Keywords

1 Introduction

Theoretical equations mathematically developed through summarized empirical studies or finite element simulations provide a convenient and simplified approach to estimate the complex behavior of highway bridges [1]. However, for bridges with parameters not considered in design and with atypical features, the estimations obtained using these equations can produce inaccurate results. For example, the AASHTO specification live load distribution factor equations neglect the influence of parapets which therefore reduce the predicted stresses induced in exterior and interior girders. In one study this simplification produced a 36% and 13% underestimation for exterior and interior girders, respectively [2]. Several other factors also affect the accuracy of analytical methods. Friction at bridge bearings due to accumulation of debris has been shown to increase the load rating capacity estimated in the AASHTO specification by 4% for a straight steel girder bridge [3]. The actual normal stress impact factor for spans larger than 49.5 m (162 ft) in a curved steel box-girder bridge can also be significantly smaller (approximately 60%) than the values predicted in the AASHTO specification [4].

A particularly important feature that is not commonly addressed in equations is the effect of bridge skew. Bridge skew occurs when the direction of the bridge span is not orthogonal to the bridge supports. The degree of bridge skew is generally dictated by space limitations and constraints from anthropogenic or natural obstacles that do not allow the bridge to be built straight [5]. In skewed bridges, the load path is oriented toward the corners of the bridge span with an angle greater than 90°, unlike the straight bridges whose orientation is toward the direction of the span [6]. As a consequence, compared to an equivalent straight bridge a skewed bridge generates larger shear forces at exterior girders [7], higher reactions at corners with an angle greater than 90° [8], and larger transverse moments [9]. As a result, the behavior of a skewed bridge differs significantly from a straight bridge. In particular, the effect of bridge skew on the bridge load rating and on the natural frequencies of the bridge can be significant. However, prior to this study these aspects have not been examined jointly in a single experimental study.

This study investigates the effect of skew by examining the responses of a four-span skewed highway bridge. Analytical load ratings at the critical moment locations were calculated using a line-girder model and the AASHTO Standard Specifications for Highway Bridges [10], and the experimental response was obtained through a field test. In the field test, after instrumenting the critical moment locations for every girder with strain gages, a truck with a calibrated weight was driven at crawl speed across the entire bridge at distinct transverse locations in a series of truck runs. Critical responses were used to complete the experimental load rating. The analytical natural frequency was estimated based on a simple span bridge using the Ontario Highway Bridge Design Code (OHBDC) [11] and then corrected for a continuous span bridge based on the recommendations given in Barth and Wu [12]. The experimental natural frequency was determined based on responses due to the calibrated truck driven at high speed. A Fast Fourier Transform (FFT) was used to transform to the portion of the response in free vibration condition (time domain response) into a frequency domain plot where the first spike was determined to be the experimental natural frequency. Although the study focuses on skewed bridges with steel girders, the methodology could also be applied to straight bridges and to bridges made of different materials (reinforced concrete, prestressed concrete, wood, etc.) and cross-sections (box-section, plate girder section, rectangular section, and so forth).

2 Bridge Description

The four-span skewed highway bridge examined in this study (Fig. 1) is located on the eastbound direction of Interstate 80 over the Bear River in Evanston, Wyoming, USA. The bridge is symmetric with continuous steel girders supporting a nominally 191 mm-thick (7.5 in.-thick) reinforced concrete deck. The total length of the bridge is 124.4 m (408 ft), with outer spans of 25.6 m (84 ft) each and inner spans of 36.6 m (120 ft) each. The outer eastern pier is supported by a pin bearing. Rocker bearings are used at the remaining four supports. The bridge has an angle of skew of 47° relative to the abutments. The bridge has a width of approximately 12.2 m (40 ft) and has a shoulder on both ends. The bridge was originally built with four nominally identical I-shaped steel girders that were designed to act non-compositely with the deck. The bridge was later widened and a fifth steel girder was added. The fifth girder was designed to act compositely with the deck. The bridge cross-section is shown in Fig. 2. The girders are haunched at the pier locations. The height of the girder web increases parabolically and the web and flange are thicker at the pier locations. The dimensions of the girders at selected cross-sectional locations are described in Lu et al. [13]. The minimum specified yield strength of the structural steel is 250 MPa (36 ksi). The steel girders were relatively free of corrosion. The specified minimum concrete compressive strengths of the original and added concrete deck are 22.4 MPa (3.25 ksi) and 27.6 MPa (4 ksi), respectively.

