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ICCE-96, The Second International Conference in Civil Engineering

on Computer Applications, Research and Practice, 6-8 April 1996, Bahrain

COMPUTER SIMULATION AS A DESIGN TOOL FOR CONCRETE STRUCTURES

by Vladimir Cervenka and Jan Cervenka

Abstract

Present advances in computational mechanics make possible the application of nonlinear finite element analysis in practical design. In this way an inconsistency, which exists in the currently used design procedures between the internal force distribution determined from the linear elastic analysis and the section proportioning based on the nonlinear material behavior can be reduced. Realistic constitutive model of concrete behavior enables a simulation of the real structural behavior in service as well as ultimate loading conditions. An application of such approach is demonstrated in this article using the computer code SBETA (ATENA) developed by the authors. Extensive experience with the use of this software in research and design shows that it can significantly improve the quality of structural analysis and serve as an efficient design tool.

1. Introduction

Nonlinear finite element analysis of concrete and reinforced concrete structures has been under steady development in recent decades. The world wide research effort led to the formulation of sound constitutive models as well as numerical techniques for their implementation in computer software. These advances made possible the application of nonlinear finite element analysis in the practical engineering design. The commercial computer programs featuring nonlinear material models are offered and codes of practice are reflecting this progress. Commercially available codes allow for the analysis of entire structures, or structural members with very realistic material behavior, and a simulation of structural performance under real loading conditions is possible.

Nonlinear finite element analysis can eliminate the inconsistency, which exists in the currently used design procedures between the internal force distribution determined from a linear elastic analysis and the section proportioning based on a nonlinear material behavior. The redistribution of internal forces due to a nonlinear material behavior is taken fully into account and a resulting stress and deformation state satisfies all three requirements of mechanics: equilibrium of forces, compatibility of deformations and material laws.

An application of such an analysis using the program SBETA (ATENA) [1] is the subject of this paper. The constitutive model in SBETA (ATENA) covers all important features of concrete and reinforced concrete behavior. It is based on the concept of smeared cracks, damage and fracture mechanics. The smeared concept allows to use the standard finite element technique, known well from the linear elastic analysis. Its extension to the nonlinear behavior in the pre-failure range including both material and geometrical nonlinearities is also well known and easy to implement. For this state prior to failure, SBETA (ATENA) uses a diffused damage concept.

The concrete failure in both, tension (cracking) and compression (crushing) causes discontinuities in the displacement fields, which are in basic disagreement with the assumptions of continuum mechanics. On the macro level concrete failure exhibits itself in the form of strain softening, and is of strongly localized nature. Therefore, special techniques based on localization limiters [8] in form of crack bands are required in order to handle properly the post-peak behavior of concrete within the finite element model.

The reinforcement can be included in two ways, either as smeared one, or as discrete bar elements. The efficient solution techniques, pre- and post-processing routines make it possible to monitor the structural behavior with all details. Large deformations, effects of temperature, shrinkage and prestressing can be also considered. Authors are dealing also with the three-dimensional material behavior. This paper is, however, limited to the two-dimensional plane stress state.

In the first part of the paper, the material model used in SBETA (ATENA) is briefly described.. The second, main part of the paper, is devoted to examples of program applications.

2. SBETA (ATENA) Constitutive Model

The constitutive model in SBETA (ATENA) can reproduce all important features of concrete and reinforced concrete behavior. Only the principles of the constitutive model are described here, to support and explain the results of the presented applications. The details of the model can be found in the program documentation [1], or in other publications [2, 6, 7].

Figure 1: Stress-strain diagram for concrete.	Figure 2: Biaxial failure criterion

The stress-strain law describing the damage of concrete due to a monotonic loading is shown in Figure 1. Complete range of monotonic loading behavior including the post-peak state is covered. The ascending branch in tension is linear, and in compression is a second degree parabola. The peak stresses are determined from the failure function shown in Figure 2, experimentally derived by Kupfer [9]. The post-peak softening is determined from the fracture mechanics considerations as shown in Figures 3 and 4.

