Ultra-high-performance Fiber-reinforced Concrete Research Articles

Numerical simulation for the flexural response of large-span UHPFRC prestressed composite box girder

Large-span girders serve as crucial components of contemporary infrastructure. For the design and analysis of these structures, a thorough comprehension of their structural behavior, particularly in terms of flexural response, is necessary. With its superior mechanical characteristics and greater durability, Ultra-High Performance Fiber Reinforced Concrete (UHPFRC) has significantly changed the construction business in recentyears. Due to their potential for achieving longer spans, reduced construction time, and enhanced loading capacity, UHPFRC composite box girders have attracted a lot of attention. Comprehensive investigations are required to study the flexural behavior of large-span UHPFRC composite box girders under various loadingconditions in order to ensure their safe and effective use. The study aims to provide an in-depth awareness of the loading capacity, deflection characteristics, and failure mechanisms exhibited by these girders by applyingfinite element analysis (FEA) techniques. The findings of the current investigation might inform the creation of large-span UHPFRC girder designs that are more effective and dependable, promoting sustainable and resilientinfrastructure for the future.

Mechanical Properties of Ultra-High Performance Concrete (UHPC) and Ultra-High Performance Fiber-Reinforced Concrete (UHPFRC) with Recycled Sand

Use of high-cost raw materials such as quartz sand can limit wider application of ultra-high performance concrete in concrete construction. In this experimental study, recycled sand was used to fabricate ultra-high performance concrete (UHPC) and ultra-high performance fiber-reinforced concrete (UHPFRC). Green UHPC with ordinary Portland cement and industrial by-products such as silica fume, fly ash, as well as recycled sand was first developed through two-step packing density tests to optimize the mix design. UHPFRC was then developed based on the UHPC mix designs and by using 1%, 2%, or 3% 13-mm straight steel fibers (SSF). The compressive strength, elastic modulus, and flexural tensile strength was 128 MPa, 46.9 GPa, and 11.9 MPa, respectively, after 28 days at water-to-binder ratio of 0.17 and with 2% SSFs. All high-performance concretes in this work utilized 100% commercially available recycled sand that was produced by wet processing method. Mechanical characteristics such as strength, elastic modulus, and density, absorption, and voids of the UHPC/UHPFRC were investigated. Development of autogenous shrinkage of UHPC/UHPFRC with recycled sand was monitored for 12 weeks, while mercury intrusion porosimetry test and scanning electron microscopy were performed for microstructural investigation. Finally, the environmental impacts and economical aspects of the green UHPC were evaluated by life cycle assessment (LCA) and cost analysis.

Effect of ultra‐high performance concrete repair layer thickness on the behavior of concrete columns

AbstractThe use of ultra‐high‐performance fiber reinforced concrete (UHPFRC) to repair or rehabilitate old concrete structures has been shown to be appropriate due to its properties, such as higher compressive strength, ductility, and durability than conventional concretes. One of the studied strengthening techniques is the reinforced concrete columns jacketing by UHPFRC. Some recent research evaluated some parameters, such as the repair thickness and the volumetric ratio of stirrups. However, other important parameters, such as UHPFRC jacketing thickness in different column cross‐sections and the concrete core compressive strength, need to be addressed. In addition, the analytical models presented in the literature are restricted to only one concrete grade. The present study aims to evaluate the behavior of short reinforced concrete columns strengthened with UHPFRC subjected to centered compression. Numerical analyses of three‐dimensional models were performed using the finite element method, validated, and calibrated using experimental results. The influence of the concrete core compressive strength, the UHPFRC jacketing thickness, and the shape of the cross‐section were evaluated. Numerical simulations concluded that the lower concrete core compressive strength columns showed the best relative increase performance. Moreover, the concrete core strength was inversely proportional to the resistance capacity of the repair system. In addition, the increase in repair thickness contributes to the axial strength of the column. Furthermore, the cross‐section influences the distribution of stresses and strains, showing the best performance for circular sections. Finally, regardless of cross‐section type, the proposed equations accurately predicted the numerical results.

