Page 68 - Advanced Design Examples of Seismic Retrofit of Structures

60 Advanced Design Examples of Seismic Retrofit of Structures

Determination of Center of Rigidity
The center of rigidity is the location on which if the resultant of the story shear
demand is applied, no torsion is generated. Having in mind that torsion is appli-
cable to rigid diaphragm only, determination of the center of rigidity (COR) is
performed for the case in which the example building has rigid diaphragms. For
this purpose, the stiffness of each masonry wall as the only load-bearing struc-
tural elements should be determined. Walls’ lateral in-plane stiffness is an
important parameter that not only is the basis of determination of the COR,
but also is used for determination of the natural period of masonry buildings,
which eventually leads to spectral acceleration demand. Different design codes
include the combination of the flexural and shear stiffness as the spring model in
series for stiffness of masonry walls. These equations, which are based on sim-
ple structural analysis formulae, are usually provided for walls with fixed-free
(cantilever) and fixed-fixed boundary conditions according to Eqs. (2.10) and
(2.11), respectively [2].
1
K ¼ (2.10)
H 3
w H w
+
3E m I g A v G m
1
K ¼ 3 (2.11)
H
w H w
+
12E m I g A v G m
where:
H w ¼solid wall height;
A v ¼shear area;
I g ¼moment of inertia for the gross section representing uncracked
behavior;
E m ¼masonry elastic modulus; and
G m ¼masonry shear modulus.
The cantilever walls can be regarded as the walls in single-story masonry
buildings. The walls with fixed-fixed boundary condition in the example build-
ing are located in the first story. Although ASCE 41-13 regards the piers
between openings as the fixed-fixed wall regardless of the number of story
on which the pier is located, the result of the study by the authors of this chapter
reveal the stiffness of the piers, especially those located in upper stories, can
differ significantly from that according to ASCE 41-13. This difference is orig-
inated from the flexibility of the boundary spandrels in the form of OKB shown
in Fig. 2.33 in the form of the stresses extended from the base of the pier to the
OKB. Also, when calculating the stiffness of the walls in the second story, the
design codes do not consider the flexibility of the first story walls. In response to
this need, the stiffness formulations in the ASCE 41 are modified by the authors
of this chapter to account for the boundary flexibility.