PRECON3D
PRECON3D: A Software for Three Dimensional Finite Element, Analysis of Prestressed Concrete Structures
A powerful and practical software for three dimensional linear elastic finite element analysis of prestressed concrete structures is presented. To account the cable profile realistically, it is modelled by B-spline. For finite element computations, tendon and concrete are modelled by 3 noded bar and 20 noded brick elements respectively. The cable concrete interactions are precisely accounted using vector calculus formulae. Pre-stress loss due to friction is also accounted. The proposed algorithm is coded in FORTRAN language to result in PRECON3D software. Using this software two simply supported beams are successfully analysed and validated with the standard analytical results.
K. K. Pathak, CSPM Group, AMPRI (CSIR), Bhopal, Ahmad Ali Khan, N. Dindorkar, Dept.Civil Engg., MANIT Bhopal, and D. K. Sehgal, Dept. Applied Mechanics, IIT Delhi, New Delhi.
A powerful and practical software for three dimensional linear elastic finite element analysis of prestressed concrete structures is presented. To account the cable profile realistically, it is modelled by B-spline. For finite element computations, tendon and concrete are modelled by 3 noded bar and 20 noded brick elements respectively. The cable concrete interactions are precisely accounted using vector calculus formulae. Pre-stress loss due to friction is also accounted. The proposed algorithm is coded in FORTRAN language to result in PRECON3D software. Using this software two simply supported beams are successfully analysed and validated with the standard analytical results.
K. K. Pathak, CSPM Group, AMPRI (CSIR), Bhopal, Ahmad Ali Khan, N. Dindorkar, Dept.Civil Engg., MANIT Bhopal, and D. K. Sehgal, Dept. Applied Mechanics, IIT Delhi, New Delhi.
Introduction
Prestressed concrete is one of the most widely used construction materials which can resist tension and compression efficiently. In this, external pre-stressing force is applied on the concrete to reduce or eliminate the tensile stresses and thereby control or eliminate cracking. In this way, a pre-stressed concrete section is considerably stiffer than reinforced concrete section. In curved tendons, upward force is imposed on the concrete which may reduce or eliminate the downward deflection as well. Because of above reasons, prestressed concrete structures has become extremely useful in the construction of liquid retaining structures and nuclear containment structures where absolutely no leakage is acceptable. This technique is of great use in railway sleepers, large span bridges and long span roofs. Prestressed concrete slab overcomes many of the disadvantages of reinforced concrete slab. Deflection, which is almost always the governing factor, is better controlled in post-tension slabs. Therefore, a safe and more slender slab can be obtained.
Because of complex nature of cable concrete interaction, analysis of prestressed concrete structures is a complicated task. For realistic analysis of these structures, advanced computing techniques such as finite element method is employed. Linear finite element analysis (FEA) of prestressed concrete structures have been reported by Pandey et al.1, Buragohian and Mukherjee2, Buragohian and Siddhyae3, Pathak and Sehgal4. Non linear analysis of the same are reported by Povoas and Figueiras5, Kang and Scordelis6, Roca and Mari7, Greunen and Scordelis8, Figueiras and Povoas9, Vanzyl and Scordelis10 and Elwi and Hurdey11. Jirousek et al.12, Buragohian and Mukherjee2 and Buragohian and Siddhyae3 have considered cable as parabolic and cubic curve in shell and semilo of shell elements, whereas Pandey et al.1 considered the cable as parabola in 20 node brick element. Pathak and Sehgal4 considered the cable as cubic spline curve in nine node Lagrangean element. Saleem Akhtar et al.13 modelled cable as B-spline for two dimensional finite element analysis.
