Large scale non-linear finite element analyses of
reinforced concrete structures
1,2 1,3 1 2
Morten Engen , Max A. N. Hendriks , Jan Arve Øverli , Erik Åldstedt
1Department of Structural Engineering,
Norwegian University of Science and Technology, 7491 Trondheim, Norway
2
Multiconsult AS, Oslo, Norway
3
Delft University of Technology, Delft, The Netherlands
Abstract
Due to practical and economic reasons the design of complex concrete structures is often based on linear FEA with rather large elements. The inelastic behavior of reinforced concrete can lead to sig-nificant load redistributions which can only be predicted using non-linear FEA (NLFEA). This work aims at developing a method for performing large scale NLFEA of concrete structures, both with respect to material behavior and assessment of the structural safety. The present paper compares results from a simple beam experiment with NLFEA on two different element size scales. The NLFEA on both scales predict consistent ULS capacity. A localized crack pattern was only observed at medium scale, but the corresponding SLS crack widths were conservative due to a non-localized equilibrium path. A scale transition procedure should preserve the non-localized crack pattern.
1 Introduction
During design of large concrete structures, e.g. offshore oil and gas platforms or concrete gravity dams, a list of requirements regarding mandatory rules and regulations, safety and constructability etc. has to be met [1]. Such requirements combined with harsh environmental loads, often lead to a complex geometry, and structural analyses thus become comprehensive and complicated. Due to the high number of load combinations linear Finite Element Analyses (LFEA) of separate load cases are normally carried out in order to take advantage of the principle of linear superposition. The results of LFEA can be combined in in-house developed postprocessor programs to construct load combinations which form the basis for the design.
Due to cracking of concrete and yielding of reinforcement, reinforced concrete behaves inelastic, and significant load redistribution can take place when the structure approaches the ultimate limit state (ULS). The design process described above based on LFEA is not able to predict such redistri-butions, but is considered safe when equilibrium is satisfied for all structural parts, the material strengths are not exceeded and sufficient ductility is achieved, thus satisfying the lower bound theo-rem.
Due to time restrictions and the large number of analyses to be performed, FEA of large concrete structures are performed using rather large finite elements. The present work aims at developing a consistent method for performing large scale non-linear Finite Element Analyses (NLFEA) of con-crete structures, both with respect to choices regarding material behavior and the assessment of the safety of a structure. An essential part of the work consists of benchmark analyses and sensitivity studies of experiments on a wide range of geometries and stress states. As a first step the present paper compares results from NLFEA of a simple beam experiment, performed on two different scales in order to investigate the effect of such a scale transition and particularly the effect of large elements. Estimates regarding both serviceability limit state (SLS) and ULS are presented and discussed briefly.
2 Scale transition
FEA can be categorized based on the size of the finite elements, or size scales [2]. Elements in the order of magnitude of the ribs of the reinforcement are termed small scale or rib scale. Elements in the order of magnitude of the reinforcement bar diameter can be termed medium scale, meso scale or
bar scale, and element sizes in the order of the member dimensions, can be termed large scale or
member scale. The present paper deals with the medium and large scale.
NLFEA on different scales require different material properties of concrete [3]. The medium scale requires a softening relation in tension and a bond-slip relation for the interface between concrete and reinforcement. On the other hand, due to the size of the elements, a bond-slip relation would be
mean-Proc. of the 10th fib International PhD Symposium in Civil Engineering
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ingless at large scale, and perfect bond should be assumed and the tension stiffening effect should be properly assigned to the concrete.
A transition from a smaller to a larger scale involves some kind of homogenization. From medi-um to large scale, the homogenization should implicitly take into account the effect of bond-slip and the capacity of the intact concrete between cracks to carry tensile stresses: tension stiffening.
Constitutive relations for concrete based on the fracture energy in compression and tension and the crack bandwidth [4] are considered mesh size objective when the crack bandwidth is made de-pendent on the finite element size. A simple way of introducing the tension stiffening effect at large scale would be to set the crack bandwidth equal to the crack spacing [5] which could be calculated using formulas from e.g. Model Code 2010 [6].