Flexural bond behavior and development length of ribbed CFRP bars in UHPFRC

Ribbed carbon fiber reinforced polymer (CFRP) bars in ultra-high-performance fiber reinforced concrete (UHPFRC) are a high-efficiency combination for its sustainable substitute and high strength to weight ratio. This paper aims to investigate the bond stress and development length of ribbed CFRP bar embedded in UHPFRC using the hinged beam test. A total of 12 groups of hinged beam specimens were tested with different parameters, such as bar type (ribbed CFRP, sand coated CFRP, ribbed glass fiber reinforced polymer (GFRP), and steel bar), bonded length (4db, 10db, 20 db, and 40 db), fibers (with and without fibers in ultra-high-performance concrete (UHPC)), and presence of shear reinforcement for assessing the bonding behavior and development length of each bar. Tensile failure occurred in the steel, ribbed GFRP, and ribbed CFRP bars with bonded lengths of 10db, 20db, and 40db, respectively, and pullout failure occurred in sand coated CFRP bars regardless of bonded length. The ultimate load for the bonded length of 10 db was in order of ribbed CFRP, ribbed GFRP, steel, and sand coated CFRP bars. Compared with the bonded length, the bond stress of ribbed CFRP bars is larger than that of the ribbed GFRP and sand coated CFRP bar and these trends became more remarkable as the bonded length increased. There was no difference in the bond stress with presence of the reinforcement cage, and the bond stress increased by 20 % when the fibers were incorporated. The code-based equation for development length of ribbed CFRP bar was found to be underestimated, and ribbed CFRP, sand coated CFRP, ribbed GFRP bar of development length was proposed as 24db, 20db, and 107db.

Electrical properties of ultra-high-performance concrete with various reinforcing fibers

Understanding the electrical properties of ultra-high-performance concrete (UHPC) is of paramount importance in the realm of smart concrete as it unlocks the potential for creating advanced, intelligent, and resilient infrastructure systems. This study focused on systematically assessing the electrical behaviors of UHPC with two commonly-used reinforcing conductive fillers, i.e. carbon fibers (CFs) and steel fibers (SFs). The effects of fiber type, fiber length, fiber content, and curing age on the alternating current (AC) resistivities and AC electrochemical impedance spectroscopy (AC-EIS) spectra of fiber-reinforced UHPC were investigated, and the equivalent circuit models of fiber-reinforced UHPC were established. Experimental results showed that the AC resistivities of UHPC with CFs and with SFs both exhibited a faster growth rate during the 14–28 d of curing, but subsequently decelerated after the completion of hydration, and stabilized at 90–120 d. Compared with the control sample, the addition of both CFs and SFs resulted in a reduction of the electrical resistivity of UHPC, with a more pronounced decrease observed with higher fiber content. In particular, the addition of SFs demonstrated a more significant reduction in UHPC’s AC resistivity in relative to CFs, with the addition of 4 vol.% copper-plated end-hook SFs remarkably lowering the resistivity by up to 87.5%. Furthermore, the introduction of different types of fibers caused remarkably different AC-EIS topologies of UHPC. The proposed equivalent circuit models reveal that compared to the control sample, the introduction of fibers can provide the fiber-fiber conductive paths and fiber-wrapped hydration products (Q F R F) within UHPC matrix. The role of UHPC matrix (Q 1 (R 1 W 1) in the conductive path of SFs-reinforced UHPC is weakened compared to that of CFs-reinforced UHPC as reflected by the differences in the impedance values of Nyquist plots.

Residual strength of ultrahigh-performance hybrid fibre-reinforced concrete columns subjected to high temperatures

The main goal of this research is to experimentally investigate the effects of hybrid fibres on the residual strength of ultrahigh-performance fibre-reinforced concrete (UHPFRC) columns subjected to high temperatures. Eight UHPFRC mixtures incorporating various types of fibres, namely, steel fibre (SF), polypropylene fibre (PP), polyvinyl alcohol fibre (PVA) and natural jute fibre (JF), were designed. Sixteen UHPFRC columns (two columns for each mixture) were tested under axial loading. Eight columns were tested at room temperature (unheated), and the other eight were subjected to 500 °C for 30 min. Results indicated that the hybrid fibres were effective in overcoming explosive spalling at high temperatures. The combination of polymer fibres (PP, PVA and JF) in addition to SF effectively improved the residual strength and fire resistance of heated UHPFRC columns. Moreover, the ultimate load capacity of the tested columns was compared with those calculated by the ACI 318–2014 design code and an existing theoretical approach.

