{"paper_id":"2ca4d7a5-63ea-44b4-a189-a788f4fbb3f7","body_text":"Experimental investigation and simulation analysis of cast-steel joints under vertical pressure | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Experimental investigation and simulation analysis of cast-steel joints under vertical pressure Zhihao Li, Yizhong Zhang, Wenfeng Du, Liming Zhu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4061078/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract The joint made of cast steel is frequently utilized within a treelike column structure to ensure a smooth transition. It is of great significance in ensuring the overall structural safety, but currently, the mechanical property and bearing capacity of this type of joint cannot be fully understood. This study delves into the load-bearing characteristics of such a cast-steel joint featuring three branches. Initially, a comprehensive model of the cast-steel joint, sourced from a practical engineering, underwent vertical load testing. Detailed scrutiny of stress distribution and vertical displacement of the tested joint was conducted based on the experimental outcomes. Subsequently, a finite element model of the tested joint was constructed using SolidWorks and subjected to analysis via ANSYS. The numerical findings were juxtaposed with experimental data and extrapolated to encompass other parametric scenarios. Ultimately, a regression analysis method was employed to derive a calculation formula for the load-carrying capacity of branch-bearing cast-steel joints. This formula aids in estimating geometric parameters and load-bearing capacity during the preliminary design phase. Comparative analysis reveals a substantial concurrence among experimental, finite element analysis, and formula-based predictive outcomes. Cast-steel joint Tree-like column structures Full-scale model test Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Introduction The tree-like column has been widely utilized in engineering owing to its visually appealing structure and remarkable mechanical characteristics [ 1 ]. Since its introduction at Stuttgart Airport in Germany in 1991, several significant undertakings, such as Stansted Airport in London, the ION Orchard Center in Singapore, Detroit Airport in the United States, and the Railway Station in Changsha, China, have adopted this pioneering architectural structure [ 2 – 5 ]. It is obvious that the joints connecting the backbone and branches play a crucial role in the treelike structure. Firstly, the entire upper framework relies solely on a singular connection, and the arboreal assembly risks collapse in the event of this connection's demise. Secondly, numerous elements, encompassing both the main stem and lateral tubes, converge at the connection with seamless shifts, rendering the mechanical properties of such connection intricate. Tree-like column structures commonly employ cast-steel joints with branches [ 6 ]. In contrast to welded tubular joints, the use of cast-steel joints eliminates the residual stress caused by intricate welds at intersections. This not only simplifies construction but also accelerates the building process of tree-like structures [ 7 – 8 ]. While the utilization of cast steel in structural applications has garnered consistent interest over recent decades, research regarding cast-steel joints remains in its nascent phase [ 9 ]. Conducting full-scale experiments serves as a direct and efficient means to comprehend the performance of cast-steel joints. Consequently, several experiments assessing cast-steel joints in significant large-span steel roof structures have been conducted, including those at Shanghai South Railway Station [ 10 ], Beijing National Stadium [ 11 ], and Chongqing Olympic Stadium in China [ 12 ]. Additionally, numerical analysis has emerged as another viable approach to delineate the stress distribution within cast-steel joints comprehensively. Finite element analysis results offer invaluable insights for cast-steel joint design, as evidenced by projects such as the Cycling Gymnasium for Beijing Olympic Games [ 13 ], Guangzhou New Railway Station [ 14 ], and Tianjin Convention and Exhibition Center [ 15 ]. Eurocode 3 presents the component method, facilitating the evaluation of joint stiffness and resistance characteristics by aggregating those of all constituent components [ 16 , 17 ]. The geometric arrangement of cast-steel joints with branches typically consists of one primary pipe and two, three, or four secondary branching pipes. It is anticipated that the utilization of these joints will become increasingly widespread as tree-like column structures continue to evolve. However, as of yet, existing building standards or codes lack relevant provisions, and research on the mechanical performance, calculation formula, and structural optimization of cast-steel joints with branches remains limited. This paper presents a comprehensive investigation involving a typical full-scale experiment on a cast-steel joint featuring one trunk and three branches. The primary objective was to validate the accuracy of numerical simulation analysis. Utilizing ANSYS, stress distribution and the ultimate bearing capacity of the joint were meticulously calculated, yielding results that closely aligned with experimental findings. Furthermore, employing regression analysis, a calculation formula for the load-bearing capacity of cast-steel joints with branches was derived. Model test of the three-branch cast-steel joint Engineering background and design philosophy The design specifications for the exhibition hall at Zhonghong Hotel in Kaifeng City, China, outline dimensions of 24 m × 34 m. Illustrated in Fig. 1 , the roof structure employs a single-layer latticed shell, supported by a tree-like column and four additional supports from reinforced concrete frames. Notably, a cast-steel joint was employed at the intersection point of the trunk and the first-level branches. The decision to utilize a cast-steel joint was driven by three primary considerations. Firstly, the joint needed to be fabricated from thin-walled pipes to minimize weight. Given the complexity of achieving hollow joints with intricate shapes through conventional methods, casting was deemed the most viable option. The thickness of the joint matches that of the main pipe, set at 1/20 of the pipe diameter to meet production and load-bearing requirements. Secondly, to ensure smooth stress transmission across different parts of the joint, the lengths of both the main pipe and branch pipes within the joint zone were carefully designed to avoid being excessively short. Thirdly, for ease of production and installation, the branch pipes were horizontally cut flat. To fulfill these criteria, the cast-steel joint was meticulously designed, as depicted in Fig. 1 . The joint configuration comprises one main pipe and three symmetric branch pipes. The main pipe has dimensions of length ( L ) = 800 mm, diameter ( D ) = 500 mm, and wall thickness ( T ) = 40 mm. Each branch pipe shares identical dimensions: length ( l ) = 500 mm, diameter ( d ) = 350 mm, and wall thickness ( t ) = 35 mm. The chamfer radius ( R 1 ) between the main pipe and the branch pipes is 1000 mm, while the outside chamfer radius ( R 2 ) between branch pipes is 50 mm. Additionally, the inside chamfer radius ( R 3 ) among branch pipes is 20 mm. Other essential parameters include θ = 30° (angle between main pipe and branch pipe), β = 0.7 (ratio of outer diameter of branch pipe to outer diameter of main pipe), and γ = 20 (ratio of outer diameter of main pipe to main pipe wall thickness). Manufacturing and material properties of the joint specimen The joint is fabricated from ZG20SiMn cast steel, characterized by the following chemical composition: Carbon (C) − 0.18%, Silicon (Si) − 0.60%, Manganese (Mn) − 1.50%, Phosphorus (P) − 0.020%, Sulfur (S) − 0.015%, Chromium (Cr) − 0.30%, Molybdenum (Mo) − 0.15%, and Nickel (Ni) − 0.40%. The joint specimen was cast using a sand mold at Tengfei Factory in Xinxiang City, China. Following heat treatment and polishing, the specimen was transported to the structural laboratory of Henan University for testing. A standard dog-bone flat specimen was also prepared to evaluate the material properties [ 18 ]. The material test results, including the typical stress-strain relationship, are illustrated in Fig. 2 . The stress-strain curve for the cast-steel coupon exhibits a distinct yielding plateau, indicating favorable material properties. The yield strength is measured at 235.7 MPa, and the ultimate strength reaches 353.6 MPa. The percentage elongation at fracture is found to be 25.9%, and the ultimate-to-yield strength ratio is calculated at 1.50. These results demonstrate that ZG20SiMn cast steel possesses notable ductility and strain hardening ability, exceeding the basic requirements stipulated in building codes such as Eurocode 3. Specifically, the percentage elongation at fracture surpasses the minimum of 15%, and the ultimate-to-yield strength ratio exceeds the required minimum of 1.1. To ensure precise adherence to design specifications, the size of the actual joint was meticulously measured, and the relative errors between the actual and design siz-es are detailed in Table 1 . Notably, the maximum relative error recorded is 2.80%, comfortably below the acceptable threshold of 5.00%. This demonstrates that the pro-duction of the joint specimen successfully meets the stringent design requirements. Table 1 The comparative analysis of the test-specimen dimensions Parts of the joint Measured sizes Design sizes Errors The main pipe length( L ) 806 mm 800 mm 0.75% The main pipe diameter( D ) 495 mm 500 mm 0.10% The main pipe wall thickness( T ) 39.5 mm 40 mm 1.25% The branch pipes length( l ) The branch pipes diameter( d ) 492 mm 345 mm 500 mm 350 mm 0.16% 1.71% The branch pipes wall thickness( t ) 34.6 mm 35 mm 1.14% The chamfer ( R 1 ) 994 mm 1000 mm 0.60% The chamfer ( R 2 ) 20.46 mm 20 mm 2.30% The chamfer ( R 3 ) 51.4 mm 50 mm 2.80% Loading procedure The experimental setup involved applying a vertical load, as depicted in Fig. 3 , with a maximum load capacity set at 5000 kN. To facilitate contact between the experimental joint and the load piston, an initial preloading stage was implemented, determining a preloading value of approximately 30 kN through several trials. Following preloading, the regular loading phase unfolded in five stages, each incrementing by 1000 kN at a constant rate of 10 kN/s. At the conclusion of each stage, the load was maintained for a duration of 30 seconds. Subsequently, the unloading process commenced, mirroring the loading stages with five steps, each decreasing by 1000 kN at a rate of 10 kN/s. Similar to the loading stages, a 30-second holding period followed the completion of each unloading stage. Strain measurement Resistance strain gauges were employed to measure the strain in the cast-steel joint during testing. These strain gauges and strain rosettes were manufactured by the Giant Star Electric Measuring Element Factory in Taizhou, China. The specifications for the strain gauges are as follows: a resistance value of 120 ± 0.2 (Ω), a sensitivity factor of 2.08 ± 1%, and a sensitive grid size of 2 mm (gate length) × 1 mm (width) on a substrate measuring 4.5 mm (gate length) × 2.4 mm (width). Similarly, the strain rosettes possess a resistance value of 120 ± 0.3 (Ω), a sensitivity factor of 2.08 ± 1%, and a sensitive grid size of 3 mm (gate length) × 2 mm (width) on a substrate measuring 11.5 mm (gate length) × 11.5 mm (width). The surfaces of the joint were pre-polished to facilitate the attachment of strain gauges and strain rosettes. The placement of measuring points was primarily determined by the characteristics of joint stress calculated through finite element analysis. Analysis results indicated elevated stress levels near the joint core area, contrasting with lower stresses in the main pipe and branch pipes. Consequently, measuring points near the joint core area were densely positioned. Overall, measuring points were concentrated in four areas: (A) upper part of the main pipe, (B) lower part of the main pipe, (C) vicinity of the joint core area, and (D) branch pipes. Strain gauges were affixed to the upper and lower parts of the main pipe, where stress distribution is simpler. Conversely, strain rosettes were placed near the joint core area, characterized by unknown principal stress directions and complex stress distribution patterns. Measuring point positions were uniformly distributed and symmetrical, with three copies along the joint circumference. Figure 4 illustrates the facade of measuring points on the joint, with identical placement on the other two symmetrical sides. The specific locations and quantities of measuring points are outlined in Table 2 . Table 2 The stresses of the measuring points (MPa) No. 1000 kN 2000 kN 3000 kN 4000 kN 5000 kN Locations A1 26.7 53.6 80.4 112.5 130.4 The upper part of the main pipe A2 23.2 51.3 78.6 109.3 126.3 A3 28.3 54.7 84.3 110.7 121.4 B1 34.5 72.8 112.3 145.7 183.3 The lower part of the main pipe B2 31.6 69.3 114.2 148.6 178.7 B3 38.7 75.4 115.3 147.3 176.9 C11 42.7 88.2 128.3 183.4 235.4 The vicinity of the joint core area C12 44.3 87.6 130.1 179.5 235.3 C13 44.7 87.4 126.8 182.4 235.7 C21 44.8 85.7 127.2 178.6 235.2 C22 45.3 86.3 128.4 185.1 235.3 C23 41.9 86.7 129.1 179.3 235.6 C31 42.4 89.2 128.6 180.1 235.4 C32 46.3 84.9 131.2 185.3 235.3 C33 44.7 85.4 127.9 176.1 235.1 D1 35.2 71.7 110.6 144.3 181.4 The branch pipes D2 34.6 70.8 117.3 151.5 179.4 D3 32.7 68.7 108.4 156.3 183.5 Experiment observation At the onset of preloading, a faint sound was discernible, attributed to the adjustment of the experimental specimen to close the gap between the piston and itself. This sound dissipated as the load reached 20 kN. Following the completion of preloading, the regular loading test proceeded in accordance with the prescribed loading procedure. Figure 5 illustrates a photograph capturing the joint under load. Throughout the loading process, both the measured displacement and strain exhibited consistent, uninterrupted development without any notable jumps. Analysis and discussion of test results (1) Analysis of the load-displacement curve Figure 6 depicts the load-displacement curve obtained from the experimental specimen. Broadly, the load-displacement curve of the joint can be delineated into three distinct stages. In stage I, characterized by relatively low loads (less than 500 kN), the slope of the load-displacement curve is minimal. This signifies a significant increase in joint displacement at the initial stage of loading, attributable to the non-tight contact between the joint and the testing equipment. With the continuous increase in load, the load becomes proportional to displacement, but the slope of the load-displacement curve experiences a noticeable augmentation. This phase is defined as stage II, suggesting that the joint operates within the elastic state as the load varies between 