Strut-and-tie Model Research Articles

STM-based symbolic regression for strength prediction of RC deep beams and corbels

This study uses symbolic regression with a strut-and-tie model to predict the shear strength of reinforced concrete deep beams (RCDBs) and corbels (RCCs). Previous studies have proposed two distinct types of models for estimating shear capacity: explainable models based on theoretical derivations and black-box models derived from machine learning (ML) methods. This study proposes a hybrid model derived from the strut-and-tie model (STM), where the performance of STM is enhanced through the ML approach using genetic programming. This model is based on a comprehensive experimental database of 810 tests for the shear strength of RC deep beams and 371 tests for RC corbels from various research papers. The developed STM-based symbolic regression (SR-STM) integrates two distinct force-transferring mechanisms: the diagonal strut mechanism utilizing concrete strength and the truss mechanism utilizing orthogonal web reinforcement. The SR-STM model is both robust and interpretable, demonstrating high prediction accuracy with mean values of the prediction-to-actual ratios of 0.999 and 1.004 and coefficient of determination values of 0.913 and 0.862 for RCDBs and RCCs, respectively, while providing explainable mathematical expressions that align with the mechanical principles of STM. The developed SR-STM model is benchmarked against several state-of-the-art models and evaluated against the CatBoost ML technique, demonstrating acceptable performance. The results highlight the SR-STM model’s effectiveness in providing reliable predictions and valuable insights for practical engineering applications. Furthermore, a SHAP (Shapley Additive Explanations) analysis was performed, and its results align with the SR-STM model, confirming the model’s effectiveness in accurately capturing the key factors influencing the shear strength of RCDBs and RCCs.

Research on topology optimization strut-and-tie model and prestress construction for cable-pylon anchorage zone of Fengshuba Bridge

To address the challenges associated with accurate load-bearing capacity calculation and prestress design for the cable-pylon anchorage zone of cable-stayed bridges, a case study focusing on the Fengshuba Bridge is conducted. Utilizing the Solid Isotropic Material with Penalization (SIMP) variable density topology optimization method, a strut-and-tie model (STM) is constructed for the cable-pylon anchorage zone. This model is subsequently employed to verify load-bearing capacity of the struts and nodes within the cable-pylon anchorage zone, as well as to calculate the required quantity of circumferential prestressing tendons. Additionally, a spatial segmental finite element model is established for the cable-pylon anchorage zone to perform a detailed analysis of prestressed structural optimization. This analysis determines various structural details, including the arrangement of prestressing tendons along the height, the design for prestressing blind zones, and the quantity of prestressing tendons. Subsequently, the optimized model is subjected to stress verification. The research results indicate that the adoption of dispersed tendon arrangement results in reduced principal tensile stress within the cable-pylon anchorage zone, in contrast to the concentrated tendon arrangement. The introduction of short prestressing steel bundles within the conduit of prestressing blind zones uniformly enhances the pre-compressive stress on the cable-pylon side wall. Due to the Poisson effect of materials, it is necessary to arrange prestressing tendons appropriately within the cable-pylon anchorage zone. Arranging prestressing steel bundles according to the proposed STM and structural optimization method adequately fulfills the structural force requirements of the cable-pylon anchorage zone.

The direct shear behavior of the ultra-high performance concrete-filled socket connections between the bridge piers and footings

Ultra-high performance concrete (UHPC)-filled socket connections (UFSCs) provide a novel method for connecting bridge piers and footings, and allow for high matching tolerance and accelerated construction. At present, the lack of related research on the direct shear behavior of UFSCs hindered the promotion of this technology. This paper carried out the push-out test to explore the influences of embedded depths, horizontal confining pressures, and the shear key shapes on the direct shear behavior of the UFSCs. Monolithic specimens were also tested as a reference. The experimental results showed that increasing the socket depth and horizontal confining pressure has a positive impact on the direct shear capacity. With the same key width, escalating the key root height could also enhance the direct shear capacity. The experimental results also indicated that the load-bearing capacity of UHPC-connected specimens was generally lower than those of corresponding monolithic specimens. AASHTO and JSCE formulas were used to predict the direct shear capacity of joints, which relatively underestimated the experimental results. The average ratio between the calculated values of the two formulas and experimental results was 0.70 and 0.81, respectively. Based on the strut-and-tie model (STM), the shear capacity formula for UFSC was proposed, exhibiting a high degree of agreement with the test results, with an average ratio between calculated values and experimental results of 0.98.

