Finite element analysis of prestressed concrete beams considering realistic cable profile.
Introduction
Prestressed concrete is a particular form of reinforced concrete in which external prestressing force is applied on the concrete to reduce or eliminate the tensile stresses and thereby control or eliminate cracking. It is a typical set up of cable and concrete which makes a prestressed concrete section considerably stiffer than reinforced concrete section. Due to this feature prestressed concrete is being used in small as well as large structures like beams, nuclear containment vessels, bridges etc. Linear finite element analysis of prestressed concrete structures have been reported by Pandey, Kumar and Trikha (1996), Buragohian and Mukherjee (1993), Buragohian, and Siddhaye (1997), Pathak, Sontakke and Kumar (1999). Non linear analysis of the same are reported by Povoas and Figueiras (1990), Kang and Scordelis (1980), Roca and Mari (1993), Greunen and Scordelis (1983), Figueiras and Povoas (1994), Vanzyl and Scordelis (1979), Elwi and Hrudey (1987). In prestressed concrete, cable layout plays an important role in reducing tension from the concrete. Due to curvature, cable exerts forces on the concrete to counterbalance the forces causing tension. Cables are laid as a continuous curve but for analysis purpose they are modeled by some mathematical curve. The prestressing cable is modeled as parabola while analyzing prestressed concrete beam in the text books ?Raju (1995), Lin and Burns(1995)?. The reason behind this assumption is that for parabolic profile curvature becomes constant and cable force can be represented as an equivalent uniformly distributed load acting in the opposite direction to the working loads. Although parabolic assumption simplifies analysis, the cable profile becomes discontinuous at intermediate supports i.e. at the juncture of two parabolas. Actual cable profile is a smooth curve passing through all the spans. Jirousek, Bouberguig and Saygun (1979) and Buragohain et al. (1993,1997) have considered cable as parabolic and cubic curve in shell and semiloof shell elements whereas Pandey et al. (1997) considered the cable as parabola in 20 node brick element. Pathak et al. (1999) considered the cable as cubic spline curve in nine node Lagrangean element. In this study, prestressing cable is modeled by Bspline curve which not only avoids discontinuity problem but also paves the way for cable layout design in much simplified manner. Bspline A Bspline is a typical curve of the CAD philosophy ?Qing and Ding (1989); Rogers and Adams (1990)?. It models a smooth curve between the given ordinates. Braibant and Fleury (1984), Pourazady and Fu (1996), Ghoddosian (1998) have used this curve to define moving boundaries in shape optimization problems. The theory of the Bspline was first suggested by Schoenberg (1946). A recursive definition useful for numerical computation was independently discovered by Cox and by de Boor ?Rogers and Adams (1990)?. After that Gordon and Riesenfeld (1974) applied the Bspline basis to curve definition. The brief definition of Bspline curve is being given below and detailed account of this can be found in Rogers and Adams (1990). If P(t) is the position vector along the Bspline curve then it is represented by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] In above equations, Pi's are the n+1 defining polygon vertices, k is the order of the Bspline and [N.sub.i,k](t) is called the weighing function; x is the additional knot vector which is used for Bspline curves to account for the inherent added flexibility. A knot vector is simply a series of real integers [x.sub.i] such that [x.sub.i] [less than or equal to] [x.sub.i+1] for all [x.sub.i]. They are used to indicate the parameter t used to generate a Bspline. When a Bspline curve is used the geometrical regularity is automatically taken into account. Following are few important properties of Bspline curve. 1. The curve exhibits the variation diminishing properties. Thus the curve does not oscillate about any straight line more often than its defining polygon. 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. The order of the resulting curve can be changed without changing the number of defining polygon vertices. When the numbers of defining polygon vertices are equal to the order of the Bspline basis, the Bspline basis reduces to Bezier curve. 