Influence of fiber dosage, fiber type, and level of prestressing on the shear behaviour of UHPFRC I-girders

Ultra-High-Performance Fiber-Reinforced Concrete (UHPFRC) has excellent mechanical and durability properties compared to conventional concrete. It offers lightweight structures compared to reinforced concrete. In this study, the shear behaviour of post-tensioned UHPFRC girders is experimentally studied. The parameters considered in this study are (i) the level of prestressing, i.e., 0%, 35%, and 70% of ultimate tensile strength of strand (fpu), (ii) the volume fraction of fibers (Vf), i.e., 1.0%, and 2.0%, and (iii) different types of steel fibers; straight fibers, hybrid fibers (straight and hooked ended). The principal tension and compression strain of beams along and across the cracks were measured throughout the loading and compared with localization strains from the direct tension test. Due to increase in prestress from 0% to 70% of fpu, the ultimate load-carrying capacity increased from 18% to 39% in girders with a 1% fiber volume fraction. Similarly, it increased from 24% to 50% for 2% fiber reinforced girders. This is due to increased active stress transfer across the concrete cross-section, which reduces the crack angle from 44° to 34°. Similarly, the shear cracking load increased, and the post-cracking reserve capacity reduced with an increase in fiber dosage. Digital image correlation analysis reveals that strain distribution across the crack is uniformly distributed at all loading stages.

Dynamic performance of ultra-high performance fiber-reinforced concrete panel exposed to explosive loading

Ultra-high performance fiber reinforced concrete (UHPFRC) is a cement-based composite material mixing with reactive powder and steel fibers. It is characterized by its high strength, high ductility, and high toughness and such characteristics enable its great potential in protective engineering against severe dynamic loads. In the current research, the dynamic performance of the concrete panel made with ultra-high performance fiber subjected to explosive loading was investigated. For this purpose, several concrete panel samples were considered and modeled in ABAQUS finite element software. The accuracy of the numerical model is verified by comparing the numerical simulation results with available testing data. First, the considered panel was modeled with normal concrete then it was modeled with UHPFRC concrete, and the effect of using this type of concrete on the behavior of concrete panels was investigated. After analyzing and examining the models, their behavior such as the degree of vulnerability, more vulnerable points and changes in the locations that occurred in each of the models were obtained and compared. The results demonstrate that the use of UHPFRC significantly improves the blast performance of RC panels by reducing maximum and residual displacements, enhancing damage tolerance, and increasing energy absorption. The results also indicate that the increase in the intensity of explosion has increased the base reaction force in all panels.

Influence of Type and Volume Content of Steel Fiber on the Compressive Strength of Ultra-High Performance Fiber Reinforced Concrete

The paper presents the results of determining the compressive strength of various compositions of Ultra-High Performance Fiber Reinforced concrete (UHPFRC) with fiber volume content from 1 to 3%. Four types of fibers were used: corrugated fiber of 15/0.3 mm and 22/0.3 mm, straight fiber of 13/0.3 mm and hooked-end fiber of 30/0.5 mm. It was found that corrugated and hooked-end fiber leads to an increase in compressive strength by 10–30 MPa when its content is increased from 1 to 3%. Straight fiber has no noticeable effect on the mechanical properties of UHPFRC. Empirical equations for predicting the compressive strength of UHPFRC depending on the strength of concrete matrix, geometric dimensions and content of dispersed reinforcement for compositions with corrugated and hooked-end fibers have been obtained. It is found that increasing the volume content of aggregate in the concrete matrix composition from 0.2 to 0.4 m3/m3 leads to an increase in the strength of UHPFRC. The strength of specimens with corrugated fiber increased by 5.3–13.3 MPa, with hooked-end fiber – by 14–19.3 MPa, with straight fiber – 5–7 MPa. It is also noted that the most intensive growth of mechanical characteristics due to the increase in the aggregate volume content in the composition is observed at a higher percentage of fiber volume fraction.

