A model is a compromise between simplicity and accuracy. A simple model is easy to understand, easy to handle and it can be analysed fast by software. On the other hand the model may not be too simple because it should accurately describe structural behaviour. 1
In this chapter a number of computations are made with stringer-panel models and the results are compared with experimental data. The accuracy of the model is demonstrated and at the same time the correctness of the software is shown.
An easy way to check the stringer and show its behaviour is a simulation with a longitudinal load. We choose a stringer with a length of 1 m and cross-section dimensions of 100 x 200 mm (see Figure 13). The stringer is rather heavily reinforced with 4 bars with a diameter of 16 mm each (4% reinforcement). It is loaded with two forces F at both ends. The material properties are assembled in Table 1.
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Table 1: Material properties of the test stringer
Figure 13: The stringer is loaded with a force F at both ends.
Figure 14 shows the tensile behaviour of the stringer simulated by SPanCAD. The figure also shows the influence of a safety factor of 1.15 at the reinforcement strength. A safety factor for the concrete tensile strength does not change the curve.
Figure 14: The load-displacement behaviour of the stringer loaded in tension simulated by SPanCAD.
Figure 15 shows the compression behaviour of the stringer simulated by SPanCAD. In the same figure the safe behaviour for design is plotted using a safety factor 1.15 for the bar strength and 1.20 for the concrete strength. As can be seen the stringer stiffness and ductility are not influenced by the factors.
Figure 15: Load-displacement behaviour of the stringer in compression.
In 1980, Collins built a large experimental set-up for tests on reinforced concrete panels. The first series of 30 panels was used to derive the modified compression field theory which is implemented in the stringer-panel model [Vecchio 1986]. In this section, one of the panels of this test series is used to check the panel implementation and show its behaviour (see Figure 16).
Figure 16: The reinforced concrete panel tested by Vecchio
The specimen - referred to as PV20 - is 890 mm square and 70 mm thick. It has two layers of welded wire mesh with a clear cover of 6 mm. The compression strength of the concrete is -19.6 MPa, its Young’s modulus 21778 MPa and its cracking strength 1.47 MPa. The yield strength of the bars is fyx = 460 MPa in the x direction and fyy = 297 MPa in the y direction. 2 The reinforcement ratio is rx = 0.0179 in the x direction and ry = 0.0089 in the y direction while its Young’s modulus is estimated as 210000 MPa.
Figure 17 shows the response that was simulated by SPanCAD together with the experimental result. It can be seen that the agreement is quite close which is no surprise since the constitutive relation of the panel was derived from this very experiment. In fact, the comparison is a test on the correctness of the implemented relations.
Figure 17: Comparison of the simulated panel behaviour with the test result, shows that tension-stiffening and ductility are somewhat underestimated.
In 1984 Thürlimann and colleagues did a series of tests on shear walls with and without flanges [Maier 1985]. The results have frequently been used to validate finite element models and strut-and-tie models [Wang 1988] [Maier 1988] [Welleman 1995]. The tests consisted of ten panels from which the first - referred to as S1 - is selected for computation with a stringer-panel model. Some of the other tests were modelled with stringers and panels too [Heeres 1996]. The computation results show a good agreement with the experiments as demonstrated in this section for wall S1.
The wall geometry and dimensions are shown in Figure 18. It is bordered by two vertical flanges, a solid bottom part and top part. The top is vertically loaded with 433 kN and subsequently with a lateral force F. Table 2 shows the material parameters that were found in the experiment. The reinforcement showed a substantial hardening after the onset of yielding and an excellent ductility. 3
Figure 18: Shear wall S1 tested by Thürlimann
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Table 2: Material Properties of the Test Wall
The stringer-panel model has 1 panel and 4 stringers as shown in Figure 19. The model is hold up with 3 fixed supports. The top load is swept to the vertical stringers and the middle of the panel edge. The lateral load is applied as a described displacement so that softening of the model can be observed. Table 3 shows the dimensions and reinforcement of the stringers. Only the vertical stringers behave nonlinearly since the horizontal stringers are compressed far below their ultimate load. The panel has a thickness of 100 mm and a reinforcement of 1005 mm2/m in the horizontal direction and 838 mm2/m in the vertical direction. Panel margins due to the overlap with the stringers do not occur in this model.
