 Original article
 Open Access
Modelling of Stirrup Confinement Effects in RC Layered Beam Finite Elements Using a 3D Yield Criterion and Transversal Equilibrium Constraints
 Péter Zoltán Berke^{1}Email authorView ORCID ID profile and
 Thierry Jacques Massart^{1}
https://doi.org/10.1186/s400690180278z
© The Author(s) 2018
 Received: 28 August 2017
 Accepted: 25 April 2018
 Published: 26 July 2018
Abstract
Apart from its recognized strengthening effect for shear loading, the presence of stirrups in reinforced concrete results in an increase of the ductility of structural members and in the capacity of reaching higher longitudinal compressive stress levels provided by transversal confinement. These effects are usually represented phenomenologically in fibre beam models by artificially increasing the compressive strength and the ultimate compressive strain of concrete. Two numerical formulations for layered beam descriptions accounting explicitly for transversal confinement are implemented and assessed in this contribution. The influence of stirrups is incorporated by means of a multidimensional yield surface for concrete, combined with equilibrium constraints for the transversal direction involving concrete and steel stirrups, and with a concrete ultimate strain dependent on the hydrostatic stress. This contribution focuses on the numerical formulations of both frameworks, and on their assessment against experimental results available in the literature.
Keywords
 reinforced concrete confinement
 stirrups
 layered beam model
 transversal equilibrium constraint
 multiaxial yield criterion
1 Introduction
The robustness of reinforced concrete structures is highly dependent on their capacity to deform in a ductile manner. One of the most important design considerations for ensuring ductility is the provision of transverse reinforcements (stirrups) in order to postpone shear failure and to prevent buckling in columns. Even though stirrups have been used for decades, several questions, such as obtaining the most efficient geometrical stirrup distribution, is still subject of ongoing research (Ding et al. 2011; Corte and Boel 2013; Breveglieri et al. 2015). Stirrups exert lateral compressive stresses on concrete as it expands in the transversal directions due to loading, which improves the structural members strength and ductility. In several scenarios the ductility of the reinforced concrete members is of utmost importance, such as in progressive collapse (Menchel et al. 2009; Hou and Song 2016; Petrone et al. 2016; Rashidian et al. 2016) and resistance against blast loading (Lim et al. 2016; Codina et al. 2016).
The material response of confined concrete is a research topic with a long history, dating back to the 1920s. Research on concrete cylinders confined by hydrostatic pressure or by spiral stirrups was conducted in Richart et al. (1928, 1929), corresponding to one of the pioneering works in the field. Different researchers, such as in Kent and Park (1971), Sargin (1971), Sheikh and Uzumeri (1980), Ahmad and Shah (1982), Park et al. (1982), Scott et al. (1982), Mander et al. (1988a, b), Xiao (1989), Saatcioglu and Razvi (1992), Cusson and Paultre (1995), Hong and Han (2005), carried out experimental and theoretical work on the behaviour of confined concrete and developed several analytical models. Some constitutive models derived from these experiments are employed in building codes for confined concrete behaviour. Yet such models do not allow considering all situations for numerical nonlinear analyses that involve multiaxial loading, since they do not cover all possible stress conditions.
The behaviour of concrete under multiaxial stress conditions was intensively studied to develop a general criterion for irreversible material response (Buyukozturk and Tseng 1984; Imran and Pantazopoulou 1996; Sfer et al. 2002; Tan 2005; Lu 2005; Gabet et al. 2008; Malécot et al. 2010; Hammoud et al. 2014; Zhou et al. 2014) (plasticity and/or failure). Numerous types of concrete failure criteria have been developed, aimed at defining an adequate shape of the limit surface (Chen 1982; Comi 2001; Cho and Park 2003; Dede and Ayvaz 2010; Comi et al. 2012; Bao et al. 2013). For quasibrittle materials like concrete, the failure criterion is affected by the hydrostatic stress, including the effect of the lateral stresses generated by stirrups in reinforced concrete members. The fact that the strength of concrete with active confinement by fluid pressure was observed to be similar to that of concrete with stirrups (Richart et al. 1929) also confirms that the consideration of the multiaxial behaviour of concrete should be taken into account in the representation of the effects of stirrups. Several contributions addressed the multiaxial behaviour of concrete in the modelling of the structural response of reinforced concrete members with stirrups (Cho and Park 2003; Saritas and Filippou 2009; GarzónRoca et al. 2012). Other existing formulations proposed in Petrangeli et al. (1999), Mullapudi and Ayoub (2010), Stramandinoli and Rovere (2012), Mullapudi and Ayoub (2013) used layered (2D)/fibre (3D) beam models. These works focused on the incorporation of shear deformation, and used a forcebased approach with the softened membrane model (Mullapudi and Ayoub 2010, 2013) or modified compression field theory (Stramandinoli and Rovere 2012). In Petrangeli et al. (1999), Mullapudi and Ayoub (2010), Stramandinoli and Rovere (2012) a biaxial stress state was assumed for concrete in beam elements for planar frames with a single transversal equilibrium condition, while (Mullapudi and Ayoub 2013) takes the triaxial stress state into account in a 3D formulation.
