Rajeev Goel, Scientist, Bridges and Structures, CRRI, New Delhi, Ram Kumar, Advisor and HOD, Bridges and Structures, D. K. Paul, Professor, Department of Earthquake Engineering, Indian Institute of Technology, Roorkee
Behavior of prestressed concrete structures mainly depends on the performance of the prestressing force over a design period of these structures. A generalized three-dimensional failure criterion for concrete and constitutive relationships of reinforcing steel and prestressing tendons are implemented into the finite element codes. Subsequently, computer software DAPCAS has been developed for the finite element analysis of prestressed concrete structures.
Prestressed concrete sections are economical compared to reinforced concrete and hence, have a wide use in bridges, buildings and other structures. Reliability of these structures mainly depends on the performance of the prestressing force over a design period of these structures. Prediction of the response of prestressed concrete structures requires three-dimensional structural idealization and true modelling of nonlinear behavior of concrete, reinforcing steel and prestressing tendons.
All these effects have been considered for the analysis of the prestressed concrete structures. Based on the above, computer software has been developed for the three-dimensional finite element analysis of prestressed concrete structures. Validation of the software has been done by analyzing experimental as well as analytical studies, reported in the literature. The results of these validation studies have been compared with the reported results in the literature and found to be in a good agreement.
Finite Element MethodFinite element method is accepted as the most powerful technique for the numerical solution of a variety of engineering problems and hence, the same has been used to model and analyze the nonlinear behavior of prestressed concrete structures.
Modelling of ConcreteThe finite element formulation of three-dimensional solid element is well in literature. In the present study, concrete has been modelled by quadratic serendipity 20-noded three-dimensional solid-element, Figure 1.
Concrete and steel reinforcement are represented with a composite single element, Figure 2. Each set of reinforcing bars is smeared as two-dimensional membrane layer of equivalent thickness and hence, equal area inside the solid element. In this formulation, it is required that the reinforcement membrane should be perpendicular to one of the local axes. Perfect bond is assumed between the steel einforcement and the surrounding concrete. The stiffness and internal forces associated with the steel reinforcement are integrated and added to those of concrete to get the total stiffness and internal forces in the element.
Prestressing TendonA discrete formulation of prestressing tendons using 20-noded solid-element has been developed. The influence of prestressing tendon on the concrete is modelled by distributed normal and tangential load acting along the tendon length, Figure 3 and concentrated point load at anchorages. Prestressing tendon is modelled using one-dimensional curved element embedded in the solid-element. Geometry of the tendon segment, lying inside the solid-element, is defined by three nodal points, Figure 4. Prestressing tendon may be located anywhere in the depth of the section. Full strain compatibility is assumed between concrete and prestressing tendon. The stiffness of the tendon is added to the stiffness of the solid element to get the total stiffness.
Modelling of LoadsFinite element formulations for application of loads such as concentrated nodal loads, concentrated point loads, self–weight, pressure over a surface and prestress load on the structures have been developed and implemented into finite element code.
Concrete is highly nonlinear material and its nonlinear response is mainly due to progressive cracking and nonlinear deformations. Triaxial tests performed on concrete shows that concrete is pressure sensitive material and under high hydrostatic compression, concrete flows like metals in a limited manner. The compressive, tensile and flexural strengths of concrete depend upon the rate of loading. An approach based on elasto-viscoplasticity theory of Perzyna  is adopted here to take into account the loading rate effects.
A generalized three-dimensional failure criterion for concrete of Fan and Wang  is implemented into the finite element codes. The semi-theoretical strength criterion for concrete is employed to govern the elastic limit envelope and plastic limit (ultimate strength) envelope. Prior to damage, the behavior of the concrete is assumed to be isotropic elasto-plastic hardening. The stiffness degradation in the plastic range is simulated by a plastic hardening formula with two hardening parameters: plastic hydrostatic strain and plastic deviatoric strain. When the stress state reaches the ultimate strength envelope, damage occurs in the direction of maximum principal stress. This criterion requires only one parameter, the uniaxial compressive strength of concrete.
Tension stiffening effect, resulting from the interaction between the cracked concrete and the steel reinforcement, given by Belarbi and Hsu  has also been modelled. Non–linear behavior of reinforcing steel as given by Belarbi and Hsu  and prestressing tendon given by Kwak and Kim  has been modelled. The strain-rate dependent behavior of concrete, reinforcing steel and prestressing tendons is also taken into account.
