The study focuses on centrally compressed reinforced concrete columns with various cross-sectional shapes, including square, rectangular, cruciform, T-shaped, and L-shaped sections. In structural engineering, the load-bearing capacity of columns is typically enhanced through transverse reinforcement, external jackets, or composite confinements that create a triaxial compression effect. However, the influence of cross-sectional geometry on the formation of a 3D stress state has not been sufficiently investigated. A numerical analysis of centrally compressed reinforced concrete columns with different cross-sectional configurations was carried out using the finite element method in the LIRA-SAPR software package, accounting for the nonlinear behavior of concrete. The results indicate that the shape of the cross section has only a minor effect on strength characteristics (differences below 3 %), but significantly affects the distribution of volumetric stress zones, which are pronounced in T- and L-shaped columns when dilation effects are considered. It was found that volumetric confinement primarily develops in the support regions and has little influence on the overall structural performance of the columns. Future research will focus on eccentrically loaded columns and on the effect of external confinement systems, including concrete-filled steel tubes, steel jackets, and indirect reinforcement, on the formation and evolution of the 3D stress-strain state of reinforced concrete elements. The proposed research direction confirms the practical relevance of the study and the potential for further investigation in this field.
1.Introduction
Reinforced concrete columns with various cross-sectional shapes are widely used in building and civil engineering, particularly when the layout configuration or architectural requirements call for efficient use of interior space. In addition to conventional rectangular and square columns, cross-shaped, T-shaped, and L-shaped columns are used in structural systems because they enable structural members to be integrated into walls and beams and allow multiple functions to be combined within a single element. The complex geometry of the cross-section has a significant impact on the structural stress-strain state. Local stress concentrations in massive members lead to the formation of a 3D stress-strain state. Considering this 3D stress-strain state in such zones greatly influences the overall strength, crack resistance, and durability of the structures.
A number of studies have shown that a 3D stress state in reinforced concrete columns can develop under the influence of various factors, including the level and eccentricity of loading, end-restraint conditions, the presence of longitudinal and transverse reinforcement, as well as the geometry of the cross-section [1–17]. For rectangular and square columns, the stress state is generally close to plane stress conditions, whereas in members with complex shapes, such as cross-shaped, L-shaped, and T-shaped sections, local zones of spatial stress interaction may arise due to the mutual influence of the flanges and webs. These characteristics lead to non-uniform deformation, asymmetry in crack development, and differences in failure modes, all of which should be taken into account when analyzing the load-bearing capacity and stability of such elements.
Under uniaxial compression, concrete exhibits relatively brittle failure, whereas under biaxial or triaxial confinement its load-carrying capacity and deformability increase significantly [18, 19]. In nonstandard cross-sections (such as cross-shaped, T-shaped, and L-shaped), the distribution of lateral stresses is nonuniform, which leads to the formation of zones with different levels of confinement and alters the concrete stress-strain relationship within a single section.
In [20, 21], reinforced concrete columns with equivalent cross-sectional areas and different cross-sectional shapes were considered, including square, cross-shaped, L-shaped, and T-shaped sections. It was found that, for a slenderness ratio of less than 40, the load-bearing capacity of equivalent columns differs only slightly: an increase of approximately 4.5 % was observed for the cross-shaped section, whereas reductions of about 1.4–6.9 % were reported for the T- and L-shaped sections, which may be considered insignificant. For a slenderness ratio greater than 40, the influence of the cross-sectional shape becomes significant, and cross-shaped, L-shaped, and T-shaped columns provide an increase in load-bearing capacity of up to 12 % compared with square columns.
In [22], theoretical investigations were carried out on short reinforced concrete columns with equivalent cross-sectional areas and different cross-sectional shapes, namely circular, square, rectangular, octagonal, cross-shaped, L-shaped, and T-shaped sections. It was established that rectangular, square, and cross-shaped columns differ only slightly in terms of load-bearing capacity: the increase for the cross-shaped section was approximately 1.4 %, while the difference between square and rectangular columns was practically negligible. For L- and T-shaped columns, the reduction in ultimate load compared with the rectangular section was 6.9 % and 9.7 %, respectively.
