Rigid Body Spring Model 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.

Numerical evaluation of the internal fracture mechanism of the single PBL shear connector considering the effects of position and strength of the transverse rebar

AbstractThe perfobond (PBL) shear connector reinforced with transverse rebar effectively contributes to sufficient ductility, bearing capacity, and anti‐fatigue behavior to hybrid structures. The construction practice involves the placement of the transverse rebar at the bottom position within the concrete dowel, whereas past studies employed transverse rebar in the center of the dowel; hence, there exists a gap between literature and design practices. Furthermore, there is no specific information in the design guidelines for hybrid structures on the influence of the position of the transverse rebar. Therefore, the current research deals with the analytical investigations employing the coupled Rigid Body Spring Model (RBSM) and the nonlinear solid Finite Element Method (FEM) that encompass the internal fracture behavior of concrete with the various parameters of transverse rebar, particularly the position and strength, to recognize the failure mechanism. It was determined that the numerical model accurately captured the test shear capacity and failure modes. The internal failure process of a specimen with transverse rebar at the bottom position, identical to actual field practice, revealed that as the height of the concrete region above the rebar rises, more compressive stresses in a trapezoidal manner are being generated, leading to an enhancement of the deformation capacity. Furthermore, it was confirmed that the specimen with the transverse rebar inserted at the bottom generates the greatest axial force, which is followed by the greatest shear capacity.

Meso-scale simulation of moisture transport in concrete during wet-dry cycles using 3D RBSM conduit model with variable diffusion model

This paper introduces the 3D RBSM Conduit model, an improved simulation program based on the Rigid Body Spring Model (RBSM), designed to accurately depict the intricate phenomenon of moisture transfer within concrete. This is achieved by incorporating an expanded diffusion model into the simulation framework. The diffusion coefficient is dynamically adjusted at each location and at each time step to accurately represent the precise moisture transfer processes occurring in the concrete. This dynamic adjustment allows the program to efficiently simulate the rate of moisture transport in concrete based on the moisture degree and the dry or wet state of the material. By adapting the diffusion coefficient to the changing conditions in this way, the program achieves simulated moisture transfer rates that align closely with the real-world behavior of moisture in concrete. It offers a dynamic model of the changes in concrete transmission during the switch from drying and wetting conditions. Further, the program exhibits inherent advantages in addressing water transfer problems involving nonlinear boundary conditions and transfer equations. In this study, the program is used to simulate various processes including drying, wetting, local wetting, and combined wet-dry cycles. The results reveal the lag in wet-dry transitions between the interior and exposed surface during the wet-dry conversion process due to retained interior moisture. The simulation captures the moisture variation in the transitional zone.

Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response

This research presents a novel approach to modeling fracture propagation using a discrete lattice element model with embedded strong discontinuities. The focus is on enhancing the linear elastic response within the model followed by propagation of fractures until total failure. To achieve this, a generalized beam lattice element with an embedded strong discontinuity based on the kinematics of a rigid-body spring model is formulated. The linear elastic regime is refined by correcting the stress tensor at nodes within the domain based on the internal forces present in lattice elements, which is achieved by introducing fictitious forces into the standard internal force vectors to predict the right elastic response of the model related to Poisson’s effect. Upon initiation of the first fractures, the procedure for the computation of the fictitious stress tensor is terminated, and the embedded strong discontinuities are activated in the lattice elements for obtaining an objective fracture and failure response. This transition ensures a shift from the elastic phase to the fracture propagation phase, enhancing the predictive capabilities in capturing the full fracture processes.

Mesoscopic evaluation of the bond behavior of concrete with deformed rebar subjected to passive confinement employing 3D discrete model

The bond between concrete and steel is one of the most critical properties of reinforced concrete and affects the overall structural performance. The current research realistically evaluates the macroscopic bond behavior, quantitatively highlights the detailed internal fracture process, and clarifies the internal fracture mechanism. The numerical analyses are conducted for cover confinement effects employing the coupled Rigid Body Spring Model (RBSM) and the nonlinear solid Finite Element Method (FEM) model with varying cover thicknesses and rebar diameters. The modeling of concrete and deformed rebar considering the actual geometrical features is performed employing 3D-RBSM and solid FEM, respectively. It has been determined that the mesoscopic numerical model adequately simulates the test stress-slip relationships, demonstrates the increasing tendency of peak stress and peak slip against the increase in cover thickness, and reproduces the post-peak softening. The model enables simulation of splitting failure with a small cover thickness and the transformation of failure into pull-out and showed the enhancement of peak stress and ductility with a large cover, while reproducing the splitting as well as pull-out for medium cover. Furthermore, the model accurately reproduces the different bond and internal crack propagation caused by variations in bearing stress and radial pressure against different rebar diameters. The model precisely captured the radial fractures arising from the ribs, such as Goto cracks, which is not possible using conventional one-dimensional beam elements to represent the reinforcing bar.

