Buckling is a critical phenomenon that occurs when slender structures, such as columns, are subjected to compressive loads (Kubiak, 2013; Kołakowski and Teter, 2016; Emam and Lacarbonara, 2022; Atashipour et al., 2023). The initial concepts were established by Yoa and Pfeiffer (1983), who formulated the basic principles of column behavior under compression. The Euler equation for the critical buckling load is expressed as Equation 1: P c r = π 2 E I k L 2 (1) where P c r is the critical load, E the modulus of elasticity, I the moment of inertia, k the effective length factor, and L the column length. Timoshenko and Gere, (2012) extended these principles by incorporating bending and deformation effects, which are particularly relevant to composite materials (Timoshenko and Gere, 2012; Tsai, 2018).

Composite materials exhibit complex behavior under buckling due to their anisotropy. Tsai (2018) developed specific models to predict the buckling load of composites, considering fiber orientation and directional mechanical properties. This approach has been complemented by further work, such as that of Almeida et al. (2019) and Saiki (2024), who analyzed the effect of different fiber configurations.

Salari-Sharif et al. (2018) examined the impact of manufacturing defects on buckling resistance. They showed that even small imperfections can significantly reduce the critical load (Little, 2003; Standard, 2005; Zio, 2016; Modarres and Groth, 2023). Integrating these defects into mechanical behavior models is essential for improving the accuracy of predictions. The effect of defects can be modeled by Equation 2. R d = P a c t u a l P i d e a l (2)

Where R d is the performance reduction factor, P a c t u a l is the actual load supported, and P i d e a l is the theoretical load without defects (Salari-Sharif et al., 2018; Yan et al., 2020; Yao et al., 2023).

After critical loading, the behavior of composite materials evolves into a post-buckling phase. Vasiliev and Morozov (2013) studied this phase, indicating that load-bearing capacity can decrease non-linearly with increasing strain. The relationship between applied load P and strain δ is given by Equation 3. P = P c r 1 − δ δ c r n (3)

Where δ c r is the critical strain and n is a coefficient specific to the material properties.

Experimental work by Vasiliev and Morozov (2013) revealed that carbon/epoxy composites can suffer significant mechanical property losses of up to 40% after exposure to critical loads. These results underline the importance of assessing the residual reliability of composites after buckling. Residual load after buckling can be modeled by Equation 4. P r e s i d u a l = P c r ⋅ 1 − α (4)

Where PresidualPresidual is the residual load and α is a degradation coefficient based on experimental tests (Vasiliev and Morozov, 2013; Agarwal et al., 2017).

Advances in numerical methods, such as the finite element method (FEM), have also enabled the modeling of the behavior of composites under buckling and post-buckling with increased accuracy (Xu et al., 2013; Masood et al., 2021; Yıldırım and Ünal, 2024). Watt et al. (2006) incorporated environmental factors into their simulations, showing that conditions such as humidity and temperature can exacerbate the risk of buckling (Watt et al., 2006; Lal et al., 2011; Jaroszweski et al., 2019; Li et al., 2024).

The assessment of residual reliability has become a priority in the design of composite structures. Gioncu and Mazzolani (2013) emphasized that understanding failure mechanisms is essential for ensuring the durability of structures. Building codes also require more rigorous safety analyses to ensure that structures can withstand critical loads while maintaining their integrity (Standard, 2006; Neville, 2011; Arya, 2022).

In conclusion, the literature on buckling and post-buckling of composite materials has evolved significantly. Research has progressed from theoretical models to modern experimental and numerical approaches, allowing for a deeper understanding of failure mechanisms and residual reliability (Chawla, 2012; Gibson, 2007; Mehta and Monteiro, 2006; Neville, 2011). This knowledge is crucial for ensuring the performance and safety of composite structures under critical conditions (Bae et al., 2005; Lopes et al., 2012; Afzal et al., 2020). The reliability of composite materials, especially under critical loads, remains an active and essential research area for the development of safe and sustainable infrastructures (Hansen et al., 2011; Chawla, 2012).

