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Importance of Realistic Cable Models in the Design ofPrestressed Concrete Structures

Saleem Akhtar, Dr.K.K.Pathak, Dr. S.S.Bhadauria, Dept. Civil Engg., UIT, RGTU Bhopal CSDGroup, AMPRI (CSIR) Bhopal.

Concrete is perhaps the most important construction material due to its various added advantages.The major problem with concrete is that it is very weak in tension. To strengthen the concrete towithstand tensile stresses, reinforced and prestressed concrete have been developed. Inprestressed concrete, cable layout plays vital role in reducing tension from the concrete. Due tocurvature cable exerts forces on the concrete to counterbalance the forces causing tension. Cablesare laid as a continuous curve but for analysis purpose, they are modeled by some mathematicalcurve. The most important of them is the discrete parabolic modeling in different spans. Parabolicmodeling simplifies the analysis to large extent but in this process cable geometry at theintermediate supports becomes discontinuous. To have more realistic models, we have usedreverse parabolas and B-spline curves for cable modeling. These curves not only ensure thesmooth profile but also provide realistic stress value. In this study, on comparing the results ofthese three different models, it is found that there is significant variation in stress values atintermediate supports, which clearly indicates the importance of realistic cable modeling.

Introduction

While analysing prestressed concrete beams in the text books1-4, the prestressing cable is assumed to traverse parabolicprofile. The reason being, for parabolic profile, curvaturebecomes constant and cable force can berepresented as anequivalent uniformly distributed load acting in the oppositedirection to the working loads. Although, parabolic assumptionsimplifies analysis, thecable profile becomes discontinuous atintermediate supports. Actual cable profile is like a smoothspline. In this study, prestressed concrete beams are analysedfor parabolic, reverse parabolas and B-spline profiles (Figure 1).Using these methodologies, two prestressed concrete beamshave been analyzed. Results are compared in terms of stressesand deflections at various sections of the beam. It is observedthat cable profile plays vital role on the stresses and deflectionshence it should be properly accounted for analysis and designpurposes.

Cable Models

In this study, following three different types of cable modeling i.e. parabolic, reverse parabolas,and B-spline are considered.

(a) Parabolic Modeling

A general parabola may be defined as y = ax2 . . . . (1)

The approximate constant curvature of this curve may be defined as- K d2y/dx2 = 2a . . . . (2)

In continuous prestressed concrete structures where cable profile changes its curvature severaltimes, each time a new parabola has to be defined. This is accounted through load balancingapproach, analytically and STADD software, numerically.

(b) Reverse Parabolic Modeling

This method was proposed by Lin and Burn4. In this reverse parabolas are used at the support tomaintain smooth ness of the cable. Due to fitting of multi parabolas and mathematicalcomplexities in maintaining the continuity at the juncture of two parabolas, this approach is quiteinvolved. Since this models the true profile, results are quite close to the exact ones.

(c) B-Spline Modeling

A B-spline is a typical curve of the CAD philosophy5-6. It models a smooth curve between the givenordinates. Braibant and Fleury7, Pourazady et al8, Ghoddosian9 etc., have used this curve in shapeoptimization problems. When a B-spline curve is used the geometrical regularities are automaticallytaken care of. Following are few important properties of B-spline curve.

(1) The curve exhibits the variation diminishing properties. Thus the curve does not oscillate aboutany straight line more often than its defining polygon.

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(2) The curve generally follows the shape of the defining polygon.

(3) The curve is transformed by transforming the defining polygonal vertices.

(4). The curve lies between the convex hull of its defining polygon (Figure 2).

(5). The order of the resulting curve can be changed without changing the number of definingpolygon vertices (Figure 3).

The curvature of B spline curve, K, in a finite element is calculated using vector calculus asdescribed in next section.

Finite Element Formulation

In this study, linear elastic analysis of prestressed concrete structures is carried out. Nine nodeLagrangean element and three node curved bar element have been used to model the concrete andthe cable. Cable is assumed to be embedded in the concrete. The radius of curvature R in theelement is given by- R = 1/K . . . . (3)

Forces transferred by the cable to concrete element is shown in Figure 4. Assume curvaturebetween 1&2 and 2&3 as equal to the curvature at two Gauss points. Let the length of the cablebetween 1-2 and 2-3 be approximated as-

. . . . (4)

where (x1 ,y

1), (x

2 ,y

2) and (x

3 ,y

3) are the co-ordinates of 1,2 and 3.

The tension variation in the bar element can be expressed by isoparametric interpolation as-

Tn=3i=1

Ti N

ci . . . . . (5)

Ncis are shape functions of the cable element.

Using the formulae of vector calculus, normal and tangential forces Pn and Pt are calculated in the

following manner.

where Tn is the tension in the cable, T is the tangent vector, r is the local co-ordinate and R is the

radius of curvature.

The resultant of these forces is given by-

. . . . . (8)

Their equivalent finite element nodal forces are worked out using virtual work theorem. Perfectbond between cable and the concrete has been assumed. The detailed formulation of this can befound in Ref.10. Using these formulations, a FE software PRES2D has been developed in FORTRANprogramming language.

