Dynamic characteristic analysis and fatigue life estimation of steel-structure bridges under vehicle loading

3.1. Layout of strain gauges

Two strain gauges were arranged at the holes on one side of wind guide hole of No. 12 U-shaped rib of an upstream diaphragm plate, where the strain gauge closer to the hole edge was numbered as No. 1, while the farther strain gauge was numbered as No. 2. The position was selected because cracks appeared at corresponding positions of both the adjacent diaphragm plates in the bridge axis direction, indicating that the position bore large loads. However, the position of the selected diaphragm plate had no crack, so that the measurement was not disturbed by cracks. A strain gauge No. 3 was arranged at struts, as shown in Fig. 5. The photo of the steel-structure bridge treated by adhesion of strain gauges is shown in Fig. 6. Data collection was carried out for twice at vehicle running peaks. Collection periods were respectively 5000 s and 2000 s. Finally, stress history of the 3 strain gauges during the two periods was obtained. No. 1 strain gauge was closer to the hole edge than No. 2 strain gauge, so that its stress concentration was more obvious. Sequence of stress values: No. 1 strain gauge > No. 2 strain gauge > No. 3 strain gauge.

Fig. 5Layout of testing points of strain gauges

Fig. 6On-site picture after adhesion of strain gauges

3.2. Computation of vehicle load spectrum and dynamic stress

During measurement of stress history of the horizontal diaphragm plates, a video recorder was set on the bridge deck to record vehicle travelling courses on the first lane in real time. Recording and strain gauge measurement were carried out at the same time. Each vehicle passing the lane was corresponding to the stress caused by it. A negative stress value obtained in the measurement indicates that the strain gauge measurement position was pressed under effects of vehicle loads. Under the pressure, defects inside steel materials could not be spread easily, and fatigue damage would not take place. However, in the actual structure, a lot of fatigue cracks were found at this position because the horizontal diaphragm plate had the initial residual stress, while the initial residual stress at the hole edge was relatively large. The position was pressed under vehicle loads, but after overlaying of it and the initial tension stress, the position would suffer from the cyclic tension stress, and fatigue cracks would happen easily. According to stress history, effects of front and rear vehicle wheels could be distinguished. In general, front and rear wheels of a heavy vehicle are equipped with multiple axles. However, the vehicle axles are close to each other, so that only two big waveforms could be observed in the stress history when front and rear wheels pass. When the wheels pass right above the detected horizontal diaphragm plate, the stress history curve reached the peak, while peak value corresponding to rear wheels exceeded that of front wheels. During the detection, we found that only vehicles on the first lane could cause stress responses. Obviously, the orthotropic steel bridge deck was influenced by vehicle loads only at some local positions.

Table 1Simplified load spectrums of five types of vehicles

No	Quantity of axle	Axle weight (kN)	Wheelbase (m)	Total weight (kN)	Diagram
A	2	15	2.8	30
		15
B1	4	40	2.0 4.5 1.4	420
		60
		160
		160
B2	3	105	3.6 1.2	525
		210
		210
C1	6	30	2.9 1.4 6.9 1.4 1.4	340
		50
		50
		70
		70
		70
C2	6	50	2.9 1.4 6.9 1.4 1.4	865
		90
		125
		200
		200
		200

Vehicles with high occurrence frequencies in the experiment were selected for classification and statistics. Results show that the passing vehicles are mainly classified into 5 types: small car A, light truck B1, heavy truck B2, light platform trailer C1 and heavy platform trailer C2. According to stress values measured by the strain gauges as well as front-rear stress ratios of various types of vehicles and in combination with relevant technical specifications, the simplified vehicle load spectrums could be obtained, as shown in Table 1.

Under loads of moving vehicles, dynamic effects such as vibration and impact may be generated in a bridge structure. In bridge design specifications of different countries, the impact coefficient is used to measure the effect. The impact coefficient is influenced by conditions such as dynamic characteristics of the bridge structure, vehicle self-vibration characteristics, vehicle flow rules and deck smoothness. Impact coefficient of a bridge structure is a function of structural base frequency. As stipulated in American AASHOLRFD, the impact coefficient is the function of bridge span. Impact coefficient of 0.3 was selected in this paper. Multiplied by the coefficient of 1.3, the vehicle load spectrum was loaded into the finite element model. Stress history of units corresponding to the three strain gauge positions was computed. Stress amplitudes under each vehicle type were extracted. Results are shown in Table 2. It is shown in Table 2 that the stress amplitude: No. 1 strain gauge > No. 2 strain gauge > No. 3 strain gauge. Vehicles A and B2 were only corresponding to one stress amplitude, while other vehicle types were corresponding to two stress amplitudes. Stress history of the unit corresponding to No. 2 strain gauge position is shown in Fig. 7.

