the so-called ‘isogonic’ meridians[8,20].� The N � a method, where both the natural axis depth (d) and

orientation (h) are iteratively calculated for incrementing valuesof both axial load (N) and moment vector inclination (a) [8,20].

Fig. 12 depicts the aforementioned methods, where it is ob-served that the d � h approach produces an irregular wireframewith non-planar, non-equidistant meridians, as opposed to themore elaborate N � h and N � a methods. It is apparent, that inter-action curves (surface cuts) are produced in the same sense, specif-ically using the N � h (or N � a) approach for MX �MY contours (fora constant N) and the N � a approach for N �M contours (for aconstant a). As a result, there are two problems to be addressed:

given N; h! solution! d ð28Þgiven N;a! solution! d; h ð29Þ

For the solution of (28) and (29), the root-finding Brent method[32] will be employed, successfully introduced for section analysisin Charalampakis and Koumousis [8]. The Brent method is consid-ered superior than similar derivative-free Regula-Falsi [16,23] orbisection methods, providing guaranteed and fast convergencewith fewer iterations, for well bracketed roots. The Brent root-find-ing method may be symbolically represented as:

vt ¼ BvBvA½UðvÞ ¼ wt� ð30Þ

where vt is the solution (root), sought in bracket [vA. . .vB], whichyields the requested (target) result wt, using the evaluation proce-dure U(v). It is noted that in the current implementation, the target

Fig. 12. Various 3D interaction surface constructio

value wt may be nonzero and the evaluation procedure U is non-invertible (i.e. cannot be solved with respect to vt). The convergencecriteria for the Brent method are the requested accuracies vTOL andwTOL for v and w variables respectively. By applying definition (30)in problem (28), the solution is expressed as:

dt ¼ BdBdA½Iðd; hÞ ¼ Nt � ð31Þ

where dt is the natural axis depth solution that lies in the [dA. . .dB]bracket and Nt is the given (target) axial force. Similarly, the solu-tion to problem (29) is defined as:

ht ¼ BhBhA½Iðd ¼ BdB

dA½Iðd; hÞ ¼ N�; hÞ ¼ at � ð32Þ

a ¼ arctanðMY=MXÞ ð33Þ

where ht is the natural axis orientation solution that lies in the[hA. . .hB] bracket and at is the target moment vector inclination. Thissolution involves a double-nested application of the Brent method;the axial force (N) is equilibrated in the inner loop and the inclina-tion of the moment vector (a) is equilibrated in the outer.

A crucial element that determines the performance of the afore-mentioned solution strategy is the effective definition of the root-finding brackets [v1. . .v2]. In the trivial case where extendedbrackets are selected e.g. ½dA . . . dB� ¼ R or ½hA . . . hB� ¼ ½0 . . . 2p�,apart from the apparent computational overhead (i.e. more itera-tions are needed to reach convergence), it is possible that multipleor no roots exist, resulting to undesirable non-convergence issues[23]. For this reason, a new adaptive bracketing technique is intro-duced herein, depicted in Fig. 13 and described as follows: startingfrom an initial value v0, the corresponding w0 = U(v0) value is firstcalculated. From the initial w0 and target wt values, the searchdirection is defined as: positive (+v) if wt > w0 and negative (�v)otherwise. The value of v is then incremented towards this direc-tion, and corresponding w values are calculated in the form ofwi = U(vi), where i is the increment number. The search procedurestops when the target wt value lies between wi�1 and wi and theresulting search bracket for the Brent method is [vA. . .vB] =[vi�1. . .vi].

n methods of increasing computational cost.

Fig. 13. Preliminary bracket-search procedure before applying derivative-free solution.


The advantage of this preliminary process is that, by introduc-ing a very few additional section integrations, the target Nt andat values are always bracketed (in Eqs. (31) and (32)), and henceconvergence of the solution is guaranteed with reduced computa-tional cost, compared to the aforementioned trivial bracketingpractice. Further details on this technique are the following: ithas been measured that, for di values, an exponential incrementingscheme is providing the fastest performance, while linear incre-menting is suitable for hi values, specifically:

di ¼ d0 þ ðymax � yminÞe�bþi ð34Þhi ¼ h0 þ iðp=16Þ ð35Þ

where b ¼ 2 and d0 = 0 for solving Eq. (31) and b ¼ 3, h0 = at and d0

is equal to the previously converged value of d (inner loop), for solv-ing Eq. (32). Finally, the tolerance values used in the Brent methodare:

NTOL ¼ 10�4ðNmax � NminÞdTOL ¼ 10�4ðymax � yminÞhTOL ¼ aTOL ¼ 10�2deg

8><>: ð36Þ

where Nmax, Nmin are the section tensile and compressive capacitiesand ymax, ymin are the section coordinate limits (for the current nat-ural axis orientation h).