Flexural Performance of Hybrid Fiber-Reinforced Ultrahigh-Performance Concrete with Locally Available Materials

Flexural Performance of Hybrid Fiber-Reinforced Ultrahigh-Performance Concrete with Locally Available Materials

Experimental and analytical studies of size effects on compressive ductility response of Ultra-High-Performance Fiber-Reinforced concrete

Ultra-high-performance fiber-reinforced concrete (UHPFRC) has gained a great deal of increasing interest in structural engineering applications, particularly where high ductility, strength, and high impact resistance are of prime concern. This study focuses primarily on the size effects ductility characteristics of UHPFRC with varying fiber concentrations subjected to uniaxial compressive load. It shows how to process the data from compression cylinder tests to extract the size-dependent strain at peak stress to provide a generic size-dependent stress–strain analytical model. For a slenderness factor of 2, the predicted peak stress from the analytical model deviated 0.34%, 0.26%, and 10.5% from the experimental peak stress results for a fiber concentration of 1%, 2%, and 3%, respectively, indicating good estimate of the analytical model with the experimental results. Furthermore, a numerical flexural segmental moment-rotation approach is applied to incorporate an analytical model to quantify apparently disparate UHPFRC member strength and ductility. Tests have shown that it is not the enhancement in the material concrete compressive strength but the phenomenal brittle ductility nature, observed as a result of increasing the slenderness of the specimen; in contrast, a substantial increase in ductility was achieved after crushing of concrete due to the addition of fibers. A size-dependent analytical approach has estimated good fit with the experimental and other published results. Finally, numerical simulation using a segmental approach at the ultimate limit state of rotation dealing with flexural ductility is significantly influenced by the increase in slenderness factor of the specimens and fiber concentrations.This is evident as the rotation capacity decreased by 62% and 75% for a slenderness factor of 3 and 4, respectively, compared to the slenderness factor 2, when the fiber concentration was 1%. Furthermore, for 2% fiber concentration, the decrease was 54%, and 80%, and for 3% fiber concentration, the decrease was found to be 49%, and 79% for the slenderness factor of 3 and 4, respectively in comparison to the slenderness factor 2, suggesting both slenderness factor and fiber concentration have a significant impact on the flexural ductility.

Experimental and numerical analyses of the effect of fibre content on the close-in blast performance of a UHPFRC beam

Limited research has been conducted on the influences of fiber content on close-in blasting characteristics for ultrahigh-performance fiber-reinforced concrete (UHPFRC) beams. This paper aims to address this knowledge gap through experimental and mesoscale numerical methods. Experiments were conducted on ten UHPFRC beams built with varying steel fiber volumetric fractions subjected to close-in explosive conditions. Additionally, this study considered other parameters, such as the longitudinal reinforcement type and ratio. In the case of UHPFRC beams featuring normal-strength longitudinal reinforcement of diameters Φ12, Φ16, and Φ20, a reduction in maximum displacement by magnitudes of 19.6%, 19.5%, and 17.4% was observed, respectively, as the volumetric fractions of fiber increased from 1.0% to 2.5%. In addition, increasing the longitudinal reinforcement ratio and using high-strength steel longitudinal reinforcement both significantly reduced the deformation characteristics and increase the blasting resistances of UHPFRC beams. However, the effects on the local crushing and spalling damage were not significant. A mesoscale finite element model, which considers the impacts of fiber parameters on UHPFRC beam behaviors, was also established and well correlated with the test findings. Nevertheless, parametric analyses were further conducted to examine the impacts of the steel fiber content and length and the hybrid effects of various types of microfibers and steel fibers on the blasting performance of UHPFRC beams.

The Influence of UHPFRC Jacket Steel Fiber Content on Strengthening Damaged Columns

Steel fiber is a commonly used material to repair damaged concrete, caused by environmental or design issues. This study used various Micro-copper-coated Steel Fiber (MSF) content (0.0, 0.5, 2.0, and 2.5%) with varying aspect ratios (28, 37, and 45) as part of Ultra-High-Performance Fiber-Reinforced Concrete (UHPFRC) mixtures to repair damaged concrete columns using a 30 mm layer jacketing. Twelve columns were prepared and tested at first by loading them with roughly 90% of their ultimate axial load capacity. Damage was caused and the columns were subsequently strengthened and rebuilt using UHPFRC mixtures in 30-mm layer jacketing for a second test, to determine the effect of UHPFRC and MSF content on damaged and reinforced columns. The test results showed that the concrete properties improved as the MSF content increased to 2.0% of the volume fraction, beyond which there was a slight reduction. Additionally, the UHPFRC-strengthened columns with and without MSFs experienced higher load capacities than the corresponding unstrengthened. On the contrary, using 2.5% MSF in the UHPFRC decreased the loading capacity by 14% compared to the UHPFRC with 2.0% MSF. The strengthened column with 2.0% MSF content showed the highest load efficiency (165.7% compared to unstrengthened), along with substantial displacement and ductility.

