Longitudinal crack modelling calibration: real and equivalent straight geometries

plastic strain evolution in the arch crown. The main reason for not achieving this collapse load is related to the deficiencies in representing the joint openings between the stones in a continuous model. Thus, in the arch, higher tensile stresses are found in the masonry material (that are not so realistic) which result from this inability to simulate the joint openings in tension zones. A micro-modelling approach with a stone-joint discretization would allow a better representation of the joint openings. For example, the mesoscale models presented in section 3.4.4 are proving very effective for ultimate load assessment (Zhang et al., 2018).

Despite these limitations to achieve ultimate scenarios, this incremental static analysis with the FE non-linear continuous model presents some potentialities, such as the possibility to have large load multipliers before reaching a collapse mechanism; the possibility to identify critical zones in the bridge arch and spandrel walls; the definition of a tension cut-off that allows limiting the maximum tensile stresses in the masonry, that should be low.

Numerical methodologies for the analysis behaviour of bridges with damage under railway loading

until reaching a load factor of 2. The LM71 load pattern was placed in the rail beam elements near the quarter-span of the arch. The response of the bridge with contact models is plotted in Figure 5.23, in terms of the vertical displacements obtained at the arch crown using both real and simplified straight geometries for the longitudinal crack. The results show a very good agreement between the two models in terms of vertical response.

Figure 5.23 – Comparison between real and equivalent straight crack modelling simulations under increasing vertical loading: vertical displacement in the arch crown.

Also under incremental vertical static loading, the response of the bridge contact models is evaluated in terms of relative vertical movements between the contact surfaces, plotted in Figure 5.24a, and in terms of gap distance in the contact surfaces, plotted in Figure 5.24b. The gap distance values obtained in the contact elements (Figure 5.24b) measure the opening displacement in the normal direction to the surfaces; these values increase as the surfaces pull apart and approach zero as the surfaces become closer.

The results are obtained for both real and simplified straight geometries for the contacts and globally a good agreement is verified.

The results show that the relative movements in the vertical direction of the crack surface are not so conditioning as compared with the normal direction (opening/closing movements). For a service loading (1P), a maximum value of 0.12 mm is obtained for the vertical relative movements and a maximum opening of 0.33 mm is obtained in the direction normal to the contact surfaces. This different behaviour observed in the plots can be related to the higher stiffness found in the vertical direction in railway masonry bridges, mainly due to spandrel walls (that are thicker in depth) and to a higher strength infill material.

a) b)

Figure 5.24 – Comparison between real and equivalent straight crack modelling simulations under increasing vertical loading: a) vertical relative movements between contacts; b) Gap distance in the contact elements.

For the contact model with real geometry, Figure 5.25 shows the plots of shear stress vs. vertical load multiplier in the transversal contact elements in two arch zones, one near the crown and the other at ¼ span. For these arch zones, the graphs also indicate the normal stress and the maximum sliding values in the contacts, obtained for a load multiplier of 5P. By analysing their shear behaviour response, it is concluded that these elements remain in the linear branch during the incremental analysis and only the elements located in the arch crown start to slide in the residual branch when the load multiplier reaches a value of 3. This means that for service loading, the shear stresses remain in the linear elastic branch and, likewise, for the equivalent crack the normal stresses remain in linear elastic behaviour.

a) b)

Figure 5.25 – Shear stress in the transversal contact elements of real geometry crack modelling simulations under increasing vertical load multipliers: a) arch crown; b) arch ¼ span.

In the second loading case, the bridge contact models were evaluated under the influence of transversal loads applied along the arch near the mid-span. The response of the bridge with contact models was assessed up to a maximum load factor of 10, amounting to a total force pair of 1200 kN, which approximately corresponds to 7.5% of the bridge self-weight; in terms of seismic response, this could be seen as a horizontal seismic coefficient of 7.5%. For these conditions, the bridge with contact models’

responses are plotted in terms of the transversal displacements in the spandrel wall, Figure 5.26a, and of the gap distance in the contact elements, Figure 5.26b.

The analysis of Figure 5.26a shows that the models follow the incremental transversal loading from 0L to 10L with increasing displacements along the spandrel wall up to a maximum of 0.8 mm and a good agreement is achieved between the real and simplified models. Looking at Figure 5.26b, the gap distance along the contact elements in the arch shows a reasonable agreement between both models in terms of peak values. The values are approximately multiplied by a factor of two compared to the displacements in the spandrel wall, given that only half of the gap distance obtained in the contact elements originates that displacement. However, in the monitored points near the transversal elements in the real crack geometry model case, a more visible difference is observed compared to the straight geometry case, the former exhibiting lower gap values and thus appearing to have more resistance to opening. The results show that the models are globally equivalent, which means that with a simplified straight contact model it is possible to have a good representation of both vertical and transversal responses.