Fig. 7Stress history of No. 2 strain gauge under vehicle load spectrums

a) Stress history of Type A vehicle

b) Stress history of Type B1 vehicle

c) Stress history of Type B2 vehicle

d) Stress history of Type C1 vehicle

e) Stress history of Type C2 vehicle

4.1. Stress distribution of steel-structure bridge sub-model

In the experiment, the stress history at the horizontal diaphragm plate hole corresponding to No. 12 U-shaped rib was measured. In fact, this position does not suffer from the most serious cracks. This position was selected under consideration of some experimental feasibility factors such as strain gauge range, while finite element analysis was not limited by this. In the simplified vehicle load spectrums, C2 vehicles had the largest loads. For example, when rear wheels of a C2 vehicle passed above the middle diaphragm plate in the finite element model, the stress distribution of horizontal diaphragm was analyzed. In the sub-model, No. 11 U-shaped rib located under wheel loads suffered from the most adverse force. The right hole position of No. 11 U-shaped rib had the largest stress of the horizontal diaphragm plate, wherein its stress cloud diagram is shown in Fig. 9. It is shown in the figure that yield happened at the position with the largest stress, and the position entered a plasticity stage. The paper focuses on three positions of the orthotropic steel bridge deck, which can easily suffer from fatigue cracks. They are respectively arc hole of horizontal diaphragm, lower end of vertical connecting weld between longitudinal rib and horizontal diaphragm plate, as well as the over-welding hole end of the connecting weld between cover plate and horizontal diaphragm plate, as shown in Fig. 10. The paper will analyze fatigue performance at three positions of No. 11 U-shaped rib, which could easily suffer from fatigue cracks.

Fig. 9Stress contour diagram at most adverse position

Fig. 10Positions with easy fatigue cracking

4.2. Detailed dynamic stress history of steel-structure bridges

The simplified vehicle load spectrums were loaded into steel-structure bridge (with struts and without struts) finite element models. By moving the load DLOAD module, wheel loads of each car type could move forward for 300 mm along the lane at each loading step. The course since the first axle of front wheel entered the deck till the last axle of rear wheel exited the deck was simulated. Stress history of holes, weld positions and plate positions in two models was obtained. Stress history of 3 positions with easy fatigue cracking in the steel-structure bridge (with struts) sub-model is shown in Fig. 11. As for B2 vehicles, wheelbase was small; stress history at hole and plate positions only had one waveform. With regard to C1 and C2 vehicles, due to the large distance between front and rear wheels, the stress history was two independent waveforms. In the figure, the interval with stress valve of zero, counted since front wheels exited the bridge deck while the rear wheels had not reached the bridge deck, is neglected. Stress variation tendencies at holes and plate positions were similar, but stress values were larger at the holes. Stress variation at weld positions was more complicated than that at other positions during vehicle passing. By the rain-flow counting method, stress amplitude spectrums at positions with easy fatigue cracking were obtained. Results are shown in Table 3 and Table 4. By comparing data in the tables, we can find that stress amplitudes at positions of the steel-structure bridge (with struts) with easy fatigue cracking were smaller than those of the steel-structure bridge (without struts). Obviously, struts could improve local load bearing. When rear wheels of B2 and C2 vehicles passed the horizontal diaphragm plate, hole and weld positions had already entered a yield stage. Therefore, the paper will take strain amplitudes as the parameter to estimate fatigue lives of structural details.

