of a concrete wall model. Large figure
ABSTRACT
In this paper, a nonlinear model is used to design reinforcement in concrete walls and deep beams with openings. It was found that a design process that directly starts with a realistic nonlinear model requires many design cycles. Therefore, a design procedure was adopted that starts with a linear-elastic model and refines both the design and the model in subsequent design cycles.
In an example it is shown that large redistributions of forces can occur between the uncracked stage and the ultimate load stage. This can have a substantial influence on the optimal layout of reinforcement in a structural concrete wall.
INTRODUCTION
An important phase in structural design is to find a suitable distribution of forces. In current practise two methods are used to estimate the force flow in reinforced concrete walls; the strut-and-tie method and a linear stress analysis with the finite element method. Recently, Blaauwendraad proposed alternative method for design of reinforced concrete walls and deep beams with openings (Blaauwendraad 1996). This method - referred to as stringer-panel method - is a nonlinear extension of the plastic method of Nielsen (Nielsen 1984) and the elastic method applied by Malerba is a subset of it (Bontempi 1998). The stringer-panel method is simple and yet sufficiently accurate for engineering design. The method makes it feasible to include real nonlinear structural behaviour in an early phase of the design process of structural concrete walls.
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Figure 1: Stringers and panels as building blocks of a concrete wall model. Large figure |
STRINGER-PANEL MODEL
As can be observed in everyday practice, the reinforcement of walls and deep beams consists often of net reinforcement at the surfaces and reinforcing bars along the edges and around holes. Starting from this geometry, a model was developed that consists of stringers which contain main reinforcing bundles and panels which contain a distributed reinforcing net (see Figure 1). A wall can be modelled as an assembly of these stringers and panels, which are in perfect equilibrium (see Figure 3). A stringer is loaded by a constant shear force along its length and discrete normal forces at both ends. A panel is loaded by constant shear forces and normal forces on the edges. The elements of the model are connected by nodes, which takes into account compatibility of deformation.
Two versions of the model were developed: a linear version and a nonlinear version (Hoogenboom 1998). In the linear version the material behaviour of the stringers and panels is linear-elastic and the panels carry only shear forces at the edges. In the nonlinear version, the reinforced concrete material can both crack and crush and the reinforcement can yield and break. In the nonlinear model, normal forces are included at the panel edges.
A graphical user-interface with the name SPanCAD has been developed to handle the model interactively. In SPanCAD, a stringer-panel model can be easily built out of a number of components. Loads can be added and various computations can be made.
Stringer-panel models have been compared with several structures for which experimental data is available. This includes slender beams, deep beams, continuous deep beam and shear walls. The agreement between the behaviour of the model and the experiments was found to be sufficient for engineering design (Hoogenboom 1998). The advantage of the stringer-panel method compared to the nonlinear finite element method is that it can be used without special education and training. In addition, its behaviour can be computed quickly, which is essential for implementation as a design tool.
PERFORMANCE-BASED DESIGN
In performance-based design the actual way in which the design is obtained is immaterial as long as can be shown that the final result does not fail. This is opposite to traditional design in structural engineering, which uses codes of practice with rules that are deemed to satisfy. In a performance-based design process the structural behaviour is simulated for each load combination in order to evaluate if a limit state occurs. If the performance is not sufficient, the design needs to be improved in a subsequent design cycle.
A performance-based design process can be time consuming because it is not always obvious how to improve a failing structure. For example the software can show that a simple deep beam cracks heavily and fails prematurely but it is not clear if the vertical shear reinforcement or the horizontal bending reinforcement should be strengthened. Even if we can identify a cause of failure, it is not clear how much extra reinforcement is needed to prevent failure. As a consequence the number of successive improvements can become large.
DESIGN PROCEDURE
A design procedure was developed for structural concrete walls. It consists of the 8 steps that are shown below. In this sequence, 3 cycles can be distinguished, in which both the structural design and the model itself are improved. The design procedure starts with a simple linear-elastic model. Subsequently, when we have set some dimensions, parts of the model are turned nonlinear. Finally, when the design is ready, the model is completely nonlinear and its behaviour is a simulation of the real structural performance.
