Delft University of Technology, The Netherlands
ABSTRACT
A computer program has been written to design reinforced concrete walls and deep beams with holes. For this purpose a special discrete element model is developed that combines the advantages of the popular strut-and-tie method and the standard finite element method. The model shows a good agreement with experiments and finite element computations. A design example shows that the program can be used advantageously in engineering practice.
INTRODUCTION
The computational power of desktop computers increases every year. This
brings the advanced analysis programs of research institutes closer to
the designer in engineering practice. Currently, non-linear analysis programs
are only used sometimes to check parts of completed designs. To the best
of our knowledge they are not used to really design structures from the
very start. However, including more accurate structural behaviour will
enable engineers to design less conservatively than is done nowadays.
To date two approaches are commonly used to design structural concrete:
The strut-and-tie method and the linear elastic finite element method.
The strut-and-tie method is simple, economical and safe. However, since
it is essentially a plasticity approach it gives no information on crack
widths in serviceability conditions. As a consequence it is difficult to
convince the responsible building authorities of the durability of a design.
The finite element method on the other hand is very suitable for designing
for serviceability conditions but leads to an uneconomical reinforcement
layout since redistributions of the flow of forces at ultimate limit states
or before are not taken into account.
For an accurate description of cracks and redistribution of forces a non-linear
model is necessary. Non-linear finite element analysis is still very time
consuming and it requires an expert to operate a finite element package.
This is obviously why non-linear analysis is not common in a normal consultancy
company. In order to introduce non-linear techniques in practise we need
a simplified model that can be evaluated in about one minute on a modern
desktop computer. This is the subject of this paper.
Figure 1: Stringers and panels are the building blocks of a concrete wall model.
STRINGER-PANEL MODEL
As can be observed in every day practise, the resulting reinforcement
of walls and deep beams is often concentrated in bundles along the edges
and around holes. Starting from this geometry we developed a discrete element
model in which some elements called stringers contain main reinforcing
bundles and others called panels contain a distributed reinforcing mesh
(see figure 1).
The beam in figure 2 is modelled as an assembly of stringers and panels.
Panels and stringers are in perfect equilibrium and as such the method
is similar to the popular strut-and-tie method (Schlaich et al 1987). However,
a stringer-panel model takes compatibility conditions into account because
the stringers and panels have grips that are connected with nodes. As figure
4 shows, a panel has four grips with which it can be connected to adjacent
stringers. A stringer has three grips with which it can be connected to
other stringers, panels, supports or forces.
The stringer and panel relations were derived with complementary potential
energy and a hybrid method. Both the stringer and the panel have four integration
points. The modified compression field theory (Collins et al 1986) was
adopted to evaluate the material behaviour. In this simple model for membrane
stresses the concrete can both crack and crush and the reinforcement can
yield.
The model is used for both linear and non-linear analysis. In the linear
model the panels carry only shear stresses while the stringers carry all
normal stresses. In the non-linear model, however, shear stresses only
proved to be not enough to describe the wall behaviour accurately. Therefore,
in a non-linear analysis the panels can carry both shear and normal stresses
as figure 2 shows.
Figure 2: An exploded view of a stringer-panel model of a beam shows that all elements are perfectly in equilibrium.
VALIDATION
The stringer-panel model results have been compared with experiments like bending and shear failure of a slender beam, shear walls and continuous deep beams. So far the results show that the accuracy is satisfactory from a design point of view. Here, the computation results are presented and discussed of shear failure of a slender beam.
Figure 3: Results of 178 shear test (dots) compared with stringer-panel computations (line) of slender beams
Figure 3 shows the results of 178 shear test, reported in the literature
(Bræstrup et al 1997) together with the results of a number of stringer-panel
computations. At the horizontal axis the amount of stirrup reinforcement
is depicted and at the vertical axis the shear strength. As the graph shows,
the model gives a conservative prediction of the beam ultimate load. This
result can be understood if one considers that the model is in essence
a equilibrium system. According to the plasticity theory this results in
an underestimate of the strength. In other words, a real structure somehow
finds ways to carry load that are not included in the model. Of course
this is only valid as long as proper detailing is provided since a stringer-panel
model is too course for an accurate prediction of local failure at support
or load platens.
It is emphasised that it is not our intention to add another method for
design of stirrups to the many methods that already exists. The shear tests
are only used to validate the model.
Figure 4: A stringer-panel model can be assembled from simple components.
COMPUTER PROGRAM
The program which is developed is called SPanCAD (see figure 5). Its basic philosophy is that the designer is in control and makes decisions as to how to model the structure and where to place reinforcement and how much. The program assists with computations and gives warnings if mistakes are made.
Figure 5: With the graphical user interface SPanCAD a model can be easily drawn and modified. The computation results can be displayed in several ways.
SPanCAD is an AutoCAD application for Windows that can be used to draw a simple model of a wall or deep beam including supports and loads. The components of the model (see figure 4) can be inserted with a few mouse clicks and moved or reshaped easily. The program displays linear elastic forces due to multiple load combinations in order to assist in the first dimensioning of the reinforcement. Subsequently, for dominant load combinations, the non-linear behaviour of the structure can be simulated including redistributions, crack widths, collapse load and ductility. The results can be used to check and improve the design interactively. The model can also be used three dimensionally for assemblies of walls. For example caissons and box-girder bridges.
DESIGN EXAMPLE
The deep beam of figure 6 was first designed by Dr. Z. Despot at the ETH in Zürich as an example of plastic optimisation (Despot 1995). In this text we give an outline of our design process for this beam using a stringer-panel model.
Figure 6: A stringer-panel model is drawn on top of a deep beam with a hole. The figure shows that there is little choice for the positions of stringers and panels.
The design process can be summarised in 7 steps:
Compared to a traditional design, step 5 and 6 are added which allow the structural engineer to see how his designed wall or deep beam behaves (see figure 7) and to make improvements to the reinforcement. Since the design is checked with simulations, the steps 3 and 4 do not have to be treated very accurately which compensates for the extra time involved in step 5 and 6. However, a first elastic design appears to be necessary to get a fast convergence of the design process. The total amount of reinforcement designed in this way is approximately the same as that of the plastic optimisation due to the large amount of distributed reinforcement required for crack control (Despot 1995). The reinforcement is about 40 % less than that of a design with the linear finite element method.
Figure 7: A simulation (non-linear analysis) of a stringer-panel model of the wall shows that it can carry 10 % more than the load combination requires.
CONCLUSIONS
Non-linear behaviour can be included in a design process of concrete walls and deep beams. This requires a simple model, a new design process and a special graphical user interface. For concrete walls a reduction of the amount of reinforcing steel can be obtained. However, more important is that it results in a reliable design for both ultimate and serviceability conditions.
REFERENCES