500 kN and 4750 kN. As the load approaches the maximum design value (stage III), the load-displacement curve exhibits a non-linear behavior, indicating that the joint transitions into the elastic-plastic state. Notably, a significant increase in displacement corresponds to a minor increment in load. The slope of the load-displacement curve in stage III indicates a rapid expansion of the plastic zone within the joint. Therefore, it is advisable to consider the proportional limit observed in stage II as the control value for the design of the cast-steel joint with branches. From stage II, it is evident that the ultimate bearing capacity of the joint specimen is 4750 kN. During the unloading process, the joint's deformation exhibited linear recovery in tandem with the decrease in load. The maximum vertical displacement recorded throughout the entire process amounted to 8.305 mm. However, upon unloading, the joint did not revert to the zero point due to residual deformation. This residual deformation can be attributed to two factors. Firstly, non-tight contact between the joint and the testing equipment, estimated to be approximately 4.435 mm (refer to Fig. 6 ). Secondly, residual plastic deformation, which amounts to about 0.890 mm. Therefore, the total residual deformation is calculated to be 5.325 mm. (2) Analysis of the stress Upon completion of each load stage, the strain data for each measuring point was collected. Subsequently, the stress values for each measuring point were calculated and are presented in Table 2 . Notably, when the load reached its maximum value of 5000 kN, the maximum stress recorded was 235.7 MPa, observed at the joint core area. This finding underscores the joint core area as the most critical location. In contrast, the minimum stress among the measuring points was 121.4 MPa, situated at the upper part of the main pipe. As the load increased, the stress at each measuring point exhibited a linear increase during the initial four loading stages. However, upon reaching a load of 4750 kN, the stresses at measuring points situated within the joint core area stabilized at approximately 235 MPa. This phenomenon indicates the onset of a plastic zone within the joint core area. Additionally, when examining the measuring points arranged symmetrically along the circumference of the joint, it is evident that the stress levels at these points were largely consistent. Numerical Simulation Analysis Analysis model To comprehensively assess the performance of the test joint, a corresponding finite element model was constructed. Initially, as depicted in Fig. 7 (a), the joint model was established using the default ANSYS preprocessor. However, challenges arose in achieving a seamless transition between the main pipe and the branches. Consequently, the joint was accurately modeled using 3D modeling software SolidWorks to ensure consistency with the actual joint configuration, as illustrated in Fig. 7 (b). The established joint model was imported into the finite element software ANSYS for analysis. The material properties used in the model were obtained from the material test. The elastic modulus of the material ( E ) was determined to be 2.0×10^5 N/mm², the yield strength ( f y ) was 235 MPa, and the Poisson's ratio ( µ ) was 0.3. The constitutive behavior of the material was selected to be the ideal elastic-plastic model, utilizing the Von-Mises yield criterion and associated flow rule for the elastic-plastic analysis [ 20 ]. To accurately simulate the real-world scenario, the boundary conditions of the joint were set as follows: the end part of the main pipe was fixed, while the ends of the branch pipes were fixed vertically. Additionally, the load was applied to the ends of the branch pipes in the form of surface pressure. After extensive research and comparison, the three-dimensional solid element Solid65 was selected from the ANSYS element type library for modeling the joint. This element type features quadratic displacement and is well-suited for irregular grid division [21]. The finite element mesh of the joint, depicted in Fig. 8 , was meticulously crafted to ensure accurate representation and analysis of the structural behavior. Analysis results of the testing joint The finite element analysis was conducted for the test joint under varying load conditions. Figure 9 illustrates the stress contours of the joint corresponding to load levels of 1000 kN, 2000 kN, 3000 kN, 4000 kN, and 5000 kN, respectively. From Fig. 9 , it is evident that the overall stress level of the joint under a 1000 kN load is relatively low. The maximum stress value is predominantly concentrated in the vicinity of the joint core area. Furthermore, the stress observed in the main pipe and branch pipes is notably lower compared to that in the core area of the joint. Specifically, the stress in the former accounts for only 11.5% of the stress observed in the latter. As the load increases to 2000 kN, the stress level of the joint exhibits a gradual linear increase, reaching a maximum stress value of 219 MPa. Upon reaching a load of 3000 kN, signs of yielding in the steel become apparent. However, the yield region is primarily concentrated at three points in the chamfer between the branch pipes. Subsequently, as the load further increases to 4000 kN, the yield region expands outward. This expansion manifests in two ways: firstly, the plastic region enlarges, and secondly, the chamfer between the main pipe and branch pipes also enters the plastic zone. Upon reaching the maximum load of 5000 kN, the expansion of the plastic zone within the joint intensifies, although it remains primarily concentrated in the vicinity of the joint core area. At this stage, nearly the entire core area of the joint enters the yield state, indicating the formation of a plastic hinge and marking the entry of the load-displacement curve into stage III (as illustrated in Fig. 6 ). Despite this, the stresses observed in the main pipe and branch pipes remain relatively low, approximately 106 MPa. This underscores the concentration of stress within the core region of the cast-steel joint with branches, which significantly influences its ultimate load-carrying capacity. Figure 10 illustrates the vertical displacement of the joint under the maximum load of 5000 kN. It is observed that the maximum vertical displacement obtained from the finite element analysis (4.366 mm) is smaller than that obtained from the test (8.064 mm). This disparity can be primarily attributed to the non-tight contact between the joint and the test equipment piston during the experimental testing process. Verification of the finite element model through the experiment results To validate the numerical model of the cast-steel joint with branches, a comparison is made between the results of finite element analysis and those of the verification experiment. The stress values obtained from representative measuring points in the finite element model are compared with the experimentally derived stress values, and their relative differences are listed in Table 3 . Table 3 The joints analysis results with different parameters under axial loading Joint number θ ( o ) L (mm) β γ R 1 (mm) R 2 (mm) R 3 (mm) Ultimate load-carrying capacity (kN) J1 20 800 0.7 20 1000 0 0 2000.24 J2 30 800 0.7 20 1000 0 0 2301.79 J3 40 800 0.7 20 1000 0 0 2317.40 J4 50 800 0.7 20 1000 0 0 1660.00 J5 30 800 0.7 10 1000 0 0 5720.70 J6 30 800 0.7 15.2 1000 0 0 3344.50 J7 30 800 0.7 25 1000 0 0 1705.06 J8 30 800 0.7 29.9 1000 0 0 1334.73 J9 30 800 0.7 20 500 0 0 2056.27 J10 30 800 0.7 20 1500 0 0 2344.98 J11 30 800 0.7 20 2000 0 0 2233.56 J12 30 800 0.7 20 1000 10 0 2298.95 J13 30 800 0.7 20 1000 20 0 2268.51 J14 30 800 0.7 20 1000 30 0 2208.79 J15 30 800 0.7 20 1000 0 50 2824.84 J16 30 800 0.7 20 1000 0 100 3275.00 J17 30 800 0.7 20 1000 0 150 3296.32 J18 30 800 0.6 20 1000 0 0 2223.88 J19 30 800 0.66 20 1000 0 0 2281.90 J20 30 800 0.74 20 1000 0 0 2352.50 J21 30 800 0.8 20 1000 0 0 2418.69 J22 30 500 0.7 20 1000 0 0 2310.59 J23 30 600 0.7 20 1000 0 0 2312.61 J24 30 700 0.7 20 1000 0 0 2314.97 Analyzing the data in Table 3 reveals that: 1) The stress distribution of the cast-steel joint with branches obtained from the experiment aligns closely with that calculated from the finite element model. The calculated stresses from the finite element analysis and the experimental values of measuring points exhibit consistency. The maximum error between the calculated stresses and the experimental results is 9.02%, substantiating the validity of the finite element model utilized in this study. 2) Both the finite element modeling and the experiment confirm that the area of large stress is concentrated in the core area of the joint. The stresses in the main pipe and the branch pipes are comparatively small, approximately half of the largest stress observed in the core area of the joint. 3) The casting precision of cast steel joints presents challenges in control. In this study, it was observed that the chamfer between the main pipe and the branch pipes was slightly larger than the design value. Additionally, the wall thickness exceeded the design specifications, resulting in a smaller diameter thickness ratio. Consequently, the stresses predicted by the finite element model tend to be generally higher than those observed in the experiment. 4) In summary, the results obtained from the finite element model align closely with those from the experiment. The numerical model effectively captures the actual stress and deformation states of the cast-steel joint with branches. Therefore, it can serve as a reliable tool for investigating the load-carrying capacity of such joints in further research. Effect of joint parameters on compression behavior of the cast-steel joint To investigate the influence of different parameters on the ultimate load-carrying capacity of the joint, a parametric study is conducted wherein individual variables are varied while keeping other parameters constant. This approach allows for a systematic analysis of how each variable impacts the joint's performance independently. The modeling results are shown in Table 4 , which could be summarized as: 1) During compression testing, the fifth joint (J5) demonstrates the highest load-bearing capacity, reaching 5720.7 kN. Conversely, the eighth joint (J8) exhibits the lowest load-bearing capacity, registering only 1334.73 kN. Although both J8 and J5 share similar geometric characteristics, J8 boasts the greatest diameter thickness ratio ( γ ), while J5 possesses the smallest. Thus, the diameter thickness ratio ( γ ) significantly impacts the load-bearing capacity of joints. 2) Increasing only θ while holding other variables constant substantially decreases the joint's ultimate load-bearing capacity. This observation underscores θ's substantial influence on the joint's load-carrying capability. 3) Gradual increments in β and R 3 , while keeping other factors constant, substantially enhance the joint's ultimate load-carrying capacity. This finding highlights the significant impact of β and R 3 on the joint's load-bearing capability. 4) When dimensions L , R 1 , and R 2 undergo gradual increments while all other variables remain constant, the ultimate load-carrying capacity of the joint exhibits minimal variation, with the largest observed change being less than 5%. This indicates that dimensions L , R 1 , and R 2 exert negligible influence on the ultimate load-carrying capacity of the joint. 5) Based on the findings of finite element modeling, it is deduced that thorough consideration of geometric parameters is imperative when analyzing the load-carrying capacity of cast-steel joints with branches. Careful selection of dimensional parameters for the joint is essential to ensure the structural safety and reliability. Load-carrying capacity estimation of the three-branch cast-steel joint In existing literature [ 19 – 20 ], load-carrying capacity formulas for welded tubular T-joints, steel tubular XK-joints, and multi-planar KX and KT-joints under axial loads are consistently represented as the product of the material yield strength and the square of the pipe wall thickness. Accordingly, the estimation of load-carrying capacity for cast-steel joints with branches can be similarly expressed as: $${F_u}=K{T^2}{f_y}$$ 1 where Fu is load-carrying capacity of the joint; K is a parameter that contains the geometric parameters such as θ , γ and β of the joint; T is the pipe wall thickness; and f y is the material yield strength of the joint. In Eq. ( 1 ), the expression of parameter K serves as the primary research focus across various types of joints. Given that K encompasses a range of geometric parameters affecting the load-carrying capacity of the joint, the focus has shifted from solely examining the relationship between K and material yield strength ( F u ) to conducting multiple studies on the correlation between each individual parameter and K . The finite element analysis results indicate that dimensions L , R 1 , and R 2 of the joint exert minimal influence on the ultimate load-carrying capacity. Consequently, these parameters are disregarded during the analysis of the comprehensive index K . Utilizing line charts depicting the relationships between θ , γ , β , and R 3 with K , as illustrated in Fig. 11, a regression analysis is performed. Following the regression analysis, the relationship between θ and Kθ is initially examined. Through this analysis, the relationship between the sine value of θ and Kθ can be expressed as: $${K_\\theta }=0.60022 - 2.5311\\sin \\theta +6.59681{\\sin ^2}\\theta - 5.07388{\\sin ^3}\\theta$$ 2 For the relationship between γ and Kγ, it could be expressed as a power function: $${K_\\gamma }=4.3725{\\gamma ^{0.66242}}$$ 3 From the observations in Fig. 10 , it is apparent that a linear relationship exists between β and Kβ. This relationship can be expressed as: $${K_\\beta }=1+0.58856\\beta$$ 4 Finally, the regression analysis is performed between R3 and KR3. According to the principle of dimensional analysis, it is essential that the parameter R3 in the formula is dimensionless. To account for the influence of R3, a dimensionless chamfer coefficient ρ is defined as: $$\\rho =\\frac{{{R_3}}}{{\\sqrt {dt} }}$$ 5 where d is the outer diameter of the branch pipe; t is the wall thickness of the branch pipe. The finite element model shows that the joint ultimate load-carrying ca-pacity is very small when R 3 is greater than or equal to 100mm. So R 3 is limited less than or equal to 100mm on the ultimate load-carrying capacity calculation formula for the cast-steel joint with three branches. Through regression analysis, the relationship between the chamfer coefficient ρ and K R3 follows a linear relation, which is expressed: $${K_{{R_3}}}=1+0.33738\\frac{{{R_3}}}{{\\sqrt {dt} }}$$ 6 Because these four parameters are independent of each other, the overall formula for the joint load-carrying capacity can be obtained by multiplying them, following the method of establishing the load-carrying