Figure 3: Crack opening law	Figure 4: Smeared crack stress-strain law
Figure 5: Compressive stress-displacement law

The cracking of concrete is governed by the nonlinear fracture mechanics. The crack opening law of Hordijk [3] shown in Figure 3 for a fictitious crack is adopted and modified for the smeared crack model, Figure 4. Here, the crack band length Lb is introduced as a localization limiter. Its purpose is to equal the fracture energy Gf used for the propagation of a unit crack length in the fictitious crack and the smeared crack models. The crack band is the product of the finite element dimension projection Lt, Figure 7, and the orientation factor g: Lb=gLt. The orientation factor takes into account the increase of the crack band when crack is inclined with respect to the element sides [10].

The compressive failure is treated in a similar way. Instead of energy as a measure of failure the plastic displacement wco is introduced, Figure 6. The crushing band is defined as Ld=gLc, , in which Lc is the projection of the element size in direction normal to the principal compressive stress, Figure 7.

Figure 6: Compressive stress-strain law	Figure 7: Crack and crush bands

Reinforcement law is defined as multi-linear according to Figure 8. It can describe perfect plasticity as well as hardening. Limited ductility can be also modeled by prescribing a limit strain. The smeared reinforcement is formulated as a material component of the composite material.

The bar reinforcement is modeled by bars embedded in quadrilateral elements, which are displacement-compatible with element boundaries, Figure 9.

Figure 8: Reinforcement stress-strain law	Figure 9: Quadrilateral finite element with embedded reinforcing bar

For the finite element discretization, a quadrilateral element composed of two triangular elements is used. The triangular elements have additional mid-side node as shown in Figure 9. The two degrees of freedom associated with the internal node are condensed-out. The final quadrilateral element has externally only 8 degrees of freedom. The advantage of this element formulation is in case of irregular and distorted shapes. In case of square element it has the same properties as the isoparametric four noded quadrilateral element.

Figure 10: Simulation of tension test of R.C. bar

4. Analysis of Cracked State.

The first example shows the ability of the program to model reinforced concrete structures in serviceability states, where deformation and crack width are important design parameters. They are influenced by two effects, namely, by the reduction of stiffness due to concrete cracking and by the increase of stiffness due to the contribution of the uncracked concrete (so called tension stiffening). This complex behavior is in the standard design methods described only approximately by formulas derived for simple stress states. It will be shown that a model based on fracture mechanics can describe this behavior very realistically and can extend the application to the wide range of more general stress states.

Figure 11: The crack patterns from SBETA (ATENA) analysis

Table 1. Crack width in tension test.

data from	number of cracks	average crack width [mm]
SBETA (ATENA) fixed	6,00	0,12
SBETA (ATENA) rotated	4,00	0,19
experiment	4,00	0,16

The shown examples are analyzed for the purpose of the validation of the FE program. The first case is the simulation of tests performed by Hartl [11], which were used to derive code formulas for tension stiffening. The single reinforcing bar with diameter of 12 mm is embedded in a concrete prism 80x80 mm and subjected to axial tension. The comparison of the load-displacement diagrams from SBETA (ATENA) analysis and experiments is shown in Figure 10. The analyses were made with two crack models: fixed crack and rotated crack. The agreement is very good, and it is found that both crack models provide bounding solutions for the experimental results.

In this analysis, the advantage of the symmetry was used and only 1/4 of the specimen was modeled as shown in Figure 10. The calculated crack patterns are shown in Figure 11 without the FE mesh. The crack widths are summarized in Table 1. It can be seen that the analysis simulates well both, the cracking and the bond failure near the reinforcement. The bond failure is modeled by the cracking of the interface between the concrete and reinforcement. In the case of rotated crack model, the bond failure is more extensive if compared to the fixed crack model leading to larger displacements, greater crack spacing and wider cracks.