Figure 19: Stringer-panel model of the shear wall with 1 panel and 4 stringers
| Stringer Number | Stringer Cross-section Area | Reinforcement Cross-section Area | Mode |
| 1 2 3 4 |
35000 mm2 35000 168000 168000 |
553 mm2 553 0 0 |
nonlinear nonlinear linear linear |
Table 3: Stringer Dimensions of the Shear Wall.
The weight of the structure was neglected. First the top load was put onto the model and afterward the horizontal load was applied as an increasing displacement. Figure 20 shows the load-displacement curve of the experiment and the stringer-panel computation together with the results of a model with 4 small panels and a model with 9 even smaller panels. The models give a good description of the initial wall stiffness. The model with 1 panel loses much stiffness at about 370 kN and it underestimates the failure load with 12%. Failure is due to crushing of the concrete in a panel together with yielding of its reinforcement. 4
Figure 20: Behaviour of the experiment compared to that of stringer-panel models. The models include 1 panel, 4 smaller panels and 9 even smaller panels.
A striking result is that the ductility of the models decreases with an increasing number of panels. This phenomena is known as mesh dependence in finite element literature: Failure is concentrated in one element and therefore the failure region depends on the element size. Solutions to this problem include relating the failure energy to the element size and higher order continua. 5 However, since no small elements are used in stringer-panel models, the modified compression theory is not extended. The model with 1 panel gives a good approximation for design computations for both stiffness, ultimate load and ductility.
Figure 21 shows the deformation of the model with 4 panels at a load of 161 kN and 628 kN. Figure 22 shows the redistribution of forces that takes place from before cracking to the ultimate load.
Figure 21: Deformation of the model with 4 panels at a horizontal load of 161 and 628 kN. The stringers are drawn blue and the panels yellow. Note that no stringers are present between the panels. In the figures the largest displacement is scaled to the same size.
Figure 22: Redistribution of forces during the simulation. Tensile forces are shown red and compression is green. The width of the colour is proportional to the size of the stringer force.
Figure 23 shows the development of cracks during simulation of the model behaviour. The cracks in the right-hand bottom panel rotates which is consistent with the experiment: In the tested wall the first cracks occurred in the diagonal direction but close to failure new cracks developed almost horizontally at the lower right-hand side of the shear wall [Maier 1985, p. 119].
Figure 23: Formation of cracks during the simulation. Note that only one crack is displayed in each panel and in each stringer end. Clearly, in the real structure many more cracks occurred.
Slender beams are part of almost every reinforced concrete structure, so, many tests have been performed on these structural elements. Bending failure, shear failure and their interaction are well understood and can be described with limit analysis [for example Hsu 1993, pp. 89-93]. The stringer-panel model gives a good description of bending failure because the position and area of the compressed stringers are chosen in agreement with the classical beam theory (see page 19). It was shown that the deflection of beam models and the strains in tensioned stringers are in good agreement with experiments [Hoogenboom 1993]. 6
In this section a series of simulations is presented of slender beams that fail in pure shear. The stringer-panel model has one panel over the height as Figure 24 shows. The panels have no horizontal reinforcement and the vertical reinforcement (stirrups) is varied. The reinforcement in the stringers is chosen such that it does not yield.
Figure 24: A stringer-panel model of the left-hand side of a slender beam.
The model dimensions are not really relevant in this section because only dimension-less quantities are compared. Nevertheless, some model properties are provided with which most of the simulations are performed: The model height is 500 mm and the horizontal distance from the load to the support (shear span) is 1500 mm. The panels have a thickness of 150 mm and all top stringers have a cross-section area of 75000 mm2. The tensioned stringers have up to 12000 mm2 reinforcement which only fits in the beam cross-section when a bottom flange is applied. The effective height of the beam h is 550 mm. The yield strength of the reinforcement fy is 500 MPa and the compression strength of the concrete fc' is 40 MPa.