The primary goals of the present contribution are the implementation and assessment of two novel stirrup confinement formulations with different assumptions of the transversal equilibrium. This is achieved in a physicallybased layered beam finite element model involving the axial and the lateral reinforcements, and a triaxial constitutive law for concrete. The present work is based on a novel combination of numerical ingredients not yet applied to the study of stirrups effects. A displacementbased 2D layered beam formulation (Santafé Iribarren et al. 2011; Zendaoui et al. 2016) is used with Bernoulli kinematics in a corotational framework. It involves a triaxial stress state for concrete, solving two simultaneous transversal equilibrium constraints in the beam cross section. Additionally, a dependency of concrete ultimate strain on the hydrostatic stress is postulated, which governs the failure by crushing of the concrete layers in the model. A special emphasis is given to the validation of the implemented numerical model through qualitative and quantitative comparison with experimental data reported in the literature.
2 Computational Framework
2.1 Corotational Layered Beam Finite Element
A Bernoulli layered beam finite element for planar frames is used here, without loss of generality for the applicability of the concepts to other beam kinematics. For clarity, the main governing equations are briefly recalled, with more details available in Crisfield (1995), Battini (2002), Oliveira et al. (2014), Oliveira (2015).
The beams are assumed contained in the \((x,\,y)\) plane, z is the outofplane direction. A corotational framework is used to incorporate large rotations and catenary actions. Strains are therefore computed in a rotating reference frame attached to the finite element to uncouple the rigid body rotation from physical strains. Assuming that strains remain small in the local frame, the axial, \(u_{l\,a}\), and transversal displacements, \(v_{l\,t}\), in the rotated element axes are interpolated using linear and cubic shape functions, respectively. The average axial strain, \(\overline{\epsilon }(x) = \displaystyle {\frac{\partial u_{l\,a}(x)}{\partial x}}\) and the beam curvature \(\chi (x) = \displaystyle {\frac{\partial ^2 v_{l\,t}(x)}{\partial x^2}}\), i.e., the generalized strains \({\mathbf{E}} = \left\{ \overline{\epsilon },\,\chi \right\} ^T\) are evaluated with a three point Gauss integration scheme.
2.2 Constitutive Behaviour for Steel and Concrete
2.3 Concrete Ultimate Strain in Confined Concrete
2.4 Layerwise Transversal Equilibrium Formulation for Confined Concrete
Although the transversal equilibrium equations used here are similar to the ones of Petrangeli et al. (1999), Mullapudi and Ayoub (2010), the numerical framework in which they are embedded is different.
The formulation implemented in this work shows similarity to the equilibrium constraint used for 2D beams in Petrangeli et al. (1999), Mullapudi and Ayoub (2010). However, as originality it includes the outof plane transversal stress component as well, in a corotational framework. The corresponding system of equations is nonlinear in the general case, since both materials exhibit a nonlinear behaviour. The problem of finding the set of strains satisfying Eqs. (10) and (11) is solved using a NewtonRaphson iterative scheme on the level of a single layer, as explained in Sect. 2.6.
2.5 Constant OutofPlane Strain Formulation for Confined Concrete
This subsection is dedicated to a second possible formulation based on different kinematical assumptions. The layerwise equality of strain in concrete and steel in the inplane transversal direction y is maintained, but the same strain is assumed in the outofplane direction, z, for all of the confined concrete layers and stirrups. This new assumption avoids defining different out of plane strains for different layers along the height. The main equations of this formulation are as follows.
2.6 Numerical Solution of the Transversal Equilibrium Problems
3 Assessment with Respect to Experimental Results
This section is dedicated to the simulation of the experimental response of structural members reported in the literature. The performance of the proposed stirrup representations is assessed in terms of the strength and ductility enhancement effects and of the correlation to experimental results when available.
3.1 Column Compression
Model parameters used in the column compression and beam bending simulations when \(\epsilon _{u}^{c}\) depends on the hydrostatic stress.
