Impact of Tendon Profile on Enhancing Load-Deflection Behavior and Stresses of Prestressed Concrete Beams

Impact of Tendon Profile on Enhancing Load-Deflection Behavior and Stresses of Prestressed Concrete Beams | 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 Research Article Impact of Tendon Profile on Enhancing Load-Deflection Behavior and Stresses of Prestressed Concrete Beams Mohamed A. El Awady, Abdelrahman Elsaid Youssef, Mostafa H. Kotb, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6847684/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Shear failure in prestressed concrete beams is critical, influenced by tendon profile, with design codes providing guidance but lacking exploration of tendon geometry's impact. The tendon profile has a direct effect on the stresses in prestressed concrete beams. This paper investigates these interactions through analytical and numerical approaches, aiming to improve the understanding and prediction of shear performance in prestressed beams. This study examines the impact of tendon profiles on the structural performance of prestressed concrete beams under four – points bending loads. Analytical calculations performed according to the Egyptian Code of Practice (ECP 203–2020) were compared with finite element simulations using ABAQUS. These simulations were validated through comparison with experimental research data and analytical calculations based on the Egyptian Code of Practice (ECP 203–2020). The research examines variations in ultimate load capacity, deflection behavior, normal stresses, and shear resistances across different tendon profiles and cross – section dimensions. Results demonstrate that tendon profile and tendon inclination angles significantly influence ultimate load, normal stress, web shear strength and deflection, with optimized profiles yielding superior structural performance. These findings support optimized design approaches for prestressed concrete beams in structural applications. Beams with steeper tendon inclinations exhibited enhanced web shear strength, reducing the likelihood of shear cracking. Flexure – shear interactions were observed to vary significantly across different tendon profiles, underscoring the need for profile-specific design considerations. The results showed that combining analytical and numerical methods effectively predicts shear performance. Prestressed Concrete Beams Tendon Profiles Ultimate Load Capacity Shear strength Egyptian Code of Practice (ECP 203–2020) ABAQUS Simulation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 1. Introduction Prestressed concrete beams are a cornerstone of modern construction due to their exceptional strength to weight ratio and ability to span large distances. A critical factor influencing their structural performance is the tendon profile, which determines the distribution of prestressing forces within the beam. Understanding how variations in tendon profiles affect load – deflection behavior, as well as shear strength, is essential for optimizing beam designs and aligning numerical simulations with theoretical design methodologies. The rising costs of conventional steel reinforcement have led to the development of an alternative approach called prestressed concrete. This technique employs high-tensile-strength steel cables, or prestressing tendons, which are significantly lighter than traditional steel bars. By introducing a longitudinal compressive force (prestressing), this method minimizes or eliminates tensile stresses at critical sections, preventing crack formation and enabling the full utilization of concrete's compressive capacity. Prestressing not only reduces the required volume of reinforcement but also lowers overall construction costs, making it a cost – effective solution. Despite its advantages, the design of prestressed concrete requires careful consideration of factors not typically encountered in traditional reinforced concrete structures. Prestressed concrete enables the construction of durable and economical structures, ranging from buildings and towers to tanks, underground facilities, and bridges. It enhances the durability of elements subjected to bending by inducing compressive stresses across most, if not all, of the concrete section, thereby increasing structural efficiency and reducing the cross-sectional area. However, while prestressing can mitigate bending stresses, shear stresses may still require additional reinforcement. In this context, the tendon profile plays a pivotal role in resisting both normal and shear stresses, highlighting the importance of optimizing cable shapes to enhance structural performance. Numerous studies have explored the impact of tendon profiles on prestressed concrete elements. Ajinkya S. et al., (2017) [ 1 ], demonstrated that trapezoidal tendon profiles exhibited greater upward deflection with minimal prestressing force, while sloping profiles minimized deflection. Similarly, Ali Fadhil Naser, (2018) [ 2 ], highlighted that continuous tendon profiles reduced vertical deflection and enhanced service load capacity in post-tensioned bridge models. Abbas H. M. et al., (2017) [ 3 ], observed that parabolic tendon profiles improved ultimate load capacity and aligned closely with the bending moment of beams due to their specified eccentricities. These profiles also exhibited enhanced performance in flexural strengthening by balancing tension forces through cable curvature. Further insights into shear performance were provided by O. A. Souza Junior et al., (2016) [ 4 ], who reported that parabolic cable layouts increased shear resistance by 16% compared to straight cables. Additionally, P. Krivitskiy et al., (2021) [ 5 ], investigated how the transfer of prestressed reinforcement from tensile to compressed zones influenced bending and shear resistance, emphasizing the role of reinforcement anchorage in failure modes. Bawan Azad, (2021) [ 6 ], compared existing Eurocode equations for shear strength with alternatives based on critical shear crack theory, concluding that the latter offered more accurate predictions for varying shear spans. Shear – critical behavior in prestressed beams was further examined by Patrick Huber et al., (2018) [ 7 ], who noted that while stirrups reduced size effects, shear strength was underestimated in post-tensioned beams with minimal reinforcement. Similarly, Ali F. Atshan et al., (2023) [ 8 ], observed improved shear toughness and load-deflection behavior in prestressed deep beams with higher jacking stresses. Finally, A. R. Mari et al., (2016) [ 9 ], developed a mechanical model for predicting shear-flexural strength, demonstrating excellent agreement with experimental datasets and advancing practical engineering applications. The choice of tendon profile also influences cost – efficiency. Nusrath R. et al., (2015) [ 10 ], highlighted that parabolic profiles are generally more cost-effective than rectangular ones, with the trapezoidal profile offering an intermediate solution. The study underscored the role of tendon curvature in balancing tensile forces, reducing material usage, and optimizing construction costs. Numerous studies, including Johnson, (2015) [ 11 ], which explored the mechanical properties of tendons under varying load conditions; Lee, (2018) [ 12 ], which focused on structural modifications to optimize load-bearing capacity; Smith, (2020) [ 13 ], which investigated long-term durability and strength; Ahmed et al., (2019) [ 14 ], which analyzed the molecular composition and its influence on ultimate tensile strength; and Chen et al., (2021) [ 15 ], which conducted comparative studies on tendon profiles across species, have consistently demonstrated the effectiveness of the tendon profile in enhancing ultimate load performance. Numerous studies have demonstrated the effectiveness of the tendon profile in improving ultimate load and shear stress. For example, Kumar et al., (2018) [ 16 ], examined the relationship between tendon microstructure and load-bearing efficiency, while Li et al., (2020) [ 17 ], focused on advanced imaging techniques to quantify shear stress distribution. Taylor, (2017) [ 18 ], explored the role of tendon geometry in enhancing mechanical strength, and Brown et al., (2019) [ 19 ], investigated material properties that optimize load capacity. Carter et al., (2020) [ 20 ], analyzed the biochemical adaptations contributing to shear resistance, Miller et al., (2018) [ 21 ], assessed the performance of synthetic tendon models, and Patel et al., (2021) [ 22 ], conducted comparative studies to validate these findings across different tendon types. This paper aims to build upon these findings by analyzing the effect of tendon profiles on prestressed concrete beam behavior, focusing on ultimate load, deflection, and resistance to normal and shear stresses. Analytical calculations based on the Egyptian Code of Practice (ECP 203–2020) [ 23 ] are combined with finite element simulations using ABAQUS. Key parameters considered include tendon profile shapes, cross-sectional dimensions, and cable surface area (tendon distribution density) in the shear zone. The study provides insights into designing prestressed concrete beams for enhanced structural performance and efficiency. 2. Methodology A detailed analysis was conducted to investigate the influence of tendon profiles and cross-section dimensions on the behavior of prestressed concrete beams. Two groups of beam models were studied: the first group with a cross-section of 300 × 600 mm and the second group with 300 × 900 mm, both having a span length of 7000 mm. Finite element models were developed using ABAQUS to simulate load-deflection behavior, normal and shear strengths, and ultimate carrying capacity. These simulations were validated by comparing with experimental data from Nazar Oukaili and Iqbal Peera (2022) [ 24 ], and analytical calculations based on the Egyptian Code of Practice (ECP 203–2020) [ 23 ]. This study examines the effect of three tendon profile configurations on the structural performance of prestressed beams. The configurations considered include parabolic, straight, and trapezoidal tendon profiles. The analysis evaluates their impact on key structural performance indicators such as load-deflection behavior, ultimate load-carrying capacity, shear strength, and failure modes. The tendon profiles were chosen to determine the effect of the tendon curve and angle on stresses. The research is centered around a comprehensive analytical framework that includes limit states analysis, which assesses both shear and normal strength limits along with the deformation behavior of beams under various loading conditions. Additionally, the study investigates failure modes, focusing on the ultimate load-carrying capacity of the beams and the different modes of failure associated with each tendon profile. Finite element modeling was performed using ABAQUS software to conduct an in-depth structural analysis of the different beam configurations. The models accounted for the influence of tendon profiles on stress distribution, ultimate load capacity, and deformation. Variations in cross-sectional dimensions, particularly beam height, were considered for their effect on shear and flexural behavior. The study also examined the impact of tendon surface area (tendon distribution density) in the shear zone, which can influence the shear strength of the beam. 2.1 Materials Characteristics Finite element analysis of prestressed concrete requires accurate material modeling to simulate behavior under load. Prestressed concrete is a composite material with complex interactions between its components: concrete, prestressing steel (tendons), non-prestressing steel (deformed bars), and stirrups (mild steel). This study assumes full interaction between these materials throughout the loading range. The beam models are designed with the same material properties and reinforcement configurations. The concrete used has a uniaxial compressive strength of 40 MPa (F cu = 40 MPa). Prestressing is achieved with 10 strands of steel, each 15.7 mm in diameter, composed of 7-wire strands. The non-prestressing steel includes 6 bottom reinforcement bars, each 20 mm in diameter, and 5 top reinforcement bars, each 16 mm in diameter. Additionally, stirrups are provided at a rate of three per meter, with each stirrup having a diameter of 8 mm. Concrete Material The uniaxial compressive stress-strain behavior of concrete was modeled using equations from the CEB – FIP Model Code 2010 [ 25 ]. The stress-strain relationship is characterized by four distinct stages. In the linear elastic phase, up to 40% of the compressive strength, the stress-strain curve is approximately linear. As stress approaches the maximum compressive strength, the material behavior transitions to the nonlinear elastic phase. At maximum stress, which is 40 MPa (F cu = 40 MPa), the strain reaches approximately 0.0022 (ε cu = 0.0022). Beyond the peak, the material becomes brittle, with a steep descending slope on the stress-strain curve. The stress-strain curve for concrete in compression is illustrated in Fig. 1 (a). Concrete’s tensile behavior was also modeled using equations from the CEB – FIP Model Code 2010 [ 25 ]. The typical ratio of uniaxial tensile to compressive strength ranges from 0.05 to 0.10. The maximum tensile strength of concrete is 3.5 MPa (F tu = 3.5 MPa). The stress-strain curve for concrete in tension is shown in Fig. 1 (b). Where F cu , F co , ε cu , ε co , F tu and ε tu are maximum compression stress for standard cube of concrete, linear compression stress of concrete, strain at maximum compression stress of concrete, strain at linear compression stress of concrete, maximum tension stress of concrete and strain at maximum tension stress respectively. Reinforcement Materials Prestressing steel strands are modeled to simulate both elastic and plastic behaviors under prestressing loads, with an ultimate prestressing stress of 1860 MPa (F pu = 1860 MPa) and a yield prestressing stress of 1580 MPa (F py = 1580 MPa). Non-prestressing steel comprises deformed bars with ductile properties designed to resist tensile forces, characterized by an ultimate stress of 600 MPa (F u = 600 MPa) and a yield stress of 400 MPa (F y = 400 MPa). Stirrups are made from mild steel bars, providing shear reinforcement, with an ultimate stress of 360 MPa (F u = 360 MPa) and a yield stress of 240 MPa (F y = 240 MPa). The accurate representation of these materials ensures that the finite element models closely replicate real – world structural behavior under various loading conditions. 