The Effects of Corrugated Steel Fiber on the Properties of Ultra-High Performance Concrete of Different Strength Levels

This article describes the influence of corrugated steel fiber on the mechanical properties and fracture energy of Ultra-High Performance Concretes (UHPC) of various strength levels. Three UHPC formulations with compressive strengths of 143, 152, and 177 MPa were tested. The following parameters for the formulations without fiber and those containing 2% steel fiber by volume were determined: compressive strength, splitting tensile strength, flexural strength, modulus of elasticity, Poisson’s ratio and critical stress intensity factor. From the axial tensile test results, the following parameters were obtained: the cracking stress, tensile strength, and fracture energy of Ultra-High Performance Fiber Reinforced Concrete (UHPFRC) of different strength levels. With the introduction of steel fiber, an increase in all the investigated parameters is observed regardless of the strength of the concrete matrix. The most remarkable influence the fiber has on the splitting tensile strength, flexural strength and critical stress intensity coefficient, the increase is up to 1.6–3.2 times. There was a slight increase in compressive strength and elastic modulus—up to 5.0–7.4% depending on the composition. Poisson’s ratio was equal to 0.2 regardless of the strength of the concrete matrix and the presence of steel fiber. Based on the test results, equations were proposed to predict the properties of UHPC and UHPFRC depending on the water–cement ratio, silica fume content, cement compressive strength and the volumetric content of corrugated steel fiber. The calculated and experimental values showed good convergence with a correlation coefficient in the range of 0.885–0.997.

Optimizing ultra-high performance cementitious functionally graded composite slab towards bullet impact resistance

This study develops functionally graded composite slab (FGCS) by a one-time grouting approach based on the design concepts of ultra-high performance fibre reinforced concrete (UHPFRC) and two-stage concrete (TSC). The anti-penetration performance of the FGCS targets is investigated against 7.62 mm in-service bullets with shooting velocity of 843 m/s. The results show that while the TSC target presents superior bullet penetration resistance, it concurrently poses a significant threat of widespread cracking damage. Upon weighing the factors of bullet penetration and risk of cracking, the FGCS emerges as a more viable candidate for protective structures. The FGCS target is cast through a one-time grouting method, and there is no interface delamination under the projectile impact. The optimized FGCS exhibits enhanced in-service bullet impact resistance and a reduced safety thickness of 80 mm compared to the normal UHPFRC.

Influence of steel fiber types on residual mechanical properties and explosive spalling of hybrid fiber reinforced ultra-high performance concrete: Optimization and evaluations

To investigate the influence of steel fiber on improving the high-temperature resistance of hybrid fiber-reinforced ultra-high performance concrete (UHPC), four types of UHPC specimens were prepared. The residual strength and fracture energy of all UHPCs were determined after exposure to varying high temperatures, and internal temperature in the specimens were also measured. Additionally, explosive spalling tests were performed on specimens with different moisture contents. The results indicate that the residual mechanical properties of UHPC initially increase and subsequently decrease as the target temperature increases. Notable differences in thermal conductivity were observed among the tested UHPC specimens. As the moisture content of the specimens increased, the likelihood and severity of explosive spalling in the specimens increased. Corrugated high-strength steel fibers, measuring 35 mm in length and 1 mm in diameter, are highly effective at preventing explosive spalling of UHPC, and high-temperature treated recycled steel fiber from tires also exhibits better performance than other two types of steel fiber. High tensile strength of steel fiber and its distinctive corrugated appearance can be the primary contributing factors. Furthermore, two established computational models which demonstrated a significant level of congruence were employed for evaluating explosive spalling and anti-spalling properties of UHPC specimens. This suggests that cracking resistance provided by steel fiber plays an important role in preventing the explosive spalling of UHPC, emphasizing the importance of exploring specific steel fiber based on its appearance and tensile strength.