E\(^s\) (GPa)  \(\sigma _{0}^{s}\) (MPa)  K (MPa)  m  \(\epsilon _{u}^{s}\)  \(\Theta _y\)  \(\Theta _z\)  \(\Phi _z\) (m) 

200  469  250  0.1  0.14  0.00899  0.00899  0.00498 
E\(^c\) (GPa)  \(\sigma _{0}^{c}\) (MPa)  \(\alpha \) (1/MPa)  \(\beta \)  \(h_c\)  \(\epsilon _{u\,0}^c\)  k  n 

27  32  0.0962  12.7435  − 5  0.0035  0.00062  0.4 
In Fig. 6, showing the axial reaction force vs. axial displacement curves, the first linear elastic phase is followed by a plastic plateau when stirrups are not considered. The reaction force remains practically constant from 2.8 mm axial displacement until concrete failure, shown by the sudden decrease in the reaction force at 5.3 mm axial displacement. The almost constant normal force in the column without stirrups is explained by a sectional level balance between the exponential softening behaviour of concrete and the nonlinear hardening behaviour of steel in the uniaxial constitutive laws of the layers. Concrete fails in crushing when its ultimate strain (\(\epsilon _{u}^{c} = 0.0053\)) is reached and only the steel reinforcement then contributes to the axial reaction force from this point on. Due to the uniaxial loading condition concrete failure occurs simultaneously in all layers, which also corresponds to the final structural failure. At 5.3 mm axial displacement the reaction force suddenly drops from 19.9 to 4.2 MN due to concrete crushing.
At peak reaction force the axial stress level in concrete is − 31.6 MPa in the unconfined layers and − 39.4 MPa in the confined concrete layers (24.6% relative increase), as shown in Fig. 7. This higher axial stress level in the confined concrete core can be reached due to the − 3.9 MPa transversal confining stresses (same in both transversal directions because the stirrup ratio is the same).
3.2 NumericalExperimental Correlation in a FourPoint Bending Test
Material properties for steel and for the three concrete grades used in the four point bending simulations.
E\(^s\) (GPa)  \(\sigma _{0}^{s}\) (MPa)  K (MPa)  m  \(\epsilon _{u}^{s}\)  

Steel  200  516.5  1.445  1  0.15 
E\(^c\) (GPa)  \(\sigma _{0}^{c}\) (MPa)  \(\alpha \) (1/MPa)  \(\beta \)  \(\epsilon _{u\,0}^c\)  

40 MPa  37.4  55.47  0.1099  23.5574  0.0010 
75 MPa  39.2  74.56  0.1111  32.1400  0.0010 
90 MPa  41.5  81.61  0.1113  35.2978  0.0007 
While several of the material parameters have been measured in Biolzi et al. (2014), such as the elastic modulus and yield strength of steel and of the three concrete grades, the parameters governing the ultimate strain in concrete needed to be determined. The ultimate strain of concrete without confinement effects, \(\epsilon _{u\,0}^c\), was identified in order to have a good match between the experimental and the numerical load vs. displacement curves without stirrups. Its value was observed to be the lowest for concrete with a 90 MPa nominal compressive yield strength. The ultimate concrete strain parameter, \(K=0.00059\), was identified by matching the computationally obtained loaddisplacement curve for the 40 MPa concrete with stirrups with the experimental data, using the previously determined \(\epsilon _{u\,0}^c\) and \(n=0.4\). K and n were subsequently kept the same in the simulations for other grades of concrete. A physically acceptable steel ultimate strain of over 10% was assumed.
The experimental failure mode of all studied beams involved systematically concrete crushing on the top of the beams at midspan and the initiation of the beam failure occurred at less than 3% of axial strain in tension in the reinforcements, therefore no fine tuning of the steel ultimate strain parameter was required. The softening parameter for concrete was kept \(h = 5\), corresponding to a slight softening, as in Oliveira (2015).
Using a parameter set with the majority of the material parameters taken from measurements in Biolzi et al. (2014), the computed structural behaviour for all six considered cases matches well the experimental data, both in terms of load vs. displacement curves and failure mechanism.
First, the response of the numerical beam models without stirrups are examined. All three beams with different concretes exhibit a similar structural behaviour characterized by a single peak in the reaction force (shown by point \(A_0\) in Fig. 10 only for the 40 MPa grade concrete for the sake of good visibility), followed by a steep, monotonous decrease leading to failure by concrete crushing. The slightly lower structural stiffness in the simulations at the start of the loading is potentially due to the zero tension assumption in concrete. The postpeak steep decrease in the reaction force is explained by the continuous progression of failure of the concrete layers in compression starting from the top of the section. The main material parameter controlling the start of failure is thus the ultimate strain in concrete. The single case presenting a discrepancy is the 75 MPa concrete grade, where experimentally a significantly more ductile structural behaviour, similar to the one of beams with stirrups (formation of a plateau at constant reaction force level) was observed. This corresponds to an unexpected structural response that cannot be explained based on the simulations.