Numerical methodologies for the analysis behaviour of bridges with damage under railway loading

a) b)

Figure 5.26 – Comparison between real and equivalent straight crack modelling simulations under transversal loading: a) Transversal displacement in the spandrel wall; b) Gap distance in the contact elements.

5.5.2.2 MOVING LOAD DYNAMIC ANALYSIS

The Leça bridge contact models were also assessed through dynamic non-linear analyses with the freight train load model, for both the materials scenarios “A-ct” and “B-ct”. For “A-ct”, which represents a damage scenario resulting from the introduction of contact elements (in the calibrated model) in order to simulate the presence of a longitudinal crack in the arch, Figure 5.27 shows the vertical and transversal responses of the bridge models in the arch crown for train passage at 80km/h. By analysing Figure 5.27a, it can be seen that, as expected, there is an increase (though very slight) of the vertical displacement between the calibrated scenario (the continuous model A, representing the current condition of the Leça bridge calibrated with modal identification) and the models with damage (i.e. the models with contacts A-ct) from a maximum value of 3.6mm to 3.8mm. In Figure 5.27b (for the transversal displacements) the lateral movements of the arch are noticed in the contact models ranging between 0.1mm (due to self-weight) and 0.2mm (due to moving loads). For both vertical and transversal displacements, the responses from the real and simplified straight crack modelling strategies are quite similar as evidenced also in both the Figure 5.27a and Figure 5.27b.

a)

b)

Figure 5.27 – Leça bridge model with contacts (A-ct) – response at arch crown: a) vertical displacements and b) transversal displacements.

For scenario B (which represents a global damage scenario resulting from the strength reduction of the masonry) and scenario B-ct (resulting from the introduction of contact elements to simulate a longitudinal crack), Figure 5.28 shows the vertical and transversal responses of the bridge models in the arch crown for train passage at 80km/h. By adopting scenario B, the vertical displacement in the arch crown increases to 8.2mm, when compared with the calibrated scenario (A) for which the result was 3.6mm. Figure 5.28a, shows a comparison between the continuous and the contact models (the latter evidencing slightly higher values than the former), with a good agreement between the different strategies for damage representation. Regarding transversal displacements in Figure 5.28b, although they are very small, a clear difference is found in the maximum values obtained, i.e. 0.1mm in the continuous model and 0.3mm in the contact models. The continuous model shows a deficiency in representing the lateral movements observed in the response of the bridge model, although, as already referred, such movements are quite small.

In all scenarios, when comparing the two contact models, there is no significant difference between a real and an equivalent straight representation of the longitudinal crack, in both vertical and transversal responses.

a)

Numerical methodologies for the analysis behaviour of bridges with damage under railway loading

b)

Figure 5.28 – Leça bridge model with contacts (B-ct) – response at arch crown: a) vertical displacements and b) transversal displacements.

Looking at the contact elements in the models with contacts “B-ct” and plotting the gap distance as shown in Figure 5.29 for train passage at 80km/h, it is possible to observe the opening values’ range, the dynamic amplitudes and the difference of crack representation (real or straight). The gap distance values range from around 0.15mm due to self-weight to around 0.30mm due to the moving loads. Again, there is no significant difference between a real and an equivalent straight representation of the longitudinal crack, since both in the arch and in the contacts the equivalence is verified. There is also a high dynamic content in terms of frequencies and amplitudes in the response obtained in the contact elements, which is evidenced by comparing the static responses due to moving loads with the corresponding dynamic responses (both illustrated in Figure 5.29).

Figure 5.29 – Gap distance in the contact elements in the arch crown for train passage at 80km/h. Comparison between real and simplified straight crack modelling (both from static and dynamic responses).

The gap distance in points of the contact elements along the arch span is plotted in Figure 5.30, for the equivalent straight contact model and for four train passages at different speeds. All the contact elements are immediately open after the application of the self-weight, and remain open during the train passages, except for areas located in the arch springing, which remain closed and in contact for almost all scenarios. The plots in Figure 5.30 show the deformation growth, increasing from around 0.15 mm, due to self-weight, to a maximum 0.4 mm, for train passage at 120 km/h. The deformation increased 2.5 times when comparing the effects of dead load with those of the train passage at 120 km/h. The dynamic

amplification caused by a 30km/h train passage is higher than for 80 km/h or 50 km/h ones, resulting in higher deformation values. In fact, this is a resonant speed for the transversal response of the bridge (as already shown) in the linear dynamic analyses and, therefore, one critical speed for assessing this type of damage that affects particularly the lateral movements.

Figure 5.30 – Gap distance in the contact elements along the arch span due to self-weight and train passages at 30, 50, 80 and 120km/h.