PSI - Issue 44

Mauro Mezzina et al. / Procedia Structural Integrity 44 (2023) 566–573 Author name / Structural Integrity Procedia 00 (2022) 000–000

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When the value of tensile strength is exceeded in the masonry, the resistant mechanism of the cracked material is only provided by compressive stress: the response can be simulated by a spatial truss lying on the faces of the solid and composed of plane trusses, one on each face of the building, according to the Rausch truss [Lehonardt and Mönnig (1976)]. Let ϕ be the flow of tangential stresses on the generic side of the horizontal section of the solid, induced by the torsional stress T . The flow ϕ is a force per unit length, which, because of the admitted small thickness b , can be considered constant along the midline of the generic wall. The flow ϕ , as shown in the figure 7, can be decomposed into the two forces ϕ C and ϕ L , the former in the direction of the compressed strut inclined 45° from the vertical, and the latter in the z-direction. The value of ϕ can be obtained from the rotation equilibrium of the horizontal section of the solid. In the case of a floor with a rectangular plan, denoted by a i the length of the generic side of the section midline, by d i its distance from any point in the plane of the same section (chosen, for example, inside the section), and denoted by Ω the area enclosed by the middle fiber of the horizontal section, it can be written: = � = � = 2 Ω from which: = ∑ = 2 Ω ; = √ 2 ; = It should be noted that ϕ is derived from Bredt's first formula, as is logical, having considered the resistant section as a hollow section of small thickness. 1 st Effect of torsion: variation of the normal stress in the walls The resultant of all ϕ L acting on the rods of the multiple truss is a tensile vertical force that induces a variation of

the normal stress in the walls of the building (Fig. 7a): = � = � = ∑ � = 2 Ω where p is the perimeter of the midline of the horizontal section.

Fig. 7. Effects of torsion. Left: variation of normal stress; right: unbalanced thrust on cornerstone. ϕ is the flow of tangential stresses; ϕ L is the component of ϕ in the z-direction ; ϕ C is the compressive stress acting on the generic diagonal of the Rausch truss at the generic node A; ϕ or is the horizontal component of ϕ C ; ϕ vert is the vertical component of ϕ C ; S is the outward thrust (for a unit height h=1). 2 nd Effect of torsion: thrust at the cornerstones. The compressive stress ϕ C acting on the generic diagonal of the Rausch truss, at the generic node A, combines with the normal stress from the other truss rod (fig. 7b). On each wall it is decomposed into a horizontal stress ϕ or and a vertical stress ϕ vert : = √ 2 ⁄ ; = √ 2 ⁄ .