5. In this study Bspline is used to represent cable profile. Finite Element Modeling Modeling of the cable as a discrete parabola in different spans does not hold true in continuous span structures as this creates discontinuity at the juncture of different spans. Cable profile modeled by Bspline gives a very smooth shape (Fig.1). Force transfer from cable to concrete for Bspline and parabolic profile are shown in Figure 2. It can be seen that parabola model is erroneous and it results in opposite forces. Because of the convex hull properties of the Bspline, it is very suitable to represent cable profile in continuous span beams. The cable is considered to be embedded in the concrete and there exists perfect bond between them. The cable due to its curvature exerts thrust on the concrete which counter balances the effects of dead and live load. Cable and concrete is modeled by 3 node curved bar element and 9 node lagrangean elements respectively (Figure 3). Force transfer in a 9 node element is shown in Figure 4. [FIGURE 1 OMITTED] [FIGURE 2 OMITTED] [FIGURE 3 OMITTED] [FIGURE 4 OMITTED] The shape functions of a 3 node curved bar element, in terms of local coordinate axis [rho], are given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] The global coordinates inside the curved bar element can be defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] The tangent and normal vectors along [rho] axis for the cable are given by. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Now the unit tangent and normal vectors can be expressed as: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] The curvature at any point on the curve can be given by [Piskunov (1981)] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Finally the radius of curvature R can be obtained by R 1/K The cable exerts normal and tangential forces on the concrete due to curvature and friction. These are expressed as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [T.sub.n] is the tension in the cable. The resultant force is calculated by: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] These forces can be transferred to concrete nodes, using the principle of virtual work. The equivalent nodal force vector for the concrete element is expressed as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1) Numerical integration is carried out considering two Gauss points. At the faces of the end elements, where cable is anchored to the concrete, cable reaction act as concentrated loads on the concrete. The anchorage end point forces can be transferred at the nodes in the ratio of shape functions and are given by {[P.sub.A]} = [[N].sup.T] {[T.sub.end]} (2) where, [T.sub.end] is the tension at the cable ends. [N] in Eq. 1& 2 are the shape functions of nine noded elements. In calculation of Eq.2, local coordinates of the anchorage points are required for their known global coordinates. This is calculated by NewtonRaphson iterative procedure. Let (x,y) be the global coordinate and ([xi],[eta]) corresponding local coordinate then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] In above equation inverse matrix is nothing but jacobian matrix. ([x.sub.i+1],[y.sub.i+1]) is the computed value and ([x.sub.i],[y.sub.i]) are the known value. Initial value of ([xi],[eta]) are taken as zero. The computation is carried out iteratively till the difference of two consecutive value of ([xi],[eta]) becomes less than tolerance, which is taken as 0.001 in this study. Using Eq.1 &2, total load vector due to cable concrete interaction is obtained by {[P.sub.T]} = {[P.sub.L]} + {[P.sub.A]} This nodal load vector is applied on the structure along with live and dead load vectors to include prestressing effects. The above described formulation is incorporated in a modular based FORTRAN code named PRES2D. Using this code several validation and new problems are analyzed. In analysis Young's modulus and Poisson's ratio for all the problems are 2x[10.sup.4] MPa and 0.3 respectively. Effect of friction has been ignored. Validation Problems Following two problems of known results are analysed. Two span beam with uniformly distributed loading A two span beam of 30 m span and 300mmX750mm cross section with 23 KN/m uniformly distributed loading (including self weight) and prestressing force of 938 KN has been analyzed by load balancing method with idealized parabolic profile and by Lin's