Nonetheless, in the process of constructing the 3D failure sur-face using the most demanding N � a method, there were observedcertain instability issues for unsymmetrical sections, which can beexplained as follows (Fig. 14): although it has been reported in pre-vious studies [16] that the employment of the plastic center (C)

Fig. 14. Convergence issues in constructing N � a in

provides guaranteed convergence, because the origin of theMX �MY axes always stays within the respective interaction curve(isoload contour), it is observed that, especially for high tensile ax-ial loads, this consideration does not hold. This is because anymaterials without tensile strength (e.g. concrete) abruptly stopcontributing under tensile loads and hence the ‘updated’ plasticcenter of the still active materials in tension (e.g. steel) deviatessignificantly from the one originally considered. As a result, the ori-gin of the MX �MY axes actually falls outside the interaction curve,thus there are certain moment vector inclination angles (a) whichare not available. Hence, the requested solution cannot be possiblybracketed (Eq. (32)). For this reason, a new self-correcting centroidis herein suggested, in order to establish convergence, as follows.

It has been observed that failure surfaces for symmetrical sec-tions are surfaces of revolution about a vertical axis connectingthe apexes of ultimate axial capacity (Nmin,Nmax). This vertical axiscoincides with the locus of MX �MY origins (i.e. a vertical line pass-ing from 0, 0) and these origins are always contained in the respec-tive horizontal contours (for constant N). On the contrary, forunsymmetrical sections, the failure shape is skewed with an in-clined axis connecting these apexes. It is therefore apparent thatthere may be regions where the MX �MY origins, still lying on avertical line, fall outside the respective isoload contours and hencelead to non-convergence. The concept of the self-correcting cen-troid is to define proper eccentricities for the plastic center (C) that,when imposed on the section geometry, will move the respectiveMX �MY origin inside the isoload contour. This technique is de-picted in Fig. 15 and is described as follows: initially, the momentskewness of the failure surface (i.e. the slope of the inclined axis) isdefined for each direction as:

teraction surfaces for unsymmetrical sections.

Fig. 15. Definition of interaction surface skewness in unsymmetrical sections.


MX;skew ¼MX;max �MX;min

Nmax � Nminð37Þ

MY;skew ¼MY ;max �MY ;min

Nmax � Nminð38Þ

where (MX,max,MY,max) and (MX,min,MYmin) are the nonzero corre-sponding moment values for Nmax and Nmin (at surface apexes).The plastic center eccentricities (eX,eY) for a given axial force Nare calculated as:

eXðNÞ ¼ �MY;min þMY ;skewðN � NminÞ

Nð39Þ

eY ðNÞ ¼MX;min þMX;skewðN � NminÞ

Nð40Þ

Having calculated the above eccentricities, the section isaccordingly shifted before applying the solution procedure (Eq.(32)). The outcome of this procedure is that any potential solutioninstabilities during the construction of the N � a failure surface (orinteraction curves) for unsymmetrical sections are eliminated, asshown in the right of Fig. 14.

2.4.2. moment–curvature curvesThe derivation of uniaxial moment–curvature curves for a given

axial force in the context of the present methodology is straightfor-ward; initially, the uniaxial direction i.e. the neutral axis orienta-tion h is selected and a series of section integrations for linearly

incrementing values of curvature u are performed as follows: foreach curvature step u, the given external axial load is equilibratedto the respective internal resultant N, by iterating the value ofstrain-at-origin eo [8]. As a result, the corresponding moment(MX or MY) is plotted vs. the curvature value. This iterative schemeis performed exactly in the same way as the ultimate strengthsolution, using the Brent method as follows:

eot ¼ BeoBeoA½Iðeo;u; hÞ ¼ Nt� ð41Þ

where eot is the strain-at-origin solution that lies in the ½eoA . . . eoB�bracket and Nt is the target axial force. For adaptively deriving theapplication bracket, an exponential incrementing scheme was ap-plied (Fig. 13):

eoi ¼ eo0 þ jeo0je�3þi ð42Þ

where eo0 was set equal to its previously converged value and thetolerance value eoTOL was set equal to 10�7. It is noted that theabove process is not stopped when a predefined failure criterionis reached (e.g. ultimate strain), but is rather terminated when sec-tion collapse occurs (zero calculated moments). This provides acomplete picture of the section response (including post-peak re-sponse), where any failure criteria may be marked a posteriori (fromthe stress–strain history).


2.4.3. SynopsisThe suggested methodology for arbitrary section analysis, con-

sisting of geometry and material definitions, the integrationscheme, the derivation of ultimate limit states and solution strate-gies, all bound under a object-oriented implementation, hasemerged the following points of novelty:

(a) Adaptive linearization.(b) Adaptive strain-mapped integration.(c) Integration of linearly distributed materials.(d) Derivation of multicriteria limit states.(e) Adaptive bracketing for derivative-free methods.(f) Self-correcting centroid for unsymmetrical sections.

In the following chapters, the performance of the suggestedmethodology will be demonstrated through various case studies,benchmarks and comparisons with previous studies.

3. Validation and case studies

The object-oriented analysis engine described in the previouschapter was integrated into a user-friendly graphical front-endsoftware (Fig. 16), which contains all the necessary features toconduct the forthcoming validation tests, comparisons and bench-marks. Before proceeding, it is herein introduced a testbed con-sisting of five R/C and composite sections of increasingcomplexity, for future reference. Section (A) is a simple R/C sym-metrical column section, (B) is similar to (A), but with an unsym-metrical placement of reinforcement, (C) is a bilaterally jacketedR/C section with different concrete and steel materials for the coreand the jacket, (D) is the well-known composite section examplereported in Chen et al. [16] and finally, (E) is a conceptual, com-plex bridge deck composite section consisting of two differentconcrete grades, openings, fiber and linearly distributed reinforce-ment, structural steel sections of two different grades and FRPstrips. Detailed geometry and material definitions are providedin Fig. 17.

Fig. 16. Graphical front-end

The first validation test investigates the applicability of theadaptive linearization procedure, as it is applied on the plain con-crete curved section of Fig. 3. The analysis results are presented inFig. 18 as a series of moment interaction curves for an increasingnumber of linearization sweeps. It is observed that section re-sponse rapidly converges to a stable solution, for an adaptive line-arization process of 1% tolerance, demonstrating the effectivenessof the method.

The second test investigates the applicability of the adaptivestrain-mapped integration scheme, especially on non-polynomialfunctions. In Fig. 19, section (A) is analyzed using different concretestress–strain relationships, namely the parabolic-linear and non-linear models of EN1992 [9] and the Popovics model [12], withonly the former holding a piecewise polynomial form. It is ob-served that continuous and smooth moment interaction curvesare derived for all cases, also demonstrating the expected non-con-vexity for softening laws (as reported in [18]) for high levels of ax-ial compression ðm ¼ N=Atot ¼ �0:80Þ. Therefore, the robustness ofthe integration scheme and the accompanying solution algorithmis confirmed.

The third test investigates the validity of line integration, bycomparing a linearly distributed (perimetric) reinforcement modelto three equivalent fiber discretizations of increasing density. Fromthe moment interaction curves depicted in Fig. 20, it is observedthat fiber modeling rapidly converges to the linear distributionmodel for higher rebar density, which verifies the applicability ofline integration. Note that using line distributed reinforcement isslightly conservative.

In Fig. 21, moment interaction curves for composite sections (C)and (E) under varying axial loads are presented. For the same sec-tions, full moment–curvature curves (up to collapse) for variousnatural axis orientations (h) are depicted in Fig. 22. From the aboveanalyses and especially for section (E), it is concluded that the pres-ent methodology can successfully handle exceptionally complexcomposite sections without any instability issues. However, the ac-tual accuracy of the analysis results will be investigated by variouscomparisons to the existing literature, presented in the subsequentchapter.

for the analysis engine.

Fig. 18. Section response during adaptive linearization.

