Probabilistic Analysis of Prestressed Inverted T-beam with ... · PDF fileBased on their researches, the design procedures for prestressed concrete beams with web ... prestressed T-beam

Research Journal of Applied Sciences, Engineering and Technology 4(4): 350-366, 2012ISSN: 2040-7467© Maxwell Scientific Organization, 2012Submitted: September 23, 2011 Accepted: November 04, 2011 Published: February 15, 2012

Corresponding Author: Tie Ming, Civil Engineering and Earth Resources Faculty, University Malaysia Pahang, Malaysia350

Application of Probabilistic Analysis in Finite Element Modeling of Prestressed Inverted T-Beam with Web Openings

Tie Ming, Cheng Hock Tian, Zahrizan bin Zakaria and Ideris bin ZakariaCivil Engineering and Earth Resources Faculty, University Malaysia Pahang, Malaysia

Abstract: Recent trends of structural mechanics applications in finite element analysis demonstrate anincreasing demand for efficient analysis tools. This paper presents a probabilistic analysis approach applied infinite element analysis for modeling prestressed inverted T-beams with web openings structure used in buildingservice system (mechanical, electrical, communications, and plumbing). The experimental program reportedin this paper tested four prestressed inverted T-beams with circular web openings to failure to evaluate theopenings' effect on various beam behaviors. Using ANSYS, finite element models were developed to simulatebeam deflection behavior. Comparison of analytical results with the available experimental results for load-deflection relationships showed good agreement between both results. Probabilistic analysis methodology couldpredict the response (i.e., deflection, stress, strain etc) due to various combination of input variables (i.e.,Poisson`s ratio, modulus of elasticity, etc). In reality, uncertainties exist in a system and environment that maymake the application of deterministic design unreliable which causes the values of the variables that are actingon the system cannot be predicted with certainty. As such, probabilistic approach was applied to the model afterdeterministic analysis. In this study, the probabilistic analysis approach was applied to account for thevariability in fabrication. Probabilistic methodology applied in finite element modeling provides anotheralternative ways of structural analysis of preststressed inverted T-beams with web openings to achieve a robustand reliable design in a more efficient way. In this study, Monte Carlo simulation was used to analyze the effectof parameter uncertainty for the prestressed inverted T-beams with web openings. From the analysis results,it was observed that the changes in prestressing force, elastic modulus of prestressing steel, ultimate tensilestrength of prestressing steel and beam width tend to be the most influencing parameters, which need to betightly controlled. As a result, from deterministic analysis and probabilistic analysis, it was found thatprobabilistic analysis tends to be closer to reality than deterministic methods and gives a way of designing forquality.

Key words: Finite element modeling, monte carlo simulation, prestressed inverted t-beams, probabilisticanalysis

INTRODUCTION

A precast prestressed inverted T-beam with circularweb openings allow building services (mechanical,electrical, communication and plumbing) to cross thebeam, reducing a building`s floor to floor height andoverall height of the structure. Those height reductionshave a potential to improve the competitiveness of totalprecast concrete structures versus other type of buildingsystem. However, introducing an opening into the web ofa prestressed concrete beam reduces stiffness and leads tomore complicated behavior. Therefore, the effect ofopenings on strength and service ability must beconsidered in the design process. Numerous investigationssuch as (Mansur, 1988) have been carried out onreinforced concrete beams with opening. The firstpublished work on prestressed beams with web openingswas conducted by (Regan and Warwaruk, 1967). Since

then, several other researchers (Sauve, 1970; Kennedyand Abdalla, 1992) have investigated prestressed beamswith web openings. Based on their researches, the designprocedures for prestressed concrete beams with webopenings were developed. While they have not consideredsimplifications and probabilistic response of prestressedbeams with web openings, therefore an experimental andanalytical study on effect of opening and strength of theprestressed inverted T-beam with web openings isrequired to provide data and probabilistic analysis whichwill be used in the design guideline of the beams. Hence,an understanding of the structural behavior of invertedprestressed T-beam with circular web opening underinfluence of locations and size of the openings, the effectof multiple web opening in both constant moment andconstant shear region as well as probabilistic analysis areuseful to provide data which could be helpful to developsatisfactory design guideline of the beams into use.

Res. J. Appl. Sci. Eng. Technol., 4(4): 350-366, 2012

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Uncertainty exists in most engineering structures,such as bridges and buildings which are essential parts ofour modern lives. Uncertainty of structural parametersmay be resulted from manufacturing, constructiontolerances or caused by the aggressive environmentalagents such as progressive deterioration of concrete andcorrosion of steel. In engineering applications, uncertaintyalso exists in determining external loads, such as windforces, earthquake motions and automobiles on highways.Nondeterministic analysis of engineering structures withuncertainties in system parameters and inputs has beenattracted considerable attention in the last two decades(Chen et al., 1992; Rao and Berke, 1997; Ma et al., 2006;Falsone and Ferro, 2007; Stefanou, 2009). Usuallyprobabilistic analysis is used to deal with the uncertainty.In probabilistic methods, uncertain parameters aremodeled as random variables and uncertainties of loadsare described by random processes. As very useful tools,probabilistic methods have been widely used to predictthe static, dynamic and random responses of structuralsystems with uncertainty (Stefanou and Papadrakakis,2004; Val and Stewart, 2005; Gao and Kessissoglou,2007; Singh et al., 2009). These methods can provide notonly the mean value but also the standard deviation andeven the probability density for structural responses.Monte Carlo simulation method (Figiel and Kaminski,2009; Yu et al., 2010) has been developed to analyzerandom data structures. The probabilistic methods,however, are only applicable when information about anuncertain parameter in the form of a preferenceprobability function is available. If the uncertaininformation or statistical data of system parameters arenot sufficient to give their satisfactory stochasticcharacterization, only the non-probabilistic approachescan be applied (Falsone and Ferro, 2007). On thecontrary, if it is possible to characterize the uncertainparameters stochastically, then the probabilistic methodare the most suitable ones. In this study, the probabilisticanalysis approach is applied to account for the variabilityin fabrication. Probabilistic methodology applied in finiteelement analysis (Fazilat et al., 2002; Giuseppe, 2011;Taejun and Tae, 2008) provides another alternative waysof structural analysis of prestressed inverted T-beams withweb openings to achieve a robust and reliable design in amore efficient way.

