Softened Membrane Model Research Articles

Effect of concrete cover on the pure torsional behavior of reinforced concrete beams: Analytical model

Experimental data on RC members subjected to pure torsion is compiled from published studies so far. Interestingly, the past studies have concentrated largely on members with small to moderate concrete covers, with handful data on members with thick concrete cover. Widely accepted torsion models including the diagonal compression field theory (CFT), softened truss model (STM) and softened membrane model for torsion (SMMT) are verified on the domains of experimental data missing-in specimens with thick concrete cover. With increasing demand for durability design of RC structures use of thick concrete cover, say up to 75 mm, is being introduced in practice. Recent experimental investigation by the authors demonstrated that spalling of concrete cover, particularly in cases of thick covers, significantly impacts the torsional behavior of RC members. The present study uses these set of experiments to look at the response prediction capability of the existing advanced models and assess the reliability of existing concrete cover spalling theories. For cases with thick cover, the existing models either did not adequately capture the ultimate capacity or erroneously predicted the torsional behavior of the members due to the different mechanics between RC beams with thin and thick covers. Similarly, the spalling theories failed to fully explain the physically observed spalling behavior. Guided by the recent experimental observations and the apparent gaps, the study provides a rational theory for the spalling of concrete cover. The initiation and gradual evolution of concrete cover spalling is traced by observing the acting spalling moment and spalling resistance formulated in the proposed spalling model. The proposed spalling model inherently assumes members susceptibility to spalling of cover concrete and is governed by, the thickness of the concrete cover, tensile strength of concrete, presence of rebar cage and their location, and size of the member. The theory is unified and can explain the spalling of cover due to torsion as well as shear. Its capability is examined by integrating the approach into existing truss model. When compared with state-of-the-art models, the proposed method provides consistent prediction with a relatively smaller scatter, for cases with thick concrete cover as well.

New OpenSees material model for simulating reinforced concrete shear walls subjected to coupled axial tension and cyclic lateral loads

In high-rise buildings, reinforced concrete (RC) shear walls, particularly those forming part of coupled wall systems or core wall systems, may be subjected to coupled axial tension and cyclic lateral loads during strong seismic events. A new two-dimensional material model named the “Fixed Angle Model Considering Crack Sliding” (FAM-CS) was proposed for simulating the nonlinear behavior of RC walls subjected to axial tension and cyclic lateral loads. The proposed model was implemented in the OpenSees platform. Nonlinear constitutive models for shear aggregate interlock effects of concrete and dowel action of rebars were incorporated into the proposed model to represent the shear transfer mechanisms along concrete cracks. The proposed model was validated against the test data of six wall specimens subjected to coupled axial tension and cyclic lateral loads. The results indicated that the proposed model reasonably predicted the lateral strength capacity (an average error of 6.2 %) and the residual strength at the maximum drift loading (an average error of 12.3 %) of the specimens. For comparison, three other commonly used models, the Shear-Flexure Interaction Multiple-Vertical-Line-Element-Model, the multi-layer shell element model and the Cyclic Softened Membrane Model were adopted for the simulation of the specimens. Compared with the three existing OpenSees models, the newly implemented material model provided improved predictions of the wall hysteretic behavior with respect to lateral strength capacity, lateral strength degradation, pinching characteristics and cyclic stiffness degradation. In addition, the proposed model reasonably captured the localized failure pattern as well as the energy dissipation characteristics of the test specimens. Finally, a mesh sensitivity study was conducted to assess the robustness of the proposed model. The results revealed that the proposed model was robust regarding the lateral load-drift responses of RC walls, with a mesh size ranging from approximately one to three times the average crack spacing.

Softened membrane model for shear of ultra-high-performance concrete (SMM-UHPC)

Understanding the shear behavior of Ultra-High-Performance (UHPC) is crucial to prevent undesired responses that could be induced by diagonal shear cracks. Hence, the ability of analytical models to predict shear responses of UHPC members is of paramount importance, which is addressed in this paper by the Softened Membrane Model (SMM). Since the SMM is a two-dimensional shear modeling framework developed for reinforced concrete, a parametric investigation was conducted to assess its suitability for UHPC, namely SMM-UHPC. Through this assessment, a softening coefficient for the biaxial behavior of UHPC was developed and new constitutive laws were integrated in SMM. Comparisons with shear panel tests were used to evaluate the accuracy of SMM-UHPC and quantify the significance of key model parameters. Sensitivity analyses showed that the decomposition between biaxial and uniaxial strain fields, which is used in SMM to capture the Poisson effect, has a negligible influence on UHPC shear. Combining experimental and analytical data, the tensile localization strain was identified to be of most significance for predicting the shear stress-strain response of UHPC.