Fig. 17. Testbed including five R/C and composite sections of increasing complexity.


-150

-100

-50

0

50

100

150

-200 -150 -100 -50 0 50 100 150 200

EC2 Parabolic-Linear

EC2 Nonlinear (Sargin)

Popovics

MX (kNm)

MY (kNm)N = -3000 kN

(ν = -0.80)

Bilinear steelfy = 500 MPa

-25

-20

-15

-10

-5

0

-0.0035-0.0030-0.0025-0.0020-0.0015-0.0010-0.00050.0000

σ (ΜPa)

ε

EC2 Parabolic-Linear

-25

-20

-15

-10

-5

0

-0.0035-0.0030-0.0025-0.0020-0.0015-0.0010-0.00050.0000

σ (ΜPa)

ε

EC2 Nonlinear

-25

-20

-15

-10

-5

0

-0.0035-0.0030-0.0025-0.0020-0.0015-0.0010-0.00050.0000

σ (ΜPa)

ε

Popovics

Fig. 19. Section response using various concrete constitutive models.

-400

-300

-200

-100

0

100

200

300

400

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

4 bars

8 bars

16 bars

Linear

MX (kNm)

MY (kNm)N = -1000 kN

Section 30×70 cm, cover 5 cmfc = 20 MPa, EC2 parabolic-linear

fy = 500 MPa, bilinear, Atot = 40 cm2

-50

50

056-009-

Fig. 20. Section response using fiber and linear reinforcement modeling.


4. Comparisons with previous studies

The first comparison refers to the analysis of steel equal anglesections, reported in Charalampakis [33], which is considered a

computationally demanding task due to the section nonconvexand curved geometry. In Fig. 23, moment interaction curves for asteel section L40 � 40 � 4 rotated by 45�, for zero and high tensileand compressive axial loads (m = 0, m = ±0.8) are shown. It is ob-

-320

-240

-160

-80

0

80

160

240

320

-320 -240 -160 -80 0 80 160 240 320

N = 500 kN

N = 0 kN N = -1000 kN

N = -2000 kN N = -3000 kN

N = -4000 kN

MX (kNm)

MY (kNm)Section C

Nmax = 626.3 kN

Nmin = -4232.9 kN

with respect to plastic center

-100000

-80000

-60000

-40000

-20000

0

20000

40000

60000

80000

100000

-60000 -50000 -40000 -30000 -20000 -10000 0 10000 20000 30000 40000

N = 50000 kN N = 25000 kN N = 0 N = -50000 kN N = -100000 kN N = -150000 kN

MX (kNm)

MY (kNm)Section E

Nmax = 56419 kN

Nmin = -156348 kN

with respect to plastic center

Fig. 21. Moment interaction curves for sections C and E under various axial loads.

Fig. 22. Moment–curvature response of sections C and E for various natural axis orientations.


served that the results from both studies are absolutely coinci-dent, which confirms the accuracy of the present method for sin-gle-material sections of curved shape. Moreover, perfectsymmetry in numerical results is observed for equal axial loadingof opposite signs, which demonstrates the reliability of thesolution.

The next comparison refers to reinforced concrete sections (i.e.two materials) of irregular geometry, reported in Rosati et al. [5]. InFig. 24, moment interaction curves for a G-shaped (left) and multi-cell (right) R/C sections, under various tensile and compressive ax-ial loads are shown. It is observed that the results from bothstudies are almost coincident, albeit it is pointed out that in Rosatiet al. [5], the concrete area under steel fibers is not subtracted,resulting to minor differences (more pronounced for high com-pressive axial loads).

Fig. 25 shows comparisons of moment interaction curves for acomposite section reported in Chen et al. [16] (section D), betweenthe present study and various researchers [8,15,16]

(N = �4120 kN). Similar results are also reported in [18], whichwas not included in the plots for simplicity. It is again shown thatthe differences between all studies are marginal, which verifies theaccuracy of the present methodology.