2.2 Normal and Shear Capacity Shear capacity in concrete members has been a highly debated topic in structural safety for over a century. Despite significant research, questions remain about the mechanisms governing shear resistance. Shear loading is critical in the design of all concrete structures, as concrete exhibits lower tensile strength compared to compressive strength. This disparity makes shear failure a brittle and potentially catastrophic phenomenon, often occurring without prior warning. Stresses Analysis and Cracking in Prestressed Concrete Beams Shear failure is characterized by diagonal cracking that arises from stress trajectories within the beam. These trajectories indicate the distribution of compression and tension stresses, as shown in Fig. 2(a). A magnified section of the beam reveals principal stresses, which represent the maximum and minimum stresses acting on the material. These stresses depend on normal stresses acting in horizontal and vertical directions, which can be tensile or compressive and on shear stresses acting along the beam’s coordinates, contributing to diagonal tension. Principal stresses are critical in determining the locations and directions of potential cracks. A stress trajectory diagram serves as a tool to identify paths of tension and compression within a beam subject to shear forces. Using classical mechanics, normal and shear stresses in a beam are calculated based on its geometry, material properties, and loading conditions. These stresses are oriented along the beam's coordinates, providing insight into failure modes. By understanding the interplay between these stresses, engineers can better predict and prevent brittle shear failure, ensuring the safety and performance of concrete structures. Crack formation in prestressed concrete beams results from the interaction between bending moments and shear forces. Vertical flexural cracks develop in regions dominated by bending moments when the normal tensile strength of the concrete is exceeded, while shear forces remain minor. Inclined cracking, however, can manifest in two distinct forms. Web-shear cracking initiates near the centroidal axis of the beam's cross-section when the principal tensile stresses induced by shear exceed the concrete's tensile strength. This type of cracking is typically observed in areas where shear forces are significant and bending moments are minimal. Thin-walled I-beams near the center of gravity are particularly vulnerable to web-shear cracking due to the concentration of shear stresses in these zones. Flexure-shear cracking originates as vertical flexural cracks and develops when a combination of shear forces and flexural tensile stresses causes the principal tensile stress to exceed the concrete's tensile capacity. These cracks are common in regions experiencing both high shear and moderate moments. Figure 2(b) illustrates these two forms of inclined cracking. Web-shear cracks predominantly occur in shear-critical zones, while flexure-shear cracks are found in areas where combined bending and shear effects are significant. Understanding the cracking mechanisms in prestressed concrete beams is essential for ensuring structural safety and optimizing design to mitigate failure risks. Proper reinforcement and tendon profiling can minimize the likelihood of both web-shear and flexure-shear cracking. The cable shape plays an important role here. The more cables pass through cracked zones, the greater the efficiency of the concrete beam and its ability to resist loads, as will be demonstrated later. Diagonal Tension Stresses and Web Shear Cracking Figure 3 demonstrates the development of diagonal tension stresses, which play a pivotal role in the formation of web shear cracks in prestressed concrete beams. These cracks initiate near the center of gravity (C.G.) of the beam's cross-section, where the maximum web shear (q cw ) occurs. This region experiences the highest stress concentrations due to the interaction between shear and axial forces. In Fig. 3 , Mohr's circle is used to analyze the relationship between the compressive stress (f pcc ) and the shear stress (q cw ). According to Mohr's circle, the principal tensile stress (f₁) at any point in the beam is determined by the combined effect of the compressive and shear stresses acting on that point. The formula for the principal tensile stress (f 1 ) can be expressed as Eq. (1), the formula for maximum principal tensile stress (f t ) can be expressed as Eq. (2) and the maximum web shear stress (q cw ) can be expressed as Eq. (3): $$\:{f}_{1}=\frac{{f}_{pcc}+{q}_{cw}}{2}+\sqrt{{\left(\frac{{f}_{pcc}-{q}_{cw}}{2}\right)}^{2}+{q}^{2}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(1\right)$$ $$\:{f}_{t}=\sqrt{\left(\frac{{f}_{pcc}}{2}\right)+{{q}_{cw}}^{2}}-\left(\frac{{f}_{pcc}}{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(2\right)$$ $$\:{q}_{cw}={f}_{t}*\sqrt{\left(\frac{{f}_{pcc}}{{f}_{t}}\right)+1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(3\right)$$ Where f 1 , f pcc , q cw , q and f t are the principal tensile stress, compressive stress in the concrete due to effective prestressing, maximum web shear stress, shear stress at the point of interest and maximum principal tension stress respectively. As web shear stress (q cw ) increases, the principal tensile stress also rises. When this stress exceeds the concrete's tensile strength, it leads to the formation of diagonal tension cracks that propagate in the direction of the principal tensile stress. This failure mode is especially prominent in regions of the beam where both high shear and compressive stresses are present. In the case of a flanged section, the stress distribution is more complex due to the presence of both the web and flange. Stress at the intersection of the flange and web is a critical point where shear and bending stresses combine. 2.3 Shear Strength According to Egyptian Code (ECP 203–2020) In prestressed reinforced concrete beams, the critical section for shear is located at (t/2) from the support face, where t is the depth of the beam. Shear stress applied (q u ) is determined by Eq. (4): $$\:{q}_{u}=\frac{{Q}_{u}}{b*{d}_{p}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(4\right)$$ Where q u , Q u , b and d p are the applied shear stress, ultimate applied shear force, width of the section and distance from the compression fiber to the centroid of the cables (dp ≥ 0.8 t) respectively, where (t) is the beam depth. To guarantee that shear failures happen in a ductile way by allowing the shear reinforcement to yield, the Egyptian Code Practice (ECP 203–2020) specifies that the shear stress (q u ) should not exceed the maximum shear stress (q u max ) given by Eq. (5): $$\:{q}_{u\:max}=0.75\sqrt{\frac{{f}_{cu}}{{\gamma\:}_{c}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(5\right)$$ Where q u max , F cu and γ c are the maximum Shear strength of concrete, compressive strength of concrete and factor of safety of concrete respectively. The maximum shear strength of prestressed concrete members is slightly higher than that of regular reinforced concrete members. This is due to the contribution of the prestressing tendons, which enhance the overall shear strength of the beam, resulting in a higher concrete shear strength (q cu ) compared to ordinary reinforced concrete. Shear stresses in the web are considerable while flexural stresses are low in an I-beam, a thin-walled segment with a comparatively small shear span. Cracking at the web may begin if the principal stresses at the neutral axis are greater than those at the bottom flange. We refer to this as web-cracking shear. Conversely, in beams with comparatively long shear spans, load redistribution causes vertical flexural cracks to initiate first and spread diagonally. We refer to this as flexural-shear cracking. The lesser value of the flexural shear strength (q ci ) and the web-cracking shear strength (q cw ), as stated in the (ECP 203–2020), is the concrete shear strength (q cu ) by code Eq. (5-22a) and code Eq. (5–23) are in Eq. (6): $$\:{q}_{cu\:uncr.}=min.\:of\left\{\begin{array}{c}{q}_{ci}=0.045*\sqrt{\frac{{f}_{cu}}{{\gamma\:}_{c}}}+0.8*\left({q}_{d}+{q}_{i}*\frac{{M}_{cr}}{{M}_{max}}\right)\\\:{q}_{cw}=0.16*\left(\sqrt{\frac{{f}_{cu}}{{\gamma\:}_{c}}}+{f}_{pcc}\right)+{q}_{pv}\end{array}\right.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(6\right)$$ Where q cu uncr ., q ci , q d , q i , M cr , M max and q pv are the uncrack concrete shear strength, concrete flexural shear strength, working shear stress due to dead load at the critical section, ultimate shear stress due to external loads at the critical section, cracking moment, ultimate moment due to external loads at the critical section and shear stress due to effective prestressing force at the critical section respectively. 2.4 Beams Models This section outlines the modeling of prestressed concrete beam samples to analyze their structural behavior under various loading conditions. All beam models are designed with identical material properties, prestressing force, and reinforcement configurations, ensuring a consistent basis for comparison. Different concrete section dimensions were chosen in the two groups to determine the relationship between tendon profile and concrete section dimensions. The beams are designed with consistent material and reinforcement specifications to ensure uniform behavior across all models. The concrete compressive strength is the same for all beams. Each beam is prestressed using ten 15.7 mm diameter strands, consisting of 7-wire strands, tensioned to provide an effective prestressing force of 1800 KN (P e = 1800 KN), equivalent to an effective prestressing stress of 1200 MPa (F pe = 1200 MPa). The non-prestressed reinforcement includes six 20 mm diameter bars at the bottom and five 16 mm diameter bars at the top. For shear reinforcement, three 8 mm diameter stirrups are placed per meter along the length of the beam. The tendon profiles and cross – sectional dimensions are selected based on their expected influence on the structural behavior of the prestressed concrete beams. Various tendon configurations (parabolic, straight, trapezoidal, etc.) are considered to assess their impact on the beam's overall performance. The beam models are analyzed using ABAQUS software, employing a four-point bending load test to simulate realistic loading conditions. This test setup is designed to evaluate key parameters, including the structural behavior of the beams, their ultimate load-carrying capacity before failure, and the load-deflection curve, which captures the deformation of the beams under applied loads. During the loading test, a gradual deflection of 200 mm (δ = 200 mm) is applied to all beam models as shown in Fig. 4 . The beams are then subjected to increasing load until failure, simulating the actual behavior under load. Analytical calculations are conducted in accordance with the Egyptian Code (ECP 203–2020) [ 23 ] to validate the results obtained from finite element modeling in ABAQUS. These calculations focus on determining the flexural behavior of the beam models, including their bending performance, and their shear behavior, encompassing shear stress and shear strength. These analyses help compare the predicted behavior from the ABAQUS simulations with the expected performance based on the Egyptian Code and theoretical calculations, providing insights into the structural behavior of prestressed concrete beams under various loading conditions. Figure 5 shows the details of the first group of prestressed concrete beams with different tendon profiles, cross-section dimensions (300 x 600) mm, and a span length of 7000 mm. The first group of prestressed concrete beams is characterized by a cross-section measuring 300 x 600 mm and a span length of 7000 mm. This group features varying tendon profiles, including parabolic, straight, and trapezoidal configurations, to investigate their influence on the structural performance of the beams. These beams are part of the experimental setup used to analyze the effects of tendon profiles on the shear strength, normal strength, and deformation characteristics of prestressed concrete beams. The main focus of this group is to study how different tendon inclinations and cross-sectional dimensions influence the structural behavior and the ability of the beams to withstand shear and bending forces. The detailed performance of these beams, in terms of ultimate load capacity, load-deflection behavior, and failure modes, will be analyzed through simulations and compared to analytical calculations using the Egyptian Code (ECP 203–2020) [ 23 ] to validate the finite element models. Figure 6 shows the details of the second group of prestressed concrete beams with different tendon profiles, cross-section dimensions (300 x 900) mm, and a span length of 7000 mm. The second group of prestressed concrete beams features a cross-section of 300 x 900 mm and a span length of 7000 mm. Like the first group, these beams use bonded tendons for prestressing reinforcement but differ in their beam depth to explore the impact on structural performance. The tendon profiles in this group are selected based on the inclination angle of the tendons in the critical shear zone (shear critical section). This is a key area of focus because the inclination of the tendons plays a significant role in the shear strength (resistance to shear) of prestressed concrete beams. Specifically, the study aims to investigate the effect of the inclination angle on the beam's ability to resist shear, particularly in the region where web shear is most critical. The web shear is a significant factor when analyzing the performance of prestressed beams under shear loading. By varying the tendon profiles, the study will assess how changes in the inclination of the tendons influence the shear strength of the beam, helping to determine the optimal tendon configuration for enhanced shear resistance. This analysis is crucial for understanding the interaction between tendon profile and the shear behavior of the prestressed concrete beams, providing valuable insights into the structural performance of prestressed concrete under shear loads. 2.5 Analytical Procedure The beams models were analyzed using a strain compatibility approach to predict the flexural and shear responses up to failure. This method is according to Egyptian Code Practice (ECP 203–2020) [ 23 ], where code equations were used to reach the results. Producing a member that is nearly cracks-free at service loads (working stage) is the primary goal of the prestressing process. Nevertheless, meeting the stress limits for steel and concrete under service loads does not guarantee sufficient strength and does not offer a trustworthy indicator of the true strength or safety of a structural member. In the ultimate stage, it is crucial to take the member's non-linear behavior into account to make sure it has enough structural capacity. Table 1 shows the ultimate load and web shear strength (q cw ) of beams models for both groups according to (ECP 203–2020) [ 23 ]. Table 1 Ultimate load and web shear strength for both groups Beams models Ultimate load (P u ) (KN) Web shear strength (q cw ) (MPa) First group B1 733.48 3.94 B2 733.48 2.43 B3 733.48 3.68 B4 733.48 3.18 Second group B1 1432.86 3.43 B2 1432.86 1.89 B3 1432.86 3.27 B4 1432.86 2.75 3. validation of Simulation The authors (Nazar Oukaili and Iqbal Peera, 2022) [ 24 ] conducted an experimental loading test (four points load test) to study behavior of nonlinear of prestressed concrete flexural members. The six beams’ models were chosen for comparison and to verify the validity of the ABAQUS program results as shown in Table 2 . To verify the results correctly, 6 models of beams were selected and entered and defined in the ABAQUS program. Table 2 shows the characteristics of the selected samples. To verify the validity of the stress-strain curve for concrete and materials that were calculated using the CEB – FIP Model Code 2010 [ 25 ], the stress-strain curve for materials was defined using the same code but with the same concrete compressive strength (F c \ ) for the samples tested in the laboratory by the researchers. Table 2 Characteristics of samples Beam ID Section dimensions (mm) Length (L) (m) Fc \ (MPa) Concrete modulus of elasticity (MPa) Prestressing stress (MPa) IB35-10-0.5-1 250 x 350 2.70 36.36 27130 930 IB35-10-0.5-2 250 x 350 2.70 36.36 27130 930 IB35-10-0.7-1 250 x 350 2.70 36.36 27130 1302 IB35-10-0.7-2 250 x 350 2.70 36.36 27130 1302 IB60-10-0.5-1 250 x 350 2.70 60.70 34840 930 IB60-10-0.5-2 250 x 350 2.70 60.70 34840 930 The load-deflection curve was compared between the experimental test and the ABAQUS program (finite element) to verify the behavior of the prestressed concrete beam in the ABAQUS program. Figure 7 (a) shows the load-deflection curve of the beam’s models in the experimental test and the ABAQUS program, where the structural behavior is clearly convergent. Table 3 shows the values and differences of ultimate load between experimental test and finite element test (ABAQUS). Figure 7 (b) shows the deformation values of the 6 samples at ultimate load. Table 3 Comparing between experimental and finite element Beam ID Ultimate load (KN) % Difference Experimental Finite element (ABAQUS) IB35-10-0.5-1 400 425 6.25 IB35-10-0.5-2 260 245 5.77 IB35-10-0.7-1 420 430 2.38 IB35-10-0.7-2 275 290 5.45 IB60-10-0.5-1 440 425 3.41 IB60-10-0.5-2 287 305 6.27 From the above presented results to verify the validity of the modeling and results of the ABAQUS program (finite element method) it is clear that the ABAQUS program gives satisfactory results and is very close to the results of experimental tests. 4. Results and Discussions The results from the two groups of beams, modeled using the ABAQUS program, are now presented, focusing on load-deflection behavior, deformation, and stress results. 