Although considerable literatures on prestressed concrete are available, most of them deal with specific aspects namely, realistic modeling of the cable, frictional effects and cable concrete interaction. Since a design engineer needs all these in an integrated environment, he has to resort to separate calculation for each aspect. Also most of the software available on the topic are two dimensional. For accurate analyses, there is need of a three dimensional software to analyse prestressed concrete structures. The proposed software aims to fulfil these lacunas. The main features of this package are:
Because of complex nature of cable concrete interaction, analysis of prestressed concrete structures is a complicated task. For realistic analysis of these structures, advanced computing techniques such as finite element method is employed. Linear finite element analysis (FEA) of prestressed concrete structures have been reported by Pandey et al.1, Buragohian and Mukherjee2, Buragohian and Siddhyae3, Pathak and Sehgal4. Non linear analysis of the same are reported by Povoas and Figueiras5, Kang and Scordelis6, Roca and Mari7, Greunen and Scordelis8, Figueiras and Povoas9, Vanzyl and Scordelis10 and Elwi and Hurdey11. Jirousek et al.12, Buragohian and Mukherjee2 and Buragohian and Siddhyae3 have considered cable as parabolic and cubic curve in shell and semilo of shell elements, whereas Pandey et al.1 considered the cable as parabola in 20 node brick element. Pathak and Sehgal4 considered the cable as cubic spline curve in nine node Lagrangean element. Saleem Akhtar et al.13 modelled cable as B-spline for two dimensional finite element analysis.
Although considerable literatures on prestressed concrete are available, most of them deal with specific aspects namely, realistic modeling of the cable, frictional effects and cable concrete interaction. Since a design engineer needs all these in an integrated environment, he has to resort to separate calculation for each aspect. Also most of the software available on the topic are two dimensional. For accurate analyses, there is need of a three dimensional software to analyse prestressed concrete structures. The proposed software aims to fulfil these lacunas. The main features of this package are:
- Realistic modeling of the cable using B-spline.
- Discrete modeling of the cable using bar element in the brick element.
- Accurate calculation of the curvatures using vector calculus approach.
- Consideration of friction loss.
- Easy input and debugging.
In this paper, theoretical aspects of cable modelling, finite element formulation, friction loss etc., used in the software development, are described. The proposed software is validated with analytical results of a simply supported beam and found to be in close match.
Cable Modeling
 |
 |
| Figure 1: Actual cable and parabolic profile | Figure 2: Equivalent cable force |
 |
....(1) |
and
 |
....(2) |
In above equations, Pi’s are the n+1 defining polygon vertices, k is the order of the B spline and Ni,k(t) is called the weighing function. x is the additional knot vector which is used for B- spline curve to account for the inherent added flexibility. The curve generally follows the shape of the defining polygon and is transformed by transforming the defining polygonal vertices. The order of the resulting curve can be changed without changing the number of defining polygon vertices. In Figure 2(a,b) cable reaction considering parabolic and B-spline modeling are shown. It can be observed that B-spline model represents the realistic forces.
Finite Element Modeling
The scope of this study is restricted to linear elastic analysis. The weakening of the section due to cable duct is ignored and complete section is considered to be made of concrete. Due to this assumption, stiffness of the cable and compatibility between cable and concrete has not been considered. Cable modelling is carried out to calculate equivalent cable forces on the concrete.
Figure 3: Curved bar element
In finite element analysis of prestressed concrete, modeling of pre-stressing cable is a tedious task. The cable profile plays very important role during analysis as pressure on concrete depends on the cable profile. The cable counterbalances the effects of the live and dead loads due to its layout and curvature. Hence major task is to obtain curvature along the cable profile. For finite element analysis, cable is modeled by three noded bar element (Figure 3) and concrete by 20 noded brick element (Zienkiewicz and Tayler17). In Figure 4, a 3 node curved bar element is shown embedded in three dimensional 20 node concrete element.
The shape functions of a 3 node curved bar elements are given by
where P is the local co-ordinate axis shown in Figure 3.
The global coordinates on the curved bar element is given by
The tangent vector and normal vector along r axis for the cable is given by
Where, a is the dot product.
The unit tangent and normal vector can be given by
The curvature at any point on the cable is expressed as -
where the numerator is a cross product.
Now radius of curvature can be calculated by:
Figure 3: Curved bar element
The shape functions of a 3 node curved bar elements are given by
 |
....(3) |
where P is the local co-ordinate axis shown in Figure 3.
The global coordinates on the curved bar element is given by
 |
....(4) |
The tangent vector and normal vector along r axis for the cable is given by
 |
....(5) |
 |
....(6) |
Where, a is the dot product.
The unit tangent and normal vector can be given by
 |
....(7) |
 |
....(8) |
The curvature at any point on the cable is expressed as -
 |
....(9) |
where the numerator is a cross product.