Indeed, no mechanism potentially responsible for an increase in the reaction force after the initiation of gradual concrete crushing could be identified in the case of beams without top reinforcements and stirrups.
Between points B and C the reaction force is monotonously increasing with a slope similar to the experimental one. On the sectional level, the stress in the confined concrete core and in the steel reinforcements is increasing with the top reinforcements already working in the plastic regime. Point C is the start of a visible decrease in the tangent of the loaddisplacement curve. This corresponds to the start of the plastic response of the whole cross section, i.e. including the bottom reinforcement in tension. Figure \(C'\) in Fig. 11 shows that all the layers that discretize the cross section of the bottom reinforcement are plastic (black colour) in the increment directly following point C.
Between points C and D the reaction forces of the structure evolve smoothly, their magnitude being governed by the axial behaviour of confined concrete and the hardening behaviour of the top and bottom steel reinforcements. The number of layers developing stresses in the cross section varies smoothly and no additional layers fail until point D is reached.
Point D in Fig. 10 corresponds to the initiation of the crushing of the confined core in the numerical model. It matches the final drop in the experimental loaddisplacement curves associated to a concrete crushing failure mode. In the simulations point D is the last increment in which all the layers in the confined concrete core are unbroken, i.e. in the following increments a gradual failure of confined concrete layers by crushing propagates from the top reinforcement to the bottom (point D’ in Fig. 11). As opposed to the experimental observation, in the numerical model this results in a gradual decrease of the reaction force instead of a large drop. This can be explained by the simplified modelling that does not reproduce the breakdown of the confinement conditions in the concrete core when cracks penetrate it. Such effects are very complex to tackle using a layered beam formulation. Experimentally, this phenomenon can result in various structural responses, potentially depending on the material microstructure of concrete and the generated complex local stress states around the cracks. In the experimental work (Biolzi et al. 2014), both a strong drop in the reaction forces (90 MPa concrete grade) and a more ductile structural response (75 MPa concrete grade) were observed. Therefore, even though no “sudden” structural failure shown by a drop in the reaction force appears in the numerical loaddisplacement curves at point D, it can be considered as the last increment before initiating structural collapse.
Both numerical formulations for stirrups show a similar structural response. The main difference is the magnitude of the vertical midspan displacement at point D. The layerwise stirrup formulation predicts a slightly earlier failure than the formulation based on the constant outofplane strain assumption. This can be explained by the difference in the ultimate strain of the confined concrete layers (Fig. 12) as a result of different stress states in the models. At point D the inplane transversal stress component gives close values in both stirrup formulations (in the order of some tenth of MPa to 1 MPa) but the outof plane transversal stress is significantly higher in the constant outofplane strain (around 9.5–10 MPa) than in the layerwise formulation (some tenth of MPa). This is a natural outcome of the fact that in this formulation only the concrete layers in compression carry stresses that equilibrate the force generated in the outofplane direction by the steel stirrup. The result generally is a smaller effective area (and its evolution) over which concrete stresses can act in the z direction, compared to the layerwise formulation. Because of the nature of the considered concrete yield surface a higher value of the confining stress allows for the development of a higher axial stress. The increased capacity of the confined concrete layers to carry stress results in the development of a higher hydrostatic stress, which influences beneficially the ultimate strain of confined concrete. Considering the experimentallyinspired asymptotic nature of the increase in concrete ultimate strain as a function of the confining hydrostatic stress, the peak value of ultimate strain at point D remains close in both stirrup formulations, as shown in Fig. 12. On the other hand, this particular test was observed to be extremely sensitive to the ultimate strain in concrete, therefore even a small variation in its value is apparent in the loaddisplacement curves.
The vertical displacement at midspan at structural failure matches best the experimental values with the constant outofplane strain stirrup formulation. The general trend is that a higher initial yield strength in concrete leads to a larger increase in the vertical displacement at failure (point D) when stirrups are considered, because the capacity to develop higher stresses in the material (both in the axial and transversal directions) influences beneficially the concrete ultimate strain. The reason why the vertical displacement at failure is close for 75 and 90 MPa concrete grades is that the initial ultimate strain in concrete was taken lower for the 90 MPa concrete (Table 2), based on the fit performed on the loaddisplacement curves without stirrups.
