4.2 - VSL

2. Fundamentals of the
design process
2.1. General
The objective of calculations and detailed
design is to dimension a structure so that it
will satisfactorily undertake the function for
which it is intended in the service state, will
possess the required safety against failure,
and will be economical to construct and
maintain. Recent specifications therefore
demand a design for the «ultimate» and
«serviceability» limit states.
Ultimate limit state: This occurs when the
ultimate load is reached; this load may be
limited by yielding of the steel, compression
failure of the concrete, instability of the
structure or material fatigue The ultimate
load should be determined by calculation as
accurately as possible, since the ultimate
limit state is usually the determining criterion
Serviceability limit state: Here rules must
be complied with, which limit cracking,
deflections and vibrations so that the normal
use of a structure Is assured. The rules
should also result in satisfactory fatigue
strength.
The calculation guidelines given in the
following chapters are based upon this
concept They can be used for flat slabs
with or without column head drops or
flares. They can be converted
appropriately also for slabs with main
beams, waffle slabs etc.
3. Ultimate limit state
3.1. Flexure
3.1.1. General principles of calculation
Bonded and unbonded post-tensioned
slabs can be designed according to the
known methods of the theories of elasticity
and plasticity in an analogous manner to
ordinarily reinforced slabs [31], [32], [33].
A distinction Is made between the following
methods:
A. Calculation of moments and shear forces
according to the theory of elastimry; the
sections are designed for ultimate load.
B. Calculation and design according to the
theory of plasticity.
Method A
In this method, still frequently chosen today,
moments and shear forces resulting from
applied loads are calculated according to
the elastic theory for thin plates by the
method of equivalent frames, by the beam
method or by numerical methods (finite
differences,finite elements).
6
2.2. Research
The use of post-tensioned concrete and thus
also its theoretical and experimental
development goes back to the last century.
From the start, both post-tensioned beam
and slab structures were investigated. No
independent research has therefore been
carried out for slabs with bonded postensioning.
Slabs with unbonded posttensioning,
on the other hand, have been
thoroughly researched, especially since the
introduction of monostrands.
The first experiments on unhonded posttensioned
single-span and multi-span flat
slabs were carried out in the fifties [1], [2].
They were followed, after the introduction of
monostrands, by systematic investigations
into the load-bearing performance of slabs
with unbonded post-tensioning [3], [4], [5],
[6], [7], [8], [9], [10] The results of these
investigations were to some extent embodied
in the American, British, Swiss and German,
standard [11], [12], [13], [14], [15] and in the
FIP recommendations [16].
Various investigations into beam structures
are also worthy of mention in regard to the
development of unbonded post-tensioning
[17], [18], [19], [20],[21], [22], [23].
The majority of the publications listed are
concerned predominantly with bending
behaviour. Shear behaviour and in particular
punching shear in flat slabs has also been
thoroughly researched A summary of
punching shear investigations into normally
The prestress should not be considered as
an applied load. It should intentionally be
taken into account only in the determination
of the ultimate strength. No moments and
shear forces due to prestress and therefore
also no secondary moments should be
calculated.
The moments and shear forces due to
applied loads multiplied by the load factor
must be smaller at every section than the
ultimate strength divided by the cross-section
factor.
The ultimate limit state condition to be met
may therefore be expressed as follows [34]:
S ⋅ γ f ≤ R (3.1.)
γ m
This apparently simple and frequently
encoutered procedure is not without its
problems. Care should be taken to ensure
that both flexure and torsion are allowed for
at all sections (and not only the section of
maximum loading). It carefully applied this
method, which is similar to the static
method of the theory of plasticity,
reinforced slabs will be found in [24]. The
influence of post-tensioning on punching
shear behaviour has in recent years been the
subject of various experimental and
theoretical investigations [7], [25], [26], [27].
Other research work relates to the fire
resistance of post-tensioned structures,
including bonded and unbonded posttensioned
slabs Information on this field will
be found, for example, in [28] and [29].
In slabs with unbonded post-tensioning, the
protection of the tendons against corrosion is
of extreme importance. Extensive research
has therefore also been carried out in this
field [30].
2.3. Standards
Bonded post-tensioned slabs can be
designed with regard to the specifications on
post-tensioned concrete structures that exist
in almost all countries.
For unbonded post-tensioned slabs, on the
other hand, only very few specifications and
recommendations at present exist [12], [13],
[15]. Appropriate regulations are in course of
preparation in various countries. Where no
corresponding national standards are in
existence yet, the FIP recommendations [16]
may be applied. Appendix 2 gives a
summary of some important specifications,
either already in existence or in preparation,
on slabs with unbonded post-tensioning.
gives an ultimate load which lies on the sate
side.
In certain countries, the forces resulting from
the curvature of prestressing tendons
(transverse components) are also treated as
applied loads. This is not advisable for the
ultimate load calculation, since in slabs the
determining of the secondary moment and
therefore a correct ultimate load calculation
is difficult.
The consideration of transverse components
does however illustrate very well the effect of
prestressing in service state. It is therefore
highly suitable in the form of the load
balancing method proposed by T.Y. Lin [35]
for calculating the deflections (see Chapter
4.2).
Method B
In practice, the theory of plasticity, is being
increasingly used for calculation and design
The following explanations show how its
application to flat slabs leads to a stole
ultimate load calculation which will be easily
understood by the reader.