FRP-strengthened Reinforced Concrete Beams Research Articles

Mesoscale discrete simulation of flexural behavior of FRP-strengthened RC beams using 3D RBSM

A widely adopted method of strengthening existing reinforced concrete (RC) structures, especially for marine concrete structures, is to use external bonded fiber-reinforced polymer (FRP). However, although numerous studies in this field have been reported, accurate simulation of localized material behaviors such as crack opening and interfacial sliding remains challenging. Compared to other simulation methods, mesoscale models offer the promise of capturing these complexities. Nevertheless, there has been very little application of mesoscale simulations in analysis of FRP-strengthened RC structures due to a lack of relevant constitutive models. To fill this gap, the authors have recently developed local bond models for the FRP-concrete interface in the three-dimensional rigid body spring model (3D RBSM). The model’s parameters are defined with easily interpretable physical implications and straightforward computational methods. In this work, this mesoscale model is extended and applied to simulate the flexural behavior of FRP-strengthened RC beams. The simulation results closely align with experiments in both global and local behaviors. The parametric study conducted in this work highlights the critical role of interfacial normal bond strength between FRP and concrete in dictating flexural behavior. The capability of 3D RBSM in simulating the strain distribution along the FRP and capturing intricate localized behaviors aligned with flexural and shear forces is demonstrated. Overall, this paper reinforces the potential of 3D RBSM as a robust tool for analyzing the complex behaviors of FRP-strengthened concrete structures.

Analytical study on flexural strengthening of narrow RC beams with hybrid anchored CFRP sheets

Premature debonding of fiber-reinforced polymer (FRP) sheets commonly occurs in purely bonded reinforced concrete (RC) beams. A hybrid strengthening method for narrow FRP-strengthened RC beams has been proposed to solve this problem. Experimental results have verified the effectiveness of the novel strengthening system in preventing the end debonding and impeding the intermediate debonding of the FRP sheet. However, the mechanism of its strengthening efficiency has not been studied deeply enough. Therefore, this paper developed a non-linear three-dimensional finite element (FE) model to study the flexural behavior of the FRP-strengthened narrow RC beams hybrid anchored with end self-locking system. The FE model was verified against the available experimental studies and showed good agreement, with a R2 of 0.97. Based on the verified model, parametric studies were performed to study further the influences of four major factors: FRP sheet length, steel rebar ratio, concrete compressive strength, and number of FRP sheet layers. The efficacy of hybrid strengthened beams was reduced with longer FRP bond lengths and higher steel rebar ratios. Conversely, increasing concrete compressive strength and FRP sheet layers to a certain degree enhanced the efficacy. Finally, an analytical model to predict the load-deflection curve of the beam was proposed.

Intelligent prediction modeling for flexural capacity of FRP-strengthened reinforced concrete beams using machine learning algorithms

Fiber-reinforced polymers (FRP) are widely utilized to improve the efficiency and durability of concrete structures, either through external bonding or internal reinforcement. However, the response of FRP-strengthened reinforced concrete (RC) members, both in field applications and experimental settings, often deviates from the estimation based on existing code provisions. This discrepancy can be attributed to the limitations of code provisions in fully capturing the nature of FRP-strengthened RC members. Accordingly, machine learning methods, including gene expression programming (GEP) and multi-expression programming (MEP), were utilized in this study to predict the flexural capacity of the FRP-strengthened RC beam. To develop data-driven estimation models, an extensive collection of experimental data on FRP-strengthened RC beams was compiled from the experimental studies. For the assessment of the accuracy of developed models, various statistical indicators were utilized. The machine learning (ML) based models were compared with empirical and conventional linear regression models to substantiate their superiority, providing evidence of enhanced performance. The GEP model demonstrated outstanding predictive performance with a correlation coefficient (R) of 0.98 for both the training and validation phases, accompanied by minimal mean absolute errors (MAE) of 4.08 and 5.39, respectively. In contrast, the MEP model achieved a slightly lower accuracy, with an R of 0.96 in both the training and validation phases. Moreover, the ML-based models exhibited notably superior performances compared to the empirical models. Hence, the ML-based models presented in this study demonstrated promising prospects for practical implementation in engineering applications. Moreover, the SHapley Additive exPlanation (SHAP) method was used to interpret the feature's importance and influence on the flexural capacity. It was observed that beam width, section effective depth, and the tensile longitudinal bars reinforcement ratio significantly contribute to the prediction of the flexural capacity of the FRP-strengthened reinforced concrete beam.

