VALIDATION OF ANALYTICAL AND CONTINUUM NUMERICAL METHODS FOR ESTIMATING THE COMPRESSIVE STRENGTH OF

MASONRY

The present research work, carried out by (Lourenço and Piña Henriques, 2006) [7], analyses the capability of numerical models to reproduce the experimental behaviour of masonry subjected to compression loads. The experimental campaign is presented (Binda et al., 1988) ) [8], performing compressive tests, in displacement control, on 9 masonry panels composed by 9 layers of clay blocks connected by means of 10 mm thick mortar bed joints. The dimensions of single blocks are 250 x 120 x 55 mm, used to construct masonry prisms with dimensions of 600 x 500 x 250 mm. Three types of mortar were used (M1, M2 and M3), and each of them was employed to build 3 panels. In Figure 3.9 the test set-up and the position of LVDTs is reported.

Figure 3.9. Test set-up and positions of LVDTs [7].

Vertical deformations of masonry panels and elastic modulus were calculated on the basis of data read from transducers. In Table 3.9 the mechanical characteristics of employed materials are reported, in terms of compressive strength f_c, tensile strength f_t, Young modulus E and Poisson’s coefficient ν; while in Table 3.10 the results obtained from panels P1, P2 and P3, made with M1, M2 and M3 mortar respectively, are reported.

Table 3.9. Mechanical properties of employed materials [7].

Table 3.10. Mechanical properties of panels [7].

In order to reduce the computational effort for the analyses, only a part of the specimens was modelled, as it is shown in Figure 3.10. The model is represented by a portion of the wall, which is identically repeated in the panel, and giving to it appropriate boundary conditions in order to reproduce the actual behaviour of the whole specimen.

Figure 3.10. Definition of the base cell used in the analyses [7].

The previously described assumptions assumed for modelling do not allow to carry out a comparison of numerical results with experimental collapse modalities due to non-symmetric conditions, stress or strain localization, or effect of constraints.

Moreover, since the combination of such elements lead almost totally the post-elastic behaviour of the test, it is not possible to analyse the plastic behaviour of panels.

In order to investigate the out-of-plane of the panels, different numerical approaches were adopted: plane stress state (PS) using bidimensional finite elements, plane strain state (PE), and an intermediate approach, namely an

‘enhanced-plane strain state’ (EPE) using three-dimensional finite elements constrained to have the same displacement on two opposite faces of the panel. In the plane stress state the out-of-plane stresses are null, thus the specimen is free to deform in this direction. In the plane strain state, out-of-plane strains are null. The last case simulate an interesting behaviour between the two considered formerly. The code employed for the analysis was TNO Diana, and the non-linear parameter are reported in Table 3.11.

Table 3.11. Inelastic properties of materials [7].

Despite of the effort made in the research in last decades, information for a correct evaluation of materials parameter, such as the friction angle ɸ, the dilatancy angle ψ, and particularly the fracture energy in compression Gfc, and in tension Gft, are still lacking. In particular, the values assumed for last two properties are based on experimental evidences (Lourenço, 1996) [9] and on the necessity to assume a satisfactory value in order to reach numeric convergence of the model.

In Figure 3.11 it is shown the comparison of numerical results with the experimental ones, with reference to the P2 specimen, in terms of stress-strain diagram.

Figure 3.11. Numerical and experimental stress-strain diagram [7].

It is noticed that the response obtained from the EPE model is intermediate compared to the ones obtained from the PS and PE models. The numerical strength

of the specimen significantly overestimate the experimental one, even if the value of strain corresponding to the peak stress is very close to the experimental one. Another important difference is in the values of stiffness, much higher than the numerical ones. This can be explained because the stiffness used to simulate the numerical behaviour of the mortar was taken from experimental tests on prisms with different dimensions, constraint conditions and manufacture from the one employed in the composite model of the panel. The difficulties in the evaluation of stiffness of the mortar used in the panel represents an important disadvantage of micro-modelling strategy. Such a deficiency can be overcome by means of an inverse parametric adaptation. Considering the vertical displacement of the panel ( ) composed by the sum of the contribution given by the mortar ( and the contribution given by the blocks can be written:

that by means of some calculations can be rewritten as:

where:

is the correct modulus of elasticity of the mortar;

is the modulus of elasticity of the masonry unit;

is the elasticity modulus of the composite panel;

is the thickness of the mortar joint;

is the height of the panel.

In the Figure 3.12 are reported the results obtained correcting the values of the modulus of elasticity of the mortar.

Figure 3.12. Comparison between the stress-strain diagram obtained using an adjusted value of the modulus of elasticity of the mortar [7].

It is noticed that, compared to the case of previous analysis, even though similar values of strength were obtained, they are reached in correspondence to higher values of strain. Nevertheless, the possibility to adjust also the values of the peak strain was not considered in this work.

Failure mechanism obtained from the analysis depend, of course, from the adopted modelling strategy, but even if they are numerically correct, they do not reflect the actual physical behaviour of the element.

In Figure 3.13 is reported the behaviour of the strains along different cross-section of the element. Three level of load were considered, and each of them is representative of a particular branch of the stress-strain diagram. The specimen P1 was considered, since it is composed by a weaker mortar compared to blocks, and it represent better the conditions of ancient buildings.

Figure 3.13. Plots of the strain versus the load for different cross-section of the element: S1 (a), S2 (b), S3 (c) [7].

As can be seen from the figure, the mortar is in a three-axial compression state, while the blocks are subjected to bi-axial compression-tension. A reduction of vertical compression can be observed in proximity to head joints due to the low stiffness of mortar. This unloading effect is bigger in proximity to the failure point because of the inelastic behaviour of head joints. Moreover, it is possible to observe a stress concentration at the external borders when the load increases, causing the crisis of head joints.

Furthermore, it was performed a comparison between the experimental data and results given by formulations from the literature, in terms of compressive strength. The following equations were used:

1. Francis et al.

2. Khoo and Hendry

3. Ohler

4. Eurocode 6

5. ACI Specification for masonry units

The obtained results are summarized in Table 3.12.

Table 3.12. Comparison between analytical and experimental strength [7].

The first three equations were obtained from equilibrium methods, under the hypothesis that the blocks are subjected to uniaxial compression and biaxial tension, while the mortar joints are subjected to triaxial compression. It is evident that these formulas overestimate the actual strength of specimens, while the empirical formulas of Eurocode 6 and ACI conservatively evaluate the experimental results. It is also important to notice that the formulation by (Francis et al., 1971) [10] reduces the value of strength when the mortar stiffness is incremented. This is mainly due to the high sensibility of such formulation to the Poisson’s coefficient. This represents an important disadvantage for this method because of the objective difficulties in the

evaluation of this value. Finally, as a result, it can be said that the continuous finite elements modelling, based on plasticity and cracking, cannot be used to adequately describe masonry failure mechanisms or for assessment of strength of masonry starting from mechanical characteristics of its constitutive materials. In order to progress in this direction it is thus necessary to look for alternative models for the representation of the microstructure of masonry panels, and carry out deep experimentations for characterization of mechanical behaviour of mortar present in masonry elements joints.

3.4 MECHANICAL BEHAVIOUR OF POST-MEDIEVAL TUFF