Lackner & Mang [2] proposed an alternative method based on a bond-slip relation and the expo-nential softening relation for concrete. A one-dimensional composite model at medium scale consist-ing of concrete and tensile reinforcement is constructed. The composite model represents the tensile zone of a typical beam in bending, with length equal to the crack spacing and height equal to the effective height of the tensile zone before cracking calculated according to e.g. Model Code 2010. The first step of the proposed method is to estimate the crack spacing, where it is assumed that the tensile strength reduction of concrete due to cracking is equal to the total force transferred from con-crete to reinforcement due to bond-slip between the crack and midspan of the composite bar model.
The second step of the method is to analytically calculate the tensile response of the composite bar model. The difference between the response of the composite bar model and the bare bars is assigned to the concrete, and the corresponding deformation energy can be used to calculate the modi-fied tensile fracture energy.
3 Case study 3.1 Introduction
A set of beams were tested by Pérez Caldentey et al. [7] in order to investigate the influence of sever-al parameters on the crack pattern. Most of the results can be found in [7] and [8]. The beams were loaded as reversed two-point bending tests until failure, with length of the overhangs and constant moment region equal to 750 mm and 3420 mm, respectively.
In the present study, a beam section with height 450 mm, width 350 mm and a nominal cover of 20 mm was chosen. The longitudinal reinforcement consisted of one layer of 4ø12 as main tensile reinforcement, a layer of 2ø12 in the center of the cross section and 2ø12 as compressive reinforce-ment. 8 mm stirrups with spacing 100 mm were installed in the constant moment region.
The mean and maximum crack spacing was reported to be 182 mm and 320 mm. The reported crack widths were calculated assuming a fully cracked section, and are therefore not included here.
3.2 FE models at different scales
NLFEA were carried out with medium and large scale elements. According to [4] the fracture process zone could be smeared over a width of approximately 2-3 times the maximum aggregate size. This size is confirmed in an independent study [9], where a minimum element size of 3 times the maxi-mum aggregate size was chosen due to the size of typical strain gauges used in experiments forming the basis of constitutive relations for concrete. In order to maintain the idea of homogeneity, the minimum element size should never be chosen less than this value. The maximum aggregate size was assumed to be 16 mm, and the element size for the medium scale analysis was chosen to be 50 mm.
A parametric study was performed in order to find the lowest number of elements over the mem-ber height which could yield correct moment and shear force distribution for a LFEA compared to beam theory. With two elements over the height of the beam, the shear force and bending moment were estimated with a sufficient accuracy. The element meshes are shown in Figure 3. Eight-node plane stress elements with reduced 2x2 and full 3x3 integration were chosen for the medium and large scale analysis, respectively.
The concrete behavior was modelled using a parabolic expression in compression and an expo-nential softening expression in tension, both based on fracture energy, following the recommenda-tions in [5] for the effect of lateral strains. The constant shear retention factor was given a low value equal to 0.1 for numerical reasons [10]. All material parameters were calculated according to [6] and [5] based on an average cylinder strength of 26.9 MPa. The yield strength of the reinforcement was reported to be 500 MPa. The local bond-slip relation proposed in Model Code 2010 was used. Good bond conditions were assumed, and recommended minimum values were chosen for all parameters.
Large scale non-linear finite element analyses of reinforced concrete structures
The point loads were applied as prescribed displacements with constant increments 0.8 mm. A Secant-Newton method in combination with line search was found to give the fastest and most robust convergence rate. A combination of energy and force convergence criteria was chosen, with toleranc-es of 10-3 and 10-2, respectively.
3.3 Scale transition according to Lackner & Mang
An effective height of 100 mm was estimated using expressions in Model Code 2010. Using the two-step procedure outlined in section 2, the modified tensile fracture energy was estimated at 0.159 N/mm, compared to 0.132 N/mm based directly on mean material parameters. The corresponding crack spacing was found to be 256 mm.