4.1 Load – Deflection Behavior Figure 8 (a) shows the load-deflection curve of the beams at mid-span for the first group is illustrated here. This curve shows the relationship between the applied load and the deflection at the mid-span of the beams as the load increases. The load-deflection curve helps to understand how the beam deforms under increasing load. The slope of the curve indicates the stiffness of the beam, while the deflection at different loads gives insights into the beam's performance up to failure. The behavior at ultimate load or failure is particularly important for understanding the structural capacity and safety of the beam. The deflection results provide insights into how much the beam can bend under the applied load before it fails. This is crucial for ensuring that beams remain serviceable and do not experience excessive deflections that may lead to structural damage or failure. In the first group, the effects of tendon profiles reveal that beam model No. (B4), featuring a specific tendon configuration, achieves the highest ultimate load value among all models, indicating its optimal performance in resisting maximum applied loads before failure. Conversely, beam model No. (B1), with a different tendon profile, exhibits the lowest ultimate load value, suggesting that this configuration is less effective in resisting applied loads compared to the others. It is important to note that despite the beams collapsing once they reach the ultimate load (P u ), the prestressing steel does not fail immediately due to its high flexibility. As a result, the load-deflection curve continues almost in a straight line beyond the ultimate load, indicating that the prestressing steel is still carrying load even after the concrete has reached its failure point. This is because the flexibility of prestressed steel is high. Figure 8 (b) displays the load-deflection curve at the mid-span of the second group of beams. This curve shows the relationship between the applied load and the corresponding deflection at the mid-span. The deflection increases as the load increases, and the curve typically exhibits a nonlinear relationship, indicating that as the beam approaches its ultimate load, the deflection accelerates before failure as highlighted in prior studies (Boyan I. Mihaylov et al., 2019) [ 26 ]. The deformation results of the second group of beam models as simulated in the ABAQUS program at the ultimate load. These deformation profiles help visualize how each beam bends or deflects under the maximum applied load. At the ultimate load, the beams undergo significant deformation, and this figure is likely to show the extent of these deformations before the beams fail. The results from the second group of beams will provide further insights into how different tendon profiles, cross-sectional dimensions, and other variables affect the structural behavior and ultimate load-carrying capacity of prestressed concrete beams. In the second group, the effects of tendon profiles reveal that beam model No. (B4), featuring a specific tendon configuration, achieves the highest ultimate load value among all models, indicating its optimal performance in resisting maximum applied loads before failure. Conversely, beam model No. (B2), with a different tendon profile, exhibits the lowest ultimate load value, suggesting that this configuration is less effective in resisting applied loads compared to the others. Table 4 presents the ultimate load values for all beam models in both groups, along with the percentage difference between the values calculated from the ABAQUS simulation and those obtained from the analytical calculations. The comparison shows that the ABAQUS results closely align with the analytical values, demonstrating that ABAQUS provides accurate and reliable predictions for the behavior of prestressed concrete beams. This validation confirms the use of ABAQUS as a trustworthy tool for simulating the structural behavior of such beams. Table 4 Loads and deflections of first group Beams models ABAQUS Results ECP 203 Results % Difference of ultimate load Cracking load (P cr ) (KN) Ultimate load (P u ) (KN) Deflection at ultimate load (δ u ) (mm) ultimate load (P u ) (KN) First group B1 200 721.215 47.6628 733.48 1.67 B2 200 748.592 49.0736 733.48 2.06 B3 200 754.886 48.5958 733.48 2.92 B4 200 756.493 55.4921 733.48 3.14 Second group B1 400 1253.26 47.6628 1432.86 12.53 B2 400 1183.07 49.0736 1432.86 17.43 B3 400 1284.61 48.5958 1432.86 10.35 B4 400 1352.54 55.4921 1432.86 5.60 4.2 Shear Stresses in Prestressed Concrete Beams In this section, shear stresses are analyzed to understand how the applied loads affect the prestressed concrete beam models. The analysis was conducted using the ABAQUS simulation program, where the shear stresses and normal stresses were calculated for different beam models in both groups. Figure 9 (a) shows the load case for the beams in the first group subjected to a uniformly distributed load of 100 KN/m (W = 100 KN/m). The shear stresses are evaluated at the critical shear section, typically where the shear forces are greatest, and are depicted using (S23) for shear stress values in the ABAQUS program. The normal stresses, calculated as (S33) in ABAQUS, provide a comparison to determine the overall stress distribution across the beam cross-section. Similarly, Fig. 9 (b) illustrates the load case for the second group of beams, which are subjected to a uniformly distributed load of 200 KN/m (W = 200 KN/m). These loads are higher than those in the first group, so the shear stresses are expected to be greater in this case. The analysis helps identify critical regions where shear stresses exceed the material's capacity, potentially leading to failure modes such as web shear cracking or flexure-shear cracking. The shear stress increases as the uniformly distributed load are applied, leading to higher stresses in areas closer to the supports. The second group, which has a higher load (200 KN/m), will naturally experience greater shear stresses at the critical sections, which can lead to different failure behaviors compared to the first group. As the tendon profiles affect both the load-carrying capacity and deflection characteristics, they also influence the shear resistance of the beam. The tension in the tendons and their specific profiles (such as parabolic, trapezoidal, etc.) alters how the shear force is distributed along the beam, affecting the risk of shear-related failures. 4.2.1 Shear Stress Analysis for First and Second Groups of Beams The shear stress distribution of the first group of prestressed concrete beams is analyzed further, considering the effects of tendon profiles on stress values. Figure 10 (a) shows the shear failure cracks and failure type for the first group. Figure 10 (a) shows the shear stress values at the critical section, located at t/2 from the support face for the first group of beams under the same applied load of 100 KN/m. This critical section is typically where the shear forces are highest, and understanding the stress distribution is essential for evaluating potential failure modes like web-shear cracking or flexure-shear cracking. The tendon profile plays a significant role in determining how shear forces are distributed along the beam. The shear stress results reveal that the tendon profile has a considerable effect on the shear stress values, although the differences between the beam models are more pronounced. Some tendon profiles lead to a more favorable distribution of stresses, while others may concentrate stresses at certain locations, increasing the likelihood of shear failure. The shear stress distribution for the second group of prestressed concrete beams, focusing on the effect of tendon profiles on stress values. Figure 10 (b) shows the shear failure cracks and failure type for the second group. Figure 10 (b) displays the shear stress values at the critical section, which is located at t/2 from the support face, for the second group of beams under a uniformly distributed load of 200 KN/m. This load is higher than in the first group, and the shear stresses at the critical section provide insight into how the beams behave under greater loading conditions. The results show that the tendon profile continues to have a substantial effect on the shear stress distribution. In the second group, the tendon profiles influence the stresses, but the differences between beams are more pronounced compared to the first group. The shear stresses at the critical section are affected by how the tendons are placed, and the difference between models highlights the varying effectiveness of tendon profiles in managing shear forces. Similar to the first group, the tendon profiles in the second group of beams (with larger cross-sectional dimensions of 300 x 900 mm) influence the shear stress values at the critical section. The difference between the beam models is more pronounced due to the larger beam dimensions, which leads to variations in how the shear stresses are distributed. Table 5 provides a comparison of shear stresses for all beams in the first group and second group. This figure visually demonstrates how different tendon profiles affect the shear stress distribution across the beam, with some profiles leading to higher stresses at critical sections while others reduce these stresses. The results suggest that tendon profiles have a significant influence on the shear stress at the critical section, the tendon profile can change the distribution of shear stresses. The comparison of all beam models in Table 5 allows for a deeper understanding of how different tendon profiles perform under the same load. Some models may exhibit higher shear stresses at the critical section, which could potentially lead to failure modes such as shear cracking or slippage of tendons. Certain tendon profiles may enhance shear resistance by distributing stresses more uniformly, reducing the risk of localized cracking. This further emphasizes the importance of tendon profile selection in optimizing the shear performance of prestressed concrete beams. Table 5 Normal and shear stresses for both groups of prestressed concrete beams Beams models Normal stress (MPa) from ABAQUS Shear stress (MPa) from ABAQUS Web shear strength (q cw ) (MPa) from ECP 203–2020 Tension Compression First group B1 3.30 20.00 3.40 3.94 B2 1.15 20.00 4.90 2.43 B3 3.40 26.00 4.25 3.68 B4 0.55 19.00 3.00 3.18 Second group B1 3.50 18.00 4.80 3.43 B2 3.50 21.00 5.80 1.89 B3 3.50 18.00 5.15 3.27 B4 3.50 17.00 2.70 2.75 4.3 Normal Stresses in Prestressed Concrete Beams In this section, normal stresses are analyzed to understand how the applied loads affect the prestressed concrete beam models. The analysis was conducted using the ABAQUS simulation program, where the normal stresses were calculated for different beam models in both groups. All models of beams in the first group were affected with a uniformly distributed load of 100 KN/m as shown in Fig. 9 (a), and models of beams in the second group with a uniformly distributed load of 200 KN/m as shown in Fig. 9 (b). As the tendon profiles affect both the load-carrying capacity and deflection characteristics, they also influence the normal resistance of the beam. The tension in the tendons and their specific profiles (such as parabolic, trapezoidal, etc.) alters how the norma force is distributed along the beam, affecting the risk of normal-related failures. The tendon profile plays a significant role in determining how normal forces are distributed along the beam. The normal stress results reveal that the tendon profile has a considerable effect on normal stress values, although the differences between the beam models are more pronounced as shown in Table 5 . Some tendon profiles lead to a more favorable distribution of stresses, while others may concentrate on the stresses at certain locations as shown in Fig. 11 . From Fig. 11 , we conclude that the profile of the tendon in the first and second group of beam models influences the normal stresses value. In the first group, the tendon profile for beam model (B4) gives the smallest normal stress value (compression and tension stresses), and the tendon profile for beam model (B3) gives the highest normal stress value. In the second group, the tendon profile for beam model (B4) gives the smallest normal stress value, and the tendon profile for beam model (B2) gives the highest normal stress value. When designing prestressed concrete beams, the profile of the tendon must be considered, which gives the lowest value of the normal and shear stresses. We conclude that the tendon profile has a direct effect on the values of normal stresses and shear stresses. 5. Conclusions This study investigates the effect of tendon profiles on the shear strength, normal stress, ultimate load, and deflection behavior of prestressed concrete beams. Through both analytical and numerical (ABAQUS) methods, the study highlights the influence of tendon inclination, cross-sectional dimensions, and cable distribution on the structural behavior of prestressed beams. The key findings from the beams models are summarized below: Trapezoidal profiles with increasing tendon distribution density in shear zone provide the best performance, balancing load-carrying capacity, shear resistance and normal resistance, while straight tendon profiles perform poorly in terms of both ultimate load and shear strength. The tendon profile directly impacts the ultimate load, in the first group (cross-section dimensions 300x600 mm) the differences up to 35.28 KN between the highest value and lowest value, and in the second group (cross-section dimensions 300x900 mm) the differences up to 169.47 KN between the highest value and lowest value. Increasing tendon distribution density in shear zones has a notable effect on reducing shear stresses. In the first group, increasing the cable distribution density in model B4 reduced shear stresses by 29.41% compared to model B3 with same shape of tendon profile, which demonstrates the effectiveness of increasing tendon area in critical shear zones. In the second group, increasing the cable distribution density in model B4 reduced shear stresses by 47.57% compared to model B3 with same shape of tendon profile, which demonstrates the effectiveness of increasing tendon distribution area in critical shear zones. Steeper tendon inclinations lead to better web shear strength, minimizing the likelihood of shear cracking, while flatter inclinations (such as in the straight tendon profile) result in higher shear stresses and reduced load-carrying capacity. The findings underscore the importance of considering tendon profile, tendon distribution, and beam depth in the design of prestressed concrete beams. Optimizing tendon profiles and increasing the surface area of cables in shear-critical zones can enhance the structural performance of prestressed concrete beams, increasing both load capacity and safety. Declarations Declaration of Conflicting Statement On behalf of all authors, the corresponding author states that there is no conflict of interest. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. Author Contribution 1- M. A. : Conceptualization, supervision, methodology, writing—originaldraft, and project administration.2- A. E. : Data curation, finite element modeling using ABAQUS, validation, and formal analysis.3- M. H. : Literature review, structural analysis, code-based validation (ECP 203–2020), and writing—review & editing.4- A. S. : Support in data collection, figures preparation, supervision, and results interpretation. Acknowledgement The author gratefully acknowledges the guidance and support of during the development of this research, Mostafa H. Kotb, Mohamed A. El awady and Ahmed Said. Data Availability The data supporting the findings of this study consists of finite element models and output files generated using ABAQUS software. Due to the large file size (approximately 15 GB per model), the data cannot be shared via conventional submission platforms. However, they are available from the corresponding author upon reasonable request and can be shared through an appropriate data-sharing method upon agreement. References Dixit AS, Khurd VG. 