Numerical Simulations and Simplified Design Approaches for Large-Rupture-Strain FRP-Strengthened Reinforced Concrete Beams under Impact

Numerical Simulations and Simplified Design Approaches for Large-Rupture-Strain FRP-Strengthened Reinforced Concrete Beams under Impact

Torsional behavior of FRP-strengthened reinforced concrete members considering various wrapping configurations: Theoretical analysis and modeling

Reinforced concrete (RC) structural members such as spandrel, flanged and curved beams, slab beams, and bridge girders may be subjected to significant torsional loading. Despite extensive research on the strengthening of RC members, very little is known about the torsional behavior of RC members strengthened with fiber-reinforced polymer (FRP) sheets. Therefore, very limited guidelines apply to the design of externally bonded FRP systems for torsional strengthening, and no design provisions exist for the U-wrapped and 2-sided strengthening systems in torsion. This study aims to contribute to this growing area of research by developing a new analytical model to predict the full torsional response of strengthened RC members with many conventional configurations such as spiral, transverse and longitudinal FRP, continuous and intermittent strips along the beam length, side bonded, and full-wrapped systems. Moreover, this study provided an analytical model accurately predicting the contribution, efficiency, and failure mode of side-bonded strengthening systems. Additionally, the influence of various parameters such as FRP stiffness and transverse steel reinforcement ratio is taken into account in the constitutive law of concrete and internal reinforcement for the first time. The good agreement between the test results of 32 collected FRP-strengthened RC beams from the literature and the theoretical ones, which were predicted through programming proves the validity and accuracy of the presented theory.

Effect of wetting/drying cycling on flexural behavior of RC beams strengthened by prestressed CFRP plate: Experimental and analytical investigations

Using prestressed carbon fiber reinforced polymers (CFRPs) is a promising technique for strengthening reinforced concrete (RC) structures. This study aimed to investigate the prestress losses and flexural behavior of strengthened RC beams under wetting/drying cycling (WDC) in a chloride-containing environment. Two prestress levels (20% and 40%) and two WDC durations (90 and 180 cycles) were considered. The time-dependent prestress loss under WDC was first monitored by fiber bragg grating (FBG) for 90 days, and the composition of the prestress loss was theoretically analyzed. A four-point bending test was conducted on the strengthened beams after the WDC exposure, and the intermediate crack (IC) debonding strain and the ultimate moment of the strengthened beams were analytically evaluated. The results show that although WDC exposure aggravated the time-dependent prestress losses, the maximum time-dependent prestress losses after 90 WDCs were only 1.75%, indicating the effectiveness of the end anchorage system. After WDC exposure, the failure mode of the strengthened beams shifted from CFRP rupture to IC debonding, accompanied by the reduction of cracking and the ultimate load, CFRP utilization, and the increase of the maximum crack width. Besides, the model of Lu et al. and Said and Wu could be applied to the IC debonding strain prediction after the WDC. When the environmental reduction coefficient was incorporated, ACI 440.2 R and CECS-146 resulted in a precise IC debonding strain. The proposed method could accurately predict the ultimate moment of the prestressed FRP-strengthened RC beams with different failure modes after WDC.

Machine learning-based model for recognizing the failure modes of FRP-strengthened RC beams in flexure

Plate end (PE) debonding and intermediate crack (IC) debonding are the main failure modes of fiber reinforced polymer (FRP) strengthened reinforced concrete (RC) beams in flexure. Different failure modes exhibit different failure characteristics. Therefore, accurately identifying the failure mode is of great significance in selecting methods to prevent the debonding of the strengthened beams. This study first established a primary indicator system through literature research. Then one hundred and eighty-eight FRP-strengthened RC beams containing PE and IC debonding were collected from the published literature of forty-eight researchers. After that, correlation analysis and grey correlation analysis were used to study the data of two failure modes. Finally, the indicator system for the prediction of failure modes was established. After the indicator system was established, six machine learning algorithms, including K-nearest neighbor algorithm (KNN), decision tree (DT), random forest (RF), back propagation neural network (BPNN), logistic regression (LR), and support vector machine (SVM), were used to build the prediction model of failure modes. The evaluation of the models shows that the coefficient of variation of the accuracy of the decision tree is merely 5.4%, which has the best robustness; the average accuracy of the random forest for the two failure modes reaches 93% and 98%, which has the highest precision.

