Strut Efficiency Factor Research Articles

Influence of web reinforcement on shear behavior and size effect of lightweight aggregate concrete deep beams: Experimental and theoretical studies

In practice, deep beams with web reinforcement are the predominant form of deep beams. Understanding the effect of web reinforcement in deep beams made of lightweight aggregate concrete (LWAC) is essential for the structural application of LWAC deep beams, but it is still a pending subject. In this study, eight LWAC deep beams and two normal weight concrete deep beams were tested to investigate the influence of web reinforcement and beam depth on the shear behavior and size effect of LWAC deep beams. The experimental results showed that all deep beams failed in shear–compression mode, which is independent of the web reinforcement ratio and beam depth. The web reinforcement confined the concrete within the strut area and limited the concrete spalling, which positively affected the normalized serviceability/ultimate shear strength. Moreover, web reinforcement effectively mitigated the size effect by resisting the transverse tensile stress and improving the boundary confinement of the concrete struts, but could not eliminate the size effect in LWAC deep beams. Based on these results, a modified strut-and-tie model (MSTM) was developed to quantify the contribution of web reinforcements to the shear capacity more accurately than the conventional strut-and-tie model. The predictions obtained using the MSTM with the Warwick-Foster’s strut efficiency factor were in excellent agreement with the experimental results.

A proposed strut efficiency factor for strut-and-tie design of steel fibre-reinforced concrete

Strut-and-tie model (STM) is a lower-bound plastic theory to achieve force equilibrium. It uses a simplified truss analogy method to define the load transfer mechanism in the disturbed region (D-region) in concrete structures. STM was mostly applied in the design of conventional reinforced concrete (RC), and there were few discussions on the use of STM for newer concrete technology, e.g., steel fibre-reinforced concrete (SFRC). SFRC presents opportunities in reducing web reinforcements in serviceability checks and in improving the ductility of D-regions. In this paper, a strut efficiency factor for STM is proposed for the design of SFRC by collecting a database of SFRC deep beams. It was discovered that the strut efficiency factor best fits with a function of the product of the fibre factor and concrete cylinder compressive strength. Baseline deep beam numerical models were developed and calibrated with the experimental load-displacement graphs collected through literature. A parametric study with variables such as concrete cylinder compressive strengths, fibre volumes and strut angles were investigated. The proposed strut efficiency factor was found to have a peak capacity lower than the ultimate load capacity computed in the calibrated numerical models, which justified the safe use of STM as a lower-bound plastic method in SFRC.

Introducing strut efficiency factor in the softened strut and tie model for the ultimate shear strength prediction of steel RC deep beams based on experimental study

In this paper, an empirical model for a strut efficiency factor is proposed and introduced in to the softened strut and tie model to predict the ultimate shear strength and the corresponding ultimate load of simply supported steel reinforced concrete (RC) deep beams under 4-point bending without transverse reinforcements. The proposed model accounts for the influence of shear span to depth ratio concrete compressive strength area of steel reinforcement diagonal strut angle and width of loading plate In addition, an empirical model for a stiffness degradation coefficient accounting for the influence of overall depth of the beam and the diagonal strut angle is proposed and introduced in to a deflection prediction model to predict the corresponding ultimate deflection. In this study, 12 numbers of steel RC deep beams are cast and tested. The empirical models are proposed for the strength and the deflection prediction using regression analysis based on the test results of 373 and 86 steel RC deep beams respectively, which includes the 12 test beams of the present study. The proposed models are validated by accounting for the maximum range of the parameters involved in the database. The predictions based on the proposed models are compared with the predictions based on the other models. The comparison shows that the proposed models can accurately predict the ultimate load and the mid span deflection of the steel RC deep beams.

Improved empirical model for the strut efficiency factor and the stiffness degradation coefficient for the strength and the deflection prediction of FRP RC deep beams

In this paper, an improved empirical model for the strut efficiency factor βsis proposed in the softened strut and tie model to predict the ultimate load of the concrete deep beams longitudinally reinforced with fiber-reinforced polymer (FRP) bars that failed in shear. The proposed model accounts for the influence of the shear span to depth ratio a/d, the concrete compressive strength fc', the area of the FRP reinforcement Af, the diagonal strut angle θ and the width of the loading plate lb. In addition, an improved empirical model for the stiffness degradation coefficient kf, accounting for the influence of the overall depth of the beam h and the strut angle θ is proposed to predict the corresponding ultimate deflection. The proposed models are validated by accounting for the maximum range of the parameters involved in the database. The engineering significance of the various parameters involved in the proposed models are illustrated by carrying out parametric study. In addition, the proposed model is assessed with the other models for the strength and the deflection prediction using the data base used in this study. The prediction shows that the proposed models can accurately predict the ultimate load and the mid span deflection of FRP bar reinforced concrete deep beams for the various range of values of the parameters involved.

