4.2 - VSL
4.2 - VSL
4.2 - VSL
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Figure 29: Determining failure mechanisms for two-span beam<br />
in Fig 29 with reference to a two-span beam.<br />
It has been assumed here that the top layer<br />
column head reinforcement is protruding<br />
beyond the column by at least<br />
Ia min ≥ I . (1 - 1 ) (3.16)<br />
√<br />
1 + λ<br />
2<br />
in an edge span and by at least<br />
Ia min ≥ 1 . (1 − 1 ) (3.17)<br />
2 √<br />
1 + λ<br />
in an internal span. It must be noted that Ia min<br />
does not include the anchoring length of the<br />
reinforcement.<br />
In particular, it must be noted that, if I 1 = I 2,<br />
the plastic moment over the internal column<br />
will be different depending upon whether<br />
span 1 or span 2 is investigated.<br />
Example of the calculation of a tendon<br />
extension:<br />
According to [14], which is substantially in<br />
line with the above considerations, the<br />
nominal failure state is reached when with a<br />
determining mechanism a deflection a u of<br />
1/40th of the relevant span I is present.<br />
Therefore equations (3.13) and (3.14) for the<br />
tendon extension can be simplified as<br />
follows:<br />
Without lateral restraint, e.g. for edge spans<br />
of flat slabs:<br />
∆I=0.075 . d p<br />
(3.18.)<br />
With a rigid lateral restraint, e g. for internal<br />
spans of flat slabs:<br />
∆I=0.05 . (0.025 . 1 + 2 . hp) (319.)<br />
Figure 30: Portion of slab in column area; transverse components due to prestress in critical<br />
shear contrary<br />
3.2. Punching shear<br />
32.1. General<br />
Punching shear has a position of special<br />
importance in the design of flat slabs. Slabs, which<br />
are practically always under-reinforced against<br />
flexure, exhibit pronounced ductile bending failure.<br />
In beams, due to the usually present shear<br />
reinforcement, a ductile failure is usually assured in<br />
shear also. Since slabs, by contrast, are provided<br />
with punching shear reinforcement only in very<br />
exceptional cases,because such reinforcement is<br />
avoided if at all possible for practical reasons,<br />
punching shear is associated with a brittle failure of<br />
the concrete.<br />
This report cannot attempt to provide generally valid<br />
solutions for the punching problem. Instead, one<br />
possibile solution will be illustrated. In particular we<br />
shall discuss how the prestress can be taken into<br />
account in the existing design specifications, which<br />
have usually been developed for ordinarily<br />
reinforced flat slabs.<br />
In the last twenty years, numerous design formulae<br />
have been developed, which were obtained from<br />
empirical investigations and, in a few practical<br />
cases, by model represtation. The calculation<br />
methods and specifications in most common use<br />
today limit the nominal shear stress in a critical<br />
section around the column in relation to a design<br />
value as follows [9]:<br />
(3.20.)<br />
The design shear stress value T ud is<br />
established from shear tests carried out on<br />
portions of slabs. It is dependent upon the<br />
concrete strength f c’ the bending reinforcement<br />
content pm’, the shear reinforcement content<br />
pv’,the slab slenderness ratio h/l, the ratio of<br />
column dimension to slab thickness ζ, bond<br />
properties and others. In the various<br />
specifications and standards, only some of<br />
these influences are taken into account.<br />
3.2.2. Influence of post tensioning<br />
Post-tensioning can substantially alleviate<br />
the punching shear problem in flat slabs if<br />
the tendon layout is correct.<br />
A portion of the load is transferred by the transverse<br />
components resulting from prestressing directly to<br />
the column. The tendons located inside the critical<br />
shear periphery (Fig. 30) can still carry loads in the<br />
form of a cable system even after the concrete<br />
compressive zone has failed and can thus prevent<br />
the collapse of the slab. The zone in which the<br />
prestress has a loadrelieving effect is here<br />
intentionally assumed to be smaller than the<br />
punching cone. Recent tests [27] have<br />
demonstrated that, after the shear cracks have<br />
appeared, the tendons located outside the crlncal<br />
shear periphery rupture the concrete vertically<br />
unless heavy ordinary reinforcement is present,<br />
and they can therefore no longer provide a loadbearing<br />
function.<br />
If for constructional reasons it is not possible to<br />
arrange the tendons over the column within the<br />
critical shear periphery or column strip b ck defined<br />
in Fig. 30 then the transfer of the transverse<br />
components resulting<br />
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