Parametric study for post-tensioned composite beams with external tendons

Abstract

Strengthening of bridge superstructures composite beams with external post-tensioned tendons is a good technique for strengthening the existing structures. In this study, a numerical model is illustrated to study the nonlinear simulation of composite beams stiffened with externally post-tensioned tendons. The accuracy of the developed numerical model is validated using comparisons between the numerical and existing test data. The influence of various strengthening parameters is investigated, which include draped versus straight tendons, tendon length, the effect of post-tensioning on reinstating the flexural behavior of an overloaded beam, tendon eccentricity, and the degree of shear connection. A good agreement between the proposed model and the test data is obtained. The results demonstrate that at the same tendon eccentricity, the trapezoidal profile shows better behavior for the strengthened beams. However, more ductility is obtained when using the straight tendon profile. Applying post-tensioning through the beam of full length helps to reduce the creation of fatigue cracks, which always start at stress raisers, and subsequently increases the fatigue life of the composite beam. Also, the external post-tensioning effectively maintains the flexural behavior of the overloaded strengthened beam after unloading in comparison to the un-strengthened beam. It is observed that 80% degree of shear connection or higher is recommended to obtain the desired performance of the external post-tensioning force for strengthening composite beams.

Keywords

composite beam draped tendon finite element parametric study post-tensioning shear connection tendon length

Introduction

External post-tensioning (PT) is achieved using post-tensioned tendons located outside the composite section. Applying external PT force to the composite beam is considered an active strengthening technique that can create permanent internal stresses in the beam opposite to the internal straining actions due to the service loads. The most benefits of using this system of strengthening are an elastic performance to higher loads, higher ultimate capacity, and reduction in deformation under the applied loads (Chen and Gu, 2005).

Previous studies have been done on composite beams with external PT force in sagging moment regions. Ayyub et al. (1990) tested three specimens to investigate different types of post-tensioned composite beams, including tendon material and shape. Higher capacity in case of draped tendons in comparison to the straight tendons was the main conclusion. However, the straight profile is preferable because of lower construction cost. To improve bridge performance, Daly and Witarnawan (1997) used external PT for strengthening, where anchorage and deviator details were presented. Craine and Uy (2004) provided a set of tests on steel–concrete composite beams with and without PT. In addition, a finite element (FE) model was presented and calibrated with the test data through the load–deflection relationships. Dall’Asta and Zona (2005) proposed a numerical model for the nonlinear behavior of post-tensioned composite beams by external tendons. Furthermore, a novel method was presented to calculate the incremental stress in the tendon at the ultimate load and subsequently the beam flexural capacity. Lorenc and Kubica (2006) carried out experimental tests on six strengthened beams with straight and draped profiles as well as plain composite beams. The effect of degrees of the shear connection was considered and the slippage along the beam length was recorded. Choi et al. (2008) used external PT force in strengthening steel–concrete composite bridges and analytic formulas for the increment of the initial PT force were derived. Furthermore, a novel formula for composite beams was then presented considering the initial PT force and its increment. Park et al. (2010) investigated the bending performance and stiffening effect of a bridge using post-tensioned steel beam with externally unbonded tendons. The authors tested 11 steel beams with different parameters to study the effect of the tendon material, the amount of PT force, using a deviator, and the draped profiles of tendons. A nonlinear FE analysis was presented by Ibrahim et al. (2012) to study the behavior up to ultimate load for composite beams with external PT force. The load–deformation relationship, strains in the concrete flange and the steel beam, and types of mode failure were covered by this analysis. Taoum et al. (2015) investigated the behavior of steel beams stiffened with locally prestressed reinforcing steel rebars by testing seven beams under three-point bending. The reinforcing rebars were anchored to the both ends of the beam and prestressed to increase the beam stiffness and ultimate strength. Lou et al. (2016) proposed a FE model to investigate the prestressed composite beams by external tendons considering the time factor by applying short- and long-term loads. Geometric nonlinearity was taken into account and the presented FE model was validated using existing test data.

