Page 13 - Riem-Vol6_nº3

365 IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 3
S. J. P. J. MARQUES FILHO | B. HOROWITZ
Substituting VN in Equation (2) for its value in Equation (1), on gets:
(3)
Using the unit dummy load method for computing the horizontal
displacement of the upper section of the column one has that the
shear stress corresponding to VC =1 is:
(4)
Thus, the contribution, ΔN , to the horizontal displacement at the top
of the column due to shear stress at the joint is given by:
(5)
Where G is the transverse modulus of elasticity of concrete.
The two different alternatives for the modeling of joint flexibility dis-
cussed in the following sections are:
Adjusted rigid links model.
Scissors model.
3.2 Adjusted rigid links model
As previously discussed, the simple adoption of rigid links in
the interior of the beam/column joint results in stiffness over-
estimation. A frequently adopted alternative is to adjust the
length of the rigid links by multiplying by a factor η ≤ 1, so as
to shorten them, as can be seen in Figure 8(a). This model
implicitly assumes that flexibility is due to bending deformation
inside the joint.
Parametric study conducted by Horowitz and Marques[6]
shows that η values varies between 0.44 and 0.75, and is high-
ly dependent on the ratio of beam to column section heights,
thus not being convenient for general use. This finding is cor-
roborated by the recent study of Birely et al [7] where the ad-
justed rigid links model is used for seismic design and evalu-
ation, based on a data base of experimental results including
45 cross type interior joints. Depending on frame effective
stiffness approach and joint reinforcement design criterion η
values vary all the way from 0 to 1. Therefore this model is not
pursued any further in this study
3.3 Scissors model
Consider the model shown in Figure 8(b) consisting of rigid links at
the ends of the members inside the joint, a hinge, and a torsional
spring. The stiffness of the torsional spring represents the shear
stiffness of the joint. This simple model was proposed by Krawin-
kler and Mohasseb [8] to consider the effect of panel zones of steel
moment-resisting frames.
For the scissors model the moment in the spring is VC H. The dis-
placement at the top of the column only due to the flexibility of the
joint region is given by:
(6)
Where: KNT = stiffness of the spring of the scissors model.
Equating the displacements given by Equations (5) and (6):
(7)
Comparative studies of steel interior joint subassemblages with ex-
perimental results and full planar frames, including P-Delta effects,
reported in Charney and Downs [5] shows that the scissors model
is effective for steel building structures in general.
3.4 Parametric study of complete frame joints
Since simplifying assumptions were adopted regarding the distri-
bution of shear stresses and lever arm of the normal stress resul-
tants, a correction factor, g, is needed in Equation (7). The resulting
expression for the torsional spring is then given by:
(8)
A parametric study was conducted keeping constant the width
of all members at 20 cm and beam spans of 5m between col-
Figure 9 – Distribution of shear stresses
in the interior of the joint