Numerical Examples

To compare the effects of parabolic, reverse parabolas andB-spline modeling of the prestressing cable, two PC beams areanalysed using PRES2D software. Following data are used forthe analysis purpose:

Wobble coefficient = 1x10-5a.

Youngs modulus (concrete) = 2x104 N/mm2b.

Youngs modulus (steel) = 2x105 N/mm2c.

Density (concrete) = 2500 Kg/m3d.

Poissons ratio = 0.15e.

Cross sectional area of cable = 1600 mm2 Negligiblefriction is considered in this study.

f.

(a) Two Span Beam with Uniformly Distributed Loading

A two span beam of 30 m length and 300mm x 750mm crosssection with 23 KN/m uniformly distributed load (including selfweight) and prestressing force of 938 KN has been analyzed byload balancing method with idealized parabolic profile, Linsmethod [Lin and Burns (1995)] with actual cable profileconsidering two parabolic segments at midspan and two reversed parabolic segments at thesupport and proposed B-spline model. The idealized parabolic model and realistic model is shown inFigure 5(a) & (b). Parabolic model is accounted through load balancing approach analytically andthrough STADD software numerically. For FE analysis using B-spline model, the beam is discretisedinto 20 plane stress nine noded elements making total of 123 nodes as shown in Figure 5(c).Stresses at various locations due to different approaches are given in Table 1. Negative signindicates compressive stresses. Deflections at middle of span are given in Table 2. It can be seenthat conventional parabola approach results are in close match with STAAD results while the resultsof Lins approach are in good match with B-spline approach; particularly at the supports. It can alsobe observed that the top fibre stresses are compressive while the bottom stresses are morecompressive in case of Lins method when compared to conventional parabolic layout, which is alsoobserved from B-spline results. Deflection at mid span is less in case of Lins cable layout as well asB-spline cable profile compared to conventional approach (Table 2). It should be noted that designbased on parabolic model would be erroneous as it results in lesser tensile stresses than the actualones. The analysis considering two parabolic segments at mid span and two reversed parabolic

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segments at support, is very complex. The proposed approach using B-spline model of the cableovercomes these difficulties.

(b) Two Span Beam with Concentrated Loading

A two span beam of 24 m length and cross section of 300mm x600mm with centrally applied concentrated loads of 150 KN andprestressing force of 1200 KN is taken up for comparision(Figure 6a). The beam is analyzed using load balancing methodwith two parabolas, Lins approach using 6 parabolic segments,STAAD software and by our approach using PRES2D software.For FE analysis the beam is divided into 20 nine nodded planestress elements making a total of 123 nodes as shown in Figure6b. Stresses and deflections due to different approaches aregiven in Table 3 and Table 4. It can be observed that stressesobtained om parabolic modeling and STAAD Pro software areclose because both use same formulation. Realistic cablemodeling by B-spline gives stresses comparable to Linsapproach. But stress analysis by 6 parabolic segments model isvery complex as the very modeling of these parabolas istedious.Designbased on B-spline model will be more accurate. It can beobserved that tensile stresses due to Lins approach are onhigher side. Hence design based on it will be conservative.Deflections at arious sections due to different approaches comeout to be approximately the same.

Conclusion

In this study, a novel approach to analyse prestressed concretebeams is presented. Cables are modeled as B-spline. To make ituserfriendly, this approach is coded in a 2D FE software whereconcrete is modeled by 9 node Lagrangian and cable by 3 nodecurved bar elements. Using this software two representativeproblems are analysed and results obtained from variousmethods are compared. It is observed that realistic cable modeling should be carried out to obtaintrue stresses and deflections for design considerations. Stresses are found to be more sensitive tothe cable model than the deflections.

References

Hurst MK. Prestressed Concrete Design, First Edition,Chapman and Hall, London (1988).

Gilbert RI. Time Effects in Concrete Structures. Elsevier New York (1988).

Jain AK. Reinforced Concrete- Limit State Design. Nem Chand & Bros. Roorkee, (1989).

Lin TY, Burns NH. Design ofprestressed concrete structures, Third Edition, John Wiley andSons. Inc.,New York (1982)

Qing SB, Liu Ding Yuan. Computational Geometry-Curve and Surface Modeling. AcademicPress London (1989).

Rogers DF, Adams JA. Mathematical Elements for Computers Graphics. Second EditionMcGraw Hill New York (1990).

Braibant V, Fleury C. Shape optimal design using B-splines. Comp. Meth. Appl. Meth. andEngg, 44,247-267, (1984).

Pourazady M, Fu Z. An integrated approach to structural shape optimization. Computers &Structures, 60(2), 274-289, (1996).

Ghoddosian A. Improved Zero Order Techniques for Structural Shape Optimization UsingFEM. Ph.D Dissertation Applied Mechanics Dept. IIT Delhi (1998).

Pathak, K.K., Gradientless shape optimization of concrete structures using artificial neuralnetworks, Ph.D Thesis, Deptt. of Applied Mechanics, IIT Delhi, March 2001.

NBMCW August 2008

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