Fig. 11Stress history of steel-structure bridge under vehicle loads

a) Stress history of Type A vehicle

b) Stress history of Type B1 vehicle

c) Stress history of Type B2 vehicle

d) Stress history of Type C1 vehicle

e) Stress history of Type C2 vehicle

5.1. Analysis of detailed stress and strain history of steel-structure bridges

With regard to actual car flows, vehicles of various types pass the steel-structure bridge continuously. Multiple C2 vehicle loads were continuously loaded into the steel-structure bridge (with struts) finite element model so as to analyze its stress and strain history. When real wheels of C2 vehicles passed the middle horizontal diaphragm plate for the first time, holes, weld and plate positions entered a yield stage. Their stress and strain relations are shown in Fig. 12. When rear wheels of C2 vehicle got closer to the middle horizontal diaphragm plate gradually, the pressure borne at the holes increased, while elasticity was transformed into plasticity. When rear wheels reached right above the horizontal diaphragm plate, the sum of elastic strain and plastic strain reached the maximum value of –0.24 %. When rear wheels went far, the pressure stress decreased; elastic strain decreased gradually to zero. When the pressure stress decreased to zero, residual strain existed at the holes. Elastic unloading was continued around the holes. The holes bore the tension force. After unloading of wheel loads, the tension stress at the holes reached the maximum value of 120 MPa. Residual strain was generated after a cycle. Due to the subsequent vehicle load effects, force borne at the holes would be recycled according to the intensified stress and strain relations, wherein the strain amplitude was 0.2 % at this moment. In other words, the cyclic stress and strain curve of the holes is a straight line. In fact, steel materials will suffer from cyclic hardening or softening under cyclic stress, but such influence was not considered in the finite element model. In the subsequent fatigue life estimation, the intensified strain amplitude was taken as the parameter. Variation of stress and strain curve of the plate position was similar with that of the holes. Variation of stress and strain curve at the weld position was similar with the holes form, but their symbols were opposite. In other words, the weld position was pulled under wheel loads, while it was pressed after unloading.

Fig. 12Stress and strain relations at concerned point under C2 vehicle effects

a) Stress and strain relations at hole position

b) Stress and strain relations at plate position

c) Stress and strain relations at weld position

Strain history of holes, plates and weld positions of the steel-structure bridge (with struts) diaphragm plate under the simplified vehicle load spectrums is shown in Fig. 13. It is shown in the figure that pressure-bearing strain values at holes and plates are negative, while tension-bearing strain values at the weld position are positive. Variation rules of strain history are consistent with those of stress history. Strain value at the holes exceeded that at the plate position. Strain variation at the weld position was more complicated. Effects of front and rear wheels can be distinguished from the strain history.

Strain amplitude spectrums at three positions with easy fatigue cracking in two steel-structure bridge (with stay struts and without struts) models under the simplified vehicle load spectrums are analyzed, as shown in Table 5 and Table 7. Equivalent strain amplitudes are shown in Table 6 and Table 8. If strain amplitudes caused by all the vehicles are considered in the equivalent computation, the quantity of equivalent times at hole and plate positions is 295; there are two corresponding strain amplitudes at the weld position due to B2 vehicle, and the quantity of equivalent times is 312. When the equivalent calculation only takes into account the strain amplitudes caused by heavy vehicle rear wheels, the total quantity of equivalent times is 29.

Fig. 13Strain history of steel-structure bridge under vehicle loads

a) Strain history of B1 vehicle

b) Strain history of B2 vehicle

c) Strain history of C1 vehicle

d) Strain history of C2 vehicle

It is found in data comparison of Table 6 and Table 8 that equivalent stress amplitudes at hole and plate positions of the steel-structure bridge (with struts) model were smaller than computational results of the steel-structure bridge (without struts), but variation at the weld position was not obvious. This result shows that setting of struts and strengthening of horizontal diaphragm plate rigidity could effectively improve force borne by hole and plate positions, but cannot influence the weld position obviously.

5.2. Computation of fatigue life

Expression of the strain-life curve used by local stress-strain approach is as follows:

where: ${\sigma }_f^{\text{'}}$ denotes a fatigue intensity coefficient; $E$ denotes an elastic modulus; $N$ denotes a fatigue life; ${\epsilon }_f^{\text{'}}$ denotes a fatigue continuity coefficient; $b$ denotes a fatigue intensity index; $C$ denotes a fatigue continuity index.

The Q345 fatigue experimental results given in Reference [16] are substituted into Eq. (1). The obtained fatigue life formula is as follows:

Initial residual stress was not considered in the finite element model, while average stress in the computational stress history did not satisfy actual situations, so that this amendment is not considered in this paper. Equivalent strain amplitudes in Table 6 and Table 8 are substituted into Eq. (2). The estimated fatigue lives are shown in Table 9 and Table 10. Data in the tables is analyzed. When the 29 times are considered during computation of equivalent strain amplitudes, namely the stress amplitude was small, the estimated life would not be reasonable. In the steel-structure bridge (without struts) model, the weld position is the most adverse and has the shortest fatigue life of 15 years. In the steel-structure bridge (with struts), the fatigue life is still 15 years at this position. Obviously, struts cannot improve fatigue performance at this position. In the steel-structure bridge (without struts) model, the hole position is the most adverse, and has the shortest fatigue life of 13 years.