capacity of joints in the existing standards, which is expressed as: $$\\begin{gathered} F=4.37251{\\gamma ^{0.66242}}(1+0.58856\\beta )(1+0.33738\\frac{{{R_3}}}{{dt}})(0.60022 - 2.5311\\sin \\theta +6.59681{\\sin ^2}\\theta - \\hfill \\\\ 5.07388{\\sin ^3}\\theta ){f_y}{T^2} \\hfill \\\\ \\end{gathered}$$ 7 To validate the accuracy of Eq. ( 7 ), a comparison between the results obtained from finite element modeling and those derived from the regression formula is conducted. The comparative results are presented in Table 5 . Notably, the disparity between the calculated values obtained from the formula and those from finite element analysis is minimal, with the maximum error amounting to only 1.9%. Consequently, it is deduced that the proposed formula effectively predicts the ultimate load-carrying capacity of the cast-steel joint with three branches with a high level of accuracy. Table 5 Calculation formula error table Joint number θ ( o ) γ β R 3 (mm) F/ ( fyo*T² ) Regression formula results Difference percentage(%) J5 30 10 0.7 0 9.74 9.92 1.90 J6 30 15.2 0.7 0 13.07 13.09 0.19 J2 30 20 0.7 0 15.67 15.70 0.21 J7 30 25 0.7 0 18.14 18.21 0.37 J8 30 29.9 0.7 0 20.37 20.50 0.65 J1 20 20 0.7 0 13.62 13.62 0.00 J3 40 20 0.7 0 15.78 15.78 0.00 J4 50 20 0.7 0 11.30 11.30 0.00 J18 30 20 0.6 0 15.14 15.05 -0.61 J19 30 20 0.66 0 15.54 15.44 -0.61 J20 30 20 0.74 0 16.02 15.97 -0.32 J21 30 20 0.8 0 16.47 16.36 -0.66 J15 30 20 0.7 50 19.23 19.09 -0.75 J16 30 20 0.7 100 22.40 22.47 0.32 Conclusion The study presents a comprehensive numerical simulation and experimental investigation of a full-scale cast-steel joint with three branches. Initially, a representative full-scale model of the cast-steel joint was constructed using SolidWorks, followed by rigorous analysis using ANSYS, which accounted for geometric and material nonlinearity. To validate the accuracy of the numerical simulation, a corresponding verification experiment was conducted. Furthermore, a formula for determining the load-carrying capacity of the cast-steel joint with three branches was proposed, fulfilling the requirements of engineering design. Through the extensive research conducted in this paper, the following conclusions were drawn: 1) Analysis of the full-scale joint experimental results revealed that stress distribution under compression primarily concentrates on the core area of the joint, with minimal stress observed in the main pipe and branch pipes. This insight serves as a basis for evaluating the joint's strength and stiffness to meet design requirements. 2) SolidWorks proved to be effective in modeling cast-steel joints with branches, successfully addressing the challenge of modeling tube-to-tube intersections with smooth transitions to match actual joint configurations. 3) The finite element model of the test joint was imported into ANSYS for analysis, and the results were compared with experimental findings, demonstrating consistency. The verified finite element model is deemed reliable for evaluating the impact of joint geometry parameters on the behavior of three-branch cast-steel joints. 4) Finite element analysis was conducted on joints with various geometric parameters to determine their ultimate load-carrying capacities. The error analysis revealed a maximum error of 1.9% when comparing prediction results with finite element results, indicating that the proposed formula accurately predicts ultimate load-carrying capacity for engineering design requirements. This formula serves as a valuable tool for selecting geometry parameters in joint structural design. Declarations Data Availability Statement The datasets used and analyzed during the current study are available from the corresponding author on reasonable request. Acknowledgments The authors would like to express sincere thanks to Qi Liu, Fan Zhang, and Hao Zhang for their help during the article modification process. References Xue G, Bao W, Jiang J, et al. Hysteretic Behavior of Beam-to-Column Joints with Cast Steel Connectors[J]. Shock and Vibration, 2019, 2019: 1-20. Song Y Y, Ying L U. Decision tree methods: applications for classification and prediction[J]. Shanghai archives of psychiatry, 2015, 27(2): 130. Arslan Selçuk S, Gülle N B, Mutlu Avinç G. Tree-Like Structures in Architecture: Revisiting Frei Otto’s Branching Columns Through Parametric Tools[J]. SAGE Open, 2022, 12(3): 21582440221119479. Du W, Xia Z, Han L, et al. 3D solid model generation method based on a generative adversarial network[J]. Applied Intelligence, 2023, 53(13): 17035-17060. Wang H, Du W, Zhao Y, et al. Optimization and experimental research on treelike joints based on generative design and powder bed fusion[J]. Engineering Structures, 2023, 278: 115564. Wang H, Du W, Zhao Y, et al. Joints for treelike column structures based on generative design and additive manufacturing[J]. Journal of Constructional Steel Research, 2021, 184: 106794. C. Fang, B.A. Izzuddin, A.Y. Elghazouli. Modeling of semi-rigid beam-to-column steel joints under extreme loading, J. Frontiers of Structural and Civil Engineering. 7(3) (2013) 245–263. C. Brett, Y. Lu. Assessment of robustness of structures: current state of research, J. Frontiers of Structural and Civil Engineering. 7(4) (2013) 356–368. M.S. Aziz, Y.A.E. Sheriff. Biomimicry as an approach for bio-inspired structure with the aid of computation, J. Alexandria Engineering Journal. 55(1) (2016) 707-714. Zhang B, Yang B, Wu T, et al. Experimental and numerical study on the capability behavior of a thick-walled spatial cast-steel joint under complex load conditions[J]. Advances in Civil Engineering, 2019, 2019: 1-19. Bokhari I. Experimental Investigation of Structural Performance of Welded Interfaces Between Steel Castings and Steel Hollow Structural Sections[J]. 2022. Wang L, Jin H, Dong H, et al. Balance fatigue design of cast steel nodes in tubular steel structures[J]. The Scientific World Journal, 2013, 2013. Papatheocharis T, Sarvanis G C, Perdikaris P C, et al. Fatigue resistance of welded steel tubular X-joints[J]. Marine Structures, 2020, 74: 102809. Xiong Y, Lin K, Wu D, et al. The role of a novel coating of SFRCR-ECC in enhancing the fire performance of CFST columns: Development, characteristic and ISO-834 standard fire test[J]. Engineering Structures, 2023, 294: 116629. F. Bouafia, S.Boualem, MME. Amin, B. Benali. 3-D finite element analysis of stress concentration factor in spot-welded joints of steel: The effect of process-induced porosity, J. Computational Materials Science, 50(4) (2011) 1450-1459. Eurocode 3: Design of steel structures. Part 1.8 (design of joints) [S]. 2005. A. Loureiro, R Gutierrez, JM Reinosa, A Moreno. Axial stiffness prediction of non-preloaded T-stubs: an analytical frame approach, J. Journal of Construction Steel Research, 66(12) (2011) 613-622. International Standards Organization: ISO 6892-1:2009. Metallic materials tensile testing-part 1: Method of test at room temperature. Brussels, Belgium,2009. CECS235: Technical specification for application of connections of structural steel casting [S]. 2008. ANSI/AISC 360-05, Specification for Structural Steel Buildings [S].2005. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 11 Apr, 2024 Reviews received at journal 03 Apr, 2024 Reviewers agreed at journal 26 Mar, 2024 Reviewers invited by journal 26 Mar, 2024 Editor assigned by journal 26 Mar, 2024 Editor invited by journal 21 Mar, 2024 Submission checks completed at journal 21 Mar, 2024 First submitted to journal 09 Mar, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {\"props\":{\"pageProps\":{\"initialData\":{\"identity\":\"rs-4061078\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Article\",\"associatedPublications\":[],\"authors\":[{\"id\":283177757,\"identity\":\"0ecb108f-cf60-4009-8045-3be937463f0a\",\"order_by\":0,\"name\":\"Zhihao Li\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Henan University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Zhihao\",\"middleName\":\"\",\"lastName\":\"Li\",\"suffix\":\"\"},{\"id\":283177759,\"identity\":\"4b5a3df9-a0b4-42de-a9f9-2c9596db5ba1\",\"order_by\":1,\"name\":\"Yizhong 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Zhu\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyElEQVRIiWNgGAWjYBACA4YExgMMBjZybOyNjQ8/EKmF4QBDQZoxH8/hZmMJ4rV8OJw4TyK9TYCHGC3m7OkPDt0wOGzMJvmwjUGCwU5Ot4GAFsueBwmHcwzS5dikE9seFDAkG5sdIOSwGwkHgFqsjYFa2g0kGA4kbiOsJbEBqIU5sU3yYJsED3FakhmAWpwT2yQYidVy5hlIS5oxG08iMJANiPHL8fSHj3P+2MjJtx9/+PBDhZ0cQS3oJpCmfBSMglEwCkYBDgAAoctF6Asegk8AAAAASUVORK5CYII=\",\"orcid\":\"\",\"institution\":\"Henan University\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Liming\",\"middleName\":\"\",\"lastName\":\"Zhu\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2024-03-10 02:06:23\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-4061078/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-4061078/v1\",\"draftVersion\":[],\"editorialEvents\":[],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":53367961,\"identity\":\"7693efaf-942f-482a-af75-6d62c0f7a5ac\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":1,\"title\":\"Figure 1\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":505413,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eStructural and joint model for the exhibition hall of Zhonghong Hotel\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"1.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/490c0f3e092e9cb2748660d3.png\"},{\"id\":53367960,\"identity\":\"e889e7e5-5fea-457d-ae3e-326e40b34d4a\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":2,\"title\":\"Figure 2\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":13469,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eStress-strain relationship of the cast-steel material\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"2.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/15e433df861f7923b24bce32.png\"},{\"id\":53367963,\"identity\":\"832dafad-6921-4414-9b92-f6c531052d55\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":3,\"title\":\"Figure 3\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":22149,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eForce diagram of the joint\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"3.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/83219527ea719654d2b2084c.png\"},{\"id\":53367965,\"identity\":\"6153448d-a73f-4d90-afa8-070fc6739c43\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":4,\"title\":\"Figure 4\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":20023,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eLayout of measuring points\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"4.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/b9ec4adec008547f7bd48af3.png\"},{\"id\":53367968,\"identity\":\"056af8b4-1a05-4894-a47a-00dab49ed533\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":5,\"title\":\"Figure 5\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":144800,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eTest-specimen and loading device for the experiment\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"5.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/d7c09b786f28237bfe1acf43.png\"},{\"id\":53367962,\"identity\":\"b96555de-be00-40af-9877-27d7526858e4\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":6,\"title\":\"Figure 6\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":73494,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eLoad-deformation curves of specimens\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"6.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/aa561882cde830a10db150dc.png\"},{\"id\":53367966,\"identity\":\"b26e7659-4082-45e9-b3eb-ad430638345b\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":7,\"title\":\"Figure 7\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":79724,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eThe comparison of node model: (a)The model established by ANSY; (b) The model established by SolidWorks.\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"7.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/769759769b0d058af3620128.png\"},{\"id\":53367969,\"identity\":\"fb1e5734-35b9-4b6e-994f-2fc805dcb8ed\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":8,\"title\":\"Figure 8\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":109440,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eFinite element meshes\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"8.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/10ad2f53524813182ee6e488.png\"},{\"id\":53367964,\"identity\":\"99bf0b11-63af-4e57-acf0-a960ed51a6a8\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":9,\"title\":\"Figure 9\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":277455,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eStress contours of the joint: (a) The stress contour of the joint under 1000 kN; (b)The stress contour of the joint under 2000 kN; (c)The stress contour of the joint under 3000 kN; (d)The stress contour of the joint under 4000 kN; (e)The stress contour of the joint under 5000 kN.\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"9.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/e4e3369dbc77da066f1304f4.png\"},{\"id\":53367967,\"identity\":\"70f74966-f763-4230-a482-b206ed99dbf2\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":10,\"title\":\"Figure 10\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":84538,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eThe vertical displacement contours of joint.\\u003c/u\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"10.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/c31bb38638c7bc40ef17dd7f.png\"},{\"id\":53367970,\"identity\":\"4deb58ea-fbbd-494d-bbf4-d38d462aa592\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:05:09\",\"extension\":\"png\",\"order_by\":11,\"title\":\"Figure 11\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":350477,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e\\u003cu\\u003eFig.10 The relationship line chart between geometric parameters (\\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eθ\\u003c/em\\u003e\\u003c/u\\u003e\\u003cu\\u003e, \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eγ\\u003c/em\\u003e\\u003c/u\\u003e\\u003cu\\u003e, \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eβ\\u003c/em\\u003e\\u003c/u\\u003e\\u003cu\\u003e,\\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003e R\\u003c/em\\u003e\\u003c/u\\u003e\\u003csub\\u003e\\u003cu\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/u\\u003e\\u003c/sub\\u003e\\u003cu\\u003e) of the joint and the \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eK\\u003c/em\\u003e\\u003c/u\\u003e\\u003cu\\u003e: (a) Relationship line chart between \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eθ\\u003c/em\\u003e\\u003c/u\\u003e\\u003cu\\u003e and \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eK\\u003c/em\\u003e\\u003c/u\\u003e\\u003csub\\u003e\\u003cu\\u003e\\u003cem\\u003eθ\\u003c/em\\u003e\\u003c/u\\u003e\\u003c/sub\\u003e\\u003cu\\u003e; (b) Relationship line chart between \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eγ\\u003c/em\\u003e\\u003c/u\\u003e\\u003cu\\u003e and \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eKγ\\u003c/em\\u003e\\u003c/u\\u003e\\u003cu\\u003e; (c) Relationship line chart between\\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003e β \\u003c/em\\u003e\\u003c/u\\u003e\\u003cu\\u003eand \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eK\\u003c/em\\u003e\\u003c/u\\u003e\\u003csub\\u003e\\u003cu\\u003e\\u003cem\\u003eβ\\u003c/em\\u003e\\u003c/u\\u003e\\u003c/sub\\u003e\\u003cu\\u003e; (d) Relationship line chart between \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eR\\u003c/em\\u003e\\u003c/u\\u003e\\u003csub\\u003e\\u003cu\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/u\\u003e\\u003c/sub\\u003e\\u003cu\\u003e and \\u003c/u\\u003e\\u003cu\\u003e\\u003cem\\u003eK\\u003c/em\\u003e\\u003c/u\\u003e\\u003csub\\u003e\\u003cu\\u003e\\u003cem\\u003eR3\\u003c/em\\u003e\\u003c/u\\u003e\\u003c/sub\\u003e\\u003csub\\u003e\\u003cu\\u003e\\u003cem\\u003e。