Analysis model of a double broken line prestressed composite box girder with corrugated steel webs is shown in Fig. (1).

Fig. (1). Analysis model.

where e is the eccentricity distance from neutral axis; a is the horizontal distance between steering block and anchor point; θ is the steering angle of the tendon; y is the vibration displacement; P_t ⁰ is the initial prestressing force; P_tc ⁰ is the horizontal component of initial prestressing force; P_ts ⁰ is the vertical component of initial prestressing force; ΔP_t is the increment of prestressing force due to flexural vibration; l is the span length; q_v is the distributed mass; x is the horizontal distance between calculation section and anchor point; E is the elastic modulus of concrete; I is the moment of inertia of composite girder, A is the cross-sectional area. The prestressing force P_t is expressed as:

(6)

The horizontal component of prestressing force P_tc is represented as:

(7)

The vertical component of prestressing force P_ts is represented as:

(8)

The bending moment due to prestressing force M_t is represented as:

(9)

The following basic assumptions are adopted in the calculation:

1. As the effect of shear slip between the concrete slab and corrugated steel web is negligible, it is assumed that the cross section of the composite girder is uniform [27].
2. The vibration displacement of the composite girder is minimal, so the curvature of the girder can be rationally expressed as based on the small deformation theory.
3. Due to the “folding effect” in longitudinal direction of the corrugate steel webs, the longitudinal stress and strain in the corrugated steel webs is approximate to 0, so a quasi-plane assumption [28] (given the corrugated steel webs were removed away and the concrete roof and floor were linked up to establish a virtual plane) is adopted for strain distribution across the cross-section of the girder.
4. Axial deformation of the beam is ignored; the compressive stress and the corresponding strain are negligible.

The vibration differential equation of the box girder with corrugated steel webs subjected to external prestressing force is expressed as follows:

(10)

where is the mass of composite girder per unit length.

Substituting Eq.7-Eq.9 into Eq.10, the following equation can be obtained:

(11)

where P_tc ⁰ and P_ts ⁰ are independent of x. Since the maximum value of vibration displacement y_max e and therefore ΔP_tc ∙ y ΔP_tc ∙ e, the value of ΔP_tc ∙ y is negligibly small. Hence, Eq.11 can be rewritten as:

(12)

As the increment of prestressing force due to flexural vibration, ΔP_t is proportional to the maximum value of y. Therefore, if it is assumed that the vibrational displacement is infinitesimal, ΔP_t can be calculated from diagram multiplication:

(13)

where E_s is the elastic modulus of external tendons; A_s is the cross-sectional area of external tendons; and l_s is the length of the tendon; λ is given as:

(14)

In order to calculate the increment of prestressing force due to flexural vibration, a concentrated load was exerted at midspan of the model girder herein. The influence of shear deformation on the deflection of the box girder with corrugated steel webs was considered.

J. Cafolla et al. reported a decrease in the shear modulus of corrugated steel webs under shear force and established the formula for equivalent shear stiffness of box girder with corrugated steel webs [29]:

(15)

where G is shear elastic modulus of steel; b, d and α are illustrated in Fig. (2).

Fig. (2). Geometric size of corrugated steel webs.

Then the midspan deflection y_p (the maximum value of y) due to the applied load P can be calculated from the equation in [30]:

(16)

where y₁ is the mid-span deflection produced by shear; y₂ is the mid-span deflection produced by bending moment; A_G is the shear area of corrugated steel webs. K is the stress non-uniform coefficient. Based on the mechanical characteristics of the box girder with corrugated steel webs, the shear force is totally born by the thin steel webs, so K is fixed as 1.

Given , Eq.16 can be expressed as:

(17)

Substituting Eq.13 into Eq.17:

(18)

The upward movement of the girder yΔp_t due to ΔP_t can be calculated as:

(19)

Based on the relationship y = y_p - y_ΔPt , Eq.18 and Eq.19, y can be calculated as:

(20)

Therefore ΔP_t can be expressed as the function of y according to Eq.20:

(21)

where μ is given as:

(22)

Substituting Eq.21 into Eq.12:

(23)

where ν is given as:

(24)

Eq. 23 is the equation for undamped free vibration of the girder, which can be solved by separation variable method.