Behavior of Modified Strut-and-Tie Model (STM) on Concrete Beams

Reinforced concrete components are generally designed to withhold shear and bending based on the assumption at strain varies linearly in a section where the applied force is a combination of shear with bending, torsion, or normal forces. The behavior of shear failure in reinforced concrete beams is very different from that in flexural failure. Shear failure is brittle without warning in the form of significant deflection, resulting in diagonal cracks in the beam then the shear force mechanism will be contributed by an arching action where this action can provide a reserve capacity large enough for the beam to carry the load. Nonlinear analysis using the strut-and-tie method is particularly useful for shear-critical structures where classical beam theory is not valid due to significant shear deformations. Strut-and-Tie Model research which is applied to concrete (25 MPa), also uses the optimal configuration FEM software tool from Strut-and-Tie which leads to the efficiency of Strut-and-Tie. The results of the optimization and modification of the Strut-and-Tie Model on concrete will also be applied to the experimental models tested until they fail so that the optimal conditions for numerical models will be obtained.Based on the analysis result of element model using ANSYS computational assistance program, the ultimate flexural capacity of the beam model will increase depending on the used STM model and inclination angle (Φ), at the STM type 2 high < 1000 mm, inclination angle (Φ) 45° having a decrease in ultimate shear capacity (Vu) of 49,31% against type 1, at the STM model 3 high < 1000 mm, with inclination angle (Φ) 45° having an increase ultimate shear (Vu) of 4,14% against type 1. Stress pattern performed bottle shape in line with diagonal strut. Ductility capacity will be decreased at inclination angle <45° of 27,11%, at inclination angle >45° will be decreased of 55,67%,

Flexural performance of UHPC-filled narrow joints between precast concrete bridge slabs

The deck slabs narrow joints in concrete bridges filled with Ultra-high performance concrete (UHPC) have the potential for wide application and high-efficiency promotion, but there is limited research available on this topic. This paper conducted flexural tests on the new joints. Ten precast concrete deck panels were designed and manufactured for concrete bridges. These panels were connected using UHPC-filled narrow joints. Combined with stress analysis, this study investigates the influence and mechanism of various factors on the mechanical performance of UHPC joints. These factors include the geometric shape of the joint, the roughness of the joint interface, the filling material used in the joint, the width of the joint, and the configuration of joint reinforcement (including the shape and length of lap reinforcement). The test results show that the panel with the UHPC joint filled with steel fibers has excellent flexural and bonding capacity. The jointed panel with a lap length of 140 mm and 120 mm has better flexural performance than the one with a lap length of 90 mm. The straight-jointed panel has a higher flexural capacity compared to the complex triangular-jointed panel. Additionally, reducing the joint width can achieve similar mechanical properties as before. A method has been proposed for calculating the flexural capacity of a jointed panel, utilizing the strut-and-tie model (STM). This method can accurately predict experimental results.

Shear strength of prestressed concrete beams with short shear span: Full‐scale experiments and evaluation of design methods

AbstractPrestressed concrete (PC) is increasingly used in civil engineering construction. Notably, research on shear resistance in PC beams with short shear span remains relatively limited, necessitating further comprehensive exploration. In the study, tests were conducted on bonded post‐tensioned PC T‐beams with short shear span (a/h = 1.0), subjecting them to monotonic loading destructive tests. These beams measured 24.9 m in length, 1.605 m in height, and had a top flange width of 2.0 m. They featured a uniform cross‐section with a 0.22 m web thickness. Given the size of the full‐scale test beams and space constraints for loading, an innovative loading scheme was devised to minimize the counterweight. Test results demonstrate the safety and feasibility of the loading scheme. The tests were conducted five times and the main test variables were the draped tendons and the quantity of horizontal web reinforcement. The tests revealed that horizontal web reinforcement significantly influenced crack propagation but had a minimal effect on shear strength. For the parameter values taken for the test beams, the prestressing level has less effect on the shear strength of the beam. Simultaneously, an investigation was conducted involving a total of 119 PC beams. This investigation aimed to evaluate the degree of conservatism and accuracy in shear design code provisions and shear strength equations derived from various standards, including the strut‐and‐tie model (STM) of ACI, AASHTO, and Tan, as well as the critical crack model of Yuan and the truss model of Laskar. Comparative analysis indicates that, the design method provided by Yuan exhibit relatively superior predictive capabilities.