Figure 25: Results of 178 shear test (dots) compared with stringer-panel computations (line) of slender beams. Horizontally is depicted the degree of shear reinforcement and vertically the maximum shear capacity.
Figure 25 shows the results of 178 shear tests, reported in the literature [Bræstrup 1977], together with the results of stringer-panel computations. At the horizontal axis of Figure 25 the amount of stirrup reinforcement is depicted and at the vertical axis the nominal shear strength. In the figure, V is the shear force in a beam section and r is the stirrup reinforcement ratio. As the graph shows, the stringer-panel model follows the experiments very well with a conservative prediction of the ultimate beam load.
The reason for the conservative strength can be that both dowel action (bottom stringers) and the contribution of the compression zone (top stringers) to the shear strength, are not included in the model.
Figure 26 shows a typical force distribution just before collapse. In some panels the second principal stress appears to cross the crack. This illustrates that the combined reinforcement and concrete model is not coaxial and that aggregate interlock is included. In the simulations it was observed that the cracks in the panel at failure have a smaller angle with the beam axis when the amount of stirrups is reduced. This is consistent with the predictions of plasticity theory [Nielsen 1984, p. 207]. Note that in Figure 26 the right-hand panel is compressed due to the large amount of reinforcement in the bottom stringer. When less stringer reinforcement would be applied the panel would be tensioned instead and a bending failure could occur.
Figure 26: Forces in a beam model just before collapse. This beam is substantially reinforced with 7000 mm2 of stirrups per m beam length.
It is emphasised that it is not our intention to add another method for design of stirrups to the many methods that already exist. The shear tests are only used to validate the model and to show its credibility.
At Cambridge University in the United Kingdom, a series of 8 reinforced concrete deep beams over 3 supports was carefully tested [Ashour 1996] [Ashour 1997]. 7 The first of these tests (CDB1) is modelled with a stringer-panel model and the results are presented in this section.
The continuous beam is shown in Figure 27. The length is 3000 mm, the depth 625 mm and the width 120 mm. The main reinforcement at the bottom edge consists of 4 bars with a diameter of 12 mm. At the top edge the reinforcement is 4 bars with a diameter of 12 mm plus 2 bars with a diameter of 10 mm. 29 Stirrups with a diameter of 8 mm are applied over the hole beam length. 8 Horizontal bars with a diameter of 8 mm over the height complete the reinforcement layout.
Figure 27: The continuous deep beam tested by Ashour
The concrete compression strength measured from cylinder tests was 30.0 MPa and the bending tensile strength 4.24 MPa. The tensile strength of the longitudinal bottom and top reinforcement was 500 MPa while the stirrups and flange reinforcement had a strength of 360 MPa. The elasticity modulus of the reinforcing steel was about 200000 MPa and the concrete elasticity modulus was assumed to be 34000 MPa. Special provisions were made to exclude differential support settlements and friction at the load and support platens.
The experiment shows that the beam collapses at a load of 525 kN at each span. The failure mechanism is typical for continuous deep beams (see Figure 28).
The structure and the load are symmetrical so only half the beam is modelled with a stringer-panel model (see Figure 29). The nonlinear model is (internally) statically undetermined of the third degree.
Figure 28: In continuous deep beams the failure zone typically goes from the middle support to one of the load platens.
Figure 29: The stringer-panel model of half the continuous deep beam consists of 7 stringers, 2 panels, 3 rolling supports, a fixed support and a force.
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Table 4: The dimensions of stringers and panels of the model in Figure 29. As is the reinforcement cross-section area, A is the total stringer cross-section area and t is the panel thickness.
Figure 30 shows the load-displacement curve of the model compared with that of the experiment. The initial stiffness of the model is in good agreement with the experiment. The model fails at 472 kN which is 10% less than the 525 kN collapse load of the experiment. The model behaves more ductile than the experiment.