\\u003c/em\\u003e\\u003c/u\\u003e\\u003c/sub\\u003e\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"11.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/4102758761d7b95cd3e6440a.png\"},{\"id\":53368447,\"identity\":\"c58058ad-f716-42b3-bc61-c59c250d8962\",\"added_by\":\"auto\",\"created_at\":\"2024-03-25 07:13:11\",\"extension\":\"pdf\",\"order_by\":0,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":1735685,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"manuscript.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4061078/v1/d7410a4b-808d-47fa-b47c-bfa799182595.pdf\"}],\"financialInterests\":\"No competing interests reported.\",\"formattedTitle\":\"Experimental investigation and simulation analysis of cast-steel joints under vertical pressure\",\"fulltext\":[{\"header\":\"Introduction\",\"content\":\"\\u003cp\\u003eThe tree-like column has been widely utilized in engineering owing to its visually appealing structure and remarkable mechanical characteristics [\\u003cspan citationid=\\\"CR1\\\" class=\\\"CitationRef\\\"\\u003e1\\u003c/span\\u003e]. Since its introduction at Stuttgart Airport in Germany in 1991, several significant undertakings, such as Stansted Airport in London, the ION Orchard Center in Singapore, Detroit Airport in the United States, and the Railway Station in Changsha, China, have adopted this pioneering architectural structure [\\u003cspan additionalcitationids=\\\"CR3 CR4\\\" citationid=\\\"CR2\\\" class=\\\"CitationRef\\\"\\u003e2\\u003c/span\\u003e\\u0026ndash;\\u003cspan citationid=\\\"CR5\\\" class=\\\"CitationRef\\\"\\u003e5\\u003c/span\\u003e].\\u003c/p\\u003e \\u003cp\\u003eIt is obvious that the joints connecting the backbone and branches play a crucial role in the treelike structure. Firstly, the entire upper framework relies solely on a singular connection, and the arboreal assembly risks collapse in the event of this connection's demise. Secondly, numerous elements, encompassing both the main stem and lateral tubes, converge at the connection with seamless shifts, rendering the mechanical properties of such connection intricate.\\u003c/p\\u003e \\u003cp\\u003eTree-like column structures commonly employ cast-steel joints with branches [\\u003cspan citationid=\\\"CR6\\\" class=\\\"CitationRef\\\"\\u003e6\\u003c/span\\u003e]. In contrast to welded tubular joints, the use of cast-steel joints eliminates the residual stress caused by intricate welds at intersections. This not only simplifies construction but also accelerates the building process of tree-like structures [\\u003cspan citationid=\\\"CR7\\\" class=\\\"CitationRef\\\"\\u003e7\\u003c/span\\u003e\\u0026ndash;\\u003cspan citationid=\\\"CR8\\\" class=\\\"CitationRef\\\"\\u003e8\\u003c/span\\u003e].\\u003c/p\\u003e \\u003cp\\u003eWhile the utilization of cast steel in structural applications has garnered consistent interest over recent decades, research regarding cast-steel joints remains in its nascent phase [\\u003cspan citationid=\\\"CR9\\\" class=\\\"CitationRef\\\"\\u003e9\\u003c/span\\u003e]. Conducting full-scale experiments serves as a direct and efficient means to comprehend the performance of cast-steel joints. Consequently, several experiments assessing cast-steel joints in significant large-span steel roof structures have been conducted, including those at Shanghai South Railway Station [\\u003cspan citationid=\\\"CR10\\\" class=\\\"CitationRef\\\"\\u003e10\\u003c/span\\u003e], Beijing National Stadium [\\u003cspan citationid=\\\"CR11\\\" class=\\\"CitationRef\\\"\\u003e11\\u003c/span\\u003e], and Chongqing Olympic Stadium in China [\\u003cspan citationid=\\\"CR12\\\" class=\\\"CitationRef\\\"\\u003e12\\u003c/span\\u003e]. Additionally, numerical analysis has emerged as another viable approach to delineate the stress distribution within cast-steel joints comprehensively. Finite element analysis results offer invaluable insights for cast-steel joint design, as evidenced by projects such as the Cycling Gymnasium for Beijing Olympic Games [\\u003cspan citationid=\\\"CR13\\\" class=\\\"CitationRef\\\"\\u003e13\\u003c/span\\u003e], Guangzhou New Railway Station [\\u003cspan citationid=\\\"CR14\\\" class=\\\"CitationRef\\\"\\u003e14\\u003c/span\\u003e], and Tianjin Convention and Exhibition Center [\\u003cspan citationid=\\\"CR15\\\" class=\\\"CitationRef\\\"\\u003e15\\u003c/span\\u003e]. Eurocode 3 presents the component method, facilitating the evaluation of joint stiffness and resistance characteristics by aggregating those of all constituent components [\\u003cspan citationid=\\\"CR16\\\" class=\\\"CitationRef\\\"\\u003e16\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR17\\\" class=\\\"CitationRef\\\"\\u003e17\\u003c/span\\u003e].\\u003c/p\\u003e \\u003cp\\u003eThe geometric arrangement of cast-steel joints with branches typically consists of one primary pipe and two, three, or four secondary branching pipes. It is anticipated that the utilization of these joints will become increasingly widespread as tree-like column structures continue to evolve. However, as of yet, existing building standards or codes lack relevant provisions, and research on the mechanical performance, calculation formula, and structural optimization of cast-steel joints with branches remains limited.\\u003c/p\\u003e \\u003cp\\u003eThis paper presents a comprehensive investigation involving a typical full-scale experiment on a cast-steel joint featuring one trunk and three branches. The primary objective was to validate the accuracy of numerical simulation analysis. Utilizing ANSYS, stress distribution and the ultimate bearing capacity of the joint were meticulously calculated, yielding results that closely aligned with experimental findings. Furthermore, employing regression analysis, a calculation formula for the load-bearing capacity of cast-steel joints with branches was derived.\\u003c/p\\u003e\"},{\"header\":\"Model test of the three-branch cast-steel joint\",\"content\":\"\\u003cdiv id=\\\"Sec3\\\" class=\\\"Section2\\\"\\u003e \\u003ch2\\u003eEngineering background and design philosophy\\u003c/h2\\u003e \\u003cp\\u003eThe design specifications for the exhibition hall at Zhonghong Hotel in Kaifeng City, China, outline dimensions of 24 m \\u0026times; 34 m. Illustrated in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig1\\\" class=\\\"InternalRef\\\"\\u003e1\\u003c/span\\u003e, the roof structure employs a single-layer latticed shell, supported by a tree-like column and four additional supports from reinforced concrete frames. Notably, a cast-steel joint was employed at the intersection point of the trunk and the first-level branches.\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003cp\\u003eThe decision to utilize a cast-steel joint was driven by three primary considerations. Firstly, the joint needed to be fabricated from thin-walled pipes to minimize weight. Given the complexity of achieving hollow joints with intricate shapes through conventional methods, casting was deemed the most viable option. The thickness of the joint matches that of the main pipe, set at 1/20 of the pipe diameter to meet production and load-bearing requirements. Secondly, to ensure smooth stress transmission across different parts of the joint, the lengths of both the main pipe and branch pipes within the joint zone were carefully designed to avoid being excessively short. Thirdly, for ease of production and installation, the branch pipes were horizontally cut flat. To fulfill these criteria, the cast-steel joint was meticulously designed, as depicted in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig1\\\" class=\\\"InternalRef\\\"\\u003e1\\u003c/span\\u003e.\\u003c/p\\u003e \\u003cp\\u003eThe joint configuration comprises one main pipe and three symmetric branch pipes. The main pipe has dimensions of length (\\u003cem\\u003eL\\u003c/em\\u003e)\\u0026thinsp;=\\u0026thinsp;800 mm, diameter (\\u003cem\\u003eD\\u003c/em\\u003e)\\u0026thinsp;=\\u0026thinsp;500 mm, and wall thickness (\\u003cem\\u003eT\\u003c/em\\u003e)\\u0026thinsp;=\\u0026thinsp;40 mm. Each branch pipe shares identical dimensions: length (\\u003cem\\u003el\\u003c/em\\u003e)\\u0026thinsp;=\\u0026thinsp;500 mm, diameter (\\u003cem\\u003ed\\u003c/em\\u003e)\\u0026thinsp;=\\u0026thinsp;350 mm, and wall thickness (\\u003cem\\u003et\\u003c/em\\u003e)\\u0026thinsp;=\\u0026thinsp;35 mm. The chamfer radius (\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e1\\u003c/em\\u003e\\u003c/sub\\u003e) between the main pipe and the branch pipes is 1000 mm, while the outside chamfer radius (\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e2\\u003c/em\\u003e\\u003c/sub\\u003e) between branch pipes is 50 mm. Additionally, the inside chamfer radius (\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/sub\\u003e) among branch pipes is 20 mm. Other essential parameters include \\u003cem\\u003eθ\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;30\\u0026deg; (angle between main pipe and branch pipe), \\u003cem\\u003eβ\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;0.7 (ratio of outer diameter of branch pipe to outer diameter of main pipe), and \\u003cem\\u003eγ\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;20 (ratio of outer diameter of main pipe to main pipe wall thickness).\\u003c/p\\u003e \\u003c/div\\u003e \\u003cdiv id=\\\"Sec4\\\" class=\\\"Section2\\\"\\u003e \\u003ch2\\u003eManufacturing and material properties of the joint specimen\\u003c/h2\\u003e \\u003cp\\u003eThe joint is fabricated from ZG20SiMn cast steel, characterized by the following chemical composition: Carbon (C) \\u0026minus;\\u0026thinsp;0.18%, Silicon (Si) \\u0026minus;\\u0026thinsp;0.60%, Manganese (Mn) \\u0026minus;\\u0026thinsp;1.50%, Phosphorus (P) \\u0026minus;\\u0026thinsp;0.020%, Sulfur (S) \\u0026minus;\\u0026thinsp;0.015%, Chromium (Cr) \\u0026minus;\\u0026thinsp;0.30%, Molybdenum (Mo) \\u0026minus;\\u0026thinsp;0.15%, and Nickel (Ni) \\u0026minus;\\u0026thinsp;0.40%.\\u003c/p\\u003e \\u003cp\\u003eThe joint specimen was cast using a sand mold at Tengfei Factory in Xinxiang City, China. Following heat treatment and polishing, the specimen was transported to the structural laboratory of Henan University for testing. A standard dog-bone flat specimen was also prepared to evaluate the material properties [\\u003cspan citationid=\\\"CR18\\\" class=\\\"CitationRef\\\"\\u003e18\\u003c/span\\u003e]. The material test results, including the typical stress-strain relationship, are illustrated in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003e. The stress-strain curve for the cast-steel coupon exhibits a distinct yielding plateau, indicating favorable material properties. The yield strength is measured at 235.7 MPa, and the ultimate strength reaches 353.6 MPa. The percentage elongation at fracture is found to be 25.9%, and the ultimate-to-yield strength ratio is calculated at 1.50. These results demonstrate that ZG20SiMn cast steel possesses notable ductility and strain hardening ability, exceeding the basic requirements stipulated in building codes such as Eurocode 3. Specifically, the percentage elongation at fracture surpasses the minimum of 15%, and the ultimate-to-yield strength ratio exceeds the required minimum of 1.1.\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003cp\\u003eTo ensure precise adherence to design specifications, the size of the actual joint was meticulously measured, and the relative errors between the actual and design siz-es are detailed in Table\\u0026nbsp;\\u003cspan refid=\\\"Tab1\\\" class=\\\"InternalRef\\\"\\u003e1\\u003c/span\\u003e. Notably, the maximum relative error recorded is 2.80%, comfortably below the acceptable threshold of 5.00%. This demonstrates that the pro-duction of the joint specimen successfully meets the stringent design requirements.