method [Lin and Burns (1995)] with actual cable profile considering two parabolic segments at midspan and two reversed parabolic segments at the support (Fig.5). [FIGURE 5 OMITTED] The beam is analyzed with Bspline model of cable profile using PRES2D software. It is discretised into 20 plane stress elements with a total of 123 nodes. Nine node Lagrangean elements are used for FE modelling. The problem is also analyzed by the commercial STAAD Pro software [2002] which accounts cable profile as parabola. Stresses due to different approaches are given in (Table 1). Negative sign shows compressive stresses. Deflections at middle of span are given in (Table 2). It can be seen that conventional parabola approach results are in close match with STAAD results while the results of Lin's approach are in good match with Bspline approach; particularly at the supports. It can also be observed that the top fibre stresses are compressive while the bottom stresses are more compressive in case of Lin's method when compared to conventional parabolic layout, which is also observed from Bspline results (Fig.6 & 7). Deflection at midspan is less in case of Lin's cable layout as well as Bspline cable profile compared to conventional approach (Fig. 8). [FIGURE 6 OMITTED] [FIGURE 7 OMITTED] [FIGURE 8 OMITTED] It should be noted that design based on parabolic model would be erroneous as it results in lesser tensile stresses than the actual ones. The analysis considering two parabolic segments at midspan and two reversed parabolic segments at support, is very complex. Our approach overcomes these difficulties. Two span beam with concentrated loading A two span beam of 24 m and cross section 300mmX600mm with centrally applied concentrated loads of 150 KN and prestressing force of 1200 KN (Fig. 9) is taken up for validation. The beam is analyzed using load balancing method with parabolic and Lin's approach (6 parabolic segments), by STAAD Pro software and by our approach using PRES2D software. For FE analysis the beam is divided into 20 plane stress elements making a total of 123 nodes. Nine node Lagrangian elements are used for analysis. Stresses and deflections due to different approaches are given in Table 3 and Table 4. It can be observed that stresses obtained from parabolic modeling and STAAD Pro software are close because both use same formulation. Stresses obtained by modeling the cable by 6 parabolic segments (Lin's approach) are different than due to two parabola. Realistic cable modeling by Bspline gives stresses comparable to Lin's approach. But stress analysis by 6 parabolic segments model is very complex as the very modeling of these parabolas is tedious. Design based on Bspline model will be more accurate. It can be observed that tensile stresses due to Lin's approach are on higher side. Hence design based on it will be conservative. [FIGURE 9 OMITTED] [FIGURE 10 OMITTED] [FIGURE 11 OMITTED] [FIGURE 12 OMITTED] [FIGURE 13 OMITTED] [FIGURE 14 OMITTED] New Problems Having validated the Bspline model with other models, we analysed two new continuous span beams to show the efficacy of this approach. Three span beam A three span beam of 24 m and cross section 250 mm x 500 mm with centrally applied concentrated loads of 300 kN and prestressing force of 2000 kN is taken for analysis (Fig.13). Cable is modeled by Bspline through 7 points. Bspline ordinates at these points are given in Table 5. For FE analysis, the beam is divided into 24 nine node Lagrangian elements making a total of 147 nodes. Stresses and deflections at different sections of the beam are given in Table 6. Five span beam A five span beam of 40 m and cross section 250mmx500mm with centrally applied concentrated loads and prestressing force of 500 kN is taken for analysis (Fig.14). Cable is modeled by Bspline through 11 points. Bspline ordinates at these points are given in Table 7. For FE analysis the beam is divided into 40 nine node Lagrangian elements making a total of 243 nodes. Stresses and deflections at different sections of the beam are given in Table 8. Conclusion In this study a novel approach to analyse prestressed concrete beams is presented. Cables are modeled as Bspline. To match with realistic profile this approach is coded in a 2D FE software where concrete is modeled by 9 node Lagrangian and cable by 3 node curved bar elements. Using this software various validation and other representative problems are analysed. It is found that proposed approach can take into account the realistic cable profile in much simplified manner when compared to parabolic model where several parabolas are to model the actual profile. It is observed that proposed method is very powerful for design of prestressed concrete beams. It will prove very useful and handy for structural engineers. References [1] Braibant V, Fleury C, Beckers P. Shape optimal design: An approach matching CAD and optimization concept. Report SA109, Aerospace Laboratory of University of Liege, Belgium; 1983. [2] Buragohian DN, Mukherjee A. 