Fixed-angle Softened-truss Model Research Articles

Explicit calculation method for shear bearing capacity of RC beam with rectangular section based on the FASTM

The traditional fixed-angle softened truss model (FASTM) procedure to compute the shear capacity of RC beams requires multiple iterations. The present paper analyzes the mechanical behaviours under the critical state of shear capacity of 367 RC simply supported beams with rectangular sections. Some significant regressions have been captured in the statistical analysis of these 367 beams. A straightforward method to calculate the shear capacity of RC beams with rectangular section is proposed by introducing an additional regression relation into FASTM. The calculation results corresponding to this explicit calculation method are in good agreement with the test values and traditional FASTM.

A Quick Prediction for Shear Behaviour of RC Membrane Elements by Fixed-Angle Softened Truss Model with Tension-Stiffening

A Quick Prediction for Shear Behaviour of RC Membrane Elements by Fixed-Angle Softened Truss Model with Tension-Stiffening

Application of direct tension force transfer model with modified fixed-angle softened-truss model to finite element analysis of steel fiber-reinforced concrete members subjected to Shear

Steel fiber-reinforced concrete (SFRC) is known as one of the efficient modern composites that can greatly enhance the material performance of cracked concrete in tension. Such improved tensile resistance mechanism at crack interfaces in SFRC members can be heavily influenced by methodologies of treatments of crack direction. While most existing studies have focused on developing the numerical analysis model with the rotating-angle theory, there are only few studies on finite element analysis models with the fixed-angle model approach. According to many existing experimental studies, the direction of principal stress rotated after the formation of initial fixed-cracks, but it was also observed that new cracks with completely different angles relative to the initial crack direction very rarely occurred. Therefore, this study introduced the direct tension force transfer model (DTFTM), in which tensile resistance of the fibers at the crack interface can be easily estimated, to the nonlinear finite element analysis algorithm with the fixed-angle theory, and the proposed model was also verified by comparing the analysis results to the SFRC shear panel test results. The secant modulus method adopted in this study for iterative calculations in nonlinear finite element analysis showed highly stable and fast convergence capability when it was applied to the fixed-angle theory. The deviation angle between the principal stress direction and the fixed-crack direction significantly increased as the tensile stresses in the steel fibers at crack interfaces increased, which implies that the deviation angle is very important in the estimation of the shear behavior of SFRC members.

Torsional behavior model of steel-fiber-reinforced concrete members modifying fixed-angle softened-truss model

Steel-fiber-reinforced concrete (SFRC) is an efficient cement-based composite material that can compensate for the drawbacks of the material properties of conventional concrete and has better structural performances than conventional concrete. It can improve torsional behavior as well as flexural and shear behavior. However, analysis of the torsional behavior of SFRC members is quite complicated because force equilibrium and strain compatibility in a three-dimensional space should be satisfied. Accordingly, many studies proposed empirical evaluation equations for the torsional strength of SFRC members based on experimental results. Therefore, this study derived a constitutive model of SFRC in tension, which greatly influences the torsional behavior of SFRC, based on the test results of SFRC shear panels under biaxial stress, and this tensile behavior model was introduced to a fixed-angle softened-truss model. A theoretical evaluation model based on the modified fixed-angle model for torsional behavior of SFRC was developed, and the performance of the analytical model was also evaluated compared to test results obtained from literature.

Calculation for Torsion Strength of Prestressed Concrete Beams Based on Fixed-Angle Softened Truss Model

The calculation model for torsion strength of prestressed concrete beams (PCs) under pure torsion is deduced theoretically based on Fixed-Angle Softened Truss Model (FA-STM) and an efficient algorithm for the model is proposed. The calculated values of torsion strength and the Torque-twist curves predicted by the proposed method agree well with these obtained from tests for the two specimens available in literature. These indicate that the proposed method can not only calculate the torsion strength efficiently but can also predict the load—deformation curve of the PCs.

MODIFIED FIXED-ANGLE STRUT-AND-TIE MODEL FOR HIGH STRENGTH REINFORCED CONCRETE BEAMS

Nonlinear finite element analysis was applied to various reinforced concrete beams using a set of constitutive models established in the modified fixed-angle softened-truss model (MFA-STM). The model was implemented by modifying the general-purpose program FEAPpv. The model can take account of the six important characteristics of cracked reinforced concrete: (1) the softening effect of concrete in tension-compression; (2) the tension-stiffening effect of concrete in tension; (3) the average stress-strain curve of steel bars embedded in concrete; (4) the shear modulus of concrete; (5) the aggregate interlock; and (6) dowel action. The comparison shows the aggregate interlock and dowel action can reduce the overestimation of the shear capacity of high strength reinforced beam, especially the high strength reinforced deep beam without web reinforcement. Moreover, the model is suitable for being implemented numerical procedures due its simplicity.