Although experimental tests have been done to evaluate the performance of steel–concrete composite beams stiffened with external PT force, most of these past works dealt only with few parameters that affect the performance of the beams. Furthermore, these experimental works were not sufficient to introduce various results in different parts of the composite section to investigate the response of the strengthened composite beam in detail. Therefore, this work aims to develop a reliable nonlinear FE model that is capable of simulating the material and geometric nonlinearity of composite beams strengthened with externally post-tensioned tendons. Furthermore, an evaluation of various significant parameters that affect the behavior of the upgraded composite beams is presented.

In this study, a numerical model is illustrated to study the nonlinear simulation of composite beams stiffened with externally post-tensioned tendons at the sagging moment region. The reliability of the FE model is validated using comparisons between the numerical and existing tests results. The influence of various strengthening parameters is investigated, which include draped versus straight tendons, tendon length, the effect of PT on reinstating the flexural behavior of an overloaded beam, tendon eccentricity, and degrees of the shear connection. A significant analysis of the global performance of the retrofitted beam is presented throughout the influence of these parameters on the deflection, stresses in the steel beam, slippage between the concrete flange and the steel beam, moments on the shear connectors, and tendon stress.

FE model

Element selection

The commercial FE package ANSYS (2008) is used to implement the FE investigation. The steel beam, web and flange, is modeled using a brick eight-node element, SOLID45, which has plasticity, stress stiffening, large deflection, and large strain capabilities. The concrete flange is idealized using a solid element, SOLID65, that is able to crack in tension and crush in compression. The slippage between the steel flange and concrete induced by the shear stud as a deformable shear connector is simulated using a beam element and a spring element. The uniaxial beam element, BEAM23, is used to simulate the shank of the shear connector. The unidirectional spring element, COMBINE39, is used to resist the longitudinal shear force and simulate the relative movement between the steel beam and the concrete flange. The post-tensioned tendons and the steel rebars are simulated using a link element, LINK8. The interaction between the two surfaces of the steel beam flange and the concrete flange is simulated using contact element, CONTAC52, to maintain or break the physical contact and to allow the sliding between the two surfaces. The element is defined by two nodes: normal stiffness and tangential stiffness. The normal stiffness is the only real constant that defined for the contact element and is calculated based on the lowest stiffness of the two surfaces in contact. So, the contact slip between the steel and concrete is influenced only by the shear stud connection which is specified by a load–deformation curve for the spring element.

Material modeling

Figure 1(a) shows the simplified stress–strain relationship for concrete in compression that is composed of few points joined together by straight lines. The first point on the curve is at stress 0.3 f_c and the strain is computed from equation (1) (Gere and Timoshenko, 1997), while other points are obtained from equation (2) (Desayi and Krishnan, 1964). The last point in the relationship is the strain at the maximum concrete strength that is computed by equation (3). Perfectly plastic behavior after the last point on the curve is assumed. On the other hand, the performance of concrete in tension is approximately linear up to the tensile strength of concrete. After this point, the concrete starts to crack and the tensile strength reduces progressively to zero (Bangash, 1989)

E_{c} = \frac{f}{ε}

(1)

f = \frac{E_{c} ε}{1 + {(ε / ε_{0})}^{2}}

(2)

ε_{0} = \frac{2 f_{c}}{E_{c}}

(3)

Figure 1.

Stress–strain relationship for (a) concrete and (b) steel.

In contrast to concrete, bi-linear relationships are used in this study to demonstrate the stress–strain relationship for the web, flanges, steel rebars, and tendons, as illustrated in Figure 1(b). The steel model in the FE simulation is supposed to be an elastic linear strain hardening material and conformable in tension and compression. Modulus of elasticity E, yield strength f_y, and ultimate strength f_u for the steel beam, tendons, and steel rebars used in this analysis are the same for the laboratory works.