Optimum Strut-and-Tie Models for Reinforced Concrete Beams

In this study, the authors suggested three optimum strut-and-tie models (STMs), which consider all reinforced concrete (RC) beams' failure behavior characteristics. The recommended models are valid for the RC beams with a shear span ratio (a/d) of less than 4.5. For each model, the authors also proposed a force distribution ratio, described by the portion of force carried by a truss mechanism, to assist structural designers in conducting the appropriate design using the current design codes' STM methods. In determining the force distribution ratios, the authors incorporated the influences of the main design variables, such as the a/d ratio, flexural rebar ratio, and concrete strength. The authors verified the appropriateness of the suggested STMs and force distribution ratios by using them for the strength verification of 758 RC beams tested to failure. The proposed STMs and force distribution ratios predicted the RC beams' failure strength quite well. The average test-to-predicted strength ratios (and coefficient of variations) by the present study and conventional STM models of design codes were 1.07 (21.3%) and 1.75 – 1.82 (27.6 – 45.7%), respectively.

Resultados experimentais e análise de carga limite pelo Método de Bielas e Tirantes (MBT) de vigas-parede com entalhes e aberturas

Abstract This article presents experimental results of eight non-aligned, non-geometrically symmetrical deep beams with notches and openings. The variables considered in the study were the presence of web reinforcement, and the geometry of the deep beams. Along with the experimental investigation, a study of strut-and-tie models (STM) representing the behavior (failure mode and maximum load) of reinforced concrete deep beams with complex geometries was performed. The experimental results showed that the ultimate load capacity of deep beams is significantly affected by the presence of reinforcement in the web. Furthermore, the study showed that the STM is conservative and tends to underestimate the results, reducing its ability to predict failure load.

Shear resistance mechanism of prefabricated large-scale gerber girder

In this study, shear tests were conducted to investigate the shear resistance mechanisms of Gerber precast concrete (GPC) girders consisting of PC end units, PC girder units, and cast-in-place concrete. To this end, individual PC end-member and mid-girder specimens were fabricated along with five GPC girder specimen with CIP concrete. The key test variables were overlapping (hanging) length and connection details of the mid-girder and end units. The test results showed that the shear performance of the composite GPC girder was significantly influenced by the connection details of the end-member and mid-girder. In the specimen where the mid-girder was simply assembled with the end-member without bolts, the failure was dominated by the damage concentrated on the bottom flange of the end-member. In contrast, in the specimen with the end-member and mid-girder assembled with bolts, there was no significant damage to the connection region, resulting in exerting full shear capacity. Additionally, based on the load–displacement response, crack pattern, and strain behavior measured from the experiment, the load-transfer mechanism of the GPC girder was estimated and reflected to propose a strut-and-tie model (STM) for the shear design of the GPC girder. The proposed STM was found to evaluate the shear strengths of the GPC girder specimens with a higher degree of accuracy than the shear strength evaluation method based on design code equations. Therefore, tied-arch mechanism can be assumed to dominate the main shear transfer mechanism when designing the GPC girder.

Equilibrium-based analysis of diagonal tension failure from a re-entrant corner of an RC half-joint

For ascertaining the structural longevity of commonly used half-joint girder-to-girder connections in ageing concrete bridges, accurate strength assessments are crucially important. Existing strut-and-tie models (STM) as well as code provisions underestimate resistance to diagonal tension failure from a re-entrant corner crack, perhaps leading to unnecessary maintenance expense. We develop a more accurate equilibrium-based strength analysis of diagonal tension failure based on a faithful representation of the failure surface. We formulate equilibrium equations to represent the internal force system; strength derivation is then optimised for these equations and yield criteria. The accuracy of the new analytical method was validated through comparison with experimentally obtained strengths from 30 specimens. The analytical predictions show good agreement with the test results, with an average error of less than 2%, which is superior to STM-based prediction. Not only strength capacities but also internal force-resisting components are quantified using this analytical method. Results indicate a dependency among all internal force-resisting components and a non-dimensional parameter of a1/hd.