3.4 Results of FE analyses
The results of three analyses, all with mean material parameters, will be presented: medium scale, large scale and large scale with modified tensile fracture energy.
The load-displacement plots for the analyses are shown in Figure 1 and compared to the experi-mental result. The circular black dots indicate yielding of the main tensile reinforcement and central tensile reinforcement, respectively. Figure 2 shows the longitudinal strain along the tensile face of the beam. Note the localized strains at medium scale.
Fig. 1 Load-displacement plots for medium scale, large scale and modified large scale analysis. The load is measured at one of the supports, and the displacement is measured at midspan
a) b)
Fig. 2 Longitudinal strain profile along tensile face of beam from midspan to support for load levels up to yield of central longitudinal reinforcement, a) medium scale analysis and b) large scale analysis.
The large scale analyses show a softer behavior. As can be seen in Figure 3 the cracking process takes place over a larger area in the large scale analysis compared to the medium scale analysis. Figure 4 shows the limits of the grey scales used in Figure 3. Note that for the large scale analysis the tensile capacity of the concrete drops to zero for a smaller tensile strain compared to the medium scale
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sis due to the larger crack bandwidth. As can be seen in Figure 1, all analyses predict yielding for the same load level, thus the mentioned differences should only apply for SLS estimates.
The contributions to the total strain energy of the system was calculated and compared to the out-er work pout-erformed by the testing equipment. It was found that the total strain enout-ergy of the system was somewhat lower for both the large scale analyses compared to the medium scale analysis. This was primarily due to a lower energy absorption by the concrete. For the large scale analyses, the total strain energy was almost unaffected by the scale transition procedure.
a)
b)
c)
Fig. 3 Crack patterns: a) experimental [8], b) maximum principal concrete strain at yield of central longitudinal reinforcement for medium scale analysis and c) maximum principal concrete strain at yield of central longitudinal reinforcement for large scale
a) b)
Fig. 4 Softening relations for the a) medium scale and b) large scale, also showing limits of the grey scales used in Figure 3
3.5 SLS predictions
The experimental and predicted crack patterns are shown in Figure 3. Due to the included bond-slip relation, a localized crack pattern can be observed in the medium scale analysis with mean and maxi-mum crack spacing of 1-2 and 4 element widths. The estimated crack spacing is somewhat lower than what was observed in the experiment. No localization is predicted by the large scale analysis.
The maximum crack width for a reference load of 70 kN was calculated using three different methods: 1) the product of the average tensile strain in the reinforcement and the maximum crack spacing, both taken from the NLFEA, 2) the product of the average tensile strain in the reinforcement taken from the NLFEA and the maximum crack spacing according to Model Code 2010 and 3) a calculation according to Model Code 2010 based on a reinforcement stress in the crack taken from the NLFEA. The estimated crack widths are given in Table 1. As expected, method 1 for the medium scale analysis gave the lowest results. Method 3 seems to give the most correct estimate close to the average of method 2 for the two scales. Note that the results are almost unaffected by the scale transi-tion.
3.6 ULS predictions
The ULS capacity of the beam was estimated using three different methods: 1) the Global Resistance Factor Method (GRFM), 2) the method of Estimation of Coefficient of Variation (ECOV) and 3) hand calculation of bending moment and level III shear force resistance as proposed in Model Code 2010. For a description of GRFM and ECOV reference is made to [11] and [6]. The estimated ULS
capaci-Large scale non-linear finite element analyses of reinforced concrete structures
ties at yielding of the longitudinal reinforcement are shown in Table 2, where numbers in parentheses indicate capacity at yield of outer longitudinal reinforcement. It is evident that the scale transition procedure has no influence on the estimated ULS capacity.
Using ECOV, the results are based on two analyses, with the mean and slightly lower characteris-tic material parameters. Only when applying this method a significant change of the estimated capaci-ty for the large scale compared to the medium scale analysis was observed.