Real constants

The relative movement between the steel beam and the concrete flange is taken into account using the load–deformation data as inputs for the real constants of the spring element in ANSYS (2008). This analytical load–deformation relationship for headed shear connectors (given in equation (4)) was first presented by Slutter et al. (1971) and then extensively used by other authors, such as Aribert and Labib (1982), Abdel Aziz (1986), and Johnson and Molenstra (1991)

P_{j} = F_{m a x} \cdot {(1 - e^{- β \cdot Δ j})}^{α}

(4)

where P_j is the horizontal shear force in the shear connector, F_max is the ultimate capacity of the shear connector, Δ_j is the slippage between the concrete flange and the steel beam, and α and β are constants that control the initial start and the shape of the slippage relationship. Reasonable FE results were obtained, in comparison to experimental results, when α and β were 0.8 and 0.7, respectively. In addition, the initial PT force is defined by an initial strain (Kim et al., 2010), which is calculated as given by equation (5)

Initial strain = \frac{F}{E A_{p}}

(5)

where F is the initial PT force, E is the elastic modulus of the tendon, and A_p is the tendon cross-sectional area.

Prestress losses

The FE modeling applies the PT forces, which are assumed to be the value after all initial losses occurred. Additional losses during loading were considered constant in the FE model along the length of the PT tendons.

FE validation

To ascertain the accuracy of the proposed FE model, three FE models were built using previous laboratory works. For Model 1, a post-tensioned composite beam, tested by Chen and Gu (2005), was selected to inspect the behavior of post-tensioned composite beams by proposing a numerical model that is able to illustrate the nonlinear response of the beam. The full length of the beam was 5150 mm and tested over a clear span of 5000 mm. The composite section details and dimensions are shown in Figure 2(a). The shear connectors were designed to attain full shear connection according to BS5400 Part 5 (BSI, 1979). Beam B was one of the three post-tensioned composite beams that tested by Ayyub et al. (1990) and was chosen to validate Model 2. The beam length was 4830 mm and was tested over 4570 mm as a clear span. The concrete flange was attached to the steel beam using shear studs, as shown in Figure 2(b). Lin et al. (2013) tested three continuous composite beams subjected to cyclic loads. The initial static test in this investigation was selected to validate Model 3. The total length of each specimen was 4600 mm and was tested over a clear span of 4000 mm. The composite cross-section details and dimensions are shown in Figure 2(c). A summary of the materials properties for the validated specimens is listed in Table 1. Due to symmetry, only longitudinal halves of the beams were modeled, as shown in Figure 3.

Figure 2.

Cross sections of the analyzed beams: (a) Model 1, (b) Model 2, and (c) Model 3.

Table 1.

Summary of the materials properties for the validated specimens.

Model	Post-tensioned tendons				Concrete	Steel beam
	f_y (MPa)	f_u (MPa)	A_p (mm²)	F (kN)	f_c (MPa)	f_y (MPa)		f_u (MPa)
	f_y (MPa)	f_u (MPa)	A_p (mm²)	F (kN)	f_c (MPa)	Web	Flange	Web	Flange
Model 1	1680	1860	137.4	112.6	40	327.7	406.5	492.6	593.6
Model 2	1620	1860	279	289	40	411.6	411.6	565.4	565.4
Model 3	NA	NA	NA	NA	27	372.4	375	527	535

Figure 3.

FE model showing (a) FE mesh and (b) different parts of the model.

In Figure 4, the numerical results for Model 1 and the test data are compared. The results demonstrate that the FE measures are in a good agreement with the test data based on the beam deflection and the tendon stress. From Figure 4(a), the expected FE ultimate capacity is 300 kN m, 3.5% less than the maximum strength measured from the experimental findings. Figure 4(b) demonstrates the relationship of moment versus incremental stress in the post-tensioned tendons for the experimental and FE results. The two relationships are shifted from the origin by 819 MPa, which corresponds to the initial PT stress.

Figure 4.

Comparisons for Model 1: (a) deflection and (b) tendon stress.

The results for Model 2 are compared to the test data in Figure 5. This comparison is based on the load–deformation, the strain in the concrete flange, the strain in the steel beam, and the incremental increasing in the strand force. Another comparison between the results for Model 3 and the experimental data is illustrated in Figure 6. This comparison depends on the load–deflection response and the strains in shear connectors at two locations through the beam length. Based on these comparisons, a good agreement is achieved between the proposed FE model and the test data. Consequently, the validated Model 1 is used to introduce a comparative parametric study.

Figure 5.