With the adopted Bernoulli beam assumption the reaction forces/moments are generated by the axial stress component in the layers. An explanation for having very similar force levels between points C and D in the load versus displacement curves when using different concrete grades can be given based on the axial stress distribution in concrete in the beam cross section (Fig. 13). As expected, the stress levels are higher when concrete with a higher initial yield strength is considered. The reason why the structural response is practically the same for the three concrete grades, even though the axial stress levels are different, stems from their distribution in the cross section. For the 40 MPa concrete grade the axial stress distribution in the layerwise formulation takes a low peak stress value (− 56.6 MPa) with a larger compressed zone (i.e. in more layers). Conversely the 90 MPa concrete stress distribution has a higher peak stress (− 82.5 MPa) on a smaller number of layers. The integration (finite summation) of these different axial stress contributions in the cross section results in similar generalized stresses on the sectional level.
It is worth mentioning that in all examples above the axial stress in confined concrete reached higher values than the uniaxial yield strength although a softening behaviour is considered (up to 12% higher at point D for 90 MPa concrete grade and the constant outofplane strain formulation) due to the multidimensional stress state in concrete.
4 Discussion
The formulations developed in this contribution perform well in the structural computations in capturing the experimentally observed effects of stirrups on the strength and ultimate strain of confined concrete. The uniaxial compression of a column corresponds to a limit case in which the influence of stirrups is maximized, since the increase in peak concrete axial stress and in ultimate strain due to confinement is present in all layers of the confined core. The good agreement with experimental data in Sect. 3.2 shows that the numerical models remain valid for more complex stress distributions in the beam cross section. The impact of the simplifying assumptions used in the simulations is discussed here.
Even though the structural response of both stirrup formulations are close to each other, the local stress state in the confined concrete layers are quite different. The outof plane transversal stress component can be up to an order of magnitude lower in the case of the layerwise with respect to the constant outofplane strain stirrup formulation in the same increment. In Eq. (11) the total concrete core projected cross section (which are usually similar in both transversal directions) is used as area on which the confined concrete’s stresses counteract the forces generated in the steel stirrups. The layerwise transversal equilibrium conditions and kinematic ties in the y and z directions are similar, leading to stresses in the inplane and in the outofplane transversal directions of the same order of magnitude. In the equilibrium equation of the constant outofplane strain condition the effective concrete area that carries stresses in the outofplane direction is not assumed to be constant. It corresponds to the crosssectional area formed by the layers in compression in this model, which can be small compared to the total concrete core’s crosssectional area in some relevant practical loading cases. Which one of these simplifying modelling assumptions for stirrups and resulting stress states in confined concrete is more realistic should be further investigated in future work, possibly linking it to experimental measurements or full 3D simulations.
Depending on the application the variation of the ultimate strain of confined concrete can be dominant on the structural response (Sects. 3.1 and 3.2) or less important. This depends on the final structural failure mode and, in cases where concrete crushing is a dominant feature in the failure evolution, the parameters \(\epsilon _{u\,0}^c\), K in Eq. (8) should be determined carefully. Maintaining \(n=0.4\) is proposed, since the resulting relationship reflects well the experimentally observed first rapid, than gradual variation in the ultimate strain as a function of increasing hydrostatic stress (Imran and Pantazopoulou 1996). Solving the additional transversal equilibrium constraints obviously results in an increase in computational time compared to the model using uniaxial material representations only: up to 1.5 and 2.5 times for the layerwise and the constant outofplane strain formulations, respectively.
The layerwise stirrup formulation is computationally cheaper, because local iterations are performed separately for each layer. The number of operations depends on the degree of nonlinearity of the single layer problem. In the constant outofplane strain stirrup formulation, iterations are required on the confined core level when only a few fibres exhibit a nonlinear behaviour (Fig. 4b).
5 Conclusion and Perspectives
This contribution presented two numerical formulations for a Bernoulli layered beam element for planar RC frames, allowing for a physicallybased representation of stirrup confinement effects in concrete. In the proposed scheme, the stresses and their nonlinear, multidimensional evolution in concrete are determined as a function of the beam geometry, the yield function and the hardening/softening behaviour of each material. Additionally, the ultimate strain in concrete is taken as function of the hydrostatic stress, following an experimentallyinspired relationship (Imran and Pantazopoulou 1996).