METHODOLOGY

Probabilistic design is an analysis technique forassessing the effect of uncertain input parameters andassumptions on deterministic model. In this study,ANSYS probabilistic design system help to determine theextent to which uncertainties in the model affect theresults of a finite element analysis. An uncertainty is aparameter whose value is impossible to determine at a

given point in time (if it is time-dependent) or at a givenlocation (if it is location-dependent) (SAS, 2009).

Monte Carlo simulation and Latin hypercubesimulation: Monte Carlo Simulation is the most commonand traditional method for a probabilistic analysis. Afundamental characteristic of the Monte Carlo Simulationmethod is the fact that sampling points are located atrandom locations in the space of the random inputvariables. There are various techniques available inliterature that can be used to evaluate the randomlocations of the sampling points (Hammersley andHandscomb, 1964). The direct Monte Carlo simulationand Latin Hypercube Sampling are two approaches. Thedirect Monte Carlo Simulation method is not usedbecause its random sampling has no memory. Instead ofthat the Latin Hypercube Sampling technique isimplemented and the range of all random input variablesis divided into n intervals with equal probability.

Simulation is the process of replicating the real worldphenomenon based on a set of assumptions and conceivedrealistic models. It may be performed theoretically orexperimentally. For engineering purpose simulation maybe applied to predict or study the performance or responseof a structure. With a prescribed set of values for thedesign variable, the simulation process yields a specificmeasure of performance or response. A conventionalapproach to this process is Monte Carlo simulationtechnique. However, in practice, Monte Carlo simulationmay be limited by constrains, computer capability, andsignificant expense of computer runs in such complexstructural system as prestressed inverted T-beam with webopenings. An alternative approach is to use a constrainedsampling scheme. One such scheme developed by (Imanand Conover, 1980; Iman et al., 1981a; Iman et al.,1981b) was Latin Hypercube sampling (LHS) method. Bysampling from the assumed probability density functionof X`s and evaluating Y for each sample, the distributionof Y, its mean, standard deviation, percentiles and so on,can be estimated.

The LHS methods consist of two steps to obtain a Ndesign matrix. The first step is dividing each inputvariable into N intervals. The second step is the couplingof input variables with tables of random permutations ofrank numbers. Every input variable, Xk, (where, k =1, 2,K) is described by its known Cumulative DistributionFunction (CDF), Fxk(x) with appropriate statisticalparameters. The range of the known CDF, Fxk(x) of eachinput variable, Xk is partitioned into N intervals with equalprobability of 1/N.

The representative value in each interval is used justonce during the simulation procedure. Therefore there areN observations on each of the K input variables. They areordered in the table of random permutations of ranknumbers which have N rows and K columns. For such a


352

sample, one can evaluate the corresponding value Yn of theoutput variable. From N simulations, one can obtain a setof statistical data. {Y} = [Y1, Y2, Yn]T. This set isstatistically assessed and thus the estimations of somestatistically parameters, such as the mean value andstandard deviation, are obtained. A more detaileddiscussion of this sampling method can be found in thepapers by some researchers (Nov et al., 1998; Iman andConover, 1982).

Monte Carlo simulation: Monte Carlo Simulation(MCS) results established prestressed inverted T-beamresponse statistics. Three graphical tools are utilized todetermine a best-fit distribution model for each response:probability distribution, empirical CumulativeDistribution Function (CDF), and histogram. ConfidenceInterval (CI) with 95% confidence level and p-value werealso included in the probability distribution. CI representsthe intervals covering the estimation. p-value representsthe probability of obtaining an observed result and itssignificance level. The smaller the p-value, the moresignificant the results are.