Table 1 Crack widths calculated using three different methods [mm]
Analysis Method 1 Method 2 Method 3
Medium scale 0.05 0.08
-Large scale - 0.36 0.19
Large scale modified - 0.35 0.19
Table 2 Design capacities [kN]
Analysis GRFM ECOV MC2010 Medium scale 131 (121) 151 (142)
122 (105) Large scale 131 (119) 141 (123)
Large scale modified 131 (120) 142 (121)
4 Discussion
Cracking is a local process, which should lead to a local increase in concrete crack strains, and elastic unloading in neighboring integration points. Materials which are modelled using a softening relation after the peak stress may tend to stay on an unstable, non-localized equilibrium path. This leads to strains, and thus stresses, which are approximately equal in neighboring integration points, and hence no elastic unloading. This phenomenon would result in a too stiff solution and overestimated energy absorption [12]. It is shown in Figure 2 that this is the case for the analyses on both scales up to the yield of the outer longitudinal reinforcement, and in fact beyond this point for the large scale analysis. This explains the overestimated post-cracking stiffness in the medium scale analysis. The more rapid drop to tensile capacity for the large scale analysis could explain the soft behavior despite the non-localized, unstable equilibrium path.
The chosen scale transition procedure was only partly successful. The initial stiffness after crack-ing in the large scale analysis was only slightly improved, and the estimated crack widths were practi-cally unaffected. On the other hand, the ULS capacity remained unchanged as expected.
When comparing the crack patterns in Figure 3, two obvious differences can be observed, 1) the lack of localized cracking at large scale and 2) the larger area over which the cracking process takes place at large scale. An important effect which should be transferred to the large scale is thus the localization which is triggered by the bond-slip relation at medium scale, and not necessarily the energy dissipation associated with bond-slip which was found to be in the order of magnitude of the energy difference caused by the out-of-balance force due to convergence tolerances. Corresponding results from analyses carried out using the single crack approach [10] would be interesting to investi-gate closer. These effects seem only to have impact on SLS estimates, but if the beam was part of a larger structure, the changed stiffness prior to yielding could result in global effects.
The predicted ULS capacity based on NLFEA is approximately 10-20 % higher than the corre-sponding hand calculated capacities, thus the beam possesses a significant residual capacity compared to the capacity based on beam theory. The different medium scale and large scale estimates of the capacity provided by the ECOV method invites for a further investigation. A possible influence of using inhomogeneous properties might be a part of this investigation.
5 Conclusion
Both the medium and the large scale NLFEA succeed in predicting the ULS capacity of the beam, based on the proposed safety formats in Model Code 2010, both showing significant residual capacity compared to critical section design.
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A localized crack pattern can be estimated by using a medium scale NLFEA, but crack widths calculated at SLS load levels could be rather non-conservative given the case of a non-localized, unstable equilibrium path. Calculation of the crack widths using results from the large scale analysis as input to the formulas in Model Code 2010 seems to yield reasonable results.
As bond-slip is not included in large scale analyses, a scale transition procedure should preserve the localized crack pattern, and not necessarily the energy associated with bond-slip. The procedure investigated in the present paper did not succeed in improving the SLS estimates.
Acknowledgements
The research is carried out as an industrial PhD project funded by Multiconsult AS and the Research Council of Norway. The author would like to thank his supervisors and PhD-candidate Egil Møen for reviewing the manuscript, all colleagues in Multiconsult, department for Marine Structures, and spe-cially Per Horn, Senior Vice President of Multiconsult, for having the courage to initiate the research project.