Residual reliability refers to a material’s ability to retain its performance after experiencing extreme loads or damage. According to Wu et al. (2019) and Podolak et al. (2021), while composite materials have been widely studied, little effort has been devoted to understanding their post-buckling behavior in terms of residual reliability. Most studies focus on initial mechanical properties and critical load, often neglecting the analysis of long-term performance after buckling events.

Another overlooked aspect is research on traditional composite materials, such as reinforced concrete. Mallick (2007) and Chawla (2012) highlight that most work concentrates on advanced composites, like carbon/epoxy composites, leaving aside materials that are widely used in construction. This creates an urgent need for specific research on the behavior of reinforced concrete under buckling and post-buckling, considering its unique characteristics such as ductility and compressive strength.

Bamboo-reinforced composite concrete is emerging as a promising alternative in modern construction, offering advantages in terms of durability, lightness and strength. Recent studies, such as that presented by Sreadha and Pany (2020), Sreadha and Pany (2021), highlight the superior mechanical performance of this material, as well as its potential for reducing the environmental impact of construction. This research opens up new avenues for the integration of composite materials in a variety of structural applications.

Numerical modeling of buckling and post-buckling phenomena is also an area with existing gaps. While advanced models have been developed for specific composites, few studies have been conducted to incorporate environmental factors and manufacturing defects into the modeling of reinforced concrete. It is crucial to develop models that account for these factors to improve the accuracy of behavior predictions.

Current building codes emphasize durability and safety but often lack specific guidelines for assessing the residual reliability of composite materials under buckling. Gioncu and Mazzolani (2013) indicate that safety requirements should include buckling and post-buckling analyses, yet few standards provide clear procedures for this.

In summary, the gaps in current knowledge regarding the residual reliability of composite materials, particularly reinforced concrete, highlight the need for thorough research in this area. Special attention should be given to post-buckling assessment, integration of experimental data, and improvement of simulation models to develop more robust and accurate safety standards.

For this study, reinforced concrete specimens were designed and manufactured following rigorous construction standards. The concrete mix was composed of cement, aggregates, water, and additives to enhance mechanical properties and durability. The water-cement ratio was carefully controlled to ensure the desired strength.

The concrete was prepared with a water-cement ratio of 0.45, using 400 kg cement, 800 kg fine aggregate and 1,200 kg coarse aggregate per cubic meter.

The materials were mixed for 5 min in a concrete mixer, followed by vibration for 15 min to eliminate air bubbles.

The reinforcing bars used in the specimens have a diameter of 12 mm. These steel bars are essential for strengthening the concrete against tensile forces and bending moments.

A total of 50 cylindrical specimens were prepared, each with a diameter of 150 mm and a height of 600 mm. These dimensions were chosen to ensure representative behavior under buckling conditions, as presented in Table 1.

The manufacturing process of the specimens followed these steps:

1. Mold Preparation: Steel molds were used to ensure precise dimensions.

The prepared specimens were then subjected to buckling tests. The results were recorded to evaluate the critical load at which each specimen failed, as well as to analyze post-buckling behavior.

Figure 1 illustrates a reinforced concrete specimen prepared for the buckling tests.

The selected materials and specimens will provide reliable data on the behavior of reinforced concrete under buckling and post-buckling, which is essential for assessing its residual reliability.

Compression tests on the specimens were conducted at ambient temperature and pressure, using an Instron universal compression/bending machine (available at the Mechanical Engineering laboratory of the University Institute of Technology, University of Ngaoundéré, Cameroon) (Figure 2), at a speed of 2 mm/min. The technical specifications of the compression machine are provided in Table 2.

Compressive strength is one of the most fundamental measures for assessing the reliability of a material. It is determined by the standard test ASTM C39/C39M. The compressive strength f c can be calculated using Equation 5. f c = P A (5)

Where P represents the load applied at the moment of failure and A is the cross-sectional area of the specimen. A significant decrease in this value after the buckling tests indicates a loss of load-bearing capacity.