Figure 12: Crack patterns of Beam 7 tested by Braam

Figure 13: Average crack width in beam web

The second example is an investigation of crack widths in a reinforced concrete beam subjected to constant bending moment, which was tested by Braam [5]. Beam No. 7 tested by Braam is selected in this paper. It is a four-point bend beam with T-shape cross-section, which means that the central part is subjected to a constant moment. The calculated crack patterns with those obtained from the experiment are compared in Figure 12. In addition, it is possible to determine average crack width along the beam web (Figure 13).

5. Application to Concrete Tunnel Design

This section presents an application of nonlinear analysis to a practical engineering problem, which in this case is a pre-cast reinforced concrete tunnel under a highway Pilsen-Nuernberg. The geometry of this tunnel is shown in Figure 14.

The tunnel is 8 m wide and 5.6 m high with the wall thickness of 0.3 m. The tunnel walls are reinforced along inner and outer surfaces, and the tunnel is loaded with the weight of the soil filling above the tunnel, active soil pressure from both sides and a temporary loading on top of the soil filling due to a vibration machine. The finite element model, which was used in the analysis can be seen on Figure 16. It consists of 550 quadrilateral elements with 8 degrees of freedom. There are 5 elements through the wall thickness. The tunnel is built of pre-cast segments and there are construction joints at approximately third height of the tunnel walls. For comparison, the problem was analyzed using both linear and nonlinear finite element analyses by program SBETA (ATENA). The results are summarized in the form of moment, normal and shear force diagrams in Figures 17 to 22. Linear analyses are always on the left side and nonlinear on the right. In this way it is possible to compare the results, and observe the redistribution of internal forces due to the nonlinear behavior, which in this case can be mainly attributed to concrete cracking (Figure 15). The moment, normal and shear force diagrams are obtained automatically in the program by integration of normal and tangential stresses respectively. The stresses are integrated through the wall thickness along lines normal to the wall central axis in the finite element centers.

Figure 14: Geometry and loading for the highway tunnel.

Figure 16: Finite element model with cracks from nonlinear analysis

Figure 15: Cracked regions in the nonlinear analysis of the highway tunnel.

This analysis demonstrates some of the advantages of nonlinear analysis for structural design. Due to the nonlinear behavior of reinforced concrete where cracks can develop even for relatively low loading levels a significant stress redistribution takes place. In this case, this redistribution decreases the internal moments and increases normal and shear forces, which is a change in the favorable direction, and can result in savings of reinforcement. Generally, in statically indeterminate structures, this redistribution does not have to be always favorable for the design, and nonlinear analysis is the only possibility how to recognize and quantify the redistribution effects.

Figure 17: Moments in the tunnel walls from linear analysis.	Figure 20: Moments in the tunnel wals from nonlinear analysis.
Figure 18: Normal forces in the tunnel wals from linear analysis	Figure 21: Normal forces in the tunnel wals from nonlinear analysis
Figure 19: Shear forces in the tunnel wals from linear analysis	Figure 22: Shear forces in the tunnel wals from nonlinear analysis

6. Offshore Platform

An offshore platform is a large complex structure requiring the employment of advanced design methods. The internal force distribution is statically indeterminate and thus depends on the nonlinear material behavior. As an example a simulation of the behavior of the tri-cell wall from an offshore platform is presented . The failure of the tri-cell wall was the cause of the total loss of the structure as reported in ref. [4].