Investigating equivalency of different element models in discrete mesoscale analysis of ASR-damaged concrete

The alkali-silica reaction (ASR) is one of the most serious durability issues affecting the safety of concrete structures. Numerical simulation is useful for monitoring the internal condition of an RC structure with ASR damage and predicting its long-term behavior. A mesoscale three-dimensional rigid body spring model (3D RBSM), a type of discrete analysis model, has previously been developed to simulate ASR expansion by modeling aggregate and mortar separately. The element size in simulations using this model (known as the aggregate model) is 2–3 mm, so the number of elements is huge and computation time correspondingly long. An equivalent model with lower number of elements is required, so in this study a concrete model based on RBSM is proposed. The aggregate is not separately modeled and expansive elements are included to simulate ASR expansion, allowing a larger element size (1–2 cm). The number of elements and computation time are less than 1 % compared with the aggregate model. This proposed model is used to simulate free ASR expansion, ASR expansion under confinement, and the mechanical properties of ASR-damaged concrete, as well as pullout behavior after ASR expansion. The simulation results are compared against previously reported results obtained using the aggregate model, as well as against experimental results. This comparison confirms equivalency between the concrete model and the aggregate model, opening the possibility of simulating ASR damage in RC structures in the future.

Simulation of moisture variations in concrete during prolonged alternate wetting and drying cycles using 3D discrete network model

The precise prediction of moisture variations within concrete is of great significance because various deterioration phenomena spanning corrosion to carbonation are induced by mass transfer facilitated by the presence of moisture. Under actual environmental conditions, concrete is exposed to numerous wetting and drying cycles. Thus, to accurately predict moisture variations it is essential to consider the effect of these conditions. This study investigates the effect of alternate wetting and drying cycles on moisture penetration through concrete at the mesoscale using a 3D discrete network model based on a Rigid Body Spring Model (RBSM). For modeling moisture transport during wetting, the non-linear diffusion equation is used and the effects of capillary suction in moisture transport are also considered. On the other hand, moisture transport during drying is modeled as an evaporation-diffusion process. The simulation results obtained using this discrete network model are validated against the available experimental outcomes. During the wetting cycle, the moisture content in regions close to the exposed surface rapidly increases and approaches the maximum saturation. The effect of subsequent drying is seen only until a certain depth below the exposed surface, known as the influential depth. It is also observed that moisture continues to penetrate deeper into a specimen even during a drying cycle.

Evaluation of seismic performance of aged massive RC wall structure restrained by adjacent members using RBSM

Aging management and maintenance are essential processes to ensure the safety of reinforced concrete (RC) buildings in nuclear power plants (NPPs). This is because several phenomena induce cracking behavior and mechanical property changes (MPC) in concrete that might change the structural performance and durability of massive RC structures. Although massive RC structures have not been encountered with excessive loading or severe environmental conditions, structural performance and durability have changed owing to cracks regarding restraint volumetric changes (e.g., thermal and drying-shrinkage cracks) and MPC of concrete throughout the service life. This study numerically investigated the whole history of massive RC structures observed under the general conditions of NPPs in Japan. The alternation in seismic performance according to the whole history of the structure was investigated using a rigid body-spring model. The impacts of restrained cracks, the interaction with structural cracks, and the MPC of concrete were determined to clarify their significance, which is generally neglected for the design and inspection process in the engineering practice.