Capturing the stirrup confinement effects is achieved by the coupling of (i) the transversal equilibrium conditions using a multidimensional yield surface for concrete (responsible for the gain in stress levels) and (ii) the increase of the ultimate strain of concrete, defined here as a function of the hydrostatic stress. These result in higher structural strength and ductility and the capacity of the confined concrete volume to dissipate more energy during degradation.

A satisfying agreement between the predictions of the numerical formulations and experimental structural response for several concrete grades was obtained using a physically sound modelling parameter set in Sect. 3.2.

An increase of around 25% in the concrete axial stress in the confined core has been observed numerically, compared to layers without confinement.

Matching the experimental data, an increase factor of 5 in the vertical displacement at midspan at failure was observed when stirrups were considered in the simulations of the four point bending test.

Coupled with the increase in the stress levels this leads to a higher capacity of the structural member to dissipate energy until failure (more than 80% higher than in the uniaxial compression case).

With the values used for the material parameters governing the evolution of the ultimate strain in concrete, both developed stirrup models produce similar structural responses. The local stress state in the confined concrete core is then however different, with the outofplane transversal stress taking significantly higher values in the constant outofplane strain stirrup formulation.
Notes
Declarations
Authors’ contributions
TJM contributed to the formulation of the incorporation of the 3D yield criterion and to the writing, reviewing and enhancement of the manuscript. PZB contributed to the writing and performed the numerical developments and the simulations, including the comparison with experimental data. Both authors read and approved the final manuscript.
Acknowledgements
The authors also acknowledge the support of F.R.S.FNRS Belgium (Grant No. 1.5.032.09.F) for the intensive computational facilities used for this work.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Ahmad, S. M., & Shah, S. P. (1982). Complete triaxial stress–strain curves of concrete confined by spiral reinforcement. Journal of the Structural Division ASCE, 108, 728–742.Google Scholar
 Bao, J. Q., Long, X., Tan, K. H., & Lee, C. K. (2013). A new generalized Drucker–Prager flow rule for concrete under compression. Engineering Structures, 56, 2076–2082.View ArticleGoogle Scholar
 Battini, J.M. (2002) Corotational beam elements in instability problems (PhD thesis, Royal Institute of Technology).Google Scholar
 Biolzi, L., Cattaneo, S., & Mola, F. (2014). Bendingshear response of selfconsolidating and highperformance reinforced concrete beams. Engineering Structures, 59, 399–410.View ArticleGoogle Scholar
 Breveglieri, M., Aprile, A., & Barros, J. A. O. (2015). Embedded throughsection shear strengthening technique using steel and CFRP bars in RC beams of different percentage of existing stirrup. Computers and Structures, 126, 101–113.View ArticleGoogle Scholar
 Buyukozturk, O., & Tseng, T.M. (1984). Concrete in biaxial cyclic compression. Journal of Structural Engineering ASCE, 110, 461–476.View ArticleGoogle Scholar
 Chen, W. F. (1982). Plasticity in reinforced concrete. New York: McArawHill Book Company.Google Scholar
 Cho, C.G., & Park, M.H. (2003). Finite element prediction of the influence of confinement on RC beamcolumns under single or double curvature bending. Engineering Structures, 25, 1525–1536.View ArticleGoogle Scholar
 Codina, R., Ambrosini, D., & de Borbón, F. (2016). Alternatives to prevent the failure of RC members under closein blast loadings. Engineering Failure Analysis, 60, 96–106.View ArticleGoogle Scholar
 Comi, C. (2001). A nonlocal model with tension and compression damage mechanisms. European Journal of Mechanics A/Solids, 20, 1–22.View ArticleMATHGoogle Scholar
 Comi, C., Kirchmayr, B., & Pignatelli, R. (2012). Twophase damage modeling of concrete affected by alkali–silica reaction under variable temperature and humidity conditions. International Journal of Solids and Structures, 49, 3367–3380.View ArticleGoogle Scholar
 Crisfield, M. A. (1995). Non linear finite elements analysis of solids and structuresvolume 1: The essentials. London: Wiley.Google Scholar