References
[1] Brekke, D.-E.; Åldstedt, E. & Grosch, H., Design of Offshore Structures Based on Postpro-cessing of Results from Finite Element Analysis (FEA): Methods, Limitations and Accuracy, Proceedings of the Fourth (1994) International Offshore and Polar Engineering Conference, 1994
[2] Lackner, R. & Mang, H. A., Scale Transition in Steel-Concrete Interaction. I: Model, Journal of Engineering Mechanics, 2003, 129(4), 393-402
[3] fib, Bulletin 45: Practitioner’s guide to finite element modelling of reinforced concrete struc-tures, International Federation for Structural Concrete (fib), 2008
[4] Bazant, Z. P. & Oh, B. H., Crack band theory for fracture of concrete, Materials and Struc-tures, 1983, 16(3), 155-177
[5] Hendriks, M. A. N.; den Uijl, J. A.; de Boer, A.; Feenstra, P. H.; Belletti, B. & Damoni, C., Guidelines for Nonlinear Finite Element Analysis of Concrete Structures, RTD 1016:2012, Rijkswaterstaat – Ministerie van Infrastructuur en Milieu, 2012
[6] fib, fib Model Code for Concrete Structures 2010, Ernst & Sohn, 2013
[7] Pérez Caldentey, A.; Corres Peiretti, H.; Iribarren, J. P. & Soto, A. G., Cracking of RC mem-bers revisited: influence of cover, ࢥs,ef and stirrup spacing – an experimental and theoretical
study, Structural Concrete, 2013, 14(1), 69-78
[8] Pérez Caldentey, A.; Corres Peiretti, H. & Peset, J., Estudio de fisuración en muros pantallas. Final report, research project No. IDI-20080937, funded by CDTI, Ministry of Science &Technology, Spain, 2010
[9] Kotsovos, M. D.; Pavlovic, M. N. & Arnout, S., Nonlinear finite element analysis of concrete structures: A model based on fundamental material properties, NUMETA 85: Numerical Methods in Engineering: Theory and Applications. Vol. 2, 1985
[10] Bédard, C. & Kotsovos, M. D., Fracture Processes of Concrete for NLFEA Methods, Journal of Structural Engineering, 1986, 112(3), 573-587
[11] Cervenka, V., Reliability-based non-linear analysis according to fib Model Code 2010, Struc-tural Concrete, 2013, 14(1), 19-28
[12] Crisfield, M. A. & Wills, J., Solution strategies and softening materials, Computer Methods in Applied Mechanics and Engineering, 1988, 66(3), 267-289
1
1D and 3D analysis of anchorage in naturally corroded
specimens
Ignasi Fernandez Perez, Mohammad Tahershamsi, Antonio R. Marí, Jesús
Bairán, Karin Lundgren, Kamyab Zandi, Mario Plos
Dept. of Construction Engineering, Division of Structure Technology, Concrete Structures, Universitat Politècnica de Catalunya-BarcelonaTech,
Jordi Girona 31, Barcelona (08034), Spain
Dept. of Civil and Environmental Engineering, Division of Structural Engineering, Concrete Structures, Chalmers University of Technology,
Sven Hultins gata 8, Gothenburg (412 96), Sweden
Abstract
Corrosion of reinforcement causes cracking and spalling of concrete cover which affects the bond; this is a crucial factor in deterioration of concrete structures. Earlier, tests have been carried out on specimens with naturally corroded reinforcements; in this study, the focus is given to the modelling of these specimens. The aim was to evaluate the scope of simpler and more complex bond models to assess the structural behaviour. A comparison of two approaches to model the anchorage behaviour was done: (a) a one-dimensional analysis, where the bond-slip differential equation with a non-linear bond-slip constitutive model is numerically solved, and the mean bond strength as well as the re-quired anchorage length to anchor the yield force are computed. (b) Finite element (FE) analyses were performed using 3D solid elements for concrete, and beam elements for reinforcement, where the interaction was explicitly described using the same bond-slip constitutive model as in approach (a). The results show differences between the two approaches. Each of the modelling alternatives had both drawbacks and advantages; while the more complicated model accounting for more variables led to more realistic results in comparison with observations, the simpler 1D analysis was very fast and efficient.