Experimental Study and Finite Element Modelling of Squat Shear Walls under Combined Cyclic Loads and High Axial Loads

Experimental observations on three reinforced concrete shear walls with small shear span-to-depth ratio (SDR) under combined high vertical axial load and horizontal cyclic loads are presented. The influence of high axial load ratio (ALR) on the failure mode, hysteretic behaviour, displacement ductility, shear strength and stiffness of the squat shear walls is investigated. In addition, a novel built-in strain gauges measuring system is employed for measuring the strain conditions in the reinforcements during the whole test process. Test results indicate that high axial load restrains the development of cracks and improves the shear load capacity, but that it also decreases ductility and energy dissipation and aggravates stiffness degradation. Concrete crush and out-of-plane buckling were observed in all specimens, resulting in the final failure of the specimens. According to the strain analysis, the section of the squat walls coincided well with the assumption of plane section under the condition of high ALR. With the increase of ALR, the depth of the compression zone of members increases, while the length of plastic hinge decreases. When the axial load is relatively small, the vertical and horizontal reinforcements provided almost equal contribution to the shear capacity of squat shear walls. However, under extremely high axial load, both vertical and horizontal reinforcements cannot provide full contribution to the shear capacity. The hysteretic behaviours of the tested shear walls were simulated by a cyclic softened membrane model (CSMM). Simulation results indicate that CSMM captured well the nonlinear characteristics of the squat shear wall under high axial load.

Experimental investigation of shear-extension coupling effect in anisotropic reinforced concrete membrane elements

Performance based analysis under seismic loads using the finite element method for wall-type reinforced concrete (RC) members in buildings and in important structures like liquid retaining structures, nuclear containment structures, offshore concrete gravity structures etc., necessitates the understanding of the non-linear behaviour of the constituent membrane elements. The current orthotropic formulation of the softened membrane model (SMM) can be strictly used only when the reinforcement is symmetric to the principal axes of applied stresses. When the reinforcement is asymmetric, shear strain is generated due to the normal stresses in the principal axes of applied stresses, which is referred to as shear-extension coupling. An anisotropic formulation is required to capture the generated shear strain. The current study quantifies the shear strain due to asymmetry in reinforcement, by testing panels under biaxial tension-compression using a large-scale panel testing facility. A model for the shear strain is proposed based on the tests data. The paper presents the experimental programme, important test results and the modelling of shear strain. Expression developed for the shear strain can be incorporated in the solution algorithm of the SMM for improved prediction of the shear behaviour of a membrane element. This further aids in accurate prediction of the seismic performance of the important structures mentioned earlier.

Modified softened membrane model for ultra-high-performance fiber-reinforced concrete solid and hollow beams under pure torsion

Limited research has focused on the development of rational theoretical models for evaluating the pure-torsional behavior of ultra-high-performance fiber-reinforced concrete (UHPFRC) beams. To fill this research gap, this study aimed to develop a modified softened membrane model for torsion (SMMT). Based on the direct tension force transfer model, a UHPFRC tensile constitutive model was derived by considering the fiber pull-out behavior. The proposed constitutive model was implemented in the SMMT, and the modified SMMT was found to be useful in predicting the pure-torsional behavior of UHPFRC beams with solid and hollow cross-sections. Moreover, the theoretical torsional responses were compared with the experimental results, and the experimental data reported in previous studies were used to verify the applicability of the proposed model. The results indicate that the estimated analytical responses matched well with the experimental outcomes, and the modified SMMT considered the influences of both the wall thickness and excellent UHPFRC material properties. The proposed model is expected to be effective for further investigation of the torsional mechanism and design methodology of UHPFRC members under pure torsion.