\\u003c/p\\u003e \\u003cp\\u003e \\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"Yes\\\" id=\\\"Tab1\\\" border=\\\"1\\\"\\u003e \\u003ccaption language=\\\"En\\\"\\u003e \\u003cdiv class=\\\"CaptionNumber\\\"\\u003eTable 1\\u003c/div\\u003e \\u003cdiv class=\\\"CaptionContent\\\"\\u003e \\u003cp\\u003eThe comparative analysis of the test-specimen dimensions\\u003c/p\\u003e \\u003c/div\\u003e \\u003c/caption\\u003e \\u003ccolgroup cols=\\\"4\\\"\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c3\\\" colnum=\\\"3\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c4\\\" colnum=\\\"4\\\"\\u003e\\u003c/div\\u003e \\u003cthead\\u003e \\u003ctr\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eParts of the joint\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003eMeasured sizes\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003eDesign sizes\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003eErrors\\u003c/p\\u003e \\u003c/th\\u003e \\u003c/tr\\u003e \\u003c/thead\\u003e \\u003ctbody\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eThe main pipe length(\\u003cem\\u003eL\\u003c/em\\u003e)\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e806 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e800 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e0.75%\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eThe main pipe diameter(\\u003cem\\u003eD\\u003c/em\\u003e)\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e495 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e500 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e0.10%\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eThe main pipe wall thickness(\\u003cem\\u003eT\\u003c/em\\u003e)\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e39.5 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e40 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e1.25%\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eThe branch pipes length(\\u003cem\\u003el\\u003c/em\\u003e)\\u003c/p\\u003e \\u003cp\\u003eThe branch pipes diameter(\\u003cem\\u003ed\\u003c/em\\u003e)\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e492 mm\\u003c/p\\u003e \\u003cp\\u003e345 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e500 mm\\u003c/p\\u003e \\u003cp\\u003e350 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e0.16%\\u003c/p\\u003e \\u003cp\\u003e1.71%\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eThe branch pipes wall thickness(\\u003cem\\u003et\\u003c/em\\u003e)\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e34.6 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e35 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e1.14%\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eThe chamfer (\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e1\\u003c/em\\u003e\\u003c/sub\\u003e)\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e994 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e1000 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e0.60%\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eThe chamfer (\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e2\\u003c/em\\u003e\\u003c/sub\\u003e)\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e20.46 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e20 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e2.30%\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eThe chamfer (\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/sub\\u003e)\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e51.4 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e50 mm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e2.80%\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003c/tbody\\u003e \\u003c/colgroup\\u003e \\u003c/table\\u003e\\u003c/div\\u003e \\u003c/p\\u003e \\u003c/div\\u003e \\u003cdiv id=\\\"Sec5\\\" class=\\\"Section2\\\"\\u003e \\u003ch2\\u003eLoading procedure\\u003c/h2\\u003e \\u003cp\\u003eThe experimental setup involved applying a vertical load, as depicted in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003e, with a maximum load capacity set at 5000 kN. To facilitate contact between the experimental joint and the load piston, an initial preloading stage was implemented, determining a preloading value of approximately 30 kN through several trials. Following preloading, the regular loading phase unfolded in five stages, each incrementing by 1000 kN at a constant rate of 10 kN/s. At the conclusion of each stage, the load was maintained for a duration of 30 seconds. Subsequently, the unloading process commenced, mirroring the loading stages with five steps, each decreasing by 1000 kN at a rate of 10 kN/s. Similar to the loading stages, a 30-second holding period followed the completion of each unloading stage.\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003cdiv id=\\\"Sec6\\\" class=\\\"Section3\\\"\\u003e \\u003ch2\\u003eStrain measurement\\u003c/h2\\u003e \\u003cp\\u003eResistance strain gauges were employed to measure the strain in the cast-steel joint during testing. These strain gauges and strain rosettes were manufactured by the Giant Star Electric Measuring Element Factory in Taizhou, China. The specifications for the strain gauges are as follows: a resistance value of 120\\u0026thinsp;\\u0026plusmn;\\u0026thinsp;0.2 (Ω), a sensitivity factor of 2.08\\u0026thinsp;\\u0026plusmn;\\u0026thinsp;1%, and a sensitive grid size of 2 mm (gate length) \\u0026times; 1 mm (width) on a substrate measuring 4.5 mm (gate length) \\u0026times; 2.4 mm (width). Similarly, the strain rosettes possess a resistance value of 120\\u0026thinsp;\\u0026plusmn;\\u0026thinsp;0.3 (Ω), a sensitivity factor of 2.08\\u0026thinsp;\\u0026plusmn;\\u0026thinsp;1%, and a sensitive grid size of 3 mm (gate length) \\u0026times; 2 mm (width) on a substrate measuring 11.5 mm (gate length) \\u0026times; 11.5 mm (width).\\u003c/p\\u003e \\u003cp\\u003eThe surfaces of the joint were pre-polished to facilitate the attachment of strain gauges and strain rosettes. The placement of measuring points was primarily determined by the characteristics of joint stress calculated through finite element analysis. Analysis results indicated elevated stress levels near the joint core area, contrasting with lower stresses in the main pipe and branch pipes. Consequently, measuring points near the joint core area were densely positioned. Overall, measuring points were concentrated in four areas: (A) upper part of the main pipe, (B) lower part of the main pipe, (C) vicinity of the joint core area, and (D) branch pipes. Strain gauges were affixed to the upper and lower parts of the main pipe, where stress distribution is simpler. Conversely, strain rosettes were placed near the joint core area, characterized by unknown principal stress directions and complex stress distribution patterns. Measuring point positions were uniformly distributed and symmetrical, with three copies along the joint circumference. Figure\\u0026nbsp;\\u003cspan refid=\\\"Fig4\\\" class=\\\"InternalRef\\\"\\u003e4\\u003c/span\\u003e illustrates the facade of measuring points on the joint, with identical placement on the other two symmetrical sides. The specific locations and quantities of measuring points are outlined in Table\\u0026nbsp;\\u003cspan refid=\\\"Tab2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003e.\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003cp\\u003e \\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"Yes\\\" id=\\\"Tab2\\\" border=\\\"1\\\"\\u003e \\u003ccaption language=\\\"En\\\"\\u003e \\u003cdiv class=\\\"CaptionNumber\\\"\\u003eTable 2\\u003c/div\\u003e \\u003cdiv class=\\\"CaptionContent\\\"\\u003e \\u003cp\\u003eThe stresses of the measuring points (MPa)\\u003c/p\\u003e \\u003c/div\\u003e \\u003c/caption\\u003e \\u003ccolgroup cols=\\\"7\\\"\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c3\\\" colnum=\\\"3\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c4\\\" colnum=\\\"4\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c5\\\" colnum=\\\"5\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c6\\\" colnum=\\\"6\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c7\\\" colnum=\\\"7\\\"\\u003e\\u003c/div\\u003e \\u003cthead\\u003e \\u003ctr\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eNo.\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e1000 kN\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e2000 kN\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e3000 kN\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e4000 kN\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e5000 kN\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colname=\\\"c7\\\"\\u003e \\u003cp\\u003eLocations\\u003c/p\\u003e \\u003c/th\\u003e \\u003c/tr\\u003e \\u003c/thead\\u003e \\u003ctbody\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eA1\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e26.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e53.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e80.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e112.5\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e130.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\" morerows=\\\"2\\\" rowspan=\\\"3\\\"\\u003e \\u003cp\\u003eThe upper part of the main pipe\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eA2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e23.2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e51.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e78.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e109.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e126.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eA3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e28.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e54.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e84.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e110.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e121.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eB1\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e34.5\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e72.8\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e112.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e145.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e183.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\" morerows=\\\"2\\\" rowspan=\\\"3\\\"\\u003e \\u003cp\\u003eThe lower part of the main pipe\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eB2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e31.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e69.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e114.2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e148.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e178.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eB3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e38.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e75.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e115.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e147.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e176.9\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eC11\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e42.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e88.2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e128.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e183.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e235.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\" morerows=\\\"8\\\" rowspan=\\\"9\\\"\\u003e \\u003cp\\u003eThe vicinity of the joint core area\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eC12\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e44.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e87.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e130.1\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e179.5\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e235.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eC13\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e44.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e87.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e126.8\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e182.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e235.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eC21\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e44.8\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e85.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e127.2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e178.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e235.2\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eC22\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e45.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e86.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e128.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e185.1\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e235.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eC23\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e41.9\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e86.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e129.1\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e179.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e235.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eC31\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e42.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e89.2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e128.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e180.1\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e235.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eC32\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e46.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e84.9\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e131.2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e185.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e235.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eC33\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e44.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e85.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e127.9\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e176.1\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e235.1\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eD1\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e35.2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e71.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e110.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e144.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e181.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\" morerows=\\\"2\\\" rowspan=\\\"3\\\"\\u003e \\u003cp\\u003eThe branch pipes\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eD2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e34.6\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e70.8\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e117.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e151.5\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e179.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eD3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e32.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e68.7\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e108.4\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e156.3\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e183.5\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003c/tbody\\u003e \\u003c/colgroup\\u003e \\u003c/table\\u003e\\u003c/div\\u003e \\u003c/p\\u003e \\u003c/div\\u003e \\u003c/div\\u003e \\u003cdiv id=\\\"Sec7\\\" class=\\\"Section2\\\"\\u003e \\u003ch2\\u003eExperiment observation\\u003c/h2\\u003e \\u003cp\\u003eAt the onset of preloading, a faint sound was discernible, attributed to the adjustment of the experimental specimen to close the gap between the piston and itself. This sound dissipated as the load reached 20 kN. Following the completion of preloading, the regular loading test proceeded in accordance with the prescribed loading procedure. Figure\\u0026nbsp;\\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003e illustrates a photograph capturing the joint under load. Throughout the loading process, both the measured displacement and strain exhibited consistent, uninterrupted development without any notable jumps.\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003c/div\\u003e \\u003cdiv id=\\\"Sec8\\\" class=\\\"Section2\\\"\\u003e \\u003ch2\\u003eAnalysis and discussion of test results\\u003c/h2\\u003e \\u003cp\\u003e(1) Analysis of the load-displacement curve\\u003c/p\\u003e \\u003cp\\u003eFigure\\u0026nbsp;\\u003cspan refid=\\\"Fig6\\\" class=\\\"InternalRef\\\"\\u003e6\\u003c/span\\u003e depicts the load-displacement curve obtained from the experimental specimen. Broadly, the load-displacement curve of the joint can be delineated into three distinct stages. In stage I, characterized by relatively low loads (less than 500 kN), the slope of the load-displacement curve is minimal. This signifies a significant increase in joint displacement at the initial stage of loading, attributable to the non-tight contact between the joint and the testing equipment.