Introduction
Corrosion of steel reinforcement is one of the main problems in reinforced concrete structures. Study of corrosion effects is crucial for a better understanding of the structural behaviour of existing im-paired concrete structures. The most severe effect of reinforcement corrosion is the change in bond properties between the steel and concrete. Volumetric expansion of corrosion products causes split-ting stresses along corroded reinforcement which might be harmful to the surrounding material. Gen-erally, the splitting stresses are not tolerated by concrete, and that leads to cracking and eventually spalling of the cover. As the reinforcement becomes more exposed, the corrosion rate may increase and facilitate the deterioration process.
The effect of corrosion process on bond deterioration has been studied extensively by many re-searchers. Several studies have investigated parameters which may influence bond and anchorage capacity of corroded structures, see [1-4]. Even though tests of artificially corroded specimens with low corrosion rates indicated a closer relation to the natural corrosion conditions, literature shows that accelerated corrosion methods may still result in spurious bond deterioration and change the anchor-age behaviour, [5], [6]. Thus, there was a strong need for experiments on naturally corroded speci-mens in order to facilitate the evaluation of the methods and models which were mostly developed based on accelerated-corrosion tests.
In the present paper, different approaches of modelling the anchorage capacity of naturally cor-roded specimens were used. A one-dimensional analytical model was used to calculate the local and global bond-slip behaviour along the naturally corroded reinforcements of the tested beams. Further-more, three-dimensional non-linear finite element analyses were performed to describe the anchorage behaviour of the specimens. The results of the analytical and numerical models with different corro-sion levels were compared with the experimental data.
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Experiments
The tests have been carried out as a part of an experimental campaign at Chalmers University of Technology. The test setup and the test results are described in detail in [7] and [8]. The specimens were extracted from the edge beams of an existing girder bridge with a concrete slab; the edge beams showed different levels of corrosion-induced damage. Based on the damage patterns, the specimens were categorized in three different groups: Reference (R) beams with minor or no-visible damage, Medium (M) damaged specimens with only spalling cracks, and Highly (H) damaged specimens with spalling of the cover. A total of 21 beams were tested in two test series. This work is focused on the second test series consisting of 13 tests described in [8], three of them are presented in this paper. The designed test set-up and specimens geometrical specifications are shown in Figure 1(a) and (b), re-spectively. The edge beams were tested upside down compared to their placement on the bridge.
a) b)
Fig. 1 (a) Test-setup, and (b) cross-section of tested beams
An indirectly supported four point bending test configuration was used for the experiments. The load was applied by means of two hydraulic jacks defining a central constant moment zone and two shear spans. The beams were suspended by means of a frame which at the same time was used to fix the jacks. The support settlements as well as the mid-span deflection were measured by means of dis-placement transducers. The end-slip behaviour of the reinforcement bundles was recorded at both ends. The support zones were strengthened to avoid undesirable failure at these locations.
3 1D Bond-Slip Model 3.1 Bond model
An analytical 1D bond-slip model, developed in [9], was used to analyse the bond-slip behaviour of corroded and uncorroded ribbed steel reinforcement. Accordingly, the differential equation expressing equilibrium conditions along the reinforcement in tension can be defined as in Eq.1:
గήௗమ ௗఙ (1) ή ߬ ൌ Ͳ ή ݀ െ ߨ ௗ௫ ή ସ
where d is the reinforcement diameter, ı is the stress in the reinforcement and IJ is the bond stress. The local bond-slip behaviour is computed based on the CEB-FIB Model Code 10. Corrosion ef-fect of the reinforcement is taken into account by shifting the uncorroded local bond-slip relationship along the slip axis, as suggested in [9]. The model includes a modification of the first branch of the bond-slip curves compared to Model Code 10. To provide enough stiffness in the beginning, the parameter ъ was in this case modified to a value of 18, which is significantly higher than the original value of 2. For further information on how the bond-slip is obtained and the description of the equa-tions used refer to [9]. The equivalent perimeter of the reinforcement bundles was taken following [10], considering the average value between the values given in Figure 2b
The model was developed to analyse the bond-slip behaviour of steel reinforcement within the anchorage length, and the stress in the reinforcement is assumed to be in the elastic range. The defor-mation of the surrounded concrete is neglected, thus all the slip is assumed to be caused due to the reinforcement deformation. By solving the differential equation, the load versus end-slip curves for a
1D and 3D analysis of anchorage in naturally corroded specimens
given embedded length as well as the distribution of slip, bond and steel stresses along the bar are obtained.