Simplified biaxial tension stiffening model for the analysis of shear panels using softened membrane model

Many of the existing tension stress–strain relations for concrete used in smeared shear models consider the contribution of concrete in tension even after steel yielding. Although the contribution of concrete in tension is less significant than its contribution in compression, the use of these existing models often leads to overestimating the shear response of reinforced concrete (RC) panels. An improved concrete tension stiffening model incorporating the effect of biaxial stresses is proposed in this study. The effect of tension stiffening on the behaviour of shear panels under normal and shear stresses is investigated by incorporating tension stiffening in the softened membrane model (SMM). The existing SMM algorithm can be used to estimate the behaviour of RC panels subjected only to pure shear. In the proposed analysis using SMM, a new secant-based algorithm is used to estimate the behaviour of RC panels under combined normal and shear loads. Estimates from the proposed model are compared with estimates from original SMM and experimental results. The comparison shows that the proposed model provides better estimates of shear response for RC panels that undergo high shear strains than the original SMM. Furthermore, the proposed secant-based algorithm estimates panel behaviour under combined loading in reduced computational time.

A softened membrane model for single-box multi-cell composite box-girders with corrugated steel webs under pure torsion

To analyze the nonlinear torsional behavior of single-box multi-cell prestressed concrete (PC) or reinforced concrete (RC) composite box-girders with corrugated steel webs (BGCSWs), a new theoretical model called unified softened membrane model for torsion (USMMT) is proposed in this study. It is developed from the softened membrane model for torsion which was formerly proposed for the torsional analysis of single-box single-cell PC/RC composite BGCSWs. This proposed model incorporates the different contributions of inner and outer corrugated steel webs (CSWs) in the multi-cell BGCSW to the torsional capacity by introducing a rational relationship of shear strains between inner CSWs and outer CSWs. The satisfactory accuracy of the proposed model is obtained when comparing with the experimental and the finite element analysis results, including the torque-twist curves and the smeared shear strains in concrete slabs and CSWs in the full-range torsional analysis. The comparison indicates that the proposed model is capable of predicting the overall torsional behavior of single-box multi-cell BGCSWs.

Predicting the behavior of FRP-strengthened RC membrane elements with efficient rotating-angle softened truss model procedure

This article presents an analytical model, based on a refinement of the rotating-angle softened truss model (RA-STM) with efficient solution procedure, to predict the full response of reinforced concrete (RC) membrane elements strengthened with fiber reinforced polymers (FRP). To extend the RA-STM, equations from equilibrium conditions and smeared constitutive relationships for the materials are modified in order to account for the tensile FRP reinforcement and its interactions with the other material components. In addition, an efficient algorithm is proposed for the calculation procedure to avoid using the classical trial and error technique to compute the solution points. This new algorithm provides higher numerical efficiency and stability. The reliability of the efficient RA-STM FRP solution procedure is checked against some experimental data related with FRP-strengthened RC panels tested under in-plane shear and found in the literature, and also with the predictions from the softened membrane model (SMM-FRP) for comparison. In general, reasonably good agreement is observed between the efficient RA-STM FRP procedure and the SMM-FRP, and also with the experimental response of the reference test panels.

Mixed formulation for geometric and material nonlinearity of shear-critical reinforced concrete columns

This study presents a new beam element formulation following a Hellinger-Reissner variational principle for shear critical reinforced concrete members, and considering large displacement and rotation effects. The corotational formulation is used to describe the large displacement at the element nodal level and degenerated Green-Lagrange strain measures are used at the basic element level. A new displacement shape function has been developed considering section kinematics and compatibility conditions to satisfy stability criteria. New explicit equations for the internal geometric stiffness matrix to consider large deformation effect has been developed. A robust state determination algorithm for the large deformation mixed-based formulation is proposed. The Timoshenko beam theory has been adopted to consider shear deformations in deriving the section kinematics equations. The multi-axial stress state in reinforced concrete fibre has been simulated through the fixed crack smeared softened membrane model. The developed shear fibre beam element is capable of accurately reproducing the experimentally-observed large displacement effect on the post-peak shear strength degradation in the softening region. The accuracy and efficiency of the mixed-based formulation was evaluated by examining the responses at local and global levels for both monotonic and reverse cyclic loading conditions along with the process of reproducing experimentally-observed failure modes.