Application of Graphic Statics and Strut-and-Tie Models Optimization Algorithm in Innovative Timber Structure Design

Timber has long been extensively employed within the construction industry as a famous, environmentally friendly, and low-carbon material. Considering that construction constitutes one of the most significant contributors to carbon emissions throughout the entire life-cycle of a building, there is an urgent desire to incorporate timber into this domain. Nevertheless, the use of timber faces inherent challenges stemming from its anisotropic nature, a result of the natural growth of timber fibers, which makes it challenging for it to function as a primary load-bearing material in coping with the various complex stresses inherent in architectural applications. Numerous designers have attempted to address this limitation through over-sized members and reinforcement at joints; however, none have satisfactorily resolved this issue in an economical manner. In this article, we introduce the Strut-and-Tie models (STM) from Graphic Statics (GS) and a topological optimization algorithm. This algorithm has the capability to generate a ‘load-minimizing path’ STM based on external load support conditions and the maximum structural path span. Regardless of the complexity of the initial external loads, each load transfer path in the optimized STM bears loads in only one direction, representing an optimal solution with minimal internal loads that align seamlessly with the characteristics of timber. Consequently, we endeavor to adopt this optimization algorithm to propose a structural design methodology, with the aspiration of designing structural systems that harness the unique attributes of timber perfectly and applying them to various architectural scenarios. Ultimately, we conclude that structural systems designed based on optimized STM are adaptable to diverse architectural contexts, and when applied to small-scale buildings, this method can save approximately 20% of material consumption compared to conventional timber frame structures, while in the case of mid-rise to high-rise buildings, it can lead to a material savings of approximately 5%.

Topology Optimization: Applications of VFLSM and SESO in the Generation of Three-Dimensional Strut-and-Tie Models

Topology Optimization: Applications of VFLSM and SESO in the Generation of Three-Dimensional Strut-and-Tie Models

The shear strength of existing non-seismic RC beam-column joints strengthened with CFRP Sheets: Numerical and analytical study

A key part of RC structure seismic performance is the effective distribution of earthquake forces across all resistance elements through reinforced concrete (RC) beam-column (B-C) joints. By virtue of their ease of implementation and convenient material properties, Carbon-Fiber-Reinforced Polymers (CFRP) are highly utilized for strengthening existing structures with inadequate reinforcement or insufficient confinement. Previous experimental research revealed that debonding failure widely occurs on CFRP shear-strengthened B-C joints. The experimental research as well showed the value of column axial load makes a great contribution to such joint’s failure mode. Using CFRP U-jacketing to strengthen non-seismically B-C joints, this article presents a non-linear, two-dimensional FE model for investigating shear behavior. The developed FE model considered the adhesive layer between CFRP sheets and concrete to capture the debonding and a bond-slip model was implemented between concrete-to-CFRP sheets using connector elements. The FE results were verified versus experimental works from literature to prove the proposed FE model's accuracy with appreciate to load-deformation relation and failure modes. Depending on the verified FE model, a parametric study on shear-strengthened RC B-C joints with CFRP U-jacketing with free and fixed ends was performed below monotonic loading. In the existing study, several parameters, including column-to-beam flexural strength ratio (over strength ratio), concrete strength, and value of column axial load, were investigated on the structure's strength and ductility. Using the developed strut-and-tie model (STM), new design equations were proposed to calculate strengthened RC B-C joint's shear strength using CFRP. The proposed equations incorporate the interfacial bond among concrete and CFRP effect, as well CFRP sheet end boundary (free or fixed). The developed design equation’s ability to estimate shear strengths of strengthened B-C joints with CFRP was verified through experimental works and FE results. It was evident from verification results that the produced design equations were compatible with experimental and FE model results.