Figure 30: The load-displacement curves of the experiment and the stringer-panel simulation show that the model collapses at 90% of the experimental ultimate load.
For comparison, the ACI code predicts 10% too much strength for this beam while the CIRIA guide gives 3% too little strength. 8
Figure 31: Force distribution in the stringer-panel model at a load of 152 kN. The model is still uncracked at this small load. The stringer forces have the unit kN while the principal stresses in the panels are in MPa.
Figure 31 and 32 show the force distribution in the model at 152 kN and 455 kN. A considerable redistribution can be observed from the uncracked to the cracked state which was also observed in the experiment. The left-hand support of the model carries initially 32% of the load which increases to 43% close to failure. The experiment showed the same support reaction at a low load but just before failure the left-hand support carried only 37% of the load at one span. So, this stringer-panel model redistributes 14% more than the experimental beam.
Figure 32: Force distribution in the stringer-panel model close to the ultimate load. Note that the largest force and the largest stress are scaled to the same size in this and the previous figure.
The simulations show that a stringer-panel model with large panels results in an acceptable accuracy for design. This is important since large panels have the advantage that they are conveniently manipulated in the graphical user-interface of SPanCAD. Moreover, when the largest possible panel is accurate in any situation, the designer does not have to worry about the accuracy of his model (mesh refinements).
The strength of a stringer-panel model appears conservative in nature. This can be understood if we consider that the model is in essence an equilibrium system. According to plasticity theory this results in an underestimation of the strength. In other words, a real structure somehow finds extra ways to carry forces that are not included in the model.
The stringer-panel model does not produce reliable information on the ductility of a structure yet. This deserves further attention. Also size dependence of the ductility has to be introduced in the panel formulation.
The structural elements of this chapter can be recognised as parts of more complicated structures. Consequently, the validation covers a wide area of applications. Nevertheless, these comparisons of the stringer-panel model with experiments are not enough to establish its validity in all situations. Much more test data is available which should be used to validate or falsify the method and obtain more information on its accuracy.
It can be beneficial to use an accurate model despite that much of its input data is not known accurately. This is because every uncertainty contributes to the safety factors used in design. So, when the model is more accurate we can use smaller material factors or a larger resistance factor. This results in less structural material which can represent a substantial amount of money.
SPanCAD does not support different yield strengths of the reinforcement in a model for the obvious reason that this would lead to much confusion and mistakes at a construction site. In order to make the panel computation of this Section the program was temporarily adapted.
Usually, data on reinforcement hardening is not available in design. Nevertheless, for accurate modelling of this experiment it is necessary to include hardening of the stringer reinforcement.
The order of loading in this experiment is important for the stiffness of the response. The SPanCAD interface does not allow the user to specify the order of loading. All load cases of a combination are simultaneously increased until the full load combination is at the model. The author feels that an order of loading unnecessarily complicates the design process. However, this can be an interesting subject for future research. The SPanCAD computational kernel does support a specific order of loading which enabled us to make the computations of this section.
The ductility of the stringers is formulated independently of the model size in Appendix 1. In the computations of this section, the ductility limit of the stringers was disabled in order to investigate the panel behaviour independently. When the ductility limits of the stringers are active the model with 1 panel fails at a 27.5 mm displacement due to excessive yielding at the bottom end of the right-hand vertical stringer. The ductility limit has no influence on the behaviour of the models with 4 and 9 panels.
The strains are related to crack widths. Since the strains are described well, it can be expected that also crack widths at service load are predicted well. However, because of the random nature of crack widths it is difficult to validate this aspect of the model. The implemented formulas for computing average crack widths in the stringer-panel model are according to the Eurocode [Eurocode 1991].
At Delft University of Technology similar tests on 14 continuous beams were performed [Asin 1994]. The results were only partly published when this text was being prepared.
For the series of 8 continuous deep beam tests the ACI code gives strengths varying from 18% too much to 75% too little. The CIRIA Guide 2 gives strengths varying form 36% too much to 3% too little [Ashour 1996].