2.1 Initial jacking force

The initial jacking force per tendon introduced into the standard section of the EX girder is designed so that the initial prestress $P_{i}$ introduced at the center of the girder is the same. This is because lateral displacement does not occur when the initial prestress of each tendon introduced into the central section is all the same. For this, friction loss, elastic shorting loss, and anchorage setting loss are considered in consideration of the order of introduction of jacking force. Based on the tendon with the largest initial jacking force, the number of strands per tendon is determined.

The initial jacking force of the tendon is introduced to $0.90f_{py}$. Only in the case of 2,360 MPa high-strength strand of SWPC 7DL, the initial allowable jacking force is applied separately to the case of $0.9f_{py}$ and the case of $0.95f_{py}$. This is because, in the case of a 2,360 MPa high-strength strand, the number of strands per tendon decreases, which has the advantage of reducing the depth of the plastic neutral axis ^{(Cui and Youn 2022)}.

2.2 Allowable tensile stress

When designing a PSC composite girder bridge, the magnitude of stress occurring at the bottom surface of the PSC girder is reviewed. The cracking is reviewed based on the flexural tensile strength (see Eq. ) of the concrete applied to the PSC girder.

where, $f_{r}$ : flexural tensile strength of concrete

$f_{ck}$ : design compressive strength of concrete

When evaluating flexural stress by service load, live load applies the design truck load KL-510 and live load impact factor 0.25 specified in the road bridge design standard (limit state design method). The effective prestress $P_{e}$ of the tendon considering the long-term loss is calculated under the assumption that the construction stages of each analysis variable are the same. In this study, it is assumed that the introduction of initial jacking force is 28 days, the concrete pouring for the bridge deck is 60 days, and the asphalt pavement is 90 days.

2.3 Long-term behavior analysis

The effective prestress of the tendon is calculated through long-term behavior analysis considering the drying shrinkage and creep of concrete and relaxation of the tendon ^{(Youn et al. 2006)}. In this study, AEMM is applied as a long-term behavior analysis method, and the age-adjusted effective modulus considering the aging coefficient $\chi$ of concrete is shown in Eq. . Unlike the effective modulus method (EMM), which does not consider the age of concrete, AEMM considers the aging coefficient in the effective modulus $E_{e}$ (see Eq. ). As the age of concrete increases, the concrete creep coefficient decreases, and AEMM uses the aging coefficient to control the creep effect.

where, $\chi$ : aging coefficient of concrete

$\phi$ : creep coefficient of concrete

$E_{c}$ : elastic modulus of concrete

$\overline{E_{e}}$ : age-adjusted effective modulus

$E_{e}$ : effective modulus

2.4 Design flexural strength

The design flexural strength of the PSC composite girder is calculated using the strain-compatibility analysis method based on the force equilibrium condition and those stress-strain curves of concrete, prestressing steel, and reinforcing rebar. The material resistance factor for each material is $\phi_{c}$=0.65 for concrete and $\phi_{s}$=0.9 for prestressing tendon and reinforcing rebar. The stress-strain curves of the strand for the cross-section design are assumed to be a bi-linear curve as shown in Fig. 1, and the sloped straight line expression exceeding the yield strain is shown in Eq. ^{(CEN 2004; Lee 2015)}.

where, $f_{ps}$ : failure stress in strand

$f_{py}$ : yield strength of strand

$f_{pu}$ : ultimate strength of strand

$\varepsilon_{pu}$ : ultimate strain of strand

$\varepsilon_{py}$ : yield strain of strand

$\varepsilon_{ps}$ : failure strain of strand

Concrete stress-strain curves for section design are shown in Fig. 2. It is assumed as a non-linear curve and Eqs. and specified in the road bridge design standard (limit state design method) are applied in the analysis. In the figure, it can be seen that the ultimate strain $\varepsilon_{cu}$ of concrete changes according to the design compressive strength.

where, $\alpha_{cc}$ : coefficient taking account of long term effects, and 0.85 is applied

$n$ : exponent relate to the shape of curve

$f_{c}$ : stress of concrete

$\phi_{c}$ : concrete material coefficient

$\varepsilon_{c}$ : strain of concrete

$\varepsilon_{co}$ : concrete strain corresponding to compressive strength

$\varepsilon_{cu}$ : ultimate strain of concrete

Fig. 1 Stress/strain relationship of the prestressing strand for the section design(KIBSE 2021a)

Fig. 2 Stress/strain relationship of concrete with different concrete strengths for the section design(KIBSE 2021a)

2.5 Depth of plastic neutral axis

The road bridge design standard (limit state design method) adopts the regulation on the maximum depth of plastic neutral axis when bending fracture to induce ductile failure of PSC composite girder (see Eq. ).