Now radius of curvature can be calculated by:
| R = 1 / K | ....(10) |
Friction Loss
 |
 |
| Figure 4: Brick element with curved bar element | Figure 5: Forces on concrete due to cable tension |
The friction between the cable and duct gradually reduces tension along the length of the cable. To incorporate this in the finite element analysis, following methodology is adopted. The bar element is assumed to have two Gauss points GP1 and GP2 (Figure 4). For calculation of friction loss, the radius of curvature R is required. If radius of curvatures at GP1 and GP2 be R1 and R2, cable tensions at B and C after friction loss will be –
TB = TAe(-µα1 -KLA-B) . . . . . . . . . (11)
TC = TBe(-µα2 –KLB-C)
Where,
TA = Tension at jacking end for first element
For subsequent element TA will become TC of their previous element.
α1 = LA-B/R1 . . . . . . . . . (12)
α2 = LB-C/R2
µ = coefficient of friction
K= wobble coefficient
The length of the cable between A-B and B-C are calculated as
LA-B = √(xA-xB)2+(yA-yB)2+(zA-zB)2 ........(13)
LB-C = (xB-xC)2 +(yB-yC)2 + (zB-zC)2
Normal and Tangential Forces
The cable exerts tangential and normal forces on concrete due to friction between contacting surfaces and curvature of the cable as shown in Figure 5.
Tangential and normal forces can be given by
Now the resultant force can be computed by
Where Tn is the tension in the cable and T is the tangent vector. t‾ and n‾ are unit tangent and normal vector.
Using the principle of virtual work, these loads can be transferred to the nodes of brick elements. The equivalent nodal force vector for brick element is given by-
Cable reactions at the end of the brick elements, act as concentrated loads. Corresponding equivalent finite element nodal forces can be calculated by-
where, Tend is the cable tension at the end points and [N] is the shape functions of 20 node brick elements. Now total load vector due to interaction of concrete and cable is obtained by-
This nodal load vector is applied on the three dimensional finite element model along with live and dead load vectors to include pre-stressing effects.
Tangential and normal forces can be given by
 |
....(14) |
 |
....(15) |
Now the resultant force can be computed by
 |
....(16) |
Where Tn is the tension in the cable and T is the tangent vector. t‾ and n‾ are unit tangent and normal vector.
Using the principle of virtual work, these loads can be transferred to the nodes of brick elements. The equivalent nodal force vector for brick element is given by-
 |
....(17) |
Cable reactions at the end of the brick elements, act as concentrated loads. Corresponding equivalent finite element nodal forces can be calculated by-
 |
....(18) |
where, Tend is the cable tension at the end points and [N] is the shape functions of 20 node brick elements. Now total load vector due to interaction of concrete and cable is obtained by-
 |
....(19) |
This nodal load vector is applied on the three dimensional finite element model along with live and dead load vectors to include pre-stressing effects.
Local Coordinates Computation
In order to calculate the shape functions at the cable ends, local coordinates are required for known global coordinates. Evaluation of local coordinate corresponding to known global coordinates is an inverse nonlinear problem which can be solved by Newton-Raphson method. The computation is carried out iteratively till the difference of two consecutive values becomes less than the prescribed tolerance. Let (x,y,z) is the global coordinate and (ξ,η,ζ) be the corresponding local co-ordinate. Numerical computations of local coordinates can be obtained using following iterative relationship-
In above equation, inverse matrix is the jacobian matrix. (xi+1, yi+1, zi+1) and (xi, yi, zi) are the known and computed values of the global co-ordinates. Starting values of ξ, η, ζ are considered as 0,0,0. The prescribed tolerance is considered as 0.01.
 |
....(20) |
In above equation, inverse matrix is the jacobian matrix. (xi+1, yi+1, zi+1) and (xi, yi, zi) are the known and computed values of the global co-ordinates. Starting values of ξ, η, ζ are considered as 0,0,0. The prescribed tolerance is considered as 0.01.