New algorithm for fixed-angle softened-truss model

The present paper proposes a solution methodology to predict the shear stress–shear strain curves of reinforced concrete (RC) elements (panels) using a truss model. The proposed solution procedure calculates the concrete constitutive relationships by transforming the concrete stresses and strains from the principal direction of concrete stresses to the principal direction of the applied stresses. As a result, the algorithm predicts the shear responses of RC elements having a large difference between the amounts of steel in two perpendicular directions. To check the success of the proposed methodology, the experimental shear stress–strain and shear stress–normal strain curves of 12 RC elements as well as the shear strengths of 170 beams reported in the literature are compared with their corresponding predicted results using the new algorithm of the truss model. The results show that the proposed algorithm accurately predicts the variation of the concrete strain of 12 reinforced elements and the shear strength of 170 beams.

Nonlinear finite element analysis of concrete structures using new constitutive models

Nonlinear finite element analysis was applied to various types of reinforced concrete structures using a new set of constitutive models established in the fixed-angle softened-truss model (FA-STM). A computer code FEAPRC was developed specifically for application to reinforced concrete structures by modifying the general-purpose program FEAP. FEAPRC can take care of the four important characteristics of cracked reinforced concrete: (1) the softening effect of concrete in compression, (2) the tension-stiffening effect by concrete in tension, (3) the average (or smeared) stress–strain curve of steel bars embedded in concrete, and (4) the new, rational shear modulus of concrete. The predictions made by FEAPRC are in good agreement with the experimental results of beams, panels, and framed shear walls.

Shear Behavior Prediction of Reinforced Concrete Beams by Transformation Angle Truss Model

This paper presents on the shear behavior prediction of reinforced concrete beams using Transformation Angle Truss Model (TATM). The TATM can evaluate the stress-strain relationships for cracked concrete by transforming stresses and strains for principal plane into those over the crack surfaces. This proposed analytical method simplifies the Fixed Angle Softened Truss Model (FA-STM) and removes the limitation of applicability of the FA-STM. The shear strength and strain of reinforced concrete beams are predicted by using the TATM. For the verification of proposed method, experimental results of reinforced concrete beams were compared with theoretical results by the TATM, FA-STM and Rotating Angle Softened Truss Model (RA-STM).

Rational Shear Modulus for Smeared-Crack Analysis of Reinforced Concrete

This paper presents a new rational shear modulus for the fixed-angle softened truss model (FA-STM) based on the smeared-crack concept. This rational, yet very simple, modulus not only greatly simplifies the solution of the FA-STM but also improves its accuracy. Moreover, this new shear modulus can be used widely to replace the many complicated empirical constitutive laws for shear of cracked concrete, as employed in the various nonlinear analyses of reinforced concrete structures. A notable example is the finite-element analysis.

Unified approach to shear analysis and design

The theory of reinforced concrete originated in the late 19th century with the development of the linear theory for bending. As the first rational theory in reinforced concrete behavior, it satisfied Navier's three principles of mechanics of materials: stress equilibrium; strain compatibility; and constitutive law of material. By the 1950s this linear bending theory was generalized to become the Bernoulli compatibility truss model which covered both the linear and non-linear theories for bending, axial load and their combination. For shear and torsion, however, a rational theory that satisfies Navier's principles took the entire 20th century to develop. By the 1990s the theory for shear and torsion contained four truss models: the equilibrium (plasticity) truss model; the Mohr compatibility truss model; the rotating-angle softened-truss model; and the fixed-angle softened-truss model. Together with the strut-and-tie model for local regions and the Bernoulli compatibility truss model for bending and axial load, a unified theory of six rational models for reinforced concrete was established.The two non-linear softened-truss models have capacities for predicting the behavior of reinforced concrete elements subjected to in-plane shear and normal stresses. These two models satisfy the two-dimensional stress equilibrium, Mohr's circular strain compatibility and the softened biaxial constitutive laws of concrete. As a result, they can be used to predict the strength as well as the load-deformation history of a membrane element. The rotating-angle softened-truss model is simple to use and is adequate for situation that does not require the ‘contribution of concrete’ ( V c ), while the fixed-angle softened-truss model is more complex but is capable of predicting V c . The rotating-angle softened-truss model has been used to analyze four types of structures that are subjected predominantly to shear: low-rise shear walls, framed shear walls, deep beams and shear transfer zones. In addition, both softened truss models are used to construct the concrete stiffness matrices used in the finite element analysis. These matrices contain only three moduli in the diagonal elements. The assumption that Poisson ratios are zero in the non-diagonal elements leads to undesirable limitations of applicability. To overcome this weakness, a general model of the stiffness matrix is proposed that includes two additional Poisson ratios. All five material properties in the general matrix are currently being established by new biaxial tests of panels using proportional loading and strain-control procedures.