This is different from the method of limiting the prestressing steel within a certain range by introducing a prestressing steel index for inducing ductile failure in the strength design method ^{(KIBSE 2015)}.

where, $c_{\max}$ : maximum depth of plastic neutral axis

$\delta$ : ratio of factored bending moment and the elastic bending moment after moment redistribution, and if the moment is not redistributed, $\delta$=1

$d_{p}$ : effective depth of the cross-section

$\varepsilon_{cu}$ : ultimate strain of concrete, and 0.0033 is applied for the concrete bridge deck

In this study, the maximum depth of the plastic neutral axis $c_{\max}$ is $0.4d_{p}$ by applying the design compressive strength of the concrete deck to 35 MPa or less. Also, only simple span bridges are studied, and bending moment redistribution is not considered.

3.1 Analysis section and analysis variables

The Korea Expressway Corporation EX girders with a span of 50 m are selected for the analysis section. The cross-sectional size of the EX girder changes according to the design compressive strength of the concrete (see Fig. 3). In the figure, it can be seen that the height of the girder decreases when the design compressive strength of the concrete increases. The girder spacing is 3.1 m, the thickness of the concrete bridge deck is 240 mm, and the thickness of the asphalt pavement is 80 mm, and parabolic tendon profile is used in the section as shown in Fig. 4. In the initial parameter analysis, the design compressive strength of the concrete bridge deck is assumed to be 27 MPa.

Analysis variables are the types of the strand, the initial allowable jacking force, and the design compressive strength of the PSC girder.

For the strand, SWPC 7BL and SWPC 7DL with a nominal diameter of 15.2 mm are applied. In the case of SWPC 7DL, $0.9f_{py}$ and $0.95f_{py}$ are applied for the initial allowable jacking force. Prestressing tendons are arranged in a parabola, and jacking forces are introduced at both ends. For the design compressive strength of the PSC girder, 40 MPa, 50 MPa, and 60 MPa are applied. There are 9 types of analysis sections in total.

The construction steps required for the parametric analysis are assumed to be 28 days for the introduction of jacking force, 60 days for pouring concrete for the concrete bridge deck, and 90 days for the asphalt pavement construction. For live load, the design truck load KL-510 and the standard lane load $\omega$=12.7 kN/m specified in the Road Bridge Design Standard (limit state design act) are applied ^{(KIBSE 2021b)}. For the lateral distribution coefficient, the numerical value 1.4 calculated through 2-dimensional grid analysis is equally applied ^{(Cui and Youn 2022)}.

Table 1 shows the names of the analysis variables for the case of span length of 50 m. For example, in EXH90-50-C60, H90 indicates the initial allowable jacking force of SWPC 7DL high- strength strand, and 50-C60 indicates the span length of 50 m and the design compressive strength of 60 MPa. The variable L means SWPC 7BL.

Fig. 3 Cross-section of 50-m EX-girders in the middle of the pre-stressed concrete girder bridge with different girder strengths

Fig. 4 Tendon profile of 50-m EX-girders with concrete strengths of 40 MPa, 50 MPa, and 60 MPa

3.2 Initial jacking force and number of strands

Table 1 shows the design parametric analysis results for the serviceability limit state. As the design compressive strength of the PSC girder concrete increases, the number of strands per tendon increases.

Conversely, when SWPC 7DL strands are applied, the number of strands per tendon decreased by about 5 compared to the case where SWPC 7BL strands are applied. In addition, by increasing the initial allowable jacking force to $0.95f_{py}$, it is possible to further reduce one strand (see Fig. 5).