Considering various distribution types, a best-fitdistribution for each random input variable wasdetermined. The input variable, however, were fit usingtwo distribution types: normal (Gaussian) and lognormaldistribution. A normal distribution is represented by mean(:) and standard deviation (F) to describe the distributionshape. PDF and CDF for the normal distribution are:

Probability Density Function (PDF) equation:

(1)fx xx

x( ) exp ( )= −−⎡

⎣⎢⎤⎦⎥

⎛

⎝⎜⎜

⎞

⎠⎟⎟ −∞ < < +∞

12

12

2

σμ

σ

CDF:

(2)Fx xx

dxx

( ) exp= −−⎡

⎣⎢⎤⎦⎥

⎛

⎝⎜⎜

⎞

⎠⎟⎟−∞∫

12

12

2

σ πμ

σ

where, : = mean value; F = standard deviation.PDF and CDF for the lognormal distribution is:

PDF:

(3)f xx

xy

xxy

( ) exp ln ( )= −⎡

⎣⎢

⎤

⎦⎥

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ < < +∞

12

12

02

σ σ

CDF:

(4)F xx y

x yy

dxxx

( ) expln

= −−⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜⎜

⎞

⎠⎟⎟−∞∫

12

12

2

σ πμ

σ

where,

:x = Mean value; σ σμ

22

1= +⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ln xx

and μ μ σy x yIn= −( )12

2

It is convenient to describe location, and scaleparameters for lognormal distribution. Location and scalehave a relationship with mean and standard deviation:

(5)Location InIn

= −+

22

2 2( )

( )μ

μ σ

(6)Scale In In= + −( ) ( )μ σ μ2 2 2

where,µ = The desired mean of the lognormal dataF = The desired standard

Deviation of the lognormal data from Eq. (6), : andF can be derived from location and scale parameters as:

(7) μ = +⎛⎝⎜

⎞⎠⎟exp location

scale2

2

(8)σ μ= + −exp[ ( )]2 2 2location scale

Sensitivity analysis: An important step in theprobabilistic structural analysis is the sensitivity analysisof structural responses. This helps identify the importantparameters. On other hand, sensitivity analysis is alsouseful in reducing the size of problems with a largenumbers of random variables. This is because that, ingeneral, only a few variables have a significant effect onthe probabilistic structural response. In this paper, theresults of the Latin Hypercube simulations can be used todetermine which of the model parameters are mostsignificant in affecting the uncertainty of the design. Theso-called Spearman rank-order correlation rs arefrequently applied within the framework of a simulationmethod (Oh and Yang, 2000). The Spearman rank-ordercorrelation can be defined as:


353

Fig. 1: Configuration of web openings inside the prestressed inverted T-beam and inverted T-beam overall detailed dimensions(Cheng et al., 2009)

(9)rm n

N Nri

j ji ji= −

−

−∈ −1

61

1 12

2

Σ ( )( )

, [ , ]

where ri is the order representing the value of randomvariable Xi in an ordered sample among N simulatedvalues applied in the j th simulation (the order mi equalsthe permutation at LHS); and nj is the order of an orderedsample of the resulting variable for the j th run of thesimulation process (mij - nj is the different between theranks of two samples). If the coefficient ri had a valuenear to 1 or -1, it would suggest a very strong dependenceof the output on the input. Opposite to this, the coefficientwith its value near to zero will signalize a low influence.The sensitivity coefficients ki, defined on behalf ofvariation coefficients by the relation:

(10)kiyi

y= 100

2

2

υυ

[%]

where vyi is the variation coefficient of the output quantity,assuming that all the input quantities except the ith one (i= 1, 2..., M) where, M is number of input variables) are

considered to be deterministic (during the simulation, theyare equal to the mean value); vyi is the variation coefficientof the output quantity, assuming that all the inputquantities are considered to be random ones.

Based on Eq. (9) and (10) the random input variablesthat have significant effects on the output variables can beidentified. It should be pointed out that the correlationcoefficients, defined in Eq. (9) and (10), have been usedto evaluate the probabilistic sensitivities in the ANSYSProbabilistic Design System.

Experiment program: The experimental program wasdesigned to investigate the failure behavior of prestressedinverted T-beams with circular web openings under staticloading conditions. Testing was intended to evaluate theflexural strength. Therefore, all beams were designed suchthat shear strength exceeded flexural strength.Accordingly, flexural failure was expected. Figure 1 andFig. 2 show the configuration of web openings in theprestressed beams as well as cross-section prestressingsteel and shear reinforcement detailing. The materialsused, design and fabrication of prestressed inverted T-beams are described below (Cheng et al., 2009). Theexperimental program was carried out at the Hume


354

Fig. 2: Detailing and location of prestressing steels and shear reinforcements (Cheng et al., 2009)

Fig. 3: Prestrssing steel and shear reinforcement locations (Cheng et al., 2009)

Concrete Product Research Centre (HCPRC) and TenagaNasional Research Berhad (TNBR) for testing facilitiesand analysis assistance of prestressed beam with webopenings which started from 2009 to 2010. The extendedwork of this research study, finite element modeling andprobabilistic analysis, was studied at University MalaysiaPahang 2011.

Materials: The average 28 days concrete cube strength incompression was 55 MPa for all four beams, as evaluatedby tests on three cubes specimens for each beam.Figure 3 show the prestressing steel and shearreinforcement location in the prestressed concrete beams.Straight, 3 show the prestressing steel and shearreinforcement location in the prestressed concrete


355

Fig. 4: Prestressed inverted T-beams with circular web openings fabricated at factory (Cheng et al., 2009)

Fig. 5: Four points bending test of prestressed inverted T-beam (Cheng et al., 2009)

beams. Straight, bonded, seven-wired super high tensilestrand with 12.9 mm diameter were used as prestressingtendons, with ultimate strength of 1,860 MPa. The stirrupsfor shear reinforcement were made from 10 mm rebarswith minimum specified yield strength of 250 N/mm2.Theelastic modulus of the prestressing steel is taken as 195103 N/mm2. Figure 4 shows the fabricated prestressedbeams used in the test program.