\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003cp\\u003eWith the continuous increase in load, the load becomes proportional to displacement, but the slope of the load-displacement curve experiences a noticeable augmentation. This phase is defined as stage II, suggesting that the joint operates within the elastic state as the load varies between 500 kN and 4750 kN.\\u003c/p\\u003e \\u003cp\\u003eAs the load approaches the maximum design value (stage III), the load-displacement curve exhibits a non-linear behavior, indicating that the joint transitions into the elastic-plastic state. Notably, a significant increase in displacement corresponds to a minor increment in load. The slope of the load-displacement curve in stage III indicates a rapid expansion of the plastic zone within the joint. Therefore, it is advisable to consider the proportional limit observed in stage II as the control value for the design of the cast-steel joint with branches. From stage II, it is evident that the ultimate bearing capacity of the joint specimen is 4750 kN.\\u003c/p\\u003e \\u003cp\\u003eDuring the unloading process, the joint's deformation exhibited linear recovery in tandem with the decrease in load. The maximum vertical displacement recorded throughout the entire process amounted to 8.305 mm. However, upon unloading, the joint did not revert to the zero point due to residual deformation. This residual deformation can be attributed to two factors. Firstly, non-tight contact between the joint and the testing equipment, estimated to be approximately 4.435 mm (refer to Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig6\\\" class=\\\"InternalRef\\\"\\u003e6\\u003c/span\\u003e). Secondly, residual plastic deformation, which amounts to about 0.890 mm. Therefore, the total residual deformation is calculated to be 5.325 mm.\\u003c/p\\u003e \\u003cp\\u003e(2) Analysis of the stress\\u003c/p\\u003e \\u003cp\\u003eUpon completion of each load stage, the strain data for each measuring point was collected. Subsequently, the stress values for each measuring point were calculated and are presented in Table\\u0026nbsp;\\u003cspan refid=\\\"Tab2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003e. Notably, when the load reached its maximum value of 5000 kN, the maximum stress recorded was 235.7 MPa, observed at the joint core area. This finding underscores the joint core area as the most critical location. In contrast, the minimum stress among the measuring points was 121.4 MPa, situated at the upper part of the main pipe.\\u003c/p\\u003e \\u003cp\\u003eAs the load increased, the stress at each measuring point exhibited a linear increase during the initial four loading stages. However, upon reaching a load of 4750 kN, the stresses at measuring points situated within the joint core area stabilized at approximately 235 MPa. This phenomenon indicates the onset of a plastic zone within the joint core area. Additionally, when examining the measuring points arranged symmetrically along the circumference of the joint, it is evident that the stress levels at these points were largely consistent.\\u003c/p\\u003e \\u003c/div\\u003e\"},{\"header\":\"Numerical Simulation Analysis\",\"content\":\"\\u003cdiv id=\\\"Sec10\\\"\\u003e\\n \\u003ch2\\u003eAnalysis model\\u003c/h2\\u003e\\n \\u003cp\\u003eTo comprehensively assess the performance of the test joint, a corresponding finite element model was constructed. Initially, as depicted in Fig.\\u0026nbsp;\\u003cspan\\u003e7\\u003c/span\\u003e(a), the joint model was established using the default ANSYS preprocessor. However, challenges arose in achieving a seamless transition between the main pipe and the branches. Consequently, the joint was accurately modeled using 3D modeling software SolidWorks to ensure consistency with the actual joint configuration, as illustrated in Fig.\\u0026nbsp;\\u003cspan\\u003e7\\u003c/span\\u003e (b).\\u003c/p\\u003e\\n \\u003cp\\u003eThe established joint model was imported into the finite element software ANSYS for analysis. The material properties used in the model were obtained from the material test. The elastic modulus of the material (\\u003cem\\u003eE\\u003c/em\\u003e) was determined to be 2.0\\u0026times;10^5 N/mm\\u0026sup2;, the yield strength (\\u003cem\\u003ef\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ey\\u003c/em\\u003e\\u003c/sub\\u003e) was 235 MPa, and the Poisson\\u0026apos;s ratio (\\u003cem\\u003e\\u0026micro;\\u003c/em\\u003e) was 0.3. The constitutive behavior of the material was selected to be the ideal elastic-plastic model, utilizing the Von-Mises yield criterion and associated flow rule for the elastic-plastic analysis [\\u003cspan\\u003e20\\u003c/span\\u003e]. To accurately simulate the real-world scenario, the boundary conditions of the joint were set as follows: the end part of the main pipe was fixed, while the ends of the branch pipes were fixed vertically. Additionally, the load was applied to the ends of the branch pipes in the form of surface pressure.\\u003c/p\\u003e\\n \\u003cp\\u003eAfter extensive research and comparison, the three-dimensional solid element Solid65 was selected from the ANSYS element type library for modeling the joint. This element type features quadratic displacement and is well-suited for irregular grid division [21]. The finite element mesh of the joint, depicted in Fig.\\u0026nbsp;\\u003cspan\\u003e8\\u003c/span\\u003e, was meticulously crafted to ensure accurate representation and analysis of the structural behavior.\\u003c/p\\u003e\\n\\u003c/div\\u003e\\n\\u003cdiv id=\\\"Sec11\\\"\\u003e\\n \\u003ch2\\u003eAnalysis results of the testing joint\\u003c/h2\\u003e\\n \\u003cp\\u003eThe finite element analysis was conducted for the test joint under varying load conditions. Figure\\u0026nbsp;\\u003cspan\\u003e9\\u003c/span\\u003e illustrates the stress contours of the joint corresponding to load levels of 1000 kN, 2000 kN, 3000 kN, 4000 kN, and 5000 kN, respectively. From Fig.\\u0026nbsp;\\u003cspan\\u003e9\\u003c/span\\u003e, it is evident that the overall stress level of the joint under a 1000 kN load is relatively low. The maximum stress value is predominantly concentrated in the vicinity of the joint core area. Furthermore, the stress observed in the main pipe and branch pipes is notably lower compared to that in the core area of the joint. Specifically, the stress in the former accounts for only 11.5% of the stress observed in the latter.\\u003c/p\\u003e\\n \\u003cp\\u003eAs the load increases to 2000 kN, the stress level of the joint exhibits a gradual linear increase, reaching a maximum stress value of 219 MPa. Upon reaching a load of 3000 kN, signs of yielding in the steel become apparent. However, the yield region is primarily concentrated at three points in the chamfer between the branch pipes. Subsequently, as the load further increases to 4000 kN, the yield region expands outward. This expansion manifests in two ways: firstly, the plastic region enlarges, and secondly, the chamfer between the main pipe and branch pipes also enters the plastic zone. Upon reaching the maximum load of 5000 kN, the expansion of the plastic zone within the joint intensifies, although it remains primarily concentrated in the vicinity of the joint core area. At this stage, nearly the entire core area of the joint enters the yield state, indicating the formation of a plastic hinge and marking the entry of the load-displacement curve into stage III (as illustrated in Fig.\\u0026nbsp;\\u003cspan\\u003e6\\u003c/span\\u003e). Despite this, the stresses observed in the main pipe and branch pipes remain relatively low, approximately 106 MPa. This underscores the concentration of stress within the core region of the cast-steel joint with branches, which significantly influences its ultimate load-carrying capacity.\\u003c/p\\u003e\\n \\u003cp\\u003eFigure\\u0026nbsp;\\u003cspan\\u003e10\\u003c/span\\u003e illustrates the vertical displacement of the joint under the maximum load of 5000 kN. It is observed that the maximum vertical displacement obtained from the finite element analysis (4.366 mm) is smaller than that obtained from the test (8.064 mm). This disparity can be primarily attributed to the non-tight contact between the joint and the test equipment piston during the experimental testing process.\\u003c/p\\u003e\\n\\u003c/div\\u003e\\n\\u003cdiv id=\\\"Sec12\\\"\\u003e\\n \\u003ch2\\u003eVerification of the finite element model through the experiment results\\u003c/h2\\u003e\\n \\u003cp\\u003eTo validate the numerical model of the cast-steel joint with branches, a comparison is made between the results of finite element analysis and those of the verification experiment. The stress values obtained from representative measuring points in the finite element model are compared with the experimentally derived stress values, and their relative differences are listed in Table\\u0026nbsp;\\u003cspan\\u003e3\\u003c/span\\u003e.\\u003c/p\\u003e\\n \\u003cdiv\\u003e\\n \\u003ctable id=\\\"Tab3\\\" border=\\\"1\\\"\\u003e\\n \\u003ccaption language=\\\"En\\\"\\u003e\\n \\u003cdiv\\u003eTable 3\\u003c/div\\u003e\\n \\u003cdiv\\u003e\\n \\u003cp\\u003eThe joints analysis results with different parameters under axial loading\\u003c/p\\u003e\\n \\u003c/div\\u003e\\n \\u003c/caption\\u003e\\n \\u003cthead\\u003e\\n \\u003ctr\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJoint\\u003c/p\\u003e\\n \\u003cp\\u003enumber\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003e\\u0026theta;\\u003c/em\\u003e\\u003c/p\\u003e\\n \\u003cp\\u003e(\\u003csup\\u003eo\\u003c/sup\\u003e)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003eL\\u003c/em\\u003e\\u003c/p\\u003e\\n \\u003cp\\u003e(mm)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003e\\u0026beta;\\u003c/em\\u003e\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003e\\u0026gamma;\\u003c/em\\u003e\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e1\\u003c/em\\u003e\\u003c/sub\\u003e\\u003c/p\\u003e\\n \\u003cp\\u003e(mm)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e2\\u003c/em\\u003e\\u003c/sub\\u003e\\u003c/p\\u003e\\n \\u003cp\\u003e(mm)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/sub\\u003e\\u003c/p\\u003e\\n \\u003cp\\u003e(mm)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eUltimate load-carrying capacity (kN)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003c/tr\\u003e\\n \\u003c/thead\\u003e\\n \\u003ctbody\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ1\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2000.24\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ2\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2301.79\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ3\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e40\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2317.40\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ4\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e50\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n 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\\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e25\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1705.06\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ8\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e29.9\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1334.73\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ9\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e500\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2056.27\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ10\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1500\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2344.98\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ11\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2233.56\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ12\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e10\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2298.95\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ13\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2268.51\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ14\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2208.79\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ15\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e50\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2824.84\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ16\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e100\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e3275.00\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ17\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e150\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e3296.32\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ18\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.6\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2223.88\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ19\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.66\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2281.90\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.74\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2352.50\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ21\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e800\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.8\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2418.69\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ22\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e500\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2310.59\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ23\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e600\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2312.61\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ24\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e700\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1000\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e2314.97\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003c/tbody\\u003e\\n \\u003c/table\\u003e\\n \\u003c/div\\u003e\\n \\u003cp\\u003eAnalyzing the data in Table 3 reveals that:\\u003c/p\\u003e\\n \\u003cp\\u003e1) The stress distribution of the cast-steel joint with branches obtained from the experiment aligns closely with that calculated from the finite element model. The calculated stresses from the finite element analysis and the experimental values of measuring points exhibit consistency. The maximum error between the calculated stresses and the experimental results is 9.02%, substantiating the validity of the finite element model utilized in this study.