Probabilistic models of the strut efficiency factor for RC deep beams with MCMC method

AbstractThe probabilistic model using Markov Chain Monte Carlo (MCMC) sampling procedure was proposed for estimating the value of strut efficiency factor in the strut‐and‐tie model (STM). The STMs according to various proposals from different researchers were compared with the database containing 691 tests on the shear strength of RC deep beams. Strut efficiency factor models were collected and compared against the experimental results in the database. Considering prior information, established database, and existing models, the probabilistic model was improved by considering the shear span‐to‐depth ratio, web reinforcement ratio, longitudinal reinforcement ratio, and dimension of specimen. The proposed model was evaluated by the results in the database, which shows better accuracy and discreteness with mean and CoV of 1.047 and 0.248, respectively.

Development of Models for Predicting Strut Efficiency Factors for Conventional and Steel Fibre Reinforced Concrete Deep Beams

Commonly used analysis and design methods are applicable only to Bernoulli’s region or B-region where strain distribution is linear. Due to the peculiarity of the geometry for certain structural members like deep beams, corbels, etc, their entire strain distribution is nonlinear. Hence, they will act as discontinuous regions or D-regions where strain distribution is nonlinear. The designers are forced to depend on empirical relations and their past experience for designing such elements. Since, deep beams form an important structural element in bridges and high-rise buildings as pier caps and transfer girders, their design deserves much more importance. Strut-and-Tie method (STM) is a unified approach that can be used for both regions. The main parameter that influence the accuracy of results produced by STM is the strength reduction factor (βs). In this paper, new relations are developed for predicting strength reduction factor for concrete struts with and without steel fibres. The effect of steel fibres content and percentage of vertical and horizontal reinforcement were considered for developing this relation.

Prediction of the load and deflection response of concrete deep beams reinforced with FRP bars

The load and deflection response of six concrete deep beams longitudinally reinforced with glass fiber-reinforced polymer (GFRP) bars having shear span to depth ratio of 0.5 to 1.0 were presented in this paper. The existing model originally proposed for the prediction of shear strength and the deflection at the ultimate stage of concrete deep beams longitudinally reinforced with steel reinforcement, using the Softened Strut- and-Tie (SST) concept is modified to account for the influence of FRP longitudinal bars in the deep beam. The magnitude of strut efficiency factor was proposed based on the regression analysis of 47 test data. The model is useful for the prediction of load and deflection response of FRP reinforced concrete deep beams up to ultimate stage. The degradation of the stiffness due to the progressive cracking of the concrete is accounted for in this model. The predicted load-deflection response of the six test beams in this study was found to be comparable to the corresponding experimental data. The load and the mid-span deflection at ultimate stage of 53 simply supported deep beams longitudinally reinforced with FRP bars were also predicted using the proposed model. The mean of the ratio of the experimental load to the predicted load and deflection at ultimate stage for the 53 beams is found to be 1.05 and 1.03 respectively with the corresponding coefficient of variation (CoV) of 29.5% and 40.8% respectively.

Behavior and Efficiency of Corrosion Damaged Half-bottle Shaped Struts

This study investigated the behaviors, strut efficiency factor, and the effective width of corrosion-damaged half-bottle shaped struts. Six half-bottle shaped struts were subjected to varying aimed corrosion levels (0%, 3%, 6%) with different reinforcement mat rotation angles (0°, 45°), and their behaviors and typical failure modes ascertained. Strut efficiency factors and effective width were calculated and evaluated. Average strength of 6% corroded struts was 30% less than those of uncorroded struts. The AASHTO LRFD 2013 process predicted strut efficiency factor of corrosion damaged struts better than ACI 318–14 and EN 1992. Strut effective width also reduced with increasing reinforcement corrosion. The theoretical effective width near the load bearing plate was underestimated compared to the measured effective width, while the effective width far from the load bearing plate was overestimated.

