7.8 Cantilever beams and corbels
The effective span of a cantilever is either (a) the length to the face of the support plus half the beam's overall depth. It or (b) the distance to the centre of the support if the beam is continuous.
The moments, shears and deflections for a cantilever beam are substantially greater than those for a beam that is supported at both ends with an equivalent load. Also the moments in a cantilever can never be redistributed to other parts of the structure - the beam must always be capable of resisting the full static moment. Because of these factors and the problems that often occur with increased deflections due to creep, the design and detailing of a cantilever beam should be done with care.
Particular attention should be paid to the anchorage into the support of the top tension reinforcement. The steel should he anchored at the support by, at the very least, a full maximum anchorage length beyond the end of its effective span. Some design offices specify an anchorage length equal to the length of the cantilever, mostly to avoid steel fixing errors on site.
Loads on a cantilever can cause the adjacent interior span to be subjected to a hogging moment over all or most of its span. The critical loading pattern for this condition should be as shown in figure 7.21 where the maximum load on the cantilever together with minimum load on the interior span could cause a hogging moment to occur in the interior span.
1.0Ck -1 Figure 7.21
Cantilever loading pattern
7.8.1 Design of corbels
A corbel, as shown in figure 7.22 is considered to be a short cantilever when
0.4.1, < ac < where hc is the depth of the corbel at its junction with the column and ac is the distance from the face of the column to the bearing of the vertical force,
When the vertical load has a stiff bearing ac may be measured to the edge of the bearing but where a flexible bearing is used at is measured to the vertical force.
Corbels can be designed as a strut-and-tie system as illustrated in figure 7.22. In the figure the vertical load F^a at point B is resisted by the force FC(j in the inclined concrete stmt C'B and the force FU| in the horizontal steel tie AB.
The design and detailing of a corbel has the following requirements:
1. The bearing stress of the load on the corbel directly under the load should not exceed 0.48(1 -/ck/250)fck.
2. A horizontal force H&\ 0.2F|.j must also be resisted. This force acts at the level of the top of the bearing, a distance an above the horizontal tie.
3. The main tension steel. /\vmain must be fully anchored into the column and the other end of these bars must be welded to an anchorage device or loops of reinforcing bars.
4. The angle of inclination, 0 of the compression strut must be within the limits
5. The design stress. JCd of the concrete strut must not exceed (<>cc/ck/7c)l'i where:
\ = 1.5. the partial factor of safety for concrete in compression.
Therefore/cj must not exceed 0.34/i.k(l —./ck/250)
6. 1 lorizontal links of total area As, i,nk should be provided to confine the concrete in the compression strut and X^s.iink ^ 0.5/ts „ui„.
The forces on a corbel produce a complex combination of stresses due to bearing, shear, direct compression, direct tension and bending concentrated into a small area. The strut and tie system combined with good detailing is able to simplify the design to produce a workable and safe design.
Figure 7.22 shows the corbel with the inclined strut BC at an angle 0 to the horizontal tie AB. The force in the strut is Fcd and Fld in the horizontal tic respectively. Point B is distance a' (ac + 0.2«h) from the face of the column because of the effect of the horizontal force, H^ ( = 0.2Fk,i).
From the geometry of the triangle ABC. the lever-arm depth is given by 2 = (at. + 0.2an) tan 0.
(a) Force in the concrete strut, FC(i
The design stress for the concrete strut is/C(| = 0.34/^(1 -fck/250). From the geometry of figure 7.22 the width of the concrete strut measured vertically is 2(d -). Hence, the width of the strut measured at right angles to its axis is given by tvsmi| 2(d - z)cos0.
Figure 7.22
Strut and tie system in a corbel
Thus the force Ft.d in the concrete strut is
where is the width of the corbel.
(b) Angle of inclination, 0 of the concrete strut Resolving vertically at point B:
Eej = f'\d sin 6 = Jed x 2(d ■ z) x bw x cos 0 x sin 0
This equation cannot be solved directly for 0 but table 7.2 (overleaf), which has been developed directly from equation 7.16, can be used.
(c) Main tension steel, Av main
Resolving horizontally at B, the force F,d in the steel tie is given by
The total force F[d in the steel tie, including the effect of the horizontal force of 0.2/-'hi, is given by
Rearranging
Table 7.2 Values of ti to satisfy equation 7.16
0 fEd
Table 7.2 Values of ti to satisfy equation 7.16
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