Testing set-up: All tests were conducted with a close-loop hydraulic servo-controlled MTS testing system. The360 KN jack was capable of both displacement and loadcontrol for monotonic or cyclic loading. A four-pointloading scheme, with an effective span of 4,000 mm anda distance of 1,200 mm between the loads points was usedto limit the presence of shear stress in the mid-span zone.Figure 5 shows the layout of four points bending test ofprestressed inverted T-beams (Cheng et al., 2009).

Finite element models: In this study, the finite elementanalysis of the model was set up to examine threedifferent behaviors: initial cracking of the beam, yieldingof the steel reinforcement, and the strength limit state ofthe beam. The Newton-Raphson method of analysis wasused to compute the nonlinear response. The applicationof the loads up to failure was done incrementally asrequired by the Newton-Raphson procedure. After eachload increment was applied, the restart option was used to

go to the next step after convergence. The twoconvergence criteria used for the analysis were force anddisplacement.

Element types: Concrete part was modeled using a three-dimensional solid element, SOLID65, which has thematerial model to predict the failure of brittle elements.SOLID65 is defined with eight nodes each with threedegrees of freedom: translations in nodal x, y and zdirections. This element is capable of cracking in tensionand crushing in compression. Plastic deformation andcreep can also be captured. The cracking is determined bythe criterion of maximum tensile stress, called `tensioncutoff ̀ . Concrete crushes when the compressive principalstress (Von Mises stress) on the failure surface surpassesthe Willam-Warnke failure criterion dependent on fivematerials parameters (Willam and Warnke, 1974). TheSOLID 45 element was used for the supports for thebeam. This element has eight nodes with three degrees offreedom at each node with translations in the nodal x, yand z directions. To simulate the behaviors of prestressingsteel, a truss element, LINK8, were used to withstand theinitial strain attributed to prestressing forces, by assumingperfect bond between these elements and concrete.LINK8 requires users to input `real constants` to definereinforcement geometry, material behavior, andprestressing strain. Note that this truss element cannotresist neither bending moments nor shear forces. The


356

descriptions for each element type were laid out in theANSYS element library (SAS, 2009).

Material properties:Concrete: The SOLID65 element requires linear isotropicand multi-linear isotropic materials properties to properlymodel concrete. Concrete is a quasi-brittle material andhas very different behaviors in compression and tension.The tensile strength of concrete is typically 8-15% of thecompressive strength. The ultimate concrete compressiveand tensile strengths for each beam model were calculatedby Eq. (11) and (12), respectively (ACI Committee 318-99, 1999).

(11)′ = ⎛⎝⎜

⎞⎠⎟

fE

cc

4730

2

(12)f fr c= ′0 623. where,C Ec = Elastic modulus of concrete C fc

! = Ultimate compressive strength C fr = Ultimate tensile strength (modulus of rupture)

The following Eq. (13) and (14) (Desayi andKrishnan, 1964) are used along with Eq. (15) to constructthe uniaxial compressive stress-strain curve for concretein this study:

(13)fEc=

+⎛

⎝⎜

⎞

⎠⎟

ε

εε

10

2

(14)ε02

=′f

Ec

c

(15)Ef

c =ε

where,f = Stress at any strain , , = Strain at stress fEc = Concrete elastic modulus ,0 = Strain at the ultimate compressive strength fc

!

strength. After this point, the concrete cracks and thestrength decreases to zero. Figure 6 shows the simplifieduniaxial stress-strain relationship that is used in this study.

Fig. 6: Simplified uniaxial stress-strain curve for concrete

In tension, the stress-strain curve for concrete isassumed to be linearly elastic up to the ultimate tensilePoisson`s ration for concrete was assumed to be 0.3 andwas used for all beams. The value of a shear transfercoefficient, representing conditions of the crack face, usedin many studies of reinforced concrete structures variedbetween 0.05 and 0.25 (Bangash, 1989; Hemmaty, 1998;Huyse et al., 1994). The shear transfer coefficient used inthis study is equal to 0.2.

Steel reinforcement and prestrssing steel: Steelreinforcement in the experiment beams was constructedwith typical steel reinforcing bars fy = 1000 MPa. Elasticmodulus and yield stress for the steel reinforcement usedin this FEM study follow the design material propertiesused for the experimental investigation. The steel for thefinite element models is assumed to be an elastic-perfectlyplastic material and identical in tension and compression.A Poisson`s ratio of 0.3 is used for the steelreinforcement.

The SOLID 45 element is being used for supports onthe beam. Therefore, this element is modeled as linearisotropic element with a modulus of elasticity for the steelEs = 1000 MPa and Poisson`s ratio v = 0.3. The LINK8element is being used for all the steel reinforcement in thebeam and it is assumed to be bilinear isotropic. Bilinearisotropic material was also based on the Von Mises failurecriteria.