At this point it should be noted that during the comparison pro-cess some discrepancies were also detected. Fig. 26 shows moment(left) and axial-moment (right) interaction curves for a compositesection reported in Chen et al. [16]. Initially, this section was ana-lyzed using the present methodology and the one by Charalampa-kis and Koumousis [8] (using the pertinent software by the sameAuthors [34]), producing only marginal differences (forN = �3000 kN). These are attributed to the present fiber modelingof steel bars as opposed to the area representation assumed in[8]. However, it is noted that when the above curves are comparedto the respective ones in Chen et al. [16] and Chiorean [18], signif-icant differences are observed (Fig. 26, left). On the contrary, whenthe present method is directly compared to Chiorean [18] in termsof axial-moment capacity curve (Fig. 26, right), only marginal dif-

Fig. 23. Comparison of moment interaction curves for steel angle section, between present method and Charalampakis [33].

Fig. 24. Comparison of moment interaction curves for G-shaped and multicell section, between present method and Rosati et al. [5].


ferences are observed. This is a controversial issue, since the uniax-ial moment capacity range corresponding to a moment interactioncurve (here: MX,max �MX,min = 153.2 kNm for MY = 0 andN = �3000 kN) directly maps on a horizontal cut of the respectiveaxial-moment interaction curve for the same axial load(N = �3000 kN, red arrows in Fig. 26, right), irrespective of the sec-tion origin type considered (plastic or geometric). It is therefore be-lieved, that the above discrepancies are simply attributed to amodeling oversight. Actually, if the structural steel web is removedfrom the section, all the above differences are vanishing.

At this point, the challenging computational task of handlingconcrete softening in the ultimate section response was investi-

gated. In Fig. 27, moment capacity curves are compared betweenthe present study and Charalampakis and Koumousis [34] softwarefor varying degrees of concrete softening (s), using the aforemen-tioned composite section. It is observed that the analysis resultsfrom both methods are coincident, which demonstrates the robust-ness of the present methodology even for softening functions,where non-convex interaction curves may result under highercompressive axial loads (see also Fig. 19). In Fig. 28, the analysisis taken further producing full 3D failure surfaces with and withoutconsidering the softening effect. It is shown that the present meth-od produces smooth surfaces without non-convergence issues,where increasing softening leads to smaller ultimate strength vol-

Fig. 25. Comparison of moment interaction curves for composite section reported in Chen et al. [16], between present method and various researchers [8,15,16].

Fig. 26. Comparison of moment and axial-moment interaction curves for composite section reported in Chen et al. [16], between present method and various researchers[8,16,18].


umes, with more pronounced pinching under higher compressionlevels. It should be noted here that the effect of concrete softeningwould be more pronounced (as in [18]) if the C-restriction crite-rion, which limits the quasi-concentric compressive strain, was(improperly) ignored.

The final case is investigating whether the present method cansuccessfully analyze large-scale geometries and simplistic materiallaws, which may possibly lead to numerical (e.g. floating point) is-sues, if not properly handled in computer code. In Fig. 29 (left), acomparison is presented between axial-moment interaction curvesfor a large reinforced spillway pier section, reported in Stefan andLéger [35], using linear elastic materials for concrete and steel. It isobserved that the results between the two studies are coincident.

Moreover, the 3D interaction surface of a powerhouse section (con-sisting of two non-contiguous concrete regions) is depicted inFig. 29 (right) with exactly the same shape as the one reported in[35].

From the above comparisons presented in this chapter, it is con-cluded that the present methodology not only exhibits a robustnumerical behavior but also provides accurate results for a widerange of geometry and material characteristics.

5. Benchmarks

Apart from the indispensable numerical stability that shouldcharacterize any section analysis methodology, it is deemed that

Fig. 27. Comparison of moment interaction curves for composite section reported in Chen et al. [16], between present method and Charalampakis and Koumousis [8], underdifferent levels of concrete softening.

Fig. 28. Effect of concrete softening level on section interaction surfaces.


execution speed is also of great importance, since it is not only arepetitive task inside various structural analysis computationalframeworks but also an indication of a successful computer codeimplementation. However, this issue is usually overlooked in sim-ilar studies, rendering any pertinent speed comparison difficult orimpossible. In this study, a consistent benchmarking procedure ispresented for the first time, including various benchmarking sys-tems for executing various algorithmic tasks. Specifically, four rep-resentative personal computers of the last decade were employed,the detailed characteristics and performance indices of which aredescribed in Table 3.