Multiscale viscoelastic analysis of FRP-strengthened concrete beams

A multiscale framework with an elastic reinforced concrete (RC) beam bonded to a fiber reinforced polymer (FRP) plate (connected by an adhesive layer with viscoelastic properties) is established in this research; this framework is proved to be fast and efficient in avoiding time integration and complex meshing in existing FE models. The FRP plate is composed of elastic fibers and surrounded by viscoelastic polymers whose effective properties are generated by the locally exact homogenization theory. According to the correspondence principle, the viscoelasticity problem can be solved in the Laplace domain as an elasticity problem. This leads to solutions that can be transformed back to the time domain through Zakian's numerical method. To validate the present method, comparisons with the simulations reported in literature are presented. The effects of cracks, prestress in FRP plates, microscale fiber content, and temperature on the long-term behavior of FRP-strengthened RC beams are effectively investigated using the present method. Through the recovery of stress relaxation within the adhesive layer, the present technique is found capable of conveniently predicting the long-term behavior of FRP-strengthened RC beams, providing a robust tool for structural design and maintenance.

Flexural Performance and End Debonding Prediction of NSM Carbon FRP-Strengthened Reinforced Concrete Beams under Different Service Temperatures

This paper aims to evaluate the influence of relatively high service temperatures (near or beyond the glass transition temperature (Tg) of epoxy adhesive) on the flexural performance and end debonding phenomenon in near-surface mounted (NSM) carbon fiber-reinforced polymer (CFRP)-strengthened, reinforced concrete (RC) beams. To this end, an experimental program consisting of 24 beams (divided into four groups) was performed, where different parameters was combined (i.e., service temperature, steel reinforcement ratio, CFRP ratio, and concrete compressive strength). In addition, the effect of the testing temperature on the end debonding phenomenon was investigated with an analytical procedure according to fib Bulletin 90, and the predictions were compared to experimental results. Taking specimens tested at 20 °C as a reference, no considerable change was observed in the ultimate load of the specimens tested below 60 °C (being in the range of epoxy Tg), and all specimens failed by FRP rupture. On the other hand, the increase in testing temperature up to 70 and 85 °C was followed by a decrease in the capacity of the strengthened beams and a change in failure mode, moving from FRP rupture to end debonding and concrete crushing. The analytical procedure successfully predicted the occurrence of premature end debonding failure and demonstrated that the effect of temperature on the mechanical properties of materials can be a key factor when predicting the premature end debonding in a NSM joint.

Novel Embedded FRP Anchor for RC Beams Strengthened in Flexure with NSM FRP Bars: Concept and Behavior

The near-surface-mounted (NSM) fiber-reinforced polymer (FRP) strengthening technique for reinforced-concrete (RC) beams has attracted worldwide research attention and application in the past two decades. In spite of improved bond strength between the NSM FRP and concrete, compared with the externally bonded (EB) FRP method, premature debonding failure still occurs, and, hence, the efficiency of the NSM FRP intervention is limited. An intuitive means to enhance efficiency is to apply anchorage devices to the RC beam; however, the availability of viable anchorage is limited for NSM applications. A novel anchorage device is therefore presented in this paper, which is referred to as an embedded FRP anchor (EFA). The concept, manufacture, and installation of the EFAs were initially presented. The effectiveness of EFAs in NSM FRP-strengthened RC beams was then ascertained through full-scale beam tests and comparison with an NSM FRP-strengthened beam anchored with a U-jacket. Moreover, the effects of EFA parameters on the behavior of anchored beam were also investigated. The test results not only prove the high anchoring efficiency of EFA but also reveal the significant effects of EFA parameters on the failure mode, load–deflection response, and NSM FRP strain of the EFA anchored beam.