Validation of MethodologyBased on the above modelling, non–linear finite element software DAPCAS has been developed . For the validation of the methodology and software, three examples have been taken from the literature.
Two-Span Continuous Prestressed Concrete Rectangular Beam
Lin  carried out experimental study on series of two-span continuous rectangular prestressed concrete beams (203 x 406 x 15000 mm) which were prestressed with one prestressing tendon. Longitudinal section and cross-section of the prestressed concrete beam are shown in Figure 5. The hinge support was furnished at the centre, while rollers were employed at both the ends. Profile of the prestressing tendon is shown in Figure 6. Prestressing tendon consists of 32 parallel wires of 5mm diameter with an effective prestress of 612 kN. The loading consists of two equal point loads (Figure 5) which were gradually increased to about 150 kN each in various loading stages and at the end of each loading stage, deflection at the mid-span of the beams were measured.
Considering the symmetry of the beam about central support, only one span of the beam has been analysed by discretizing it into twelve numbers of elements. The stiffness of the tendon as well as the prestress load exerted by the tendon has been considered. The initiation of viscoplastic behavior is assumed at 30% of the ultimate strength of concrete. Material properties of beam, considered in the study, are given in Table 1. One end of the beam is assumed fixed, while other end simply supported. The beam is analysed for self weight and a variable point load, increasing up to 150 kN at an interval of 10kN.
Deflections at the mid-span of the beam were reported in the experimental study. Hence, the same are obtained in the present study for the applied loads. Predicted deflections at the mid-span of beam are shown in Figure 7 along with the experimental results.
Mid-span deflections and variation of the stresses in the prestressing tendon along its length are given in Table 2. Stresses in prestressing tendon at the anchorage end are found to decrease with the increase in point load (at 150 kN point load, reduction is about 2 percent). At location of end of straight tendon profile, stresses are found to increase with the increase in point load (at 150 kN point load, increase is about 2 percent). At location near zero eccentricity of tendon, stresses are found to decrease significantly with increase in point load (at 150 kN point load, reduction is about 22 percent). At the fixed end, reduction in stresses in the tendon is observed but to a lesser extent (at 150 kN point load, reduction is about 1 percent).
In the experimental study, initiation of first crack was observed in the concrete near the top surface of the central support at a point load of 73.4 kN and thereafter under the point load at a point load of 117.9 kN. In the present numerical study, first crack is found at the top surface of the fixed end (i.e. central support) at a point load of 80 kN. Thereafter, cracks are found under the point load at a load of 110 kN.
Simply Supported Prestressed Concrete T-Beam
Popovic and Anderson  carried out experimental studies on two dentical simply supported 32 300mm long precast prestressed concrete T-beams (named as A-27D and A-33D). The compressive strength of beams was about 40 MPa. The x-sections of the beam at the supports and at mid-span are shown in Figure 8. Each of these beams was prestressed with 18 Nos. of the prestressing tendon of 12.7mm diameter with yield strength of 1862 MPa, each consisting of 7 wires of low relaxation strands initially stressed up to 135.7 kN approximately. Sixteen numbers of prestressing tendons were single point depressed by 300mm at the mid-span. Remaining two tendons were having straight profile.
In the experimental studies, these two beams were placed side by side (Figure 9). Supplemental dead load simulating the additional dead load was applied over the beam before the load testing operation. The locations of supplemental dead load and test load are shown in Figure 10. During load testing, test load on these beams were applied in four stages and at the end of each load stages, deflections of the beam at the mid-span were observed. Four loading stages are given below:
Stage-I Supplemental dead load + (Load-1 at locations shown in Figure 10)
Stage-II Stage-I + (Load-2 at locations shown in Figure 10)
Stage-III Stage-II + (Load-3 at locations shown in Figure 10)
Stage-IV Stage-III + (Load-4 at locations shown in Figure 10)
In each stage including supplemental dead load, concrete sleepers each weighing 14 kN were placed over the beams at selected locations and sequences shown in Figure 10.
