Viu
(c)
Figure 5.1 Actions on a beam-column joint and joint core.
JOINT BEHAVIOUR 79
Nc
Figure 5.2 A daptation of EC8 exam ple of simplified determ ination of shear forces acting on the concrete core of a joint; equations for V)h and Vjv are given in the text (equations (5.3)-(5.5)).
shown in Figure 5.1 (a). The forces acting on the joint are shown in Figure 5.1(b) and the joint core shears in Figure 5.1(c). The following simplified approach after Cheung et al (1993) assumes that for building fram eworks of regular configuration the seismic shear forces from the beams at the opposite sides of the joint core are similar and equal to Vb and those for the columns equal to Vc. The beam and column shears arise from the change in bending m om ent over their lengths and heights respectively.
Thus:
Vr (Mqa + M ao) IL C
V B ~ (M oc + M co)/L h
Noting that T2 - Cc2 + CS2 and T x = Cc] + CS], then
Vjh = Tx + T2 - Vc (5.1)
Ignoring the influence of axial load, the vertical shear on the joint core may be approxim ated as
Vly = Vih h j h c (5.2)
EC8 adopts a similar approach to the above, and Figure 5.2 is an adaptation of an example given in
the Code of a simplified determ ination of the shear forces acting on the core of a joint. The difference in notation and sense of the applied m om ents should be noted. The values of V]h and V]y are
= YRd [2/3 0 4 sl + A a q/5) f yd] - Vc (5.3) Vjv = yRd [2/3 + A J f yd] - V w + N J2 (5.4) (the factor q/5 was introduced in the 1993 draft of EC8), or, as a simplification
v JV = Kjh h j h c
It can be seen that equation (5.1) is similar to equation (5.3) and equation (5.2) is identical to equation (5.4), simplified version. The factor yRd is introduced to balance the ys value (fyd = / yk/ys) and to com pensate for strain hardening of the rein
forcem ent. The reduction factor of 2/3 is to allow for part of the inclined bond forces flowing out of the core of the joint. Conventionally, the shear force transfer across a joint core can be effected by two mechanisms. These are shown in Figure 5.3. In EC8, these are referred to as the diagonal strut (a) and trusses and struts (b).
5.2.1 Diagonal struts
In the mechanism of Figure 5.3(a) it is assum ed that narrow flexural cracks at the beam ends, caused by previous reversal of m oderate seismic actions, are subsequently closed. H orizontal compressive forces
are transferred through the concrete com pression zone and are com bined with the vertical forces of the com pressed zone of the column. Thus a diag
onal compressive strut is form ed, self-equilibrated within the joint. In this case, it is assum ed that the compressive strength of the concrete, under simul
taneous transverse tension, is governed by the bearing capacity of the joint. In EC8, the integrity of the diagonal strut is assum ed to be m aintained if
V-h ^ 20xRdb-hc for interior joints (5.5) 15xRdb^hc for exterior joints
w here bj is the effective joint width as defined in Figure 5.4.
5.2.2 Trusses and struts
In this m echanism (Figure 5.3 (b)) it is assum ed that if wide flexural cracks at beam ends, caused by previous reversal of m ajor seismic actions, are not closed, the horizontal compressive forces may be transferred only through reinforcem ent of the beam.
It is assumed that a com plete diagonal strut cannot develop and that there is penetration of yield of the reinforcem ent into the joint, resulting in high bond stresses. Thus diagonal cracks within the core of the joint cannot be avoided. Thus an additional m ech
anism is necessary for shear transfer requiring verti
cal and horizontal reinforcem ent. W ith the provision of this reinforcem ent, EC 8 allows the maximum tensile stress in the concrete to be limited to
Ccol
Strut action
(a)
Figure 5.3 M echanisms for effecting shear force transfer across a joint core: (a) diagonal strut; (b) trusses and struts.
BEAM COLUMN
bc > bằ
(a) bj = bc or
bj — bw ^ 0.5 he Whichever is smaller
(b) bj = bw or
bj = be + 0 .5 he In the case of an eccentricity " e "
between the centre lines of the Beam and Column
bi - I bw + bc + 2 be - e Figure 5.4 Effective joint width, b}, in two different cases of beam and column widths: (a) bc > hw; (b) bc < bw.
A svi • fyd
Nc 2
v j
Figure 5.5 Simplified m odel of EC8 for m ean values of shear and norm al forces acting on a concrete joint; these values are given in equations (5.7)-(5.9) in the text.
CTct ^ f c J l c (5-6) where / ctm is the m ean concrete tensile strength; see Table 1.6.
To determ ine the value of crct, EC8 (1988) gives a simplified model, with notation as in Figure 5.5.
M ean values of the shear and norm al stresses acting on the concrete joint may be obtained from the following equations:
= [YRd x 2/3 ( A sl + A s2) / yd - VJ/6j/ijw (5.7) ctv = (N J2 + A J , f yd) l b f e (5.8)
<Th = A h / y d ^ A w ( 5 . 9 )
In equation (5.8), X allows for the precom pression of the longitudinal bars in the column. The principal tensile stress a ct (see Figure 5.6) is obtained from
°ct = K + < 0/2 ± {[(ah - a v)/2]2 + V}>/2 (5-10)
1. The horizontal confinem ent reinforcem ent in beam -colum n joints shall be equal to that provided at critical regions of the column (see C hapter 6).
2. A t least one interm ediate vertical bar is provided betw een column corners at each side of the joint.