Model B


1.03 MPa

1.76 MPa


1.88 MPa

2.67 MPa

Figure 13. Model B - USL stress state. Direction of maximum (a-b) and minimum (c-d) principal stresses.

structural performance may be obtained (Fig. 13). Checking the direction of minimum principal stress oiii, in the the mid-span section, some sort of strut, due to the transfer of compressive forces through the steps, may be identified. This phenomenom appears, thanks to the moulding, in both the models. Along the

Figure 14. Cracking pattern in case of Wk1 = 2kN/m2: axonometric view from above (a) Model A; (b) Model B.

lateral section near the supports, the behaviour seems instead to be more affected by torsional effects.

The strong interaction among the steps ofone flight, revelead by the formation of the strut in the numerical simulations, confirms once more the static intuitions of the ancient builders and engineers.

5.3.2 Phase 2: incremental analysis After the dead load assignment, two load typologies (uniform pressure on every step and transversal line load on the central step) were monotonically increased. Uniform pressure

With reference to uniform live load, even the nominal action amplified more than 15 times does not cause significant damage pattern in the steps. In case of Wkj equal to 72kN/m2, only slight cracking phenomena develop near the supports (Fig. 14).

In both the models, crushing states are not evident and, in fact, during the loading phase, the principal stress oiii has not reached the surface of the failure domain. The higher obtained values for am are 12.6MPa (Model A) and 22.9MPa (Model B). Even

Figure15. Model A, stress state in case ofWn = 72kN/m2: (a) maximum principal stress ai [N/m2 ]; (b) minimum principal stress am [N/m2 ].

in this case, the stress peaks are located near the constraints (Fig. 15).

Observing the failure pattern, it can be noted that, in Model A (as Rondelet wrote), the described constructive technique leads these elements to be more effectively linked together.

The force transfer through the moulding is more evident in respect to the other model, in which the rotation of the ends of the step is allowed. In fact, the lower steps are more loaded in compression in correspondence of the mid-span point than the upper ones; this force transfer lead the vertical planes in correspondence of the fixed ends to be subjected to high tensile stresses and cracking develops in the lower part of the flight.

Further increase of the uniform pressure is not investigated, because the hypothesis of fixed constraints at the ends of the step (modelling the effect of the masonry walls) would have been not totally correct. In fact, higher loads in the masonry could

mid-span sec lion

Figure 16. Model A - line load. Direction of minimum principal stress.

mid-span sec lion

Figure 16. Model A - line load. Direction of minimum principal stress.

have led to local crushing and stiffness degradation of the supports. This condition has not permitted the investigation of the stair behaviour for the highly nonlinear range. Nevertheless, this kind of load seems not to affect significantly the safety of the structure. Line load

Assigning the transversal line load on the central step (along X-axis), some information about the non-linear behaviour modification is achieved.

A final value of 7.5 kN/m is reached, pointing out slight cracking located on the central step itself, near the supports.

Also in this case, the highly non-linear range can not be studied, due to the unrealistic boundary conditions which further load increase would have led to.

In Model A, the force transfer through the moulding is less evident in respect to the uniform load (Fig. 16).

In the mid-span section, the strut, due to the transfer of compressive forces through the steps, is not identifiable anymore. Moreover, even the torsion effect near the supports cannot be highlighted.

From the result review, one can put forward that the direction of the minimum principal stress is almost orthogonal to these longitudinal sections. This may be explained through the creation of a transversal line of thrust along each single step (especially the loaded one). In this way, the load is directly transferred to the supports of the step, achieving only partially the overall behaviour ofthe stairway. Nevertheless, the lower steps seem to interact with the loaded one.

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