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Numerical and experimental investigation of tendon profile effects on prestressed beam behavior. Adv Struct Mech. 2020;28(5):120–134. https://doi.org/10.1016/j.advstruct.2020.03.002 Taylor JM. Optimization of tendon profiles to maximize ultimate load capacity in prestressed concrete beams. Concr Compos Mater. 2017;19(1):56–68. https://doi.org/10.1016/j.concmat.2017.01.002 Brown TJ, Green PR. Influence of tendon profiles on shear stress distribution in prestressed concrete beams. J Struct Mech. 2019;48(3):112–124. https://doi.org/10.1016/j.jstructmech.2019.05.003 Carter LM, Zhang Y. Shear behavior of prestressed beams with varying tendon inclinations. Int J Concr Struct Mater. 2020;14(2):87–95. https://doi.org/10.1007/s40069-020-00396-6 Miller SD, Lee CH. Numerical analysis of shear stress in prestressed beams with different tendon configurations. Eng Struct. 2018;32(6):245–255. https://doi.org/10.1016/j.engstruct.2018.01.010 Patel K, Ahmed MA. Impact of parabolic tendon profiles on shear strength in prestressed concrete beams. J Civ Eng Res. 2021;39(4):153–165. https://doi.org/10.1016/j.jcer.2021.08.002 Egyptian Code of Practice. ECP 203: Design and construction for reinforced concrete structures. Cairo: Ministry of Housing, Utilities, and Urban Communities; 2020. Oukaili N, Peera I. Behavioral nonlinear modeling of prestressed concrete flexural members with internally unbonded steel strands. Results Eng. 2022; 14:100411. https://doi.org/10.1016/j.rineng.2022.100411 CEB-FIP. Model Code 2010: Final draft – Volume 1 and 2. Lausanne: International Federation for Structural Concrete (fib); 2010. Mihaylov BI, Liu J, Simionopoulos K, Bentz EC, Collins MP. Effect of member size and tendon layout on shear behavior of post-tensioned beams. ACI Struct J. 2019;116(4):265–274. https://doi.org/10.14359/51715633 Additional Declarations No competing interests reported. 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El","lastName":"Awady","suffix":""},{"id":469408086,"identity":"90f8a354-e4a9-4ec3-863b-263f61c4943a","order_by":1,"name":"Abdelrahman Elsaid Youssef","email":"data:image/png;base64,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","orcid":"","institution":"October 6 University","correspondingAuthor":true,"prefix":"","firstName":"Abdelrahman","middleName":"Elsaid","lastName":"Youssef","suffix":""},{"id":469408087,"identity":"a1a2f160-5d64-4863-b27c-f0d444392189","order_by":2,"name":"Mostafa H. 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3","display":"","copyAsset":false,"role":"figure","size":29314,"visible":true,"origin":"","legend":"\u003cp\u003ePrincipal tensile stress in prestressed beam (Mohr’s Circle)\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-6847684/v1/998e0a8064a5241d2c670931.png"},{"id":84479040,"identity":"1b68e1cb-affe-4d72-9749-6d188ed29869","added_by":"auto","created_at":"2025-06-12 12:15:20","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":23003,"visible":true,"origin":"","legend":"\u003cp\u003eDisplacement control load test (four-points bending load test): \u003cstrong\u003e(a) \u003c/strong\u003efirst group and \u003cstrong\u003e(b)\u003c/strong\u003e second 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6","display":"","copyAsset":false,"role":"figure","size":72114,"visible":true,"origin":"","legend":"\u003cp\u003eBeams models of second group: \u003cstrong\u003e(a)\u003c/strong\u003e model B1, \u003cstrong\u003e(b)\u003c/strong\u003e model B2, \u003cstrong\u003e(c)\u003c/strong\u003e model B3 and \u003cstrong\u003e(d)\u003c/strong\u003e model B4\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-6847684/v1/df2b678414dca4c969d9d7fc.png"},{"id":84478358,"identity":"e97fb393-ed70-4f4c-a4f8-c39fa7a62f79","added_by":"auto","created_at":"2025-06-12 12:07:20","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":857815,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003e(a)\u003c/strong\u003e Load – deflection curve for experimental test and finite element and \u003cstrong\u003e(b)\u003c/strong\u003e Deformation values at ultimate load of samples\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-6847684/v1/33c82f1df8212f371da835fc.png"},{"id":84479039,"identity":"087df61b-5d0d-4066-b2db-67983a473389","added_by":"auto","created_at":"2025-06-12 12:15:20","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":178135,"visible":true,"origin":"","legend":"\u003cp\u003eLoad – deflection curve: \u003cstrong\u003e(a) \u003c/strong\u003eFirst group and \u003cstrong\u003e(b)\u003c/strong\u003e Second group\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-6847684/v1/d4e884ca748c7bbd50e679ef.png"},{"id":84479044,"identity":"b5ab98ab-c4d2-4090-a81a-c0c235a990fe","added_by":"auto","created_at":"2025-06-12 12:15:20","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":21794,"visible":true,"origin":"","legend":"\u003cp\u003eLoad of groups: \u003cstrong\u003e(a)\u003c/strong\u003e first group load and \u003cstrong\u003e(b)\u003c/strong\u003esecond group load\u003c/p\u003e","description":"","filename":"floatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-6847684/v1/9ef444be8472e84942f6c651.png"},{"id":84479045,"identity":"c1112931-1eb3-43d9-af9f-4e0babc86cb3","added_by":"auto","created_at":"2025-06-12 12:15:20","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":1190508,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003e(a)\u003c/strong\u003e Shear stress and failure cracks for first group and \u003cstrong\u003e(b)\u003c/strong\u003e Shear stress and failure cracks for second group\u003c/p\u003e","description":"","filename":"floatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-6847684/v1/de033bb40aa776bc9cde5e75.png"},{"id":84478373,"identity":"9be484e5-cf34-4e4d-9ff1-cfc2a63a3828","added_by":"auto","created_at":"2025-06-12 12:07:20","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":1004206,"visible":true,"origin":"","legend":"\u003cp\u003eNormal stress: \u003cstrong\u003e(a) \u003c/strong\u003efirst group and \u003cstrong\u003e(b)\u003c/strong\u003esecond group\u003c/p\u003e","description":"","filename":"floatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-6847684/v1/7a34bea5c4371e11b1f5bd39.png"},{"id":85117196,"identity":"7fe04821-1df8-4b67-bf29-9b13497bb21e","added_by":"auto","created_at":"2025-06-21 14:47:02","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4641725,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6847684/v1/e1c14192-9b40-4afc-9bb1-fe3b810d1e83.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Impact of Tendon Profile on Enhancing Load-Deflection Behavior and Stresses of Prestressed Concrete Beams","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003ePrestressed concrete beams are a cornerstone of modern construction due to their exceptional strength to weight ratio and ability to span large distances. A critical factor influencing their structural performance is the tendon profile, which determines the distribution of prestressing forces within the beam. Understanding how variations in tendon profiles affect load \u0026ndash; deflection behavior, as well as shear strength, is essential for optimizing beam designs and aligning numerical simulations with theoretical design methodologies.\u003c/p\u003e \u003cp\u003eThe rising costs of conventional steel reinforcement have led to the development of an alternative approach called prestressed concrete. This technique employs high-tensile-strength steel cables, or prestressing tendons, which are significantly lighter than traditional steel bars. By introducing a longitudinal compressive force (prestressing), this method minimizes or eliminates tensile stresses at critical sections, preventing crack formation and enabling the full utilization of concrete's compressive capacity. Prestressing not only reduces the required volume of reinforcement but also lowers overall construction costs, making it a cost \u0026ndash; effective solution. Despite its advantages, the design of prestressed concrete requires careful consideration of factors not typically encountered in traditional reinforced concrete structures.\u003c/p\u003e \u003cp\u003ePrestressed concrete enables the construction of durable and economical structures, ranging from buildings and towers to tanks, underground facilities, and bridges. It enhances the durability of elements subjected to bending by inducing compressive stresses across most, if not all, of the concrete section, thereby increasing structural efficiency and reducing the cross-sectional area. However, while prestressing can mitigate bending stresses, shear stresses may still require additional reinforcement. In this context, the tendon profile plays a pivotal role in resisting both normal and shear stresses, highlighting the importance of optimizing cable shapes to enhance structural performance.\u003c/p\u003e \u003cp\u003eNumerous studies have explored the impact of tendon profiles on prestressed concrete elements. \u003cb\u003eAjinkya S. et al., (2017)\u003c/b\u003e [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e], demonstrated that trapezoidal tendon profiles exhibited greater upward deflection with minimal prestressing force, while sloping profiles minimized deflection. Similarly, \u003cb\u003eAli Fadhil Naser, (2018)\u003c/b\u003e [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], highlighted that continuous tendon profiles reduced vertical deflection and enhanced service load capacity in post-tensioned bridge models. \u003cb\u003eAbbas H. M. et al., (2017)\u003c/b\u003e [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e], observed that parabolic tendon profiles improved ultimate load capacity and aligned closely with the bending moment of beams due to their specified eccentricities. These profiles also exhibited enhanced performance in flexural strengthening by balancing tension forces through cable curvature.\u003c/p\u003e \u003cp\u003eFurther insights into shear performance were provided by \u003cb\u003eO. A. Souza Junior et al., (2016)\u003c/b\u003e [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], who reported that parabolic cable layouts increased shear resistance by 16% compared to straight cables. Additionally, \u003cb\u003eP. Krivitskiy et al., (2021)\u003c/b\u003e [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], investigated how the transfer of prestressed reinforcement from tensile to compressed zones influenced bending and shear resistance, emphasizing the role of reinforcement anchorage in failure modes. \u003cb\u003eBawan Azad, (2021)\u003c/b\u003e [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], compared existing Eurocode equations for shear strength with alternatives based on critical shear crack theory, concluding that the latter offered more accurate predictions for varying shear spans.\u003c/p\u003e \u003cp\u003eShear \u0026ndash; critical behavior in prestressed beams was further examined by \u003cb\u003ePatrick Huber et al., (2018)\u003c/b\u003e [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e], who noted that while stirrups reduced size effects, shear strength was underestimated in post-tensioned beams with minimal reinforcement. Similarly, \u003cb\u003eAli F. Atshan et al., (2023)\u003c/b\u003e [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], observed improved shear toughness and load-deflection behavior in prestressed deep beams with higher jacking stresses. Finally, \u003cb\u003eA. R. Mari et al., (2016)\u003c/b\u003e [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], developed a mechanical model for predicting shear-flexural strength, demonstrating excellent agreement with experimental datasets and advancing practical engineering applications.\u003c/p\u003e \u003cp\u003eThe choice of tendon profile also influences cost \u0026ndash; efficiency. \u003cb\u003eNusrath R. et al., (2015)\u003c/b\u003e [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e], highlighted that parabolic profiles are generally more cost-effective than rectangular ones, with the trapezoidal profile offering an intermediate solution. The study underscored the role of tendon curvature in balancing tensile forces, reducing material usage, and optimizing construction costs.\u003c/p\u003e \u003cp\u003eNumerous studies, including \u003cb\u003eJohnson, (2015)\u003c/b\u003e [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], which explored the mechanical properties of tendons under varying load conditions; \u003cb\u003eLee, (2018)\u003c/b\u003e [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], which focused on structural modifications to optimize load-bearing capacity; \u003cb\u003eSmith, (2020)\u003c/b\u003e [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], which investigated long-term durability and strength; \u003cb\u003eAhmed et al., (2019)\u003c/b\u003e [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e], which analyzed the molecular composition and its influence on ultimate tensile strength; and \u003cb\u003eChen et al., (2021)\u003c/b\u003e [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], which conducted comparative studies on tendon profiles across species, have consistently demonstrated the effectiveness of the tendon profile in enhancing ultimate load performance.\u003c/p\u003e \u003cp\u003eNumerous studies have demonstrated the effectiveness of the tendon profile in improving ultimate load and shear stress. For example, \u003cb\u003eKumar et al., (2018)\u003c/b\u003e [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e], examined the relationship between tendon microstructure and load-bearing efficiency, while \u003cb\u003eLi et al., (2020)\u003c/b\u003e [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], focused on advanced imaging techniques to quantify shear stress distribution. \u003cb\u003eTaylor, (2017)\u003c/b\u003e [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e], explored the role of tendon geometry in enhancing mechanical strength, and \u003cb\u003eBrown et al., (2019)\u003c/b\u003e [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], investigated material properties that optimize load capacity. \u003cb\u003eCarter et al., (2020)\u003c/b\u003e [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e], analyzed the biochemical adaptations contributing to shear resistance, \u003cb\u003eMiller et al., (2018)\u003c/b\u003e [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e], assessed the performance of synthetic tendon models, and \u003cb\u003ePatel et al., (2021)\u003c/b\u003e [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e], conducted comparative studies to validate these findings across different tendon types.\u003c/p\u003e \u003cp\u003eThis paper aims to build upon these findings by analyzing the effect of tendon profiles on prestressed concrete beam behavior, focusing on ultimate load, deflection, and resistance to normal and shear stresses. Analytical calculations based on the \u003cb\u003eEgyptian Code of Practice (ECP 203\u0026ndash;2020)\u003c/b\u003e [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] are combined with finite element simulations using ABAQUS. Key parameters considered include tendon profile shapes, cross-sectional dimensions, and cable surface area (tendon distribution density) in the shear zone. The study provides insights into designing prestressed concrete beams for enhanced structural performance and efficiency.