For prestressing steel, the bi-linear elastic-plasticmaterial models can be used as well as the multi-linearisotropic model from the manufacture`s data. In thepresent study a multi-linear isotropic stress-strainrelationship is adopted for prestressing strands by usingEq. (16) or (17) (Mohamad and Akram, 2010):

(16)ε εps p psf≤ =0 008 28 000. : ,

where:

,ps = Strain of prestressing steel


357

200018001600140012001000800600400200

00 0.006 0.010 0.015 0.020 0.025 0.030 0.035

Strain (mm/mm)

Stre

ss (M

pa)

Fig. 7: Stress-strain curves of steel strands n12.7mm) ()(Mohamad and Akram, 2010)

(17)εεps p

pspf f> = −

−<0 008 268

0 0750 0065

0 98. ,.

..

where,

C ,ps = Strain of prestressing steelC fp = Ultimate strength of prestressing steel

The ultimate stress fp =1, 860 MPa and the elasticityof modulus of prestressing steel Ep = 195,000 MPa.Poisson`s ration for prestressing steel was also assumedto be 0.3. The adopted stress-strain curve is show inFig. 7. The initial prestressing strain is taken equal to0.006 mm.

Analytical model: The purpose of the analysis was toobtain the load-deflection curve of prestressed beams. The

Fig. 8: FEA model of prestressed inverted T-beam with web opening

Fig. 9: FEA model of iso-viewing of tendon and stirrup


358

Fig. 10: FEA model of 1/4 beam of tendon and stirrup

beam geometry was modeled and meshed, then thetendons were defined and finally the loads area applied toit. The analyzed beam was 5 m long and had variousdifferent types openings configurations along it. It wasprestressed by 2 straight tendons localized on the top andbottom sides of it. Figure 8 to 10 show the finite elementmodels for the prestressed inverted T-beam.

Due to the symmetry in cross-section of theprestressed concrete beam and loading, symmetry wasutilized in the finite element analysis; only one quarter ofthe beam was model. This approach reducedcomputational time and computer disk space requirementssignificantly. The steel reinforcement is simplified in themodel by ignoring the inclined portion of the steel barpresent in the test beams. Ideally, the bond strengthbetween the concrete and steel reinforcement should beconsidered. However, in this study, perfect bond betweenmaterials is assumed.

Numerical results: The experimental and numerical load-deflection curves obtained for the beams are illustrated inFig. 11 to 14. The load-deflection plots for all four beamsfrom the finite element analysis agree quite well with theexperimental data. First cracking loads for all four modelsfrom the finite element analysis are higher than thosefrom experimental results by 3-8%. After first cracking,the stiffness of the finite element models is agreed wellwith the experimental beams by 1-2% differences. Thereare several factors that may cause the higher stiffness inthe finite element models at the initial loading periods.Micro-cracks produced by drying shrinkage and handling

are present in the concrete to some degree. These wouldreduce the stiffness of the actual beams, while the finiteelement models do not include micro-cracks. Perfect bondbetween the concrete and steel reinforcing is assumed inthe finite element analysis, but the assumption would notbe true for the actual beams. When bond slip occurs, thecomposite action between the concrete and steelreinforcing is lost. The overall stiffness of the actualbeams could be lower than what the finite element modelspredict, due to the factors that are not incorporated intothe models.

Results from the non-linear finite element analysiswere compared to those from the test on the fourprestressed inverted T-beam with web openings as well assolid beam. The stress and strain distributions around theopening were determined. The influence of the openingon deflection, cracking and shear distribution was studied.Figure 11 to 14 show a typical comparison between thetheoretical and experimental test results for the deflectionin the vicinity of web openings in prestressed beam, whenthe external vertical load is applied at web openingslocated at constant moment area as well as otherlocations. The agreement between experiment and finiteelement analysis is acceptable in all cases.

From the experimental results of beam with 9 webopenings analysis data, the maximum value of deflectionis 37.9 mm, maximum Von Mises strain is 71.4 :, andmaximum Von Mises stress is 131.733 KN have a quitesmall difference with solid beams analysis data.Meanwhile, as shown in Table 1, the deterministicanalysis data also showed a quite difference between


359

250

200

150

100

50

00 10 20 30 40

Deflection (mm)

Load

(KN

)

Experimental results FEM results

250

200

150

100

50

00 10 20 30 40

Deflection (mm)

Load

(KN

)


250

200

150

100

50

00 10 20 30 40

Deflection (mm)

Load

(KN

)


Fig. 11: FE modeling and load-deflection plots for beam with nine web openings

Fig. 12: FE modeling and load-deflection plots for beam with six web openings

Fig. 13: FE modeling and load-deflection plots for beam with three web openings


360

250

200

150

100

50

0

Load

(KN

)

0 10 20 30 40Deflection (mm)


Fig. 14: FE modeling and load-deflection plots for solid beam

Table 1:Comparison of FEM results and experimental results ofprestressed beams with different number of web openings

Deterministic analysis Experimental dataParameter maximum value maximum valueBeam with 9 web openingsDeflection (mm) 40 37.9Von Mises strain (:,) 68.1 71.4Von Mises stress (KN ) 123.776 131.733Beam with 6 web openingsDeflection (mm) 35.3 35.1Von Mises strain (:,) 68.2 68.2Von Mises stress (KN ) 123.966 125.830Beam with 3 web openingsDeflection (mm) 33.8 30.3Von Mises strain (:,) 68.1 66.3Von Mises stress (KN ) 123.859 122.324Solid beamDeflection (mm) 31 35.7Von Mises strain (:,) 68.2 68.4Von Mises stress (KN ) 123.979 125.829

numbers of web openings and solid beam. In consequenceof data result from Table 1, the numbers of web openingsand solid beam were almost having identical strengthproperty.

