*3.3. Results of Laboratory Tests*

The model without treads (M1) was loaded with concentrated forces P1 = P3 = 1.5 kN. The value of the force at point P2 was increased until the first crack occurred at the load value of 4.5 kN. This moment was considered to be the end of elastic work of the structure. Failure was caused by the opening of the crack within the joint located under the concentrated force at point P2.

The model with masonry treads (M2) was loaded at the same locations as model M1 by applying concentrated forces on the masonry treads. At points P1 and P3, a structural load of P1 = P3 = 6 kN was applied. At the middle point, P2, the value of the load was increased until failure due to cracking, which occurred at a force of P = 59.8 kN. The results of the measurements of the displacement and support reactions of the researched models are presented in Figure 10.

By including the treads in the stair flight curve analysis, the load capacity was considerably higher than previously assumed. The load-bearing capacity for the model with treads was 13.3 times higher than in the model without treads. The deflection for the same load level P = 4 kN at point P2 for model M1 was u = 1.9 mm and, for model M2, u = 0.2 mm. The deflection value of model M2 was 9.5 times smaller than the deflection of model M1. The steps masoned above the staircase significantly increased the load-bearing capacity of the structure, as well as its stiffness. Figure 11 shows the failure mode of the tested models of stair arches and the results of the measurement of the deformations for M2 made with the ARAMIS system at the moment of destruction.

The model of structural failure changed, which, in the case of model M2, occurred as a result of cracking along the arch, at the interface of the arch with the brick treads. First of all, the masonry above the arch detached from the rest of the structure (masoned brick treads). This proved the significant importance of the stair treads above the arch in its load-bearing capacity. Once the crack formed between the arch and the treads, the rest of the structure exhibited rapid failure.

For model M1 (without treads), the deflection measured vertically in the load range up to 0.9 kN developed similarly for points P1 and P2. The deflection curve for P3 was significantly different from that for P1 and P2. Comparing the curves for a load level P = 1 kN, it was found that the smallest deflection of the structure was recorded for the load at point P1 (48.2 cm from the lower support), for which a vertical deflection of the structure u = 0.3 mm was measured. At point P2 (100.7 cm from the lower support), the vertical deflection u = 0.4 mm was obtained. At point P3 (153.2 cm from the lower support and 57.4 cm from the upper support), the highest vertical deflection of the structure u = 0.8 mm was achieved. At point P2, the deflection was 1.3 times higher in relation to point P1, while, at point P3, it was 2.7 times higher than at P1. That is, the lower the location of force along the staircase was, the lower the influence on the deformation of the structure.

**Figure 10.** Experimental results of masonry staircases: relation P–u (vertical displacement in the middle of an arch in point P2) for model M1 (**a**); force P–reaction R for model M1 (**b**); force P– displacement u (vertical displacement in the middle of an arch in point P2) for model M2 (**c**); force P–reaction R for model M2 (**d**).

Changing the location of the force had also a significant effect on the recorded reaction R in the upper part of the staircase. For model M1, in the case of force applied to the middle and lower parts of the staircase (P1 and P2), the reaction was positive, which means that it was directed vertically downwards and balanced the horizontal forces in the arch. However, for the force at point P3, the situation was completely different. Due to the different geometry of the arch compared to typical vaults, the reaction had the opposite direction. The arch in this place did not generate compressive force on the upper support, but, rather, a tension. Comparing the values of the reaction R between different force points at the force level P = 1 kN, it should be noted that the reaction for points P1 and P2 was similar and was about R = 0.8 kN. For point P3, the reaction was R = −0.1 kN, which was eight times lower. The reactions for points P1 and P2 had different values until the inflection point on the P2 curve, at force P = 0.9 kN.

**Figure 11.** Registered failure mode of tested models of stair arches: model M1 (**a**), model M2 (**b**) and measurements from ARAMIS of deformation during failure of model M2 (**c**).

Different results were obtained for model M2 (with treads). The differences in deflection u were less visible between P1 and P3. For the load level P = 5 kN at points P1 and P3, the deflection had a value of u = 0.2 mm. In the investigated load range, the addition of stair treads increased the stiffness of the structure regardless of force location.

For model M2, the reaction forces in the support differed from each other. Negative values for the support were registered again for point P3 but also partially for point P2. The reaction in the case of the force at the center (P2) operated in a tension range, up to P = 9 kN, after which it started to work in compression. At the force level P = 5 kN, the reaction at point P1 was R = 0.4 kN; at point P2, it was R = −0.2 kN; and, at point P3, it was R = −1.4 kN. The spreading force for the force P = 5 kN was recorded only for the force at point P1. In the case of force at points P2 and P3, tension was observed. The value of the reaction for P3 was seven times higher than for P2.

The results measured using the ARAMIS system allowed us to analyze the displacement with higher precision. For the reference load level P = 6 kN, there were visible differences in the operation of the structure depending on the load application point. Analyzing the displacement maps (Figure 12), it is visible that there were local deformations. Those were caused by the movement of the bricks toward each other due to the changes in the joints.

For model M2, regardless of the load application point, the values of arch displacement were of the same sign. This is different from a typical operation of a masonry arch. The biggest displacement of the staircase was achieved when the force was applied at point P2. The results for points P1 (value for E4, virtual point, in the middle, d = 0.186 mm) and P3 (also value for E4, virtual point, in the middle, d = 0.189 mm) were similar. The values of deflections at the points directly under the applied forces differed for P1 (virtual point E2, d = 0.217 mm) and P3 (virtual point E6, d = 0.144 mm), even though the total deflection at point E2 was similar. For the load level P = 6 kN, the highest resultant deflection for point P2 was d = 0.274 mm. This is a value that can be hardly seen with a naked eye. The failure of the structure was also at a small resultant displacement of d = 6.714 mm. These results confirm a significant influence of masonry treads on the rigidity of the whole structure. The treads are responsible for increased stiffness of a lean masonry arch. Due to the low values of the displacement of treads, the diagnostic of the structure should be performed with precise measurement equipment, such as 3D scanners (TLS).

**Figure 12.** *Cont*.

(**c**)

**Figure 12.** *Cont*.

**Figure 12.** Results of experimental test of masonry staircases—displacement measured by ARAMIS: load P = 6 kN at point P1 (bottom) (**a**), load P = 6 kN at point P2 (middle) (**b**), load P = 6 kN at point P3 (top) (**c**) and load at failure P = 59.8 kN in point P2 (**d**). d—resultant displacement; dX—horizontal displacement; dY—vertical displacement.