(25-1)

(25-2)

where f_n is the natural frequency of the girder(Hz); n is the order of natural frequency, if n=1, f₁ is fundamental frequency; ζ is the correction coefficient.

in Eq.25-1 is the equation based on general beam theory [31], manifesting that the natural frequency of simply supported beam with uniform rectangular cross-section only relates to the mass of the girder per unit length, span length and cross-section stiffness. ζ is a correction coefficient taking account of the characteristics of box girder with corrugated steel webs, the shear deformation of composite girder and the influence of both prestress force and arrangement of external tendons on section stiffness.

It is well established that the corrugated steel webs of composite box-girder can deform freely in longitudinal direction, which is called “folding effect”, therefore its axial and bending stiffness are negligible. Considering this effect, the cross-section stiffness of the composite girder can be simplified as the stiffness of concrete roofs and floors only, as expressed in the following:

(26-1)

(26-2)

where E_c is the elastic modulus of concrete; A_c is the cross-sectional area of concrete in roof and floor, I_c is the moment of inertia of the concrete components of cross-section.

In addition, during the tensioning process, the cross section remains uniform and no cracking occurs if the tensioning control stress is not large (P_tc≤0.3f_pk, where f_pk is the tensile strength of the external tendons).

With a series of concrete compressive strength, the papers of Saiidi [6], Liu [12], Xie [13], Zhang [14] and Xiong [15] show a remarkable difference in the change rate of effective stiffness. The parameter η was defined as the effective stiffness ratio which accounts for the influence of diverse axial forces on the section stiffness. The lower the concrete compressive strength is, the faster η changes. For the beam with the compressive strength of concrete was 50.09MPa in Xie’s paper, η ranged from 1 to 1.038 with N/f_cA varied from 0 to 0.0599, which indicates the slightest variation in η. Whereas in Saiidi’s paper, the 28-day compressive strength of concrete was 20.3 MPa, and η ranged from 1 to 2.352 with N/f_cA varied from 0 to 0.501, manifesting the most significant impact on cross-section stiffness. With the test results of all five papers integrated, it is reasonable to propose that the rate of η change correlates with prestressing force as well as the compressive strength of concrete. So the regression parameter N/f_c A (axial compression ratio) was selected to develop a regression equation for effective stiffness, which can be written in the form:

(27-1)

(27-2)

The related coefficient is 0.989, which means that the equation fits well with the data. The regression curve of effective stiffness ratio is shown in Fig. (3).

Fig. (3). Effective stiffness ratio versus axial compression ratio.

Substituting Eq.27 into Eq.25:

(28-1)

(28-2)

(28-3)

When external tendons are arranged in i different configurations, Eq.28 is extended as:

(29-1)

(29-2)

(29-3)

Compared with general beam theory, the shear deformation of box girder with corrugated steel webs, “folding effect” in longitudinal direction of the corrugate steel webs, the effect of the existing prestressing force on stiffness and arrangement of external tendons are all considered in this paper.

Fig. (4) shows San Dao-he Bridge in Qinghai Province, which is a simply supported concrete box girder bridge with corrugated steel webs and effective span length is 49.92m. Fig. (5) presents the cross section of San Dao-he Bridge. The strength grade of concrete slabs is C50, whose mechanical properties can be referred to the Code for Design of Concrete Structures [32]. Q345 steel was used as the corrugated steel webs with the thickness of 12mm and its properties were in accordance with the standard of Carbon Structural Steels [33]. The lengths of horizontal and inclined panels are 430mm and 370mm, respectively, and the wave height is 220mm.

Fig. (4). Photograph of San Dao-he Bridge.

Fig. (5). Cross section of San Dao-he Bridge (mm).

There are 14 bundles of external tendons; each of them consists of 13 strips of steel strand with 1395MPa tensioning control stress. The layout of external tendons is double broken line. External tendons are turned at the inner diaphragms which are closest to side diaphragms and anchored at side diaphragms. Detail information for external tendons is listed in Table 1.

The San Dao-he Bridge was subjected to ambient vibration test to attain its fundamental frequency. Without traffic load on the bridge deck and regular vibration source near the bridge site, the microseism of the bridge excited by stochastic loads, e.g. wind load, micro-tremor and flow, was catch by the data acquisition instrument INV306U-5160 made by China Orient Institute of Noise & Vibration. Eleven tested cross sections symmetrical to the mid-span were arranged on the bridge, and there were two measuring points for each cross section, as presents in Fig. (6).

Fig. (6). San Dao-he Bridge and measuring points arrangement (cm).

DASP (Data Acquisition & Signal Processing) was employed as the analysis system. The theoretical results and the measured values of the fundamental frequency of the bridge are ​compared in Table 2.