Shear strength behavior of high-strength fly ash concrete beams with low-bond reinforcement

This study aims to investigate the shear behavior of high-strength fly ash (HSFA) concrete beams with low-bond reinforcement (LBR). With this purpose, twenty-three HSFA concrete beam specimens were constructed and tested. The experimental results showed that owing to the high-strength property of LBR, the tested beams easily failed in shear. The shear strength of the HSFA beams was mainly controlled by the shear span to depth ratio (a/d) and transverse reinforcement ratio. The shear strength Vs carried by the transverse reinforcement started to increase when the shear strength Vc carried by the concrete started reducing or growing at a lower rate. Formulas in design provisions, except for strut-and-tie model (STM) and those in the “AIJ standard for structural calculation of reinforced concrete structures” proposed by Architectural Institute of Japan (AIJ), could not accurately assess the shear strength of HSFA concrete beams because they do not reasonably consider the effect of a/d on the Vc. Generally, the current design models underestimate the Vs. Based on the test results, a new model that can reasonably consider the effect of a/d on Vc was proposed to accurately evaluate the shear strength of HSFA concrete beams reinforced by LBRs. This study deepens the understanding of researchers and engineers of the shear behavior of HSFA concrete beams with LBRs.

Graphical modeling of discontinuous stress field for RC deep beams under uniform load

AbstractThe fanning compression stress pattern in RC deep beams under uniform load presents a challenge to formulating the configuration of strut‐and‐tie model (STM). To this end, this paper proposes an alternative method for creating a graphical discontinuous stress field model (DSFM) to fill up the whole beam continuum. It comprises a certain number of trapezoidal stress‐bands with varying bandwidths, and their stress values can be calculated based on force equilibrium conditions. Several stress subfields are defined through a certain stress combination from several directions. The DSFM is implementable for both service limit state (SLS) and ultimate limit state (ULS) in that its configuration is primarily oriented by elastic stress field and only a limited plastic redistribution is allowed for plastic failure. The graphical procedure of DSFM is demonstrated and examined by a test deep beam example, which shows that the predictions are fairly in good agreement with the test results.

Investigation of the strut‐and‐tie model of a single‐plane cable‐stayed bridge tower under vertical component of the cable force

AbstractThe design of reinforcement of concrete pylons in cable‐stayed bridges is a challenging task. Usually, only the influence of the strong horizontal component of the cable force is considered during the reinforcement design of the concrete tower, while the bursting forces caused by the vertical components of the cable forces are ignored. However, the latter is also an important factor for concrete cracking in bridge towers. In this paper, the concrete tower of a single cable plane cable‐stayed bridge is considered. The strut‐and‐tie model (STM) of the pylon wall subject to the vertical components of one single cable force was derived. Based on the minimum strain energy principle, an equation for predicting the bursting force was given. Then, the STMs of a concrete pylon with two cable forces with different vertical distances between cables were proposed. A series of equations for predicting the bursting force in different STMs were presented. The STMs of a concrete pylon with multiple cables were subsequently obtained from the STM for two cable forces. The analysis results show that the vertical distance between cables has a great influence on the STM configuration of the tower. When the cable distance is less than 0.7 times the width of the tower wall, the bursting force is larger than that when the cable distance is larger than 0.7 times the width of the tower wall. This study was expected to provide a reference in the design and applications of pylons of concrete cable‐stayed bridges.

Strength prediction of a bottle-shaped strut for evaluating the shear capacity of reinforced concrete deep beam

For the analysis and design of reinforced concrete (RC) deep beams, the strut-and-tie model is typically employed to simulate the flow of stresses. A bottle-shaped strut develops between the loading point and support, in which compressive stress spreads laterally and induces tensile strain in the transverse direction, which reduces the compressive strength of the strut significantly. Web reinforcements are provided in deep RC beams to prevent splitting of the diagonal strut due to the presence of these transverse strains. In this article, a simplified design procedure based on the strut-and-tie model (STM) is proposed for the evaluation of the shear capacity of deep beams with web reinforcement. The suggested model is based on modified compression field theory and considers the impact of shear reinforcement ratio, transverse tensile strain, post-cracking strength of concrete, and the actual dispersion of compressive stress based on strut geometry (aspect ratio) for strength prediction of bottle-shaped struts. The experimental results of RC panels and deep RC beams reported in the literature are used to validate the results of the suggested model. The findings demonstrate that the proposed STM evaluates the shear capacity of deep RC beams more precisely than capacities estimated by ACI 318-19, AASHTO LRDF, and Eurocode 2 (EC2). In addition, a comprehensive parametric study has been carried out to investigate the effect of various design parameters on the shear capacity of deep RC beams.