Thus, the results reported in [20–22] are comparable: the advantage of the cross-shaped section is relatively small, while the reduction in load-bearing capacity for L- and T-shaped sections remains within the range of a moderate calculated difference.
In [1, 2], the behavior of cross-shaped concrete-filled steel tubular (CFST) columns and their steel counterparts was investigated. The composite columns, consisting of concrete confined within a steel tube, had a load-bearing capacity 75–80 % higher than that of the steel columns. This increase is attributed to the composite action of the materials: the steel shell restrains lateral expansion and prevents cracking of the concrete, creating a confinement effect, while the concrete core, in turn, enhances the local stability of the steel walls and delays buckling.
In [3, 4], cross-shaped and L-shaped steel-reinforced concrete (SRC) columns subjected to seismic loading were examined. Experimental results and finite element simulations showed that encasing the steel profile in concrete provides improved structural stability.
In [5, 6], the focus is on enhancing the confinement effect in T-shaped CFST columns by introducing stiffeners and tie elements, as well as on finite element modelling (FEM) of such members considering the strengthening of the concrete core.
One of the aspects examined in [7] was the influence of cross-section geometry on the load-bearing capacity of special-shaped CFST columns. A comparison of L- and T-shaped specimens showed that the cross-section configuration has a noticeable, although not decisive, effect on the behavior of the columns under axial compression. T-shaped columns exhibited slightly higher load-bearing capacity and stiffness, which can be attributed to a more uniform stress distribution and the effective composite resistance provided by the flange and the web. L-shaped members showed greater longitudinal deformability and stress localization in the corner regions, resulting in a less symmetric failure pattern.
The influence of the cross-section geometry of CFST columns was also addressed in [8], which presents an analysis of various special-shaped configurations (L-shaped, T-shaped, cross-shaped, and other combined forms), along with corresponding beam-to-column connection details. The authors note that the cross-section shape significantly affects the stress distribution within both the steel tube and the concrete core, which in turn influences the overall load-bearing capacity and stiffness of the members.
In [9], the behavior of L-shaped CFST columns with internal stiffening plates was simulated. The analysis showed that the asymmetry of the cross-section leads to a non-uniform stress distribution between the flanges and the interior corner regions, as well as to the formation of local zones with a 3D stress-strain state. The introduction of internal plates reduced the effects of geometric asymmetry and enhanced the composite action between the concrete and the steel tube. Similar conclusions were reported in [10], which showed that the configuration of an L-shaped cross-section, including the flange length ratio, has a noticeable influence on the stiffness, failure pattern, and stability of the columns.
In [11], FEM in ABAQUS showed that the angle between the branches of L-shaped CFST columns has a significant effect on the stress distribution: as the angle decreases, the edge effect intensifies and stress concentrations increase in the outer corner regions, whereas increasing the angle to 135° leads to a reduction of stresses in the inner zones and a redistribution of forces along the flanges of the section.
In rectangular columns, a 3D stress-strain state can be achieved by installing confinement jackets made of various materials [12–14] or by applying indirect reinforcement [15–17].
In [13], the axial compressive behavior of square and rectangular reinforced concrete columns strengthened with a composite fiber-reinforced polymer (FRP) jacket was investigated experimentally. It was shown that the effectiveness of confinement decreases as the aspect ratio of the cross-section increases; however, even at h/b = 2, a significant strength enhancement can still be achieved with an increased number of FRP layers.
The results of calculations for short columns showed an increase in load-bearing capacity of up to 30–50 % compared with unconfined specimens [15]. Experimental studies [16] demonstrate that reducing the spacing of closed stirrups and using welded or zigzag reinforcement meshes can increase the ultimate load by approximately 7–22 %, depending on the reinforcement configuration. This effect is pronounced when the stirrup spacing is small. Replacing traditional stirrups with reinforcement meshes increases the concrete strength by 15–28 % and reduces lateral deformations, providing a more ductile failure mode [17].