Nonlinear shear analysis of I-shaped beams with arbitrary inclination angles of shear reinforcement

We studied the effect of the inclination angle of stirrups and wire mesh on the web-crushing strength of reinforced concrete I-beams beams. Experimental studies were carried out in Kuban State Technological University. The variable parameters of the tested specimens included concrete compressive strength, level of prestressing, cross-sectional shape, shear span, shear reinforcement ratio and angles of inclination of stirrups or wire mesh. The strength of beams was calculated with the Cyclic Softened Membrane Model for reinforced concrete (CSMM), developed at the University of Houston and implemented in the OpenSees software package. The modeling of the web of the beams was fulfilled with quadrilateral finite elements with the assigned material FAReinforcedConcretePlaneStress. The flanges of the beams resisting the bending moment were modeled with nonlinear volumetric elements having fiber section. Uniaxial Kent-Scott-Park concrete material and uniaxial bilinear steel material were assigned to the flanges. Calculation results are close to experimental data having wide variation of the materials properties and structural parameters of beams.

Robustness evaluation of CSMM based finite element for simulation of shear critical hollow RC bridge piers

PurposeThe purpose of this study is to evaluate the effectiveness of two-dimensional (2D) cyclic softened membrane model (CSMM)-based non-linear finite element (NLFE) model in predicting the complete non-linear response of shear critical bridge piers (with walls having aspect ratios greater than 2.5) under combined axial and reversed cyclic uniaxial bending loads. The effectiveness of the 2D CSMM-based NLFE model has been compared with the widely used one-dimensional (1D) fiber-based NLFE models.Design/methodology/approachThree reinforced concrete (RC) hollow rectangular bridge piers tested under reversed cyclic uniaxial bending and sustained axial loads at the National Centre for Research on Earthquake Engineering (NCREE) Taiwan have been simulated using both 1D and 2D models in the present study. The non-linear behavior of the bridge piers has been studied through various parameters such as hysteretic loops, energy dissipation, residual drift, yield load and corresponding drift, peak load and corresponding drift, ultimate loads, ductility, specimen stiffness and critical strains in concrete and steel. The results obtained from CSMM-based NLFE model have been critically compared with the test results and results obtained from the 1D fiber-based NLFE models.FindingsIt has been observed from the analysis results that both 1D and 2D simulation models performed well in predicting the response of flexure critical bridge pier. However, in the case of shear critical bridge piers, predictions from 2D CSMM-based NLFE simulation model are more accurate. It has, thus, been concluded that CSMM-based NLFE model is more accurate and robust to simulate the complete non-linear behavior of shear critical RC hollow rectangular bridge piers.Originality/valueIn this study, a novel attempt has been made to provide a rational and robust FE model for analyzing shear critical hollow RC bridge piers (with walls having aspect ratios greater than 2.5).

FE simulation of cylindrical RC containment structures under reserved cyclic loading

The nuclear containment structure is one of the most important infrastructure systems ensuring the safety of a nuclear power plant. The structural behavior of a cylindrical containment structure made of reinforced concrete (RC) with large dimensions and numerous rebars is complex and difficult to predict. The complex behavior of the RC containment structure has been investigated in an international collaboration project between the National Center for Research on Earthquake Engineering (NCREE) in Taipei, Taiwan and the University of Houston (UH), Houston, Texas. At NCREE two 1/13 scaled cylindrical RC containment specimens were tested under reversed cyclic loads [1]. At UH, a finite element simulation of the two tested specimens was developed using a finite element analysis (FEA) program SCS [2]. In the program, a new shell element, the so-called CSMM-based shell element, was developed based on the Cyclic Softened Membrane Model [3] and the formulation of an 8-node Serendipity curved shell element [4] with a multi-layer approach [5]. The UH simulated seismic behavior was close to the NCREE experimental results. This paper presents the theoretical development of the FEA program SCS and the comparisons of its predictions with the experimental structural behavior of the two RC containment specimens. This simulation model and the FEA program are excellent tools to develop effective performance-based design provisions.