Data-driven prediction of FRP strengthened reinforced concrete beam capacity based on interpretable ensemble learning algorithms

Fiber-reinforced polymer (FRP) materials are one of the commonly used materials for strengthening aged reinforced concrete (RC) beams. However, it is still challenging to accurately predict the flexural capacity of an FRP-strengthened RC beam due to the intricate mechanism. To overcome the limitation of mechanical-based models, a comprehensive database of FRP-strengthened RC beam experiments was collected to develop data-driven prediction models. Four different ensemble learning (EL) algorithms, namely random forest, adaptive boosting, gradient boosting decision tree, and extreme gradient boosting were used to realize this model based on this database. To demonstrate their superiority, these models were compared with representative empirical models and the ones based on single machine learning (ML) algorithms. The performances of the EL-based models were significantly better than those of the empirical models and single ML-based models. Thus, the EL-based models proposed in this study demonstrate potential for use in engineering applications. In addition, the Shapley additive explanation (SHAP) was introduced to interpret the importance of input features in the prediction process from local and global perspectives. Finally, reliability analysis was performed to calibrate the reduction coefficient of bearing capacity.

Mitigation of Concrete Cover Separation in Concrete Beams Strengthened with Fiber-reinforced Polymer Composites

Strengthening of reinforced concrete (RC) beams using fiber reinforced polymers (FRP) is an effective method that is gaining more acceptance worldwide. However, failure of the strengthened beams at the plate-end region by concrete cover separation (CCS) compromises the effectiveness of such a strengthening system. Among available anchorage systems to remedy such problem, the use of FRP U-wraps at the end region of the externally bonded FRP has the potential to prevent CCS failure. Most studies in the literature suffer from poor design of test specimens and the use of random amounts of FRP anchorage. This resulted in nonconclusive findings and contradicting results on the real effectiveness of FRP U-wraps as an anchorage to prevent plate-end debonding failure. That in turn has been reflected in the lack of reliable design methods for the design of FRP U-wrap anchorage. In this paper, the effect of FRP U-wraps on the mitigation of CCS failure in FRP-strengthened RC beams was experimentally examined and quantified. The study included full-scale RC beams strengthened in flexure using carbon FRP (CFRP) sheets and anchored at the plate-end region using vertical CFRP U-wraps. The test variable was the amount of CFRP U-wrap that is required to achieve complete mitigation of CCS. The test results indicated the effectiveness of FRP U-wraps on mitigating CCS failure once the required area of CFRP U-wrap was placed. Relevant design models were presented and assessed against the experimental results of the current study. The comparisons indicated inaccurate predictions for the area of FRP U-wrap that is required to prevent CCS. This sheds the light on the need for developing a more reliable and accurate method for the design of an end anchorage to prevent CCS failure.

Flexural response of FRP-strengthened lightweight RC beams: hybrid bond efficiency of L‐shape ribbed bars and NSM technique

This paper presents the results of an experimental study on employing near surface mounted (NSM) fiber-reinforced polymer (FRP) reinforcement technique, and L‐shape ribbed bars, for flexural strengthening of lightweight reinforced concrete (RC) beams. 18 RC beams including 14 lightweight RC beams and four normal-weight concrete beams were designed. The beams were strengthened with glass fiber-reinforced polymer (GFRP) bars and carbon fiber-reinforced polymer (CFRP) laminate in bending tests. Test parameters included: (1) different FRP materials (glass bars and carbon sheets), (2) longitudinal steel reinforcement ratio, and (3) type of strengthening technique used (NSM reinforcement or hybrid). The ultimate tensile strength, deflection, compressive and tensile strain of concrete, and failure mode of the beams were examined under four-point flexural test. Results showed that the ultimate strength of all RC beams increased between 33 and 105% compared to the control beam. The ultimate strength of beams reinforced with CFRP in the mid-span region was 10% higher than that of beams strengthened at both ends, although the former exhibited 28% lower ultimate deflection. The ultimate strength and deflection of RC beams strengthened with combined steel reinforcing bars and GFRP bars were 10% and 108% higher, respectively, compared to those of RC beams strengthened with GFRP bars only. Hybrid L‐shape ribbed bars beams showed a considerably higher ductility (up to 170% increase in the ultimate deflection) compared to other beams. The comparison of the experimental results of the ultimate strength of the beams with ACI440-2R guidelines indicated a reasonable and conservative prediction of the code expression.

