Introduction - Unified Theory of Concrete Structures - page 18

as well as the compression failure of the diagonal struts near the point o. According to an
extensive series of full-scale tests by Bai et al. (1991), the reinforcement index, f c ,

of the tension steel at the end section of the beam should be a function of the radius ratio r / d ,
and should be limited to:

ω = ρ f y /

2 r
d

ω = 2 . (1.1)

and

2 r
d

r
d

ω = 0 . 33 + 0 . for ≤ 0 . 6 (1.2)

where d is the effective depth at the end section of the beam. The compressive strength of the
standard cylinder f c has been taken as 77% of the compressive strength of the 20 cm cubes
used in the tests to define the reinforcement index

. Equation (1.1) is governed by the local
crushing of concrete directly under the curved portion of the tension rebars, and Equation (1.2)
by the compression failure of the diagonal struts near the point o.
In addition to the primary tension rebars, the two compression rebars should first follow
the compression struts, og and fo, and then each be extended into the joint region for a length
sufficient to satisfy the compression anchorage requirement, Figure 1.5(b).
A comparison of Figure 1.4(c) with Figure 1.5(b) reveals why the rebar arrangement in
Figure 1.4(c) is deficient. Instead of connecting the top rebar of the beam and the outer rebar
of the column by a single steel bar, these two rebars in Figure 1.4(c) are actually spliced
together along the edge of the outer corner. This kind of splicing is notoriously weak because
the splice is unconfined along the edge of the outer corner and its length is limited. If splices
are desired, they should be located away from the joint region and be placed in a well-confined
region of the member, either inside the column or inside the beam.
In addition to the primary rebars, we must now consider the secondary rebar arrangement.
Secondary rebars are provided for two purposes. First, they are designed to control cracks, and
second, they are added to prevent premature failure. The crack pattern of a knee joint under
a closing moment is shown in Figure 1.5(c). The direction of the cracks is determined by the
stress state in the joint region. To understand this stress state, we notice that the four tension
and compression rebars introduce, through bonding, the shear stresses

ω

τ

around the core area.

σ 1 , which determines the direction of
the diagonal cracks as indicated. To control these diagonal cracks, a set of opposing diagonal
rebars perpendicular to the diagonal cracks are added, as shown in Figure 1.5(d).
Figure 1.5(d) also includes a set of inclined closed stirrups which radiate from the inner
corner toward the outer corner. This set of closed stirrups is added to prevent two possible types
of premature failure. First, since the curved portion of a primary tension rebar in Figure 1.5(b)
exerts a severe bearing pressure on the concrete, it could split the concrete directly beneath,
along the plane of the frame. The outer horizontal branches of the inclined stirrups, which
are perpendicular to the plane of the frame, would serve to prevent such premature failures.
Second, concrete in the vicinity of point o is subjected to extremely high compression stresses
from all directions. The three remaining branches of the inclined closed stirrups would serve
to confine the concrete in this area.

The shear stress produces the principal tensile stress