Probabilistic analysis design: In probabilistic designsystem, all these input parameters could be anythingranging from geometry and material properties todifferent boundary conditions. These parameters aredefined as random input variables and are characterizedby their distribution types and variables (e.g., mean,

standard deviation, etc). The key outputs of the simulationare defined as random output parameters. During aprobabilistic analysis, multiple analysis loops areexecuted to compute the random output parameters as afunction of the set of random input variables.The valuesfor the input variables are generated randomly usingMonte Carlo simulation.

To illustrate the application of the probabilisticmethod, four group of experiment prestressed beam withnumbers of web opening as describle in experimentprogram previously was analyzed. The geometric andsectional parameters of the pretressed beam and layoutparameters of the prestressing steel are assumed todeterministic since they are relatively predictable incomparison to other parameters such as material strength.The applied load (P), elastic modulus of concrete (Ec),elastic modulus of prestressing steel (Ep), beam depth (l),beam width (b), prestressing force (Ps ), Poisson`s ratio(v), ultimate compressive strength of concrete (fc

`) andultimate tensile strength of prestressing steel (fp) werechosen as the random input variables of interest for thisstudy. The statistical descriptions of these randomvariables are shown in Table 2.

Probabilistic analysis results: As the objective of thisresearch was to study probabilistic response analysis ofpretressed concrete beam with web openings, all randomparameters in the analysis were based on arbitrary but

Table 2: Statistical analysis of various input random variables for probabilistic designRandom variables Distribution types Mean value SD Coefficients of variationP (N) Gaussian 100000 10000 10%Ec (Mpa) Gaussian 45000 6750 15%Ep (Mpa) Gaussian 195000 9750 5%l (mm) Gaussian 523 10.46 2%b (mm) Gaussian 485 9.7 2%Ps (N) Lognormal 111600 5580 5%v Lognormal 0.3 0.015 5%fc

!( MPa ) Gaussian 55 9.25 15%fp ( MPa ) Gaussian 1860 93 5%


361

0.2500.2250.2000.1750.1500.1250.1000.0750.0500.025

0

Rel

ativ

e fr

eque

ncy

0.357024 0.377255 0.397487 0.417718 0.437949

Mean Stdev Skew

Kurt Min Max

0.40020E+000.16463E-010.10208E+00

0.97961E-010.35702E+000.44806E+00

0.400.360.320.280.240.200.160.120.080.0400.0

Rel

ativ

e fr

eque

ncy

Mean Stdev Skew

Kurt Min Max

0.68805E-030.77448E-040.68210E+00

0.13992E+010.54614E-030.10006E-02

0.00

0546

0.00

0647

0.00

0748

0.00

0849

0.00

0950

Table 3: Statistical analysis of three output parametersName Mean SD Min. Max.Beam with 9 web openingsMAXIMUM_DEFLECTION (mm) 40 1.6 35.7 44.8MAX_VON_MISES_STRAIN (:,) 68.8 7.7 54.6 100MAX_VON_MISES_STRESS(KN ) 123.8 6.332 107 142.1Beam with 6 web openingsMAXIMUM_DEFLECTION (mm) 35.3 1.7 31.5 40.3MAX_VON_MISES_STRAIN (:,) 68.8 8 53 97MAX_VON_MISES_STRESS(KN ) 123.8 6.271 109.3 143.1Beam with 3 web openingsMAXIMUM_DEFLECTION (mm) 33.8 1.6 30.1 38.7MAX_VON_MISES_STRAIN (:,) 68.8 8 49.2 92.5MAX_VON_MISES_STRESS(KN ) 123.9 6.238 109.3 143.4Solid beamMAXIMUM_DEFLECTION (mm) 31 1.7 26.7 38.3MAX_VON_MISES_STRAIN (:,) 69 8.4 50.2 100MAX_VON_MISES_STRESS(KN ) 124.1 6.670 105.8 152

Table 4: Comparative studies of results obtained from deterministicand probabilistic analysis

Deterministic analysis Probabilistic analysisOutput variables maximum value maximum value9 web openingsDeflection (mm) 40 44.8Von Mises strain (:,) 68.1 100Von Mises stress (KN ) 123.776 142.16 web openingsDeflection (mm) 35.3 40.3Von Mises strain (:,) 68.2 97Von Mises stress (KN ) 123.966 143.13 web openingsDeflection (mm) 33.8 38.7Von Mises strain (:,) 68.1 92.5Von Mises stress (KN ) 124.038 143.4Solid beamDeflection (mm) 31 38.3Von Mises strain (:,) 68.2 100Von Mises stress (KN ) 123.979 152

typical values. On the other hand, since the determinationof the correlation of the random parameters is a difficulttask, using the independence assumption cans greatlysimply the probabilistic analysis. Therefore, all randomparameters in the paper were treated as stochasticallyindependent from each other. Unless stated otherwise, thestatistics of random variables given in Table 2 were usedin the present studies.

For the probabilistic analysis, the ANSYSprobabilistic design system analysis was looped through1000 sample points considering the variations defined inthe input variables, and the corresponding statisticalanalysis of the output parameters are given in Table 3.

