Page 16 - Riem-Vol6_nº3

368 IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 3
Flexibility modeling of reinforced concrete concentric frame joints
4.1.1 Torsional stiffness
Consider a column connected to a torsional member and subject-
ed to a unit torsional moment, as shown in Figure 15(a). Assuming
a linear distribution of moment per unit length and considering that
the maximum value of the torsional moment at the center of joint
is such as to provide a unit area under the diagram (see Figure
15(b)), it follows that the function that expresses the change of the
applied moment to only one of the two arms of the torsion member,
as shown in the discussion of the equivalent frame method in Mac-
Gregor’s textbook [12], is:
(12)
where bC is the width of the column (see Figure 14(a)).
From the loading distribution one can obtain the torsional moment
diagram T(x) by integrating the function t (x), as shown in Figure 15(c):
(13)
Function T(x) expresses the variation of the torsional moment
along the member.
From Figure 15(c) one has that the value of T at the point of fram-
ing of the beam on the column, point A of strip A-A’ in Figure 14(a),
is given by:
(14)
Where bB is the width of the beams.
The value of the derivative of the rotation in each section along
the torsional member with respect to x is obtained by dividing the
torsional moment at each section by CG, as shown in Figure 15(d),
is given by:
(15)
Where G is the transverse modulus of elasticity of the mate-
rial and C is the torsional constant used in the ACI code [11],
given by:
(16)
Where x is the least value between bC and bB, and y being the
largest.
The rotation at the end of the member is the integral of the deriva-
tive of the rotation along the member, and is given by:
(17)
Poisson’s ratio for concrete is taken as 0,2, thus the transverse
modulus of elasticity is 0,42E. Substituting G for E/2, as suggested
in the ACI code [11], and assuming that the average rotation of the
member is one-third of the rotation at its end, one concludes that:
(18)
Torsional stiffness Ktor is the inverse of flexibility and therefore is
given by:
(19)
Figure 15 – Side view of the joint showing the
torsional member; (b) torsional moment applied to the
member; (c) torsional moment diagram in the torsional
member; (d) derivative of rotation along the member
A
B
C
D