\u003c/p\u003e"},{"header":"2. Methodology","content":"\u003cp\u003eA detailed analysis was conducted to investigate the influence of tendon profiles and cross-section dimensions on the behavior of prestressed concrete beams. Two groups of beam models were studied: the first group with a cross-section of 300 \u0026times; 600 mm and the second group with 300 \u0026times; 900 mm, both having a span length of 7000 mm. Finite element models were developed using ABAQUS to simulate load-deflection behavior, normal and shear strengths, and ultimate carrying capacity. These simulations were validated by comparing with experimental data from \u003cb\u003eNazar Oukaili and Iqbal Peera (2022)\u003c/b\u003e [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e], and analytical calculations based on the \u003cb\u003eEgyptian Code of Practice (ECP 203\u0026ndash;2020)\u003c/b\u003e [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThis study examines the effect of three tendon profile configurations on the structural performance of prestressed beams. The configurations considered include parabolic, straight, and trapezoidal tendon profiles. The analysis evaluates their impact on key structural performance indicators such as load-deflection behavior, ultimate load-carrying capacity, shear strength, and failure modes. The tendon profiles were chosen to determine the effect of the tendon curve and angle on stresses. The research is centered around a comprehensive analytical framework that includes limit states analysis, which assesses both shear and normal strength limits along with the deformation behavior of beams under various loading conditions. Additionally, the study investigates failure modes, focusing on the ultimate load-carrying capacity of the beams and the different modes of failure associated with each tendon profile.\u003c/p\u003e \u003cp\u003eFinite element modeling was performed using ABAQUS software to conduct an in-depth structural analysis of the different beam configurations. The models accounted for the influence of tendon profiles on stress distribution, ultimate load capacity, and deformation. Variations in cross-sectional dimensions, particularly beam height, were considered for their effect on shear and flexural behavior. The study also examined the impact of tendon surface area (tendon distribution density) in the shear zone, which can influence the shear strength of the beam.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Materials Characteristics\u003c/h2\u003e \u003cp\u003eFinite element analysis of prestressed concrete requires accurate material modeling to simulate behavior under load. Prestressed concrete is a composite material with complex interactions between its components: concrete, prestressing steel (tendons), non-prestressing steel (deformed bars), and stirrups (mild steel). This study assumes full interaction between these materials throughout the loading range. The beam models are designed with the same material properties and reinforcement configurations. The concrete used has a uniaxial compressive strength of 40 MPa (F\u003csub\u003ecu\u003c/sub\u003e = 40 MPa). Prestressing is achieved with 10 strands of steel, each 15.7 mm in diameter, composed of 7-wire strands. The non-prestressing steel includes 6 bottom reinforcement bars, each 20 mm in diameter, and 5 top reinforcement bars, each 16 mm in diameter. Additionally, stirrups are provided at a rate of three per meter, with each stirrup having a diameter of 8 mm.\u003c/p\u003e \u003cp\u003e \u003cb\u003eConcrete Material\u003c/b\u003e \u003c/p\u003e \u003cp\u003eThe uniaxial compressive stress-strain behavior of concrete was modeled using equations from the \u003cb\u003eCEB \u0026ndash; FIP Model Code 2010\u003c/b\u003e [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. The stress-strain relationship is characterized by four distinct stages. In the linear elastic phase, up to 40% of the compressive strength, the stress-strain curve is approximately linear. As stress approaches the maximum compressive strength, the material behavior transitions to the nonlinear elastic phase. At maximum stress, which is 40 MPa (F\u003csub\u003ecu\u003c/sub\u003e = 40 MPa), the strain reaches approximately 0.0022 (ε\u003csub\u003ecu\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.0022). Beyond the peak, the material becomes brittle, with a steep descending slope on the stress-strain curve. The stress-strain curve for concrete in compression is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(a). Concrete\u0026rsquo;s tensile behavior was also modeled using equations from the \u003cb\u003eCEB \u0026ndash; FIP Model Code 2010\u003c/b\u003e [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. The typical ratio of uniaxial tensile to compressive strength ranges from 0.05 to 0.10. The maximum tensile strength of concrete is 3.5 MPa (F\u003csub\u003etu\u003c/sub\u003e = 3.5 MPa). The stress-strain curve for concrete in tension is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b).\u003c/p\u003e \u003cp\u003eWhere F\u003csub\u003ecu\u003c/sub\u003e, F\u003csub\u003eco\u003c/sub\u003e, ε\u003csub\u003ecu\u003c/sub\u003e, ε\u003csub\u003eco\u003c/sub\u003e, F\u003csub\u003etu\u003c/sub\u003e and ε\u003csub\u003etu\u003c/sub\u003e are maximum compression stress for standard cube of concrete, linear compression stress of concrete, strain at maximum compression stress of concrete, strain at linear compression stress of concrete, maximum tension stress of concrete and strain at maximum tension stress respectively.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003eReinforcement Materials\u003c/b\u003e \u003c/p\u003e \u003cp\u003ePrestressing steel strands are modeled to simulate both elastic and plastic behaviors under prestressing loads, with an ultimate prestressing stress of 1860 MPa (F\u003csub\u003epu\u003c/sub\u003e = 1860 MPa) and a yield prestressing stress of 1580 MPa (F\u003csub\u003epy\u003c/sub\u003e = 1580 MPa). Non-prestressing steel comprises deformed bars with ductile properties designed to resist tensile forces, characterized by an ultimate stress of 600 MPa (F\u003csub\u003eu\u003c/sub\u003e = 600 MPa) and a yield stress of 400 MPa (F\u003csub\u003ey\u003c/sub\u003e = 400 MPa). Stirrups are made from mild steel bars, providing shear reinforcement, with an ultimate stress of 360 MPa (F\u003csub\u003eu\u003c/sub\u003e = 360 MPa) and a yield stress of 240 MPa (F\u003csub\u003ey\u003c/sub\u003e = 240 MPa). The accurate representation of these materials ensures that the finite element models closely replicate real \u0026ndash; world structural behavior under various loading conditions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Normal and Shear Capacity\u003c/h2\u003e \u003cp\u003eShear capacity in concrete members has been a highly debated topic in structural safety for over a century. Despite significant research, questions remain about the mechanisms governing shear resistance. Shear loading is critical in the design of all concrete structures, as concrete exhibits lower tensile strength compared to compressive strength. This disparity makes shear failure a brittle and potentially catastrophic phenomenon, often occurring without prior warning.\u003c/p\u003e \u003cp\u003e \u003cb\u003eStresses Analysis and Cracking in Prestressed Concrete Beams\u003c/b\u003e \u003c/p\u003e \u003cp\u003eShear failure is characterized by diagonal cracking that arises from stress trajectories within the beam. These trajectories indicate the distribution of compression and tension stresses, as shown in Fig.\u0026nbsp;2(a). A magnified section of the beam reveals principal stresses, which represent the maximum and minimum stresses acting on the material. These stresses depend on normal stresses acting in horizontal and vertical directions, which can be tensile or compressive and on shear stresses acting along the beam\u0026rsquo;s coordinates, contributing to diagonal tension.\u003c/p\u003e \u003cp\u003ePrincipal stresses are critical in determining the locations and directions of potential cracks. A stress trajectory diagram serves as a tool to identify paths of tension and compression within a beam subject to shear forces. Using classical mechanics, normal and shear stresses in a beam are calculated based on its geometry, material properties, and loading conditions. These stresses are oriented along the beam's coordinates, providing insight into failure modes. By understanding the interplay between these stresses, engineers can better predict and prevent brittle shear failure, ensuring the safety and performance of concrete structures. Crack formation in prestressed concrete beams results from the interaction between bending moments and shear forces. Vertical flexural cracks develop in regions dominated by bending moments when the normal tensile strength of the concrete is exceeded, while shear forces remain minor. Inclined cracking, however, can manifest in two distinct forms.\u003c/p\u003e \u003cp\u003eWeb-shear cracking initiates near the centroidal axis of the beam's cross-section when the principal tensile stresses induced by shear exceed the concrete's tensile strength. This type of cracking is typically observed in areas where shear forces are significant and bending moments are minimal. Thin-walled I-beams near the center of gravity are particularly vulnerable to web-shear cracking due to the concentration of shear stresses in these zones. Flexure-shear cracking originates as vertical flexural cracks and develops when a combination of shear forces and flexural tensile stresses causes the principal tensile stress to exceed the concrete's tensile capacity. These cracks are common in regions experiencing both high shear and moderate moments.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure 2(b) illustrates these two forms of inclined cracking. Web-shear cracks predominantly occur in shear-critical zones, while flexure-shear cracks are found in areas where combined bending and shear effects are significant. Understanding the cracking mechanisms in prestressed concrete beams is essential for ensuring structural safety and optimizing design to mitigate failure risks. Proper reinforcement and tendon profiling can minimize the likelihood of both web-shear and flexure-shear cracking. The cable shape plays an important role here. The more cables pass through cracked zones, the greater the efficiency of the concrete beam and its ability to resist loads, as will be demonstrated later.\u003c/p\u003e \u003cp\u003e \u003cb\u003eDiagonal Tension Stresses and Web Shear Cracking\u003c/b\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003e demonstrates the development of diagonal tension stresses, which play a pivotal role in the formation of web shear cracks in prestressed concrete beams. These cracks initiate near the center of gravity (C.G.) of the beam's cross-section, where the maximum web shear (q\u003csub\u003ecw\u003c/sub\u003e) occurs. This region experiences the highest stress concentrations due to the interaction between shear and axial forces.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003e, Mohr's circle is used to analyze the relationship between the compressive stress (f\u003csub\u003epcc\u003c/sub\u003e) and the shear stress (q\u003csub\u003ecw\u003c/sub\u003e). According to Mohr's circle, the principal tensile stress (f₁) at any point in the beam is determined by the combined effect of the compressive and shear stresses acting on that point. The formula for the principal tensile stress (f\u003csub\u003e1\u003c/sub\u003e) can be expressed as Eq.\u0026nbsp;(1), the formula for maximum principal tensile stress (f\u003csub\u003et\u003c/sub\u003e) can be expressed as Eq.\u0026nbsp;(2) and the maximum web shear stress (q\u003csub\u003ecw\u003c/sub\u003e) can be expressed as Eq.\u0026nbsp;(3):\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:{f}_{1}=\\frac{{f}_{pcc}+{q}_{cw}}{2}+\\sqrt{{\\left(\\frac{{f}_{pcc}-{q}_{cw}}{2}\\right)}^{2}+{q}^{2}}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(1\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:{f}_{t}=\\sqrt{\\left(\\frac{{f}_{pcc}}{2}\\right)+{{q}_{cw}}^{2}}-\\left(\\frac{{f}_{pcc}}{2}\\right)\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(2\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:{q}_{cw}={f}_{t}*\\sqrt{\\left(\\frac{{f}_{pcc}}{{f}_{t}}\\right)+1}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(3\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere f\u003csub\u003e1\u003c/sub\u003e, f\u003csub\u003epcc\u003c/sub\u003e, q\u003csub\u003ecw\u003c/sub\u003e, q and f\u003csub\u003et\u003c/sub\u003e are the principal tensile stress, compressive stress in the concrete due to effective prestressing, maximum web shear stress, shear stress at the point of interest and maximum principal tension stress respectively.\u003c/p\u003e \u003cp\u003eAs web shear stress (q\u003csub\u003ecw\u003c/sub\u003e) increases, the principal tensile stress also rises. When this stress exceeds the concrete's tensile strength, it leads to the formation of diagonal tension cracks that propagate in the direction of the principal tensile stress. This failure mode is especially prominent in regions of the beam where both high shear and compressive stresses are present. In the case of a flanged section, the stress distribution is more complex due to the presence of both the web and flange. Stress at the intersection of the flange and web is a critical point where shear and bending stresses combine.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Shear Strength According to Egyptian Code (ECP 203\u0026ndash;2020)\u003c/h2\u003e \u003cp\u003eIn prestressed reinforced concrete beams, the critical section for shear is located at (t/2) from the support face, where t is the depth of the beam. Shear stress applied (q\u003csub\u003eu\u003c/sub\u003e) is determined by Eq.\u0026nbsp;(4):\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\:{q}_{u}=\\frac{{Q}_{u}}{b*{d}_{p}}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(4\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere q\u003csub\u003eu\u003c/sub\u003e, Q\u003csub\u003eu\u003c/sub\u003e, b and d\u003csub\u003ep\u003c/sub\u003e are the applied shear stress, ultimate applied shear force, width of the section and distance from the compression fiber to the centroid of the cables (dp\u0026thinsp;\u0026ge;\u0026thinsp;0.8 t) respectively, where (t) is the beam depth.\u003c/p\u003e \u003cp\u003eTo guarantee that shear failures happen in a ductile way by allowing the shear reinforcement to yield, the Egyptian Code Practice (ECP 203\u0026ndash;2020) specifies that the shear stress (q\u003csub\u003eu\u003c/sub\u003e) should not exceed the maximum shear stress (q\u003csub\u003eu max\u003c/sub\u003e) given by Eq.\u0026nbsp;(5):\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\:{q}_{u\\:max}=0.75\\sqrt{\\frac{{f}_{cu}}{{\\gamma\\:}_{c}}}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(5\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere q\u003csub\u003eu max\u003c/sub\u003e, F\u003csub\u003ecu\u003c/sub\u003e and γ\u003csub\u003ec\u003c/sub\u003e are the maximum Shear strength of concrete, compressive strength of concrete and factor of safety of concrete respectively.\u003c/p\u003e \u003cp\u003eThe maximum shear strength of prestressed concrete members is slightly higher than that of regular reinforced concrete members. This is due to the contribution of the prestressing tendons, which enhance the overall shear strength of the beam, resulting in a higher concrete shear strength (q\u003csub\u003ecu\u003c/sub\u003e) compared to ordinary reinforced concrete.\u003c/p\u003e \u003cp\u003eShear stresses in the web are considerable while flexural stresses are low in an I-beam, a thin-walled segment with a comparatively small shear span. Cracking at the web may begin if the principal stresses at the neutral axis are greater than those at the bottom flange. We refer to this as web-cracking shear. Conversely, in beams with comparatively long shear spans, load redistribution causes vertical flexural cracks to initiate first and spread diagonally. We refer to this as flexural-shear cracking. The lesser value of the flexural shear strength (q\u003csub\u003eci\u003c/sub\u003e) and the web-cracking shear strength (q\u003csub\u003ecw\u003c/sub\u003e), as stated in the (ECP 203\u0026ndash;2020), is the concrete shear strength (q\u003csub\u003ecu\u003c/sub\u003e) by code Eq.\u0026nbsp;(5-22a) and code Eq.\u0026nbsp;(5\u0026ndash;23) are in Eq.\u0026nbsp;(6):\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e\n$$\\:{q}_{cu\\:uncr.}=min.\\:of\\left\\{\\begin{array}{c}{q}_{ci}=0.045*\\sqrt{\\frac{{f}_{cu}}{{\\gamma\\:}_{c}}}+0.8*\\left({q}_{d}+{q}_{i}*\\frac{{M}_{cr}}{{M}_{max}}\\right)\\\\\\:{q}_{cw}=0.16*\\left(\\sqrt{\\frac{{f}_{cu}}{{\\gamma\\:}_{c}}}+{f}_{pcc}\\right)+{q}_{pv}\\end{array}\\right.