In the numerical analysis of reinforced concrete structures, numerous finite element method-based approaches are employed that allow both the physical and geometric nonlinear behavior of concrete and reinforcement to be taken into account. A significant contribution to the advancement of numerical modelling has been made by studies conducted by domestic and international researchers using various software packages, including LIRA-SAPR, ANSYS, SCAD Office, ABAQUS, and ATENA.
In [15], the nonlinear behavior of concrete was modeled in LIRA-SAPR using 3D materially nonlinear universal 8-node isoparametric solid finite elements (FE type 236), while the nonlinear behavior of reinforcement was represented by materially nonlinear truss-bar finite elements (FE type 204).
The failure criterion was defined as the attainment of ultimate concrete stresses considering multiaxial stress conditions, based on the strength criteria of Klovanich–Bezushko [23] and Willam–Warnke [24], as well as the attainment of the yield stress in the reinforcement. The difference in strains obtained from comparison between the experimental results and the numerical simulations based on these models did not exceed 5 %.
In [25], a method for modelling reinforced concrete structures in LIRA-SAPR was presented, followed by comparison of the numerical results with experimental data. Concrete and longitudinal reinforcement were modelled using 3D materially nonlinear finite elements of types 231 and 233, while transverse reinforcement was represented by bar elements of type 210. The interaction between concrete and reinforcement was simulated through link elements based on bar-type elements 295 and 296. The calculated results for cracking, strength, and stiffness showed a deviation from the experimental data of no more than 10 %.
A similar approach was implemented in [26], where comprehensive nonlinear modelling of reinforced concrete structures was performed using the ESPRI and LIRA-SAPR software packages. The analysis was carried out in a materially nonlinear formulation, accounting for concrete creep, temperature effects, and time-dependent stress redistribution. The deformation behavior was described using the 25th nonlinear constitutive law for concrete implemented in LIRA-SAPR.
In [27], an analytical-numerical approach for determining the stress-strain state of CFST columns under eccentric compression was presented, based on the deformation theory of concrete plasticity. The method allows for consideration of the triaxial stress state of the concrete core by analytically describing the relationship between the principal stresses and strains.
In [28], an algorithm for modelling the behavior of reinforced concrete in LIRA was developed, based on the principles of fracture mechanics, which takes into account crack opening, concrete discontinuity, and reinforcement response. This approach provides a reliable representation of stress redistribution processes during cracking, allows refinement of element stiffness characteristics, and improves calculation accuracy compared with traditional linear-elastic models.
In [29, 30], the development of a 3D finite element model of reinforced concrete columns implemented in the ATENA software environment was presented. In [29], the modelling was carried out using 3D solid elements and the nonlinear Kent–Park concrete material model, which accounts for the material response under compression, tension, and cyclic loading. In contrast, in [30], ATENA’s built-in concrete model was employed, based on automatic parameter identification from cube strength and an exponential tensile fracture law. This approach made it possible to reproduce various failure modes of the columns and to validate the numerical results against experimental data.
In structural systems, the load-bearing capacity of columns is commonly enhanced through the use of transverse reinforcement [15–18], external jackets [1–11, 14], or composite wraps [12, 13], which provide a confinement effect. An analysis of [20–22] shows that, for reinforced concrete columns with complex cross-sectional shapes and a slenderness ratio of less than 40, where failure is governed by the material load-bearing capacity rather than by loss of stability, the influence of the cross-sectional shape on the ultimate load is generally limited. However, the published studies do not establish whether the cross-sectional geometry itself, without additional confinement measures, can lead to the formation of a 3D stress state within the concrete volume. This issue is particularly relevant for columns with complex configurations, such as cross-shaped, L-shaped, and T-shaped sections, which are used in frame and wall structural systems, where the interaction between flanges and webs may potentially cause stress redistribution within the concrete volume.
In this context, there is a need for a comprehensive assessment of how the cross-sectional geometry influences the stress-strain state parameters of concentrically compressed reinforced concrete columns under 3D stress conditions.
The aim of the present study is to evaluate the influence of cross-sectional geometry on the development of zones with a 3D stress-strain state in concentrically compressed reinforced concrete columns and to determine the feasibility of accounting for this effect in the strength design of such elements.