Fig. 1.
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Bridge examined in this study.

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Fig. 2.
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Cross-section.

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3 Load Rating

The load rating capacity of the bridge was obtained based on the rating factor, RF, of each individual girder. In the AASHTO rating method, the girder with the lowest RF controls the load rating of the bridge [14]. This rating method was applied. Among the different limits used to load rate bridges, the strength inventory rating factor is generally the most critical, and therefore this limit was adopted here.

Since the bridge was originally designed and evaluated using the Load Factor Design (LFD) method and Load Factor Rating (LFR), respectively, the calculation of RF was based on the LFD method. Although the HS20 design truck in the LFD method overestimates the bridge capacity compared to the HL-93 design truck in the Load and Resistance Factor Design (LRFD) method [15], the overestimation cancels out when taking the ratio of the analytical and experimental load ratings.

In this study, RF was defined as the ratio between the reserve capacity for live load and the limit state design live load and expressed as the number of standardized HS20 design truck loads:

$$RF=\frac{{R}_{n}-1.3D}{2.17LL\left(1+I\right)}$$
(1)

where Rn is the capacity of the member, D is the dead load effect on the member, LL is the live load effect of the member, and I is the dynamic impact factor. The value of Rn was calculated as the product of the specified yield strength of the structural steel and the elastic section modulus of the girder. The value of D was calculated as the summation of the contributions due to self-weight of the girder, self-weight of the deck and diaphragms, self-weight of the future wearing surface, distributed to a single girder. When RF is greater than 1.0, the bridge is regarded as structurally adequate for the design load. On the other hand, when RF is less than 1.0, the bridge is inadequate for the design load.

3.1 Analytical Load Rating

The analytical live load, LLA, accounting for dynamic impact was calculated as follows:

$${LL}_{A}\left(1+{I}_{A}\right)={M}_{HS20}{DF}_{A}{m}_{A}\left(1+{I}_{A}\right)$$
(2)

where MHS20 is the maximum analytical moment due to the HS20 design truck at the location of interest, DFA is the analytical live load distribution factor, mA is the analytical live load multi-presence factor, and IA is the analytical impact factor. DFA is calculated using the AASHTO Standard Specifications for Highway Bridges in section 3.23.2.3.1.5. The AASHTO equations provide the lateral distribution for only a single wheel-line. Since the design truck has two wheel-lines, the DFA needs to be divided by two. The value of IA was calculated using the AASHTO Standard Specifications for Highway Bridges equation in section 3.8.2.1.

It was not feasible to instrument the inner span of the bridge. Therefore, Eq. (1) and Eq. (2) were applied to the critical moment locations of the outer span. At the outer span, the critical positive longitudinal moment was located at 0.4 times the span length from the abutment, and the critical negative longitudinal moment was located at the pier between inner and outer spans. The corresponding values of MHS20 were determined using a one-dimensional line-girder bridge model, where the bridge was modeled as if it consisted of only a single girder. The single girder was modeled by attributing the value of the nominal non-composite moment of inertia without the concrete deck. This approach is consistent with values of Rn and D calculated for a single girder. To produce the maximum positive moment, MHS20 was modeled using three concentrated loads. To produce the maximum negative moment, MHS20 was replaced by the uniform-plus-two-concentrated-load configuration in AASHTO specification section 3.7.6. As defined in AASHTO specification section 3.12.1, the live load multi-presence factor accounts for the probability of coincident maximum loading. The multi-presence factor is 1.0 for single or 2-truck loadings, and 0.9 for 3-truck loadings. Since one truck was placed on the model, mA was equal to 1.0.