Construction of strut‐and‐tie models for members under complex stress states based on topology optimization

AbstractThe construction of strut‐and‐tie models (STMs) is the key, but also the difficulty when the model is used to describe the force mechanism and guide the reinforcement layout design of concrete disturbed regions (D‐regions). Topology optimization is a promising way to guide STMs construction. Different from the existing construction methods of predetermining nodes, the angle of inclination of the truss elements is first determined to ensure that it is consistent with that of the principal stresses in a 2D finite element model of the structure. The constructed model can well reflect the force transmission path of structural members, and it is better than the existing methods when it is evaluated through the principle of minimum strain energy. To verify the validity of the STM, a reinforcement layout scheme is adopted completely according to the position of the tension ties. On this basis, another reinforcement layout scheme adds additional transverse reinforcements in the bottle‐shaped stress field. The nonlinear finite element comparative analysis of an irregular deep beam designed by the two schemes is carried out. The results show that the two ultimate loads are close. Brittle failure occurs in the former, while the latter has higher deformability and eventually ductile failure occurs.

A Strut-and-Tie Model for Predicting the Shear Strength of Exterior Beam–Column Joints Strengthened with Fiber-Reinforced Polymers

This paper presents a new strut-and-tie model (STM) based on provisions of ACI 318-19 and ACI 440.2R-17 for estimating the shear strength of fiber-reinforced polymer (FRP)-strengthened exterior joints. Beam–column joints are vulnerable elements in reinforced-concrete structures subjected to seismic loadings and similar extreme demands. The FRP strengthening technique offers numerous advantages for retrofitting joints in existing structures with inadequate reinforcement or confinement. The design of FRP-strengthened joints requires the development of reliable procedures, including equations to predict the shear strength of the joint. Existing literature and research on STM warrant the development of simple, reliable, and practical specifications applied to a wide range of design criteria. The proposed method involves an STM model containing horizontal and vertical mechanisms. The methodology compares predicted joint shear strength results with existing experimental results to obtain a reliable design approach. This comparison, along with verification of alternative analytical results, indicates the appropriateness of the presented model for predicting the shear strength of FRP-strengthened joints. Results contribute to the enhancement and efficiency of the practical design of these joints.

Shear strengthening of concrete deep beams using pre-fabricated strain-hardening cementitious composite plates

This study investigates the efficiency of near surface embedded (NSE) pre-fabricated strain-hardening cementitious composite (P-SHCC) plates for shear strengthening of reinforced concrete deep beams (RC-DBs) through testing 12 identical specimens. For this purpose, P-SHCC plates of 20 mm thickness have been prepared according to the test variables, cured for about 30 days, and then dipped into the concrete cover of the strengthened DBs at the critical shear span. The use of pre-fabricated plates is intended to alleviate any volumetric change issues associated with SHCC material. The studied parameters were the net spacing between the P-SHCC plates (0, 75, 112, and 150 mm), the width of the strengthening plates (60, 75, and 90 mm), and the ratio of steel reinforcement embedded inside the P-SHCC plates (0 %, 0.67 %, 1.50 %, and 3.00 %). Furthermore, the prestressing level relative to the yielding strain of the steel reinforcement embedded inside the P-SHCC plates was of interest. Experimental results exhibited a successful employment of the P-SHCC plates in enhancing the ultimate load capacity of RC deep beams, especially when prestressing was introduced. The gain in the ultimate capacity reached up to 68.5 %, relative to the un-strengthened control beam (BC), depending on the test parameters. Besides, incorporating steel reinforcement inside the P-SHCC plates had a pronounced effect on both the serviceability and ductility characteristics of the strengthened DBs. Specimens strengthened by reinforced P-SHCC plates demonstrated around 1.57-times improvement in energy absorption capacity when compared to their counterparts strengthened by plain P-SHCC plates. Finally, a strut-and-tie model (STM) in addition to a simpler equation based on regression analysis were used to assess the shear capacity of the strengthened DBs, considering the contribution of P-SHCC plates. The proposed two equations were in good accord with the experimental results.