2.Methods
For the analysis of the stress-strain state of reinforced concrete columns, five cross-section types with equivalent area and a height of 3300 mm were considered:
square; cross-shaped; T-shaped; L-shaped; rectangular.
Longitudinal reinforcement consisted of 12 bars with a diameter of 10 mm, while transverse reinforcement was represented by 6 mm stirrups spaced at 150 mm along the entire column height. Concrete of class B25 and reinforcement of classes A500 (longitudinal) and A240 (transverse) were used. These material grades were selected due to their wide application in conventional construction practice, ensuring the comparability of the analysis results with real structures and the practical relevance of the conclusions.
The overall dimensions of the cross-section and the height of the element were selected based on the applicability condition of a bar-type calculation scheme for a centrally compressed reinforced concrete member. The geometric characteristics are shown in Fig. 1.
a
b
c
d
e
Figure 1. Geometric characteristics of the square (a), cross-shaped (b), T-shaped (c), L-shaped (d), and rectangular (e) cross-sections.
Table 1. Strength and deformation characteristics of reinforcing steel
No.
Rebar grade
Diameter, mm
Yield strength fy, MPa
Ultimate strength
fu, MPa
Ultimate strain components
Compression, εsc2·103
Tension, εs2·103
1
A500
10
500
600
–25
25
2
A240
6
240
380
–25
25
Table 2. Mechanical and deformation properties of concrete
No.
Concrete grade
Prism strength fc,pr, MPa
Initial elastic modulus E0, MPa
Ultimate strain components
Poisson’s ratio, µ
Compression, εc2·103
Tension, εt2·105
1
В25
19.4
26000
–2.6
15
0.2
The modelling of reinforced concrete columns was performed using the licensed version of the LIRA-SAPR 2022 R1.1 software package. The software package has a verification certificate issued by the Russian Academy of Architecture and Construction Sciences (RAACS)[1]. The computational model consists of solid finite elements (type 236) with dimensions of 50×50 mm used to model the concrete, and bar elements (type 204) used to model the reinforcing bars. Material nonlinearity was taken into account through a piecewise-linear stress-strain law. The concrete stress-strain diagrams were adopted in accordance with the experimental data. The behavior of reinforcement was described using a bilinear stress-strain diagram, which is widely used in FEM of reinforced concrete members [3, 7, 15, 25]. The mechanical properties of concrete and reinforcing steel used in the analysis are summarized in Table 1 and Table 2, respectively.
The columns were loaded incrementally, in steps of 10 % of the ultimate load. Based on the condition that the concrete and reinforcement simultaneously reach their respective ultimate stress levels, the ultimate load was determined as:
(1)
where is the prism compressive strength of concrete, is the yield strength of the reinforcing steel, and are the areas of concrete and longitudinal reinforcement, respectively.
A subsequent analysis was performed to determine the theoretical ultimate load for each cross-sectional geometry, based on the value obtained from equation (1). The final theoretical ultimate load was taken as the magnitude of the external force at which the stresses in the concrete 3D finite elements reached the prism compressive strength, or the stresses in the bar elements of the reinforcement reached the yield strength.
The theoretical ultimate load was defined as the value of the applied load at which the stresses in the concrete solid elements reached the prism compressive strength (in regions of biaxial and triaxial compression, according to the failure criterion of K.J. Willam and E.P. Warnke [18, 24]), while in the reinforcement bar elements the stresses reached the yield strength of steel. Once the specified limit states were attained in any finite element, that element was excluded from the structural resistance by artificially reducing its elastic modulus by a factor of 1000. If, at this reduced stiffness, the stresses in the adjacent elements did not exceed their ultimate values after the redistribution of internal forces, the failure was considered local and not significantly affecting the overall load-bearing capacity of the member.
In the computational model, the bottom nodes of the column were fully restrained in the X, Y, and Z directions, providing a fixed support, while the top nodes were restrained only in the X and Y directions, allowing free vertical displacement.
