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IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
Automation of the evaluation of bonded and unbonded prestressed concrete beams, according to brazilian
and french code specifications
1. Introduction
Prestressed concrete started to be scientifically developed in the
beginning of the 20th century, and its use was consolidated in the
1940s. According to Rudloff [1], the prestressing of concrete struc-
tures is an intelligent, efficient, and enduring technology. It is intel-
ligent because it profits from the mechanical strength both of iron
and concrete, its main materials. It is efficient because it is techni-
cally better than conventional solutions, producing safer and more
comfortable structures. And it is enduring because it promotes a
long service life of its elements, and therefore the structures may
need low or no maintenance.
Characteristics, such as covering long spans, better control and
reduction of deflections and cracking, application in pre-cast el-
ements, structural recovery and reinforcement, and the use of
prestressing in slender elements, including beamless flat slabs,
are some of the advantages of the widespread application of this
technology both in conventional and bold architectural design, as
well as in small, medium, and large works. The main economic
advantages are reducing the amount of concrete and steel due to
the efficient utilization of more resistant material, and the possibility
of covering longer spans than conventional reinforced concrete by
using slender elements.
This article presents the calculation routines implemented to
automate the evaluation of the compliance of bonded and un-
bonded prestressed concrete flexural members to Brazilian (NBR
6118:2007) and French (Règles BPEL 91) specifications.
The program takes into consideration complete, limited and partial
prestressing, and evaluates, in each case, decompression, crack-
ing formation and width limit states, as well as final prestressing
and ultimate limit states.
The structural analysis is made by employing the hybrid finite ele-
ment for planar frames proposed by Barbieri [2], which allows us-
ing single finite long elements in the span of a beam or column.
The implemented numerical model assumes a non-linear mate-
rial behavior and concrete cracking, geometric nonlinearity, cyclic
loading, and composite construction.
In this article, only some of the evaluation procedures recommend-
ed by the Brazilian and the French norms will be discussed. The
complete text on these criteria was published by Lazzari [3]. In this
article, two design situations, using bonded or unbonded partial
prestressing and considering the use of bonded straight tendons
or unbonded curved tendons, are analyzed. Crack width service
limit state and ultimate limit state results are analyzed according
to both norms.
2. Numerical model
The numerical model based on hybrid finite element formulation,
proposed by Barbieri et al. [4], was used for the analysis of pre-
stressed planar frames. Considering that the interpolation func-
tions of this element are the equations describing force variation
along a tendon, and that forces are interpolation variables, it is
possible to use long elements and a single finite element to de-
scribe a member. This considerably reduces computational efforts.
According to Barbieri [2], the theoretical exact character of the
formulation, that is, the use of the equilibrium condition, indepen-
dently from arbitrary hypothesis in the interpolation function, con-
tributes for adequate modeling of unbonded tendons as the strain
on these structures depends on the curvature of all sections of the
prestressed element.
The adopted numerical model allows considering geometric non-
linearity, cyclic loading, and composite construction, which takes
into account concrete placement in stages. With regards to the
materials, nonlinear constitutive models for concrete, passive-rein-
forcement steel, and prestressing steel were employed, according
to the literature. A five-element Maxwell chain was used to repre-
sent concrete and prestressing steel rheological behavior, taking
into account the characteristics of each material.
2.1 Structure discretization
In the adopted numerical model, each tendon of the planar frame
is represented by its reference longitudinal axis, which matches
the finite element axis. Each finite element consists of two nodes:
an initial and a final node. One or more finite elements presenting
three degrees of freedom per node can be used to model each
frame tendon.
An uneven discrete number of cross sections is defined for each
finite element. The cross sections are used as integration points
inside the element and define the properties along a tendon.
When cross sections are evenly distributed in the element, the in-
tegration techniques of Simpson or Gauss-Lobatto can be used.
Otherwise, that is, when cross section distribution is uneven, only
the integration technique of Gauss-Lobatto can be applied. The
results of this technique are more accurate compared with those
obtained using Simpson’s integration technique. Finite element
stiffness and loading matrices are assembled by integrating cross
sectional properties along its axis. Responses along the element,
such as forces, strains, and displacements are obtained at those
cross sections.
Integration modules along the element are subintervals with con-
stant properties that, together, determine total integration interval,
which corresponds to the element’s length. These modules are
used to confer numerical accuracy to the integration of discontinu-
ous functions, such as long elements with geometrical, material or
loading discontinuities along the axis. These discontinuities may
be geometrical, when sections with different shapes are used, or
material, when different materials are used in the same element,
such as two concrete types. Passive or prestressing reinforce-
ments and loadings are often discontinuous, such as in segmental
prestressing or concentrated loadings, respectively.
An uneven number of thin horizontal planes of any width is dis-
tributed along the cross-sectional symmetry vertical axis. These
horizontal planes represent integration points along the section
height. Regarding the longitudinal axis, each cross section can be
intercepted at any arbitrary point along the vertical axis, and not
necessarily at the section’s barycenter.
In addition to element integration models, there are also sectional
integration models that are used as subintervals bearing homog-
enous properties in order to represent the discontinuities along the
integration axis. Sudden changes in horizontal plane width, as in the
case of sections I or T or in parts of the section containing different
concrete types, which are common in beams and slabs, are some
examples of possible discontinuities along section vertical axis.
Bonded or unbonded passive or active reinforcements are mod-