Damage simulation with discrete cracks

A localized longitudinal crack in the bridge arch barrel can be simulated with contact elements inserted in a global homogeneous FE model of the bridge. Rather than simulating crack initiation, this study aims at representing longitudinal cracks that are already open and with zero tensile strength due to transverse effects of the bridge response. Although, at an initial stage of this cracking mechanism with cohesive mortared masonry joints, the crack opening resistance is determined by the normal tensile strength in the head joints (represented by an elastic phase followed by a softening branch) and shear strength in the bed joints (represented by an elastic phase up to a peak value followed by a softening branch and a residual phase) which depends on cohesion, friction angle and normal stress. At an advanced stage, which is the simulated scenario presented in this study, with the head joints already open (and with zero tensile strength) it is verified that the opening resistance is only determined by the contribution of the bed joints with shear strength determined by residual shear behaviour.

These cracks follow the path of the interfaces between the stones and, therefore, its correct simulation requires the model geometry to follow that irregular path which was considered in the present study.

However, beyond that strategy, an alternative procedure is proposed hereafter for the geometric definition of open cracks in the bridge model, consisting of a simplified straight longitudinal equivalent crack (or joint). Such procedure was calibrated and validated by comparing the corresponding results with those obtained with the real crack geometry simulation (see Figure 5.1).

For this purpose, 3D surface-to-surface contact elements with four-nodes (CONTA173), available in Ansys (ANSYS, 2017), were adopted. These elements can be located on the surfaces of solid or shell elements, with target surfaces defined by specific 3D target elements (TARGE170). The element has the same geometric characteristics as the solid or shell element face with which it is connected. The contact stiffness is defined based on the stiffness in the normal direction (Kn) and the stiffness in the tangential (shear) direction (Ks). In order to represent the shear of masonry joints, a Coulomb friction model is combined and associated with the contact elements. It defines an equivalent shear stress, at which surface sliding starts, as a function of a friction coefficient and a cohesion parameter. In this case, an isotropic friction model was used, specifying a single friction coefficient.

For the herein studied models with contacts, the stiffness and friction input parameters were based on the values estimated from the experimental tests conducted in joint samples extracted from PK124 bridge, and presented in Chapter 4. These tests allowed to estimate the shear response using split joint samples (with residual behaviour) and considering three different normal stress levels. In tangential direction, the adopted model consists of two phases, a linear elastic phase until reaching the maximum shear stress values, followed by a residual strength phase. In the normal direction, a normal stiffness is defined in compression and no strength exist in tension, representing a null resistance for opening. When under compression, the recontact of the two surfaces is always possible defined by the normal stiffness parameter.

Aiming at simulating the cracking evolution in masonry components using the previously described discrete crack models, a simple numerical example was built in Ansys (shown in Figure 5.2, where the global axis system XYZ is included) in order to assess and validate the represented numerical behaviour.

This material model is used for a rectangular prism with dimensions 1.5 m, 1.0 m and 1.5 m, along X, Y and Z directions, respectively, loaded by uniform pressure on both top faces and laterally imposed displacements. Two different models were defined, one based on the real crack geometry (Figure 5.2a) having head joints (parallel to the YZ plane) and bed joints (parallel to the XY plane), which is used for comparison and calibration of a second model having a simplified straight geometry of the crack (Figure 5.2b) only with a head joint (YZ plane). The load history consists of a first stage in which the normal pressure is applied in Z-direction (longitudinal, see Figure 5.2c) and then incremental displacements are imposed in the X-direction (transversal), in order to make both blocks sliding along the crack. For the normal pressure, three values were considered, namely: 100, 200 and 300 kPa, aiming to represent normal stress values expected for the arch in typical stone bridges like the case study presented next.

a) b) c)

Figure 5.2 – Example of contact joint numerical representation: a) real geometry; b) equivalent straight; c) normal pressure and induced displacements.

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

For the real geometry model (Figure 5.2a) there are contact elements in the XY (transversal) and YZ (longitudinal) interfaces. The transversal contact elements are controlled by shear behaviour: a value of 0.1 MPa/mm was assigned to the shear-stress stiffness Ks, combined with a friction model having friction angle of 36º and null cohesion, according to PK124 experimental parameters. For the longitudinal contact elements, which are controlled by the normal behaviour (opening), no tensile strength is assigned and the normal-stress stiffness Kn is null. Concerning the model with a simplified straight longitudinal equivalent crack (Figure 5.2b), from now on briefly referred to as fictitious equivalent crack, the longitudinal contact elements were defined also with 0.1 MPa/mm for Kn and no shear strength was activated, because this model does not show sliding in the longitudinal joints.

The results obtained for the tangential and normal directions on both models in terms of resultant forces are shown in Figure 5.3. In particular, for the real crack geometry model, the response in terms of force vs. sliding displacement is plotted in Figure 5.3a, showing the linear elastic branch according to the prescribed shear-stress stiffness and the shear strength in agreement with the applied pressure and the Coulomb law for the adopted parameters; the response in terms of force vs. normal displacement is plotted in Figure 5.3b, exhibiting the expected null strength for the whole displacement range. In the graphs, both normal and sliding distances represent the same distance, which is the longitudinal crack opening. For the fictitious equivalent crack model, the force vs. displacement response is plotted in Figure 5.3c, depicting the linear behaviour with the same prescribed normal-stress stiffness of the real tooth-path geometry model.

a) Transversal contacts b) Longitudinal contacts

c) Equivalent straight contacts

Figure 5.3 – Results of contact numerical example: a) shear and b) and normal force - displacement plots in real geometry; and c) normal force - displacement plots in equivalent fictitious straight geometry.