Dynamic analysis of RC beams externally bonded with FRP plates using state space method

The state space method is used to investigate the in-plane linear dynamic behavior of reinforced concrete (RC) beams externally bonded with fiber-reinforced polymer (FRP) plates. The state space formulae are properly set up by selecting the appropriate generalized displacements and their conjugated forces as state variables. By adopting the numerically stable method to solve the homogeneous state equations with respect to the mode function vectors, the frequencies and corresponding modal shapes for the free vibration of the FRP-strengthened RC beams under generalized boundary conditions are subsequently obtained. The orthogonality of the vibration modes is established based on the concept of symplectic inner product. By utilizing the modal superposition method, the analytical solutions of the inhomogeneous governing equations for forced vibration and the dynamic responses of the FRP-strengthened RC beams subjected to a uniformly distributed step load, concentrated triangular impulse load at the mid-span, moving concentrated load, and vertical seismic load are obtained. Finally, several numerical examples are presented to validate the results against those obtained by the general-purpose finite element software ABAQUS. Good agreement between the results is observed. Moreover, parametric studies are conducted to investigate the effects of the cross-sectional geometric and material properties of the FRP plate and adhesive layer on the peak interfacial shear stress at the plate end when the strengthened beam is subjected to a uniformly distributed step load or a moving concentrated load.

Effect of load distribution on the behaviour of RC beams strengthened in flexure with near-surface mounted (NSM) FRP

Debonding failures of FRP have been frequently observed in laboratory tests of reinforced concrete (RC) beams flexurally-strengthened with near-surface mounted (NSM) fibre-reinforced polymer (FRP). A number of numerical and theoretical studies have been carried out to predict debonding failures in NSM FRP-strengthened beams, and several strength models have also been proposed. The existing studies, however, were all based on the scenario of a simply supported beam tested under one or two-point loading, while the influence of load distribution has not yet been investigated. This paper presents the first ever study into the effect of load distribution on the behaviour of NSM FRP-strengthened RC beams. A series of large-scale RC beams flexurally-strengthened with NSM FRP strips were first tested under different load uniformities; then a finite element (FE) model, which can give close predictions to the behaviour of such strengthened beams, was developed; finally, the proposed FE model was utilized to investigate the influence of bond length of NSM FRP on the load uniformity effect. It was found that the load uniformity has a significant effect on the beam behaviour, and the degree of this effect varies with the bond length of NSM FRP.

Effect of Stirrups on Plate End Debonding in Reinforced Concrete Beams Strengthened with Fiber Reinforced Polymers.

Plate end (PE) debonding is one of the critical debonding failure modes that may occur in reinforced concrete (RC) beams strengthened with externally bonded fiber reinforced polymers (FRPs). This study investigated the effect of internal steel stirrups on the PE debonding failure load of FRP-strengthened RC beams. The dimensions of the beams were 3400 × 400 × 200 mm. The beams were strengthened with carbon FRP (CFRP) sheets bonded to the soffit of the beams. The beams were divided into two series based on the distance of the cutoff point of the CFRP sheets from the nearest support. This distance was 50 mm or 300 mm, and the amount of steel stirrups was varied in terms of varying the stirrup diameter and spacing. The beams were simply supported and tested under four-point bending. The test results indicate that the effect of stirrups on the load carrying capacity of the beams was more pronounced for the beams with CFRP sheets extended close to the supports. It was also indicated that beams with larger amounts of stirrups failed in PE debonding by concrete cover separation while beams with lower amounts of stirrups failed in PE by either PE interfacial debonding or critical diagonal crack-induced debonding. The beams were analyzed using several analytical models from design guidelines and the literature. The result of analysis indicates that most of the available models failed to reflect the effect of stirrups in predicting PE debonding failure load of the beams. On the other hand, the models of El-Sayed et al. and Teng and Yao were able to capture such an effect with the best predictions provided by El-Sayed et al. model.

Design of ductile H-anchorage for strengthening reinforced concrete beams with prestressed FRP

Most FRP strengthening methods increased load capacity at the cost of decreased ductility. A review of 551 specimens from 64 literatures shows that the ductility of FRP strengthened RC beams experienced an average decrease of ductility by 37.2%. To achieve balanced improvement of ductility and load capacity, a previously proposed H-anchorage is adopted to prestress FRP laminates and to strengthen reinforced concrete (RC) beams in flexure. A total of ten specimens were cast and tested under four-point bending, the test results are presented and discussed. The effects of concrete strength, bond scheme, and prestress level of FRP were investigated. The H-anchorage was found to improve the utilization of FRP in tension, achieve great increase of load capacity, and maintain satisfactory ductility. Partial bonding of prestressed FRP effectively improved the flexural performance within the elastic range, and the load capacity generally increased with increasing prestress level of FRP laminates. The increase of ductility was more prominent in the specimens using high-strength concrete than those using low-strength concrete. During the loading process, the deformed concrete section remained planar. Based on this observation, an analytical model is proposed to estimate the load capacity, reaching good agreement with the experimental results. Finally, a design method is proposed for optimizing the configuration of H-anchorage to ensure optimum performance in terms of load capacity and ductility.