On the above systems, three different numerical procedureswere assigned, namely (a) one million section integrations I(d,h)with random values of d = [�2h. . .2h] and h = [0. . .2p] (h is the sec-tion height), (b) the construction of a 3D failure surface with36,000 vertices using the N � h method and (c) the same surfaceusing the more demanding N � a method. These procedures wereexecuted for all five reference sections depicted in Fig. 17, and exe-

cution times are reported in Table 4. It is pointed out that for eachnumerical procedure, only the corresponding object-oriented pro-gramming (OOP) method was profiled, i.e. the reported times donot contain any processing overhead from graphics, logs etc. It isdeemed that the efficiency of the present methodology in termsof speed is satisfactory, performing thousands of section integra-tions or 3D point evaluations per second. The main scope of thisbenchmarking report is to establish a solid reference basis for fu-ture studies, simply by applying (at least) linear extrapolationbased on the above known system performance characteristics(Table 3).

To the best of the Author’s knowledge, the only consistentbenchmarking report in the literature is presented in Bonet et al.[1], where three bare concrete sections (solid rectangular, hollowrectangular and double-T) are integrated for different neutral axisdepths and inclinations (d,h). Integration is performed using vari-ous integration schemes, such as the Fafitis method [17], alreadydescribed in paragraph 2.2 and the Modified Thick Layer Integra-

Fig. 29. Comparison of axial-moment interaction curves for reinforced spillway pier section (left) and 3D interaction surface for powerhouse section (right), between presentmethod and Stefan and Léger [35].

Table 3Characteristics of benchmarking systems.

Feature/index Obsolete PC Older PC Regular PC High-end PC

Processor Intel Pentium 4 AMD Athlon64 4400+ Intel Core 2 Duo E6600 Intel Core i7 920CPU/FSB (MHz) 2800/200 2640/240 3000/333 3800/200SuperPIa 50.78 s 33.59 s 20.55 s 11.43 sLinXb (Linpack) 1.79 GFlops 3.60 GFlops 9.23 GFlops 12.48 GFlops7-Zipc 1307 MIPS 2367 MIPS 3011 MIPS 3823 MIPS

a v1.5 mod XS, 1M digits (www.overclock.net/downloads/138140-super-pi.html).b v0.64, 32 bit, 1 thread, size 5000 (www.xtremesystems.org/forums/showthread.php?201670-LinX-A-simple-Linpack-interface).c v9.20, 32 bit, 32M dictionary, 1 thread, total rating (www.7-zip.org). All tools extracted on 7th September 2011.

Table 4Benchmarks of stress integration and interaction surface construction on varioussystems.

Section Obsolete PC Older PC Regular PC High-end PC

1 million integrations with random d, h (s)

A 16.80 8.72 6.89 4.55B 16.71 8.71 6.55 4.54C 49.93 27.97 20.51 14.08D 204.77 113.02 83.38 59.05E 435.97 247.93 175.65 126.85

N � h interaction surface of 100 � 360 grid (s)

A 3.76 2.13 2.00 1.13B 3.86 2.22 2.11 1.16C 9.78 5.32 5.11 2.97D 43.30 22.99 20.70 12.72E 112.64 59.50 52.21 32.49

N � a interaction surface of 100 � 360 grid (s)

A 17.22 9.04 7.62 5.00B 20.82 10.97 9.20 6.00C 32.98 17.69 14.67 9.55D 146.76 75.99 64.59 42.23E 670.68 338.76 284.75 183.68


tion (MTLI) method [1]. Since both the present and the MTLI meth-ods are improved integration schemes originating from the Gauss/Green concept introduced by Fafitis [17], it is deemed interestingto attempt a comparison between them in terms of executionspeed. Of course, a direct comparison in terms of running time isnot evenhanded, because system details in [1] are not available.For reference only, a direct comparison shows that the presentmethod is at least 50 times faster than MTLI (using the ‘Older PC’system of Table 3). However, a just alternative is to perform anindirect comparison procedure as follows: for both present andMTLI methods, a ‘speed index’ is defined as the time ratio betweenthe method under consideration and the reference Fafits scheme(using three Gauss points), i.e. the percentage increase in computa-tional time. This index will be evaluated against the reported accu-racy for each method, which is expressed as the maximumnormalized error of all resultant actions with respect to exact inte-gration (using 48 or 100 Gauss points respectively). In Table 5, theresults of this indirect comparison are presented. For the presentmethod, a series of 21 section depth/orientation combinations (de-scribed in Table 3 in [1]) with 100,000 integrations per combina-tion was performed and compared against the respective resultsfor the MTLI method for 2, 3 and 4 Gauss points, reported in [1].This procedure was repeated for both normal strength(fc = 20 MPa) and high strength (fc = 80 MPa) concrete, featuring anon-polynomial stress–strain law according to Model Code 90[11], including concrete tensile response.