In this research, the three output parameters valuelists in Table 3 which obtained from using probabilityanalysis have a differences value between deterministicanalyses. This differences value was examined andtabulated in Table 4. For example, as the specimen of 9web openings, the maximum deflection value is 40 mmand maximum Von Mises stress is 123.776 KN ofdeterministic results with probabilistic results themaximum deflection value is 44.8 mm and maximum Von

Fig. 15: Histogram of maximum deflection (beam with 9 webopenings)

Fig. 16: Histogram of maximum Von Mises strain (beam with9 web openings)

Mises stress is 142.1 KN, probabilistic results are higherthan deterministic ones. So from the results from Table 4,the approach of probabilistic analysis is relatively higherthan deterministic analysis.

Probabilistic post-processing results of beam with9 web opening specimen: Figure 15 to 17 graphicallydepicts a histogram of each output parameters. The values


362

107.039 114.837 122.636 130.434 138.233

0.400.360.320.280.240.200.160.120.080.0400.0

Rel

ativ

e fr

eque

ncy

Mean Stdev Skew

Kurt Min Max

0.12378E+030.63325E+010.14076E+00

0.12986E+000.10704E+030.14213E+03

99.9999.9099.0090.0070.0050.0030.0010.001.0000.1000.010

0.3520.360

0.3680.376

0.3840.392

0.4000.408

0.4160.424

0.4320.440

0.4480.456

Prob

abili

ty in

α

Mean Stdev Skew

Kurt Min Max

0.40020E+000.16463E-010.10208E+00

0.97961E-010.10704E+030.14213E+00

Confidence limit 95.00%

99.9999.9099.0090.0070.0050.0030.0010.001.0000.1000.010

Prob

abili

ty (%

)

Mean Stdev Skew

Kurt Min Max

0.68805E-030.77448E-040.68210E+00

0.13992E+010.54614E-030.10006E-02

0.000520.00056

0.00060.00064

0.000680.00072

0.000760.0008

0.000840.00088

0.000920.00096104

0.0010


Fig. 17: Histogram of output variable maximum Von Misesstress (beam with 9 web openings)

given on each distribution plot were mean value, standarddeviation, skewness, kurtosis, minimum value andmaximum value, respectively.

Descriptions below provide some technical statisticsinformation on the coefficient of skewness and coefficientof kurtosis which has been broadly used in this researchanalysis.

Coefficient of skewness, a measure of the asymmetryof the probability distribution of a real-valued randomvariable. The skewness value can be positive or negative,or even undefined. Qualitatively, a negative skewindicates that the tail on the left side of the probabilitydensity function is longer than the right side and the bulkof the values lie to the right of the mean. A positive skewindicates that the tail on the right side is longer than theleft side and bulk of the values lie to the left of the mean.A zero value indicates that the value a relatively evenlydistributed on both side of the mean, typically but notnecessarily implying a symmetric distribution. Normaldistributions produce a skewness statistic of about zero.As shown in Fig. 15 to 17, the value of skewness 0.10208,0.6821 and 0.14076 would be an acceptable skewnessvalue for normally distribution set of test scores becauseit is very close to zero and is probably just a chancefluctuation from zero.

Coefficient of kurtosis, a measure of the peakednessof the probability distribution of a real-valued randomvariable, although some sources are insistent that heavytails, and not peakedness, is what is really being measuredby kurtosis. Higher kurtosis means more of the varianceis the result of infrequent extreme deviations, as opposedto frequent modestly sized deviations. Normaldistributions produce a kurtosis statistic of about zero. Inthis analysis, a kurtosis statistic of 0.097961, 0.13992 and0.12986 from Fig. 15 to 17 would be an acceptablekurtosis value for a normal distribution because it is closeto zero.

As shown in Fig. 15 to 17, the minimum andmaximum of Von Mises strain are 54.6 and 100 :,,

Fig. 18: 95% confidence interval for maximum deflection(beam with 9 web openings)

Fig. 19: 95% confidence interval for maximum Von Misesstrain (beam with 9 web openings)

respectively, and it increases by 83.2%. While theminimum and maximum of deflection are 35.7 and 44.8mm, respectively, and it increases by 25.5%. Meanwhilethe minimum and maximum of Von Mises stress are 107.4to 142.13 KN, respectively, and it increases by 32.3%. Itmeans that output parameter of Von Mises strain has moresignificant influence by input parameter`s variations.

Technical products are typically designed to fulfillcertain design criteria based on the output parameters. Forexample, a design criterion is that the deflection will beabove or below a certain limit. The cumulativedistribution curve for maximum deflection is 44.8 mm,maximum Von Mises stress is 142.1 KN and maximumVon Mises strain is 100 :, are shown in Fig. 18 to 20.The line in middle is the probability P that the maximumdeflection, maximum Von Mises stress and maximum VonMises strain remains lower than a certain limit valueMaximum deflection is 42.6 mm, maximum Von Misesstress is 135 KN and maximum Von Mises strain is 95 :,with 95% confidence interval. The complement 1.0 - P isthe probability that Maximum deflection, maximum VonMises stress and maximum Von Mises strain exceed thislimit. For the stability of the concrete beam, the reliabilityof beams is given by the probability that the deflection,


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99.9999.9099.0090.0070.0050.0030.0010.001.0000.1000.010

Prob

abili

ty (%

)

Mean Stdev Skew Kurt Min Max

0.12378E+030.63325E+010.14076E+000.12986E+000.10704E+030.14213E+03

104108

112116

120124

128132

136140

144148

152156


Fig. 20: 95% confidence interval for maximum Von Misesstress (beam with 9 web openings)

stress and strain should falls within the range ofconfidence interval. The upper and lower curves inFig. 18 to 20 are the confidence interval using a 95%confidence level. The confidence interval quantifies theaccuracy of the probability results. After the reliability ofthe beams has been quantified, it may happen that theresulting value is not sufficient. Then, probabilisticmethods can be used to answer the following question:Which input variables should be addressed to achieve arobust design and improve the quality? The answer to thatquestion can be derived from probabilistic sensitivitydiagrams plot.