Comparisons for Model 2: (a) deflection, (b) strain in the concrete flange, (c) strain in the steel flange, and (d) increase in the PT force.

Figure 6.

Comparisons for Model 3: (a) deflection, (b) strain in steel reinforcement, (c) strain in stud near the support, and (d) strain in stud at one-sixth of the span.

Parametric study

Effect of draping tendons

To investigate the influence of draping tendons on the performance of the composite beam, six models divided into two groups are analyzed. The first group consists of three models with different tendons profiles that are parallel to the steel beam web, as shown in Figure 7. The first profile is straight tendons anchored at both ends of the beam and free to move up or down relative to the beam cross section. The second profile is triangular tendons anchored at both ends of the beam along the centroidal axis of the steel section, 135 mm over the bottom flange, and positioned at the mid-span using deviators. The third profile is trapezoidal tendons positioned under the loading points using deviators.

Figure 7.

Different profiles parallel to the steel web: (a) straight profile, (b) triangle profile, and (c) trapezoidal profile.

Figure 8 presents the comparison between the results of composite beams of the first group. First, the post-tensioned beams show better behavior and larger capacity over the control beam. The application of the externally PT force enables to create opposite straining actions on the composite beams that can help to reduce the effects of the service loads. Figure 8(a) shows the moment–deformation relationships for the three profiles as well as the control beam. According to these relationships, the initial upward movement of the strengthened beams is considered the same and their behaviors are very close during the elastic region. After yielding, the beam with the straight profile has a better ductility in comparison to the other profiles.

Figure 8.

Results of different profiles parallel to the steel web: (a) deflection, (b) slippage, (c) shear connector moment, and (d) tendon stress.

Furthermore, the relative horizontal displacement between the concrete flange and the steel beam causes moments on the shank of the shear connectors and these moments are concentrated near the base of the shank. So, the influence of tendon profile is also evaluated by monitoring the relative movement between the concrete flange and the steel beam as well as the shear connector moment, as shown in Figure 8(b) and (c), respectively. The draped tendons, triangle and trapezoidal profiles, show larger slippage and connector moment due to the smaller eccentricity at the beam ends. The incremental increasing in the tendon stresses is shown in Figure 8(d).

Figure 9 illustrates the ultimate stress distribution along the steel beam bottom flange for the beams with different profiles including the control beam. Ultimate stress in the flange takes the same behavior like the externally applied moment on the beam but with different capacities. For the straight profile, the eccentricity of the tendon is constant along the beam length including the maximum moment region. So, the stress in the steel flange along this region is uniformly distributed and equal to the yielding strength, as shown in Figure 9(b). Otherwise, the eccentricity of the triangle tendon is maximum at mid-span only and decreases toward the beam ends. So, the steel flange does not yield at mid-span but yielding occurs in the rest of the maximum moment region. On the other hand, the eccentricity of the trapezoidal tendon is constant only in the constant moment region and decreases toward the beam ends. So, the stress reaches the yield point at the edges of the constant moment region while the rest of this region does not yield, as shown in Figure 9(c). These results confirm a recommendation to use a tendon profile with constant eccentricity along the beam length or at least extended outside the maximum moment region.

Figure 9.

Ultimate stress distribution along the bottom flange: (a) control beam, (b) straight profile, (c) triangle profile, and (d) trapezoidal profile.

The second group also consists of three models with different tendons profiles that are under the bottom flange of the steel beam, as shown in Figure 10. Higher capacities and stiffness for strengthened beams over the control beam are achieved in comparison to the draping tendons parallel to the steel web. Figure 11 illustrates the difference in behavior for different tendon profiles. However, higher tendon stresses are achieved when using trapezoidal tendon, but this profile shows better behavior in deflection, stiffness, and capacity. This profile helps to maintain the tendon eccentricity to coincide with the influence of the externally applied moment and subsequently reduces the stresses in the bottom flange of the steel beam in comparison to the other two profiles. Based on these results and comparisons, the draping tendons under the bottom flange show better behavior and response than the draping tendons in parallel to the steel beam web. Furthermore, more ductility is obtained when using the straight tendon under the bottom flange.

Figure 10.