Figure 23: Offshore platform geometry

The platform was a large cellular reinforced concrete structure as shown in Figure 23. During construction it undergoes submerging for deck-mating and then it is again elevated and positioned in the site. The tri-cell wall must be designed to resist the water pressure due to submerging. The analytical model is made for a symmetrical section of the tri-cell wall. The detail of joint is the same as the one in the real structure, except the wall ends, which are slightly modified to accommodate the loading forces, which substitute the action of the middle part of the wall. This geometry corresponds to the specimens for experimental investigation of the tri-cell wall [5]. Two finite element meshes were used in the analysis, the coarse with 549 and the fine with the 1695 elements, respectively. The material parameters are: concrete compressive strength fc=57.4MPa, steel reinforcement yield stress fy=550 MPa. Additional parameters needed for the nonlinear constitutive law in SBETA (ATENA) were derived form the compressive strength. Three loading forces represent the real loading conditions: 1.axial normal force in the large cell wall, 2.excentric normal force in the inclined tri-cell wall, 3.shear force normal to the tri-cell wall (representing the water pressure inside the tri-cell; in the FE-model this inclined force is substituted by two orthogonal components). The relation of the amplitudes of these forces varied through the loading. The same load history as the one defined for experiments was used for the loading of FE-models. Due to the nonproportional loading the Newton-Raphson iteration method had to be used.

The reinforcing is shown in Figures 25 and 27. The critical shear section is reinforced by the T-headed bars. The length of the T-headed bars is important for the shear resistance. Two cases are studied: case Y2 with short bar, Y7 with long bar.

The analysis was performed for both cases of reinforcement and for two FE meshes with the prescribed non-proportional loading. The loading forces are related to the water pressure and therefore, the water depth is taken as an independent measure of loading. The analysis provided a detail insight into the tri-cell behavior during the loading history, which can be used to study the structural response at any water depth. Only selected results of the failure states are shown here, and the rotated crack model is always used.

The graphical documentation of the failure can be seen in Figures 24 and 26 for the cases Y2 and Y7, respectively. The failure is in both cases in the tri-cell wall under combination of normal and shear forces. The case with the short bar fails at the water depth 70m, Figure 24. The failure plane bypasses the bar and the failure is mainly due to concrete fracture. The case with the long bar fails at the water depth 110m, Figure 26. The bar is activated and the shear failure is not concentrated into one narrow plane. The failure is due to the reinforcement yielding and the concrete fracture. Evidently, the longer T-headed bar increases the shear resistance of the wall. Similar results were obtained for the coarse meshes, but their graphical results are not presented here.

The load-displacement diagrams are compared in Figure 28. The water level obtained in experiments [5] are also shown. The failure loads from computer simulations agree well with experimental results. The fine and coarse mesh solutions give very similar results until the shear failure crack forms. Behavior near the peak load is more influenced by the mesh size, which is probably due to the brittle behavior, and the fine mesh gives consistently lower response. However, the difference is not too large. The response curve has only ascending branch due to the use of the Newton-Raphson method. Thus, the flat plateau near the peak load need not to reflect the correct response of the structure. In reality, the response can be more brittle, with a descending branch. Such response could be simulated numerically if an alternative loading scheme is considered.

Figure 24: Crack pattern with concrete crushing for analysis with short bar	Figure 26: Crack pattern with concrete crushing for analysis with long bar
Figure 25: Reinforcement yielding for analysis with short bar	Figure 27: Reinforcement yielding for analysis with long bar

4. CONCLUDING REMARKS

The nonlinear finite element analysis is an advanced tool for design of reinforced concrete structures. If based on the smeared crack model, which is a well accepted approach, it should meet certain requirements in order to ensure a safe application. On the constitutive level the material model must be able to reproduce the failure modes important for the structural behavior under consideration. In concrete structures with extensive crack propagation the constitutive law has to describe the nonlocal nature of the tensile fracture. This can be modeled by the fracture energy.