3.2 Structural model
The tensile load, Ft, at the end of the remaining available anchorage length is obtained by integrating
the local bond-slip along the bar according to equation (1). To obtain the equivalent load, P, applied on the tested beam, the same structural model presented in [7] was used, see Figure 2a.
a) b)
Fig. 2 (a) Structural scheme to obtain the applied load from the tensile load reinforcement, and (b) Equivalent perimeter for bundled reinforcement bars proposed by [10]
Fig. 3 Top, Bond-slip curves accounting for four different corrosion levels, obtained by means of the 1D bond-slip model for beam M5 (Right figure is enlargement of left). Bottom, Applied load end-slip curves for H5 and M4 specimens accounting for 1 to 4 % corrosion compared experimental data
3.3 Results
A comparison between the analytical model and the test data for some of the specimens is shown in Figure 3. The corrosion level of the damaged specimens was estimated to be around 2-3% [7]. As can be seen in Figure 3, the agreement between the experimental and 1D analysis results is reasonably good; this is true for all specimens. Thus, a reasonable good estimation for the remaining anchorage capacity of corroded specimens can be obtained using the model provided the available anchorage length is known. It should be noted that “all other bond conditions” according to Model Code 10 was assumed to get the local bond-slip curves, taking into account that the specimens were damaged and
Ignasi Fernandez Perez, Mohammad Tahershamsi, Antonio R. Marí, Jesús Bairán, Karin Lundgren, Kamyab Zandi, Mario Plos
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taken from a real bridge; the assumption of “good bond conditions” would have overestimated the capacity. There is not experimental data for the local bond-slip behaviour from the tests.
The 1D model generally reproduces the pre-peak behaviour and the capacity accurately. The post peak behaviour however does not show the same agreement, whereas the remaining capacity in most of the specimens is estimated reasonably well.
4 3D model with 1D bond-slip relationship 4.1 FE model
3D non-linear finite element analyses were performed to describe the behaviour and capacity of the anchorage zone. The commercial software DIANA with pre- and post-processor FX+ was used for the numerical simulations.
The beams that were modelled had the same dimensions as the tested specimens accounting for the different geometrical specifications of each specimen. The exact positions of the reinforcements as well as the stirrups were taken into account in the development of each model. The symmetry of the test-setup allows for half of the span and loading to be considered in the model as shown in Figure 4. Boundary conditions were applied on the top node of the suspension drill, supporting the displace-ments in the vertical axis and out of plane one. The load was applied by means of displacement con-trol.
3D tetrahedral elements (TE12L) were used for the concrete and the reinforcement bars were em-bedded into concrete element; this allowed to describe the concrete-rebar interaction using a bond-slip relation. An analytical local bond-slip relationship for each specimen was defined according to the outputs of the 1D bond-slip; i.e. the same local bond-slip as used in the 1D analyses, Figure 2. Five different bond-slip relations were used for each specimen taking 5 different corrosion levels into account. The loading zone was modelled by means of a wood board and a steel plate using triangular-prism elements (TP18L). An overview of the model and the boundary conditions are shown in Figure 4. Tying elements with eccent properties were disposed in fixed nodes of the support to avoid un-desairble local failure. This tying fixed the upper node with two slave nodes imposing the same rota-tion respect the master node.