The values of the variables in Eq. (1) and Eq. (2) for the positive and negative critical moment locations are given in Table 1. It was determined that LLA(1 + IA) was equal to 1320 kN-m (972 k-ft) and −1820 kN-m (−1340 k-ft) at the positive and negative moment, respectively. The strength inventory RF values are equal to 0.98 and 0.91 at the positive and negative moment, respectively.

Table 1. Values of the variables for Eq. (1) and Eq. (2).
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3.2 Experimental Load Rating

The experimental live load accounting for dynamic impact is calculated as follows:

$${LL}_{E}\left(1+{I}_{E}\right)=\left(\frac{{M}_{HS20}}{{M}_{TRK\_OS}}\right){M}_{girder}\left(\frac{{M}_{TRK}}{{M}_{TRK\_OS}}\right){m}_{E}(1+{I}_{E})$$
(3)

where Mgirder is the non-composite moment in the girder, MTRK is the analytical maximum moment due to the calibrated truck used in the field test, MTRK_OS is the analytical maximum moment due to the calibrated truck used in the field test when the truck is at the outer span, mE is the live load multi-presence factor according to the loading conditions of the actual experiment, and IE is the experimental impact factor based on the field test.

The line-girder model used to obtain MHS20 was also used to determine the values of MTRK and MTRK_OS. The axle loads and axle spacings of the calibrated truck (Fig. 3) were measured and converted into three concentrated loads that were then applied to the model. MTRK_OS was determined because the bridge was only instrumented on the outer span. For the positive moment, MTRK and MTRK_OS were the same value, equal to 890 kN-m (657 k-ft), because they occurred at the outer span. For the negative moment, MTRK (at the inner span) was equal to −762 kN-m (-554 k-ft), and MTRK_OS was equal to −570 kN-m (−420 k-ft).

The values of Mgirder were obtained through a field test. In the field test, strain gages were installed at the most critical moment locations along the bridge. At the negative moment, the strain gages were installed with a 2.44-m (8-ft) offset from the support to minimize the influence of the support on the response. A larger offset was not used to reduce errors due to extrapolation [16]. Two strain gages were installed on each girder at each location. One strain gage was mounted on the bottom of the bottom flange. The other strain gage was mounted on the web 102 mm (4 in.) below the bottom of the top flange.

The calibrated truck was driven at crawl speed over the bridge in successive runs that traverse the width of the roadway according to the requirements given in the AASHTO Manual for Bridge Evaluation [17]. Figure 4 shows the run sequence and truck position. A total of fifteen runs were conducted to traverse the entire roadway, and the runs were conducted from left to right relative to the direction of travel. In this paper, the girders are numbered 1 to 5. (Girder 5 is the composite girder that was added when the bridge was widened.) In the first run (Run 1), the center of the left front wheel was positioned 0.91 m (3 ft) from the left curb. The position of each successive run was 0.61 m (2 ft) offset to the right of the previous run. The configuration of the run sequence is nearly symmetric relative to Girder 3. A total of fifteen runs were conducted.

Since the truck load was calibrated to produce a response that was within the linear elastic range, the responses for side-by-side truck loading combinations were determined by superimposing the measured results. The controlling combination was defined as the loading that produced the greatest LLE(1 + IE) at the positive or negative moment location. For this bridge, the three side-by-side truck loading combination involving Runs 3, 8 and 13 controlled the positive moment, and the two side-by-side truck loading combination involving 1 and 6 superimposed controlled the negative moment. Mgirder was equal to 317 kN-m (234 k-ft) and occurred in Girder 3 at the positive moment. Mgirder was equal to −320 kN-m (-236 k-ft) and occurred in Girder 2 at the negative moment. Field tests were not conducted to determine the actual dynamic impact factor; therefore, it was assumed that IE was equal to IA. As a result, LLE(1 + IE) was equal to 515 kN-m (380 k-ft) and −1640 kN-m (−1210 k-ft) at the positive and negative moment locations, respectively.