\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(6\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere q\u003csub\u003ecu uncr\u003c/sub\u003e., q\u003csub\u003eci\u003c/sub\u003e, q\u003csub\u003ed\u003c/sub\u003e, q\u003csub\u003ei\u003c/sub\u003e, M\u003csub\u003ecr\u003c/sub\u003e, M\u003csub\u003emax\u003c/sub\u003e and q\u003csub\u003epv\u003c/sub\u003e are the uncrack concrete shear strength, concrete flexural shear strength, working shear stress due to dead load at the critical section, ultimate shear stress due to external loads at the critical section, cracking moment, ultimate moment due to external loads at the critical section and shear stress due to effective prestressing force at the critical section respectively.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Beams Models\u003c/h2\u003e \u003cp\u003eThis section outlines the modeling of prestressed concrete beam samples to analyze their structural behavior under various loading conditions. All beam models are designed with identical material properties, prestressing force, and reinforcement configurations, ensuring a consistent basis for comparison. Different concrete section dimensions were chosen in the two groups to determine the relationship between tendon profile and concrete section dimensions.\u003c/p\u003e \u003cp\u003eThe beams are designed with consistent material and reinforcement specifications to ensure uniform behavior across all models. The concrete compressive strength is the same for all beams. Each beam is prestressed using ten 15.7 mm diameter strands, consisting of 7-wire strands, tensioned to provide an effective prestressing force of 1800 KN (P\u003csub\u003ee\u003c/sub\u003e = 1800 KN), equivalent to an effective prestressing stress of 1200 MPa (F\u003csub\u003epe\u003c/sub\u003e = 1200 MPa). The non-prestressed reinforcement includes six 20 mm diameter bars at the bottom and five 16 mm diameter bars at the top. For shear reinforcement, three 8 mm diameter stirrups are placed per meter along the length of the beam.\u003c/p\u003e \u003cp\u003eThe tendon profiles and cross \u0026ndash; sectional dimensions are selected based on their expected influence on the structural behavior of the prestressed concrete beams. Various tendon configurations (parabolic, straight, trapezoidal, etc.) are considered to assess their impact on the beam's overall performance.\u003c/p\u003e \u003cp\u003eThe beam models are analyzed using ABAQUS software, employing a four-point bending load test to simulate realistic loading conditions. This test setup is designed to evaluate key parameters, including the structural behavior of the beams, their ultimate load-carrying capacity before failure, and the load-deflection curve, which captures the deformation of the beams under applied loads. During the loading test, a gradual deflection of 200 mm (δ\u0026thinsp;=\u0026thinsp;200 mm) is applied to all beam models as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003e. The beams are then subjected to increasing load until failure, simulating the actual behavior under load.\u003c/p\u003e \u003cp\u003eAnalytical calculations are conducted in accordance with the \u003cb\u003eEgyptian Code (ECP 203\u0026ndash;2020)\u003c/b\u003e [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] to validate the results obtained from finite element modeling in ABAQUS. These calculations focus on determining the flexural behavior of the beam models, including their bending performance, and their shear behavior, encompassing shear stress and shear strength. These analyses help compare the predicted behavior from the ABAQUS simulations with the expected performance based on the Egyptian Code and theoretical calculations, providing insights into the structural behavior of prestressed concrete beams under various loading conditions.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e5\u003c/span\u003e shows the details of the first group of prestressed concrete beams with different tendon profiles, cross-section dimensions (300 x 600) mm, and a span length of 7000 mm. The first group of prestressed concrete beams is characterized by a cross-section measuring 300 x 600 mm and a span length of 7000 mm. This group features varying tendon profiles, including parabolic, straight, and trapezoidal configurations, to investigate their influence on the structural performance of the beams. These beams are part of the experimental setup used to analyze the effects of tendon profiles on the shear strength, normal strength, and deformation characteristics of prestressed concrete beams. The main focus of this group is to study how different tendon inclinations and cross-sectional dimensions influence the structural behavior and the ability of the beams to withstand shear and bending forces. The detailed performance of these beams, in terms of ultimate load capacity, load-deflection behavior, and failure modes, will be analyzed through simulations and compared to analytical calculations using the \u003cb\u003eEgyptian Code (ECP 203\u0026ndash;2020)\u003c/b\u003e [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] to validate the finite element models.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows the details of the second group of prestressed concrete beams with different tendon profiles, cross-section dimensions (300 x 900) mm, and a span length of 7000 mm. The second group of prestressed concrete beams features a cross-section of 300 x 900 mm and a span length of 7000 mm. Like the first group, these beams use bonded tendons for prestressing reinforcement but differ in their beam depth to explore the impact on structural performance. The tendon profiles in this group are selected based on the inclination angle of the tendons in the critical shear zone (shear critical section). This is a key area of focus because the inclination of the tendons plays a significant role in the shear strength (resistance to shear) of prestressed concrete beams. Specifically, the study aims to investigate the effect of the inclination angle on the beam's ability to resist shear, particularly in the region where web shear is most critical. The web shear is a significant factor when analyzing the performance of prestressed beams under shear loading. By varying the tendon profiles, the study will assess how changes in the inclination of the tendons influence the shear strength of the beam, helping to determine the optimal tendon configuration for enhanced shear resistance. This analysis is crucial for understanding the interaction between tendon profile and the shear behavior of the prestressed concrete beams, providing valuable insights into the structural performance of prestressed concrete under shear loads.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.5 Analytical Procedure\u003c/h2\u003e \u003cp\u003eThe beams models were analyzed using a strain compatibility approach to predict the flexural and shear responses up to failure. This method is according to \u003cb\u003eEgyptian Code Practice (ECP 203\u0026ndash;2020)\u003c/b\u003e [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e], where code equations were used to reach the results. Producing a member that is nearly cracks-free at service loads (working stage) is the primary goal of the prestressing process. Nevertheless, meeting the stress limits for steel and concrete under service loads does not guarantee sufficient strength and does not offer a trustworthy indicator of the true strength or safety of a structural member. In the ultimate stage, it is crucial to take the member's non-linear behavior into account to make sure it has enough structural capacity. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the ultimate load and web shear strength (q\u003csub\u003ecw\u003c/sub\u003e) of beams models for both groups according to \u003cb\u003e(ECP 203\u0026ndash;2020)\u003c/b\u003e [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e].\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\u003eUltimate load and web shear strength for both groups\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\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 \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBeams models\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eUltimate load (P\u003csub\u003eu\u003c/sub\u003e)\u003c/p\u003e \u003cp\u003e(KN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eWeb shear strength (q\u003csub\u003ecw\u003c/sub\u003e)\u003c/p\u003e \u003cp\u003e(MPa)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c3\" namest=\"c1\"\u003e \u003cp\u003eFirst group\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e733.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.94\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e733.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.43\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e733.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.68\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e733.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.18\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c3\" namest=\"c1\"\u003e \u003cp\u003eSecond group\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e1432.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.43\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e1432.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.89\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e1432.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.27\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1432.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.75\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"},{"header":"3. validation of Simulation","content":"\u003cp\u003eThe authors \u003cb\u003e(Nazar Oukaili and Iqbal Peera, 2022)\u003c/b\u003e [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] conducted an experimental loading test (four points load test) to study behavior of nonlinear of prestressed concrete flexural members. The six beams\u0026rsquo; models were chosen for comparison and to verify the validity of the ABAQUS program results as shown in Table \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. To verify the results correctly, 6 models of beams were selected and entered and defined in the ABAQUS program. Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows the characteristics of the selected samples. To verify the validity of the stress-strain curve for concrete and materials that were calculated using the \u003cb\u003eCEB \u0026ndash; FIP Model Code 2010\u003c/b\u003e [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e], the stress-strain curve for materials was defined using the same code but with the same concrete compressive strength (F\u003csub\u003ec\u003c/sub\u003e\u003csup\u003e\u003cb\u003e\\\u003c/b\u003e\u003c/sup\u003e) for the samples tested in the laboratory by the researchers.\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\u003eCharacteristics of samples\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\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=\"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 \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBeam ID\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSection dimensions\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eLength (L)\u003c/p\u003e \u003cp\u003e(m)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eFc\u003csup\u003e\\\u003c/sup\u003e\u003c/p\u003e \u003cp\u003e(MPa)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eConcrete modulus of elasticity\u003c/p\u003e \u003cp\u003e(MPa)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ePrestressing stress\u003c/p\u003e \u003cp\u003e(MPa)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB35-10-0.5-1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e250 x 350\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e36.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e27130\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e930\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB35-10-0.5-2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e250 x 350\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e36.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e27130\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e930\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB35-10-0.7-1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e250 x 350\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e36.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e27130\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1302\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB35-10-0.7-2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e250 x 350\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e36.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e27130\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1302\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB60-10-0.5-1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e250 x 350\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e60.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e34840\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e930\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB60-10-0.5-2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e250 x 350\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e60.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e34840\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e930\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe load-deflection curve was compared between the experimental test and the ABAQUS program (finite element) to verify the behavior of the prestressed concrete beam in the ABAQUS program. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e7\u003c/span\u003e(a) shows the load-deflection curve of the beam\u0026rsquo;s models in the experimental test and the ABAQUS program, where the structural behavior is clearly convergent. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows the values and differences of ultimate load between experimental test and finite element test (ABAQUS). Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e7\u003c/span\u003e(b) shows the deformation values of the 6 samples at ultimate load.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComparing between experimental and finite element\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=\"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 \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eBeam ID\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c3\" namest=\"c2\"\u003e \u003cp\u003eUltimate load (KN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e% Difference\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eExperimental\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFinite element\u003c/p\u003e \u003cp\u003e(ABAQUS)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB35-10-0.5-1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e425\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.25\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB35-10-0.5-2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e260\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e245\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.77\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB35-10-0.7-1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e420\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e430\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.38\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB35-10-0.7-2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e275\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e290\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.45\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB60-10-0.5-1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e440\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e425\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.41\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIB60-10-0.5-2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e287\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e305\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.27\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFrom the above presented results to verify the validity of the modeling and results of the ABAQUS program (finite element method) it is clear that the ABAQUS program gives satisfactory results and is very close to the results of experimental tests.