However, by comparing the results in Figure 5.3a and Figure 5.3c, an adequate equivalence between the real and the fictitious crack model response still lacks two issues to be considered: i) although similar values were adopted for shear-stress stiffness, Ks, and normal-stress stiffness, Kn, in both geometries, still it must be taken into account the difference between the contact areas in transversal and longitudinal interfaces of the crack; ii) a normal-stress fictitious limit has to be imposed in the response given by Figure 5.3c in order to make it similar to that of Figure 5.3a.

The first issue (i) can be addressed by adopting for the fictitious crack an equivalent stiffness Kn,eq

calculated as proposed in the next paragraphs based in the parallel association of springs schematized in Figure 5.4. In the real crack geometry normal-stress springs (in blue) are considered in the longitudinal interfaces, while shear-stress springs (in green) are considered in transversal interfaces, as shown Figure 5.4a-left. The resultant of stress generated by such springs over the corresponding interface areas must be, therefore, imposed equal to the stress resultant in the equivalent fictitious crack (see Figure 5.4a-right) in order to enforce equilibrium equivalence between the real and the fictitious crack modelling.

Mathematically, this is expressed by Eq. 13:

𝐾 _, 𝐴 = ∑ 𝐾 𝐴 + ∑ 𝐾 𝐴 (13) where Aeq is the area of the fictitious crack with an equivalent normal-stress stiffness 𝐾 _, (see Figure 5.4b-right), and 𝐴 and 𝐴 are the areas of the i-th longitudinal and j-th transversal interfaces, respectively, with normal-stress stiffness 𝐾 and shear-stress stiffness 𝐾 as illustrated in Figure 5.4b-left. The interface areas 𝐴 and 𝐴 are equal to the corresponding interface lengths Lli and Lti (Figure 5.4b-left), respectively, times the interface thickness and, similarly, the equivalent Aeq area is equal to the total length Ll of the fictitious crack times the same thickness.

a) b)

Figure 5.4 – Determination of the equivalent stiffness for an equivalent fictitious straight joint: a) schematic representation of forces’ transfer; b) springs’ parameters and contact lengths.

If it is assumed that 𝐾 = 𝐾 = 𝐾 = 𝐾 and 𝐾 = 𝐾 = 𝐾 = 𝐾 , and considering the same thickness for the longitudinal, transversal and equivalent fictitious interfaces, Eq. (14) allows writing:

𝐾 𝐿 = 𝐾 ∑ 𝐿 + 𝐾 ∑ 𝐿 (14)

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

from which it can be derived the Eq. (15) for 𝐾 _, as a function of the sum of all contact lengths:

𝐾 _, = ^{∑ ∑} (15) For the studied example shown in Figure 5.2, with Kn = 0 in the longitudinal contacts, Eq. (15) reduces to:

𝐾 _, = ^∑ 𝐾 (16) Using Eq. (16) with the dimensions of the studied example, an equivalent stiffness Kn,eq= (2/3)Ks is found for the equivalent fictitious interface.

The equivalence between Kn,eq and Ks remains true if the shear stress is on the linear branch of the behaviour curve. If not, the shear stress is in the residual phase and a limit has to be imposed in the equivalent normal stress curve, as mentioned before. This feature of the interface behaviour, in the normal direction of the equivalent fictitious interface, could be addressed by adopting the Cohesive Zone Model (CZM), already used in chapter 4, to simulate contact debonding (Figure 5.5). The model allows defining the normal stress-displacement behaviour, in the tensile regime, through a bilinear curve. The normal contact stress (σ) and normal displacement (δ) is plotted in Figure 5.5a, which shows linear elastic loading (branch OA) associated with a constant value of the normal stiffness (Kn) followed by a linear softening branch (AB) where the tensile stress (σ) decreases from its maximum value (σmax) until zero.

For the studied example of Figure 5.2, the resultant force in the X-direction is computed and plotted against the joint opening distance in Figure 5.5b, which shows good agreement between the response of the real geometry model (dashed yellow line) and that of the equivalent straight interface geometry with the CZM (solid blue line). The residual branch was defined for a normal stress level of 200 kPa. As evidenced in Figure 5.5b, the introduction of a tension cut-off in the CZM leads to approximate an elasto-plastic behaviour in the equivalent straight model. By adopting a long branch of the linear softening (AB), through the use of a high value for the input parameter, δ2, a good equivalence of both models is achieved, thus providing an adequate representation of the shear behaviour (exhibited by the real irregular joint) by means of a normal stress behaviour joint (equivalent straight).

a) b)

Figure 5.5 – CZM model: a) interface behaviour in the normal direction; b) resultant forces in X direction vs.

joint opening distance, for the real and equivalent straight joint geometries (without and with CZM).