Sensitivity analysis results: The result of the proposedmethod was Spearman rank-order correlation Eq. (9) and(10) to determine which random parameters are mostsignificant in affecting the uncertainty of the design. The

sensitivity analysis results obtained for nonlinear analysisare shown in Fig. 21 to 23. The sensitivities are given asabsolute values (bar chart) and relative to each other (piechart). For easy input these parameters in ANSYSprogram, here use shortened form of proper name, suchas: P = F1, l = B14, b = B1, Ec = E3, Ep = E2, Ps = PREF,v = PRXY4, fc

`= SYGJ and fp = SB. From Figures as shownbelow, the prestressing force Ps and beam depth lsignificant influence on the output parameter formaximum deflection. On the other hand, applied load P;the prestressing force Ps; elastic modulus of prestressingsteel Ep; ultimate tensile strength of prestressing steel fp;beam depth l and beam width b have a significantinfluence on for maximum deflection. On the other hand,applied load P; the prestressing force Ps; elastic modulusof prestressing steel Ep; ultimate tensile strength ofprestressing steel fp; beam depth l and beam width b havea significant influence on the output parameter formaximum Von Mises strain. Besides, the ultimate tensilestrength of prestressing steel fp; the prestressing force Ps;beam depth l and beam width b have a significantinfluence on the output parameter for maximum VonMises stress.

The sensitivity analysis results indicate that appliedload, prestressing force, elastic modulus of prestressingsteel, ultimate tensile strength of prestressing steel, beamdepth and beam width are more sensitive to other inputvariables because the impact of other input variables onthe result is not significant enough to be worthconsidering. This is a reduction of the complexity of theproblem from nine input variables down to six.

Fig. 21: Sensitivity plot for maximum deflection (beam with 9 web openings)


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Fig. 22: Sensitivity plot for maximum Von Mises strain (beam with 9 web openings)

Fig. 23: Sensitivity plot for maximum Von Mises stress (beam with 9 web openings)

CONCLUSION

A method of probabilistic and sensitivity analysis toassess the openings` effect on the prestressed inverted T-beam with web openings is proposed. Latin Hypercubesimulation technique was used to study the uncertainty ofmodel parameters. The samples are obtained according tounderlying probabilistic distributions and the outputs fromthe numerical simulation are translated into probabilisticdistributions. To conduct sensitivity analysis on therealizations of input vectors, a rank transformation wasapplied to the input and output variables. Multiple linear

regression on the ranks is then performed to obtainrelationships between the input and output variables. Thecoefficients of the regression equations are related to thecoefficient of determination and can be used to identifythe most important model parameters.

The sample of probabilistic studies that have focusedon the beam with 9 web openings is very limited. Inaddition, to enable a probabilistic analysis to beconducted, the model employed was of necessitysimplified. However, the present work has shown that theselection of different samples, variables and outputparameters leads to varying sensitivity results. Therefore,


365

future work will focus on a model incorporating beamwith 6 and 3 web openings.

LIST OF ABBREVIATIONS

Xk = input variables of Latin Hypercube samplingmethod

Fxk (x) = Fx(x) cumulative distribution function (CDF)p-value = Represents the probability of obtaining an

observed result and its significance level:, = Mean valueF, = Standard deviation fx(x) = Probability density functionri = The order representing the value of random

variablej th = Number of simulationsnj = The order of an ordered sample of the

resulting variable for the j th run on thesimulation process

ki = Sensitivity coefficientsvyi = The variation coefficient of the output

quantityP = Applied loadPs = Prestressing forcev = Poisson`s ratioEc = Elastic modulus of concretefc

` = Ultimate compressive strengthfr = Ultimate tensile strength (modulus of rupture),fy = Strength of reinforcing steel for stirrupf = Stress at any strain ,, = Strain at stress f,0 = Strain at the ultimate compressive strength fc

`

,ps = Strain of prestressing steelEs = Elastic modulus of steelfp = Ultimate strength of prestressing steelEp = Ulastic modulus of prestessing steel

ACKNOWLEDGMENT

This study was supported by the Faculty of CivilEngineering & Earth Resources at University MalaysiaPahang short-term research grant (Grant No: RDU100702) and Ministry of Science, Technology andInnovation (MOSTI) through e-science fund (Grant No:03-02-03 SF0115). Thanks are extending to HumeConcrete Product Research Centre (HCPRC) and TenagaNasional Research Berhad (TNBR) for testing facilitiesand analysis assistance of prestressed beam with webopenings.

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Probabilistic Analysis of Prestressed Inverted T-beam with ... · PDF fileBased on their researches, the design procedures for prestressed concrete beams with web ... prestressed T-beam