Importance of Realistic Cable Models in the Design of Prestressed Concrete Structures
Saleem Akhtar, Dr.K.K.Pathak, Dr. S.S.Bhadauria, Dept. Civil Engg., UIT, RGTU Bhopal CSD Group, 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 to withstand tensile stresses, reinforced and prestressed concrete have been developed. In prestressed concrete, cable layout plays vital 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 most important of them is the discrete parabolic modeling in different spans. Parabolic modeling simplifies the analysis to large extent but in this process cable geometry at the intermediate supports becomes discontinuous. To have more realistic models, we have used reverse parabolas and B-spline curves for cable modeling. These curves not only ensure the smooth profile but also provide realistic stress value. In this study, on comparing the results of these three different models, it is found that there is significant variation in stress values at intermediate supports, which clearly indicates the importance of realistic cable modeling.
Cable ModelsIn this study, following three different types of cable modeling i.e. parabolic, reverse parabolas, and B-spline are considered.
(a) Parabolic ModelingA 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 several times, each time a new parabola has to be defined. This is accounted through load balancing approach, analytically and STADD software, numerically.
(b) Reverse Parabolic ModelingThis method was proposed by Lin and Burn4. In this reverse parabolas are used at the support to maintain smooth– ness of the cable. Due to fitting of multi parabolas and mathematical complexities in maintaining the continuity at the juncture of two parabolas, this approach is quite involved. Since this models the true profile, results are quite close to the exact ones.
(c) B-Spline ModelingA B-spline is a typical curve of the CAD philosophy5-6. It models a smooth curve between the given ordinates. Braibant and Fleury7, Pourazady et al8, Ghoddosian9 etc., have used this curve in shape optimization problems. When a B-spline curve is used the geometrical regularities are automatically taken 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 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 (Figure 2).
(5). The order of the resulting curve can be changed without changing the number of defining polygon vertices (Figure 3).
The curvature of B spline curve, K, in a finite element is calculated using vector calculus as described in next section.
Finite Element FormulationIn this study, linear elastic analysis of prestressed concrete structures is carried out. Nine node Lagrangean element and three node curved bar element have been used to model the concrete and the cable. Cable is assumed to be embedded in the concrete. The radius of curvature R in the element is given by- R = 1/K . . . . (3)
Forces transferred by the cable to concrete element is shown in Figure 4. Assume curvature between 1&2 and 2&3 as equal to the curvature at two Gauss points. Let the length of the cable between 1-2 and 2-3 be approximated as-
. . . . (4)
where (x1 ,y1), (x2 ,y2) and (x3 ,y3) are the co-ordinates of 1,2 and 3.
The tension variation in the bar element can be expressed by isoparametric interpolation as-
Tn=å3i=1Ti Nci . . . . . (5)
Nci’s 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. Perfect bond between cable and the concrete has been assumed. The detailed formulation of this can be found in Ref.10. Using these formulations, a FE software PRES2D has been developed in FORTRAN program–ming language.
- Wobble coefficient = 1x10-5
- Young’s modulus (concrete) = 2x104 N/mm2
- Young’s modulus (steel) = 2x105 N/mm2
- Density (concrete) = 2500 Kg/m3
- Poisson’s ratio = 0.15
- Cross sectional area of cable = 1600 mm2 Negligible friction is considered in this study.
(a) Two Span Beam with Uniformly Distributed LoadingA two span beam of 30 m length and 300mm x 750mm cross section with 23 KN/m uniformly distributed load (including self– weight) and prestressing force of 938 KN has been analyzed by load balancing method with idealized parabolic profile, 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 and proposed B-spline model. The idealized parabolic model and realistic model is shown in Figure 5(a) & (b). Parabolic model is accounted through load balancing approach analytically and through STADD software numerically. For FE analysis using B-spline model, the beam is discretised into 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 sign indicates 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 B-spline 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 B-spline results. Deflection at mid span is less in case of Lin’s cable layout as well as B-spline cable profile compared to conventional approach (Table 2). 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 mid span and two reversed parabolic segments at support, is very complex. The proposed approach using B-spline model of the cable overcomes these difficulties.
(b) Two Span Beam with Concentrated Loading
ConclusionIn this study, a novel approach to analyse prestressed concrete beams is presented. Cables are modeled as B-spline. To make it user–friendly, 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 two representative problems are analysed and results obtained from various methods are compared. It is observed that realistic cable modeling should be carried out to obtain true stresses and deflections for design considerations. Stresses are found to be more sensitive to the cable model than the deflections.
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