Nonlinear finite-element analysis for the behavior prediction and strut efficiency factor of GFRP-reinforced concrete deep beams

The recent experimental results and proposed strut-and-tie model (STM) for deep beams reinforced entirely with glass-fiber-reinforced polymer (GFRP) bars have suggested that a comprehensive examination is required to improve the strut efficiency factor and its affecting parameters. This study uses nonlinear finite-element analysis (FEA) to perform an in-depth investigation. FEA response was compared against the experimental results in terms of crack patterns, failure modes, strains in reinforcement and concrete, and load–deflection relationships. The results show that the simulation procedures employed were stable and compliant, and that they provided reasonably accurate simulations of the behavior. FEA was used to confirm some hypotheses associated with the experimental investigations. A comprehensive parametric study was conducted to investigate the effect of web reinforcement and loading-plate size on the strut efficiency factor. It was shown that vertical web reinforcement has no clear effect on the strength, but it is required for crack control. On the other hand, horizontal web reinforcement should be accompanied with vertical reinforcement. Loading-plate size showed a clear effect on the deep-beam strength. Based on the numerical simulation results, a modification to a recently proposed STM is suggested. The modified STM was compared to available STMs in design codes and provisions, yielding better correlation with experimental results.

Determination of strut efficiency factor for concrete deep beams with and without fibre

Based on the variation of strain along the cross section, any region in a structural member can be classified into two regions namely, Bernoulli´s region (B-region) and Disturbed region (D-region). Since the variation of strain along the cross section for a B-region is linear, well-developed theories are available for their analysis and design. On the other hand, the design of D-region is carried out based on thumb rules and past experience due to the presence of nonlinear strain distribution. Strut-and-Tie method is a novel approach that can be used for the analysis and design of both B-region as well as D-region with equal importance. The strut efficiency factor (βs) is needed for the design and analysis of concrete members using Strut and Tie method. In this paper, equations for finding βs for bottle shaped struts in concrete deep beams (a D-region) with and without steel fibres are developed. The effects of transverse reinforcement on βs are also considered. Numerical studies using commercially available finite element software along with limited amount of experimental studies were used to find βs.

Strut Efficiency-Based Design for Concrete Deep Beams Reinforced with Fiber-Reinforced Polymer Bars

In this paper, a strut-and-tie-based model is proposed to predict the shear strength of fiber-reinforced polymer-reinforced (FRP) deep beams. An assessment of the available strut-and-tie models (STMs) in ACI and CSA provisions was conducted, identifying the important parameters affecting the strut efficiency factor. The tendency of each parameter (concrete compressive strength, shear span-depth ratio, and strain in longitudinal reinforcement) was assessed against the efficiency factor. The data from the 28 specimens with and without web reinforcement, including 12 tested FRP-reinforced concrete deep beams in our study and 16 FRP-reinforced deep beams taken from the literature, were used to assess the proposed model. The model was capable of capturing the failure mode and predicting the ultimate capacity of the tested FRP-reinforced deep beams. The proposed model was verified against a compilation of databases on 172 steel-reinforced deep beams, resulting in an acceptable level of adequacy.

Efficiency factors of recycled concrete aggregate concrete bottle-shaped struts

A comparison with code-based predictions of efficiency factors of bottle-shaped struts computed using strut-and-tie models from shear test results of coarse recycled concrete aggregate concrete beams reported in the literature is presented. The objective of the comparison was to ascertain whether strut efficiency factors in current design codes originally meant for natural coarse aggregate concrete can be conservatively applied to concrete made with coarse recycled concrete aggregate or not. In addition to the codal provisions, a strut efficiency factor model developed from first principles was also used in the comparative study. The results of the comparison indicate that the AASHTO LRFD code provisions and the proposed efficiency factor model gave conservative predictions for the coarse recycled concrete aggregate concrete bottle-shaped struts although the ACI 318-08 predictions were unconservative. The most accurate predictions were obtained from the Eurocode 2 model, although with an erratic degree of conservatism. The lack of conservatism vis-à-vis the ACI code recommended strut efficiency factors does not per se indicate inferior strength of coarse recycled concrete aggregate concrete but may be attributed to the insensitivity of the ACI code efficiency factors to strut inclination with adjoining tie(s).