Improved softened membrane model for prestressed composite box girders with corrugated steel webs under pure torsion

In this study, a theoretical model called the improved softened membrane model for torsion (ISMMT) is proposed for evaluating prestressed composite box girders with corrugated steel webs (PCBGCSWs) under pure torsion. First, considering the mechanical properties of PCBGCSWs, a series of equations are derived from equilibrium and compatibility conditions and constitutive relationships of the materials. In the equations, the criteria of shear flow equivalence and shear strain equivalence are adopted to determine the relationships between the concrete flanges and the corrugated steel webs (CSWs) before and after the yielding of the CSWs, respectively. Then, the solution algorithms, including a general algorithm for the entire torsional process and a simplified algorithm for the stage when both steel bars, the prestressing steel and the CSWs are in the elastic state, of the ISMMT are presented to solve all the equations. To validate the feasibility of the proposed theoretical model, the experimental results of a prismatic PCBGCSW under pure torsion available in the literature are selected for comparison. Additionally, a three-dimensional (3D) finite element (FE) model of the specimen is established using ABAQUS software. The results from the theoretical model, including the overall torque-twist curve, shear strain in the CSWs and strains in the prestressing steels and steel bars, are compared with those from the FE model and the experiment. The theoretical, numerical and experimental results are in good agreement, demonstrating that the proposed analytical model accurately predicts the torsional behaviour of the PCBGCSWs under pure torsion, including the torsional behaviour during the pre- and post-cracking stages.

A softened membrane model for prestressed concrete composite box girders with corrugated steel webs under pure torsion

Torsion can be regarded as a principal factor in some cases, such as in curved girders and eccentrically loaded girders, when conducting the structural analysis of prestressed concrete composite box girders with corrugated steel webs. Recently, a rational model, called the softened membrane model for torsion, was proposed for the torsional analysis of reinforced concrete members; thereafter, this model was extended to prestressed concrete members under pure torsion and called softened membrane model for torsion prestressed concrete. This article presents a modified model, the softened membrane model for torsion prestressed concrete for prestressed concrete composite box girders with corrugated steel webs, to analyze full torsional behavior. To build the model, the softened membrane model for torsion in reinforced concrete members is first extended to perform the torsional analysis of prestressed concrete composite box girders with corrugated steel webs by incorporating the torsional contribution of corrugated steel webs. Afterward, the initial stresses and strains due to prestressing are considered to extend the softened membrane model for torsion to softened membrane model for torsion prestressed concrete for prestressed concrete composite box girders with corrugated steel webs by modifying the equilibrium equations, convergence criteria, and constitutive laws of materials. The modified model is validated by experimental data and is proven to be capable of predicting the overall torque–twist curve, especially the precracked branch and postcracked ascending branch. In addition, a comparison between the softened membrane model for torsion and softened membrane model for torsion prestressed concrete indicates that the torque values before and after concrete cracking will be overestimated and underestimated, respectively, without considering the effect of the initial stresses and strains. Finally, another comparison shows that the softened membrane model for torsion prestressed concrete is superior to the rotating-angle truss model for torsion in its ability to predict the precracked branch of the torque–twist curve.

A softened membrane model for composite box-girders with corrugated steel webs under pure torsion

The softened membrane model for torsion (SMMT), formerly developed for the torsional analysis of reinforced concrete (RC) members, has the capability of modeling the biaxial strains of concrete with Poisson effect, the tension stiffening effect of concrete and the contribution of concrete in shear. It is extended to the composite box-girders with corrugated steel webs under pure torsion in this study. The extended SMMT for RC composite box-girders with corrugated steel webs incorporates the contribution of corrugated steel webs into the SMMT for RC members by applying the assumption of shear strain relations between concrete slabs and corrugated steel webs. Moreover, the thickness of shear flow zone in concrete slabs is reasonably corrected. Besides, the modifications of equilibrium equations and convergence criteria are conducted for the prestressed concrete (PC) composite box-girders with corrugated steel webs. The modified SMMT for RC/PC composite box-girders with corrugated steel webs is able to predict the overall torsional behavior, including the torque-twist curve and the smeared shear strains in concrete slabs and corrugated steel webs, as well as any other related variables. The accuracy and superiority of the proposed model are verified by comparing the theoretical predictions from the SMMT with the experimental results and the predictions from existing model.