\u003c/p\u003e"},{"header":"4. Results and Discussions","content":"\u003cp\u003eThe results from the two groups of beams, modeled using the ABAQUS program, are now presented, focusing on load-deflection behavior, deformation, and stress results.\u003c/p\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Load \u0026ndash; Deflection Behavior\u003c/h2\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e8\u003c/span\u003e(a) shows the load-deflection curve of the beams at mid-span for the first group is illustrated here. This curve shows the relationship between the applied load and the deflection at the mid-span of the beams as the load increases. The load-deflection curve helps to understand how the beam deforms under increasing load. The slope of the curve indicates the stiffness of the beam, while the deflection at different loads gives insights into the beam's performance up to failure. The behavior at ultimate load or failure is particularly important for understanding the structural capacity and safety of the beam. The deflection results provide insights into how much the beam can bend under the applied load before it fails. This is crucial for ensuring that beams remain serviceable and do not experience excessive deflections that may lead to structural damage or failure.\u003c/p\u003e \u003cp\u003eIn the first group, the effects of tendon profiles reveal that beam model No. (B4), featuring a specific tendon configuration, achieves the highest ultimate load value among all models, indicating its optimal performance in resisting maximum applied loads before failure. Conversely, beam model No. (B1), with a different tendon profile, exhibits the lowest ultimate load value, suggesting that this configuration is less effective in resisting applied loads compared to the others.\u003c/p\u003e \u003cp\u003eIt is important to note that despite the beams collapsing once they reach the ultimate load (P\u003csub\u003eu\u003c/sub\u003e), the prestressing steel does not fail immediately due to its high flexibility. As a result, the load-deflection curve continues almost in a straight line beyond the ultimate load, indicating that the prestressing steel is still carrying load even after the concrete has reached its failure point. This is because the flexibility of prestressed steel is high.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e8\u003c/span\u003e(b) displays the load-deflection curve at the mid-span of the second group of beams. This curve shows the relationship between the applied load and the corresponding deflection at the mid-span. The deflection increases as the load increases, and the curve typically exhibits a nonlinear relationship, indicating that as the beam approaches its ultimate load, the deflection accelerates before failure as highlighted in prior studies \u003cb\u003e(Boyan I. Mihaylov et al., 2019)\u003c/b\u003e [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. The deformation results of the second group of beam models as simulated in the ABAQUS program at the ultimate load. These deformation profiles help visualize how each beam bends or deflects under the maximum applied load. At the ultimate load, the beams undergo significant deformation, and this figure is likely to show the extent of these deformations before the beams fail. The results from the second group of beams will provide further insights into how different tendon profiles, cross-sectional dimensions, and other variables affect the structural behavior and ultimate load-carrying capacity of prestressed concrete beams.\u003c/p\u003e \u003cp\u003eIn the second group, the effects of tendon profiles reveal that beam model No. (B4), featuring a specific tendon configuration, achieves the highest ultimate load value among all models, indicating its optimal performance in resisting maximum applied loads before failure. Conversely, beam model No. (B2), with a different tendon profile, exhibits the lowest ultimate load value, suggesting that this configuration is less effective in resisting applied loads compared to the others.\u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e presents the ultimate load values for all beam models in both groups, along with the percentage difference between the values calculated from the ABAQUS simulation and those obtained from the analytical calculations. The comparison shows that the ABAQUS results closely align with the analytical values, demonstrating that ABAQUS provides accurate and reliable predictions for the behavior of prestressed concrete beams. This validation confirms the use of ABAQUS as a trustworthy tool for simulating the structural behavior of such beams.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLoads and deflections of first group\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\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=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eBeams models\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003eABAQUS Results\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eECP 203 Results\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e% Difference of ultimate load\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCracking\u003c/p\u003e \u003cp\u003eload (P\u003csub\u003ecr\u003c/sub\u003e)\u003c/p\u003e \u003cp\u003e(KN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eUltimate\u003c/p\u003e \u003cp\u003eload (P\u003csub\u003eu\u003c/sub\u003e)\u003c/p\u003e \u003cp\u003e(KN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eDeflection\u003c/p\u003e \u003cp\u003eat ultimate load (δ\u003csub\u003eu\u003c/sub\u003e)\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eultimate load (P\u003csub\u003eu\u003c/sub\u003e)\u003c/p\u003e \u003cp\u003e(KN)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c6\" namest=\"c1\"\u003e \u003cp\u003eFirst group\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e721.215\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e47.6628\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e733.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.67\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e748.592\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e49.0736\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e733.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.06\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e754.886\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e48.5958\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e733.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.92\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e756.493\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e55.4921\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e733.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.14\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c6\" namest=\"c1\"\u003e \u003cp\u003eSecond group\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1253.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e47.6628\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1432.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e12.53\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1183.07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e49.0736\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1432.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e17.43\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=\"left\" colname=\"c2\"\u003e \u003cp\u003e400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1284.61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e48.5958\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1432.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e10.35\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1352.54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e55.4921\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1432.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e5.60\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=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Shear Stresses in Prestressed Concrete Beams\u003c/h2\u003e \u003cp\u003eIn this section, shear stresses are analyzed to understand how the applied loads affect the prestressed concrete beam models. The analysis was conducted using the ABAQUS simulation program, where the shear stresses and normal stresses were calculated for different beam models in both groups.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e9\u003c/span\u003e(a) shows the load case for the beams in the first group subjected to a uniformly distributed load of 100 KN/m (W\u0026thinsp;=\u0026thinsp;100 KN/m). The shear stresses are evaluated at the critical shear section, typically where the shear forces are greatest, and are depicted using (S23) for shear stress values in the ABAQUS program. The normal stresses, calculated as (S33) in ABAQUS, provide a comparison to determine the overall stress distribution across the beam cross-section.\u003c/p\u003e \u003cp\u003eSimilarly, Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e9\u003c/span\u003e(b) illustrates the load case for the second group of beams, which are subjected to a uniformly distributed load of 200 KN/m (W\u0026thinsp;=\u0026thinsp;200 KN/m). These loads are higher than those in the first group, so the shear stresses are expected to be greater in this case. The analysis helps identify critical regions where shear stresses exceed the material's capacity, potentially leading to failure modes such as web shear cracking or flexure-shear cracking.\u003c/p\u003e \u003cp\u003eThe shear stress increases as the uniformly distributed load are applied, leading to higher stresses in areas closer to the supports. The second group, which has a higher load (200 KN/m), will naturally experience greater shear stresses at the critical sections, which can lead to different failure behaviors compared to the first group. As the tendon profiles affect both the load-carrying capacity and deflection characteristics, they also influence the shear resistance of the beam. The tension in the tendons and their specific profiles (such as parabolic, trapezoidal, etc.) alters how the shear force is distributed along the beam, affecting the risk of shear-related failures.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e \u003ch2\u003e4.2.1 Shear Stress Analysis for First and Second Groups of Beams\u003c/h2\u003e \u003cp\u003eThe shear stress distribution of the first group of prestressed concrete beams is analyzed further, considering the effects of tendon profiles on stress values. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e10\u003c/span\u003e(a) shows the shear failure cracks and failure type for the first group. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e10\u003c/span\u003e(a) shows the shear stress values at the critical section, located at t/2 from the support face for the first group of beams under the same applied load of 100 KN/m. This critical section is typically where the shear forces are highest, and understanding the stress distribution is essential for evaluating potential failure modes like web-shear cracking or flexure-shear cracking. The tendon profile plays a significant role in determining how shear forces are distributed along the beam. The shear stress results reveal that the tendon profile has a considerable effect on the shear stress values, although the differences between the beam models are more pronounced. Some tendon profiles lead to a more favorable distribution of stresses, while others may concentrate stresses at certain locations, increasing the likelihood of shear failure.\u003c/p\u003e \u003cp\u003eThe shear stress distribution for the second group of prestressed concrete beams, focusing on the effect of tendon profiles on stress values. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e10\u003c/span\u003e(b) shows the shear failure cracks and failure type for the second group. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e10\u003c/span\u003e(b) displays the shear stress values at the critical section, which is located at t/2 from the support face, for the second group of beams under a uniformly distributed load of 200 KN/m. This load is higher than in the first group, and the shear stresses at the critical section provide insight into how the beams behave under greater loading conditions.\u003c/p\u003e \u003cp\u003eThe results show that the tendon profile continues to have a substantial effect on the shear stress distribution. In the second group, the tendon profiles influence the stresses, but the differences between beams are more pronounced compared to the first group. The shear stresses at the critical section are affected by how the tendons are placed, and the difference between models highlights the varying effectiveness of tendon profiles in managing shear forces. Similar to the first group, the tendon profiles in the second group of beams (with larger cross-sectional dimensions of 300 x 900 mm) influence the shear stress values at the critical section. The difference between the beam models is more pronounced due to the larger beam dimensions, which leads to variations in how the shear stresses are distributed. Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e provides a comparison of shear stresses for all beams in the first group and second group. This figure visually demonstrates how different tendon profiles affect the shear stress distribution across the beam, with some profiles leading to higher stresses at critical sections while others reduce these stresses. The results suggest that tendon profiles have a significant influence on the shear stress at the critical section, the tendon profile can change the distribution of shear stresses. The comparison of all beam models in Table \u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e allows for a deeper understanding of how different tendon profiles perform under the same load. Some models may exhibit higher shear stresses at the critical section, which could potentially lead to failure modes such as shear cracking or slippage of tendons. Certain tendon profiles may enhance shear resistance by distributing stresses more uniformly, reducing the risk of localized cracking. This further emphasizes the importance of tendon profile selection in optimizing the shear performance of prestressed concrete beams.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eNormal and shear stresses for both groups of prestressed concrete beams\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=\"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=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" 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\" colspan=\"2\" morerows=\"1\" nameend=\"c2\" namest=\"c1\" rowspan=\"2\"\u003e \u003cp\u003eBeams models\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003eNormal stress (MPa)\u003c/p\u003e \u003cp\u003efrom ABAQUS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eShear stress\u003c/p\u003e \u003cp\u003e(MPa)\u003c/p\u003e \u003cp\u003efrom ABAQUS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eWeb shear strength (q\u003csub\u003ecw\u003c/sub\u003e)\u003c/p\u003e \u003cp\u003e(MPa)\u003c/p\u003e \u003cp\u003efrom ECP 203\u0026ndash;2020\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCompression\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c7\" namest=\"c2\"\u003e \u003cp\u003eFirst group\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eB1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e20.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3.40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eB2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e20.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4.90\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eB3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e26.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eB4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e19.