The measured natural frequency is the biggest of all and the relative errors are -10.9%,-12.7% and -5.6% for (2), (3) and (4) cases, respectively. The result calculated by the theory in this paper is nearest to measured result and the fundamental frequency increases with external prestressing taken into account.

In order to further explore the influence law of prestressing force on the natural frequency, the dynamic characteristic test on a simply supported prestressed composite girder with seven prestressing levels was conducted. Since the fundamental frequency of girder is noticeably affected by the external prestressing while there is no regularity for the high-order frequencies to follow [17], the fundamental frequency of model girder was measured merely in this research.

Fig. (7) shows the model box girder, which is a simply supported concrete box girder with corrugated steel webs and the span length is 4400mm. Fig. (8) presents the cross section of the model hybrid girder. The compressive strength of concrete slabs is 22.4 MPa, whose elastic modulus is 27000MPa. Q345 steel was used as the corrugated steel webs with the thickness of 4mm and its properties were in accordance with the standard of Carbon Structural Steels [33]. The lengths of horizontal and inclined panels are 150mm and 125mm, respectively, and the wave height is 60mm.

Fig. (7). Photograph of the model box girder.

Fig. (8). Cross section of model box girder (mm).

There are two bundles of external tendons; each of them consists of 3φ^j15.2 steel strands with 1860 MPa tensioning control stress. The sectional area of each φ^j15.2 steel strand is 139mm².The layout of external tendons is double broken line. External tendons are turned at the inner diaphragms which are closest to side diaphragms and anchored at side diaphragms. Detail information for external tendons is listed in Table 3.

The fundamental frequency was measured by impact hammer test. In general, sensors are located at the girder where deformation is relatively larger, and the nodes of mode shape should be avoided lest missing some important modal information. It is well established that the mid-span deflection is maximum for simply supported girder in the first mode of vibration, so an accelerometers was deployed at the mid-span section. WS-5924 produced by Beijing Wave spectrum & Science & Technology Co., Ltd. was employed as the acceleration data collection system, as shown in Fig. (9).

Fig. (9). Signal amplification and data collection instrument.

There were five hammering points arranged at the model girder, as illustrated in Fig. (10). Two rounds of hammering were conducted for each prestressing level. With the time-domain curves of hammering force signal and acceleration signal recorded, as shown in Figs. (11 and 12), the fundamental frequencies of the model girder with seven prestressing levels can be obtained by modal analysis software.

Fig. (10). Hammering points arrangement (mm).

Fig. (11). Time-domain curve of force signal.

Fig. (12). Time-domain curve of acceleration signal.

The measured and calculated results are listed in Table 4. Fig. (13) shows the comparison between calculated and measured results of the model box girder.

Fig. (13). Comparison between theoretical and experimental results.

With the desired prestress force applied and the prestressed steel strands anchored properly, dynamic tests were carried out on the box girder. The data (2) in Table 4 clearly indicate that the measured fundamental frequency increases nonlinearly along with the growing prestressing force, and the maximum increase is 2.35%. The results (3) calculated by general beam theory do not change, for the natural frequency only concerns with E and I, and has nothing to do with the prestressing force outside. The calculated results (7) show an increase of nearly 3.68% in the fundamental frequency as the prestressing force changed from 0 to 140kN, which is slightly larger than the maximum increase in the test. This phenomenon may be explained by the fact that the induced compressive stress in concrete dramatically affects the safety and function of prestressed composite structures, where long term effects, i.e. shrinkage and creep in concrete and relaxation of prestressing external tendons, lead to a decrease of the stress in concrete over time [1, 2], which has a softening effect on the cross-section stiffness of the composite girder based on Eq.27. Consequently, according to Eq.29, the measured fundamental frequency of the girder decreases to some degree.

In contrast, the calculated results (5) show a 0.1% reduction in the natural frequency, which is opposite to the tendency of measured results. The results calculated by the formula in this paper show the same tendency with the measured results, and the maximum relative error for (7) is 1%. This is because that the effect of the existing prestressing force on stiffness, the shear deformation of the box girder, “folding effect” in longitudinal direction of the corrugate steel webs and the arrangement of external tendons are all considered. Therefore, the proposed formula can provide a reference for the dynamic design of prestressed structures. However, due to different test programs and measurement systems, whether the results can give guidance to other concrete structures accurately should be studied further; besides, the long term effects of the concrete components of the cross-section on the loss of prestress forces need further investigations.