An inspection to determine the in-situ dead loads was not practical. Therefore, the experimental values of Rn and D were taken equal to the analytical values in Table 1. The resulting experimental strength inventory RF is equal to 2.51 and 1.01 for positive and negative moment, respectively. Based on the HS20 standardized design truck, the bridge was rated for a HS50.2 truck (20 times 2.51) at the positive moment location and HS20.2 (20 times 1.01) at the negative moment location.

Fig. 3.
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Calibrated truck.

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Fig. 4.
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Run sequence and truck positions.

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3.3 Comparison of Analytical and Experimental Load Ratings

The ratio of the experimental load rating to the analytical load rating was used to determine the effect of additional contributors to the bridge actual capacity, including the effect of bridge skew. The ratio of the experimental to the analytical load ratings is equal to 2.56 at the positive moment and equal to 1.11 at the negative moment. The ratio of load ratings can also be taken as the inverse of the ratio of the live load effects (LLA/LLE) since Rn and D were taken as constants. This conclusion was expected because the design load rating is generally conservative. It does not consider additional contributors to the bridge capacity (such as the additional stiffness due to curbs and railings, actual longitudinal and lateral distribution factor, unintended composite action, and in special, the effect of the bridge skew). The additional contributions discretized in a skewed bridge are discussed in detail in Lu et al. [13].

4 Natural Frequency

The AASHTO specification limits deflections (e.g., L/800 to L/1000, where L is the bridge span length) but it does not use natural frequency as a direct measure of serviceability. In contrast, the OHBDC recommends the calculation of natural frequency to control vibration. In the OHBDC approach, a relationship is established between the natural frequency, the live-load deflection, and the pedestrian usage of the bridge. The natural frequency of a simple span bridge (first bending frequency), fb, is

$${f}_{b}=\frac{\pi }{2{L}^{2}}\sqrt{\frac{E{I}_{b}g}{w}}$$
(4)

where E is the modulus of elasticity of the girder, Ib is the flexural composite moment of inertia of the girder (for levels of serviceability, slippage is not expected to occur even if the bridge is designed to act non-compositely), and w is the weight per unit length of the girder. Equation (4) has been extended for continuous span bridges by Barth and Wu [12] based on the results of parametric studies and regression analysis. In their approach, the value of fb is corrected (multiplied) by the “natural frequency coefficient” λ2, calculated as follows:

$${\lambda }^{2}=a\frac{{I}_{b}^{c}}{{L}_{max}^{b}}$$
(5)

where Lmax is the maximum bridge span length. In Eq. (5), when SI units are used for Lmax (m) and Ib (m4), a is equal to 1.49, b is equal to −0.033, and c is equal to 0.033 for bridges with three or more spans.

4.1 Analytical Natural Frequency

Equation (4) was applied to the bridge, and the resulting values are given in Table 2. In this study, L was equal to 36.6 m (120 ft), and Ib was calculated based on a weighted average of each girder and each span relative to its corresponding length. For the haunched portions of the bridge, it was opted to simplify the calculation so that the averages of the greatest and smallest moments of inertia were taken. Here w was determined as a summation of the self-weight of the girder (a weighted average was determined for the computation of this component), self-weight of the deck and diaphragms, composite dead load which consists of the weight of the curbs and railings, and the weight of the future wearing surface, all distributed to a single girder. The resultant value of fb is equal to 2.05 Hz. Applying Eq. (5), the value of λ2 is equal to 1.49. Thus, the analytical corrected natural frequency for the bridge is equal to 3.07 Hz.

Table 2. Variables for natural frequency per Eq. (4).
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4.2 Experimental Natural Frequency

After the field test, the natural frequency of the bridge was determined experimentally by measuring the acceleration caused by the calibrated truck crossing the bridge at high speed while the bridge was closed to ambient traffic. The truck velocity was set to the speed limit for the bridge, 121 km/h (75 mph). An accelerometer with a range between −2g and +2g was mounted on the top of the bottom flange of Girder 2. Three tests were conducted. The acceleration data was collected using 1000 samples per second. A representative acceleration history is shown in Fig. 5. Small and regular vibrations before the truck enters the bridge are evident in the history. At approximately 6 s, the accelerations increase, reaching a peak of 0.15 g at approximately 14 s (at this instant, the truck is immediately above the accelerometer). As the truck leaves the bridge, a steady decline of the vibration is observed, and the bridge enters the free vibration condition at approximately 16 s. The bridge was reopened for ambient traffic after each test. Therefore, small intermittent increases can be observed starting at approximately 22 s. As a result, the free vibration condition of the bridge between approximately 16 and 22 s was used to determine the natural frequency.