\\u003c/p\\u003e\\n \\u003cp\\u003e2) Both the finite element modeling and the experiment confirm that the area of large stress is concentrated in the core area of the joint. The stresses in the main pipe and the branch pipes are comparatively small, approximately half of the largest stress observed in the core area of the joint.\\u003c/p\\u003e\\n \\u003cp\\u003e3) The casting precision of cast steel joints presents challenges in control. In this study, it was observed that the chamfer between the main pipe and the branch pipes was slightly larger than the design value. Additionally, the wall thickness exceeded the design specifications, resulting in a smaller diameter thickness ratio. Consequently, the stresses predicted by the finite element model tend to be generally higher than those observed in the experiment.\\u003c/p\\u003e\\n \\u003cp\\u003e4) In summary, the results obtained from the finite element model align closely with those from the experiment. The numerical model effectively captures the actual stress and deformation states of the cast-steel joint with branches. Therefore, it can serve as a reliable tool for investigating the load-carrying capacity of such joints in further research.\\u003c/p\\u003e\\n\\u003c/div\\u003e\\n\\u003cdiv id=\\\"Sec13\\\"\\u003e\\n \\u003ch2\\u003eEffect of joint parameters on compression behavior of the cast-steel joint\\u003c/h2\\u003e\\n \\u003cp\\u003eTo investigate the influence of different parameters on the ultimate load-carrying capacity of the joint, a parametric study is conducted wherein individual variables are varied while keeping other parameters constant. This approach allows for a systematic analysis of how each variable impacts the joint\\u0026apos;s performance independently.\\u003c/p\\u003e\\n \\u003cp\\u003eThe modeling results are shown in Table\\u0026nbsp;\\u003cspan\\u003e4\\u003c/span\\u003e, which could be summarized as:\\u003c/p\\u003e\\n \\u003cp\\u003e1) During compression testing, the fifth joint (J5) demonstrates the highest load-bearing capacity, reaching 5720.7 kN. Conversely, the eighth joint (J8) exhibits the lowest load-bearing capacity, registering only 1334.73 kN. Although both J8 and J5 share similar geometric characteristics, J8 boasts the greatest diameter thickness ratio (\\u003cem\\u003e\\u0026gamma;\\u003c/em\\u003e), while J5 possesses the smallest. Thus, the diameter thickness ratio (\\u003cem\\u003e\\u0026gamma;\\u003c/em\\u003e) significantly impacts the load-bearing capacity of joints.\\u003c/p\\u003e\\n \\u003cp\\u003e2) Increasing only \\u003cem\\u003e\\u0026theta;\\u003c/em\\u003e while holding other variables constant substantially decreases the joint\\u0026apos;s ultimate load-bearing capacity. This observation underscores \\u003cem\\u003e\\u0026theta;\\u0026apos;s\\u003c/em\\u003e substantial influence on the joint\\u0026apos;s load-carrying capability.\\u003c/p\\u003e\\n \\u003cp\\u003e3) Gradual increments in \\u003cem\\u003e\\u0026beta;\\u003c/em\\u003e and \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/sub\\u003e, while keeping other factors constant, substantially enhance the joint\\u0026apos;s ultimate load-carrying capacity. This finding highlights the significant impact of \\u0026beta; and \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/sub\\u003e on the joint\\u0026apos;s load-bearing capability.\\u003c/p\\u003e\\n \\u003cp\\u003e4) When dimensions \\u003cem\\u003eL\\u003c/em\\u003e, \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e1\\u003c/em\\u003e\\u003c/sub\\u003e, and \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e2\\u003c/em\\u003e\\u003c/sub\\u003e undergo gradual increments while all other variables remain constant, the ultimate load-carrying capacity of the joint exhibits minimal variation, with the largest observed change being less than 5%. This indicates that dimensions \\u003cem\\u003eL\\u003c/em\\u003e, \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e1\\u003c/em\\u003e\\u003c/sub\\u003e, and \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e2\\u003c/em\\u003e\\u003c/sub\\u003e exert negligible influence on the ultimate load-carrying capacity of the joint.\\u003c/p\\u003e\\n \\u003cp\\u003e5) Based on the findings of finite element modeling, it is deduced that thorough consideration of geometric parameters is imperative when analyzing the load-carrying capacity of cast-steel joints with branches. Careful selection of dimensional parameters for the joint is essential to ensure the structural safety and reliability.\\u003c/p\\u003e\\n \\u003cdiv\\u003e\\u003cimg src=\\\"https://myfiles.space/user_files/122228_c8a1650c59388082/122228_custom_files/img1711349960.png\\\"\\u003e\\u003c/div\\u003e\\n\\u003c/div\\u003e\\n\\u003cdiv id=\\\"Sec14\\\"\\u003e\\n \\u003ch2\\u003eLoad-carrying capacity estimation of the three-branch cast-steel joint\\u003c/h2\\u003e\\n \\u003cp\\u003eIn existing literature [\\u003cspan\\u003e19\\u003c/span\\u003e\\u0026ndash;\\u003cspan\\u003e20\\u003c/span\\u003e], load-carrying capacity formulas for welded tubular T-joints, steel tubular XK-joints, and multi-planar KX and KT-joints under axial loads are consistently represented as the product of the material yield strength and the square of the pipe wall thickness. Accordingly, the estimation of load-carrying capacity for cast-steel joints with branches can be similarly expressed as:\\u003c/p\\u003e\\n \\u003cdiv id=\\\"Equ1\\\"\\u003e\\n \\u003cdiv id=\\\"FileID_Equ1\\\" name=\\\"EquationSource\\\"\\u003e$${F_u}=K{T^2}{f_y}$$\\u003c/div\\u003e\\n \\u003cdiv\\u003e1\\u003c/div\\u003e\\n \\u003c/div\\u003e\\n \\u003cp\\u003ewhere Fu is load-carrying capacity of the joint; \\u003cem\\u003eK\\u003c/em\\u003e is a parameter that contains the geometric parameters such as \\u003cem\\u003e\\u0026theta;\\u003c/em\\u003e, \\u003cem\\u003e\\u0026gamma;\\u003c/em\\u003e and \\u003cem\\u003e\\u0026beta;\\u003c/em\\u003e of the joint; \\u003cem\\u003eT\\u003c/em\\u003e is the pipe wall thickness; and \\u003cem\\u003ef\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ey\\u003c/em\\u003e\\u003c/sub\\u003e is the material yield strength of the joint.\\u003c/p\\u003e\\n \\u003cp\\u003eIn Eq.\\u0026nbsp;(\\u003cspan\\u003e1\\u003c/span\\u003e), the expression of parameter \\u003cem\\u003eK\\u003c/em\\u003e serves as the primary research focus across various types of joints. Given that K encompasses a range of geometric parameters affecting the load-carrying capacity of the joint, the focus has shifted from solely examining the relationship between \\u003cem\\u003eK\\u003c/em\\u003e and material yield strength (\\u003cem\\u003eF\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003eu\\u003c/em\\u003e\\u003c/sub\\u003e) to conducting multiple studies on the correlation between each individual parameter and \\u003cem\\u003eK\\u003c/em\\u003e.\\u003c/p\\u003e\\n \\u003cp\\u003eThe finite element analysis results indicate that dimensions \\u003cem\\u003eL\\u003c/em\\u003e, \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e1\\u003c/em\\u003e\\u003c/sub\\u003e, and \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e2\\u003c/em\\u003e\\u003c/sub\\u003e of the joint exert minimal influence on the ultimate load-carrying capacity. Consequently, these parameters are disregarded during the analysis of the comprehensive index \\u003cem\\u003eK\\u003c/em\\u003e. Utilizing line charts depicting the relationships between \\u003cem\\u003e\\u0026theta;\\u003c/em\\u003e, \\u003cem\\u003e\\u0026gamma;\\u003c/em\\u003e, \\u003cem\\u003e\\u0026beta;\\u003c/em\\u003e, and \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/sub\\u003e with \\u003cem\\u003eK\\u003c/em\\u003e, as illustrated in Fig.\\u0026nbsp;11, a regression analysis is performed.\\u003c/p\\u003e\\n \\u003cp\\u003eFollowing the regression analysis, the relationship between \\u0026theta; and K\\u0026theta; is initially examined. Through this analysis, the relationship between the sine value of \\u0026theta; and K\\u0026theta; can be expressed as:\\u003c/p\\u003e\\n \\u003cdiv id=\\\"Equ2\\\"\\u003e\\n \\u003cdiv id=\\\"FileID_Equ2\\\" name=\\\"EquationSource\\\"\\u003e$${K_\\\\theta }=0.60022 - 2.5311\\\\sin \\\\theta +6.59681{\\\\sin ^2}\\\\theta - 5.07388{\\\\sin ^3}\\\\theta$$\\u003c/div\\u003e\\n \\u003cdiv\\u003e2\\u003c/div\\u003e\\n \\u003c/div\\u003e\\n \\u003cp\\u003eFor the relationship between \\u0026gamma; and K\\u0026gamma;, it could be expressed as a power function:\\u003c/p\\u003e\\n \\u003cdiv id=\\\"Equ3\\\"\\u003e\\n \\u003cdiv id=\\\"FileID_Equ3\\\" name=\\\"EquationSource\\\"\\u003e$${K_\\\\gamma }=4.3725{\\\\gamma ^{0.66242}}$$\\u003c/div\\u003e\\n \\u003cdiv\\u003e3\\u003c/div\\u003e\\n \\u003c/div\\u003e\\n \\u003cp\\u003eFrom the observations in Fig.\\u0026nbsp;\\u003cspan\\u003e10\\u003c/span\\u003e, it is apparent that a linear relationship exists between \\u0026beta; and K\\u0026beta;. This relationship can be expressed as:\\u003c/p\\u003e\\n \\u003cdiv id=\\\"Equ4\\\"\\u003e\\n \\u003cdiv id=\\\"FileID_Equ4\\\" name=\\\"EquationSource\\\"\\u003e$${K_\\\\beta }=1+0.58856\\\\beta$$\\u003c/div\\u003e\\n \\u003cdiv\\u003e4\\u003c/div\\u003e\\n \\u003c/div\\u003e\\n \\u003cp\\u003eFinally, the regression analysis is performed between R3 and KR3. According to the principle of dimensional analysis, it is essential that the parameter R3 in the formula is dimensionless. To account for the influence of R3, a dimensionless chamfer coefficient \\u0026rho; is defined as:\\u003c/p\\u003e\\n \\u003cdiv id=\\\"Equ5\\\"\\u003e\\n \\u003cdiv id=\\\"FileID_Equ5\\\" name=\\\"EquationSource\\\"\\u003e$$\\\\rho =\\\\frac{{{R_3}}}{{\\\\sqrt {dt} }}$$\\u003c/div\\u003e\\n \\u003cdiv\\u003e5\\u003c/div\\u003e\\n \\u003c/div\\u003e\\n \\u003cp\\u003ewhere \\u003cem\\u003ed\\u003c/em\\u003e is the outer diameter of the branch pipe; t is the wall thickness of the branch pipe. The finite element model shows that the joint ultimate load-carrying ca-pacity is very small when \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/sub\\u003e is greater than or equal to 100mm. So \\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/sub\\u003e is limited less than or equal to 100mm on the ultimate load-carrying capacity calculation formula for the cast-steel joint with three branches. Through regression analysis, the relationship between the chamfer coefficient \\u0026rho; and \\u003cem\\u003eK\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003eR3\\u003c/em\\u003e\\u003c/sub\\u003e follows a linear relation, which is expressed:\\u003c/p\\u003e\\n \\u003cdiv id=\\\"Equ6\\\"\\u003e\\n \\u003cdiv id=\\\"FileID_Equ6\\\" name=\\\"EquationSource\\\"\\u003e$${K_{{R_3}}}=1+0.33738\\\\frac{{{R_3}}}{{\\\\sqrt {dt} }}$$\\u003c/div\\u003e\\n \\u003cdiv\\u003e6\\u003c/div\\u003e\\n \\u003c/div\\u003e\\n \\u003cp\\u003eBecause these four parameters are independent of each other, the overall formula for the joint load-carrying capacity can be obtained by multiplying them, following the method of establishing the load-carrying capacity of joints in the existing standards, which is expressed as:\\u003c/p\\u003e\\n \\u003cdiv id=\\\"Equ7\\\"\\u003e\\n \\u003cdiv id=\\\"FileID_Equ7\\\" name=\\\"EquationSource\\\"\\u003e$$\\\\begin{gathered} F=4.37251{\\\\gamma ^{0.66242}}(1+0.58856\\\\beta )(1+0.33738\\\\frac{{{R_3}}}{{dt}})(0.60022 - 2.5311\\\\sin \\\\theta +6.59681{\\\\sin ^2}\\\\theta - \\\\hfill \\\\\\\\ 5.07388{\\\\sin ^3}\\\\theta ){f_y}{T^2} \\\\hfill \\\\\\\\ \\\\end{gathered}$$\\u003c/div\\u003e\\n \\u003cdiv\\u003e7\\u003c/div\\u003e\\n \\u003c/div\\u003e\\n \\u003cp\\u003eTo validate the accuracy of Eq.\\u0026nbsp;(\\u003cspan\\u003e7\\u003c/span\\u003e), a comparison between the results obtained from finite element modeling and those derived from the regression formula is conducted. The comparative results are presented in Table\\u0026nbsp;\\u003cspan\\u003e5\\u003c/span\\u003e. Notably, the disparity between the calculated values obtained from the formula and those from finite element analysis is minimal, with the maximum error amounting to only 1.9%. Consequently, it is deduced that the proposed formula effectively predicts the ultimate load-carrying capacity of the cast-steel joint with three branches with a high level of accuracy.\\u003c/p\\u003e\\n \\u003cdiv\\u003e\\n \\u003ctable id=\\\"Tab5\\\" border=\\\"1\\\"\\u003e\\n \\u003ccaption language=\\\"En\\\"\\u003e\\n \\u003cdiv\\u003eTable 5\\u003c/div\\u003e\\n \\u003cdiv\\u003e\\n \\u003cp\\u003eCalculation formula error table\\u003c/p\\u003e\\n \\u003c/div\\u003e\\n \\u003c/caption\\u003e\\n \\u003cthead\\u003e\\n \\u003ctr\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJoint number\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003e\\u0026theta;\\u003c/em\\u003e\\u003c/p\\u003e\\n \\u003cp\\u003e(\\u003csup\\u003eo\\u003c/sup\\u003e)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003e\\u0026gamma;\\u003c/em\\u003e\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003e\\u0026beta;\\u003c/em\\u003e\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003eR\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003e3\\u003c/em\\u003e\\u003c/sub\\u003e\\u003c/p\\u003e\\n \\u003cp\\u003e(mm)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e\\u003cem\\u003eF/\\u003c/em\\u003e\\u003c/p\\u003e\\n \\u003cp\\u003e(\\u003cem\\u003efyo*T\\u0026sup2;\\u003c/em\\u003e)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eRegression formula results\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003cth align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eDifference percentage(%)\\u003c/p\\u003e\\n \\u003c/th\\u003e\\n \\u003c/tr\\u003e\\n \\u003c/thead\\u003e\\n \\u003ctbody\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ5\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e10\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e9.74\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e9.92\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e1.90\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ6\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e15.2\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e13.07\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e13.09\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.19\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ2\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e15.67\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e15.70\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.21\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e25\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e18.14\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e18.21\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.37\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ8\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e29.9\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e20.37\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e20.50\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.65\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ1\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e13.62\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e13.62\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.00\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ3\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e40\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e15.78\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e15.78\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.00\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ4\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e50\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e11.30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e11.30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.00\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ18\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.6\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e15.14\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e15.05\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e-0.61\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ19\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.66\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e15.54\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e15.44\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e-0.61\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.74\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e16.02\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e15.97\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e-0.32\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ21\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.8\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e16.47\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e16.36\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e-0.66\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ15\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e50\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e19.23\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e19.09\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e-0.75\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003ctr\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003eJ16\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e30\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"left\\\"\\u003e\\n \\u003cp\\u003e20\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.7\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e100\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e22.40\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e22.47\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003ctd align=\\\"char\\\"\\u003e\\n \\u003cp\\u003e0.32\\u003c/p\\u003e\\n \\u003c/td\\u003e\\n \\u003c/tr\\u003e\\n \\u003c/tbody\\u003e\\n \\u003c/table\\u003e\\n \\u003c/div\\u003e\\n\\u003c/div\\u003e\"},{\"header\":\"Conclusion\",\"content\":\"\\u003cp\\u003eThe study presents a comprehensive numerical simulation and experimental investigation of a full-scale cast-steel joint with three branches. Initially, a representative full-scale model of the cast-steel joint was constructed using SolidWorks, followed by rigorous analysis using ANSYS, which accounted for geometric and material nonlinearity. To validate the accuracy of the numerical simulation, a corresponding verification experiment was conducted. Furthermore, a formula for determining the load-carrying capacity of the cast-steel joint with three branches was proposed, fulfilling the requirements of engineering design. Through the extensive research conducted in this paper, the following conclusions were drawn:\\u003c/p\\u003e \\u003cp\\u003e1) Analysis of the full-scale joint experimental results revealed that stress distribution under compression primarily concentrates on the core area of the joint, with minimal stress observed in the main pipe and branch pipes. This insight serves as a basis for evaluating the joint's strength and stiffness to meet design requirements.\\u003c/p\\u003e \\u003cp\\u003e2) SolidWorks proved to be effective in modeling cast-steel joints with branches, successfully addressing the challenge of modeling tube-to-tube intersections with smooth transitions to match actual joint configurations.\\u003c/p\\u003e \\u003cp\\u003e3) The finite element model of the test joint was imported into ANSYS for analysis, and the results were compared with experimental findings, demonstrating consistency. The verified finite element model is deemed reliable for evaluating the impact of joint geometry parameters on the behavior of three-branch cast-steel joints.\\u003c/p\\u003e \\u003cp\\u003e4) Finite element analysis was conducted on joints with various geometric parameters to determine their ultimate load-carrying capacities. The error analysis revealed a maximum error of 1.9% when comparing prediction results with finite element results, indicating that the proposed formula accurately predicts ultimate load-carrying capacity for engineering design requirements. This formula serves as a valuable tool for selecting geometry parameters in joint structural design.\\u003c/p\\u003e \"},{\"header\":\"Declarations\",\"content\":\"\\u003cp\\u003e\\u003cstrong\\u003eData Availability Statement\\u003c/strong\\u003e\\u003c/p\\u003e\\n\\u003cp\\u003eThe datasets used and analyzed during the current study are available from the corresponding author on reasonable request.\\u003c/p\\u003e\\n\\u003cp\\u003e\\u003cstrong\\u003eAcknowledgments\\u003c/strong\\u003e\\u003c/p\\u003e\\n\\u003cp\\u003eThe authors would like to express sincere thanks to Qi Liu, Fan Zhang, and Hao Zhang for their help during the article modification process.\\u003c/p\\u003e\"},{\"header\":\"References\",\"content\":\"\\u003col\\u003e\\n\\u003cli\\u003eXue G, Bao W, Jiang J, et al. Hysteretic Behavior of Beam-to-Column Joints with Cast Steel Connectors[J]. Shock and Vibration, 2019, 2019: 1-20.\\u003c/li\\u003e\\n\\u003cli\\u003eSong Y Y, Ying L U. Decision tree methods: applications for classification and prediction[J]. Shanghai archives of psychiatry, 2015, 27(2): 130.\\u003c/li\\u003e\\n\\u003cli\\u003eArslan Sel\\u0026ccedil;uk S, G\\u0026uuml;lle N B, Mutlu Avin\\u0026ccedil; G. Tree-Like Structures in Architecture: Revisiting Frei Otto\\u0026rsquo;s Branching Columns Through Parametric Tools[J]. SAGE Open, 2022, 12(3): 21582440221119479.\\u003c/li\\u003e\\n\\u003cli\\u003eDu W, Xia Z, Han L, et al. 3D solid model generation method based on a generative adversarial network[J]. Applied Intelligence, 2023, 53(13): 17035-17060.\\u003c/li\\u003e\\n\\u003cli\\u003eWang H, Du W, Zhao Y, et al. Optimization and experimental research on treelike joints based on generative design and powder bed fusion[J]. Engineering Structures, 2023, 278: 115564.\\u003c/li\\u003e\\n\\u003cli\\u003eWang H, Du W, Zhao Y, et al. Joints for treelike column structures based on generative design and additive manufacturing[J]. Journal of Constructional Steel Research, 2021, 184: 106794.\\u003c/li\\u003e\\n\\u003cli\\u003eC. Fang, B.A. Izzuddin, A.Y. Elghazouli. Modeling of semi-rigid beam-to-column steel joints under extreme loading, J. Frontiers of Structural and Civil Engineering. 7(3) (2013) 245\\u0026ndash;263.\\u003c/li\\u003e\\n\\u003cli\\u003eC. Brett, Y. Lu. Assessment of robustness of structures: current state of research, J. Frontiers of Structural and Civil Engineering. 7(4) (2013) 356\\u0026ndash;368.\\u003c/li\\u003e\\n\\u003cli\\u003eM.S. Aziz, Y.A.E. Sheriff. Biomimicry as an approach for bio-inspired structure with the aid of computation, J. Alexandria Engineering Journal. 55(1) (2016) 707-714.\\u003c/li\\u003e\\n\\u003cli\\u003eZhang B, Yang B, Wu T, et al. Experimental and numerical study on the capability behavior of a thick-walled spatial cast-steel joint under complex load conditions[J]. Advances in Civil Engineering, 2019, 2019: 1-19.\\u003c/li\\u003e\\n\\u003cli\\u003eBokhari I. Experimental Investigation of Structural Performance of Welded Interfaces Between Steel Castings and Steel Hollow Structural Sections[J]. 2022.\\u003c/li\\u003e\\n\\u003cli\\u003eWang L, Jin H, Dong H, et al. Balance fatigue design of cast steel nodes in tubular steel structures[J]. The Scientific World Journal, 2013, 2013.\\u003c/li\\u003e\\n\\u003cli\\u003ePapatheocharis T, Sarvanis G C, Perdikaris P C, et al. Fatigue resistance of welded steel tubular X-joints[J]. Marine Structures, 2020, 74: 102809.\\u003c/li\\u003e\\n\\u003cli\\u003eXiong Y, Lin K, Wu D, et al. The role of a novel coating of SFRCR-ECC in enhancing the fire performance of CFST columns: Development, characteristic and ISO-834 standard fire test[J]. Engineering Structures, 2023, 294: 116629.\\u003c/li\\u003e\\n\\u003cli\\u003eF. Bouafia, S.Boualem, MME. Amin, B. Benali. 3-D finite element analysis of stress concentration factor in spot-welded joints of steel: The effect of process-induced porosity, J. Computational Materials Science, 50(4) (2011) 1450-1459.\\u003c/li\\u003e\\n\\u003cli\\u003eEurocode 3: Design of steel structures. Part 1.8 (design of joints) [S]. 2005.\\u003c/li\\u003e\\n\\u003cli\\u003eA. Loureiro, R Gutierrez, JM Reinosa, A Moreno. Axial stiffness prediction of non-preloaded T-stubs: an analytical frame approach, J. Journal of Construction Steel Research, 66(12) (2011) 613-622.\\u003c/li\\u003e\\n\\u003cli\\u003eInternational Standards Organization: ISO 6892-1:2009. Metallic materials tensile testing-part 1: Method of test at room temperature. Brussels, Belgium,2009.\\u003c/li\\u003e\\n\\u003cli\\u003eCECS235: Technical specification for application of connections of structural steel casting [S]. 2008.\\u003c/li\\u003e\\n\\u003cli\\u003eANSI/AISC 360-05, Specification for Structural Steel Buildings [S].2005.\\u003c/li\\u003e\\n\\u003c/ol\\u003e\"}],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":true,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":false,\"hideJournal\":false,\"highlight\":\"\",\"institution\":\"\",\"isAcceptedByJournal\":true,\"isAuthorSuppliedPdf\":false,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":false,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"scientific-reports\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":false,\"externalIdentity\":\"scirep\",\"sideBox\":\"Learn more about [Scientific Reports](http://www.nature.com/srep/)\",\"snPcode\":\"\",\"submissionUrl\":\"\",\"title\":\"Scientific Reports\",\"twitterHandle\":\"\",\"acdcEnabled\":true,\"dfaEnabled\":true,\"editorialSystem\":\"stoa\",\"reportingPortfolio\":\"Scientific Reports\",\"inReviewEnabled\":true,\"inReviewRevisionsEnabled\":true},\"keywords\":\"Cast-steel joint, Tree-like column structures, Full-scale model test\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-4061078/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-4061078/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"\\u003cp\\u003eThe joint made of cast steel is frequently utilized within a treelike column structure to ensure a smooth transition. It is of great significance in ensuring the overall structural safety, but currently, the mechanical property and bearing capacity of this type of joint cannot be fully understood. This study delves into the load-bearing characteristics of such a cast-steel joint featuring three branches. Initially, a comprehensive model of the cast-steel joint, sourced from a practical engineering, underwent vertical load testing. Detailed scrutiny of stress distribution and vertical displacement of the tested joint was conducted based on the experimental outcomes. Subsequently, a finite element model of the tested joint was constructed using SolidWorks and subjected to analysis via ANSYS. The numerical findings were juxtaposed with experimental data and extrapolated to encompass other parametric scenarios. Ultimately, a regression analysis method was employed to derive a calculation formula for the load-carrying capacity of branch-bearing cast-steel joints. This formula aids in estimating geometric parameters and load-bearing capacity during the preliminary design phase. Comparative analysis reveals a substantial concurrence among experimental, finite element analysis, and formula-based predictive outcomes.\\u003c/p\\u003e\",\"manuscriptTitle\":\"Experimental investigation and simulation analysis of cast-steel joints under vertical pressure\",\"msid\":\"\",\"msnumber\":\"\",\"nonDraftVersions\":[{\"code\":1,\"date\":\"2024-03-25 07:05:04\",\"doi\":\"10.21203/rs.3.rs-4061078/v1\",\"editorialEvents\":[{\"type\":\"communityComments\",\"content\":0},{\"type\":\"decision\",\"content\":\"Revision requested\",\"date\":\"2024-04-11T04:12:24+00:00\",\"index\":\"\",\"fulltext\":\"\"},{\"type\":\"editorInvitedReview\",\"content\":\"\",\"date\":\"2024-04-03T21:24:53+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"reviewerAgreed\",\"content\":\"e15d2433-0902-4499-ba9d-c293822c5288\",\"date\":\"2024-03-26T15:32:12+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"reviewersInvited\",\"content\":\"\",\"date\":\"2024-03-26T14:58:57+00:00\",\"index\":\"\",\"fulltext\":\"\"},{\"type\":\"editorAssigned\",\"content\":\"\",\"date\":\"2024-03-26T14:57:44+00:00\",\"index\":\"\",\"fulltext\":\"\"},{\"type\":\"editorInvited\",\"content\":\"\",\"date\":\"2024-03-21T05:26:44+00:00\",\"index\":\"\",\"fulltext\":\"\"},{\"type\":\"checksComplete\",\"content\":\"\",\"date\":\"2024-03-21T05:21:09+00:00\",\"index\":\"\",\"fulltext\":\"\"},{\"type\":\"submitted\",\"content\":\"Scientific Reports\",\"date\":\"2024-03-10T01:36:32+00:00\",\"index\":\"\",\"fulltext\":\"\"}],\"status\":\"published\",\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"scientific-reports\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":false,\"externalIdentity\":\"scirep\",\"sideBox\":\"Learn more about [Scientific Reports](http://www.nature.com/srep/)\",\"snPcode\":\"\",\"submissionUrl\":\"\",\"title\":\"Scientific Reports\",\"twitterHandle\":\"\",\"acdcEnabled\":true,\"dfaEnabled\":true,\"editorialSystem\":\"stoa\",\"reportingPortfolio\":\"Scientific Reports\",\"inReviewEnabled\":true,\"inReviewRevisionsEnabled\":true}}],\"origin\":\"\",\"ownerIdentity\":\"d444d76e-0998-4eed-8f17-79310102fa81\",\"owner\":[],\"postedDate\":\"March 25th, 2024\",\"published\":true,\"recentEditorialEvents\":[],\"rejectedJournal\":[],\"revision\":\"\",\"amendment\":\"\",\"status\":\"under-review\",\"subjectAreas\":[],\"tags\":[],\"updatedAt\":\"2024-05-14T06:33:28+00:00\",\"versionOfRecord\":[],\"versionCreatedAt\":\"2024-03-25 07:05:04\",\"video\":\"\",\"vorDoi\":\"\",\"vorDoiUrl\":\"\",\"workflowStages\":[]},\"version\":\"v1\",\"identity\":\"rs-4061078\",\"journalConfig\":\"researchsquare\"},\"__N_SSP\":true},\"page\":\"/article/[identity]/[[...version]]\",\"query\":{\"redirect\":\"/article/rs-4061078\",\"identity\":\"rs-4061078\",\"version\":[\"v1\"]},\"buildId\":\"qtupq5eGEP_6zYnWcrvyt\",\"isFallback\":false,\"isExperimentalCompile\":false,\"dynamicIds\":[84888],\"gssp\":true,\"scriptLoader\":[]}","source_license":"CC-BY-4.0","license_restricted":false}