Page 450 - Design of Reinforced Masonry Structures

7.12 CHAPTER SEVEN

Rigidity of a cantilevered pier, R , that is, the force required to cause a unit deflection at
c
its free end can be expressed as
R = 1 (7.4)
∆ c
c
For a wall of thickness t and width d, the section properties, viz., the sectional area A
and the moment of inertia I can be expressed as
A = td (7.5)

I = td 3 (7.6)
12
Substitution of Eqs. (7.5) and (7.6) in Eq. (7.3) yields

∆ = ⎛ 1 ⎞ ⎡ 4 ⎛ h ⎞ 3 + 3 ⎛ h ⎞ ⎤ ⎥ (7.7)
c ⎜ ⎝ Et ⎟ ⎢ ⎝ d ⎠ ⎝ d ⎠ ⎦
m ⎠ ⎣
The term h/d in Eq. (7.7) is called the aspect ratio. Substitution of Eq. (7.7) in Eq. (7.4)
gives the rigidity R of a cantilevered wall:
c
R = Et (7.8)
m
⎡ ⎛ h ⎞ ⎛ h ⎞ ⎤
c 3
⎢ 4 ⎝ + 3 ⎝ ⎥
⎣ d ⎠ d ⎠ ⎦
The units of rigidity are kips per inch (same as that for the stiffness of a spring, when

E is expressed in kips per square inch and t in inches). In design of a shear wall structure,
m
one is interested in only the relative rigidity of a shear wall. At a given level of a building,
all shear walls would have (most generally) the same thickness, t, and the same masonry
compressive strength, E . Therefore, the term E t appearing in the numerator of Eq. (7.8)
m
m
is dropped for computational implicity, and the resulting expression is used to express the
relative rigidity of a cantilevered shear wall:
R = 1 (7.9)
⎡ ⎛ h ⎞ 3 ⎛ h ⎞ ⎤
r
⎢ 4 ⎝ + 3 ⎝ ⎥
⎣ d ⎠ d ⎠ ⎦
A comparison of Eqs. (7.8) and (7.9) shows that
R = R c (7.10)
r
Et
m
The relative rigidity, R , given by Eq. (7.9) is a function of h/d ratio only; therefore, this
r
expression can be used to determine the relative rigidity of walls of any thickness and mate-
rial (e.g., concrete). The values of relative rigidity given by Eq. (7.9) for large aspect ratios
(i.e., large values of h/d ratio) are relatively small. Therefore, in order to preserve accuracy
of these small values to three places of decimals, the right side of Eq. (7.9) is multiplied by
an arbitrary factor of 10, the resulting expression being
R = 1 (7.11)
⎡ ⎛ h ⎞ ⎛ h ⎞ ⎤
r 3
⎢ 04 . ⎝ + 03 . ⎝ ⎥
⎣ d ⎠ d ⎠ ⎦