A Fast Fourier Transform (FFT) was conducted in which the acceleration history (time domain plot) of the free vibration condition (approximately 6 s) was transformed into a frequency domain plot (Fig. 6) and used to determine the principal frequencies of the bridge. The data for the free vibration condition was condensed to 4096 (212) data points. The frequency domain plot displays the magnitude of the Fourier Coefficient in function of frequency with noticeable spikes. Although the coefficients do not have any physical meaning for the method, the frequencies corresponding to the respective spikes are values of the different modes of vibration. The first spike represents the natural frequency, the second the second harmonic frequency, the third the third harmonic frequency, and so forth. For this bridge, the experimental natural frequency was equal to 3.42 Hz.

Fig. 5.
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Representative acceleration history.

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Fig. 6.
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Frequency domain plot.

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4.3 Comparison of Analytical and Experimental Natural Frequencies

Comparing the natural frequencies obtained analytically and experimentally, it was observed that the analytical natural frequency underestimated the experimental natural frequency by 11%. The increase of natural frequency with the bridge skew agrees with prior studies [18, 19]. The difference is partially due to the estimation of Ib in the calculation of the analytical natural frequency, but the bridge skew effect is thought to be the main cause. In a skewed bridge, if the span length is significantly greater than the cross-sectional width, flexural behavior tends to govern. The load path tends to be oriented toward the obtuse corners of the bridge, i.e., the “shortest path” across the span. Although the actual load path could vary between the “shortest path” and the longitudinal distance between abutments (Fig. 7), it is significantly shorter than the bridge longitudinal span length of an equivalent straight bridge. This observation has considerable impact on the variables of L and Lmax of Eq. (4) and Eq. (5), respectively. In particular, the application of the correction factor λ2 for the estimating natural frequencies of skewed bridges may exceed the valid range for Eq. (5).

Fig. 7.
figure 7

Shortest path, bridge span length, and actual load path.

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5 Conclusions and Future Recommendations

The analytical and experimental load ratings and natural frequencies were determined for a four-span skewed highway bridge. The analytical load rating was calculated using a line-girder model and the AASHTO Standard Specifications for Highway Bridges. The experimental load rating was determined using a field test in which a calibrated-weight truck was driven at crawl speed across the bridge at a range of transverse locations. The results indicated that the analytical load rating was equal to 0.98 and 0.91 at the positive and negative moment, respectively.

The experimental load rating was equal to 2.51 for the positive moment and 1.01 for the negative moment, which were 2.56 and 1.11 times higher than the corresponding positive and negative analytical load ratings. The increase observed was attributed to effects not considered in analytical calculation, namely additional stiffness due to curbs and railings, actual longitudinal and lateral distribution factor, unintended composite action, and, in particular, the effect of bridge skew.

The analytical natural frequency was determined using the OHBDC and an adjustment to account for a continuous span. The experimental natural frequency was determined using a FFT analysis of the free vibration response for truck crossing the bridge at high speed. The results indicate that the analytical natural frequency underestimated the experimental by 11%. The underestimation was attributed to the length of the load path, which is inversely proportional to the natural frequency. In a skewed bridge, the load path length is shorter than in an equivalent straight bridge.

The findings in this study suggest that accuracy of equations for estimating the load rating and the natural frequency of a highway bridge depend on several aspects, including the effect of bridge skew. As a result, additional research is needed to adjust the equations to better estimate the response. It is recommended that the equations account for the main contributors to the bridge load rating, including the effect of skew. Parametric studies are also needed to expand the applicability of the equations for estimating natural frequency to account for the effective length of the load path.