Assessment and enhancement of the behavior of lightly RC wall elements subjected to pure shear

Observed damage from the past earthquakes has indicated that several structural walls before the 1970s are non-seismically designed and performed poorly, exhibiting unexpected failure modes such as shear failure due to inadequate detailing and/or poor confinement of concrete core. The shear behavior alone is complex, because it influenced by a host of parameters often interdependent. This paper aims to assess and enhance the non-linear behavior of lightly reinforced concrete wall or panel elements subjected to pure shear. The existing softened membrane model was chosen as a basic model to evaluate the structural behavior and the work strategy was given as follows: first, identification of the range in which reinforced concrete panels were considered as lightly reinforced elements, based on the non-ductile behavior, Experimental data of reinforced concrete panel element A1 conducted by Pang and Hsu and the recommendation of Euro code 2. Next, the influence of the following design parameters on the shear behavior of lightly reinforced panel elements was investigated: the steel grid orientation; the diameter of longitudinal reinforcement, the type of reinforcement, and the crack angle; Finally, to enhance the shear performance, one case that reinforced with ribbed bars exhibited a non-ductile behavior has been selected to strengthen it with one, two, or three layers of fiber-reinforced polymer as an alternative to the traditional methods, using the so-called “softened membrane model-FRP”.

Effect of load history on shear strength of reinforced concrete I-Beams

A series of 32 reinforced concrete I-beams was tested to investigate the effect of existing inclined cracks on shear strength. Testing of beams consisted of two stages. At the preliminary stage, load large enough to cause a formation of a developed crack pattern in the web was applied with the primary shear span-depth ratio. After that, the load with breaking shear span-depth ratio was applied until failure of the specimen. The load applied during the preliminary loading stage was maintained with a full or partial value by a lever, or removed. The test variables include shear span, shear reinforcement ratio and angle of stirrups inclination. Pre-cracking was found to decrease the ultimate shear strength of beams by a maximum of 16 percent depending on the angle of stirrups. A corresponding reduction factor is suggested to the formula of the Russian code for the design of bridges. The finite element simulation of beams was fulfilled by using the program OpenSees. The webs of beams were simulated with a 2D element built on the Cyclic Softened Membrane Model to take into account the deterioration of concrete due to the previous loading. The accuracy of the proposed FE model was verified with the aforementioned tests and the results of several other studies.

Finite element model for predicting the shear behavior of FRP-strengthened RC members

The shear behavior of FRP strengthened reinforced concrete (FRP strengthened RC) membrane elements can be predicted by developing logical models that satisfy the principles of mechanics of materials namely stress equilibrium, strain compatibility, and constitutive relationships of concrete, steel and, FRP reinforcements. The Softened Membrane Model (SMM), which was developed for predicting the shear behavior of reinforced concrete (RC) membrane elements, is extended to FRP strengthened RC members subjected to shear. This new analytical model, referred to as the Softened Membrane Model for FRP strengthened RC members (SMM-FRP), considers new constitutive laws for each material component of the member. Similar to the case of the SMM model for RC, this new SMM-FRP model can predict the entire stress-strain curve, including pre- and post-cracking, and the ascending and descending branches. The SMM-FRP is implemented into an OpenSees-based finite element program for a membrane 2-D element that will allow structural engineers to predict the monotonic responses of FRP strengthened RC members subjected to shear. The developed program is validated in this paper by the prediction of the monotonic responses of 10 FRP strengthened RC panels subjected to pure shear stresses. The good agreement between the experimental and analytical results confirms the validity of the analytical model in predicting the shear behavior of RC members strengthened with FRP sheets.

Development of CSMM-based shell element for reinforced concrete structures

Reinforced concrete shell structures have been widely used in a variety of modern engineering applications. It is found from earthquake reconnaissance that reinforced concrete (RC) shell structures, such as nuclear containments, cooling towers, roof domes, shear walls, etc., are the key elements in resisting earthquake disturbances. This paper presents the development of a finite element analysis (FEA) program, SCS-3D, to predict the inelastic behavior of RC shell structures. In the program, a Cyclic Softened Membrane Model (CSMM)-based shell element is developed based on the degenerated shell theory with a layered approach and taking into account the CSMM developed at the University of Houston. To form the FEA program, the constitutive relation modules and the analysis procedure were implemented into a finite element program development framework, OpenSees developed at UC Berkeley. Several large-scale structural tests were employed to validate the developed FEA program, including RC panels subjected to a combination of shear and bending, three-dimensional RC shear wall and cylindrical RC tanks subjected to reversed cyclic loading.