The modulus of elasticity is a key measure of the stiffness of concrete. It is evaluated according to the ASTM C469/C469M standard. The modulus of elasticity E is calculated using Formula 6. E = σ ε (6)

Where σ is the applied stress and ε is the corresponding strain. A decrease in the modulus of elasticity after buckling may indicate a degradation of mechanical properties.

Elongation at break is an indicator of the ductility of concrete. This measurement is performed according to the ASTM C496/C496M standard. Elongation at break is calculated using Equation 7. E l o n g a t i o n = L f − L 0 L 0 × 100 (7)

Where L f ​ is the final length of the specimen after failure and L 0 ​ is the initial length. This measurement helps to understand how the material behaves before and after buckling.

The use of scanning electron microscopy (SEM) allows for the examination of the microstructure of concrete after testing. This method provides detailed images that help identify cracks, delaminations, and other forms of degradation. SEM observations can reveal crucial information about damage mechanisms, such as Equation 8: D = f S i z e   o f   c r a c k , D i s t r i b u t i o n   o f   d e f e c t s (8)

Where D represents the degree of observed degradation.

The data collected during the tests are analyzed to identify trends and relationships between the various measured properties. Statistical techniques may be applied to assess the significance of the differences observed in mechanical properties before and after the tests.

The analysis of the results reveals that reinforced concrete presents a balanced combination of strength and ductility, which is essential for structural applications. The ability of the specimens to withstand an average buckling load of 120 kN, coupled with maximum deformations of 2.1%, demonstrates the material’s robust performance under extreme loads.

The standard deviation of 7.5 kN for the buckling load indicates homogeneity in the quality of the concrete, which is crucial for ensuring the reliability of structures. The observed variations in buckling loads are relatively low, suggesting that the concrete mix and manufacturing process were well controlled. This reinforces confidence in the use of reinforced concrete for critical constructions, where material consistency is paramount.

The measured ductility, with an average of 2.1%, is particularly significant. This property allows reinforced concrete to deform without rupture, which is crucial during events such as earthquakes or sudden loads. The ability to absorb such deformations contributes to the safety of structures by allowing energy dissipation before catastrophic failure.

The analysis of the data shows that the majority of the specimens had buckling loads between 115 kN and 130 kN, as shown by the histogram in Figure 4.

This histogram reveals a normalized distribution, reinforcing the idea that the material is homogeneous and reliable. Such a distribution is desirable in practical applications, as it indicates that all specimens behave similarly under loads.

In conclusion, the results of the buckling and post-buckling tests indicate that reinforced concrete presents a good balance between strength and ductility, essential characteristics for ensuring the safety and durability of structures. These data provide a solid foundation for further studies and the development of recommendations for the use of reinforced concrete in critical structural applications.

The results of the compressive strength tests are presented in Table 4. The measurements taken before and after the tests show a marked decrease in strength.

The average compressive strength decreased from 31 MPa to 22 MPa, representing a reduction of 29%. Specimens subjected to higher loads exhibited lower residual performance, as shown in Figure 5, which illustrates the relationship between buckling load and residual compressive strength.

This figure indicates that higher buckling loads result in lower residual strength, highlighting the impact of load levels on the material’s durability.

The results for the elastic modulus are presented in Table 5.

The elastic modulus decreased from 25.4 GPa to 18 GPa, representing a loss of 29%. As illustrated in Figure 6, this reduction is more pronounced for specimens that underwent higher buckling loads.

This figure shows that the elastic modulus decreases with increasing load, indicating a loss of material rigidity.

The durability of the material was evaluated through a series of mechanical tests designed to assess its performance under varying load conditions. Specifically, the compressive strength of the reinforced concrete samples was measured before and after applying incremental loads. By monitoring the changes in the elastic modulus during these tests, we were able to determine how the material’s rigidity and structural integrity were affected by the applied stress.

Additionally, fatigue tests were conducted, where the samples were subjected to repeated loading cycles to observe how they behaved over time. This approach allowed us to identify any signs of microcracking or permanent deformation, which are critical indicators of potential failure. Furthermore, microscopic analyses were performed to examine internal defects and the development of cracks within the material, providing insights into the mechanisms of degradation. Overall, these evaluations helped us understand the material’s resilience and its ability to maintain performance under stress, thus providing a comprehensive assessment of its durability.