Software PRECON3D
Incorporating above mentioned formulations, a FORTRAN code named as PRECON3D has been developed for analysis of three dimensional prestressed concrete structures. For finite element modelling, 20 node brick element for concrete and 3 node curved bar element for the cable have been used respectively. There are total 11 main subroutines in the software i.e. Subroutine INPUT, LOADPS, GEOM, PREST, STIFFP, ASSEMBL, GREDUC, BACKSUB, DISPLAC, STCAL, OUTPUT. Out of two input files for the software, one deal with cable related data and another with finite element related data. Cable profile is defined using the B-spline ordinates. The smoothness of the curve is controlled by changing the order. In this study, constant B-spline order of three is considered. The cable lies in the same plane as that of the B-spline ordinates. The cable segment in a particular element is automatically detected by calculating the intersections of cable with brick element. An automatic mesh generator has been developed to create finite element mesh. For this, coordinates of 20 seed points of the main block are required. For efficient debugging, flags have been used between major computational steps. The output file stores results like pre-stress cable profile, pre-stress loss along the cable, displacements, strain and stresses. This software is compiled using Microsoft Power Station Compiler on a Intel Pentium Dual Core CPU T2310 processor.
Numerical Examples
A simply supported prestressed concrete beam of size 8000x800x300mm is analysed using PRECON3D software. The cable eccentricity at the mid span is 300 mm. Pre-stressing force of 200 KN is applied on the beam. The beam is discretised into 10 twenty node brick elements and there are 128 nodes in FE model. The loading conditions and finite element mesh are shown in Figure 6. Following data is accounted for the analysis purpose:
Figure 6: Simply supported beam
Although software is capable to handle self-weight, in this study, it is ignored for validation purpose. The bending stresses and displacements obtained from the software are given in Table 1. The stresses are compared with the same obtained using analytical method as calculated in Appendix. Top fibre stresses at mid section obtained from software and analytical approaches are 0.66 and 0.67 MPa. The same for bottom fibre stresses are -2.31 and -2.33 MPa. It can be observed that both are in close match.
Figure 7: Cable as circular arc
The same problem is also analysed considering friction and corresponding stresses and displacements are given in Table 1. The friction loss of the pre-stressing force is calculated using analytical formulae as given in Appendix. For this, cable is modelled as circular arc passing through three points on the cable (Figure 7). The final pre-stressing force at the anchorage end obtained from software and analytical approaches are 181621 N and 181543 N which are almost close. It can be observed that due to friction, stress reduction at quarter span on bottom and top fibres are 15% and 36% respectively. The same at mid span are 12% and 18% respectively. It is also seen that deflections at quarter span and mid span are reduced by 16% and 15% respectively. Hence, friction effect must be accounted in the analysis to get accurate forces.
- Young’s modulus of concrete = 2x104N/mm2
- Coefficient of friction = 0.20
- Poisson’s ratio = 0.20
Figure 6: Simply supported beam
| Table 1: Stresses and displacement in beams | ||||||
| Distance from left end (mm) |
Nodal points | Coordinates (mm) | Stresses (N/mm2) | Displacement in zz- direction (mm) | ||
| Without friction |
With friction | |||||
| Without friction | With friction | |||||
| 2000 | 6 | 2000,0,0 | -2.00 | -1.70 | 0.94 | 0.79 |
| 38 | 2000,300,0 | -2.00 | -1.70 | 0.94 | 0.79 | |
| 81 | 2000,0,800 | +0.39 | +0.25 | 0.94 | 0.79 | |
| 113 | 2000,300,800 | +0.39 | +0.25 | 0.94 | 0.79 | |
| 4000 | 11 | 4000,0,0 | -2.31 | -2.04 | 1.31 | 1.12 |
| 27 | 4000,150,0 | -2.31 | -2.04 | 1.31 | 1.12 | |
| 43 | 4000,300,0 | -2.31 | -2.04 | 1.31 | 1.12 | |
| 86 | 4000,0,800 | +0.66 | +0.54 | 1.31 | 1.12 | |
| 102 | 4000,150,800 | +0.66 | +0.54 | 1.31 | 1.12 | |
| 118 | 4000,300,800 | +0.66 | +0.54 | 1.31 | 1.12 | |
| 6000 | 16 | 6000,0,0 | -2.00 | -1.78 | 0.94 | 0.81 |
| 48 | 6000,300,0 | -2.00 | -1.78 | 0.94 | 0.81 | |
| 91 | 6000,0,800 | +0.39 | +0.32 | 0.94 | 0.81 | |
| 123 | 6000,300,800 | +0.39 | +0.32 | 0.94 | 0.81 | |
Figure 7: Cable as circular arc
Conclusion
In this paper, a three dimensional finite element software to analyse prestressed concrete structures is described. Realistic cable modelling is carried out using B-spline curve. Effect of friction is also accounted. The software is validated with analytical results and found to be close match. The proposed software provides a fast and efficient solution and can prove to be a handy tool for the design engineers.