One of the two beams has been analysed using the software. Due to symmetry of the beam about mid-span, half of the beam has been modelled using 144 numbers of 20-noded solid elements. Along the length, beam is discretized into sixteen equal parts. Prestressing tendons have been modelled using the 3-noded discrete line element lying inside the solid element. The stiffness of the tendons as well as the prestress load exerted by these tendons has also been considered. The initiation of viscoplastic behavior is assumed at 40% of the ultimate strength of concrete. The material properties of the concrete and prestressing tendons are given in Table 3. The deflections at the mid-span of beam are computed for all the above four loading stages and are given in Table 4 along with the experimental values.
During experimental studies, both beams were visually monitored for distress throughout the loading. No cracking or visible distress has been reported during the application of supplemental dead load and first three loading stages. Flexural cracks were reported at mid-span of the beam, extending from the bottom of the web to the flange-web junction, at the fourth loading stage.
In the present numerical study, initiation of cracks is observed at the third loading stage. These cracks might have not been visible by naked eyes during experimental studies. During the fourth loading stage, cracks are predicted at the mid-span. These cracks are found extending from the bottom of the web to the flange-web junction. At the mid-span, all the gauss points are found cracked in the elements lying between bottom of the web and flange-web junction.
The stresses in the prestressing tendons, at the gauss points of the uppermost layer of tendons, for all the four loading stages are shown in Table 5. It is observed from this Table that the maximum stress is at mid-span of the beam for all the loading stages. At the supports, stress in the tendons is lesser than the initial prestress of 1380.47 MPa for all the loading stages. In all the loading stages, stresses in tendons increase from support to mid-span. With the increase in load over the beam, stresses in tendon at supports reduce. In the straight tendons, there is no change in stresses throughout the length with increase in load over the beam.
Simply Supported Prestressed Concrete Rectangular Beam
Pandey  analyzed a simply supported prestressed concrete rectangular beam (200 x 300 x 4000mm). Due to symmetry about mid-span, half span of this beam was modelled using four numbers of 20-noded solid-element. The beam was prestressed with three prestressing tendons (of 60mm2x-sectional area each), having parabolic profile with zero eccentricity at supports and 100mm eccentricity at the mid-span (Figure 11). Each prestressing tendon was prestressed with 80 kN force. These tendons are located at a distance of 50mm, 100mm and 150mm in the x-section from one of the face across the width throughout the length of the beam. In the analysis, compressive strength of concrete is taken as 36 MPa.
Following two load cases (with and without considering stiffness of the prestressing tendon) have been considered for the analysis of the beam:
Load Case-I Prestress load only
Load Case-II Prestress load along with a UDL of 16 kN/m
Mid-span deflections of this beam and variation of stresses in the prestressing tendon were reported for these two load cases.
In the present study, considering the symmetry, only half-span of this beam is modeled using five numbers of 20-noded solid elements. Prestressing tendons are modelled using the 3-noded discrete line element lying inside the solid element. The stiffness of the tendons as well as the prestress load exerted by these tendons has also been considered. Analysis is carried out for the same two load cases with and without considering stiffness of the prestressing tendon. The material properties considered in the present study are given in Table 6. Deflections at mid-span are computed using beam theory also. Results of the analysis are given in Table 7. It is observed from this Table that the predicted deflections using the software are very close (variation is about 0.01 and 2.34 percent for Load Case-I and Load Case-II respectively) to the results obtained by beam theory. No cracks have been observed in the concrete under both load cases.
For these load cases, variation of stresses in the prestressing tendon have been obtained and are given in Table 8 and shown in Figure 12. Observed stresses in prestressing tendons are different than reported by Pandey . Difference in the stresses in prestressing tendons is mainly due to following reasons:
- Linear stress-strain relationship has been considered by Pandey  for the tendons whereas in the present study, non-linear stress-strain relationship of tendons is used.
- Pandey  externally applied the equivalent nodal loads to represent the prestress load due to prestressing tendons in the modelling whereas in the present study, the software computes these forces with in the software itself.
Computer software has been developed for three-dimensional finite element analysis of prestressed concrete structures taking into account nonlinear material properties and effects of creep and shrinkage. Validation of the software has been done with the help of available literature. A very good match between the experimental and numerical results is found.
AcknowledgmentsAuthors wish to express their gratitude to the Director, Central Road Research Institute, New Delhi for continuous encouragement and permission to publish this paper.
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