Reproducible estimations of internal corrosion distribution from surface cracks using MPC-RBSM

Rebar corrosion estimation is essential for maintenance of reinforced concrete structures against deterioration. Currently available non-destructive methods of evaluating corrosion do not clearly indicate the internal corrosion condition along the rebar and its effect within the concrete. In prior research, an inverse analysis scheme has been proposed in which the model predictive control (MPC) optimization method is utilized within a rigid body spring model (RBSM) to estimate the corrosion-induced expansion distribution along the rebar from corresponding surface crack width data. In this study, an innovative approach is introduced to verify the precision of MPC-RBSM outcomes and systematically examine the major contributing factors. To facilitate this verification, target cracking is generated by precisely controlling the corrosion expansion within the RBSM framework. In total, three sets of simulations are considered: two cases with varied expansive strain distribution profile along a straight rebar, and third in which a bent rebar arrangement is subject to corrosion. The simulated expansion distribution is compared against the original expansion used to generate the crack width data to validate the reproducibility of the MPC-RBSM results. The inverse analysis results show that the automated expansion scheme in MPC-RBSM successfully reproduces the target surface cracking. In the inverse analysis process, the necessity to account for corrosion distribution is identified. When appropriate corrosion distribution criterion is used to estimate corrosion-induced expansive strain levels along the reinforcement, the results are in good agreement with the assumed corrosion-induced expansive strain profiles (including both uniform and non-uniform profiles). Additionally, simulations of bent rebar corrosion are performed. The results indicate that surface cracking local to the bent region is mainly influenced by corrosion-induced stresses along the straight lengths of the rebar, which complicates the corrosion analysis of bent rebars via this form of inverse analysis.

Energy-based approach to predict thermal cracking characteristics of a massive RC member subjected to external restraint from its bottom

Thermal cracking is a crucial problem in massive reinforced concrete structures, particularly in terms of durability. Although several standards and numerical studies have been used to evaluate thermal crack width and number of cracks, quantitative evaluation remains challenging. This paper proposes a robust prediction model that can be used to theoretically evaluate thermal cracks using a simplified approach. A parametric study based on a rigid-body–spring model (RBSM) was conducted to clarify the factors affecting the thermal cracking characteristics of massive concrete members. The proposed energy-based prediction model based on the information from the parametric study reasonably evaluates the thermal cracks observed in an existing experiment. The crack width distribution in the cracked section was also investigated to define the equations for the gradient of the thermal crack width, which is important for evaluating the crack width on the surface with respect to durability.

The effects of various parameters in sensitivity analysis of plain concrete beam using rigid body spring model

The Rigid Body Spring Model (RBSM) is an alternative method to Finite Element Model (FEM) to solve continuum problems in civil engineering. The method has three main advantages, i.e., having a smaller local stiffness matrix, the ability to directly simulate cracks, and independent formulation of element types. Although the method has been extensively used in studying the behaviour of concrete structures, specific research on sensitivity analysis of this method has not been done nor published. In this research, sensitivity analysis of plain concrete beam using the model is performed. The parameters to be considered are boundary size, boundary element type, number of elements, crack mesh refinement, number of stages, and the effect of half modelling. Based on the sensitivity analysis, optimum parameters were selected to calibrate the model to three plain concrete samples. The result suggests that modulus elasticity of cylindrical concrete under compressive test is to be reduced around 0.2 times for simulating beam under third point loading.

Mesoscale modelling of SFRC based on 3D RBSM considering the effects of fiber shape and orientation

In this study, a mesoscale model of steel fiber reinforced concrete (SFRC) based on 3D RBSM (rigid body spring model), was proposed to take the fiber shapes (straight and hook end) and orientation into account in concrete. The steel fibers were distributed into the Voronoi mesh system of RBSM, and a zero-size spring was assigned at the interface of Voronoi mesh crossed by discrete fiber, to transfer the fiber pullout load to the nearest nodes of RBSM. The straight steel fiber was modeled based on proposed local bond-slip relationship and the mechanical hook action was proposed additionally to model the hook end fiber. The model can predict the macroscopic response and cracking of SFRC made with straight and hook end fibers under direct tensile and bending loads. The analysis results found the mechanical hook action as the governing parameter for hook end fiber then the bond-slip properties for straight fiber. The fiber orientation and matrix spalling due to friction induced pullout load component, were found prominent for steel fiber. The anisotropy induced by fiber shapes and orientations was headed by hook end fiber with superior macroscopic load capacity by delaying the crack propagation and transformation.