References
- Pandey AK, Ram Kumar and Trikha DN. Finite element analysis of prestressed concrete containment structure. Proceedings of the First National Conference on Computer Aided Structural Analysis and Design, Hyderabad, India, 1996, 373-379.
- Buragohian DN and Mukherjee A. PARCS –A pre-stressed and reinforced concrete shell element for analysis of containment structures, Transactions of the Twelth SMiRT International Conference, Stuttgart, Germany, Vol B, 1993, 141-146.
- Buragohian DN and Siddhyae VR. Finite element analysis of prestressed concrete box bridges, Journal of Structural Engineering, 24(3), (1997), 135-141.
- Pathak KK and Sehgal DK. Analysis of a prestress concrete beam using different cable models, Journal of Bridge and Structural Engineering, 33(2004), 29-40.
- Povoas RHCF and Figueiras JA. Non linear analysis of curved prestressed girder bridges, Proceedings of the Second International Conference on Computer Aided Analysis and Design, Austria, 1989, pp. 225-236.
- Kang YJ and Alexander C Scordelis. Nonlinear analysis of prestressed concrete frames, Journal of Structural Engineering, ASCE, 106(2), (1980), 445-462.
- Roca P and Mari AR. Numerical treatment of prestressing tendons in the non linear analysis of prestressed concrete structures, Computers and Structures ,46:950(1993), 905-916.
- Greunen JV and Alexander C Scordelis. Nonlinear analysis of prestressed concrete slabs, Journal of Structural Engineering, ASCE, 109 (7), (1983), 1742-1760.
- Figueiras JA and Povoas RHCF. Modeling of prestressing in non linear analysis of concrete structures, Computers and Structures, 53(1), (1994), 173-187.
- Vanzyl SF and Scordelis AC. Analysis of curved prestressed segmental bridges. Journal of Structural Engineering, ASCE, 105, (1979), 2399 -2411.
- Elwi A and Hrudey T. Finite element model for curved cable embedded reinforcement, Journal of Engineering Mechanics, ASCE, 115 (4), (1989), 740-754.
- Jirousek J and Bouberguig A, Saygun A. A macro element analysis of pre-stressed curved box girder bridges, Computers and Structures, 10(1979), 467-482.
- Saleem Akhtar, Pathak K.K. and S.S.Bhaduria, Finite element analysis of prestressed concrete beams considering realistic cable profile. International Journal of Applied Engineering Research, 3(1), (2008), 121-138.
- Raju N Krishna. Prestressed Concrete, Tata Mc -Graw Hill, Third Edition, New Delhi, India, 1995.
- Lin TY and Burns NH. Design of prestressed concrete structures, John Wiley and Sons, 3rd edition, New York, 1982.
- Rogers DF and Adams JA. Mathematical Element for computer Graphics, McGra-Hill, 2nd edition, New York, 1990.
- Zienkiewicz OC and Taylor RL. The Finite Element Method Vol.1 & 2, McGraw -Hill, Fourth Edition, London, UK, 1991.
Appendix
- Stresses
Given: Concentrated load P=6000 N
Span L= 8000 mm
Pre-stress force P = 200 KN
Cross sectional area A(b x h) = 300x800 mm
and eccentricity at mid section e =300 mm
Top and bottom stresses are-
- Pre-stress loss due to friction
Let cable profile be a circular arch passing through three points A,B,C on the cable (Figure 7). If R is its radius then-
(R -300)2 + 40002 = R2
R=26816.66 mm
Angle at the centre intended by the arc can be obtained as-
α/2 = 8.578°
α =17.1566° = 0.29943 radian
Cable length will be-
L ≈ R. α = 26816.66 x 0.29943 = 8029.71 mm
Given K = 0.0000046 /mm
µ = 0.20
TA = 20000
TC = TA e(-µa – KL) = 200000* e – (0.2*0.29943 + 0.0000046 *8029.71) = 181543.30 N
NBMCW October 2011
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