The results of Table 5 are depicted as ‘speed index vs. maximumerror’ plots in Fig. 30. It is observed that the present method is

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Table 5Performance comparison between present method and MTLI method by Bonet et al. [1].

Normal strength fc = 20 MPa Rectangular Hollow Double-T

Present method – total 21 � 100,000 = 2.1 million integrations

Simulated Fafitis method (3 Gauss points) (s) 10.21 19.77 8.81Present method (s) 20.21 (+98%) 34.20 (+73%) 11.70 (+33%)Error of present method (%) 0.087% 0.086% 0.016%

MTLI method – total 21 � 10 � 50 = 10,500 integrations

Simulated Fafitis method (3 Gauss points) (s) 1.65 3.13 5.21MTLI with 2 Gauss points (s) 4.39 (+166%) 5.66 (+81%) 12.19 (+134%)Error of MTLI with 2 Gauss points (%) 0.54% 0.54% 0.12%MTLI with 3 Gauss points (s) 5.94 (+260%) 7.25 (+132%) 15.32 (+194%)Error of MTLI with 3 Gauss points (%) 0.0068% 0.0068% 0.0016%MTLI with 4 Gauss points (s) 6.97 (+322%) 8.95 (+186%) 18.62 (+257%)Error of MTLI with 4 Gauss points (%) 0.000096% 0.000096% 0.000024%

High strength fc = 80 MPa Rectangular Hollow Double-T

Present method – total 21 � 100,000 = 2.1 million integrations

Simulated Fafitis method (3 Gauss points) (s) 10.80 20.70 8.74Present method (s) 37.12 (+244%) 55.17 (+167%) 18.44 (+111%)Error of present method (%) 0.71% 1.04% 0.70%

MTLI method – total 21 � 10 � 50 = 10,500 integrations

Simulated Fafitis method (3 Gauss points) (s) 1.65 3.07 5.21MTLI with 2 Gauss points (s) 6.21 (+276%) 9.12 (+197%) 13.34 (+156%)Error of MTLI with 2 Gauss points (%) 0.25% 0.34% 0.80%MTLI with 3 Gauss points (s) 8.02 (+386%) 11.76 (+283%) 16.75 (+221%)Error of MTLI with 3 Gauss points (%) 0.1223% 0.244% 0.39%MTLI with 4 Gauss points (s) 10.00 (+506%) 14.28 (+365%) 20.38 (+291%)Error of MTLI with 4 Gauss points (%) 0.0448% 0.10% 0.145%

Fig. 30. Performance comparison between present and MTLI method by Bonet et al. [1]for normal and high strength concrete.


clearly faster than its MTLI counterpart for normal strength con-crete, exhibiting smaller error levels with faster execution timesfor all three sections under consideration. As far as the highstrength concrete is concerned, the difference between the twomethods is less pronounced probably due to the brittle materialbehavior (very steep softening part in the concrete stress–strainlaw), albeit the present method generally performs faster underacceptable error levels, which can be further reduced if tighter tol-erance criteria are used (Eq. (16)). It is therefore concluded that thesuggested methodology exhibits a very satisfactory performance inboth terms of accuracy and execution speed, fulfilling all prerequi-site conditions imposed beforehand.

6. Closure

In this study, a new methodology was presented for the analysisof fully arbitrary composite sections in biaxial bending and axial

load, featuring various new algorithmic procedures regardinggeometry manipulation, stress integration and solution strategies.With the aid of validation tests, comparisons with the existing lit-erature and benchmarks, it was demonstrated that the suggestedmethod shows very satisfactory performance in terms of stabilityand speed. It is therefore believed that it can provide a solid basisfor further developments in the fields of structural design, assess-ment and optimization. Various extensions of the present methodare currently investigated, specifically in the directions of sectionassessment against external loading, design for reinforcement, pre-stressed tendons, jacketed sections with preloading and designoptimization.

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