Fatigue analysis of orthotropic steel-ultra-high-performance fiber-reinforced concrete (UHPFRC) composite deck considering accelerated deterioration and self-healing of fractured ultra-high-performance fiber-reinforced concrete in surface water condition

In this study, an orthotropic steel bridge deck overlaid with ultra-high-performance fiber-reinforced concrete (UHPFRC) is investigated using the finite element analysis. The composite bridge deck which is undergone moving-wheel load is examined under environmental surface water conditions. Two phases, i.e., Phases 1 and 2, are considered for the material model of the UHPFRC with stagnant water. In Phase 1, mechanical recoveries of the tensile strength and reloading stiffness are considered for the cracked UHPFRC caused by the autogenous self-healing behavior. In Phase 2, under the moving-wheel load, the crack bridging stress degradation in reinforced overlayer accelerates due to the closing–opening actions of surface cracks in water. In both phases, the deformation behaviors of the steel deck plate and UHPFRC overlayer are numerically examined. The results of the current numerical model agree with the experimental data in terms of the strain tendency, wherein the strain range of the steel deck plate and UHPFRC overlayer decreases in Phase 1 and progressively increases in Phase 2. Therefore, it can be asserted that, under the surface water condition, scenarios considering two phases of the material model of cracked UHPFRC, have governed the strain behaviors of the tested composite bridge deck.

Shock response of concrete, fibre concrete and ultrahigh performance concrete

A range of concrete types, representing commercially important formulations, were characterised under shock loading and release, using a large-bore single-stage light-gas gun. The plate impact experiments used a wave reverberation technique and an array of optical velocimetry probes to measure Hugoniot data and the subsequent release states for shocks from 2 to 11 GPa. The use of multiple velocimetry probes enabled quantification of the variation in particle velocity arising from the heterogeneous nature of concrete. Four types of concrete were investigated; a regular concrete, a fibre reinforced concrete using poly-propylene fibres, fibre concrete with steel fibres and, lastly, ultra-high performance fibre-reinforced concrete. Despite the compositional complexity, and range of quasi-static compressive strengths and reinforcement methods, the average Hugoniot data were remarkably similar between compositions.

Design oriented shear strength prediction model of UHPFRC beams

The application of Ultra-high performance fiber reinforced concrete (UHPFRC) to beams can effectively improve their strength and ductility properties. At present, there are more shear test studies on UHPFRC beams, but few theoretical studies on their strength calculation method. The fibers in UHPFRC can inhance the strength of the beams, and how to quantify the fibers’ contribution to the total strength is a research difficulty. Firstly, the shear strength database including 197 UHPFRC beam specimens is created. The key characteristics determining the shear strength, namely fiber volume fraction, stirrup configuration, and shear-span ratio, are determined by evaluating the test results. Then, considering the effects of the factors above, a truss-arch model is developed. The arch shear strength is determined by utilizing the shear deformation coordination of the arch compression structs and truss compression structs. Superimposing the shear strengths of the arch and truss models yields the shear strength of UHPFRC beams. Finally, the accuracy and reliability of the proposed method are assessed using the test database and compared to existing shear strength calculations. The study findings demonstrate that the mean value of the calculated to observed shear strength ratios produced by the suggested technique is 1.10, with a coefficient of variation of 0.28, indicating that the proposed method has superior calculation accuracy and reliability than existing methods.

In-situ nonlinear ultrasonic technique for monitoring damage in ultra-high performance fibre reinforced concrete (UHPFRC) during direct tensile test

This paper aims to employ the nonlinear ultrasonic technique for the in-situ monitoring of Ultra-High Performance Fiber Reinforced Concrete (UHPFRC) structures subject to tensile loads. The UHFPRC, characterized by specific mechanical properties of high compressive and tensile strength, provides the basis for this investigation. Particularly, the nonlinear parameter β of the nonlinear ultrasonic technique is extracted by formulating a theoretical model that accounts for the interaction of contacting rough surfaces under tensile loads. The experimental findings substantiate that β provides enhanced sensitivity and feasibility for evaluating crack conditions compared to the measurements obtained through conventional Linear Variable Differential Transformers (LVDT). Furthermore, this study explores the potential application of β to estimate tensile load condition by: i) establishing an estimation equation grounded in the relationship between tensile load and displacement (kN-mm); ii) incorporating β into this estimation equation to calculate the anticipated tensile load; and iii) comparing real and estimated tensile loads. The analysis outcomes reveal that estimated tensile loads are in good agreement with the actual loads. Although the estimation equation is based on the findings of this study, it underscores the potential applicability of β for estimating the load conditions in various structures.