Fig. 6 shows the effective ratio of prestressing force considering the long-term loss. In the figure, it can be seen that the effective ratio increases when the concrete design compressive strength is increased, and when high-strength strands is applied, the effective ratio increases. In Fig. 7, it can be seen that the stress developed at the bottom surface of the PSC girder becomes compressive stress, not tensile stress, when a service load is applied.

In addition, the compressive stress is greater in the case of applying 2,360 MPa high-strength strands than in the case of applying 1,860 MPa strands (see Fig. 7).

These results indicate that creep deformation decreases when high-strength concrete is applied and that when 2,360 MPa high- strength strands are applied, the amount of reduction in tension due to long-term behavior is reduced compared to when 1,860 MPa strands are applied.

Fig. 5 Changes in the number of strands per tendon according to the change in the design compressive strength of concrete

Fig. 6 Changes in the effective ratio according to the change in the design compressive strength of concrete

Fig. 7 Comparison of the stress developed at the bottom surface of the pre-stressed concrete girders under the service load

3.3 Design flexural strength and depth of plastic neutral axis

Table 2 show the design flexural strength, safety factor, depth of plastic neutral axis, and stress of the tendon at flexural failure calculated by applying the strain compatibility analysis method for each analysis variable. In Fig. 8, it is confirmed that the safety factor for bending moment decreased as the design compressive strength of the PSC girder concrete increased, but it is greater than 1.0 in all cases.

However, it can be seen from Fig. 9 that the EX girder with a span of 50 m uses a lot of prestressing strands, so the stress on the tendon is small during flexural failure, so it cannot sufficiently plastically behave and stay in an elastic state. For this reason, at the time of flexural failure, the depth of plastic neutral axis exceeds the maximum depth of plastic neutral axis, which may not satisfy the ductility requirement (see Fig. 10).

Fig. 8 Comparison of safety factors $M_{dl1}/M_{u}$ according to the change in the design compressive strength of concrete

Fig. 9 Comparison of the prestressing steel stress $f_{psl1}/\phi_{s}f_{py}$ at flexural failure according to the change in the design compressive strength of concrete

Fig. 10 Comparison of the depth of plastic neutral axis depth $c_{l1}/c_{\max}$ according to the change in the design compressive strength of concrete

3.4 Additional parametric analysis related to the depth of plastic neutral axis

In some cases, the depth of the plastic neutral axis of the PSC composite girder with a span of 50 m did not satisfy the design specification of $c_{\max}$. The additional parametric analysis is performed to find design conditions that can satisfy the design criteria.

First, Fig. 11 shows the parametric analysis results of the depth of the plastic neutral axis according to the design compressive strength of the concrete bridge deck under compression and the change in the longitudinal reinforcement ratio.

The depth of the plastic neutral axis can be seen in the figure when the design compressive strength of the concrete bridge deck is 27 MPa, 30 MPa, and 35 MPa, and the longitudinal reinforcement ratio is changed from 0.6 % to 1.2 %. In the figure, it can be seen that it is difficult to satisfy the ductility regulation for the maximum depth of the plastic neutral axis in the case of a span length of 50 m by increasing the design compressive strength and longitudinal reinforcement ratio of the concrete bridge deck under compression.

As a second method, additional parameter analysis is conducted to reduce the depth of the plastic neutral axis by reducing the amount of prestressing steel used. If initial jacking force is introduced by reducing the amount of prestressing steel used, it can be predicted that the tensile stress developed on the bottom surface of the PSC girder will increase when the service load is applied. Table 3 and Fig. 12 shows the change in the depth of the plastic neutral axis when the stress developed at the bottom surface of the PSC girder is changed to the allowable tensile stress level. In the figure, it can be seen that when the tensile stress is changed to the allowable tensile stress level (see Eq. ), the depth of the plastic neutral axis is rapidly reduced, and the design regulation can be satisfied.

Fig. 11 Changes in the depth of plastic neutral axis according to the design compressive strength of the concrete bridge deck and the change in the longitudinal reinforcement ratio

Fig. 12 Depth change of plastic neutral axis according to tensile stress change of the bottom surface of PSC girder under service load