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| Figure 2: Simple deep beam with opening. Large figure |
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| Figure 3: Stringer-panel model of the deep beam in Figure 2. Large figure |
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| Figure 4: The force flow in the linear stringer-panel model for load combination 4. Tension in a stringer is plotted black, while compression is grey. Stringer forces have the unit kN and panel shear flows are in kN/m. Large figure |
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| Figure 5: Envelope of the stringer and panel forces for the load combinations of the ultimate limit state. Large figure |
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| Figure 6: Reinforcement based on the force flow in Figure 5. In addition, over the whole surface a standard net is present of 7Ø-200. (95 kg panel reinforcement and 125 kg stringer reinforcement. Detailing reinforcement is not included.) Large figure |
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| Figure 7: The nonlinear force flow for load combination 4. Substantial redistributions occur in comparison with Figure 4. The principal stresses in the panels have the unit N/mm2. Large figure |
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| Figure 8: The deformed stringer-panel model and cracks of load combination 4. Of course, the deformations are plotted exaggeratedly. The largest displacement is only 3.3 mm. Large figure |
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| Figure 9: Redesigned reinforcement in the deep beam (115 kg panel reinforcement and 130 kg stringer reinforcement. Detailing reinforcement is not included.) Large figure |
EXAMPLE
The deep beam of this example is simply supported and has a large opening (see Figure 2). The stringer-panel model of this beam is shown in Figure 3. Only two independent concentrated forces F1 and F2 act on the beam. Other load cases such as dead load or thermal load are neglected for concision of the example. As presented in Table 1, the load for the service limit state is organised in 3 load combinations. The load for the ultimate limit state consists of 4 combinations. The material properties of the deep beam are assembled in table 2. The material safety factors of the ultimate limit state are 1.15 for the yield strength of reinforcement and 1.2 for the compressive strength of concrete.
Table 1: Load combinations and performance criteria of the limit states
| Service Limit State | Ultimate Limit State | ||||||||||||||||
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| Crack width < 0.4 mm | No collapse |
Figure 4 shows the force flow in the linear-elastic stringer-panel model for load combination 4. Figure 5 shows the envelope of the stringer and panel forces for all load combinations of the ultimate limit state. This figure is used to select initial reinforcement for the beam and concrete cross-section areas for the stringers (Figure 6). At both surfaces a standard net is placed of 7 mm bars with 200 mm spacing in horizontal and vertical direction. This provides a reinforcement ratio of 0.0019. In the panels with large shear forces, additional stirrups are placed. Often, stringer reinforcement is extended to the edges of the wall to include sufficient anchorage or prevent large cracks at bar tips. The size of the concrete section area of tensioned stringers is selected to represent tension-stiffening. The concrete section area of the compressed stringers represents the compression zone of the stress blocks (For more details on application of the stringer-panel method see Hoogenboom 1998).
Table 2: Material properties of the deep beam
| Concrete | Steel Reinforcement | ||||||||||||||||||||||||
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Subsequently, the nonlinear analysis is performed. Most of the cracks are sufficiently small for the load combinations of the service limit state. However, the panel below the opening has a large crack of 0.8 mm width. This shows that the reinforcement below the opening needs to be better distributed. Also some cracks at the top edge of the beam become rather wide. This is caused by a substantial redistribution of the force flow. Figure 7 shows this force flow for load combination 4. Some stringers that were compressed in the linear model are tensioned in the nonlinear model. This can be understood from the deformations in Figure 8. The part above the opening appears to push the left part of the beam outwards.
The reinforcement is redesigned according to Figure 9. Though reinforcement is selected frugally, it is not attempted to use redistributions from the load carrying system of one load combination to another. Finally, simulations of the beam behaviour for all load combinations show that service and ultimate limit states do not occur. Figure 10 shows the behaviour of the beam for load combination 4. In this graph, the vertical axis displays the load factor and the horizontal axis the displacement of the force F1. The full load combination acts on the model at a load factor of 1. The strength of the beam shows to be somewhat more than required and its ductility is sufficient. For the other load combinations of the ultimate limit state, the model shows an ultimate load factor between 0.98 and 1.18.
Detailing of the reinforcement is clearly equally important, but not included in this example. Especially, the left-hand support should be designed carefully because it may be tensioned in both directions. Strut-and-tie models can be used to quantify the detailed local force flow at the anchorage of bars. Also it may be prudent to provide inclined bars at the corners of the opening to disperse cracking. This reinforcement is drawn with the dashed lines in Figure 9.
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| Figure 10: Load displacement curve of the simulated behaviour for load combination 4. Large figure |
CONCLUDING REMARKS
This paper shows that realistic nonlinear models can be used efficiently in the design of reinforced concrete walls and deep beams with openings. The experience so far is that performance-based design does not bring about large reductions in reinforcing steel. Due to the realistic model, it does give more reliable structures, which eventually could lead to smaller safety factors in codes of practice. Perhaps the most important advantage is that a design process with a nonlinear model is more transparent because just one model is sufficient for all load combinations, limit states and construction stages. Together with good software, this can simplify checking and reduce the number of mistakes that are inevitably made in design.
LITERATURE
FOOTNOTE