Different profiles under the bottom flange: (a) straight profile, (b) triangle profile, and (c) trapezoidal profile.

Figure 11.

Results of different profiles underneath the steel flange: (a) deflection, (b) steel beam flange stress, (c) slippage, (d) shear connector moment, and (e) tendon stress.

Effect of tendon length

Fatigue cracks always begin at stress concentrations, so eliminating such imperfections helps to increase the fatigue life of the structure member. High local stresses can cause failure more quickly, so the lengths of the externally post-tensioned tendons must be designed to minimize any stress concentrations in the steel beam web or flange. Three different lengths of the external tendons as a ratio of the beam length are analyzed, as shown in Figure 12. The flexural stress distributions along the steel beams demonstrate stress concentration on the web portion that is attached to the anchorage of the post-tensioned tendons, as shown in Figure 13. This stress concentration is relieved by increasing the tendon length and vanishes when using tendons along the full length of the beam. Furthermore, there is an abrupt change in the stress in the bottom flange of the steel beam near the tendon ends which leads to stress concentration at those features, as shown in Figure 14. All of these stress raisers help to initiate the fatigue failures in these regions.

Figure 12.

Beams with different tendon lengths: (a) L = 100%, (b) L = 65%, and (c) L = 30%.

Figure 13.

The flexural stress contour along the steel beams for different tendons lengths: (a) L = 30%, (b) L = 65%, and (c) L = 100%.

Figure 14.

The flexural stress distribution along the bottom flange for different tendons lengths: (a) initial stress and (b) ultimate stress.

In addition, the results shown in Figure 15 confirm that the increasing in the tendons length helps to increase the ultimate strength of the composite section and improve the total behavior of the strengthened beam. This improvement is confirmed by decreasing the moments on the shear connectors, and the incremental stresses in the tendons.

Figure 15.

Results of different tendons lengths: (a) deflection, (b) shear connector moment, (c) slippage, and (d) tendon stress.

Effect of the PT force on restoring the ultimate strength of overloaded composite beam

The deflections for overloaded composite beams are improbable to be restored with passive techniques of strengthening, such as using steel plate and fiber-reinforced plastic (FRP) sheets (Elrefai et al, 2012). So, the efficiency of using externally post-tensioned tendons, as an active technique for strengthening, in restoring the flexural strength of the overloaded composite beam is examined by comparing results of overloaded beams with and without external PT force. Moment–deflection relationships for the analyzed beams are shown in Figure 16(a). The reference beam is loaded to a moment of 234 kN m with vertical deformation of 31.23 mm and extreme flange stress of 393 MPa. The yielding in the bottom flange of the steel beam and the numerous cracks in the concrete flange make the beam be considered as an overloaded beam. Subsequently, the beam is unloaded with the same trend like the loading process but with permanent deformation of 7.6 mm. Adding external PT force to the composite beam causes upward deflection of 4.8 mm and relieves part of the stress that experienced in the steel beam flange. So, the beam deformation is 18.5 mm with a flange stress of 307 MPa at the same loading level of the control beam. The residual deformation in the strengthened beam is an upward deflection of 0.8 mm.

Figure 16.

Effect of PT force on restoring: (a) deflection, (b) flange stress, (c) slippage, and (d) shear connector moment.

Figure 16(b) and (c) illustrates the stress in the steel beam flange and the relative movement between the concrete flange and the steel beam, respectively. These results are comparable to the load–deformation response. The results emphasize reductions in the residual flange stress and the plastic slippage by 75% and 36%, respectively. The moment on the shear connectors during the loading and unloading process is illustrated in Figure 16(d). Furthermore, a residual stress of 0.5% over the initial stress applied in the tendons is encountered when unloading occurred, as shown in Figure 16(e). All of these findings confirm that the external PT effectively maintains the flexural behavior of the overloaded strengthened beam after unloading in comparison to the un-strengthened beam.