Behavior and Analysis of 100 MPa Concrete Membrane Elements

Fourteen full-size reinforced panels (membrane elements) made of 100 MPa high-strength concrete were biaxially tested in the universal panel tester at the University of Houston. A new servocontrol system, newly fitted to the panel tester, allowed us to conduct strain-control tests and, for the first time, to correctly measure the three-dimensional (3-D) stress-strain curves of panels and the descending branches of the compressive stress-strain curves of concrete. The 3-D stress-strain curves provided new insights into the failure processes, and thus revealed the true stress-strain relationships of concrete and steel that were misinterpreted in the past using the conventional two-dimensional (2-D) stress-strain curves. Five 100 MPa concrete panels in this research were subjected to biaxial tension-compression. By comparing the behavior of these panels with those of 42 and 65 MPa tested previously, the writers were able to evaluate the effect of concrete strength on the constitutive laws of concrete in compression. Because the softening coefficient in this law was found to be inversely proportional to the square root of concrete strength (fc′), those constitutive laws obtained previously for normal strength concrete can now be generalized to include concrete strengths up to 100 MPa. The remaining nine test panels were subjected to pure shear. The load-deformation behavior of these panels was correctly predicted by both the rotating-angle softened-truss model (RA-STM) and the fixed-angle softened-truss model (FA-STM). The range of applicability of these two models previously obtained for normal strength concrete were found to be valid for high-strength concrete as well. The failure modes of high-strength concrete panels were also predicted by a theoretical failure modes diagram.

Concrete stiffness matrices for membrane elements

The concrete stiffness matrices of membrane elements used in the finite element analysis of wall-type structures are reviewed and discussed. The behavior of cracked reinforced concrete membrane elements is first described by summarizing the constitutive laws of concrete and steel established for the two softened truss models (the rotating-angle softened-truss model and the fixed-angle softened-truss model). These constitutive laws are then related to the concrete stiffness matrices of the two existing cracking models (the rotating-crack model and the fixed-crack model). In view of the weakness in the existing models, a general model of the matrix is proposed. This general matrix includes two Poisson ratios which are not clearly understood at present. It is proposed that all five material properties in the general matrix should be established by new biaxial tests of panels using proportional loading and strain-control procedures.

Nonlinear Analysis of Membrane Elements by Fixed-Angle Softened-Truss Model

Wall-type and shell-type reinforced structures, such as shear walls, box bridges, I-girders, and shell roofs, can be visualized as assemblies of membrane elements subjected to in-plane stress. The key to a rational analysis of these structures is to first thoroughly understand the behavior of reinforced membrane elements isolated from such structures. A fixed-angle softened-truss model, which assumes the cracks to be oriented in the principal compression direction of the externally applied stresses, has been proposed for nonlinear analyses of these elements. This new smeared-crack model takes into account the concrete that is produced by the shear resistance of along the initial crack direction. This contribution cannot exist in the various rotating-angle models, because they assume the cracks to be oriented in the principal compression direction of the post-cracking concrete. The fixed-angle softened-truss model is capable of predicting the entire load-deformation history, because it takes into consideration all 12 equations governing the equilibrium condition, the compatibility condition, and the constitutive laws of materials. This paper proposes an efficient algorithm for solving these 12 governing equations. This algorithm is used to analyze the behavior of reinforced panels tested at the University of Houston. The analysis shows that the new fixed-angle model is much more powerful than the prevailing rotating-angle models.

Toward A Unified Nomenclature for Reinforced-Concrete Theory

Research in the fundamental behavior of reinforced concrete has progressed greatly since the 1960s, particularly in the areas of shear and torsion. This large body of knowledge spawned many theories and created many terms that do not adhere to a centralized nomenclature—a very confusing situation for the engineering profession. This paper proposes a unified nomenclature that covers all four basic actions (bending, axial load, shear, and torsion) and is based on the angle of inclination of the concrete struts, the scope of applications, and the three fundamental principles of the mechanics of materials (stress equilibrium, strain compatibility, and the constitutive laws of materials). Six models are presented: strut-and-tie model, Bernoulli compatibility truss model, equilibrium (plasticity) truss model, Mohr compatibility truss model, softened truss model, and fixed-angle softened-truss model. This paper shows that all rational reinforced-concrete theories developed to date can be covered by these six models.

Fixed Angle Softened Truss Model for Reinforced Concrete

A softened truss model has previously been developed for reinforced and prestressed concrete membrane elements subjected to in-plane shear and normal stresses. This existing model satisfies the three principles of mechanics of materials : two-dimensional stress equilibrium, Mohr's circular strain compatibility, and the softened biaxial constitutive laws of concrete. That is, the model can predict the strength of a membrane element as well as its load-deformation history. However, this model cannot predict the contribution of concrete observed in tests, because it is based on the assumption that the direction of the cracks (and thus, the concrete struts) is inclined at the rotating angle following the postcracking principal stresses of the concrete. This paper presents a new and more general softened truss model in which the direction of the cracks is assumed to incline at the fixed angle following the principal stresses of the applied loading. This new model, although more complex, is capable of predicting the contribution of concrete. The fixed angle softened truss model requires four constitutive laws of materials. Three have been established previously for the rotating angle softened truss model (concrete in compression, concrete in tension, and steel embedded in concrete). This paper presents the fourth constitutive law relating the average shear stress of concrete to the average shear strain.