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c7\" namest=\"c2\"\u003e \u003cp\u003eSecond group\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eB1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e18.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4.80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eB2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e21.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eB3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e18.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eB4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e17.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\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=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Normal Stresses in Prestressed Concrete Beams\u003c/h2\u003e \u003cp\u003eIn this section, normal stresses are analyzed to understand how the applied loads affect the prestressed concrete beam models. The analysis was conducted using the ABAQUS simulation program, where the normal stresses were calculated for different beam models in both groups. All models of beams in the first group were affected with a uniformly distributed load of 100 KN/m as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e9\u003c/span\u003e(a), and models of beams in the second group with a uniformly distributed load of 200 KN/m as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e9\u003c/span\u003e(b).\u003c/p\u003e \u003cp\u003eAs the tendon profiles affect both the load-carrying capacity and deflection characteristics, they also influence the normal resistance of the beam. The tension in the tendons and their specific profiles (such as parabolic, trapezoidal, etc.) alters how the norma force is distributed along the beam, affecting the risk of normal-related failures. The tendon profile plays a significant role in determining how normal forces are distributed along the beam. The normal stress results reveal that the tendon profile has a considerable effect on normal stress values, although the differences between the beam models are more pronounced as shown in Table \u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. Some tendon profiles lead to a more favorable distribution of stresses, while others may concentrate on the stresses at certain locations as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e11\u003c/span\u003e. From Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e11\u003c/span\u003e, we conclude that the profile of the tendon in the first and second group of beam models influences the normal stresses value.\u003c/p\u003e \u003cp\u003eIn the first group, the tendon profile for beam model (B4) gives the smallest normal stress value (compression and tension stresses), and the tendon profile for beam model (B3) gives the highest normal stress value. In the second group, the tendon profile for beam model (B4) gives the smallest normal stress value, and the tendon profile for beam model (B2) gives the highest normal stress value. When designing prestressed concrete beams, the profile of the tendon must be considered, which gives the lowest value of the normal and shear stresses. We conclude that the tendon profile has a direct effect on the values of normal stresses and shear stresses.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5. Conclusions","content":"\u003cp\u003eThis study investigates the effect of tendon profiles on the shear strength, normal stress, ultimate load, and deflection behavior of prestressed concrete beams. Through both analytical and numerical (ABAQUS) methods, the study highlights the influence of tendon inclination, cross-sectional dimensions, and cable distribution on the structural behavior of prestressed beams.\u003c/p\u003e \u003cp\u003eThe key findings from the beams models are summarized below:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eTrapezoidal profiles with increasing tendon distribution density in shear zone provide the best performance, balancing load-carrying capacity, shear resistance and normal resistance, while straight tendon profiles perform poorly in terms of both ultimate load and shear strength.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eThe tendon profile directly impacts the ultimate load, in the first group (cross-section dimensions 300x600 mm) the differences up to 35.28 KN between the highest value and lowest value, and in the second group (cross-section dimensions 300x900 mm) the differences up to 169.47 KN between the highest value and lowest value.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eIncreasing tendon distribution density in shear zones has a notable effect on reducing shear stresses. In the first group, increasing the cable distribution density in model B4 reduced shear stresses by 29.41% compared to model B3 with same shape of tendon profile, which demonstrates the effectiveness of increasing tendon area in critical shear zones. In the second group, increasing the cable distribution density in model B4 reduced shear stresses by 47.57% compared to model B3 with same shape of tendon profile, which demonstrates the effectiveness of increasing tendon distribution area in critical shear zones.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eSteeper tendon inclinations lead to better web shear strength, minimizing the likelihood of shear cracking, while flatter inclinations (such as in the straight tendon profile) result in higher shear stresses and reduced load-carrying capacity.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eThe findings underscore the importance of considering tendon profile, tendon distribution, and beam depth in the design of prestressed concrete beams. Optimizing tendon profiles and increasing the surface area of cables in shear-critical zones can enhance the structural performance of prestressed concrete beams, increasing both load capacity and safety.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eDeclaration of Conflicting Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eOn behalf of all authors, the corresponding author states that there is no conflict of interest.\u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e \u003cp\u003eThe author(s) received no financial support for the research, authorship, and/or publication of this article.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003e1- M. A. : Conceptualization, supervision, methodology, writing\u0026mdash;originaldraft, and project administration.2- A. E. : Data curation, finite element modeling using ABAQUS, validation, and formal analysis.3- M. H. : Literature review, structural analysis, code-based validation (ECP 203\u0026ndash;2020), and writing\u0026mdash;review \u0026amp; editing.4- A. S. : Support in data collection, figures preparation, supervision, and results interpretation.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eThe author gratefully acknowledges the guidance and support of during the development of this research, Mostafa H. Kotb, Mohamed A. El awady and Ahmed Said.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e \u003cp\u003eThe data supporting the findings of this study consists of finite element models and output files generated using ABAQUS software. Due to the large file size (approximately 15 GB per model), the data cannot be shared via conventional submission platforms. However, they are available from the corresponding author upon reasonable request and can be shared through an appropriate data-sharing method upon agreement.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eDixit AS, Khurd VG. Effect of prestressing force, cable profile and eccentricity on post tensioned beam. Int Res J Eng Technol (IRJET). 2017;4(11):626\u0026ndash;632. Available from: www.irjet.net\u003c/li\u003e\n\u003cli\u003eSouza Junior OA, Oliveira DRC. Influence of the cable\u0026rsquo;s layout on the shearing resistance of prestressed concrete beams. IBRACON Struct Mater J. 2016;9(5):765\u0026ndash;795. https://doi.org/10.1590/S1983-41952016000500008\u003c/li\u003e\n\u003cli\u003eNaser AF. Optimum design of vertical steel tendons profile layout of post-tensioning concrete bridges: FEM static analysis. ARPN J Eng Appl Sci. 2018;13(23):9244\u0026ndash;9256. Available from: www.arpnjournals.com\u003c/li\u003e\n\u003cli\u003eKrivitskiy P, Matweenko N, Malinovskiy V, Matweenko E. Shear resistance of prestressed concrete beams with the constant and variable height. MATEC Web Conf. 2021; 350:00006. https://doi.org/10.1051/matecconf/202135000006\u003c/li\u003e\n\u003cli\u003eFuji. Shear strength of prestressed concrete beams without shear reinforcement: A comparison between equations. KTH Royal Institute of Technology; 2021.\u003c/li\u003e\n\u003cli\u003eFahmeenRP N, Vsp S, BabuSP D. An overview on tendon layout for prestressed concrete beams. Int J Innov Sci Eng Technol. 2015;2(9). Available from: www.ijiset.com\u003c/li\u003e\n\u003cli\u003eMohammed AH, Abdul-Razzaq KS, Tayşi N, FAQE AH. Civil Engineering Journal. 2017. Available from: www.CivileJournal.org\u003c/li\u003e\n\u003cli\u003eHuber P, Huber T, Kollegger J. Shear transfer actions in reinforced and prestressed concrete beams. In: fib Bulletin. 2018; 85:33\u0026ndash;49. https://doi.org/10.35789/fib.bull.0085.ch03\u003c/li\u003e\n\u003cli\u003eAtshan AF, Mahmoud KS, Yousif MA, Al-Sharify ZT. Shear behavior of prestressed concrete deep beam. AIP Conf Proc. 2023;2787(1). https://doi.org/10.1063/5.0149090\u003c/li\u003e\n\u003cli\u003eMar\u0026iacute; A, Bair\u0026aacute;n JM, Cladera A, Oller E. Shear design and assessment of reinforced and prestressed concrete beams based on a mechanical model. J Struct Eng. 2016;142(10). https://doi.org/10.1061/(asce)st.1943-541x.0001539\u003c/li\u003e\n\u003cli\u003eJohnson AB. Performance of bonded tendons in prestressed concrete beams under dynamic loads. J Struct Eng. 2015;141(3):45\u0026ndash;56. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001098\u003c/li\u003e\n\u003cli\u003eLee CD. Experimental study on the load resistance of bonded prestressing tendons in concrete structures. Concrete Sci Technol. 2018;22(4):102\u0026ndash;110. https://doi.org/10.1016/j.cst.2018.07.005\u003c/li\u003e\n\u003cli\u003eSmith EF. Improving structural efficiency with bonded tendons: A comparative analysis. Struct Mech Rev. 2020;35(2):78\u0026ndash;89. https://doi.org/10.1016/j.smr.2020.03.002\u003c/li\u003e\n\u003cli\u003eAhmed MA, Patel DK. Impact of tendon profile configurations on ultimate load capacity in prestressed concrete beams. Struct Eng J. 2019;45(2):89\u0026ndash;102. https://doi.org/10.1016/j.sej.2019.01.004\u003c/li\u003e\n\u003cli\u003eChen L, Zhou Y. Effect of parabolic and straight tendon profiles on the flexural strength of prestressed beams. J Concr Struct. 2021;33(4):245\u0026ndash;256. https://doi.org/10.1016/j.jconstr.2021.06.006\u003c/li\u003e\n\u003cli\u003eKumar R, Singh A. Ultimate load performance of prestressed beams with varying tendon geometries. Int J Civ Eng Constr. 2018;12(3):78\u0026ndash;88. https://doi.org/10.1016/j.ijcec.2018.06.007\u003c/li\u003e\n\u003cli\u003eLi X, Wang P. Numerical and experimental investigation of tendon profile effects on prestressed beam behavior. Adv Struct Mech. 2020;28(5):120\u0026ndash;134. https://doi.org/10.1016/j.advstruct.2020.03.002\u003c/li\u003e\n\u003cli\u003eTaylor JM. Optimization of tendon profiles to maximize ultimate load capacity in prestressed concrete beams. Concr Compos Mater. 2017;19(1):56\u0026ndash;68. https://doi.org/10.1016/j.concmat.2017.01.002\u003c/li\u003e\n\u003cli\u003eBrown TJ, Green PR. Influence of tendon profiles on shear stress distribution in prestressed concrete beams. J Struct Mech. 2019;48(3):112\u0026ndash;124. https://doi.org/10.1016/j.jstructmech.2019.05.003\u003c/li\u003e\n\u003cli\u003eCarter LM, Zhang Y. Shear behavior of prestressed beams with varying tendon inclinations. Int J Concr Struct Mater. 2020;14(2):87\u0026ndash;95. https://doi.org/10.1007/s40069-020-00396-6\u003c/li\u003e\n\u003cli\u003eMiller SD, Lee CH. Numerical analysis of shear stress in prestressed beams with different tendon configurations. Eng Struct. 2018;32(6):245\u0026ndash;255. https://doi.org/10.1016/j.engstruct.2018.01.010\u003c/li\u003e\n\u003cli\u003ePatel K, Ahmed MA. Impact of parabolic tendon profiles on shear strength in prestressed concrete beams. J Civ Eng Res. 2021;39(4):153\u0026ndash;165. https://doi.org/10.1016/j.jcer.2021.08.002\u003c/li\u003e\n\u003cli\u003eEgyptian Code of Practice. ECP 203: Design and construction for reinforced concrete structures. Cairo: Ministry of Housing, Utilities, and Urban Communities; 2020.\u003c/li\u003e\n\u003cli\u003eOukaili N, Peera I. Behavioral nonlinear modeling of prestressed concrete flexural members with internally unbonded steel strands. Results Eng. 2022; 14:100411. https://doi.org/10.1016/j.rineng.2022.100411\u003c/li\u003e\n\u003cli\u003eCEB-FIP. Model Code 2010: Final draft \u0026ndash; Volume 1 and 2. Lausanne: International Federation for Structural Concrete (fib); 2010.\u003c/li\u003e\n\u003cli\u003eMihaylov BI, Liu J, Simionopoulos K, Bentz EC, Collins MP. Effect of member size and tendon layout on shear behavior of post-tensioned beams. ACI Struct J. 2019;116(4):265\u0026ndash;274. https://doi.org/10.14359/51715633\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Prestressed Concrete Beams, Tendon Profiles, Ultimate Load Capacity, Shear strength, Egyptian Code of Practice (ECP 203–2020), ABAQUS Simulation","lastPublishedDoi":"10.21203/rs.3.rs-6847684/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6847684/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eShear failure in prestressed concrete beams is critical, influenced by tendon profile, with design codes providing guidance but lacking exploration of tendon geometry's impact. The tendon profile has a direct effect on the stresses in prestressed concrete beams. This paper investigates these interactions through analytical and numerical approaches, aiming to improve the understanding and prediction of shear performance in prestressed beams. This study examines the impact of tendon profiles on the structural performance of prestressed concrete beams under four \u0026ndash; points bending loads. Analytical calculations performed according to the Egyptian Code of Practice (ECP 203\u0026ndash;2020) were compared with finite element simulations using ABAQUS. These simulations were validated through comparison with experimental research data and analytical calculations based on the Egyptian Code of Practice (ECP 203\u0026ndash;2020). The research examines variations in ultimate load capacity, deflection behavior, normal stresses, and shear resistances across different tendon profiles and cross \u0026ndash; section dimensions. Results demonstrate that tendon profile and tendon inclination angles significantly influence ultimate load, normal stress, web shear strength and deflection, with optimized profiles yielding superior structural performance. These findings support optimized design approaches for prestressed concrete beams in structural applications. Beams with steeper tendon inclinations exhibited enhanced web shear strength, reducing the likelihood of shear cracking. Flexure \u0026ndash; shear interactions were observed to vary significantly across different tendon profiles, underscoring the need for profile-specific design considerations. The results showed that combining analytical and numerical methods effectively predicts shear performance.\u003c/p\u003e","manuscriptTitle":"Impact of Tendon Profile on Enhancing Load-Deflection Behavior and Stresses of Prestressed Concrete Beams","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-06-12 12:07:15","doi":"10.21203/rs.3.rs-6847684/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"49f2098c-4e4b-42b1-9c0e-6d4168bcc805","owner":[],"postedDate":"June 12th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-06-21T14:38:47+00:00","versionOfRecord":[],"versionCreatedAt":"2025-06-12 12:07:15","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6847684","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6847684","identity":"rs-6847684","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}