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[16] Povoas RHCF, Figueiras JA. Nonlinear analysis of curved prestressed girder bridges. Proceedings of Second International Conference on Computer Aided Analysis and Design, Austria 1990. [17] Qing SB, Liu DY. Computational geometryCurve and Surface Modeling, London: Academic Press, 1989. [18] Raju NK. Prestressed Concrete, Third Edition, New Delhi : Tata McGrawHill, 1995. [19] Roca P, Mari AR. Numerical treatment of prestressing tendons in the nonlinear analysis of prestressed concrete structures. Computers and Structures 1993; 46: 905916. [20] Rogers DF, Adams JA. Mathematical Elements for Computer Graphics, Second Edition, : McGrawHill, 1990. [21] Schoenberg IJ. Contribution to the problem of approximation of equidistant data by analytic functions. Quarterly of Applied Mathematics 1946; 4 : 4599,112141. [22] STADD Pro software. Research Engineers International, Orange County, CA, USA 2002 [23] Vanzyl SF, Scordelis AC. Analysis of curved prestressed segmental bridges. Journal of Struct. Engg. ASCE 1979; 105 : 23992411. Saleem Akhtar (a), K.K. Pathak (b), S.S. Bhadauria (a) and N. Ramakrishnan (b) (a) University Institute of Technology, RGPV, Bhopal (M.P.) INDIA 462026 (b) Advanced Materials & Processes Research Institute (CSIR), Bhopal (M.P.) INDIA 462026 Table 1 : Stresses in two span beam with uniformly distributed loading Distance Stresses against different approach (N/[mm.sup.2]) from left support Parabola STAAD Lin's method T B T B T B 7.5 m 8.15 0.18 8.15 0.14 7.88 0.44 15 m 3.83 12.16 3.83 12.12 4.52 12.84 22.5 m 8.15 0.18 8.15 0.14 7.88 0.44 Stresses against different approach (N/[mm.sup.2]) Distance from left support B spline T B 7.5 m 7.42 0.97 15 m 4.25 13.42 22.5 m 7.42 0.97 Table 2 : Deflection in two span beam with uni. dist. loading Deflection against different approaches (mm) Distance from left support STAAD Lin's method B spline 22.5 m 10 9 8.24 22.5 m 10 9 8.24 Table 3 : Stresses in two span beam with concentrated loading Stress for various cases (N/[mm.sup.2]) Distance Load balancing from left method STAAD support Top Bot. Top Bot. 6.0 m 13.97 0.63 14.96 1.65 12.0 m 2.53 10.81 2.54 10.76 18.0 m 13.97 0.63 14.96 1.65 Stress for various cases (N/[mm.sup.2]) Distance from left Lin's method B spline support Top Bot. Top Bot. 6.0 m 14.25 0.93 13.88 0.16 12.0 m 1.13 12.24 0.83 13.08 18.0 m 14.25 0.93 13.88 0.16 Table 4 : Deflection in two span beam with concentrated loading Deflection for various cases (mm) Distance from left support STAAD Actual cable layout B spline 7.5 m 7.0 7.0 7.11 22.5 m 7.0 7.0 7.23 Table 5 : Bspline ordinates in three span beam Point Distance from left Polygon ordinate Bspline ordinate support (mm) (mm) (mm) 1 0 250 250.00 2 4000 100 152.77 3 8000 600 405.55 4 12000 100 75.00 5 16000 600 405.55 6 20000 100 152.77 7 24000 250 250.00 Table 6 : Stress and Deflection in three span beam Distance from Stress (N/ left support [mm.sup.2]) Deflection (mm) (mm) Top Bottom 4000 31.95 1.17 11.9 8000 11.67 21.60 0.00 12000 7.91 25.12 +19.20 16000 11.67 21.60 0.00 20000 31.95 1.17 11.90 Table 7 : Bspline ordinates in five span beam Point Distance from left Polygon ordinate Bspline ordinate support (mm) (mm) (mm) 1 0 250 250 2 4000 150 178 3 8000 650 394 4 12000 100 105 5 16000 600 418 6 20000 100 75 7 24000 600 418 8 28000 100 105 9 32000 650 394 10 36000 150 178 11 40000 250 250 Table 8 : Stresses and Deflection in five span beam Distance from Stress (N/ Deflection (mm) left support [mm.sup.2]) (mm) Top Bottom 4000 5.70 2.48 0.16 8000 0.74 7.67 0.00 12000 8.12 0.29 2.42 16000 4.61 3.75 0.00 20000 1.27 9.47 +6.81 24000 4.61 3.75 0.00 28000 8.12 0.29 2.42 32000 0.74 7.67 0.00 36000 5.70 2.48 0.16 

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