3.Results and Discussion
As a result of the analysis of the numerical simulations of centrally compressed columns with different cross-section geometries, theoretical curves based on the nonlinear stress-strain laws of concrete were generated, as shown in Fig. 2.
|s3|
ei×103
square section*
square section**
cross-shaped section*
cross-shaped section**
T-shaped section*
T-shaped section**
– L-shaped section*
– L-shaped section**
– rectangular section*
– rectangular section**
* theoretical concrete stress–strain curves obtained from the linear analysis; ** theoretical concrete stress–strain curves obtained from the nonlinear analysis.
Figure 2. Theoretical curves based on the nonlinear stress-strain laws.
Based on Fig. 2, it can be concluded that the cross-sectional shape of a centrally compressed column does not have a significant effect on its strength or deformation characteristics. The difference between the curves obtained using the nonlinear stress-strain laws of concrete is less than 3 %.
The obtained results are comparable with those reported in [20–22]: the increase in load-bearing capacity for the cross-shaped section was approximately 1 %, whereas for the T-shaped, L-shaped, and rectangular sections, the reduction was 2 % compared with the square column. This comparison confirms that the influence of the cross-sectional shape on the load-bearing capacity of columns is insignificant when the slenderness ratio does not exceed 40.
It has been established that, at a slenderness ratio greater than 40, the influence of the cross-sectional shape on the load-bearing capacity becomes significant, and the increase in strength may reach 12 % [20, 21]. However, failure of such columns is generally associated with loss of stability, whereas, in the case of failure governed by the material load-bearing capacity, the influence of the cross-sectional shape on the ultimate load remains insignificant. Based on this distinction, further research should include a comparative analysis of eccentrically compressed columns with different cross-sectional shapes.
A comparison of the results (Fig. 2) shows that up to 30 % of the ultimate load, the concrete deforms linearly; therefore, the linear and nonlinear analyses yield nearly identical values, which is consistent with the conventional understanding of the initial elastic stage of material behavior. However, beyond this threshold, the curves begin to diverge: the nonlinear model captures the gradual reduction in stiffness, whereas the linear model preserves a constant modulus of elasticity. As a result, in the linear analysis the prism strength is reached at approximately 80 % of the ultimate load, while in the nonlinear analysis the concrete reaches its prism strength at about 95–100 % of the ultimate load calculated according to equation (1). In the nonlinear analysis, the reinforcement reaches a stress level of 87–90 %, whereas in the linear analysis it reaches only 27–30 %. This comparison confirms that the nonlinear formulation more accurately reflects the behavior of a reinforced concrete element under high load levels.
The values of the ratio were also calculated for each solid finite element according to the Willam–Warnke [24] failure criterion. Based on these results, contour plots of the ratio were generated at the mid-depth of each column cross-section (Fig. 3), as well as contour plots of the principal stress ratios at elevation +1.650 m, corresponding to the mid-height of the columns (Fig. 4).
The main analyses were performed with a Poisson’s ratio of = 0.2. To assess the possibility of developing lateral confinement due to the increased transverse deformability of concrete in the pre-failure stage, an additional analysis was performed with = 0.4. The adoption of this value of Poisson's ratio makes it possible to parametrically account for the dilatancy effects characteristic of concrete at high levels of compressive stress [31, 32].
a
b
c
d
e
Figure 3. I – segment of the analyzed specimen; II – contour plot of the s3/fc,pr ratio for μ = 0.2; III – contour plot of the s3/fc,pr ratio for μ = 0.4 (square column (a); cross-shaped section column (b); T-shaped column (c); L-shaped column (d); rectangular column (e)).
a
b
c
d
e
Figure 4. Contour plot of the σ₃:σ₂:σ₁ ratios in the cross-section at elevation +1.650 (rectangular column (a); cross-shaped section column (b); T-shaped column (c); L-shaped column (d); rectangular column (e)).
An analysis of the results obtained for = 0.2 reveals the following patterns. Within the support zones, regardless of the cross-section geometry, regions of triaxial compression are observed, where the ratio exceeds 3 over a height of 50–100 mm. Further upward, within the range of 100–200 mm, the ratio decreases to 1 (uniaxial compression) for the cross-shaped section. In the remaining sections, local zones of compression accompanied by transverse tension occur between 500 and 1150 mm, where the ratio falls within the range from 0.99 to slightly below 0.91. In the square and cross-shaped sections, localized regions of triaxial stress are observed along the entire column height, with values ranging from 1.09 to slightly below 1.11.