RBSM-based mesoscale study of frost deterioration for recycled concrete considering air-entrainment in old and new mortar

To describe or predict the frost damage of recycled aggregate concrete (RAC), one should consider the material details of both old mortar and new mortar, especially the air entrainment. In this study, a freezing pressure model with entrained air is embedded into Rigid Body Spring Model (RBSM) to simulate degradation of RAC during freezing-thawing cycles (FTCs). First, quantitative models on frost damage of RAC are developed, and basic concepts of RBSM are introduced. Then the deformation of mortar affected by air entrainment is discussed and RBSM-based model for normal concrete is verified. Furthermore, RAC models are established with the method of RBSM, and the simulated frost damage results are compared with experimental results to validate this model. Finally, the validated model is used to study more cases of RAC, providing a comprehensive picture of frost resistance of RAC with different air entrainment, water to cement ratios and replacement ratios. It is found that while an air content of 6% is enough to protect new mortar from frost damage, weak old mortar still renders RAC vulnerable to frost damage and the damage significantly becomes more severe with the increase of replacement ratio of recycled aggregate. On the contrary, replacement ratio has little influence on frost resistance of RAC with weaker new mortar than old mortar. Besides, a linear relationship between compressive strength reduction and frost-related expansion is obtained.

A mesoscale simulation of the FRP-to-concrete interfacial debonding propagation process by 3D RBSM

Although there are extensive investigations on fiber reinforced polymer (FRP)-strengthened concrete structures, reliable numerical simulation is still one of the major challenges in the analysis of such structures. This paper has successfully developed a rational FRP-to-concrete model based on the 3D rigid body spring model (RBSM). This is the first work to study the bond problem by involving FRP in the 3D RBSM. All the parameters in the developed model are proposed with clear and precise physical meanings and straightforward calculation methods. The simulation results are in good agreement with both the results of tested specimens and analytical solutions. Based on the proposed model, parametric studies are conducted to study the effects of the adhesive-concrete interfacial strength and width factor. From the simulation results, the relationship between the adhesive-concrete interfacial strength and failure mode is revealed. It is also found from the simulation results that the width factor has an upper limit, which is ignored by all the existing models. When the concrete width (bc) is greater than a certain value (e.g., FRP width (bF) + 30 mm in this work), the bond increase due to the width factor is insignificant.

Reduction mechanism of the compressive properties of concrete with primary crack of different widths and angles evaluated by 3D-RBSM

In this study, the reduction mechanism of the compressive properties of concrete with mechanically induced crack of different widths and angles was evaluated using a discrete mechanical model. Analytical investigations were conducted along with experiments to visualize the change in the stress transfer in a cross-section due to the presence of a primary crack. Based on the visualization of the stress distribution in the cross-section using a rigid body spring model (RBSM), the mechanical response of the primary crack under compressive stress and the reduction mechanism of compressive strength and elastic modulus were elucidated. Based on the results, it was concluded that the RBSM could reproduce the reduction in compressive strength and maximum tangential stiffness of cracked concrete with different crack widths and angles. Furthermore, the reduction mechanism of strength and stiffness due to the presence of a primary crack was newly elucidated considering the influence of the angle of the crack.

Numerical evaluation of the perfobond (PBL) shear connector with transverse rebar using coupled rigid Body spring model (RBSM) and solid finite element method (FEM)

Numerical analyses focused on the detailed fracture process of concrete in a reinforced PBL shear connector with the transverse rebar subjected to a simple push-out test employing coupled RBSM and solid FEM model were performed to reveal the internal failure mechanism, quantitatively. The numerical model effectively captured the shear capacity and failure modes of test results. It was investigated as the shear resistance of the concrete dowel was enhanced against the increased diameter of the transverse rebar, the shear capacity was improved consequently, and the deformed behaviors transformed from shear to splitting failure. The internal failure process of a specimen with transverse rebar was highlighted; the normal stress distribution along the transverse rebar exhibited the presence of the compressive zones in the concrete dowel region due to the formation of compression struts and compressive stresses transferred from anchored parts of the transverse rebar towards the concrete dowel region and got localized around the concrete dowel and in side concrete blocks and reproduced the enhanced shear capacity compared with specimen without transverse rebar. It was evaluated that the change in shear capacity was similar to the change in axial force of the transverse rebar. The shear capacity of the PBL shear connector increased as the axial force increased, while the inverse response against rebar bending moment was observed and determined that the main contribution towards the increase in the shear capacity of the PBL shear connector was from the axial force of the transverse rebar.