An experimental approach to the influence of partial rebar corrosion and concrete degradation on behavior of FRP-strengthened RC beams

Human factors are one of the main reasons for structural damage as they decrease the bearing capacity and also lead to an inaccurate estimation of the structure. Previous studies show that the use of CFRP in the damaged structures can significantly increase their bending and shear capacity. This study examines the capacity and cracks distribution in eight RC (reinforced concrete) beams (210 × 250 × 250 cm), each of which was rehabilitated with seven CFRP (carbon fiber-reinforced polymer) strips using the strip method. Each beam, except for the control specimen, experiences different types of concrete and rebar damages, which are finally compared with those of the control specimen. The results indicated that rebar damage in all the beams was significant and the effects of concrete damage were minimized by CFRP strips. Moreover, the force–displacement diagrams indicate the greatest force for the control specimen. Other specimens reached up to 80% of the force experienced by the control specimen. Finally, the parametric study showed that the influence of the crack width on decreasing the bearing capacity was more significant compared with the other parameters.

StructuresNet and FireNet: Benchmarking databases and machine learning algorithms in structural and fire engineering domains

Machine learning (ML) continues to rise as an effective and affordable method of tackling engineering problems. Unlike other disciplines, the integration of ML into structural and fire engineering domains remains deficient. This is due in part to the lack of benchmark databases to compare the effectiveness of ML models. In order to bridge this knowledge gap, this paper presents a benchmark examination of common supervised learning ML algorithms that can be easily deployed into structural and fire engineering problems. The selected algorithms include; Decision Trees (DT), Random Forest (RF), Extreme Gradient Boosted Trees (ExGBT), Light Gradient Boosted Trees (LGBT), TensorFlow Deep Learning (TFDL), and Keras Deep Residual Neural Network (KDP), and are used with their default values to establish a proper benchmark against six databases. The compiled datasets have been thoroughly tested and span two domains, structural engineering; 1) elemental response of concrete-filled steel tubular (CFST) circular columns at ambient conditions, 2) shear response of cold-formed steel (CFS) channels with slotted webs, 3) compressive strength of concrete, 4) fatigue life data, 5) shear strength of reinforced concrete (RC) beams and FRP-strengthened RC beams; and fire engineering, 6) fire behavior of RC concrete columns in terms of spalling occurrence and fire resistance. This study also investigates a variety of commonly used performance metrics that are applicable to regression and classification-based ML problems. We invite ML users to apply their models to the presented databases to establish a benchmark by mean of external validation and then extend their models to other problems and databases. Collectively, the presented work establishes the first step towards a unified framework that can be used to accelerate the adoption of ML into structural and fire engineering domains.

Experimental investigation on behavior of FRP-strengthened RC beams subjected to combined twisting-bending moments

Even small torsional moments can produce significant stress in beams and alter the behavior of the entire structure. Although many studies on the torsional behavior of reinforced concrete (RC) beams strengthened with fiber-reinforced polymer (FRP) sheets, there are still some uncertainties regarding the behavior of these beams under combined torsional-bending loads. In practice, torsional and bending loads usually occur together, and it is widely believed that the effects of these loads interact and interfere with each other. This study investigated the torsional and bending behavior of FRP-strengthened RC beams under pure torsion and combined torsion-bending. For this investigation, 8 half-scale RC beam specimens divided into two groups of 4 with lengths of 1300 mm and 2300 mm were subjected to pure torsion and combined torsion-bending loads, respectively. In each group, three specimens were strengthened with the sheets of carbon fiber-reinforced polymer (CFRP) with three different configurations, and one beam was left unchanged to serve as the control specimen. Finally, specimens of the two groups were analyzed and compared in terms of cracking torsional strength, ultimate torsional strength, cracking twist angle, and ultimate twist angle.