The results regarding elongation at rupture are presented in Table 6.

Elongation at rupture decreased from 2.54% to 1.24%, representing a reduction of 51%. Figure 7 illustrates this trend, highlighting the relationship between applied load and elongation at rupture.

This figure shows that specimens subjected to higher loads exhibit lower elongation at rupture, indicating increased brittleness.

The impact of the number of loading cycles on residual properties was also examined. The results are presented in Table 7.

The data show a progressive reduction in compressive strength, elastic modulus, and elongation at rupture with the increase in the number of cycles. Figure 8 illustrates this trend.

This figure indicates that as the number of cycles increases, mechanical properties degrade, emphasizing the importance of fatigue in evaluating material reliability.

In conclusion, the results clearly demonstrate that the residual reliability of reinforced concrete is strongly influenced by the applied load and the number of loading cycles. The significant decrease in compressive strength, elastic modulus, and elongation at rupture highlights the necessity for thorough material evaluations to ensure durability in critical applications. These data will serve as a reference for future recommendations on the use of reinforced concrete in structures subjected to varying loads.

Concrete is inherently brittle, and when subjected to high compressive stresses, it tends to crack. As shown in Figure 9, the relationship between the applied load and the number of cracks formed indicates a significant increase in cracking with higher loads.

This figure illustrates that as the load increases beyond a certain threshold, the number of visible cracks in the concrete increases dramatically. This cracking contributes to a reduction in structural integrity, leading to lower compressive strength and elastic modulus, as seen in the results presented in Tables 4, 5.

The local buckling of reinforcements occurs when the applied load exceeds the critical load that the reinforcement can withstand. Table 8 summarizes the impact of different loads on the elastic modulus of the specimens. As the buckling occurs, the effective load-carrying capacity of the reinforcement is diminished.

As shown in Table 8, the elastic modulus significantly decreases after testing under higher loads, indicating that local buckling has occurred. This reduction in elastic modulus correlates with a loss of stiffness, further contributing to the degradation of mechanical properties.

Debonding is another critical failure mechanism that occurs at the interface between concrete and steel reinforcement, often exacerbated by cyclic loading. Figure 10 illustrates the interface condition before and after cyclic loading.

The figure show noticeable deterioration of the bond, which leads to a reduction in the load transfer efficiency between the concrete and the steel reinforcements. This debonding can be particularly detrimental in structures subjected to dynamic loads, as it compromises the composite action intended in reinforced concrete systems.

The cumulative effects of these mechanisms—cracking, buckling, and debonding—result in a significant decline in the residual reliability of the concrete specimens. Figure 11 provides a visual summary of the overall impact of various loading conditions on the mechanical properties of the specimens.

This figure summarizes the trends observed across different mechanical tests, highlighting the interrelationship between load levels, cycle counts, and the degradation of material properties.

In summary, the physical mechanisms underlying the observed behaviors in reinforced concrete under load include concrete cracking, local buckling of reinforcements, and debonding at the concrete-steel interface. These mechanisms not only explain the decline in mechanical properties, as shown in the tables and figures, but also emphasize the importance of understanding these processes for predicting the durability and reliability of concrete structures under varying loads. Continued research and monitoring are essential for developing strategies to mitigate these issues and enhance the performance of reinforced concrete in practical applications.

The measured mechanical properties, such as the elastic modulus and compressive strength, were evaluated from the experimental results. Table 9 presents a comparison of the average values obtained in this study with those found in the literature.

The values obtained in this study show that the compressive strength and elastic modulus of reinforced concrete are significantly lower than those of advanced composites. This can be attributed to the complex microstructure and inherent variability of concrete, which impact its performance under load.

