Basic HTML Version
140
IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
The strut-and-tie models in reinforced concrete structures analysed by a numerical technique
1. Introduction
In structural engineering, most concrete linear elements are de-
signed by a simplified theory, using the Bernoulli hypothesis. How-
ever, the application of this hypothesis to any structural element
can lead to over or under sizing of certain parts of the structure.
This hypothesis is valid for parts of the frame that suffer no in-
terference from rigid regions, such as sections near the columns,
cavities or other areas where the influence of strain due to shear
efforts is not negligible.
Thus, there are structural elements or regions for which the as-
sumptions of Bernoulli hypothesis do not adequately represent the
bending structural behavior and the stress distribution. Structural
elements such as beams, walls, footings and foundation blocks,
and special areas such as beam-column connection, openings
in beams and geometric discontinuities are examples. These re-
gions, denominated
discontinuity regions
or
D-regions
, are limited
to distances of the dimension order of structural adjacent elements
(Saint Venant’s Principle), in which the shear stresses are appli-
cable and the distribution of deformations in the cross section is
not linear.
For a real physical analysis about the behavior of these elements,
the use of the strut-and-tie model, a generalization of the classical
analogy of the truss beam model, is customary. This analogy was
shown by Ritter and Morsch at the beginning of the twentieth cen-
tury, associated with a reinforced concrete beam in an equivalent
truss structure. The discrete elements (bars) represent the fields of
tensile (rods) and compression (compressed struts) stresses that
occur inside the structural element as bending effects. This anal-
ogy has been improved and is still used by technical standards
in the design of reinforced concrete beams in flexural and shear
force and devising various criteria for determining safe limits in its
procedures.
In the 1980s, a Professor at the University of Stuttgart and oth-
er collaborators presented several papers that more adequately
evaluate these D-regions. The pioneering work by Schlaich
et al.
[1] describes the strut-and-tie model more generally, covering the
equivalent truss models and including special structural elements.
The analogy used in the strut-and-tie model uses the same idea as
that of the classical theory in order to define bars representing the
flow of stress trying to create the shortest and more logical path
loads. It is a simple model, but the designer’s experience is nec-
essary to select and distribute the elements of the model in order
to better represent this flow of stresses; the use of more reliable
and automatic tools is made evident for defining its geometric and
structural configuration by graphic and visual tools.
Numerical analysis has been providing these tools for years, with
faster processing, new theories and formulations. Together with
these tools, Topology Optimization (TO) techniques have been em-
ployed via strut-and-tie models in reinforced concrete structures as
shown in Ali [2], Liang and Steven [3], Liang
et al.
[4], Liang
et al.
[5], Liang
et al
. [6], Liang [7], Reineck [8] and Brugge [9].
TO is a recent topic in the field of structural optimization. However,
the basic concepts that support the theory have been established
for over a century, as described Rozvany
et al
. [10]. The great ad-
vantage of TO, as compared to traditional optimization methods,
such as shape or parametric optimization, is that the latter are not
able to change the layout of the original structure, therefore not
helping the project conceptual framework for designing adequate
flow stress.
In topological analysis, two methodologies are important: the micro
and macro approach. The micro approach considers the existence
of a micro porous structure, depending on its geometry and on
the volumetric density of a unit cell representative of the material
properties and its constitutive relations. These properties are rep-
resented by continuous variables, successively distributed in the
space of the extended fixed domain, which is a region where the
structure can exist, (Stump, [11]). An example of this group is the
SIMP (Simple Isotropic Material with Penalization) method, Bend-
søe [12], Rozvany
et al.
[13] and [10].
In the macro approach, the topology of the structure is modified by
the insertion of holes in the field. As an example of this TO group,
ESO (Evolutionary Structural Optimization) can be mentioned,
which is based on solving the objective function when an element
is removed from the finite element mesh, and TSA (Topological
Sensitivity Analysis), based on a scalar function, called derived
topology, which provides the sensitivity of the cost function when
a small hole is created for each set point in the problem domain
(Labanowski
et al
., [14]).
In order to propose an effective tool for developing a strut-and-tie
model, this work uses the TO technique called SESO (Smoothing-
ESO) (Simonetti
et al
., [15]). This technique is a variant of ESO,
whose philosophy lies in verifying if the element is not really neces-
sary to the structure. The new contribution for the ESO technique
is the reduction of stiffness until it no longer has any influence.
That is, the removal of elements is performed smoothly, reducing
the values of the constitutive element of the array, as if it is in the
process of damage and is capable of generating ideal members of
a strut-and-tie model.
2. Evolutionary structural optimization
(ESO)
Xie and Steven [16] developed a very simple way to impose modi-
fications on the topology of a structure, using a heuristic gradual
removal in the mesh of the finite elements, corresponding to the
regions that do not effectively contribute to a better performance
of the structure.
An initial finite element mesh is defined circumscribing the entire
structure, or extended domain of the design to include the boundary
conditions (forces, displacements, cavities and other initial condi-
tions). The parameters of interest for optimization are evaluated in
an iterative process, particularly in this paper, to decrease the weight
by a maximum stress criterion of the structure. Thus, the stresses of
each element are evaluated by using the inequality [1a]:
(1a)
(1b)