Simulation of masonry joints’ shear tests

Characterization of stone masonry arch bridges’ materials: experimental tests and numerical simulations

span bridges. The more easy detectable modes are transversal, since the wind is the main environmental action and tends to excite the bridge in this direction. Particularly for the vertical modes, a more difficult detection is proved. The railway traffic induces vibrations in the bridge, but in this case, it cannot be used favourably for modal identification simply because this action is not of the "white noise" type and it is so strong that the structure's natural behaviour is masked by this vibrations.

step, a constant normal load and, in a second step, the horizontal displacements in the upper block. The deformed mesh obtained in the numerical shear testing simulation is presented in Figure 4.19b.

a) b)

Figure 4.19 – Shear test numerical model: a) solid blocks with an interface and b) deformed mesh after loading.

In order to materialize the interface between the blocks, surface-to-surface contact pair elements with four nodes for each face, were adopted namely the so-called CONTA173 and TARGE170 elements available in ANSYS. This type of contact element allows opening/closing and sliding movements between the nodes of the interface. Several different contact algorithms are available to enforce compatibility at the contact interface, namely, the selected penalty algorithm, which is based on the contact stiffness and the maximum allowed penetration between the two surfaces.

A frictional contact was established at the interface by assigning in the normal direction a normal stiffness Kn in compression and no strength in tension, as well as a shear stiffness Ks in the tangential direction, both defining the stress-displacement relationship in the normal and tangential directions, respectively. The non-linear behaviour of the masonry joints in the tangential direction is represented by a Coulomb friction model associated with the contact elements. The model allows defining an elastoplastic evolution represented in Figure 4.20a, where the maximum shear stress (τmax), at which residual sliding on the surface begins, is a function of the contact pressure (), the friction coefficient (μ=tan) and the cohesion (c), representing the usual Mohr-Coulomb envelope. In the tensile domain, a cut-off is defined assuming a null tensile strength.

For the representation of masonry joints with peak behaviour before the residual phase, it is possible to combine the Coulomb friction model with a Cohesive Zone Model (CZM), proposed by Alfano and Crisfield (2001) and available in ANSYS. The relationship between shear stress (τ) and tangential displacement (δt) in the contacts is represented in Figure 4.20b for this combined cohesive and friction model. The model shows a linear elastic loading (OA) followed by linear softening (AB) and finalising with a residual phase. The maximum shear contact stress (τpeak) is achieved at point A, for a tangential displacement equal to δt1. After the peak at point A, any unloading and subsequent reloading occur in a linear elastic manner along with line OB at a more gradual slope. The residual behaviour follows the frictional model represented by the Coulomb friction formulation as described previously.

Characterization of stone masonry arch bridges’ materials: experimental tests and numerical simulations

a)

b)

Figure 4.20 – Shear stress vs. tangential displacement model: a) Coulomb friction model and b) Combined cohesive and Coulomb friction model.

According to the above described modelling strategy, the masonry joints’ material properties assigned to the contact interfaces are listed in Table 4.5. In order to simulate the experimental shear behaviour of the joint samples, different shear stiffness (Ks) values were considered according to the normal stress of each test (2^nd row values in Table 4.5). The mean values of Ks are also included in the 3^rd row of Table 4.5.

Table 4.5 – Non-linear material properties for the stone-to-stone contacts.

Material parameter Durrães bridge PK124 bridge

Split samples Bonded samples Split samples

Normal stress (MPa) 0.2 0.6 1.2 0.6 0.1 0.2 0.6

Ks (MPa/mm) 0.6 0.7 0.8 2.1 0.1 0.2 0.6

Ks (mean) (MPa/mm) 0.7 - 0.3

Friction angle (º) 36 36 32

Residual cohesion (kPa) 128 297 38

Particularly for the bonded behaviour of Durrães samples, the values of peak shear strength (τpeak) and corresponding tangential displacement (δt1) were used to define the combined cohesive and friction model. Concerning the normal behaviour, only elastic properties were considered (as already presented in Table 4.2) and, therefore, no non-linear simulation was carried out.

ii) Results and discussion

The tangential behaviour of the split joints is shown in Figure 4.21, in terms of shear stress versus tangential displacement, considering the three levels of normal stress used in the lab tests with the split samples. The Coulomb friction model defining a frictional/residual behaviour (Figure 4.20a) and the properties listed in Table 4.5 were assigned to the contacts. The results of Figure 4.21a-b include the variation of the shear stiffness Ks with the normal stress applied in each test, while in Figure 4.21c-d a mean value for the shear stiffness Ks is defined for all tests. Both simulations show a satisfactory agreement between numerical and experimental results, by adjusting an elastoplastic behaviour where the residual strength phase represents the dependency of the shear stress with the normal stress, friction angle and cohesion defined by the Mohr-Coulomb envelope.

a) b)

c) d)

Figure 4.21 – Numerical simulation and experimental test results of the tangential behaviour of split joint samples: with different ks values for a) Durrães bridge and b) PK124 bridge; and with a mean ks value for c)

Durrães bridge and d) PK124 bridge.

In the first analysis (Figure 4.21a-b), by adjusting for each test the set of material parameters obtained for each normal stress level, the model represents the dependency of both the tangential stiffness and the residual shear strength with the normal stress. In the second analysis (Figure 4.21c-d), for which a mean value of the tangential stiffness is adopted, a good compromise is obtained in the elastic phase for all three levels of normal stress for the Durrães bridge case, though not so good for the PK124 case; also a reasonable representation is obtained for the residual phase, despite the Ks dependency with the normal stress not being accounted for. A better adjustment is obtained for Durrães bridge case than for the PK124 case wherein the experimental shear stiffness clearly dropped for lower normal stress levels. It is worth noting that the second analysis corresponds to a realistic way of defining the input material

Characterization of stone masonry arch bridges’ materials: experimental tests and numerical simulations

parameters for numerical simulation with variable load histories, as are the cases of interest for bridge modelling.

Figure 4.22a shows the tangential behaviour, obtained with the FE simulation of shear tests of the Durrães bridge bonded joints, in terms of shear stress vs. tangential displacement, for a constant normal stress of 0.6 MPa. For comparison purposes, the experimental curve is also shown in the plot, corresponding to the mean values’ curve of the two performed tests presented in Figure 4.6. The simulation is based on the combined cohesive and friction model (shown in Figure 4.20b) and the properties presented in Table 4.5 were assigned to the contacts. In general, by comparing numerical and experimental results, a good agreement is found based on the adopted numerical model which allows representing the three different phases of the experimental curves, i.e. the linear, softening and residual phases. Figure 4.22b shows the simulated response of the combined cohesive and friction model using the same properties of the Durrães bridge bonded joints for three normal stress levels: 0.2 MPa, 0.6 MPa and 1.2 MPa. For the residual strength phase, the results show the dependency of the shear stress with the normal stress, friction angle and cohesion defined by the Mohr-Coulomb envelope. For the shear stress peak value, the results show no dependency on the normal stress level. This is adequate for the case of high cohesion mortared joints, but it is a limitation in the case of weaker mortared joints where this dependency can be observed as evidenced in (Arêde et al., 2019).

a) b)

Figure 4.22 – Numerical simulation of the bonded joint samples of Durrães bridge: a) comparison with the experimental shear test results; b) for other levels of normal stress.