Although, the nonlinear analysis offers the possibility of very realistic simulation of the real structural behavior, it has been well recognized, that global analyses of large structures can be too demanding or even practically impossible. In addition, the superposition principle for the determination of the most unfavorable load combinations can not be used in nonlinear analysis. Therefore, it was suggested [5] to combine linear and nonlinear analyses. A linear elastic analysis shall be used for the determination of the internal force distribution within the global structure. The critical parts shall be selected and analyzed using nonlinear analysis. In this manner, nonlinear analysis can serve as a tool for refined dimensioning. The nonlinear FE-models of local regions (such regions are in the strut-and-tie approach called as D-regions) can take into account all real factors and effects influencing the local behavior: reinforcement detailing, multi-axial interaction of stresses, fracture properties, and more. The size of this local region can be larger than the cross-section in traditional dimensioning. In practical situations, a region for nonlinear analysis can consist of structural elements (i.e. walls, beams), joints of structural elements, anchoring details, etc.

The experience with program SBETA (ATENA) shows, that nonlinear FE simulations can be used not only for the analysis of ultimate loads as demonstrated on the above examples, but also for the analysis of the service conditions for crack widths, crack spacing and deflections (see ref.[6,7]). In the first two examples, it was shown that no special treatment of tension stiffening on the constitutive level is necessary, and the tension stiffening effect can be captured solely by strain softening based on fracture energy (Figure 4 ).

www.cervenka.cz - product ATENA

REFERENCES

1. "SBETA (ATENA) Program Documentation", Peekel Instruments, Rotterdam, 1994.

2. Cervenka, V., Pukl, R., "Computer Models Of Concrete Structures", Structural Engineering International, Vol.2, No.2, May 1992, IABSE Zürich, Switzerland, ISSN 1016-8664, pp. 103-107.

3. Hordijk, D.A., "Local Approach to Fatigue of Concrete", Doctor Dissertation, Delft University of Technology, The Netherlands, 1991, ISBN 90-9004519-8.

4. Olsen, T.O., and Olsen, O., "The Loss of the Sleipner Platform, Computer Modeling of Concrete Structures, Proceedings of EURO-C 1994 International Conference held in Innsbruck, Austria, 22nd-25th March, 1994, Edited by Mang, Bicanic, De Borst, Pineridge Press Ltd., pp.1061-1069, ISBN 0-906674-48-0.

5. Holand, I., "The Loss of the Sleipner Condeep Platform", DIANA Computational Mechanics '94, Proceedings of the First International DIANA Conference on Computational Mechanics, Edited by G.M.A.Kusters, M.A.N.Hendriks, Kulwer Academic Publishers, pp.25-36, ISBN 0-7923-3104-4.

6. Cervenka,V., Margoldova,J., "Tension Stiffening Effect in Smeared Crack Model", ACSE EMD Specialty Conference, May 21-24, 1995, University of Colorado, Boulder, USA.

7. Cervenka,V., "Nichtlineare FE-Berechnungen von ebenen Stahlbetontragwerken", Finite Elemente in der Baupraxis, FEM'95, Tagung der GACM, 23-24. Feb. 1995, Universitaet Stuttgart.

8. Bazant, Z., and Oh, B.H., "Crack Band Theory for Fracture of Concrete", Materials and Structures, RILEM, Paris, France, 1983, Vol. 16, pp. 155-177.

9. Kupfer, H., Hilsdorf, H.K., and Ruesch, H., "Behavior of Concrete Under Biaxial Stresses", Journal ACI, Proc. V. 66, No. 8, Aug. 1969, pp. 656-666.

10. Cervenka, V., Pukl, R., Ozbolt, J., and Eligehausen, R., "Mesh Sensitivity Effects in Smeared Finite Element Analysis of Concrete Fracture", FraMCoS-2, ISBN 3-905088-13-4, Zurich, Switzerland, July 25-28, 1995, pp. 1387-1396.

11. Hartl, G., "Die Arbeitslinie "Eingebetete Staehle" bei Erst- und Kurzzeitbelastung", Dissertation, Universitaet Insbruck, 1977.

12. Braam, C.R. "The Behavior of Deep Reinforced Concrete Beams. Experimental Results", Stevin Laboratory, Faculty of Civil Engineering, Delft University of Technology, Delft, The Netherlands, Report No. 25-5.90.5, Sept. 1990.