 Cusson, D., & Paultre, P. (1995). Stressstrain model for confined highstrength concrete. Journal of Structural Engineering ASCE, 121, 468–477.View ArticleGoogle Scholar
 De Corte, W., & Boel, V. (2013). Effectiveness of spirally shaped stirrups in reinforced concrete beams. Engineering Structures, 52, 667–675.View ArticleGoogle Scholar
 Dede, T., & Ayvaz, Y. (2010). Plasticity models for concrete material based on different criteria including Bresler–Pister. Materials & Design, 31, 278–286.View ArticleGoogle Scholar
 Ding, Y., You, Z., & Jalali, S. (2011). The composite effect of steel fibres and stirrups on the shear behaviour of beams using selfconsolidating concrete. Engineering Structures, 33, 107–117.View ArticleGoogle Scholar
 Gabet, T., Malécot, Y., & Daudeville, L. (2008). Triaxial behaviour of concrete under high stresses: Influence of the loading path on compaction and limit states. Cement & Concrete Research, 38, 403–412.View ArticleGoogle Scholar
 GarzónRoca, J., Adam, J. M., Calderón, P. A., & Valente, I. B. (2012). Finite element modelling of steelcaged RC columns subjected to axial force and bending moment. Engineering Structures, 40, 168–186.View ArticleGoogle Scholar
 Hammoud, R., Yahia, A., & Boukhili, R. (2014). Triaxial compressive strength of concrete subjected to high temperatures. Journal of Materials in Civil Engineering ASCE, 26, 705–712.View ArticleGoogle Scholar
 Hong, K. N., & Han, S. H. (2005). Stress–strain model of highstrength concrete confined by rectangular ties. Journal of Structural Engineering ASCE, 9, 225–232.Google Scholar
 Hou, J., & Song, L. (2016). Progressive collapse resistance of RC frames under a side column removal scenario: The mechanism explained. International Journal of Concrete Structures and Materials, 10, 237–247.View ArticleGoogle Scholar
 Imran, I., & Pantazopoulou, S. J. (1996). Experimental study of plain concrete under triaxial stress. ACI Materials Journal, 93, 589–601.Google Scholar
 Kent, D. C., & Park, R. (1971). Flexural members with confined concrete. Journal of the Structural Division ASCE, 97, 1969–1990.Google Scholar
 Lew, H. S., Bao, Y., Sadek, F., Main, J. A., Pujol, S., & Sozen, M. A. (2011). An experimental and computational study of reinforced concrete assemblies under a column removal scenario. NIST Technical Note 1720.Google Scholar
 Lim, K.M., Shin, H.O., Kim, D.J., Yoon, Y.S., & Lee, J.H. (2016). Numerical assessment of reinforcing details in beamcolumn joints on blast resistance. International Journal of Concrete Structures and Materials, 10, S87–S96.View ArticleGoogle Scholar
 Lu, X. (2005). Uniaxial and triaxial behavior of high strength concrete with and without steel fibers (PhD thesis, New Jersey Institute of Technology).Google Scholar
 Malécot, Y., Daudeville, L., Dupray, F., Poinard, C., & Buzaud, E. (2010). Strength and damage of concrete under high triaxial loading. European Journal of Environmental and Civil Engineering, 14, 777–803.View ArticleGoogle Scholar
 Mander, J. B., Priestley, M. J. N., & Park, R. (1988a). Theoretical stress–strain model for confined concrete. Journal of Structural Engineering ASCE, 114, 1804–1826.View ArticleGoogle Scholar
 Mander, J. B., Priestley, M. J. N., & Park, R. (1988b). Observed stress–strain behavior of confined concrete. Journal of Structural Engineering ASCE, 114, 1827–1849.View ArticleGoogle Scholar
 Menchel, K., Massart, T. J., Rammer, Y., & Bouillard, Ph. (2009). Comparison and study of different progressive collapse simulation techniques for RC structures. Journal of Structural Engineering ASCE, 135, 685–697.View ArticleGoogle Scholar
 Mullapudi, T. R., & Ayoub, A. (2010). Modeling of the seismic behavior of shearcritical reinforced concrete columns. Engineering Structures, 32, 3601–3615.View ArticleGoogle Scholar
 Mullapudi, T. R. S., & Ayoub, A. (2013). Analysis of reinforced concrete column subjected to combined axial, flexure, shear and torsional loads. Journal of Structural Engineering ASCE, 139, 561–573.View ArticleGoogle Scholar
 Oliveira, C. E. M. (2015). The influence of geometrically nonlinear effects on the progressive collapse of reinforced concrete structures (PhD thesis, Universidade Federal de Ouro Preto).Google Scholar
 Oliveira, C. E. M., Batelo, E. A. P., Berke, P. Z., Silveira, R. A. M., & Massart, T. J. (2014). Nonlinear analysis of the progressive collapse of reinforced concrete plane frames using a multilayered beam formulation. IBRACON Structures and Materials Journal, 7, 845–855.Google Scholar