An appraisal of design provisions for bottle-shaped struts

Ever since their inception, strut-and-tie modelling (STM) provisions in most of the current design codes have remained virtually unchanged, in spite of significant research-driven advances in the recent past. An appraisal of the ACI 318 STM provisions related to bottle-shaped struts has been carried out in the context of research findings reported in the literature. This appraisal has been used to propose a simple and robust analytical model for the efficiency factor which is used to estimate the strength of bottle-shaped struts. The proposed strut efficiency factor model, besides taking into account the strength of concrete, also considers the effect of inclination of the strut with an adjoining tie. For struts with inclination angles less than a certain threshold value that depends on the concrete strength, the efficiency factors recommended by current design codes are shown to be unconservative when compared with the predictions of the proposed model and other similar models in the literature. An alternative to the ACI minimum reinforcement formulation for bottle-shaped struts, which takes into account material strength characteristics, is discussed and the corrected form of the minimum reinforcement formulation has been used to identify a conflict in the deep beam web reinforcement provisions of ACI 318-08.

Effect of Detailing of Web Reinforcement on the Behavior of Bottle-shaped Struts

The effect of detailing of web reinforcement as per current design codes on the strength and serviceability behavior of bottle-shaped struts has been experimentally investigated, particularly in the context of codal minimum reinforcement requirements, by testing eleven scaled deep beams. The experimentally obtained efficiency factors of the bottle-shaped struts besides being significantly higher than the values given in the ACI 318-08 were also found to be sensitive to the amount of web reinforcement. The minimum web reinforcement requirements specified in current design codes were found to be adequate for controlling service load crack widths though the unreinforced bottle-shaped struts allowed in the ACI code were observed to violate the maximum crack width requirements. It has been shown that the transformation equation in the ACI code does not accurately represent the relationship between the strut efficiency factor and the amount of transverse reinforcement whereas a corrected version of the equation shows a clear dependence of the former on the latter.

An appraisal of the ACI strut efficiency factors

The strut-and-tie modelling recommendations of the American Concrete Institute (ACI) 318-08 Appendix A are widely followed for the design of disturbed regions in concrete structures. In this investigation, an experimental appraisal of the ACI 318-08-recommended efficiency factors for prismatic and bottle-shaped struts has been carried out and inconsistency in the recommendations for bottle-shaped struts highlighted. It has been demonstrated that a bottle-shaped strut is in no way inferior to a prismatic strut in terms of strength and, consequently, the efficiency factor of bottle-shaped struts needs to be revised. An error in the ACI 318-08-recommended formulation for minimum effective transverse reinforcement in reinforced bottle-shaped struts is identified and corrected.

Strength and deformation characteristics of bottle-shaped struts

Isolated strut tests were performed under monotonically increasing displacement controlled in-plane loading on thin plain, steel fibrous and reinforced concrete panels to determine the strength and deformation characteristics of bottle-shaped struts for use in strut-and-tie modelling. A total of 12 concrete panels were tested to failure. The results were used to evaluate the performance of reinforced struts and explore the possibility of using steel fibrous concrete in bottle-shaped struts. ACI 318–05 recommends a strut efficiency factor of 0·75 for an effective transverse reinforcement content greater than or equal to 0·3%. However, a recent study has indicated that the strut efficiency factor is insensitive to reinforcement in the strut. The results of this investigation reveal that reinforcement indeed has a significant influence on strut efficiency and the use of steel fibrous concrete in reinforced struts improves their toughness and post-peak response compared to that of non-fibrous-reinforced concrete struts.

Minimum Transverse Reinforcement for Bottle-Shaped Struts

This paper investigates the amount of transverse reinforcement needed to resist the tension developed in a bottle-shaped strut. Bottle-shaped struts are wider at their midpoint than at either end. As the struts widen near the midpoint, tensile stresses transverse to the direction of compression are developed. Reinforcement must be placed within the strut to carry this transverse tension. An expression for the required transverse reinforcement was developed and a database of 476 test specimens was used to evaluate these equilibrium-based equations along with a series of three tests of deep beams. The strut efficiency factors presented in Appendix A of ACI 318-05 were used in this evaluation. The shortcomings of current ACI 318-05 provisions that allow the design of a deep beam without any shear reinforcement are discussed. The amount of transverse reinforcement required to maintain equilibrium in a bottle-shaped strut is proportional to the force applied to the strut and inversely proportional to the slope of the angle of dispersion. These findings suggest that additional research regarding the serviceability of structures designed using strut-and-tie provisions is needed.