Effect of the tendon eccentricity

The tendon eccentricity is the distance between the tendon location and the centroid of the composite section. So, changing the tendon position, related to the bottom flange of the steel beam, can control the tendon eccentricity, as shown in Figure 17. Two models are analyzed with different eccentricities and the initial PT force and the other parameters are kept the same. The results confirm 32% reduction of stresses in the bottom flange of the steel beam, 25% increasing in the initial upward deflection, and 30% enhancement in the beam stiffness for the beam with higher eccentricity, as shown in Figure 18. This means that tendons with bigger eccentricity can introduce more improvement to the beam behavior, but these tendons are subjected to more stresses during loading, as shown in Figure 18(c).

Figure 17.

Beams with different eccentricities: (a) above the bottom flange and (b) underneath the bottom flange.

Figure 18.

Results of different eccentricities: (a) deflection, (b) bottom flange stress, and (c) tendon stress.

Effect of degrees of the shear connection

Effect of degrees of the composite action between the concrete flange and the steel beam on the overall performance of the post-tensioned composite beam is evaluated using four models with different degrees of shear connection. Changing the total number of connectors along the beam length can control the degree of the composite action, as listed in Table 2. The degree of composite action can be calculated according to the following equation

η = \frac{n}{N}

(6)

where η is the degree of shear connection, n is the actual number of shear connectors in the beam, and N is the required number of shear connectors to get the full composite action.

Table 2.

Models with different degrees of the shear connection.

Model no.	L_eff. (mm)	Tendon profile	F (kN)	n ^a	η^b (%)
1	5000	Straight	112.6	11	40
2	5000	Straight	112.6	16	60
3	5000	Straight	112.6	21	80
4	5000	Straight	112.6	26	100

n is the number of the actual shear connectors in the beam.

η is the degree of the shear connection.

Figure 19 shows the applied moment on the beam versus the deflection of all models, control, and strengthened beams. It is observed that the initial behaviors of the control beams are the same followed by an increase in the beam stiffness and capacity by increasing the degree of the shear connection. The same behavior for the strengthened beams is noted but with initial upward deflections. In cases of the lower degrees of the shear connection, the beam capacities are controlled by the shear connection capacity which leads to premature failure before the beams could realize their overall sectional strengths. Figure 20 illustrates the relationships between the applied moment on the beam and the predicted moment on the shear connectors for all models with and without post-tensioned tendons. The trends show larger moments on the shear connectors in case of lower degrees of the shear connection. These results confirm early failure in the shear connectors. Consequently, higher composite action enables the post-tensioned tendons to enhance the behavior and capacity of the composite beam in comparison to beams with the lower composite action. The difference in beam capacities between the control and strengthened beams for different degrees of shear connection is illustrated in Figure 21. The capacity of the strengthened beams increases by 6% and 21% for degrees of shear connection 40% and 100%, respectively. So, it is recommended to achieve 80% or higher of the degree of composite action between the concrete flange and the steel beam to obtain the full benefit of using externally post-tensioned tendons in strengthening the composite beams.

Figure 19.

Deflections of different degrees of the shear connection: (a) reference beams and (b) strengthened beams.

Figure 20.

Shear connector moment of different degrees of the shear connection: (a) reference beams and (b) strengthened beams.

Figure 21.

Effect of degrees of the shear connection on the beam capacity.

Conclusion

In this study, a nonlinear numerical analysis for composite beams externally post-tensioned with tendons has been evaluated. The FE model is validated using existing test results. In addition, a parametric study is carried out to explore the different behaviors of the strengthened composite beams. The main conclusions are as follows:

A good agreement between the presented FE model and the test data is obtained, which ensured its reliability and accuracy in expecting and analyzing the behavior of the externally post-tensioned composite beams.

Using draping tendons under the bottom flange is better than draping tendons parallel to the steel beam web.

At the same tendon eccentricity under the bottom flange, the trapezoidal profile shows better behavior. However, more ductility is obtained when using the straight tendon.

Applying external PT through the full length of the beam helps to reduce the creation of fatigue cracks that always start at stress raisers and subsequently increases the fatigue life of the composite beam.

The external PT effectively maintains the flexural behavior of the overloaded strengthened beam after unloading in comparison to the un-strengthened beam.

To achieve optimum performance of PT strengthening, it is recommended that the technique be used for bridges with at least 80% composite action between the steel beam and concrete slab.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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