An analytical investigation of bond deterioration between rebar and ASR/DEF-damaged concrete with and without stirrup confinement using 3D RBSM

The bond performance between rebar and concrete is essential for the safety of RC structures. There is a need for research on bond deterioration due to the alkali-silica reaction (ASR) and delayed ettringite formation (DEF) damage which affects real structures. In this paper, a parametric study is conducted to quantitatively study the effect of slight-to-severe ASR/DEF damage in the presence of stirrup confinement on the pullout behavior between concrete and reinforcement using three-dimensional rigid body spring model (3D RBSM) simulation. It is found through the simulation that the bond stress in ASR damaged cases increases when ASR expansion before pullout is small and then decreases as the damage level rises, while the bond stress in DEF damaged cases keeps decreasing from the beginning. Besides, higher stirrup confinement effectively mitigates bond deterioration in both ASR and DEF damaged cases when damage is serious. More importantly, stress development and crack propagation during the expansion stage and the pullout stage are visualized. Interface cracking condition varies according to the damage type (ASR or DEF) and the level of stirrup confinement due to different cracking mechanisms. However, the tendency for the number of cracks in the concrete to increase, which reflects the overall damage level in ASR and DEF damaged cases, is similar, resulting in a similar trend for bond stress to decrease.

3D RBSM Analysis of Bond Degradation in Corroded Reinforced Concrete as Observed Using Digital Image Correlation.

The buildup of corrosion products over a reinforcing bar and associated reduction in rib height lead to degradation of the bond between reinforcement and concrete. The authors have previously used digital image correlation (DIC) to visualize and quantify load-induced cracking at the interface in specimens with varying degrees of corrosion. The results obtained in that study are used here to simulate the post-corrosion local bond behavior. A bond degradation model is incorporated into the discrete analysis tool, 3D Rigid Body Spring Model (RBSM) for the simulation. This analysis method allows the shape of the reinforcing bar to be directly modeled, and concrete cracking behavior is simulated by using a randomly shaped mesh. The magnitude of opening and sliding over the tips of ribs in the simulation, in which the reduction in rib height could not be modeled, is significantly lower than observed in the experiment. The results demonstrate that reduction in rib height is an important factor in post-corrosion behavior, and needs to be included in simulation models. It is also understood that in order to gain a better understanding of local post-corrosion bond behavior, de-bonding between reinforcement and concrete needs to be modeled in a discrete analysis framework.

Evaluation of thermal crack width and crack spacing in massive reinforced concrete structures subject to external restraints using RBSM

This study aimed to develop a numerical method to evaluate thermal cracking, in terms of parameters such as thermal crack width and crack spacing, on massive reinforced concrete (RC) wall structures externally restrained by a base structure. The existing experimental study was investigated using the rigid body–spring model (RBSM) with new constitutive laws developed for this objective. Thermal cracks, which localized and propagated throughout the wall thickness, were successfully reproduced, and the good agreement with the experiment indicated the capability of the proposed numerical method. The time-dependent cracking behavior and the impacts of factors affecting thermal cracking, which are difficult to investigate experimentally, are discussed.

Systematic calibration procedure for CFRP sheets and rods strengthened RC T-beams by using RBSM

In this study, a nonlinear 2D rigid body spring model (RBSM) was developed to verify the experimental works of Carbon Fiber Reinforced Polymer (CFRP) sheets and rods strengthened reinforced concrete (RC) T-shape beam (RC T-beam) section under a combination of bending and shearing loads. Modified Mohr–Coulomb criteria were adopted for modeling the plastic damage of concrete material. The orthotropic smeared layers representing the steel and CFRP materials were laminated on the concrete surface to model the reinforcing steel bars, stirrup steel, and CFRP sheets and rods. A mechanical behavior-based approach to fit the experimental results was discussed using the numerical results at each calibration stage. This study shows how the gradient of compressive and ultimate tensile strength of concrete affects the initial flexure behavior of the load-displacement curve of an RC T-beam. In contrast with finite element modeling, the RBSM can exhibit the crack propagation processes of element separation during the simulation. The calibrations showed the agreement of the models used in predicting the flexural behavior, ultimate load, and strengthening effects of the CFRP sheets and rods to the RC T-beam under bending loads. A systematic calibration procedure combined with the recommended use of energy-based criteria to evaluate the results of calibrated load-displacement curve was proposed.