In this framework, the development of very accurate models is generally required, both in case of service loads (for which cracks formation may represent a condition to avoid) and ultimate limit state. In fact, non-linear analyses may give information, by means of incremental analysis, about the collapse, but the models should be able to reproduce the limited tensile strength and possible crushing of stone material, the frictional behaviour on the contact surface of the moulding.
To this aim, the modelling was carried out through a detailed non-linear solid model of the flights of steps, focusing on effectively describe the interface behaviour.
Brick elements (SOLID65) are used to model the stone material of the monolithic steps. The frictional behavior on the contact-planes (horizontal and vertical) of the moulding is simulated by CONTACT52 elements.
In Table 4, the basic information about the FEM model, made up of 34630 elements and 41124 nodes, are provided.
The boundary conditions of FEM model are intended to simulate two situations, related to different hyphotesis of interaction between the step and the masonry walls:
- Model A: displacements along Y and Z axes are avoided on the bottom surface of the steps, in correspondence of the insertion surface of the step (0.2 m in depth) in the wall. This condition aims to simulate structural fixed ends (Fig. 12).
- Model B: displacements along Y and Z axes are avoided on the bottom of the steps, in correspondence of the chase of the step in the wall. A line of nodes only (0.2 m in distance from each ends of the step) is constrained, in order to enable the rotation, aiming to simulate structural hinges.
The actual condition is intermediate between the analysed ones; due to the chase depth in the walls, it is probably better described by Model A.
Moreover, the steps at both ends of the flight need to be realistically constrained. On the bottom one, the vertical support of the masonry arch below (Fig. 7b) and the horizontal constraint of the floor have to be simulated. On the top step, instead, horizontal constraints are needed to model the interaction of the step and the landing at turn of stairs, made up of stone slabs having the same thickness. All this kinds of constraints behave obviously as compression-only elements.
5.1.1 Mechanical parameters
The pink granite stone mechanical characteristics are derived by the experimental test campaign previously described.
In the FEM simulations, in the elastic range, a linear isotropic constitutive law was assumed. This kind of behaviour can be appropriate to satisfy the requirements of the safety check, serviceability and ultimate limit state, of the Technical Rules (2005).
The values of the mechanical characteristics assumed correspond to the average of the statistical distribution of each quantity (in case of Young's modulus E = 30,000 MPa, the value is derived by the experimental tests on the stone of the Caffa palace).
Moreover, the non-linear behaviour is modelled through the CONCRETE model (enclosed in the ANSYS 8.0 library), whose constitutive law simulates the limited tensile and compressive strength of stone material. The failure surface in the stress domain is represented by a law that depends on the hydrostatic stress, as Mohr-Coulomb or Drucker-Prager ones. Cracking along three orthogonal axes and crushing are allowed and five failure-surface parameters have to be assigned. If only two (ft and fc) are set, the failure surface is defined as in Willam and Warnke (1975).
The non-linear parameters in terms of strength values are obtained as the 5% percentile of the stochastic distribution of the experimental results. In the model, the uniaxial tensile strength is set ft equal to 7.52 Mpa for and the uniaxial compressive strength fc to 97.39 Mpa.
The stiffness parameters of the contact element on the interface of the step moulding are assumed to simulate the interaction effect (equivalent Young's modulus equal to 30,000 MPa). Being difficult to have an estimation of these quantities, it was verified that their variation among a pre-defined range does not significatively affect the results.
The frictional coefficient on the contact surface is set equal to 0.5.
Two analysis phases were carried out: in the first one, the load cases prescribed by the technical rules (ultimate and serviceability limit state) were imposed and, in the second one, the behaviour related to incremental loads was studied.
Phase 1: on the two models (A and B), the following analyses were performed:
where Gk is the dead load and Wk1 is the live load (equal to 4 kN/m2 in case of common stairs);
- Phase 2: on the two models (A and B), the analyses carried out concerned the monotonic increase of two load typologies (uniform pressure on every step and transversal line load on the central step), after the dead load assignment.
It has to be noted that, in case of Phase 1, the behaviour is very far from the inelastic range; so, it was verified that the action assignment in two subsequent stages or the concurrent loading did not lead to different results (i.e., the effect superposition is still correct).
In both cases, live loads were assigned to the walking surface of the steps.
In the following, the main results obtained from the numerical simulations are discussed. Special emphasis is put on the evaluation of interaction effects among the steps.
5.3.1 Phase 1: technical rule requirements As previously noticed, the analyses results may be useful in order to obtain some information about the serviceability and structural safety of the stairways, according to Italian Technical Building Rules.
In case of SLS, the mid-span vertical displacement is maximum in the central steps of the flight: with reference to Models A and B, it is equal to 0.06 mm and 0.07 mm respectively. Those values are both much lower than the limit displacement L/400 (7.5 mm).
In case of ULS, the stress field in Model B is higher. The extreme values (in terms of average nodal stresses) are reported in Table 5.
From these results, it can be highlighted that the stress state is much lower than the material strength. This is much more significant if one considers that those values are concentrated in very narrow areas, near the constraints: the safety evaluation is amply conservative.
Focusing the attention on longitudinal sections of steps, corrisponding to the portion near the supports and the mid-span one, interesting remarks about the
Table 5. Values
of maximum (ffi) and
in Models A and B.
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