Accounting for dilatational effects significantly increases the horizontal stresses in the concrete. The region of triaxial compression, where the ratio exceeds 3, expands to 100–150 mm. Further along the height, a layered variation of the ratios is observed, ranging from compression-compression, with = 1.5, to compression-tension, with dropping below 0.91. The compression-compression zones are governed by the confinement effect provided by the transverse reinforcement.
The increase in horizontal stresses when concrete dilatancy is taken into account is in qualitative agreement with the results reported in [15, 33], where the increase in transverse deformations of concrete is considered one of the factors contributing to the development of lateral confinement stresses in the presence of elements restraining transverse expansion. At the same time, the obtained results show that the cross-sectional shape alone does not provide stable lateral confinement of concrete comparable to that produced by external jackets or indirect reinforcement. Increasing Poisson’s ratio to = 0.4 mainly led to the formation of a compression-tension-tension stress state over a significant part of the cross-section, whereas the compression-compression-compression zones remained local. Thus, the cross-sectional shape affects the redistribution and localization of principal stresses but does not create a stable lateral confinement effect characteristic of special confinement systems.
It can also be observed that in the T- and L-shaped cross-sections, the triaxial stress zones in the support regions are more pronounced. This phenomenon is attributed to the more massive flanges in these geometries, which enhance the restraining effect on lateral deformations in the central part of the cross-section.
4.Conclusions
The analysis of existing experimental and numerical studies has shown that the confinement effect in concrete is achieved by providing external restraining reinforcement, such as concrete-filled tubular shells, steel jackets, or indirect reinforcement systems. The use of these methods increases the load-bearing capacity of compressed members, reduces the risk of brittle failure, and thereby improves the performance of concrete under a complex stress state. Investigation of the influence of the 3D stress-strain state in concentrically and eccentrically compressed members, including the effect of external restraining reinforcement, such as concrete-filled tubular shells, steel jackets, and indirect reinforcement systems, is therefore of significant scientific and practical interest. The study of the 3D stress state of concentrically compressed members carried out by the authors shows that the cross-sectional shape of a concentrically compressed column has almost no effect on the strength and deformation properties of the structure, since the differences in the values of the investigated parameters did not exceed 3 %. At the same time, the cross-sectional shape has a significant influence on the nature and size of the zone of the 3D stress state in the support regions. With an increase in the length of the web and flanges in T- and L-shaped sections, the restraining effect on horizontal deformations becomes more pronounced, which leads to an expansion of the 3D confinement zone and an increase in the level of triaxial compression in the concrete. Accounting for dilatancy effects when determining the parameters of the 3D stress state of concentrically compressed members showed a significant increase in horizontal stresses and an increase in the height of the 3D stress-state zone up to 100–150 mm. In addition, a layer-by-layer alternation of zones characterized by compression-compression and compression-tension stress states was observed, indicating a complex spatial stress distribution. By contrast, the use of the transverse strain coefficient = 0.2, including at load levels close to the ultimate load, results in the 3D confinement effect appearing only in the support regions of the columns. In the remaining part of the section, zones of uniaxial compression and local tensile regions predominate, which indicates a relatively uniform distribution of compressive stresses in the middle part of the member height. Extending the scope of research on the influence of geometric parameters on the formation and development patterns of the 3D stress-strain state from concentrically compressed members to eccentrically compressed members is of significant scientific and practical interest.
[1] Rossiyskaya akademiya arkhitektury i stroitelnykh nauk. Svidetelstvo o verifikatsii programmnogo sredstva LIRA-SAPR No. 06/LIRA-SAPR/2015 [Verification certificate for the LIRA-SAPR software No. 06/LIRA-SAPR/2015] [Online]. URL: https://rflira.ru/files/verify/Verif_LIRA-SAPR_Sign.pdf?ysclid=mu440t1v4u582410961 (accessed 16.09.2026)