Specifically, the irregular arrangement of aggregates and the bonding characteristics between the cement matrix and reinforcement lead to a less uniform stress distribution. Additionally, the presence of microcracks and voids in the concrete can initiate failure mechanisms under high loads, resulting in reduced strength and stiffness. Comparatively, advanced composites, such as carbon or glass fiber-reinforced materials, benefit from a more consistent microstructure and superior tensile strength, allowing them to maintain better mechanical properties under similar loading conditions. This disparity highlights the need for careful consideration of design and material selection in applications where high load-bearing capacity and durability are critical, emphasizing the importance of understanding the limitations of reinforced concrete in structural engineering.

Figure 12 illustrates the results of the buckling test, showing the relationship between the applied load and the deformation of the reinforced concrete samples.

This figure demonstrates that, unlike advanced composites that exhibit better resistance to buckling, the reinforced concrete samples show significant deformation at lower loads. This confirms observations in the literature regarding the vulnerability of reinforced concrete to buckling.

The residual properties of the samples after buckling tests were also evaluated. Table 10 presents the results of the residual properties compared to advanced composites.

The data indicate that the residual properties of reinforced concrete degrade more significantly after buckling compared to advanced composites, which retain a larger portion of their initial properties. This emphasizes the importance of material design in structural applications.

In conclusion, the results of this study align with trends observed in the literature, but the absolute properties of reinforced concrete remain inferior to those of advanced composites. These differences can be attributed to the heterogeneous and less performant nature of reinforced concrete, as well as its sensitivity to buckling and failure phenomena. A deep understanding of these mechanisms is essential for improving the performance of reinforced concrete in critical structural applications. Future research should focus on developing hybrid composite materials that could combine the advantages of concrete and advanced materials to enhance the reliability and durability of structures.

This study has provided valuable insights into the performance of reinforced concrete subjected to buckling and post-buckling loads, revealing critical findings regarding the degradation of its mechanical properties. The experimental tests demonstrated a significant decrease in compressive strength and modulus of elasticity as the applied loads and number of cycles increased. These findings are consistent with previous studies, such as those by Yoo and Lee (2011) and Kashani et al. (2013), which also reported similar degradation patterns under extreme loading conditions.

Microscopic observations identified key failure mechanisms, including cracking, local buckling of the reinforcement, and delamination between concrete and steel, all of which contribute to this degradation. Scanning Electron Microscopy (SEM) observations provided crucial visual evidence of these damage mechanisms, as illustrated in Figure 13.

This image reveals the extent of cracks and delamination, underscoring the complex behavior of reinforced concrete under stress.

Overall, these results indicate that the residual reliability of reinforced concrete is lower than that of advanced composites, emphasizing the necessity for rigorous design practices to ensure the durability and safety of reinforced concrete structures under extreme loading conditions. This study contributes to a deeper understanding of the limitations of reinforced concrete and its behavior under complex stresses, aligning with the conclusions drawn in earlier research (e.g., Gioncu and Mazzolani, 2013).

This article makes a significant contribution by offering detailed experimental data on the residual reliability of reinforced concrete under buckling and post-buckling tests—an area that has often been overlooked in the literature. By identifying specific failure mechanisms and contrasting the results with those of advanced composite materials, this study enhances our understanding of the inherent limitations of reinforced concrete. Additionally, it proposes a rigorous methodology for assessing residual reliability, which includes tailored experimental protocols and advanced analytical methods. This approach could serve as a benchmark for future research and encourage the development of more robust testing standards for composite materials. The findings also highlight the importance of a multidisciplinary approach that integrates knowledge from material science, mechanics, and structural engineering.

Future research should focus on innovative strategies to enhance the performance of reinforced concrete, such as the integration of hybrid composite materials that leverage the strengths of both concrete and advanced composites. Investigating the effects of various treatments and additives, including synthetic fibers or polymers, on the failure resistance and ductility of concrete would be particularly relevant. Furthermore, the development of new testing protocols that simulate realistic and diverse loading conditions—including thermal and environmental effects—could yield valuable insights into the long-term durability of structures.

Finally, a comprehensive investigation of fatigue mechanisms in reinforced concrete, taking into account the long-term impacts of repeated loading cycles, could facilitate the design of more durable and reliable structures. Addressing these contemporary challenges in civil engineering will be crucial for meeting evolving building codes and ensuring public safety.