Fig. 4 Overview of the FE model
The concrete was modelled with a constitutive model based on non-linear fracture mechanics using a total strain based smeared-crack model with rotating crack approach. Thorenfeldt compression curve was used in order to more realistically describe the behaviour of concrete in compression. The soften-ing behaviour of this curve was adapted to the element size as described in [11]. The tensile behaviour was modelled using a stress-strain relation proposed by Hordijk for tension softening. The reinforcing steel was modelled with an isotropic plasticity model with Von Mises yielding criterion including hardening. The material properties for steel and concrete used in the analysis can be found in [7] and [8]. Compressive concrete strength, fc, was obtained in the laboratory from every specimen by means
of cylindrical specimens. The E modulus tensile concrete strength was the average from 3 of the first round specimens. The energy fracture was calculated by means of the expression proposed in the fib MC 10.
1D and 3D analysis of anchorage in naturally corroded specimens
4.2 Results
The results from 3D FE analysis, see Figure 5, show a pull-out failure in most of the cases for a corro-sion level of 2% and above. For values below 2%, the reinforcement yields. In general, the maximum load capacity is reasonably well described by these analyses. However, the overall stiffness is overes-timated in several cases, most likely because of the pre-existing internal damage of the specimens was not reproduced in the described models. For some of the tests, the initial stiffness of the local bond-slip curve used as input in the model was not enough, yielding high bond-slip values at the first load steps in the applied load versus end slip curves; however, in general the local bond-slip curves yields in a good overall behaviour as it is shown in Figure 5.
Fig. 5 Results of 3D FE analysis in comparison with experiments and 1D Bond-slip Model
Figure 6 shows the crack pattern and the remaining anchorage length for specimen M4. As can be seen, there were two shear cracks between the loading plate and the support in the experiment (marked with 2 and 3 in the figure), while there was only one shear crack in the corresponding region in the analysis. Thus, in the 3D model, all the damage tends to concentrate to the first shear crack. This results in larger remaining anchorage lengths in the analyses than in the experiments for several specimens. Accordingly, the remaining load capacity was slightly higher in comparison to the tested beams as it is directly related with the anchorage length. A probable reason for this discrepancy is that the modelling does not describe the splitting stresses and cracks in a correct way; in the tested beams the splitting cracks connected with the second shear crack. To describe this interaction, simple bond-slip input is too simplified; more sophisticated modelling including the splitting stresses of both the corrosion and the slip would be needed.
42
2
1
3
48
2
1
Fig. 6 Crack pattern comparison of M4 specimen; the anchorage length is given in mm
Ignasi Fernandez Perez, Mohammad Tahershamsi, Antonio R. Marí, Jesús Bairán, Karin Lundgren, Kamyab Zandi, Mario Plos
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Conclusions and Outlook
It was shown that simplified 1D model can be used to obtain rough estimations of the ultimate an-chorage capacity. From that point of view these models are very useful, but they are strongly depend-ent of the available data such as the available anchorage length; data that is not always easy to obtain. More complex models are necessary to describe the overall structural behaviour. Using simple bond-slip relationships between concrete and steel in complex models is useful and relatively fast to obtain a good approximation to the real behaviour. However, it was shown that also this modelling technique has shortcomings, mainly because the splitting action is not included. In future work, a frictional model where the effect of corrosion is taken into account by introducing the swelling action and the flow of rust through cracks will be used, to account for both the internal concrete damage due to steel corrosion and splitting stresses generated by the slip .
References
[1] Regan PE and Kennedy Reid IL (2010) Assessment of Concrete Structures Affected by De-lamination – 2, Graduate School in Concrete Structures – Fratelli Pesenti, Politecnico di Mi-lano, Italy
[2] Zandi Hanjari K, Coronelli D and Lundgren K (2011) Bond capacity of severely corroded bars with corroded stirrups. Magazine of concrete Research 63(12): p532-541
[3] Sæther I (2011) Bond deterioration of corroded steel bars in concrete. Structure and In-frestructure Engineering 7(6): p 415-429.
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