 Oliveira, R. S., Ramalho, M. A., & Corrêa, M. R. S. (2008). A layered finite element for reinforced concrete beams with bondslip effects. Cement & Concrete Composites, 30, 245–252.View ArticleGoogle Scholar
 Park, R., Priestley, M. J. N., & Gill, W. D. (1982). Ductility of squareconfined concrete columns. Journal of the Structural Division ASCE, 108, 929–950.Google Scholar
 Petrangeli, M., Pinto, P. E., & Ciampi, V. (1999). Fiber element for cyclic bending and shear of RC structures. I: Theory. Journal of Engineering Mechanics, 125, 994–1001.View ArticleGoogle Scholar
 Petrone, F., Shan, L., & Kunnath, S. K. (2016). Modeling of RC frame buildings for progressive collapse analysis. International Journal of Concrete Structures and Materials, 10, 1–13.View ArticleGoogle Scholar
 Rashidian, O., Abbasnia, R., Ahmadi, R., & Nav, F. M. (2016). Progressive collapse of exterior reinforced concrete beamcolumn subassemblages: Considering the effects of a transverse frame. International Journal of Concrete Structures and Materials, 10, 479–497.View ArticleGoogle Scholar
 Richart, F. E., Brantzaeg, A., & Brown, R. L. (1928) A study of the failure of concrete under combined compressive stresses. Engineering Experiment Station, University of Illinois, Urbana, Bulletin No. 185.Google Scholar
 Richart, F. E., Brantzaeg, A., & Brown, R. L. (1929). The failure of plain and spirally reinforced concrete in compression. Engineering Experiment Station, University of Illinois, Urbana, Bulletin No. 190.Google Scholar
 Saatcioglu, M., & Razvi, S. R. (1992). Strength and ductility of confined concrete. Journal of the Structural Division ASCE, 118, 1590–1607.View ArticleGoogle Scholar
 Santafé Iribarren, B., Berke, P., Bouillard, Ph, Vantomme, J., & Massart, T. J. (2011). Investigation of the influence of design and material parameters in the progressive collapse analysis of RC structures. Engineering Structures, 33, 2805–2820.View ArticleGoogle Scholar
 Santos, J., & Henriques, A. A. (2015). New finite element to model bondslip with steel strain effect for the analysis of reinforced concrete structures. Engineering Structures, 86, 72–83.View ArticleGoogle Scholar
 Sargin, M. (1971). Stress–strain relationship for concrete and the analysis of structural concrete section (PhD thesis, University of Waterloo)Google Scholar
 Saritas, A., & Filippou, F. C. (2009). Numerical integration of a class of 3D plasticdamage concrete models and condensation of 3D stress–strain relations for use in beam finite elements. Engineering Structures, 31, 2327–2336.View ArticleGoogle Scholar
 Scott, B. D., Park, R., & Priestley, M. J. N. (1982). Stress–strain behaviour of concrete confined by overlapping hoops at low and high strain rates. Journal of American Concrete Institute, 79, 13–27.Google Scholar
 Sfer, D., Carol, I., Gettu, R., & Etse, G. (2002). Study of the behavior of concrete under triaxial compression. Journal of Engineering Mechanics, 128, 156–163.View ArticleGoogle Scholar
 Sheikh, S. A., & Uzumeri, S. M. (1980). Strength and ductility of tied concrete columns. Journal of the Structural Division ASCE, 106, 1079–1102.Google Scholar
 Simo, J. C., & Taylor, R. L. (1985). Consistent tangent operators for rate independent plasticity. Computer Methods in Applied Mechanics and Engineering, 48, 101–118.View ArticleMATHGoogle Scholar
 Stramandinoli, R. S. B., & La Rovere, H. L. (2012). FE model for nonlinear analysis of reinforced concrete beams considering shear deformation. Engineering Structures, 35, 244–253.View ArticleGoogle Scholar
 Tan, T. H. (2005). Effects of triaxial stress on concrete. In 30th conference on our world in concrete & structures.Google Scholar
 Xiao, Y. (1989). Experimental study and analytical modeling of triaxial behavior of confined concrete (PhD thesis, Kyushu University).Google Scholar
 Zendaoui, A., Kadid, A., & Yahiaoui, D. (2016). Comparison of different numerical models of RC elements for predicting the seismic performance of structures. International Journal of Concrete Structures and Materials, 10, 461–478.View ArticleGoogle Scholar
 Zhou, J. J., Pan, J. L., Leung, C. K. Y., & Li, Z. J. (2014). Experimental study on mechanical behavior of high performance concrete under multiaxial compressive stress. Science China Technological Sciences, 57, 2514–2522.View ArticleGoogle Scholar